Section: New Results

Fields of applications

Bioenergy

Modelling microalgae production

Participants : Olivier Bernard, Antoine Sciandra, Frédéric Grognard, 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 [23] 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 measured and correlated to the environment fluctuations.

Other experiments were carried out to reproduce the light signal percept by a cell in a raceway pond [84] , derived from hydrodynamical studies [92] . 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 [23] . Experiments with coloured light demonstrated that the growth rate results from the absorbed light, whatever its wavelength.

A new methodology to measure cell viability has been set up. This approach is very promising to distinguish between net and gross growth rate [20] . It was used in the models to assess the impact of temperature on growth and mortality. The mortality turns out to increase exponentially with temperature. The effect of a short term temperature stress was also tested to understand the consequences of a temperature peak in a cultivation system. Finally, it was shown that microalgae can bear with temperature peaks above $T_{max}$ if they do not last too long [57] .

On top of this, we set up a new experimental platform to carry out pilot experiments with solar light. The platform includes four raceways and the equipment to inoculate and harvest the microalgae [60] . We tested the impact of coloured film mimicking possible photovoltaic material. The collected data were used to calibrate models integrating the light spectrum [64] .

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

In collaboration with the IFREMER-PBA team (Nantes) we contributed to a study on the efficiency of dyes (BODIPY and Nile red) to quantify lipid content in microalgae [38] .

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 review of the microalgal metabolism reconstruction [15] together with the associated metabolic models has been carried out [14] . 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 [72] , [1] . A model was developed to represent heterotrophic growth on a mixture of acetate and butyrate [39] . A metabolic model was set up, on the basis of the DRUM framework [1] , in order to simulate autotophic, heterotropic and mixotrophic growth, and to determine how to reduce substrate inhibition.

Modelling 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 [79] . This model is supported by the work of ANGE aiming at improving the multi-layer Saint-Venant approach to more finely represent the hydrodynamics of the raceway [54] .

Modelling the photosynthesis response to fast fluctuating light

The impact of hydrodynamics on the light perceived by a single cell was studied thanks to fluid dynamics simulations of a raceway pond [90] . 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 [34] . Single cell trajectories were simulated, and the effect on photosynthesis efficiency was assessed using models of photosynthesis [91] . These results were compared to experimental measurements where the high frequency light was reproduced [84] .

Modeling microalgae production processes

The integration of different models developed within BIOCORE [54] , [19] , [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 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 [78] , [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 [107] . This model is used to predict lipid production in the perspective of large scale biofuel production [54] . It was also used to assess the microalgal production potential in France, when taking into account the actual meteorology on a 2.5 degree grid, for 2012, the use of lands, slope, proximity of nutrients and CO${}_2$ [93] .

In the framework of the ANR project Purple Sun, we developed a thermic model of a raceway pond within a greenhouse in order to estimate the culture temperature. We also included in the microalgae model the effect of light wavelength. This model has been calibrated on experimental data from LOV and has been used to support lighting strategy in order to optimize microalgal productivity (a patent on this process has been submitted).

Nitrogen fixation by diazotrophs

The fixation of nitrogen by Croccosphera watsonii was represented with a macro metabolic model [87] quantifying te main fluxes of carbon and nitrogen in the cell. The model was calibrated and validated with the data of three experiments carried out with different duration of the light period and daily dose. Extension of the model were studied to include the effect of temperature [61] .

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

Modelling thermal adaptation in microalgae

An extended statistical analysis was carried out on a database representing the temperature response of more than 200 microalgal species. First the model proposed by [78] turned out to properly reproduce the temperature response. A model was then extracted to predict the observed link between the cardinal temperatures. This lead to the reduction of the parameter number down to 2, with still a good prediction capability.

We have used 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 studied the case where temperature is periodically fluctuating [88] . We now use this method at the scale of the global ocean, validating our approach with experimental data sets from 194 species [42] , [49] .

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 [86] . 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), [104] . Interval observers were designed for the estimation of the microalgae growth and lipid production within a production process [101] [54] and validated experimentally [100] [29] .

Optimization of the bioenergy production systems

Based on simple microalgae models, analytical optimization strategies were proposed. We assessed strategies for optimal operation in continuous mode using the detailed model for raceways [106] , [107] . We first solved 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. Then, we re-analysed the optimization problem and derived a simplified strategy to reach biomass productivities very near to the maximal productivities obtained with the optimal control. These approaches were extended to outdoor cultivation, considering a possible variable culture depth. The optimal strategy for both depth and dilution rate was proposed in order to better manage the process inertia and finally avoid over warming periods. This work was done during the doctoral stay of Riccardo de Luca (Univ. Padova).

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 [31] .

Interactions between species

Large scale culture of microalgae for bioenergy involves a large biodiversity. 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 [28] .

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 had formerly proposed an adaptive controller which regulates the light at the bottom of the reactor [102] . 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. We have also proposed a strategy based on light stresses in order to penalize the strains with a high pigment content and finally select microalgae with a low Chlorophyll content [12] . This characteristic is of particular interest for maximizing biomass production in dense culture. The strategy has been carried out at the LOV and eventually the productivity of Tisochrysis lutea was improved by 75% [62] . A patent on this strategy is under submission.

Strategies to improve the temperature response have also been proposed. First we modelled the adaptive dynamics for a population submitted to a variable temperature [88] . This was used at the LOV to design experiments with periodic temperature stresses during 200 days aiming at enlarging the thermal niche of Tisochrysis lutea. It resulted in an increase by 2 degrees of the thermal niche [12] .

