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, Francis Mairet, Rafael Muñoz-Tamayo, Elsa Rousseau.

Mathematical study of semi-discrete models

Semi-discrete models have shown their relevance in the modeling of biological phenomena whose nature presents abrupt changes over the course of their evolution [95] . We used such models and analysed their properties in several situations that are developed in 6.2.3 , 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 [59] or in the co-existence analysis of epidemiological strains [19] , [50] . 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 survives the 'winter' [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 [30] , 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.

The competition of several microalgal species was also studied in order to determine conditions that may give a competitive advantage to a species of interest. We study the competition for two species subject to photoinhibition at high light. This leads to a closed loop control strategy based on the regulation of the light intensity at the bottom of the reactor. The winning species is the one with the highest growth rate at high light. Then we show that the proposed controller allows the selection of a species of interest among n species [102] .

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 [2] . These systems can be represented by models having the general structure popularized by [69] , [74] , and based on an underlying reaction network:

We address two problems: the determination of the pseudo-stoichiometric matrix $K$ and the modeling 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 [68] , anaerobic digestion [87] , [114] and anaerobic digestion of microalgae [20] . Recently it was also used to reduce the ADM1 model in the case of winery effluent wastewater [88] .

Nonlinear observers

Participants : Jean-Luc Gouzé, Olivier Bernard, Francis Mairet.

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 [7] . 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 [105] . This approach was then extended to $n$-dimensional linear systems, leading to the design of interval observers in high dimensions [106] . Interval observers for non linear triangular systems satisfying Input to State Stability has been proposed [22] . Extension to time-delay systems have also been proposed [23] . The efficiency of the interval observer design, even with chaotic systems has been developed and applied considering parameters uncertainties of the system and biased output [108] , [105] .

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" [73] . These algorithms have been improved by the estimation of the observer gain providing the best estimate [40] , [21] . The approach has been applied to estimation of the microalgae growth and lipid production [101] .

These works are done in collaboration with Frédéric Mazenc (DISCO, Inria) and Marcelo Moisan (EMEL S.A., Chile).

Metabolic and genomic models

Participants : Jean-Luc Gouzé, Madalena Chaves, Alfonso Carta, Ismail Belgacem, Xiao Dong Li, Olivier Bernard, Wassim Abou-Jaoudé, Luis Casaccia, Caroline Baroukh, Rafael Muñoz-Tamayo, Jean-Philippe Steyer.

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. 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] .

Global stability for metabolic models and unreduced Michaelis-Menten equations

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 types of metabolic models 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 behavior in a more general genetic-metabolic loop [26] . Moreover, with the same techniques, 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 [34] , [45] , [70] .

Interconnections of Boolean modules: asymptotic and transient behavior

A biological network can be schematically described as an input/output Boolean module: that is, both the states, the outputs, and the inputs are Boolean. The dynamics of a Boolean network can be represented by an asynchronous transition graph, whose attractors describe the system's asymptotic behavior. 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 semi-attractors (states of the attractor with same output) of each module. In  [82] , the asymptotic graph was proposed, which is quite fast to compute and identifies all attractors of the interconnected system, but may also generate some spurious attractors. In [31] the cross graph is proposed, which exactly identifies the attractors of the interconnected system but is slower to compute. The asymptotic dynamics of high-dimensional biological networks can thus be predicted through the computation of the dynamics of two isolated smaller subnetworks. An application is, for instance, to interconnect four individual “cells” to obtain all the attractors of the segment polarity genes model in Drosophila.

Probabilistic approach for predicting periodic orbits in piecewise affine differential models

The state space of a piecewise affine system is partitioned into hyperrectangles which can be represented as nodes in a directed graph, so that the system's trajectories follow a path in a transition graph. Using this property we defined a transition probability between two nodes $A$ and $B$ of the graph, based on the volume of the initial conditions on the hyperrectangle $A$ whose trajectories cross to $B$ [15] . The parameters of the system can thus be compared to the observed or experimental transitions between two hyperrectangles. This definition is useful to identify sets of parameters for which the system yields a desired periodic orbit with a high probability, or to predict the most likely periodic orbit given a set of parameters, as illustrated by a gene regulatory system composed of two intertwined negative loops.

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 [35] .

E. coli modeling and control

In the framework of ANR project Gemco, we developed and analyzed a model of a minimal synthetic gene circuit, that describes part of the gene expression machinery in Escherichia coli, and enables the control of the growth rate of the cells during the exponential phase.

This model is a piecewise non-linear system with two variables (the concentrations of two gene products) and an input (an inducer). We studied the qualitative dynamics of the model and the bifurcation diagram with respect to the input. Moreover, an analytic expression of the growth rate during the exponential phase as function of the input has been derived. A relevant problem was that of parameters identifiability of this expression supposing noisy measurements of exponential growth rate. We presented such an identifiability study that we validated in silico with synthetic measurements [36] .

We also studied a model of the global cellular machinery designed by D. Ropers and collaborators (IBIS team, Grenoble). This model has 11 variables and many parameters ; we explored different techniques for reduction and simplification [56] , [57] .

Transition graph and dynamical behavior of piecewise affine systems

We investigated the links between the topology of the transition graph and the number and stability of limit cycles in a class of two-dimensional piecewise affine biological models. To derive these structure-to-dynamics principles, we use the properties of continuity, monotonicity and concavity of Poincare maps associated with transition cycles of the transition graph [64] .

Robust estimation for a hybrid model of genetic networks

State estimation problems with Boolean measurements for a classical negative loop genetic network governed by a piecewise affine (PWA) model have been studied in [39] . Observers are proposed for the cases where either full state or only partial state Boolean measurements are available. In the first case, sliding modes may occur, which leads to finite time convergence for the observer. In the second case, an algebraic computation is proposed to solve the initial condition inverse problem. The robustness of the observer for a parametric uncertain model is investigated, and we show that the error bound is proportional to the magnitude of the uncertainty.

Modeling the metabolic network in non balanced growth conditions

We have developed a new approach to represent the metabolic network of organisms for which the hypothesis of balanced growth is not satisfied [67] . This is especially true for microalgae which store carbon during the day and nitrogen during the night [44] . The proposed formalism is based on the assumption that some parts of the metabolic network satisfy the balance growth conditions, i.e. there is no accumulation of intermediate compounds. This hypothesis specifically applies to the main functions in the cell (respiration, photophosphorylation,...). Between two functions, some compounds can accumulate with storage/reuse kinetics. The resulting system is thus a slow-fast system.