A mechanical wave travels along a genetic guide to drive the formation of an epithelial furrow during Drosophila gastrulation.

Epithelial furrowing is a fundamental morphogenetic process during gastrulation, neurulation, and body shaping. A furrow often results from a fold that propagates along a line. How fold formation and propagation are controlled and driven is poorly understood. To shed light on this, we study the formation of the cephalic furrow, a fold that runs along the embryo dorsal-ventral axis during Drosophila gastrulation and the developmental role of which is still unknown. We provide evidence of its function and show that epithelial furrowing is initiated by a group of cells. This cellular cluster works as a pacemaker, triggering a bidirectional morphogenetic wave powered by actomyosin contractions and sustained by de novo medial apex-to-apex cell adhesion. The pacemaker's Cartesian position is under the crossed control of the anterior-posterior and dorsal-ventral gene patterning systems. Thus, furrow formation is driven by a mechanical trigger wave that travels under the control of a multidimensional genetic guide.

Wrinkling Instability in Unsupported Epithelial Sheets.

We investigate the elasticity of an unsupported epithelial monolayer and we discover that unlike a thin solid plate, which wrinkles if geometrically incompatible with the underlying substrate, the epithelium may do so even in the absence of the substrate. From a cell-based model, we derive an exact elasticity theory and discover wrinkling driven by the differential apico-basal surface tension. Our theory is mapped onto that for supported plates by introducing a phantom substrate whose stiffness is finite beyond a critical differential tension. This suggests a new mechanism for an autonomous control of tissues over the length scale of their surface patterns.

Morphology of vesicle triplets: shape transformation at weak and strong adhesion limits.

We investigate the morphologies of adhering vesicle triplets as a function of volume-to-area ratio encoded by the reduced volume in strong and weak adhesion regimes. In the strong adhesion regime, the morphology change of the vesicle triplet depends on the arrangement of vesicles. By decreasing the reduced volume, a triangular triplet composed of three spherical caps with a trifurcated flat contact zone deformed to a compact spherical shape with a sigmoidal contact zone, whereas a linear vesicle triplet composed of pancake-shaped vesicles sandwiched between two spherical-cap vesicles with a flat contact zone deformed into a compact spherical shape with biconvex interfaces. The morphologies of vesicle triplets with flat contact zones are reproduced by the so-called two-tension model based on the total energy consisting of bending energy, adhesion energy and surface energy, where the surface tension in the noncontact zone is different from that in the contact zone. When the flat interface deforms, the two-tension model is no longer applicable. The compact spherical triplets with curved interfaces can be reproduced by introducing geometrical constraints requiring that the total area of the non-contact zones is minimal, thereby confining the aggregate to a spherical cavity; this is referred to as the cavity model. In the weak adhesion regime, vesicle triplets with either a triangular or linear topology deform into prolate-based triplets by decreasing the reduced volume.

Intrinsic lipid curvatures of mammalian plasma membrane outer leaflet lipids and ceramides.

We developed a global X-ray data analysis method to determine the intrinsic curvatures of lipids hosted in inverted hexagonal phases. In particular, we combined compositional modelling with molecular shape-based arguments to account for non-linear mixing effects of guest-in-host lipids on intrinsic curvature. The technique was verified by all-atom molecular dynamics simulations and applied to sphingomyelin and a series of phosphatidylcholines and ceramides with differing composition of the hydrocarbon chains. We report positive lipid curvatures for sphingomyelin and all phosphatidylcholines with disaturated and monounsaturated hydrocarbons. Phosphatidylcholines with diunsaturated hydrocarbons in turn yielded intrinsic lipid curvatures with negative values. All ceramides, with chain lengths varying between C2:0 and C24:0, displayed significant negative lipid curvature values. Moreover, we report non-additive mixing for C2:0 ceramide and sphingomyelin. This suggests for sphingolipids that in addition to lipid headgroup and hydrocarbon chain volumes also lipid-specific interactions are important contributors to membrane curvature stress.

Morphologies of compressed active epithelial monolayers.

Using a three-dimensional active vertex model, we numerically study the shapes of strained unsupported epithelial monolayers subject to active junctional noise due to stochastic binding and unbinding of myosin. We find that while uniaxial, biaxial, and isotropic in-plane compressive strains do lead to the formation of longitudinal, herringbone pattern, and labyrinthine folds, respectively, the villus morphology characteristic of, e.g., the small intestine appears only if junctional tension fluctuations are strong enough to fluidize the tissue. Moreover, the fluidized epithelium features villi even in the absence of compressive strain provided that the apico-basal differential surface tension is large enough. We analyze several details of the different epithelial forms including the role of strain rate and the modulation of tissue thickness across folds. Our results show that even unsupported, non-patterned epithelia can form nontrivial morphologies.

