Luís Almeida

Coordinated waves of actomyosin flow and apical cell constriction immediately after wounding

Epithelial wounding causes waves of actomyosin flow and apical cell constriction that are dependent on calcium signaling and actin filament severing.

Topological methods for the Ginzburg-Landau equations

Abstract We consider the complex-valued Ginzburg-Landau equation on a two-dimensional domain Ω, with boundary data g, such that ∥g∥ = 1, −Δu= 1 ϵ 2 u(1−∥u∥ 2 ),u=g. We develop a variational framework for this equation: in particular we show that the topology of the level sets is related to a finite dimensional functional, the renormalized energy. As an application, we prove a multiplicity result of solutions for the equation, when e is small and the winding number of g is larger or equal to 2.

JNK Signalling Controls Remodelling of the Segment Boundary through Cell Reprogramming during Drosophila Morphogenesis

Reprogramming of a specific group of Drosophila epidermal cells allows the mixing of normally segregated populations and the release of mechanical tension that arises during morphogenesis.

A mathematical model for dorsal closure.

During embryogenesis, drosophila embryos undergo epithelial folding and unfolding, which leads to a hole in the dorsal epidermis, transiently covered by an extraembryonic tissue called the amnioserosa. Dorsal closure (DC) consists of the migration of lateral epidermis towards the midline, covering the amnioserosa. It has been extensively studied since numerous physical mechanisms and signaling pathways present in DC are conserved in other morphogenetic events and wound healing in many other species (including vertebrates). We present here a simple mathematical model for DC that involves a reduced number of parameters directly linked to the intensity of the forces in the presence and which is applicable to a wide range of geometries of the leading edge (LE). This model is a natural generalization of the very interesting model proposed in Hutson et al. (2003). Being based on an ordinary differential equation (ODE) approach, the previous model had the advantage of being even simpler, but this restricted significantly the variety of geometries that could be considered and thus the number of modified dorsal closures that could be studied. A partial differential equation (PDE) approach, as the one developed here, allows considering much more general situations that show up in genetically or physically perturbed embryos and whose study will be essential for a proper understanding of the different components of the DC process. Even for native embryos, our model has the advantage of being applicable since an early stages of DC when there is no antero-posterior symmetry (approximately verified only in the late phases of DC). We validate our model in a native setting and also test it further in embryos where the zipping force is perturbed through the expression of spastin (a microtubule severing protein). We obtain variations of the force coefficients that are consistent with what was previously described for this setting.

A few symmetry results for nonlinear elliptic PDE on noncompact manifolds

Abstract We prove some symmetry theorems for positive solutions of elliptic equations in some noncompact manifolds, which generalize and extend symmetry results known in the case of the euclidean space R n . The (variational) technique that we use relies on Sobolev inequalities available for manifolds together with the well known method of moving planes. In the particular case of the standard n-dimensional hyperbolic space H n we get the radial symmetry of positive solutions of the equation −Δ H n u=f(u) in H n , which tend to zero at infinity (or belong to the Sobolev space H 1 ( H n ) in some cases), under different hypotheses on the relationship between the behavior of the nonlinearity f in a neighborhood of zero and the summability properties of the solution. One of the main features of this work is to single out and study the connection between the geometric properties of the manifold considered and the growth conditions on the nonlinearity in order to have our symmetry results.

Symmetry Results for Positive Solutions of Some Elliptic Equations on Manifolds

We adapt the method of moving planes in order to obtainsymmetry results for positive solutions of some elliptic equations onmanifolds, in the case where our problem satisfies certain symmetryhypothesis. We also obtain monotonicity results using the slidingmethod.

Modeling actin cable contraction

Extension of an epithelial membrane to close a hole is a very widespread process both in morphogenesis and in tissue repair. In many circumstances an important component driving these movements is an actomyosin contraction which consists of meshworks of actin filaments cross-linked by Myosin II molecular motors. We introduce a mathematical model to simulate the contraction of an actin cable structure attached to an external epithelial tissue and we use this curvature-type model as a basis to build other models in more general settings. This result is obtained by adding extra terms that describe the particular process we want to model (lamellipodial crawling, granulation tissue contraction, extension of actin protrusions, epithelial resistance, etc.). Finally, we concentrate on the treatment of non-homogeneous forces, i.e. non-constant boundary terms which can be associated with a non-uniform cable, internal pull or zipping force due to the non-uniformity of the biological or physical properties of the boundary cells or of the connective tissue.

The regularity problem for generalized harmonic maps into homogeneous spaces

AbstractLet ℳ be a Riemannian surface and

Mean curvature flow with obstacles

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