Section: New Results

Fields of application

Bioenergy

Modelling of microalgae production

Participants : Olivier Bernard, Antoine Sciandra, Frédéric Grognard, Philipp Hartmann, Ghjuvan Grimaud, Quentin Béchet, David Demory, Hubert Bonnefond, Jean-Philippe Steyer, Francis Mairet.

Experimental developments

Experiments have been carried out to study the effects of nitrogen limitation on the lipid production in microalgae [28] and support model development. These experiments have been carried out in the Lagrangian simulator, under constant or periodic light and temperature, varying the total amount of light dose in the day. The response in terms of storage carbon (triglycerides and carbohydrates) has been observed.

Other experiments were carried out to reproduce the light signal perceived by a cell in a raceway pond [89] , derived from hydrodynamical studies [55] . An electronic platform was developed to reproduce this high frequency light signal. The experiments show that the microalgae adapt their pigments to the average light that they have received [28] .

The effect in the cell cycle of both the light periodic signal, the temperature and a nitrogen limitation were studied. The strong interactions between the different phases of the cell cycle through checkpoints was highlighted [106] . Temperature turned out to play a key role in modulating metabolic fluxes and synchronization.

The effect of cement flue gas on microalgae growth has been tested. It was demonstrated that this CO${}_2$ source can be used to feed microalgal industrial cultures [114] .

Finally a new methodology to measure cell viability has been set up. This approach is very promising to distinguish between net and gross growth rate [22] .

These works have been carried out in collaboration with A. Talec, S. Rabouille, E. Pruvost and C. Combe (CNRS/UPMC -Oceanographic Laboratory of Villefranche-sur-Mer).

In collaboration with the IFREMER-PBA team (Nantes) we contributed to a study of the possible associations between microalgae and bacteria to enhance overall productivity [98] .

Metabolism of carbon storage and lipid production

A macroscopic model for lipid production by oleaginous microalgae [7] has been previously proposed. This model describes the accumulation of neutral lipids (which can be turned into biofuel), carbohydrates and structural carbon. A metabolic model has been set up and validated for the microalgae Isochrysis luthea. It predicts carbohydrate and lipid accumulation, under conditions of light/dark cycles and/or nitrogen deprivation [78] , [88] , [16] .

Modeling the coupling between hydrodynamics and biology

In collaboration with the Inria ANGE team, a model coupling the hydrodynamics of the raceway (based on multilayer Saint-Venant system) with microalgae growth was developed [83] . This model is supported by the work of ANGE aiming at reproducing the hydrodynamics of the raceway, with a specific attention to the effect of the paddle wheel on the fluid [55] .

Modeling the photosynthesis response to fast fluctuating light

The impact of the hydrodynamics on the light perceived by a single cell was studied thanks to fluid dynamics simulations of a raceway pond [34] . The light signals that a cell experiences at the Lagrangian scale, depending on the fluid velocity, were then estimated. A Droop-Han model was used to assess the impact of light fluctuation on photosynthesis. A new model accounting for photoacclimation was also proposed [96] . Single cell trajectories were simulated by this software, and the effect on photosynthesis efficiency was assessed using models of photosynthesis [95] . These results were compared to experimental measurements where the high frequency light was reproduced [89] .

We also developed a model to reproduce the fluorescence of microalgae during a PAM protocol [51]

Modeling microalgae production processes

The integration of different models developed in the group [81] , [101] , [7] was performed to represent the dynamics of microalgae growth and lipid production in raceway systems, on the basis of the dynamical model developed to describe microalgal growth in a photobioreactor under light and nitrogen limitations. The strength of this model is that it takes into account the strong interactions between the biological phenomena (effects of light and nitrogen on growth, photoacclimation ...), temperature effect [82] , [111] and the radiative transfer in the culture (light attenuation due to the microalgae).

Using these approaches, we have developed a model which predicts lipid production in raceway systems under varying light, nutrients and temperature [109] . This model is used to predict lipid production in the perspective of large scale biofuel production. It was also used to assess the potential of France for microalgae, when taking into account the actual 2012 meteorology at the scale of France the use of lands, slope, proximity of nutrients and CO${}_2$ [73] .