Morphologies of Vesicle Doublets: Competition among Bending Elasticity, Surface Tension, and Adhesion.

To study the mechanical laws governing the form of multicellular organisms, we examine the morphology of adhering vesicle doublets as the simplest model system. We monitor the morphological transformations of doublets induced by changes of adhesion strength and volume/area ratio, which are controlled by intermembrane interactions and thermal area expansion, respectively. When we increase the temperature in the weak adhesion regime, a dumbbell flat-contact doublet is transformed to a parallel-prolate doublet, whereas in the strong adhesion regime, heating transforms the dumbbell flat-contact doublet into a spherical sigmoid-contact doublet. We reproduce the observed doublet morphologies by numerically minimizing the total energy, including the contact-potential adhesion term as well as the surface and bending terms, using the Surface Evolver package. From the reproduced morphologies, we extract the adhesion strength, the surface tension, and the volume/area ratio of the vesicles, which reveals the detailed mechanisms of the morphological transitions in doublets.

Collective cell mechanics of epithelial shells with organoid-like morphologies.

The study of organoids, artificially grown cell aggregates with the functionality and small-scale anatomy of real organs, is one of the most active areas of research in biology and biophysics, yet the basic physical origins of their different morphologies remain poorly understood. Here, we propose a mechanistic theory of epithelial shells which resemble small-organoid morphologies. Using a 3D surface tension-based vertex model, we reproduce the characteristic shapes from branched and budded to invaginated structures. We find that the formation of branched morphologies relies strongly on junctional activity, enabling temporary aggregations of topological defects in cell packing. To elucidate our numerical results, we develop an effective elasticity theory, which allows one to estimate the apico-basal polarity from the tissue-scale modulation of cell height. Our work provides a generic interpretation of the observed epithelial shell morphologies, highlighting the role of physical factors such as differential surface tension, cell rearrangements, and tissue growth.

Metallic-mean quasicrystals as aperiodic approximants of periodic crystals.

Ever since the discovery of quasicrystals, periodic approximants of these aperiodic structures constitute a very useful experimental and theoretical device. Characterized by packing motifs typical for quasicrystals arranged in large unit cells, these approximants bridge the gap between periodic and aperiodic positional order. Here we propose a class of sequences of 2-D quasicrystals that consist of increasingly larger periodic domains and are marked by an ever more pronounced periodicity, thereby representing aperiodic approximants of a periodic crystal. Consisting of small and large triangles and rectangles, these tilings are based on the metallic means of multiples of 3, have a 6-fold rotational symmetry, and can be viewed as a projection of a non-cubic 4-D superspace lattice. Together with the non-metallic-mean three-tile hexagonal tilings, they provide a comprehensive theoretical framework for the complex structures seen, e.g., in some binary nanoparticles, oxide films, and intermetallic alloys.

Fluidization of epithelial sheets by active cell rearrangements.

We theoretically explore fluidization of epithelial tissues by active T1 neighbor exchanges. We show that the geometry of cell-cell junctions encodes important information about the local features of the energy landscape, which we support by an elastic theory of T1 transformations. Using a 3D vertex model, we show that the degree of active noise driving forced cell rearrangements governs the stress-relaxation timescale of the tissue. We study tissue response to in-plane shear at different timescales. At short time, the tissue behaves as a solid, whereas its long-time fluid behavior can be associated with an effective viscosity which scales with the rate of active T1 transformations. Furthermore, we develop a coarse-grained theory, where we treat the tissue as an active fluid and confirm the results of the vertex model. The impact of cell rearrangements on tissue shape is illustrated by studying axial compression of an epithelial tube.

Bronze-mean hexagonal quasicrystal.

The most striking feature of conventional quasicrystals is their non-traditional symmetry characterized by icosahedral, dodecagonal, decagonal or octagonal axes. The symmetry and the aperiodicity of these materials stem from an irrational ratio of two or more length scales controlling their structure, the best-known examples being the Penrose and the Ammann-Beenker tiling as two-dimensional models related to the golden and the silver mean, respectively. Surprisingly, no other metallic-mean tilings have been discovered so far. Here we propose a self-similar bronze-mean hexagonal pattern, which may be viewed as a projection of a higher-dimensional periodic lattice with a Koch-like snowflake projection window. We use numerical simulations to demonstrate that a disordered variant of this quasicrystal can be materialized in soft polymeric colloidal particles with a core-shell architecture. Moreover, by varying the geometry of the pattern we generate a continuous sequence of structures, which provide an alternative interpretation of quasicrystalline approximants observed in several metal-silicon alloys.

Role of Inverse-Cone-Shape Lipids in Temperature-Controlled Self-Reproduction of Binary Vesicles.

