Section: New Results

Fields of application

Growth of marine plankton

Participants : Olivier Bernard, Jean-Luc Gouzé, Antoine Sciandra, Christophe Mocquet, Thomas Lacour, Eric Benoît, Jonathan Rault.

Experiments about the phytoplankton cellular cycle

We have run experiments to observe the response of a population of microalgal cells to various periodic light/dark or nitrate signals. The measurements performed with the diatom Thalassiosira weissflogii show the synchronicity of the cells for some conditions. These experiments support the hypothesis that uptake of nitrogen stops during cell division [11] .

Growth models of zooplankton

The model built in [66] was adapted to zooplankton. Some parts of the model had to be modified, most notably the death by starvation and the maintenance energy. The simulations of the model were compared to the measurements done using the Zooscan in the bay of Villefranche-sur-Mer. For this comparison, statistical tools were applied on the Villefranche data and some parameters were found in the literature. The model and the Villefranche data allowed us to identify other parameters [105] . This model was then integrated in a size-structured model of detritus (faeces of zooplankton for example) describing the sedimentation, which is a central piece of the carbon pump in the ocean. We also studied discrete size-structured models and compared them with continuous models. The discrete models are less numerically demanding, are more easily incorporated into biggest models, and can more often be studied analytically.

Zooplankton growth models were validated using data acquired in laboratory with various species of appendicularians [27] . These models represent the main pathways of filtered material transformation into somatic growth and reproduction. These models led to an estimate of optimal growth conditions depending on temperature and food, combined with ecological niches for each species. Transposed to the natural environment, these models turned out to be capable of simulating the worldwide biogeography of the species, which constitutes a further validation [28] .

Carbon fixation by coccolithophorids

A set of 18 models was developed and studied [79] , [78] to describe the coupling between photosynthesis and calcification for algae that are responsible for large carbon fluxes in the ocean. The qualitative study of this set showed that the standard hypotheses usually made by physiologists disagree with observed behaviors, since experiments have shown that an increase in the CO₂ partial pressure paradoxically leads to a decrease in the calcification rate.

One model was then included in an ocean model where a bloom of coccolithophorids was simulated [14] , [39] . It was shown that the uncertainty on the mecanisms driving calcification leads to uncertainties which are in the same range as the effect of an increase or the CO₂ partial pressure.

Modelling and optimization of lipid production

Participants : Olivier Bernard, Antoine Sciandra, Frédéric Grognard, Francis Mairet, Pierre Masci, Thomas Lacour.

In the framework of the ANR project Shamash, experiments have been carried out to study the effects of nitrogen limitation on the lipid production in a culture of microalgae (Isochrysis affinis galbana ) [90] . We have proposed a new model for lipid production by microalgae which describes accurately the complex behavior of the lipid quota [91] . This model highlights and explains the phenomenon of hysteresis in lipid production which has been experimentally verified.

On the other hand, a new dynamical model has been developped to describe microalgal growth in a photobioreactor under light and nitrogen limitations [38] [92] , [75] . The strength of this model is to take into account the strong interactions between the biological phenomena (effects of light and nitrogen on growth, photoadaptation...) and the radiative transfer in the photobioreactor (light attenuation due to the microalgae).

Using these two approaches, we have developped a model which describes lipid production in a photobioreactor under light limitation. This model is used to predict lipid production in the perspective of large scale biofuel production. Simpler models have also been developed and have been used to provide optimization strategies [94] . In this work, we focus on the effects of the reactor design (depth) and the operating conditions on the lipid productivity under day/night cycles.

Finally, an analysis of the potential environmental impacts of biodiesel production from microalgae has been realised using the life cycle assessment (LCA) methodology [26] . This study has allowed to identify the obstacles and limitations which should receive specific research efforts to make this process environmentally sustainable.

Coupling microalgae to anaerobic digestion

Participants : Olivier Bernard, Antoine Sciandra, Frédéric Grognard, Francis Mairet, Pierre Masci.

The ANR Symbiose project is aiming at coupling microalgae to anaerobic digestion in order to produce methane from CO₂ [62] , [34] . A model describing this process has been developped coupling and adapting existing models for each subsystem (microalgae growth [38] and anaerobic digestion [4] ). The mathematical analysis of this model is not straightforward due to its complexity. Nevertheless, the model's behavior has been analysed under operating conditions for which the model can be simplified [58] . This analysis has enhanced the understanding of the coupled system dynamics and guided the design of the process.

Bioprocesses

Participants : Jean-Luc Gouzé, Olivier Bernard, Frédéric Grognard.

Anaerobic digestion monitoring and control

Comore has developed models for anaerobic digestion processes and proposed dedicated methods for model selection [71] and calibration [88] . This work has mainly been focused on the design of control laws that stabilize Wastewater Treatment Plants (WWTPs), which tend to be unstable without feedback control [107] .

The new model linking the Henry constant k_La and the biogas flow-rate shows a linear relation between the dissolved CO₂ and the biogas quality (in terms of %CO₂ ), which gives new prospects for the control of the biogas quality. A strategy based on the regulation of the alkalinity has been proposed, and two feedback control laws were tested; a simple PI controller based on the measurement of the partial pressure of CO₂ , and a mixed law based on an advanced control of the alkalinity [89] . Experiments have been carried out at the LBE INRA Narbonne to validate these two controllers. The regulation strategies turn out to accurately keep the biogas quality constant so that it can be used as an energy source [88] , [89] . This result has been patented [74] .

