Phase Reduction in the Noise Induced Escape Problem for Systems close to Reversibility

We consider n-dimensional deterministic flows obtained by perturbing a gradient flow. We assume that the gradient flow admits a stable curve of stationary points, and thus if the perturbation is not too large the perturbed flow also admits an attracting curve. We show that the noise induced escape problem from a stable fixed point of this curve can be reduced to a one-dimensional problem: we can approximate the associated quasipotential by the one associated to the restricted dynamics on the stable curve. The error of this approximation is given in terms of the size of the perturbation.Comment: 25 pages, 2 figure

Transitions in active rotator systems: invariant hyperbolic manifold approach

Our main focus is on a general class of active rotators with mean field interactions, that is globally coupled large families of dynamical systems on the unit circle with non-trivial stochastic dynamics. Each isolated system is a diffusion process on a circle, with drift -delta V', where V' is a periodic function and delta is an intensity parameter. It is well known that the interacting dynamics is accurately described, in the limit of infinitely many interacting components, by a Fokker-Planck PDE and the model reduces for delta=0 to a particular case of the Kuramoto synchronization model, for which one can show the existence of a stable normally hyperbolic manifold of stationary solutions for the corresponding Fokker-Planck equation (we are interested in the case in which this manifold is non-trivial, that happens when the interaction is sufficiently strong, that is in the synchronized regime of the Kuramoto model). We use the robustness of normally hyperbolic structures to infer qualitative and quantitative results on the |delta|< delta0 cases, with delta0 a suitable threshold: as a matter of fact, we obtain an accurate description of the dynamics on the invariant manifold for delta=0 and we link it explicitly to the potential V . This approach allows to have a complete description of the phase diagram of the active rotators model, at least for |delta|< delta0, thus identifying for which values of the parameters (notably, noise intensity and/or coupling strength) the system exhibits periodic pulse waves or stabilizes at a quiescent resting state. Moreover, some of our results are very explicit and this brings a new insight into the combined effect of active rotator dynamics, noise and interaction. The links with the literature on specific systems, notably neuronal models, are discussed in detail.Comment: 29 pages, 4 figures. Version 2: some changes in introduction, added reference

LONG TIME BEHAVIOR OF MARKOV PROCESSES AND BEYOND

International audienceThis note provides several recent progresses in the study of long time behavior of Markov processes. The examples presented below are related to other scientific fields as PDE's, physics or biology. The involved mathematical tools as propagation of chaos, coupling, functional inequalities, provide a good picture of the classical methods that furnish quantitative rates of convergence to equilibrium.Cet article présente plusieurs progrés récents dans l'étude du comportement en temps long de certains processus de Markov. Les exemples présentés ci-dessous sont motivés par différentes applications issues de la physique ou de la biologie. Les outils mathématiques employés, propagation du chaos, couplage, inégalités fonctionnelles, couvrent un large spectre des techniques disponibles pour obtenir des comportements en temps long quantitatifs

Production, secretion and purification of a correctly folded staphylococcal antigen in Lactococcus lactis.

International audienceLactococcus lactis, a lactic acid bacterium traditionally used to ferment milk and manufacture cheeses, is also, in the biotechnology field, an interesting host to produce proteins of medical interest, as it is "Generally Recognized As Safe". Furthermore, as L. lactis naturally secretes only one major endogenous protein (Usp45), the secretion of heterologous proteins in this species facilitates their purification from a protein-poor culture medium. Here, we developed and optimized protein production and secretion in L. lactis to obtain proteins of high quality, both correctly folded and pure to a high extent. As proteins to be produced, we chose the two transmembrane members of the HtrA protease family in Staphylococcus aureus, an important extra-cellular pathogen, as these putative surface-exposed antigens could constitute good targets for vaccine development. A recombinant ORF encoding a C-terminal, soluble, proteolytically inactive and tagged form of each staphylococcal HtrA protein was cloned into a lactococcal expression-secretion vector. After growth and induction of recombinant gene expression, L. lactis was able to produce and secrete each recombinant rHtrA protein as a stable form that accumulated in the culture medium in similar amounts as the naturally secreted endogenous protein, Usp45. L. lactis growth in fermenters, in particular in a rich optimized medium, led to higher yields for each rHtrA protein. Protein purification from the lactococcal culture medium was easily achieved in one step and allowed recovery of highly pure and stable proteins whose identity was confirmed by mass spectrometry. Although rHtrA proteins were monomeric, they displayed the same secondary structure content, thermal stability and chaperone activity as many other HtrA family members, indicating that they were correctly folded. rHtrA protein immunogenicity was established in mice. The raised polyclonal antibodies allowed studying the expression and subcellular localization of wild type proteins in S. aureus: although both proteins were expressed, only HtrA1 was found to be, as predicted, exposed at the staphylococcal cell surface suggesting that it could be a better candidate for vaccine development. In this study, an efficient process was developed to produce and secrete putative staphylococcal surface antigens in L. lactis and to purify them to homogeneity in one step from the culture supernatant. This allowed recovering fully folded, stable and pure proteins which constitute promising vaccine candidates to be tested for protection against staphylococcal infection. L. lactis thus proved to be an efficient and competitive cell factory to produce proteins of high quality for medical applications

Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems

We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Friedlin--Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation.Comment: To appear in Journal of Statistical Physic