We investigate a temperature-driven recursive division of binary giant unilamellar vesicles (GUVs). During the heating step of the heating-cooling cycle, the spherical mother vesicle deforms to a budded limiting shape using up the excess area produced by the chain melting of the lipids and then splits off into two daughter vesicles. Upon cooling, the daughter vesicle opens a pore and recovers the spherical shape of the mother vesicle. Our GUVs are composed of DLPE (1,2-dilauroyl-sn-glycero-3-phosphoethanolamine) and DPPC (1,2-dipalmitoyl-sn-glycero-3-phosphocholine). During each cycle, vesicle deformation is monitored by a fast confocal microscope and the images are analyzed to obtain the time evolution of reduced volume and reduced monolayer area difference as the key geometric parameters that quantify vesicle shape. By interpreting the deformation pathway using the area-difference elasticity theory, we conclude that vesicle division relies on (1) a tiny asymmetric distribution of DLPE within the bilayer, which controls the observed deformation from the sphere to the budded shape; and (2) redistribution of DLPE during the deformation-division stage, which ensures that the process is recursive. The spontaneous coupling between membrane curvature and PE lipid distribution is responsible for the observed recursive division of GUVs. These results shed light on the mechanisms of vesicle self-reproduction.

Quantitative Morphology of Epithelial Folds.

The shape of spatially modulated epithelial morphologies such as villi and crypts is usually associated with the epithelium-stroma area mismatch leading to buckling. We propose an alternative mechanical model based on intraepithelial stresses generated by differential tensions of apical, lateral, and basal sides of cells as well as on the elasticity of the basement membrane. We use it to theoretically study longitudinal folds in simple epithelia and we identify four types of corrugated morphologies: compact, invaginated, evaginated, and wavy. The obtained tissue contours and thickness profiles are compared to epithelial folds observed in invertebrates and vertebrates, and for most samples, the agreement is within the estimated experimental error. Our model establishes the groove-crest modulation of tissue thickness as a morphometric parameter that can, together with the curvature profile, be used to estimate the relative differential apicobasal tension in the epithelium.

Theory of epithelial elasticity.

We propose an elastic theory of epithelial monolayers based on a two-dimensional discrete model of dropletlike cells characterized by differential surface tensions of their apical, basal, and lateral sides. We show that the effective tissue bending modulus depends on the apicobasal differential tension and changes sign at the transition from the flat to the fold morphology. We discuss three mechanisms that stabilize the finite-wavelength fold structures: Physical constraint on cell geometry, hard-core interaction between non-neighboring cells, and bending elasticity of the basement membrane. We show that the thickness of the monolayer changes along the waveform and thus needs to be considered as a variable rather than a parameter. Next we show that the coupling between the curvature and the thickness is governed by the apicobasal polarity and that the amplitude of thickness modulation along the waveform is proportional to the apicobasal differential tension. This suggests that intracellular stresses can be measured indirectly by observing easily measurable morphometric parameters. We also study the mechanics of three-dimensional structures with cylindrical symmetry.

Elasticity of polymeric nanocolloidal particles.

Softness is an essential mechanical feature of macromolecular particles such as polymer-grafted nanocolloids, polyelectrolyte networks, cross-linked microgels as well as block copolymer and dendrimer micelles. Elasticity of individual particles directly controls their swelling, wetting, and adsorption behaviour, their aggregation and self-assembly as well as structural and rheological properties of suspensions. Here we use numerical simulations and self-consistent field theory to study the deformation behaviour of a single spherical polymer brush upon diametral compression. We observe a universal response, which is rationalised using scaling arguments and interpreted in terms of two coarse-grained models. At small and intermediate compressions the deformation can be accurately reproduced by modelling the brush as a liquid drop, whereas at large compressions the brush behaves as a soft ball. Applicable far beyond the pairwise-additive small-strain regime, the models may be used to describe microelasticity of nanocolloids in severe confinement including dense disordered and crystalline phases.

Embryo-scale tissue mechanics during Drosophila gastrulation movements.

Morphogenesis of an organism requires the development of its parts to be coordinated in time and space. While past studies concentrated on defined cell populations, a synthetic view of the coordination of these events in a whole organism is needed for a full understanding. Drosophila gastrulation begins with the embryo forming a ventral furrow, which is eventually internalized. It is not understood how the rest of the embryo participates in this process. Here we use multiview selective plane illumination microscopy coupled with infrared laser manipulation and mutant analysis to dissect embryo-scale cell interactions during early gastrulation. Lateral cells have a denser medial-apical actomyosin network and shift ventrally as a compact cohort, whereas dorsal cells become stretched. We show that the behaviour of these cells affects furrow internalization. A computational model predicts different mechanical properties associated with tissue behaviour: lateral cells are stiff, whereas dorsal cells are soft. Experimental analysis confirms these properties in vivo.

