Section: New Results

Mathematical methods and methodological approach to biology

Mathematical analysis of biological models

Mathematical study of semi-discrete models

Participants : Jean-Luc Gouzé, Frédéric Grognard, Ludovic Mailleret, Pierre Bernhard, Elsa Rousseau, Nicolas Bajeux.

Semi-discrete models have shown their relevance in the modeling of biological phenomena whose nature presents abrupt changes over the course of their evolution [96] . We used such models and analyzed their properties in several practical situations that are developed in Section 7.2.3 , some of them requiring such a modeling to describe external perturbations of natural systems, and others to take seasonality into account. External perturbations of interacting populations occur when some individuals are introduced or removed from a natural system, which occurs frequently in pest control applications, either through the direct removal of pests, or through the introduction of biological control agents [71] ,[27] . Seasonality is an important property of most agricultural systems in temperate environments since the year is divided into a cropping season and a `winter' season, where the crop is absent, as in our analysis of the sustainable management of crop resistance to pathogens [25] or in the dynamics of plant pathogens [50] .

Model reduction and sensitivity analysis

Participants : Suzanne Touzeau, Jean-Luc Gouzé, Stefano Casagranda, Victor Bernal Arzola.

Analysis and reduction of biochemical models. Dynamic models representing complex biological systems with numerous interactions can reach high dimensions and include complex nonlinearities. A model reduction method based on process weighing and pruning was developed and implemented on various models (ERK signaling pathway, circadian rythms in Drosophila) [41] . A global sensitivity analysis was performed to check the method robustness against parameter uncertainty and variability. This work is part of Stefano Casagranda's ongoing PhD thesis and is also a collaboration with Bayer (Sophia-Antipolis).

Parameter identification in compartmental systems. In collaboration with F. Dayan (R&D Manager, Dassault Systèmes), we worked on practical problems of identifiability of parameters in linear pharmacokinetic models. This was the subject of the internship of V. A. Bernal [58] .

Metabolic and genomic models

Participants : Jean-Luc Gouzé, Madalena Chaves, Ismail Belgacem, Olivier Bernard, Stefano Casagranda, Francis Mairet, Sofia Almeida.

Continuous models analysis

Piecewise quadratic systems for studying growth rate in bacteria. These new systems (first introduced in  [82] ) use an expression for growth rate that may depend on any number of variables and have several quadratic modes. Relative to the “classical” piecewise affine systems, this new formulation allows the existence of sliding motion as well as oscillatory behaviour for solutions at the thresholds where the vector fields are opposing [21] .

Transcription and translation models in bacteria. We study detailed models of transcription and translation for genes in a bacterium, in particular the model of gene expression of RNA polymerase. We also study other models of the global cellular machinery. This is part of the PhD theses of Ismael Belgacem [11] and Stefano Casagranda, and done in collaboration with Inria IBIS project-team, in particular with D. Ropers.

Design of a bistable switch to control cellular uptake. In a joint work with Diego Oyarzún (Imperial College), we analyse the construction of a synthetic bistable system using an unbranched metabolic chain with a global enzyme regulator, as an application of  [109] . Bistability can be achieved by choosing an appropriate pattern of regulation. Robustness conditions are given in terms of the promoter dynamic ranges to guarantee a bistable uptake flux [35] .

A reduced model for the mammalian cell cycle. We focused on identifying and analyzing the main mechanisms behind the cell cycle and proposed a mathematical model to describe them. This reduced model successfully reproduces oscillatory behaviors including the progress towards a mitosis phase, and then mitosis itself, characterized by an increase in cyclin B. The model was the topic of a poster at the Signalife Workshop [68] . This is a collaboration with F. Delaunay (Ibv Nice) in the framework of Labex Signalife.

Hybrid models analysis

Attractor computation using interconnected Boolean networks. During the visit of Daniel Figueiredo, we have worked on an extension of the method proposed in  [83] . The idea is to not only use the attractors but also an appropriate set of strongly connected components in the computation of the asymptotic graph  [115] . Numerical simulations show a great improvement in the problem of discarding spurious attractors.

Periodic orbits in non monotonic negative feedback circuits. We study the occurrence of periodic solutions in an $n$-dimensional class of negative feedback systems defined by smooth vector fields with a window of not necessarily monotonic activity. By circumscribing the smooth system by two piecewise linear ones, we show there exists an invariant toroidal region which contains a periodic orbit of the original smooth system [37] .

Estimation and control

Optimal allocation of resources in a bacterium. We study by techniques of optimal control the optimal allocation between metabolism and gene expression during growth of bacteria [85] , in collaboration with Inria IBIS project-team.

Control of a model of synthesis of a virulence factor. In collaboration with J.-A. Sepulchre (INLN Nice), we model the production of a virulence factor by a bacterium in a continuous stirred tank reactor. The production of this enzyme is genetically regulated, and degrades a polymeric external substrate into monomers. A nonlinear control is built [48] .