Various asymptotic, interval based software sensors have also been developed based on the possible sets of measurements [100] , [99] , [88] , [101] ,[32] .

Models of ecosystems

Participants : Jean-Luc Gouzé, Frédéric Grognard, Sapna Nundloll.

Biological control

With L. Mailleret (URIH, INRA Sophia-Antipolis), we have a collaboration about biological control. The studied problem consists in evaluating the effect of periodic release of predators and periodic harvest on the population of pests. We investigate the minimal predator-budget that should be invested in order to ensure pest eradication. The model proposed in [104] has been shown to be valid for a pest-natural enemy couple present in the greenhouses of INRA and the proposed strategy has been experimentally shown to be the most efficient. More details on the methodological aspects of this problem are given in Section 6.1.4, Nonlinear Control.

Metabolic and genomic models

Participants : Olivier Bernard, Jean-Luc Gouzé, Frédéric Grognard, Wassim Abou Jaoudé, Ibrahima Ndiaye, Madalena Chaves, Eric Benoît.

Dynamics of genetic regulatory networks

We are studying a class of piecewise-linear dynamical systems, , where x is a n -vector of protein concentrations, the vector f(x) and matrix g(x) are piecewise constant and represent synthesis and degradation rates respectively. Piecewise-linear systems form the basis of an important class of models used for genetic regulatory networks, where the regulatory interactions between the genes are approximated as step functions. The use of step functions is motivated by the switch-like behavior seen experimentally in many of the interactions in gene expression and breakdown of proteins. The piecewise-linear models have the advantage that they are amenable to qualitative analysis and are well-suited to the qualitative character of the majority of experimental data from genetic regulatory networks. Current work in this project consists in continuing the work of Gouzé and Sari  [85] and de Jong et al   [82] by characterizing the equilibrium points and periodic orbits in this special class of systems. These methods and algorithms are used by the software GNA (Genetic Network Analyzer) developed by de Jong et al (HELIX, INRIA Rhône-Alpes). This year we have described and analyzed several systems made of intricated loops [60] . Moreover, larger and more realistic models for carbon growth of E. coli have been studied (thesis of I. Ndiaye).

Periodic solutions of models of genetic regulatory networks

An important family of piecewise-linear systems consists of systems that have a negative loop involving all variables in their state transition graph. Using theorems about monotone operators acting on positive variables, we have shown that this loop always corresponds to a unique, stable limit cycle [21] .

Moreover, we have generalized this result to more complex interaction graphs (multiple intricate loops of any sign, multiple thresholds...). Our main result is an alternative theorem showing that, if a sequence of regions is periodically visited by trajectories, then under some hypotheses, there exists either a unique stable periodic solution, or the origin attracts all trajectories in this sequence of regions [55] , [20] .

Control of genetic regulatory networks

Since recent biological techniques allow for the synthesis of more and more elaborate gene regulatory networks, it seems appropriate to develop control-theoretic methodologies for these networks. We have thus introduced new mathematical techniques for the control of piecewise-linear equations towards a prescribed behavior. Namely, we suppose that the piecewise constant terms f(x) and g(x) depend on an input vector u .

We elaborated on our previous work about control problems for this class of models, using also some recent results guaranteeing the existence and uniqueness of limit cycles (see above), based solely on a discrete abstraction of the system and its interaction structure. Our aim is to control the transition graph of the piecewise-affine (PWA) system to obtain oscillatory behaviour, which is of primary functional importance in numerous biological networks. We show how to control the appearance or disappearance of a unique stable limit cycle by hybrid qualitative action on the degradation rates of the PWA system, both by static and dynamic feedback [83] .

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, we analysed the controllability and stabilizability with respect to each steady state, for a piecewise-affine model of the bistable switch with a single input, and using piecewise constant control laws (constant in given regions of the state space, or constant for a given time interval)   [81] .

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 [103] .

Methods for qualitative analysis of genetic networks and model comparison

There are several methods to qualitatively study genetic models, which are adapted to the type and frequency of the experimental data available. These methods range over different degrees of description, from Boolean networks, to discrete multi-valued models, piecewise affine systems and continuous models. A brief overview can be found in [41] . To compare these different modelling approaches, we analyzed a genetic network involved in the response to carbon availability in E. coli  [54] . It is shown that a discrete multi-valued model can in general be obtained from a piecewise affine model. Then, a procedure is suggested to show that, under appropriately biological conditions, multi-valued discrete models can be written as strictly Boolean models. The advantage of a Boolean model for large systems is its tractability through graph theoretical and computational tools. Boolean models recover many of the properties of piecewise-affine models (steady states, oscillatory orbits). A special type of solutions which often appear in piecewise-affine models, namely the “sliding mode” solutions, can also be recovered, if the Boolean model is more finely analysed.

Interaction between signaling and gene expression networks

A simple model, consisting of one protein (x ) and one mRNA (y ), was developed and fully analyzed  [102] : the protein x activates transcription of the gene and also contributes to its own synthesis. The mRNA y (or its corresponding protein) contributes to the degradation of protein x . The idea that signaling or metabolic networks achieve an operational steady state much faster than the dynamics of gene expression was used to study the model in a “fast-slow” framework. It is shown how the signaling network can be regulated by switching between two operational modes (or steady states) in response to gene expression patterns. Conditions for the existence of an oscillatory cycle, as well as an estimate of its period, are provided. Using experimental data on the cell cycle of Xenopus laevis oocytes (both oscillatory response and bistability) some of the parameters were estimated, with the corresponding confidence intervals. This work is part of the thesis of I. Ndiaye.