In the framework of the ANR project Purple Sun, we develop an innovative system for microalgae production: a raceway pond under a greenhouse with semi-transparent photovoltaic panels. To this end, we include in the microalgae model the effect of light wavelength, and we develop a thermic model of the system in order to estimate the culture temperature.

Finally, we provide guidelines for the design of experiments with high informative content that allows an accurate estimation of the parameters concerning the effect of temperature and light on microalgae growth. The optimal experiment design problem was solved as an optimal control problem. E-optimal experiments were obtained by using two discretization approaches namely sequential and simultaneous. Simulation results showed the relevance of determining optimal experimental inputs for achieving an accurate parameter estimation [39] .

Nitrogen fixation by nitrogenotrophs

The fixation of nitrogen by Croccosphera watsonii was represented with a macro metabolic model [92] . The main fluxes of carbon and nitrogen are represented in the cell. The accumulation of starch during the day to fuel the nitrogenase working in the absence of oxygen during the night was the key process to explain the nitrogen fixation. The strong influence of the cell cycle was also included in the model. Finally, the model was calibrated and validated with the data of 3 experiments carried out with different duration of the light period and daily dose. The model succeeded to efficiently reproduce the experimental data.

This work is done in collaboration with Sophie Rabouille (CNRS-Oceanographic Laboratory of Villefranche-sur-Mer).

Modeling thermal adaptation in microalgae

We have used the Adaptive Dynamics theory to understand how temperature drives evolution in microalgae. For a constant temperature, we have shown that the optimal temperature trait tends to equal the environment temperature. We then study the case where the temperature is periodically fluctuating [53] . We now use this method at the scale of the global ocean, validating our approach with experimental data sets from 194 species.

Including phytoplankton photoadaptation into biogeochemical models

The complexity of the marine ecosystem models and the representation of biological processes, such as photoadaptation, is very challenging to tackle so that their representation remains an open question. We compared several marine ecosystem models with increasing complexity in the phytoplankton physiology representation in order to assess the consequences of the complexity of photoadaptation models in biogeochemical model predictions. Three models of increasing complexity were considered, and the models were calibrated to reproduce ocean data acquired at the Bermuda Atlantic Time-series Study (BATS) from in situ JGOFS (Joint Global Ocean Flux Study) data. It turns out that the more complex models are trickier to calibrate and that intermediate complexity models, with an adapted calibration procedure, have a better prediction capability [77] , [15] .

This work is done in collaboration with Sakina Ayata (UPMC-Oceanographic Laboratory of Villefranche-sur-Mer).

Control and Optimization of microalgae production

On-line monitoring

Interval observers give an interval estimation of the state variables, provided that intervals for the unknown quantities (initial conditions, parameters, inputs) are known [91] . Several developments were carried out in this direction to improve the design and performances of interval observers, and accounting for a specific structure (i.e. triangular) or property (i.e. Input to State Stable), [38] . Interval observers were designed for the estimation of the microalgae growth and lipid production within a production process [37] and validated experimentally [36] .

Optimization of the bioenergy production systems

Based on simple microalgae models, analytical optimization strategies were proposed. We first focused on the optimal operating conditions for the biomass productivity under day/night cycles using Pontryagin's maximum principle (assuming a periodic working mode) [32] .

On the other hand, we assessed strategies for optimal operation in continuous mode using the detailed model for raceways [108] , [109] . Two strategies were developed. The first one consists in solving numerically an optimal control problem in which the input flow rate of the raceway is calculated such that the productivity in microalgae biomass is maximized on a finite time horizon. In the second strategy, we translated the optimization problem into a regulation problem. We proposed a simple operational criterion that when integrated in a strategy of closed-loop control allows to attain biomass productivities very near to the maximal productivities obtained with the optimal control. We demonstrated that the practical advantages for real implementation makes our proposed controller a suitable control strategy for optimizing microalgae production in raceways.

