Lectures
2020
- titre
- Comprendre la signalisation cellulaire à l'aide de modèles mathématiques
- article
- Master. France. 2020
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2019
- titre
- Vocabulaire et formalisme de modélisation de systèmes dynamiques en biologie
- article
- Doctorat. France. 2019
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2016
- titre
- Chemical Reaction Network Theory, notations and selected results
- article
- 2016
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2015
- titre
- Piecewise deterministic Markov processes, applications in biology
- article
- 3rd cycle. Mini-cours chercheur (Laboratoire Jacques Louis-Lions, Université Paris VI), 2015, 24 p
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Other publications
2015
- titre
- Boundary value for a nonlinear transport equation emerging from a stochastic coagulation-fragmentation type model
- article
- 2015
- resume
- We investigate the connection between two classical modelsof phase transition phenomena, the(discrete size) stochastic Becker-D ̈oring, a continous time Markov chain model, and the (continu-ous size) deterministic Lifshitz-Slyozov model, a nonlinear transport partial differential equation.For general coefficients and initial data, we introduce a scaling parameter and prove that the em-pirical measure associated to the stochastic Becker-D ̈oring system converges in law to the weaksolution of the Lifshitz-Slyozov equation when the parameter goes to 0. Contrary to previousstudies, we use a weak topology that includes the boundary ofthe state space (i.e.the sizex=0)allowing us to rigorously derive a boundary value for the Lifshitz-Slyozov model in the case of in-coming characteristics. The condition reads limx→0(a(x)u(t)−b(x))f(t,x)=αu(t)2wherefis thevolume distribution function, solution of the Lifshitz-Slyozov equation,aandbthe aggregationand fragmentation rates,uthe concentration of free particles andαa nucleation constant emergingfrom the microscopic model. It is the main novelty of this work and it answers to a question thathas been conjectured or suggested by both mathematicians and physicists. We emphasize that thisboundary value depends on a particular scaling (as opposed to a modeling choice) and is the resultof a separation of time scale and an averaging of fast (fluctuating) variables.We investigate the connection between two classical modelsof phase transition phenomena, the(discrete size) stochastic Becker-D ̈oring, a continous time Markov chain model, and the (continu-ous size) deterministic Lifshitz-Slyozov model, a nonlinear transport partial differential equation.For general coefficients and initial data, we introduce a scaling parameter and prove that the em-pirical measure associated to the stochastic Becker-D ̈oring system converges in law to the weaksolution of the Lifshitz-Slyozov equation when the parameter goes to 0. Contrary to previousstudies, we use a weak topology that includes the boundary ofthe state space (i.e.the sizex=0)allowing us to rigorously derive a boundary value for the Lifshitz-Slyozov model in the case of in-coming characteristics. The condition reads limx→0(a(x)u(t)−b(x))f(t,x)=αu(t)2wherefis thevolume distribution function, solution of the Lifshitz-Slyozov equation,aandbthe aggregationand fragmentation rates,uthe concentration of free particles andαa nucleation constant emergingfrom the microscopic model. It is the main novelty of this work and it answers to a question thathas been conjectured or suggested by both mathematicians and physicists. We emphasize that thisboundary value depends on a particular scaling (as opposed to a modeling choice) and is the resultof a separation of time scale and an averaging of fast (fluctuating) variables.
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2014
- titre
- From a stochastic Becker-Döring model to the Lifschitz-Slyozov equation with boundary value
- article
- 2014
- resume
- We deal with the convergence in law of the stochastic Becker-Döring process to the Lifschitz-Slyozov partial differential equation, up to a small scaling parameter. The former is a probabilistic model for the lengthening/shrinking dynamics of a finite number and discrete size clusters, while the latter is seen as its infinite number and continuous size extension. In the Becker-Döring model, the clusters are assumed to increase or decrease their size (number of particles in a cluster) by addition or subtraction of only one single particle at a time (stepwise coagulation and fragmentation) without regarding the space structure. More precisely, in this model, the transitions are assumed to be Markovian and actually related to some random Poisson point measures. The lengthening rates depend on the size, the number of clusters of this size and the number of free particles throught a Law of Mass Action. The fragmentation rates depend on the size and the number of clusters of this size, through a spontaneous shricking (exponential law). The evolution of the configuration of the system is then described thanks to its empirical measure. It starts with a finite number of clusters and particles. So that, the state space of the model is finite (but possibly large) and bounded by the number of particles and clusters of all possible sizes up to the maximal one (given by the total number of particles in the system). Under an appropriate scaling of the rates parameters, the number of monomers and the sizes of clusters, we construct a rescaled measure-valued stochastic process from the empirical measure of the Becker-Döring model. We prove the convergence in law of this process towards a measure solution of the Lifschitz-Slyozov equation. This equation is of transport type with a nonlinear flux coupling the particle variable. The necessity of prescribing a boundary value at the minimal size naturally appears in the case of incoming characteristics. The value of the latter is still an open-debated question for this continuous model. The probabilistic approach of this work allows us to rigorously derive a boundary value as a result of a particular scaling (as opposed to a modeling choice) of the original discrete model. The proof of this result is mainly based on an adiabatic procedure, the boundary condition being the result of a separation of time scale and an averaging of a fast (fluctuating) variable.
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2013
- titre
- Adiabatic reduction of models of stochastic gene expression with bursting
- article
- 2013
- resume
- This paper considers adiabatic reduction in both discrete and continuous models of stochastic gene expression. In gene expression models, the concept of bursting is a production of several molecules simultaneously and is generally represented as a compound Poisson process of random size. In a general two-dimensional birth and death discrete model, we prove that under specific assumptions and scaling (that are characteristics of the mRNA-protein system) an adiabatic reduction leads to a one-dimensional discrete-state space model with bursting production. The burst term appears through the reduction of the first variable. In a two-dimensional continuous model, we also prove that an adiabatic reduction can be performed in a stochastic slow/fast system. In this gene expression model, the production of mRNA (the fast variable) is assumed to be bursty and the production of protein (the slow variable) is linear as a function of mRNA. When the dynamics of mRNA is assumed to be faster than the protein dynamics (due to a mRNA degradation rate larger than for the protein) we prove that, with the appropriate scaling, the bursting phenomena can be transmitted to the slow variable. We show that the reduced equation is either a stochastic differential equation with a jump Markov process or a deterministic ordinary differential equation depending on the scaling that is appropriate. These results are significant because adiabatic reduction techniques seem to have not been applied to a stochastic differential system containing a jump Markov process. Last but not least, for our particular system, the adiabatic reduction allows us to understand what are the necessary conditions for the bursting production-like of protein to occur.
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2011
- titre
- On the bursting of gene products
- article
- 2011
- resume
- In this article we demonstrate that the so-called bursting production of molecular species during gene expression may be an artifact caused by low time resolution in experimental data collection and not an actual burst in production. We reach this conclusion through an analysis of a two-stage and binary model for gene expression, and demonstrate that in the limit when mRNA degradation is much faster than protein degradation they are equivalent. The negative binomial distribution is shown to be a limiting case of the binary model for fast ''on to off'' state transitions and high values of the ratio between protein synthesis and degradation rates. The gene products population increases by unity but multiple times in a time interval orders of magnitude smaller than protein half-life or the precision of the experimental apparatus employed in its detection. This rare-and-fast one-by-one protein synthesis has been interpreted as bursting.
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