Alexis Guyot

Annabelle Gillet

Eric Leclercq
email: eric.leclercq@u-bourgogne.fr

Éric Leclercq

Detection of antagonism and polarization on social media through community boundaries

Keywords: Social networks, Graph mining, Communities, Polarization, Communities boundaries

HAL is

Nowadays, Online Social Networks (OSN) play an important role in data analytics thanks to their ability to capture a massive amount of interactions between individuals. They can be represented by graphs with people as vertices and their interactions as weighted and directed links. They usually exhibit scale-free properties [START_REF] Barabási | Scale-free networks[END_REF] and allow the detection and the study of communities [START_REF] Fortunato | Community detection in graphs[END_REF][START_REF] Lancichinetti | Community detection algorithms: a comparative analysis[END_REF][START_REF] Reichardt | Statistical mechanics of community detection[END_REF] to better understand the roles and behaviours of users.

However, for a detailed interpretation of some phenomenons, it is also necessary to adopt a broader view by studying communities themselves and their interactions. An example of phenomena is polarization, defined in social sciences by Isenberg [8] as the process whereby a social group can be divided into two opposed sub-groups having conflicting and contrasting positions on a given subject, with few individuals remaining neutral or in an intermediate position.

In our contribution [START_REF] Guyot | Frontières des communautés polarisées : application à l'étude des théories complotistes autour des vaccins[END_REF], we propose new tools to detect polarization traces and to study community boundaries [START_REF] Clauset | Finding local community structure in networks[END_REF][START_REF] Guerra | A measure of polarization on social media networks based on community boundaries[END_REF] on graphs extracted from OSN (more particularly from Twitter). Our proposal extends Guerra's work [START_REF] Guerra | A measure of polarization on social media networks based on community boundaries[END_REF], formalized for undirected and unweighted graphs. Because human interactions are not necessarily symmetrical and a lot of contextual information is contained in link direction (source/destination), we decide to exploit this feature in addition to weights, which allow a better granularity by quantifying the strength of the connection. Using the same original idea, we want to obtain a more precise tool, better suited to real data extracted from OSN.

Just like Guerra's, our method exclusively uses a structural analysis. We do not exploit any Natural Language Processing (NLP) or Sentiment Analysis method in order to overcome the language barrier and spelling approximations, unlike Alamsyah and Adityawarman's proposal [START_REF] Alamsyah | Hybrid sentiment and network analysis of social opinion polarization[END_REF]. Contrary to Morales et al. [START_REF] Morales | Measuring political polarization: Twitter shows the two sides of venezuela[END_REF] and Al Amin et al.'s [START_REF] Al Amin | Unveiling polarization in social networks: A matrix factorization approach[END_REF] methods, our analysis also does not need any additional domain knowledge.

Our method takes a graph decomposed into communities as an input and produces a more detailed structure of each community with two different areas: internals and boundaries. They are built and populated for each pair of communities with different users according to formalized structural properties, intuitively described as: 1) internals, are sets of nodes without any neighbour in the other community; and 2) boundaries, are sets of nodes with at least one neighbour located in the other community and another one in the internal area of their own community.

Interactions between community areas can be used to assess antagonism, on the assumption that more an individual is involved in his community, more likely he is committed to the group opinion, and so more likely he will be vehicle for antagonism. Thus, we estimate this value by calculating the internal interactions ratio for each node located in a boundary area. The method produces a matrix containing the average antagonism score in boundaries for each pair of communities.

Based on our formally defined boundaries, we propose a new measure: the porosity of boundaries, which is a rate estimated from the number of nodes having at least as much interactions with the other community as with the internal area of their own community. This measure can be used to comment on cohesion inside a given group.

We also supply an algorithm and an open source implementation in R1 to compute the different areas (internals and boundaries), the antagonism matrix and the porosity of boundaries from a weighted and directed graph. The algorithm uses an intermediate data structure called the Structural Matrix which allows to reduce the algorithmic complexity by precomputing each vertex role within its community compared to each other.

Finally, we apply our method and run through a way to interpret its results on a case study. We look for polarization in a corpus containing more than 9 millions tweets related to COVID-19 vaccines. Among the detected communities with the Louvain method, we highlight the absence of a welldefined anti-vaccine group whereas pro-vaccine users stand out in their own community. We suggest this could be an indicator that anti-vaccine users either don't share too much their opinion on Twitter or split themselves on different communities with other concerns, for example anti-government ones. We also point out the omnipresence of conspiracy hashtags inside all the communities main topics, such as #GreatReset, #BlanquerMent, #Plandemie, #ComplotVaccinObligatoire or #JeNeSaisPasJeDemande. 2 This observation is pretty surprising considering the fact that no keyword used to build the corpus mentions or refers to any conspiracy theory and underlines a likely mistrust in the COVID-19 vaccination campaign in France.