We also propose a nonlinear adaptive controller for light-limited microalgae culture, which regulates the light absorption factor (defined by the ratio between the incident light and the light at the bottom of the reactor). We show by numerical simulation that this adaptive controller can be used to obtain near optimal productivity under day-night cycles [103] .

Interactions between species

Large scale culture of microalgae for bioenergy involves a large biodiversity (different mutants, invasion, growth-promoting bacteria [98] ...). Control of such systems requires to consider the interactions between the different species. Such systems involve bacteria and microalgae, and the competition between these organisms can have several equilibrium points, which can be studied with Monod, Contois and Droop type models [33] .

In the framework of the ANR Facteur 4 project, we propose to drive this competition exploring different strategies in order to select species of interest.

We have proposed an adaptive controller which regulates the light at the bottom of the reactor [104] . When applied for a culture with $n$ species, the control law allows the selection of the strain with the maximum growth rate for a given range of light intensity. This is of particular interest for optimizing biomass production as species adapted to high light levels (with low photoinhibition) can be selected.

Strategies to improve the temperature response have been proposed. First we modeled the adaptive dynamics for a population submitted to a variable temperature [53] . This was then used to design experiments aiming at enlarging the thermal niche of a species. Experiments with periodic temperature stresses are currently carried out at the LOV.

Finally, in a more theoretical framework, we studied how to select as fast as possible a given species in a chemostat with two species at the initial instant. Using the Pontryagin maximum principle, we have shown that the optimal strategy is to maintain the substrate concentration to the value maximizing the difference between the growth rates of two species [17] . We now try to extend this result for $n$ species with mutations.

Design of ecologically friendly plant production systems

Controlling plant pests

Participants : Frédéric Grognard, Ludovic Mailleret, Suzanne Touzeau, Nicolas Bajeux, Bapan Ghosh.

Optimization of biological control agent introductions

The question of how many and how frequently natural enemies should be introduced into crops to most efficiently fight a pest species is an important issue of integrated pest management. The topic of natural enemies introductions optimization has been investigated for several years [6] [110] , unveiling the crucial influence of within-predator density dependent processes. Since parasitoids may be more prone to exhibit positive density dependent dynamics rather than negative ones, which are prevalent among predatory biocontrol agents,the current modeling effort consists in studying the impact of positive predator-predator interactions on the optimal introduction strategies (PhD of Nicolas Bajeux, [45] ). The influence of the spatial structure of the environment on biological control efficacy has also been investigated; first results indicate that spatial structure tends to influence it in quite a same way as intra-specific competition does [60] . An extension of that modeling framework was also studied, that considered state dependent impulsive feedback for the stabilization of a positive equilibrium [54] .

Connected research on the influence of space on the establishment capacities of biological control agents is also being pursued both through computer simulations and laboratory experiments on parasitoids of the genus Trichogramma. This is the topic of the PhD thesis of Thibaut Morel Journel (UMR ISA); in particular, we show how landscape connectivity or spatial heterogeneity shape establishment dynamics in spatially structured environments [63] , [64] , [65] .

Plant compensation, pest control and plant-pest dynamics

Introducing a plant compartment into our models, we first focused on plant-insect interactions and showed how the level and timing of the pest invasion and pests control interventions could have important effects on the plant's growth pattern and its final biomass. We then modeled plant compensation, which is the process by which some plants respond positively to recover from the effects of pest injury. We have shown that depending on plants and pests characteristics, as well as the level of pest attack, plant overcompensation may or may not happen [35] . Experiments have then been held at UMR ISA on tomato plants facing tutta absoluta invasion; tendencies to compensation have been evidenced, but need to be confirmed through larger scale experiments.

This work is part of the PhD thesis of Audrey Lebon (Cirad), supervised in collaboration with Yves Dumont (Cirad), which has been defended in December 2014.

Controlling plant pathogens

Participants : Frédéric Grognard, Ludovic Mailleret, Suzanne Touzeau, Elsa Rousseau.

