Section: New Results

Stochastic modeling and identification of gene regulatory networks in bacteria

At the single-cell level, the processes that govern gene expression are often better described by stochastic models. Modern techniques for the real-time monitoring of gene expression in single cells enable one to apply stochastic modelling to study the origins and consequences of random noise in response to various environmental stresses, and the emergence of phenotypic variability. The potential impact of single-cell stochastic analysis and modelling ranges from a better comprehension of the biochemical regulatory mechanisms underlying cellular phenotypes to the development of new strategies for the control of cell populations and even of single cells. General modeling paradigms, such as the Chemical Master Equation, exist for the description of stochastic dynamics at the single-cell level. However, due to the complexity of the interactions, current studies have often preferred to focus on specific cases of interest by ad-hoc modeling and analysis. In addition, theoretical and practical challenges inherent in the inference of stochastic models from biological experimental data have limited the development of general identification approaches.

Work in IBIS on the probabilistic modeling of gene expression and interaction dynamics at the level of individual cells is centered around two main challenges. On the one hand, we address identification from microscopy data and analysis of the arabinose uptake dynamics in E. coli upon glucose exhaustion. Starting from a reduced arabinose uptake model, Eugenio Cinquemani and Michel Page are working on methods for the estimation of unknown stochastic model parameters from statistical population snapshot data collected by fluorescence microscopy experiments. Analysis of the model focuses on the problem of model-based real-time single-cell state estimation, with feedback control applications in mind, in collaboration with Alfonso Carta (BIOCORE). Based on a stochastic model reflecting switch-like dynamics in the form of sigmoidal reaction rates, taking a Chemical Master Equation model with cell-dependent parameters as a gold standard desdcription of the system, a Chemical Langevin Equation approximation is proposed as a convenient approximation of the model for observer design purposes. On top of this model approximation, a so-called Square-Root Unscented Kalman filter (SRUKF) is designed. Based on simulations of a realistically tuned model, SRUKF is found to perform as good as much heavier particle filters based on the gold-standard model. Results were presented at the European Control Conference (ECC) in 2013 [11] , where we also showed that including extrinsic noise effects explicitly in the estimation process allows one to improve the knowledge of the hidden states.

On the other hand, we investigate the use mixed effects-modelling and identification techniques to characterize single-cell profiles in isogenic cell populations. Mixed-effects models are hierarchical models where parametric response profiles of individuals is subject to inter-individual parameter variability following a common population distribution. In collaboration with Gregory Batt (CONTRAINTES) and Giancarlo Ferrari-Trecate (University of Pavia, Italy), we are adapting and applying existing procedures from pharmacokinetics to the context of microfluidic data, with focus on the budding yeast response to osmolarity shocks. The first results of the work were presented at the European Control Conference (ECC) this year [12] . Rigorous model identification and validation steps are performed on data from real-time control experments performed in Pascal Hersen's lab at Université Paris Descartes, for both mixed-effects modelling and for the competing method of moment-based identification. Results show the tendency of mixed-effects modelling to avoid overfitting for this system, trading fitting performance for validation performance and hence predictive capabilities. The work is being further developed and the collaboration tightened by the ongoing visit of Andres Gonzalez, PhD candidate at the University of Pavia, to CONTRAINTES and IBIS. A first journal publication is in preparation, which will be followed by extensions and refinements of the method.

In parallel, work concerning the study of noise propagation in gene regulatory networks is carried out in collaboration with Irina Mihalcescu (Université Joseph Fourier). Finally, collaboration of Eugenio Cinquemani with Marianna Rapsomaniki, PhD student affiliated with Zoi Lygerou (University of Patras, Greece) and John Lygeros (ETH Zürich, Switzerland), has been devoted to the analysis of data from Fluorescence Recovery After Photobleaching (FRAP) experiments. It has given rise to a novel method for reconstructing physical diffusion and immobilization parameters at the level of single cells. The method has been applied to nuclear species of mammalian cells and results are part of a journal paper under revision.