Patrizia Bagnerini

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.

On the role of source terms in continuum traffic flow models

We introduce some models for vehicular traffic flow based on hyperbolic balance laws. We focus in particular on source terms for modeling highway entries and exits or local changes of the traffic flow due to inhomogeneities of the road. Rigorous well-posedness results and numerical investigations are presented. We show in particular how real phenomena (e.g. the formation of a queue) that are not captured by models based on systems of conservation laws are instead observable with our models.

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.

A Fast Algorithm for Determining the Propagation Path of Multiple Diffracted Rays

We present a fast algorithm for path computation of multiple diffracted rays relevant to ray tracing techniques. The focus is on double diffracted rays, but generalizations are also mentioned. The novelty of our approach is in the use of an analytical geometry procedure which permits to re-write the problem as a simple nonlinear equation. This procedure permits a convergence analysis of the algorithms involved in the numerical resolution of such nonlinear equation. Moreover, we also indicate how to choose the iteration starting point to obtain convergence of the (locally convergent) Newton method. As in previous works, explicit solutions are obtained in the relevant cases of parallel or incident diffraction edges

Optimal control of level sets dynamics

The problem of optimal tracking of level sets is investigated. More specifically, we propose a novel method to design a controller that is able to track a reference curve generated by a level set equation. Such a functional optimal control problem is nonstandard as it involves a moving domain over time, thus requiring the development of both an ad-hoc approximation scheme and an efficient numerical solver. We address the approximate solution of the resulting optimization problem by using the extended Ritz method, which has been recently applied to the solution of a number of optimal control problems in high dimensional setting. Such a method relies on the use a control law of fixed structure that depends on a number of parameters to be suitably chosen. Toward this end, we derive the adjoint equation for the optimal tracking problem. Such an equation allows one to compute the gradient of the cost function with respect to the vector of parameters of the control law. Numerical results are reported to show the effectiveness and computational effort of the proposed approach for the purpose of tracking curves generated by the normal and mean curvature flow equations.

Further results on the optimal control of fronts generated by level set methods

The control of level sets generated by partial differential equations is still a challenge because of its complexity both from the theoretical and computational points of view. Specifically, we focus on the space-dependent optimal control problem of a moving front and search for an approximate solution method that is computationally feasible. We formulate the problem in an Eulerian setting and develop an efficient approximation scheme based on the extended Ritz method. Such a method consists in adopting a control law with fixed structure that depends nonlinearly from a number of parameters to be suitably chosen by using a gradient-based technique. Toward this end, we derive the adjoint equations for optimal control problems involving the normal and mean curvature flow partial differential equations. The adjoint equations allow to compute the gradient of the cost with respect to the vector of parameters of the control law. Numerical results are reported to show the effectiveness of the proposed approach in some 2D and 3D examples.

Modeling and Identification of Amnioserosa Cell Mechanical Behavior by Using Mass-Spring Lattices

Various mechanical models of live amnioserosa cells during Drosophila melanogaster’s dorsal closure are proposed. Such models account for specific biomechanical oscillating behaviors and depend on a different set of parameters. The identification of the parameters for each of the proposed models is accomplished according to a least-squares approach in such a way to best fit the cellular dynamics extracted from live images. For the purpose of comparison, the resulting models after identification are validated to allow for the selection of the most appropriate description of such a cell dynamics. The proposed methodology is general and it may be applied to other planar biological processes.

Extended Kalman filtering to design optimal controllers of fronts generated by level set methods

The computational issues involved in the solution of optimal control problems of propagating fronts described by level sets motivate the search for effective optimization algorithms. In this paper, we attack the problem of optimal control of moving fronts by searching for an approximate solution method that is computationally feasible and robust to local minima trapping. The presence of many local minima is a crucial difficulty one encounters in dealing with such a problem. Following previous results, we use the extended Ritz method to find approximate solutions. This approach consists in adopting a control law with fixed structure that depends nonlinearly on a number of parameters to be suitably chosen. To overcome the local minima issue, we propose to optimize the weights for the level set optimal control by a recursive minimization based on the extended Kalman filter (EKF). As compared with techniques based on gradient-descent methods, the EKF optimization turns out to be successful to reduce computational burden and increase robustness with respect to local minima trapping, as shown by simulation results in a test case involving a change of topology.

Parameter identification of the normal flow equation by using adaptive estimation

We address the problem of estimating the constant parameters involved in the normal flow equation, which is a Hamilton-Jacobi PDE widely used in many different research areas. The identification of such parameters allows one to estimate the flow function, which is the velocity vector field that governs the dynamics of the level sets associated with the solution of the equation. The estimates are obtained by using a Luenberger observer and a parameter estimator based on the adaptation law proposed by Pomet and Praly in 1992. Such a law makes it possible to explicitly take into account bounds on the parameters. Conditions for the stability of the parameter estimation error are established. Simulation results are presented that confirm the theoretical achievements.