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Section: New Results

Parameter estimation for kinetic models of carbon metabolism in bacteria

Kinetic models capture the dynamics of the large and complex networks of biochemical reactions that endow bacteria with the capacity to adapt their functioning to changes in the environment. In comparison with the qualitative PL models described in Sections  6.1 and 6.3 , these more general classes of ODE models are intended to provide a quantitative description of the network dynamics, both on the genetic and metabolic level. New experimental techniques have led to the accumulation of large amounts of data, such as time-course measurements of metabolite, mRNA and protein concentrations and measurements of metabolic fluxes under different growth conditions. However, the estimation of parameter values in the kinetic models from these data remains particularly challenging in biology, mostly because of incomplete knowledge of the molecular mechanisms, noisy, indirect, heterogeneous, and partial observations, and the large size of the systems, with dynamics on different time-scales. We have addressed parameter estimation in the context of the analysis of the interactions between metabolism and gene expression in carbon metabolism in E. coli.

In collaboration with Matteo Brilli and Daniel Kahn (INRA and Université Claude Bernard in Lyon), we previously developed an approximate model of central metabolism of E. coli, as described in an article published in Bioinformatics in 2011. The model was based on the use of so-called linlog functions to approximately describe the rates of enzymatic reactions. More precisely, linlog models define reactions rates as proportional to both the enzyme concentrations and a linear combination of the logarithms of metabolite concentrations. The estimation of parameters in the linlog model from metabolomics, transcriptome, proteomics data sets required the development of a new approach, adapted to the occurrence of numerous missing values in the data sets. When applied to the above-mentioned linlog model, exploiting a high-throughput dataset published in the literature, we were able to obtain reasonable estimates of the 100 parameters.

The results of the above application also revealed the fundamental role played by the identifiability of the model parameters, an issue often overlooked in systems biology. This prompted us for a thorough investigation of the concepts of structural identifiability (in presence of perfect, idealized data), practical identifiability (in presence of noisy and limited amounts of data), and the relations among the two. In addition, we looked into the implications of this analysis for the reduction of nonidentifable to identifiable models. While having a solid mathematical basis, the study was tailored to the actual experimental practice, and resulted in a practical model reduction method that improves upon our previous approach in case of large measurement noise. This study, and the results from its application to both in-silico case studies and state-of-the-art datasets, were reported in a paper that has been accepted for publication in the Journal of Mathematical Biology [6] (see also [11] for a short version with preliminary results).

A second line of work is based on the use of classical kinetic models that are, in comparison with the above-mentioned linlog models, much reduced in scope (the focus is on the metabolic and genetic regulation of the glycolysis pathway) and granularity (individual reactions are lumped together). The models, developed by Delphine Ropers, have been calibrated using experimental data from the experimental part of the IBIS group for the gene expression measurements and the group of Jean-Charles Portais at INSA in Toulouse for the measurements of metabolism. The model with the estimated parameter values is currently being tested and used to understand some key mechanisms in the adaptation of E. coli to the exhaustion of glucose. The PhD thesis of Manon Morin, which started at the end of this year in the framework of a collaboration supported by a Contrat Jeune Scientifique INRA-Inria, will further develop these research directions.