Section: New Results

Mathematical methods and methodological approach to biology

Mathematical analysis of biological models

Participants : Jean-Luc Gouzé, Olivier Bernard, Frédéric Grognard, Ludovic Mailleret, Pierre Bernhard, Andrei Akhmetzhanov.

Mathematical study of semi-discrete models

Semi-discrete models have shown their relevance in the modelling of biological phenomena whose nature presents abrupt changes over the course of their evolution [77] . We used such models and analysed their properties in several situations that are developed in 6.2.3 , most of them requiring such a modelling 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 [58] or in the co-existence analysis of epidemiological strains [17] , [21] . Seasonality also plays a big role in the semi-discrete modelling required for the analysis of consumers' adaptive behaviour in seasonal consumer-resource dynamics, where only dormant offspring survives the 'winter' [12] , [52] .

Mathematical study of models of competing species

When several species are in competition for a single substrate in a chemostat, and when the growth rates of the different species only depend on the substrate, it is known that the generic equilibrium state for a given dilution rate consists in the survival of only one of the species. In [28] , we propose a model of competition of $n$ species in a chemostat, where we add constant inputs of some species. We achieve a thorough study of all the situations that can arise when having an arbitrary number of species in the chemostat inputs; this always results in a Globally Asymptotically Stable equilibrium where all input species are present with at most one of the other species.

Model design, identification and validation

Participants : Olivier Bernard, Jean-Luc Gouzé.

Model design and identification

One of the main families of biological systems that we have studied involves mass transfer between compartments, whether these compartments are microorganisms or substrates in a bioreactor, or species populations in an ecosystem. We have developed methods to estimate the models of such systems [62] . These systems can be represented by models having the general structure popularized by [61] , [65] , [66] , 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, most recently on activated sludge processes [60] , anaerobic digestion [71] , [87] and anaerobic digestion of microalgae [22] .

Nonlinear observers

Participants : Jean-Luc Gouzé, Olivier Bernard.

Interval observers

Interval observers give an interval estimation of the state variables, provided that intervals for the unknown quantities (initial conditions, parameters, inputs) are known [8] . We have extended the interval observer design to new classes of systems. First, we designed interval observers, even when it was not possible in the original basis, by introducing a linear, time-varying change of coordinates, [80] . This approach was then extended to $n$-dimensional linear systems, leading to the design of interval observers in high dimensions [25] . Extension to time-delay systems have also been proposed [26] . The combination of the observers has also been improved in the case where various types of interval observers are run in parallel in a so-called "bundle of observers" [64] . The approach has been applied to estimation of the microalgae growth and lipid production [39] .

In order to demonstrate the efficiency of the interval observer design, even with chaotic systems, a special application of the interval observer has been developed for Chua's chaotic systems. The interval estimation of the state variables are performed considering parameters uncertainties of the system and biased output [83] , [80] .

Metabolic and genomic models

Participants : Jean-Luc Gouzé, Madalena Chaves, Alfonso Carta, Ismail Belgacem, Xiao Dong Li, Olivier Bernard, Christian Breindl, Frédéric Grognard.

Qualitative control of piecewise affine models

In the control of genetic networks, the construction of feedback control laws is subject to many specific constraints, including positivity, appropriate bounds and forms of the input. In addition, control laws should be liable to implementation in the laboratory using gene and protein components. In this context, under the hypothesis that both the observations and control functions are qualitative (or piecewise constant), and using sliding mode solutions, we analysed the controllability and stabilizability with respect to either of the steady states, for a piecewise affine model of the bistable switch with single input. It is also possible to find a qualitative control law that leads the system to a periodic orbit passing through the unstable steady state [14] . Moreover, we designed some preliminary control laws for the negative loop with two genes, which has an oscillatory behaviour [54] .

Interconnections of Boolean modules: asymptotic behaviour

A biological network can be schematically described as an input/output Boolean module: that is, both the states, the outputs and inputs are Boolean. The dynamics of a Boolean network can be represented by an asynchronous transition graph, whose attractors describe the system's asymptotic behaviour. We have shown that the attractors of the feedback interconnection of two Boolean modules can be fully identified in terms of cross-products of the attractors of each module. Based on this result, a model reduction technique is proposed in [33] , to predict the asymptotic dynamics of high-dimensional biological networks through the computation of the dynamics of two isolated smaller subnetworks. Applications include a model of cell-fate decision (represented as an interconnection of two 3-input/3-output modules).

