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, Bapan Ghosh.

Semi-discrete models have shown their relevance in the modeling of biological phenomena whose nature presents abrupt changes over the course of their evolution [99] . We used such models and analyzed their properties in several practical situations that are developed in Section 6.3.2 , 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 [62] , or through the introduction of biological control agents [45] , [60] , [54] . 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 eco-evolutionary dynamics of plant pathogens [25] , [59]

Model reduction and sensitivity analysis

Participant : Suzanne Touzeau.

Dynamic models representing complex biological systems with numerous interactions can reach high dimensions and include complex nonlinearities. Especially if data are scarce, identifying the model parameters is then a challenge. So we designed an ad-hoc method based on global sensitivity analysis to simplify the model and determine the most influential parameters. It was applied to a within-host immunological model [30] , [61] . This application was part of Natacha Go's PhD thesis, supervised by S. Touzeau and C. Belloc (BioepAR, INRA & Oniris Nantes) [90] .

Metabolic and genomic models

Participants : Jean-Luc Gouzé, Madalena Chaves, Alfonso Carta, Ismail Belgacem, Olivier Bernard, Caroline Baroukh, Jean-Philippe Steyer, Diego de Pereda Sebastian, Francis Mairet.

Continuous models analysis

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. With techniques of monotone systems, and time scale hypotheses, we can show the stability of the fast part of these systems, and reduce them to much smaller models [49] , [48] , [47] . We also study other models of the global cellular machinery. This is part of the PhD theses of Ismael Belgacem and Alfonso Carta [12] , and done in collaboration with Inria IBIS project-team.

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.

Analysis and reduction of biochemical models In collaboration with D. Ropers (Inria IBIS project team), we address the problem of reduction of large biochemical networks, to decompose the dynamic behavior of the whole system into simpler models. This is the subject of the thesis of S. Casagranda.

Design of a bistable switch to control cellular uptake In joint work with Diego Oyarzún (Imperial College), we explore the idea of constructing a synthetic bistable system using an unbranched metabolic chain with a global enzyme regulator. Bistability can be achieved by choosing an appropriate pattern of regulation and deriving conditions on the promoter dynamic ranges to guarantee a bistable uptake flux. This work started during the visit of Diego to Biocore in October 2014.

Analysis of signaling pathways leading to apoptosis In joint work with Jérémie Roux (Marie Curie Fellow, IRCAN Nice), a cascade of signaling modules leading to apoptosis (or programmed cell death) was implemented and studied through simulations. The goal of this work is to determine whether, and at which stage in the pathway, the system may exhibit bistability. This was the work of Xiao Han's internship.

Hybrid models analysis

Piecewise quadratic systems for studying growth rate in bacteria The class of piecewise affine systems was extended to deal with dynamics dependent on dilution due to cell growth rate, leading to switched-piecewise quadratic systems  [85] . These new systems use an expression for growth rate that may depend on any number of variables and have several quadratic modes. The behavior of piecewise quadratic systems introduces new features, notably regarding solutions at the thresholds when the vector fields are opposing: not only sliding mode solutions but also oscillatory behavior may happen. Part of this work is in the PhD thesis of Alfonso Carta [12] .

Attractor computation using interconnected Boolean networks The method developed in [10] has been extended towards a better characterization of the attractors of the interconnected system in terms of invariant sets [26] . The method was used to test growth rate models in E. Coli using Boolean networks.

Analysis of circadian rhythms in cyanobacteria The model describing the system responsible for the circadian rhythm of cyanobacteria previously proposed in  [86] has been improved in [50] . Here, we have tested the robustness of the circadian rhythm with respect to the perturbations inherent to the noisy environment of the cell, including cell growth and division. The interconnection between two models was studied: circadian rhythm and a stochastic model for cell division.

Structure estimation for Boolean models of gene regulation networks The problem of estimating Boolean models of gene networks from few and noisy measurements is addressed in  [84] , joint work with C. Breindl and F. Allgöwer from the University of Stuttgart. The class of unate or canalizing Boolean functions has been further considered and represented by multi-affine polynomials, leading to a reformulation of the estimation problem as a mixed integer linear program.

Structural principles for the existence of limit cycles in two-dimensional piecewise affine models Using concavity and continuity properties of Poincaré maps, we have derived some structural principles which link the topology of the transition graph to the existence, number and stability of limit cycles in a class of two-dimensional piecewise affine biological models of genetic networks [14] .

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 [52] , in collaboration with Inria IBIS project-team.

Estimation of biological models In a joint work with Diego de Pereda (visiting PhD student), we studied observers and interval observers for models of glucose concentration in diabetes.