From discrete to continuum models of three-dimensional deformations in epithelial sheets

International audienceEpithelial tissue, in which cells adhere tightly to each other and to theunderlying substrate, is one of the four major tissue types in adultorganisms. In embryos, epithelial sheets serve as versatile substratesduring the formation of developing organs. Some aspects of epithelialmorphogenesis can be adequately described using vertex models, in which thetwo-dimensional arrangement of epithelial cells is approximated by apolygonal lattice with an energy that has contributions reflecting theproperties of individual cells and their interactions. Previous studieswith such models have largely focused on dynamics confined to two spatialdimensions and analyzed them numerically. We show how these models can beextended to account for three-dimensional deformations and studiedanalytically. Starting from the extended model, we derive a continuumplate description of cell sheets, in which the effective tissue properties,such as bending rigidity, are related explicitly to the parameters of thevertex model. To derive the continuum plate model, we duly take intoaccount a microscopic shift between the two sublattices of the hexagonalnetwork, which has been ignored in previous work. As an application of thecontinuum model, we analyze tissue buckling by a line tension applied alonga circular contour, a simplified set-up relevant to several situations inthe developmental context. The buckling thresholds predicted by thecontinuum description are in good agreement with the results of directstability calculations based on the vertex model. Our results establish adirect connection between discrete and continuum descriptions of cellsheets and can be used to probe a wide range of morphogenetic processes inepithelial tissues

Model Flames in the Boussinesq Limit: The Effects of Feedback

We have studied the fully nonlinear behavior of pre-mixed flames in a gravitationally stratified medium, subject to the Boussinesq approximation. Key results include the establishment of criterion for when such flames propagate as simple planar flames; elucidation of scaling laws for the effective flame speed; and a study of the stability properties of these flames. The simplicity of some of our scalings results suggests that analytical work may further advance our understandings of buoyant flames.Comment: 11 pages, 14 figures, RevTex, gzipped tar fil

Asymptotics for turbulent flame speeds of the viscous G-equation enhanced by cellular and shear flows

G-equations are well-known front propagation models in turbulent combustion and describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are Hamilton-Jacobi equations with convex ($L^1$ type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue $\bar H$ from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed $s_T$. An important problem in turbulent combustion theory is to study properties of $s_T$, in particular how $s_T$ depends on the flow amplitude $A$. In this paper, we will study the behavior of $\bar H=\bar H(A,d)$ as $A\to +\infty$ at any fixed diffusion constant $d > 0$. For the cellular flow, we show that $$\bar H(A,d)\leq O(\sqrt {\mathrm {log}A}) \quad \text{for all $d>0$}.$$ Compared with the inviscid G-equation ($d=0$), the diffusion dramatically slows down the front propagation. For the shear flow, the limit \nit $\lim_{A\to +\infty}{\bar H(A,d)\over A} = \lambda (d) >0$ where $\lambda (d)$ is strictly decreasing in $d$, and has zero derivative at $d=0$. The linear growth law is also valid for $s_T$ of the curvature dependent G-equation in shear flows.Comment: 27 pages. We improve the upper bound from no power growth to square root of log growt

On the folding and deployment of tape springs: a large displacements and large rotations rod model with highly flexible thin-walled cross-section

International audienceIn the framework of deployable structures, we focus on the modeling of tape springs, i.e. rod-like elastic bodies with thin-walled cross-section which develop localized folds due to a flattening of the cross-section. A rod model with highly deformable cross-section and few kinematics parameters is derived from a complete shell model, for large displacements, large rotations and dynamics. The simplicity of the model is achieved by introducing an elastica kinematics to describe the changes in the cross-section shape. This model is able to handle the formation of localized folds which can move along the rod line, merge or split, allowing to simulate complex scenarios of folding and deployment

The interplay of crack hopping, delamination and interface failure in drying nanoparticle films

Films formed through the drying of nanoparticle suspensions release the build-up of strain through a variety of different mechanisms including shear banding, crack formation and delamination. Here we show that important connections exist between these different phenomena: delamination depends on the dynamics of crack hopping, which in turn is influenced by the presence of shear bands. We also show that delamination does not occur uniformly across the film. As cracks hop they locally initiate the delamination of the film which warps with a timescale much longer than that associated with the hopping of cracks. The motion of a small region of the delamination front, where the shear component of interfacial crack propagation is believed to be enhanced, results in the deposition of a complex zig-zag pattern on the supporting substrate

Internal stress as a link between macroscale and mesoscale mechanics

The internal (or residual) stress is among the key notions to describe the state of the systems far from equilibrium. Such stress is invisible on the macroscopic scale where the system is regarded as a blackbox. Yet nonequilibrium macroscopic operations allow to create and observe the internal stress. We present in this lecture some examples of the internal stress and its operations. We describe the memory effect in some detail, the process in which the history of past operations is recalled through the relaxation of internal stress.Comment: 11pages, 2 figures, 7 equations: Refereed notes of the Lectures given at "Morphogenesis through the interplay of nonlinear chemical instabilities and elastic active media" July 2-14 2007. to be published from Springer (NATO series