Even if there is no anti-vaccine community to be polarized with the provaccine one, the antagonism values allow to detect polarization with the community supporting alternative medications to treat the disease. Finally, we introduce the hypothesis that boundary porosity could be used to measure how much a community as seen by a graph mining algorithm is close to a social community.

Overviews like the one presented in figure 1 can be used to highlight results of case studies. Each diamond represents a community. Shapes have proportional size according to: 1) community size for diamonds; 2) porosity value for light grey borders; and 3) antagonism score between communities pair for arrows. This kind of view allows analysts to easily interpret values returned by our method, for example to quickly identify communities relationships and rivalries or to compare porosity values between groups with different sizes.

In a near future, we plan to run more experiments on real data to confront our method with other subjects and problems, leaning on the interdisciplinary project Cocktail 3 . We also are working on extensions to handle overlapping communities, to better develop and use the porosity concept and to integrate our algorithm in a complete analysis workflow of communities in a graph. More experiments will also be done to precisely estimate the algorithmic complexity of our method.

Fig. 1

 1 Fig.1Overview of communities rivalries in our case study.

https://github.com/AlexisGuyot/CommunityBoundaries

#GreatReset, #BlanquerLies, #Plandemic, #MandatoryVaccineConspiracy or #Jus-tAsking

3 https://projet-cocktail.fr/


Jérémy Mascarade
email: jeremy.mascarade@cea.fr

Karine Liger

Michèle Troulay

Xavier Joulia

Xuan-Mi Meyer

Christophe Perrais

Silvano Tosti

On the study of catalytic membrane reactor for water detritiation: Membrane characterization

Keywords:

◮ Catalytic palladium based membrane reactor is studied for ITER tritium waste management. ◮ Concentration polarization effect was highlighted by two-dimensional mass transfer model. ◮ Mass transfer resistance due to concentration polarization is reduced by the increase of fluid velocity. ◮ Concentration polarization phenomenon is enhanced by the decrease of non-permeable species content in the feed stream.

Introduction

Conventional gaseous streams detritiation techniques use a two steps process of catalytic oxidation, to convert hydrogen containing gaseous species into vapor, followed by a physical getter such as adsorber [START_REF] Heinze | French experience in tritiated water management[END_REF][START_REF] Beloglazov | Configuration and operation of detritiation systems for ITER Tokamak complex[END_REF][START_REF] Wilms | Mathematical comparison of three tritium system effluent HTO cleanup systems[END_REF] or absorber [START_REF] Perevezentsev | Wet scrubber technology for tritium confinement at ITER[END_REF]. All these techniques have the same drawback; they produce great amounts of tritiated water. Admittedly, these volumes can be reduced either by concentration of gaseous stream by means of hollow fiber organic membranes [START_REF] Digabel | Application of gas separation membranes to detritiation systems[END_REF] or by recycling in the tritium plant, but using dense palladium based membrane reactor can avoid them.

The concept of coupling a hydrogen perm-selective membrane and a transition metal-coated amorphous support has already been studied for fusion applications such as the Impurity Processing module of JET Active Gas Handling System [START_REF] Glugla | A Permcat reactor for impurity processing in the JET active gas handling system[END_REF] or JET soft housekeeping detritiation [START_REF] Lefebvre | Preliminary results from a detritiation facility dedicated to soft housekeeping waste[END_REF][START_REF] Tosti | Design of Pd-based membrane reactor for gas detritiation[END_REF].

These membrane reactors, based on a counter current isotopic swamping of tritiated species (i.e. molecular hydrogen, water and hydrocarbons) with protium, are generally designed considering that thermodynamic equilibrium of isotope exchange reaction is reached [START_REF] Tosti | Design of Pd-based membrane reactor for gas detritiation[END_REF]. Nevertheless, to extend design tools to a wider range of operations, it is necessary to better understand the phenomena involved, in order to identify the rate limiting steps and thus provide a more accurate modeling of the process.

In order to provide all the parameters needed for membrane reactor scale-up, such as reaction laws and rate constants of isotope exchange reactions, an instrumented pilot was built at CEA Cadarache. After some preliminary characterization tests, presented below, parametric study will be run in order to study system response to several external perturbations such as temperature, flow rate and composition. Results will then be used to fit a model accounting for mass, thermal and momentum transfer rates as well as reaction rates.

