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 [99] . We used such models and analysed their properties in several practical situations that are developed in Section 6.2.2 , most of them requiring such a modeling in order to take seasonality into account. Such is the case when 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 [53] or in the co-existence analysis of epidemiological strains [21] . Seasonality also plays a big role in the semi-discrete modeling required for the analysis of consumers' adaptive behavior in seasonal consumer-resource dynamics, where only dormant offspring survive the 'winter' [61] .

Model design, identification and validation

Participants : Olivier Bernard, Francis Mairet.

One of the main families of biological systems that we have studied involves mass transfer between compartments, whether these compartments are microorganisms or chemical species in a bioreactor, or species populations in an ecosystem. We have developed methods to estimate the models of such systems [79] . These systems can be represented by models having the general structure popularized by [78] , [84] , and based on an underlying reaction network:

We address two problems: the determination of the pseudo-stoichiometric matrix $K$ and the modelling of the reaction rates $r(\xi ,\psi )$.

In order to identify $K$, a two-step procedure has been proposed. The first step is the identification of the minimum number of reactions to be taken into account to explain a set of data. If additional information on the process structure is available, we showed how to apply the second step: the estimation of the pseudo-stoichiometric coefficients.

This approach has been applied to various bioproduction processes, among which activated sludge processes [77] , anaerobic digestion [92] , [106] and anaerobic digestion of microalgae [100] . Recently it was also used to reduce the ADM1 model in the case of winery effluent wastewater [24] .

Metabolic and genomic models

Participants : Jean-Luc Gouzé, Madalena Chaves, Alfonso Carta, Ismail Belgacem, Olivier Bernard, Caroline Baroukh, Rafael Muñoz-Tamayo, Jean-Philippe Steyer.

Global stability for metabolic models and full Michaelis-Menten equations

With techniques of monotone and compartmental systems, we studied full (i.e. not reduced by any time-scale argument) Michelis-Menten reactions or chains of reactions: we prove global stability when the equilibrium exists, and show that it may not exist. This fact has important consequences for reduction of metabolic systems in a coupled genetic-metabolic system [17] .

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 [13] .

Probabilistic approach for predicting periodic orbits in piecewise affine models

In the state transition graph, a transition probability between two nodes can be defined in terms of the parameters of the piecewise affine models. For a cyclic transition graph, this approach can be used to predict the most likely periodic orbit for a given a set of parameters [22] .

Growth rate models in bacteria: piecewise affine systems with a dilution term

We have extended the class of piecewise affine systems to deal with dynamics dependent on dilution due to cell growth rate. Considering that growth rate is determined by two limiting factors (RNA polymerase and ribosomes), in [42] we propose and analyze a switched system with two piecewise quadratic modes. This is part of the PhD thesis of Alfonso Carta, and done in collaboration with IBIS project-team.

Transcription and translation models in bacteria

We study detailed models of transcription and translation for genes in a bacterium. 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 [40] , [39] . We also study other models of the global cellular machinery. This is part of the PhD theses of Ismael Belgacem, Alfonso Carta, and done in collaboration with IBIS project-team. Moreover, in collaboration with IBIS, we studied and experimentally validated the time scale reduction of the classical two-step model for gene expression [51]

Analysis of circadian rhythms in cyanobacteria

A hierarchy of models (from Boolean to continuous) was used in [23] to successively characterize the wiring structure, qualitative dynamical properties, and then perform parameter estimation on a model describing the system responsible for the circadian rhythm of cyanobacteria.

Interconnections of Boolean modules: asymptotic and transient behaviour

The asymptotic dynamics of high-dimensional networks (e.g., genetic) can be obtained from the interconnection of two input/output Boolean subnetworks, and the analysis of their attractors. This computational cost reducing method is described in [34] . Some extensions include the characterization of the attractors of the interconnected system in terms of invariant sets.

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 [41] , joint work with C. Breindl and F. Allgöwer from the University of Stuttgart. The class of unate or canalizing Boolean functions is considered and represented by multi-affine polynomials, leading to a reformulation of the estimation problem as a mixed integer linear program.

Analysis of dynamical systems by combining discrete and continuous formalisms

The work reviewed in the HDR of M. Chaves [11] highlights methods of analysis that use and combine techniques from discrete and piecewise affine modeling formalisms, such as construction of the transition graph and its association with the parameters of the system. Some basic methods for generating a discrete transition graph from a given continuous system are described in the internship project of F. Todoran [75] .

State estimation for gene networks

We address state estimation for gene regulatory networks with intrinsic and extrinsic noise at the level of single cells. We take the Chemical Master Equation (CME) with random parameters as a reference modeling approach, and investigate the use of stochastic differential model approximations for the construction of practical real-time filters (based on non-linear Kalman filtering) [43] . This is a collaboration with Ibis team.

Modelling the metabolic network in non balanced growth conditions

On the basis of the knowledge of the metabolic network, we propose a new methodology to go beyond the “balanced growth paradigm”( assuming that there is no storage within the cell). We have therefore a tool to represent the possible storage of some key biochemical compounds. This approach was applied to describe the effect of both a light cycle and a nitrogen starvation on the lipid accumulation [37] . The first stage of the approach consists in splitting the metabolic network into sub-networks, which are assumed to satisfy balanced growth condition. The left metabolites interconnecting the sub-networks are allowed to behave dynamically. Then, thanks to Elementary Flux Mode analysis, each sub-network is reduced to macroscopic reactions, for which simple kinetics are assumed. This approach was applied to the accumulation of lipids and carbohydrates of the microalgae Tisochrysis lutea under day/night cycles. The resulting model described accurately experimental data obtained in day/night conditions; it efficiently predicts the accumulation and consumption of lipids and carbohydrates.