Modélisation probabiliste en biologie moléculaire et cellulaire

Romain Yvinec 1
1 DRACULA - Multi-scale modelling of cell dynamics : application to hematopoiesis
CGMC - Centre de génétique moléculaire et cellulaire, Inria Grenoble - Rhône-Alpes, ICJ - Institut Camille Jordan [Villeurbanne], UCBL - Université Claude Bernard Lyon 1 : EA
Abstract : The importance of stochasticity in gene expression has been widely shown recently. We will first review the most important related work to motivate mathematical models that takes into account stochastic effects. Then, we will study two particular models where stochasticity induce interesting behavior, in accordance with experimental results: a bursting dynamic in a self-regulating gene expression model; and the emergence of heterogeneity from a homogeneous pool of protein by post-translational modification.\\ In Chapter I, we studied a standard gene expression model, at three variables: DNA, messenger RNA and protein. DNA can be in two distinct states, ''ON'' and ''OFF''. Transcription (production of mRNA) can occur uniquely in the ''ON'' state. Translation (production of protein) is proportional to the quantity of mRNA. Then, the quantity of protein can regulate in a non-linear fashion these production rates. We used convergence theorem of stochastic processes to highlight different behavior of this model. Hence, we rigorously proved the bursting phenomena of mRNA and/or protein. Limiting models are then hybrid model, piecewise deterministic with Markovian jumps. We studied the long time behavior of these models and proved convergence toward a stationary state. Finally, we studied in detail a reduced model, explicitly calculated the stationary distribution and studied its bifurcation diagram. Our two main results are 1) to highlight stochastic effects by comparison with deterministic model; 2) To give back a theoretical tool to estimate non-linear regulation function through an inverse problem.\\ In Chapter II, we studied a probabilistic version of an aggregation-fragmentation model. This version allows a definition of nucleation in agreement with biological model for Prion disease. To study the nucleation, we used a stochastic version of the Becker-Döring model. In this model, aggregation is reversible and through attachment/detachment of a monomer. The nucleation time is defined as a waiting time for a nuclei (aggregate of a fixed size, this size being a parameter of the model) to be formed. In this work, we characterized the law of the nucleation time. The probability distribution of the nucleation time can take various forms according parameter values: exponential, bimodal or Weibull. We also highlight two important phenomena for the mean nucleation time. Firstly, the mean nucleation time is a non-monotone function of the aggregation kinetic parameter. Secondly, depending of parameter values, the mean nucleation time can be strongly or very weakly correlated with the initial quantity of monomer. These characterizations are important for 1) explaining weak dependence in initial condition observed experimentally; 2) deducing some parameter values from experimental observations. Hence, this study can be directly applied to biological data. Finally, concerning a polymerization-fragmentation model, we proved a convergence theorem of a purely discrete model to hybrid model, which may be useful for numerical simulations as well as a theoretical study.
Document type :
Probabilités [math.PR]. Université Claude Bernard - Lyon I, 2012. Français
Contributor : Romain Yvinec <>
Submitted on : Wednesday, November 7, 2012 - 10:06:32 PM
Last modification on : Wednesday, July 29, 2015 - 1:29:10 AM
Document(s) archivé(s) le : Friday, February 8, 2013 - 3:45:24 AM


  • HAL Id : tel-00749633, version 1



Romain Yvinec. Modélisation probabiliste en biologie moléculaire et cellulaire. Probabilités [math.PR]. Université Claude Bernard - Lyon I, 2012. Français. <tel-00749633>




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