Physical models of mesoderm invagination in Drosophila embryo.

The invagination of the mesoderm in the Drosophila melanogaster embryo is an intensely studied example of epithelial folding. Several theoretical studies have explored the conditions and mechanisms needed to reproduce the formation of the invagination in silico. Here we discuss the aspects of epithelial folding captured by these studies, and compare the questions addressed, the approaches used, and the answers provided.

A model of epithelial invagination driven by collective mechanics of identical cells.

We propose a 2D mechanical model of a tubular epithelium resembling the early Drosophila embryo. The model consists of a single layer of identical cells with energy associated with the tension of cell cortex. Depending on the relative tension of the apical, basal, and lateral sides of the cells, tissue thickness, and the degree of external constraint, the minimal-energy states of the epithelial cross section include circular shapes as well as a range of inward-buckled shapes. Some of the solutions are characterized by a single deep groove, which shows that an epithelium consisting of cells of identical mechanical properties can infold. This is consistent with what is seen in embryos of certain Drosophila mutants. To ensure that the infolding occurs at a predetermined section of the epithelium, we extend the model by increasing the cross-sectional area of a subset of cells, which is consistent with observations in wild-type embryos. This variation of cell parameters across the epithelium is sufficient to make it fold at a specific site. The model explores previously untested minimal conditions for tissue invagination and is devoid of specificity needed to accurately describe an in vivo situation in Drosophila.

Reaction kinetics and mechanical models of liposome adhesion at charged interface.

Dynamics of adhesion of single liposome at the charged mercury interface is analyzed through its amperometric signal using a reaction kinetics model and a mechanical model. We present analytical solutions of the reaction kinetics model for decoupling and identifying temporal evolution of three distinct states: i) the initial state corresponding to an intact liposome, ii) the intermediate state where the liposome is partly deformed, and iii) the final state of a lipid monolayer. The results obtained with this model indicate that all three states simultaneously evolve from the onset of the adhesion process. The new mechanical model provides a physical interpretation of the three states and emphasizes the role of the forces involved in liposome adhesion process. The main conclusion is that the water content of the liposome is released through the pores formed in the membrane rather than through the channels parallel to the electrode. Both models reproduce the measurements well in the wide potential range and offer a complementary insight into the dynamics of single adhesion event, which can find application in studies of cell adhesion and in drug delivery.

Analysis of side chain rotational restrictions of membrane-embedded proteins by spin-label ESR spectroscopy.

Site-directed spin-labeling electron spin resonance (SDSL-ESR) is a promising tool for membrane protein structure determination. Here we propose a novel way to translate the local structural constraints gained by SDSL-ESR data into a low-resolution structure of a protein by simulating the restrictions of the local conformational spaces of the spin label attached at different protein sites along the primary structure of the membrane-embedded protein. We test the sensitivity of this approach for membrane-embedded M13 major coat protein decorated with a limited number of strategically placed spin labels employing high-throughput site-directed mutagenesis. We find a reasonably good agreement of the simulated and the experimental data taking a protein conformation close to the one determined by fluorescence resonance energy transfer analysis [P.V. Nazarov, R.B.M. Koehorst, W.L. Vos, V.V. Apanasovich, M.A. Hemminga, FRET study of membrane proteins: determination of the tilt and orientation of the N-terminal domain of M13 major coat protein, Biophys. J. 92 (2007) 1296-1305].

The cooperative role of membrane skeleton and bilayer in the mechanical behaviour of red blood cells.

Red blood cell (RBC) shape, behaviour and deformability can be consistently accounted for by a model for the elastic properties of the RBC membrane that includes the elasticity of the membrane skeleton in dilation and shear, and the local and nonlocal resistance of the bilayer to bending. The role of the corresponding energy terms in different RBC shape and deformation situations is analyzed. RBC shape transformations are compared to the shape transformations of phospholipid vesicles that are driven by the difference between the equilibrium areas of the bilayer leaflets (DeltaA0). It is deduced that the skeleton energy contributions play a crucial role in the formation of an echinocyte. The effect of a transformation of the natural biconcave RBC shape into an echinocyte on its resistance to entry into capillary-sized cylindrical tubes is analyzed. It is shown that, during the aspiration of an echinocyte into a pipette, there are two competing skeleton deformation effects, which arise due to skeleton density changes, one due to spicule formation and the other due to deformation induced by micropipette aspiration. Furthermore, the shift of the observed dependence of the projection length on the aspiration pressure of more crenated cells towards higher aspiration pressures can be accounted for by an increase of the equilibrium area difference DeltaA0 and consequent modification of the nonlocal contribution to the cell elastic energy.