In this study, deuterium has been chosen as tritium representative.

Experimental devices

The pilot, whose flowsheet is represented in Fig. 1, was designed with three modules: a tubular fixed bed reactor, a permeator and a membrane reactor. By-pass system allows working in two different configurations:

-In parallel mode, modules are studied one by one; it is also possible to evaluate separately permeation and reaction kinetics. -In series mode, larger amounts of impurities can be fed and impact of reaction products on the overall dedeuteration efficiency can be evaluated.

The membranes, supplied by ENEA Frascati, consisted of commercial Pd 77 Ag 23 tubes. This alloy was chosen for its embrittlement resistance [START_REF] Hunter | Silver-palladium films for separation and purification of hydrogen[END_REF] and its high permeability [START_REF] Uemiya | Hydrogen permeable palladium-silver alloy membrane supported on porous ceramics[END_REF]. These dense 10 mm diameter tubes have a net length of 487 mm and a wall thickness of 150 m. They are hosted in 304 L stainless steel shells of 650 mm length, 1.5 mm wall thickness and 25 mm external diameter. As shown in Fig. 2, the membrane is in a finger-like configuration; one end is welded to a flange of the reactor's shell while the other is closed by a welded cap and linked to the opposite shell's flange by a prestressed spring avoiding membrane damages during warping and hydrogenation [START_REF] Tosti | Mechanical design of a PERMCAT reactor module[END_REF].

In this configuration, the feed stream, sent to the membrane lumen, is "decontaminated" through a protium stream, sent into the shell side in counter-current mode, thanks to isotope exchange reaction:

Q 2 O + H 2 ↔ H 2 O + Q 2 (1)
where Q stands for protium or deuterium. Non-permeable gases are collected via a small stainless tube put inside the Pd-Ag one and exit the module by the retentate stream whereas Q 2 permeates the membrane and leaves it by shell's outlet.

Unlike in the permeator, whose lumen is empty, the membrane reactor's lumen is filled with a Ni-based catalyst. This catalyst contains transition metals which activate hydrogen bonds and thus promotes both the isotope exchange reaction in the membrane's reactor and the hydrogenation/dehydrogenations reactions in the fixed bed reactor. This last one consists of a 304 L stainless steel tube of 350 mm length, 2 mm wall thickness and 20 mm external diameter. It is filled with stacked glass balls/catalyst particles/glass balls layers separated by sintered stainless steel sheets.

The reactor is brought to an operating temperature between 300 • C and 400 • C to avoid mechanical stress due to ␣ and ␤ phase coexistence in membrane bulk [START_REF] Knapton | Palladium alloys for hydrogen diffusion membranes: a review of high permeabiliy materials[END_REF]. The membrane's close wall temperature is monitored by means of a K-type thermocouple.

Modules are connected to gas bottles and vapor generator through pressure and mass flow controllers. These provide gas mixtures of desired compositions and flow rates. In addition, modules outlets' streams are pressure controlled and their temperature and flow rates are monitored by a distributed control system (DCS).

Finally, all the modules' streams are linked to a mass spectrometer (MS), to get their on-line composition measurements, allowing partial mass balances calculations.

This pilot can process flow rates up to 300 mL (STP) min -1 in a wide range of gas mixtures (i.e. from 0 to 100% of each species except oxygen which composition must not exceed 2% to stay beyond the lower explosive limit of the ternary N 2 -H 2 -O 2 mixture [START_REF] Lefebvre | Preliminary results from a detritiation facility dedicated to soft housekeeping waste[END_REF]).

During the preliminary tests, only pure H 2 /D 2 and mixtures of N 2 -H 2 have been used. These tests have been carried out to measure the membrane permeability and evaluate the mass transfer limitations due to concentration polarization.

Preliminary tests

Permeation tests

Hydrogen permeation through dense Pd-based membrane follows a solution-diffusion mechanism [START_REF] Caravella | Modelling and simulation of hydrogen permeation through supported Pd-alloy membranes with a multicomponent approach[END_REF]. Permeation tests were done to assess Sieverts' law applicability. This one considers thermodynamic equilibrium state between gaseous H 2 and dissolved protium at membrane interface (i.e. kinetics of solubilization are widely faster than diffusion). It leads to the following equilibrium constant known as Sieverts' constant K S :

K S = C H P H 2 (2)
with C H : the dissolved protium concentration (mol m -3 ); P H 2 : the hydrogen partial pressure near the membrane surface (Pa).

