Gilles Wainrib Email : gilles.wainrib [at] ens.fr
Recent preprints & publicationsBranching random walks on binary strings and application to adaptive immunity I. Balelli, V. Milisic and G. Wainrib, submitted [arxiv]During the germinal center reaction, B lymphocytes proliferate, mutate and differentiate, while being submitted to a powerful selection, creating a microevolutionary mechanism at the heart of adaptive immunity. We introduce and analyze a simplified mathematical model of the divisionmutation process, by considering random walks and branching random walks on graphs, whose structure reflects the associated mutation rules. In particular, we investigate how the combination of various division and mutation models influences the typical timescales characterizing the efficiency of state space exploration for these processes, such as hitting times and cover times. Beyond the initial biological motivation, this framework is not limited to the modelling of Bcell learning process in germinal centers, as it may be relevant to model other evolutionary systems, but also information propagation in networks, gossip models or epidemic processes.
The real Ginibre ensemble with k=0(n) real eigenvalues L.C. Garcia del Molino, K. Pakdaman, J.Touboul and G. Wainrib, submitted [arxiv]
We consider the ensemble of Real Ginibre matrices with a positive fraction α > 0 of real
eigenvalues. We demonstrate a large deviations principle for the joint eigenvalue density of such
matrices and we introduce a two phase loggas whose stationary distribution coincides with the
spectral measure of the ensemble. Using these tools we provide an asymptotic expansion for the
probability pnαn that an n × n Ginibre matrix has k = αn real eigenvalues and we characterize the
spectral measures of these matrices.
Mathematical modeling of lymphocytes selection in the germinal center V.Milisic and G.Wainrib, submitted [arxiv]Lymphocyte selection is a fundamental operation of adaptive immunity. In order to produce Blymphocytes with a desired antigenic profile, a process of mutationselection occurs in the germinal center, which is part of the lymph nodes. We introduce in this article a simplified mathematical model of this process, taking into account the main mechanisms of division, mutation and selection. This model is written as a nonlinear, nonlocal, inhomogeneous second order partial differential equation, for which we develop a mathematical analysis in the case of piecewiseconstant coefficients. We assess, mathematically and numerically, the performance of the biological function by evaluating the duration of this production process as a function of several parameters such as the mutation rate or the selection profile, in various asymptotic regimes.
Network modeling of Crohn's disease JeanMarc Victor, Gaëlle Debret,
Annick Lesne, Leigh Pascoe, Pascal Carrivain,
Gilles Wainrib, JeanPierre Hugot, submitted.Crohn's Disease (CD) is a complex
genetic disorder related to genetic and environmental risk factors. We modelled
the disease as a modular network of pathophysiological functions, each summarizing
geneenvironment interactions. The disease resulted from one or few specific
combinations of the modules' functional states. Network aging dynamics was able
to reproduce agespecific CD incidence curves and their variations over the past
century in Western countries. The model allowed translating the Odds Ratios
associated to atrisk alleles in terms of disease propensities of the
functional modules. Finally, this modelling was successfully applied to other
complex genetic disorders including
ulcerative colitis, ankylosing spondylarthritis, multiple sclerosis and
schizophrenia.
Research Interests
Probability theory / Dynamical systems limit theorems  large deviations  piecewisedeterministic Markov processes  singular perturbations  averaging principles  stochastic bifurcations random matrix theory  random walk on graphs  partial differential equations  random fields
Applications in theoretical biology / computer science Hodgkin Huxley models with stochastic ion channels  Action potential generation and propagation  Information transmission  Noiseinduced phenomena  Synchronization  Synaptic plasticity and learning Random neural networks  Links with machine learning, reservoir computing  Immune system modeling  Gene regulatory networksMy collaborators
Vuk Milisic Hatem Zaag
Laboratoire Analyse Géométrie et Applications (LAGA) Institut Galilée
Université Paris 13
99, avenue JeanBaptiste Clément
93430  Villetaneuse
Bureau D 306 Tel : 01 49 40 35 83 Fax : 01 49 40 35 68
Email : wainrib [at] math.univparis13.fr
Recent Announcements

new affiliation > new email !!
my new email adress is wainrib [at] math.univparis13.fr
Posted Sep 20, 2011, 7:30 AM by Gilles Wainrib

New email adress
My new email adress is:gwainrib [at] stanford.edu
Posted Sep 8, 2010, 11:00 AM by Gilles Wainrib