The smectic order of wrinkles

A thin elastic sheet lying on a soft substrate develops wrinkled patterns when subject to an external forcing or as a result of geometric incompatibility. Thin sheet elasticity and substrate response equip such wrinkles with a global preferred wrinkle spacing length and with resistance to wrinkle curvature. These features are responsible for the liquid crystalline smectic-like behaviour of such systems at intermediate length scales. This insight allows better understanding of the wrinkling patterns seen in such systems, with which we explain pattern breaking into domains, the properties of domain walls and wrinkle undulation. We compare our predictions with numerical simulations and with experimental observations

The role of the EP receptors for prostaglandin E2 in skin and skin cancer

One of the most common features of exposure of skin to ultraviolet (UV) light is the induction of inflammation, a contributor to tumorigenesis, which is characterized by the synthesis of cytokines, growth factors and arachidonic acid metabolites, including the prostaglandins (PGs). Studies on the role of the PGs in non-melanoma skin cancer (NMSC) have shown that the cyclooxygenase-2 (COX-2) isoform of the cyclooxygenases is responsible for the majority of the pathological effects of PGE2. In mouse skin models, COX-2 deficiency significantly protects against chemical carcinogen- or UV-induced NMSC while overexpression confers endogenous tumor promoting activity. Current studies are focused on identifying which of the G protein-coupled EP receptors mediate the tumor promotion/progression activities of PGE2 and the signaling pathways involved. As reviewed here, the EP1, EP2, and EP4 receptors, but not the EP3 receptor, contribute to NMSC development, albeit through different signaling pathways and with somewhat different outcomes. The signaling pathways activated by the specific EP receptors are context specific and likely depend on the level of PGE2 synthesis, the differential levels of expression of the different EP receptors, as well as the levels of expression of other interacting receptors. Understanding the role and mechanisms of action of the EP receptors potentially offers new targets for the prevention or therapy of NMSCs

Genomes of the Most Dangerous Epidemic Bacteria Have a Virulence Repertoire Characterized by Fewer Genes but More Toxin-Antitoxin Modules

We conducted a comparative genomic study based on a neutral approach to identify genome specificities associated with the virulence capacity of pathogenic bacteria. We also determined whether virulence is dictated by rules, or if it is the result of individual evolutionary histories. We systematically compared the genomes of the 12 most dangerous pandemic bacteria for humans ("bad bugs") to their closest non-epidemic related species ("controls").We found several significantly different features in the "bad bugs", one of which was a smaller genome that likely resulted from a degraded recombination and repair system. The 10 Cluster of Orthologous Group (COG) functional categories revealed a significantly smaller number of genes in the "bad bugs", which lacked mostly transcription, signal transduction mechanisms, cell motility, energy production and conversion, and metabolic and regulatory functions. A few genes were identified as virulence factors, including secretion system proteins. Five "bad bugs" showed a greater number of poly (A) tails compared to the controls, whereas an elevated number of poly (A) tails was found to be strongly correlated to a low GC% content. The "bad bugs" had fewer tandem repeat sequences compared to controls. Moreover, the results obtained from a principal component analysis (PCA) showed that the "bad bugs" had surprisingly more toxin-antitoxin modules than did the controls.We conclude that pathogenic capacity is not the result of "virulence factors" but is the outcome of a virulent gene repertoire resulting from reduced genome repertoires. Toxin-antitoxin systems could participate in the virulence repertoire, but they may have developed independently of selfish evolution

Mathematical modelling of clostridial acetone-butanol-ethanol fermentation

Clostridial acetone-butanol-ethanol (ABE) fermentation features a remarkable shift in the cellular metabolic activity from acid formation, acidogenesis, to the production of industrial-relevant solvents, solventogensis. In recent decades, mathematical models have been employed to elucidate the complex interlinked regulation and conditions that determine these two distinct metabolic states and govern the transition between them. In this review, we discuss these models with a focus on the mechanisms controlling intra- and extracellular changes between acidogenesis and solventogenesis. In particular, we critically evaluate underlying model assumptions and predictions in the light of current experimental knowledge. Towards this end, we briefly introduce key ideas and assumptions applied in the discussed modelling approaches, but waive a comprehensive mathematical presentation. We distinguish between structural and dynamical models, which will be discussed in their chronological order to illustrate how new biological information facilitates the ‘evolution’ of mathematical models. Mathematical models and their analysis have significantly contributed to our knowledge of ABE fermentation and the underlying regulatory network which spans all levels of biological organization. However, the ties between the different levels of cellular regulation are not well understood. Furthermore, contradictory experimental and theoretical results challenge our current notion of ABE metabolic network structure. Thus, clostridial ABE fermentation still poses theoretical as well as experimental challenges which are best approached in close collaboration between modellers and experimentalists