Coupling this expression with Fick's first law of protium diffusion in the membrane leads to the Richardson expression [START_REF] Buxbaum | Hydrogen transport through tubular membranes of palladium-coated tantalum and niobium[END_REF]:

J H 2 = H 2 • ı • P H 2 ,ret -P H 2 ,perm (3)
with J H 2 : the transmembrane H 2 flux (mol m -2 s -1 ); H = K S • D H : the protium permeability (mol m -1 s -1 Pa -0.5 ); D H : the protium diffusion coefficient in the membrane bulk (m 2 s -1 ); ı: the membrane thickness (m). Investigated in the temperature range of 300-400 • C and for pressure difference between lumen and shell ranging from 0.2 to 1 bar, permeability data were collected versus the reciprocal of the absolute temperature according to the Arrhenius law (Fig. 3):

i = 0 i • exp - Ea i R • T (4)
with i : the frequency factor (mol m -1 s -1 Pa -0.5 ); Ea i : the activation energy (J mol -1 ); R: the ideal gas constant (J mol -1 K -1 ); T : the absolute temperature (K). In order to avoid mass transfer limitations in the fluid phase, permeation tests have been carried out under pure H 2 and D 2 atmospheres. The results, reported in Table 1, fall into the interval of various data reported in the literature for the same kind of membranes.

Now that permeation regime was evaluated in the pilot's range of operating conditions, one can focus on the impact of nonpermeable gases presence.

Concentration polarization

When a gas mixture is fed to the membrane's lumen, a depletion of the permeable component Q 2 is expected near the membrane surface. To highlight this phenomenon, a simple two-dimensional model was built to map species distribution in the membrane reactor's lumen.

Modeling

If one assume that:

flow regime is steady-state plug flow, the entire module is isothermal the lumen's pressure-drop can be neglected permeate's pressure and flow rate (shell side) remain constant the thermodynamic behavior is described by ideal gas equation of state membrane is defect free (infinite selectivity toward protium and deuterium) -Axis-symmetric profiles along the reactor's centerline (∂/∂ = 0) then, the mass transfer of species i can be modeled by a convectiondiffusion equation (written in partial pressures):

∇ -→ N = 1 R • T • ∇ -D • ∇P i + -→ u • P i = 0 ∀ i ∈ [H 2 , N 2 ] (5)
with -→ N : the total flux (mol m -2 s -1 ); -→ u : the fluid velocity (m s -1 ) t u 0; ε • V /S ; ε: the catalyst bed porosity, V : the actual volume flow rate (m 3 s -1 ) S: the lumen's cross sectional area (m 2 ) D: the isotropic binary diffusion coefficient of N 2 -H 2 mixture (m 2 s -1 ) calculated as [START_REF] Reid | The Properties of Gases and Liquids[END_REF].

D = 2.66 • 10 -7 • T 3/2 • Pret • 10 -5 • 2 1/MH 2 + 1/MN 2 • H 2 + N 2 2 2 • ˝D -1 (6)
where M i : the molecular weight of species i (g mol -1 ), i : the collision diameter determined from Lennard-Jones potential (Å), ˝D:

a function of k B • T • ε H 2 • ε N 2 -0.5
; ε i : the Lennard-Jones force constant of i-specie (J); k B : the Boltzmann constant (J K -1 ). Fig. 4 shows the 6 boundary conditions needed to resolve relation (5).

Results and discussion

This partial differential equations system was implemented in COMSOL Multipysics ® 4.2 commercial code and resolved by PAR-DISO solver. Fig. 5 clearly highlights the decrease of permeation flux with the increase of nitrogen partial pressure in the feed. Indeed, for feed streams containing more than 10 mol.% of hydrogen, the membrane length is not sufficient to remove all the permeable species because the transmembrane hydrogen partial pressure difference still exist at the reactor's outlet, involving a permeation flux. This reveals an additional mass transfer resistance in the fluid phase [START_REF] Catalano | Influence of the gas phase resistance on hydrogen flux through thin palladium-silver membranes[END_REF] that one can write as an average pressure gradient in the gas phase (driving force) to flux ratio:

Rg = 1 L • L P H 2 r=r b -P H 2 r=rm 1/S • S N H 2 • dr • d • dz (7)
As shown in Fig. 6, this resistance (dotted line) can be decreased by increasing fluid velocity. This phenomenon is due to nitrogen molecules brought to the membrane surface which generates a radial concentration gradient leading to a diffusive flux (from the membrane to the fluid bulk) facing the H 2 one. Increasing fluid velocity will increase nitrogen advection and smooth this gradient; it will also, by the same way, reduce this concentration polarization effect, allowing more hydrogen molecules to be brought to the membrane surface leading to a permeation flux improvement.