Sustainable management of plant resistance

Because in addition to being eaten, plants can also get sick, we studied other forms of biological control dedicated to fight plant pathogens. One such method is the introduction of plant strains that are resistant to one pathogen. This often leads to the appearance of virulent pathogenic strains that are capable of infecting the resistant plants. It is therefore necessary to find ways to protect the durability of such resistances, which are a natural exhaustible resource. Experiments were conducted in INRA Avignon, followed by high-throughput sequencing (HTS) to identify the dynamics of several virus strains in competition within host plants. Different plant genotypes were chosen for their contrasted effects on genetic drift and selection they induce on virus populations. Those two evolutionary forces can play a substantial role on the durability of plant resistance. Therefore we fitted a mechanistic-statistical model to these HTS data in order to disentangle the relative role of genetic drift and selection during within-host virus evolution [68] , [67] . This is the topic of Elsa Rousseau's PhD thesis, and is done in collaboration with Frédéric Fabre and Benoit Moury (INRA Avignon).

We also represented the pathogen spread in agricultural landscapes [40] . At this scale, we looked at how the landscape structure facilitates or impedes the disease spread among host patches. We showed that, when deploying a host with complete resistance to the pathogen along with a susceptible host, mixed landscapes were always more efficient to hamper the disease spread. However, when using a quantitatively resistant host, aggregating the hosts in different regions could result in a better control of the pathogen spread [41] . This work is part of Julien Papaïx's PhD thesis (MIA, INRA Jouy-en-Josas & BIOGER, INRA Grignon).

Eco-evolutionary dynamics of plant pathogens in seasonal environments

Understanding better pathogen evolution also requires to understand how closely related plant parasites may coexist. Indeed, such coexistence is widespread and is hardly explained through resource specialization. We showed that, in agricultural systems in temperate environments, the seasonal character of agrosystems is an important force promoting evolutionary diversification of plant pathogens [94] . Plant parasites reproduction mode may also strongly interact with seasonality. In this context, we investigated the influence of cyclical parthenogenesis, i.e. the alternation of sexual and asexual reproduction phases, on the eco-evolutionary dynamics of plant parasites [25] .

This work was part of the PhD thesis of Magda Castel (Agrocampus Ouest) and has been done in collaboration with Frédéric Hamelin (Agrocampus Ouest), Didier Andrivon (INRA Rennes) and Virginie Ravigné (CIRAD Montpellier).

Biological depollution

Control and optimization of bioprocesses for depollution

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

We have considered the problem of global stabilization of an unstable bioreactor model (e.g. for anaerobic digestion), when the measurements are discrete and in finite number ("quantized"). These measurements define regions in the state space, wherein a constant dilution rate is applied. We show that this quantized control may lead to global stabilization: trajectories have to follow some transitions between the regions, until the final region where they converge toward the reference equilibrium [71] .

Although bioprocesses involve an important biodiversity, the design of bioprocess control laws are generally based on single-species models. In [56] , we have proposed to define and study the multispecies robustness of bioprocess control laws: given a control law designed for one species, what happens when two or more species are present? We have illustrated our approach with a control law which regulates substrate concentration using measurement of growth activity. Depending on the properties of the additional species, the control law can lead to the correct objective, but also to an undesired monospecies equilibrium point, coexistence, or even a failure point. We now start to develop control laws more robust to the presence of additional species.

Moreno [107] has proposed an optimal strategy for fed-batch bioreactor with substrate inhibition. Thanks to the Pontryagin maximum principle and the Hamilton-Jacobi equation, we have shown that the same strategy is still optimal when mortality is included in the model [79] . We have also studied the problem when the singular arc is non-necessarily admissible everywhere (i.e. the singular control can take values outside the admissible control set). We have pointed out the existence of a frame point on the singular arc above which any singular trajectory is not globally optimal. Moreover, we have provided an explicit way for computing numerically the switching curves and the frame point [46] , [19] .

Coupling microalgae to anaerobic digestion

Participants : Olivier Bernard, Antoine Sciandra, Jean-Philippe Steyer, Frédéric Grognard, Philipp Hartmann, Francis Mairet.