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 [73] . We now try to extend this result for $n$ species with mutations.

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 [30] .

Although bioprocesses involve an important biodiversity, the design of bioprocess control laws are generally based on single-species models. In [98] , 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. Finally, we have shown that, for this case, the robustness can be improved by a saturation of the control.

Moreno [105] have 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 [75] . We have also studied the problem when the singular arc is non-necessary 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 [17] .

Coupling microalgae to anaerobic digestion

Participants : Olivier Bernard, Antoine Sciandra, Jean-Philippe Steyer, Frédéric Grognard, 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 optimisation [97] . On the other hand, we have tested the ability of ADM1 [114] (a reference model which considers 19 biochemical reactions) to represent the same dataset. This model, after modification of the hydrolysis step [99] has then been used to evaluate process performances (methane yield, productivity...) and stability though numerical simulations.

We have proposed and analysed a three dimensional model which represent 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 have determined numerically optimal trajectories by direct and indirect methods [74] .

Finally, we have studied the coupling between three ecosystems: an anaerobic digester, a wastewater treatment pond (with microalgae and nitrifiers) and a microalgal pond. Different possible coupling configurations were tested in simulation. A numerical optimization was carried out to identify, depending on the choice of the objective function (energy production, pollution removal) the optimal arrangement between the three processes. The optimal volume for each process was then determined. This work has been carried out in the framework of the Phycover ANR project and was the subject of the internship of Ignacio Lopez (Universidad de Chile).

Life Cycle Assessment

Participants : Olivier Bernard, Jean-Philippe Steyer.

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

In the sequel of the pioneering life cycle assessment (LCA) work of [94] , 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 [36] . 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 [77] under which microalgae are growing.

Finally, some work are aiming at normalising LCA for microalgae and proposing guidelines to make the LCA more easily comparable [22] .

These works have been carried out in collaboration with E. Latrille and B. Sialve (INRA-LBE).

Design of ecologically friendly plant production systems

Controlling plant pests

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

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 optimization of natural enemies introductions has been investigated for several years [6] [108] , 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, [70] , [71] ).

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 [27] .

Connected research on the influence of space on the establishment 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) [13] ; in particular, we showed how landscape connectivity or spatial heterogeneity shape establishment dynamics in spatially structured environments [33] , [51] , [40] . This research linked to invasion biology also led some of us to contribute with opinion or review contributions to a special issue on biological invasions, in connexion with the GdR Invabio [26] , [32] .

Controlling plant pathogens

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

Sustainable management of plant resistance

Because plants can get sick, we studied other plant protection methods 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. We looked at landscape scale spatial deployment strategies of resistant crops able to maximize crop yield [25] , allowing for the modification of the spatial arrangement of resistant crops over cropping seasons, showing dramatic increases in crop yield in particular epidemic situations [25] .

Experiments were also conducted in INRA Avignon, followed by high-throughput sequencing (HTS) to identify the dynamics of virus strains competing 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 [53] , [69] , [43] , [44] . A stochastic model was also produced to simulate the effect of drift on the virus epidemiological dynamics and on the durability of qualitative resistances [59] . This is the topic of Elsa Rousseau's PhD thesis, and is done in collaboration with Frédéric Fabre (INRA Bordeaux) and Benoît Moury (INRA Avignon).

We also developed an epidemiological model describing the dynamics of root-knot nematodes in a protected vegetable cropping system, to design optimal management strategies of crop resistance [110] . The model was fitted to experimental and field data. Preliminary results show that alternating susceptible and resistant crops not only increased the resistance durability, but reduced the disease intensity over time [63] .

Finally we developed an epidemiological model including non-conventional gene-for-gene interactions in crops, based on the phoma stem canker of oilseed rape, to assess the durability of crop resistance in the field and design efficient deployment strategies [65] . This ongoing work is part of the K-Masstec project, which also incorporates experimental and field studies in collaboration with 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. 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 [89] . The plant parasites reproduction mode may also strongly interact with seasonality. In this context, we investigated the special case of oak powdery mildew, an oak disease which is actually caused by a complex of two different species, combining original plant epidemic data with the semi-discrete seasonal plant epidemic model we introduced a few years ago [50] [95]

This work has been done in collaboration with Frédéric Hamelin (Agrocampus Ouest) during Anne Bisson's internship, Marie Laure Desprez Loustau and Frederic Fabre (INRA Bordeaux).

Optimality/games in population dynamics

Participants : Frédéric Grognard, Ludovic Mailleret, Pierre Bernhard.

Optimal foraging and residence times variations

A continued collaboration with Vincent Calcagno (UMR ISA) has yielded paper where we reanalyzed the so-called Marginal Value Theorem (MVT), first published in 1976 [80] , [81] . Ongoing work aims at pointing how this latter theorem has been misused in some biological literature.

We also investigated the problem in foraging theory of evaluating the expected harvest of an animal when conspecifics may arrive on the same patch of resource in a stochastic fashion, specificaly according to a Poisson process or a Bernoulli process. A joint article with Frédéric Hamelin (Agrocampus Ouest) has been submitted for publication.

With Marc Deschamps, similar questions were studied in theoretical economy in the context of a Cournot competition on a single market. Again, an article has been submitted for publication.

The handicap paradox

We have investigated the question of “how could evolution have reached a state characterized by the handicap paradox?” with the tools of adaptive dynamics. We have reached the conclusion that, if one accepts adaptive dynamics as a model of evolution, and our model of sexual selection, the handicap paradox equilibrium is indeed the limit state of evolution [18] .

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