Structure estimation for unate Boolean models of gene regulation networks

Estimation or identification of the network of interactions among a group of genes is a recurrent problem in the biological sciences. Together with collaborators from the University of Stuttgart, we have worked on the reconstruction of the interaction structure of a gene regulation network from qualitative data in a Boolean framework. The idea is to restrict the search space to the class of unate functions. Using sign-representations, the problem of exploring this reduced search space is transformed into a convex feasibility problem. The sign-representation furthermore allows to incorporate robustness considerations and gives rise to a new measure which can be used to further reduce the uncertainties. The proposed methodology is demonstrated with a Boolean apoptosis signaling model  [70] .

Multistability and oscillations in genetic control of metabolism

Genetic feedback is one of the mechanisms that enables metabolic adaptations to environmental changes. The stable equilibria of these feedback circuits determine the observable metabolic phenotypes. Together with D. Oyarzun from Imperial College, we considered an unbranched metabolic network with one metabolite acting as a global regulator of enzyme expression. Under switch-like regulation and exploiting the time scale separation between metabolic and genetic dynamics, we developed geometric criteria to characterize the equilibria of a given network. These results can be used to detect mono- and bistability in terms of the gene regulation parameters for any combination of activation and repression loops [40] . We also find that metabolic oscillations can emerge in the case of operon-controlled networks; further analysis reveals how nutrient-induced bistability and oscillations can emerge as a consequence of the transcriptional feedback [27] .

Uniqueness and global stability for metabolic models

We are interested in the uniqueness and stability of the equilibrium of reversible metabolic models. For biologists, it seems clear that realistic metabolic systems have a single stable equilibrium. However, it is known that some type of metabolic systems can have no or multiple equilibria. We have made some contribution to this problem, in the case of a totally reversible enzymatic system. We prove that the equilibrium is globally asymptotically stable if it exists; we give conditions for existence and behaviour in a more general genetic-metabolic loop [84] .

Birhythmicity in the p53-Mdm2 network

The p53-Mdm2 network is one of the key protein module involved in the control of proliferation of abnormal cells in mammals. Recently, a differential model of the p53-Mdm2 biochemical network which shows birhythmicity has been proposed to reproduce the two experimentally observed frequencies of oscillations of p53. Our study aimed at investigating the mechanisms at the origin of this birhythmic behaviour. To do so, we approximated this continuous non-linear model into a lower dimensional piecewise affine model and performed a first return map analysis. Based on this analysis, an experimental strategy has been proposed to test the existence of birhythmicity in the p53-Mdm2 network [31] , [11] .

E. coli modelling and control

In the framework of ANR project Gemco, with the aim of better understanding how to build a controller for Escherichia coli growth, three reduced models of E. coli gene expression machinery have been developed: the wild-type model, the open-loop model and the closed-loop model. Each one of these models is made up of two piece-wise non-linear differential equations.

Notably, the wild-type model describes the qualitative dynamics of the unmodified bacterium in terms of two relevant protein concentrations (RNAP and CRP) and a carbon source as input, which can be either glucose or maltose. Bacteria prefer glucose, which leads to a higher growth rate in wild type.

The open-loop model describes the qualitative dynamics of RNAP end CRP concentrations when the gene encoding for RNAP is controlled externally by an IPTG-inducible promoter. This control, biologically implemented by our collaborators in Grenoble—yielding different E. coli growth rates depending on IPTG concentration—shows that a controller for the bacterial growth can be built acting on the gene expression machinery level. Moreover, lumped parameters related to RNAP dynamics and IPTG regulation function have been estimated by means of E. coli growth curves.

Finally, the closed-loop model implements a possible feedback control-loop able to theoretically generate inverse diauxie, i.e. higher growth on maltose than on glucose.

Observation problems of a class of genetic regulatory networks

A state reconstruction problem with Boolean measurements is considered for a piecewise affine genetic network model. The problem has two distinct aspects with respect to classical ones: 1) the model is a hybrid system, and 2) the measurements (of genes expression) are only qualitative due to the experimental techniques. A Luenberger-like observer is proposed which can present some sliding modes and has finite-time convergence. A transition graph is given for the coupled observer-nominal system. To minimize the convergence time, different convergence scenarios are discussed for optimizing the choice of initial condition of the observer [35] , [57] . Robustness of the observer is checked for two types of parametric perturbed systems: 1) the observer is used to identify an unknown but fixed variation on the synthesis coefficient, via an adaptive dichotomy algorithm; and 2) the observer is robust in practical sense for the model with an uncertainty on the threshold value [76] .