The coupling between a microalgal pond and an anaerobic digester is a promising alternative for sustainable energy production and wastewater treatment by transforming carbon dioxide into methane using light energy. The ANR Phycover project is aiming at evaluating the potential of this process [113] , [112] .

In a first stage, we developed models for anaerobic digestion of microalgae. Two approaches were used: first, a dynamic model has been developed trying to keep a low level of complexity so that it can be mathematically tractable for optimization [100] . On the other hand, we have tested the ability of ADM1 [115] (a reference model which considers 19 biochemical reactions) to represent the same dataset. This model, after modification of the hydrolysis step [102] has then been used to evaluate process performances (methane yield, productivity...) and stability though numerical simulations.

Finally, we have proposed and analysed a three dimensional model which represents the coupling of a culture of microalgae limited by light and an anaerobic digester. We first prove the existence and attraction of periodic solutions. Applying Pontryagin's Maximum Principle, we have characterized optimal controls, including the computation of singular controls, in order to maximize methane production. Finally, we determine numerically optimal trajectories by direct and indirect methods [18] .

Life Cycle Assessment

Participants : Olivier Bernard, Jean-Philippe Steyer.

This work is the result of a collaboration with Arnaud Helias of INRA-LBE and Pierre Collet (IFPEN).

In the sequel of the pioneering life cycle assessment (LCA) work of [97] , we continued to identify the obstacles and limitations which should receive specific research efforts to make microalgae production environmentally sustainable.

The improvements due to technological breakthrough (leading to higher productivities) have been compared to the source of electricity. It turns out that the overall environmental balance can much more easily be improved when renewable electricity is produced on the plant [27] . As a consequence, a new paradigm to transform solar energy (in the large) into transportation biofuel is proposed, including a simultaneous energy production stage. This motivated the design of the purple sun ANR-project where electricity is produced by semi transparent photovoltaic panels [74] under which microalgae are growing.

These studies have allowed to identify the obstacles and limitations which should receive specific research efforts to make this process environmentally sustainable [93] .

Finally, some works are aiming at normalizing LCA for microalgae and proposing guidelines to make the LCA more easily comparable [87] .

These works have been carried out in collaboration with E. Latrille and B. Sialve (INRA - Laboratory of Environmental Biotechnology, Narbonne).

Models of ecosystems

Optimality/games in population dynamics

Participants : Frédéric Grognard, Ludovic Mailleret, Pierre Bernhard.

Optimal foraging and residence times variations

In a pair of papers [23] , [24] , we reanalyzed the so-called Marginal Value Theorem (MVT), first published in 1976. This theorem, also used in human behavior and economics, establishes how individuals should behave to optimize resource exploitation. This result has been has been routinely applied in ecology to understand the foraging strategy of animals such as insect parasitoids used for biological control purposes. We obtained the first mathematical characterization of how habitat characteristics (e.g. patch quality, or the distance between resource patches) affect the optimal foraging strategy. This allowed to confirm or refine MVT predictions, and to provide new predictions in the more realistic case of heterogeneous habitats. Some counterintuitive predictions emerged: making resource patches richer can actually make individuals move more rapidly, contradicting generally admitted earlier predictions.

This work was conducted with Vincent Calcagno (UMR ISA) and Frédéric Hamelin (Agrocampus Ouest).

The handicap paradox

We have continued our investigation of the handicap paradox of sexual selection with the tools of signaling theory. Zahavi's handicap principle, and our game theoretic analysis, explain why an equilibrium displays the “handicap” feature [21] . However, the explanation seems somewhat contrived, so the next question is “how could evolution have reached such a state ?” We have investigated that question with the tools of adaptive dynamics, and reached the conclusion that, if one accepts adaptive dynamics as a model of evolution, and our model of sexual selection, the equilibrium described in our previous article is indeed the limit state of evolution [20] .

This work was conducted with Frédéric Hamelin (Agrocampus Ouest).