These preliminary tests also brought to the light the membrane infinite selectivity toward hydrogen and, by the same token, verified defect-free membrane assumption; indeed, no traces of nitrogen were revealed during permeate stream mass spectrometer analysis.

Conclusion

An instrumented lab-scale pilot dedicated to the phenomenological study of water dedeuteration by protium isotope swamping in palladium-silver membrane reactor was presented. Preliminary tests done on the reactor's membrane show its high permeability, in adequacy with values reported in open literature, and infinite selectivity toward hydrogen and its isotopes. Presence of nonpermeable gases revealed a gas phase mass transfer resistance which could be reduced by increasing fluid velocity. Nevertheless, as it also shortened the contact time between the catalyst particles and the gas molecules, an optimization of the operating conditions is needed to maximize the conversion capabilities of the membrane reactor.

Fig. 1 .

 1 Fig. 1. Pilot flowsheet.

Fig. 2 .

 2 Fig. 2. Scheme of the membrane reactor.

Fig. 3 .

 3 Fig. 3. Arrhenius plot (ln vs 1/T) of H and D permeabilities.

Fig. 4 .

 4 Fig. 4. Schematic of the computational domain and boundary conditions.

Fig. 5 .

 5 Fig. 5. Influence of H2 inlet content on axial permeation profile P H 2 ,inlet = 0.4 bar, T = 645 K, V = 100 mL (STP) .min -1 .

P H 2

 2 Fig. 6. Influence of inlet flow rate on permeation flux P H 2 ,inlet = 1.5 bar, T = 645 K, y H 2 ,inlet = 50 mol.% .

Table 1

 1 Results of pure specie permeation: comparison with literature.

	i-Specie	0 i (mol m -1 s -1 Pa -0.5 )	Ea i (J mol -1 )	Reference
	H	6.12 × 10 -8	6876	This study
	H	3.86 × 10 -8	5752	[15]
	H	5.58 × 10 -8	6304	[16]
	D	4.93 × 10 -8	7965	This study
	D	2.52 × 10 -8	6172	[15]
	D	3.43 × 10 -8	6156	[16]


Olivier Bernard
email: olivier.bernard@inria.fr

Pierre Masci

Antoine Sciandra

A photobioreactor model in nitrogen limited conditions

We propose a new photobioreactor model that deals both with nitrogen limitation and light attenuation. On the basis of the Droop model we include the light effect, and then we relate the nitrogen status to the chlorophyll content, for a given photoadaptation light. In a second step, we compute the light distribution thanks to a Beer-Lambert model. It results in a model where biology (growth in nitrogen limited conditions) and physics (radiative transfer) are deeply coupled. The model is validated with experimental data of Isochrysis galbana.

Microalgae and cyanobacteria use light as energy source to fix carbon dioxide (CO 2 ). These microorganisms (abusively gathered under the name "microalgae") have recently received a specific attention in the framework of renewable energy. Their high actual photosynthetic yield compared to terrestrial plants (whose growth is limited by CO 2 availability) lead to large potential algal biomass productions of several tens of tons per hectare and per year. After a nitrogen limitation, this biomass can reach a very high lipid content (up to 80% of dry weight under some stress conditions [START_REF] Metting | Biodiversity and application of microalgae[END_REF]. These possibilities have led some authors to consider that microalgae could be one of the main biofuel producers in the future [START_REF] Huntley | Co2 mitigation et renewable oil from photosynthetic microbes: A new appraisal[END_REF][START_REF] Chisti | Biodiesel from microalgae[END_REF].

On top of this, the ability of microalgae to fix CO 2 in a controlled way has recently involved them in the race for mitigation systems [START_REF] Benemann | Co2 mitigation with microalgae systems[END_REF][START_REF] Olaizola | Commercial development of microalgal biotechnology: from the test tube to the marketplace[END_REF]. Thus microalgal biofuel production systems could be associated to industrial powerplants with a high CO 2 production. In the same spirit, microalgae could be used to consume inorganic nitrogen and phosphorus, and thus avoid expensive wastewater treatment plants.

These advantages put microalgae in a good position for renewable energy production at large scale [START_REF] Chisti | Biodiesel from microalgae[END_REF]. This means that in the coming years there might be large scale industrial plants to produce microalgae. However, the culture of algae is not straightforward and suffers from many limitations [START_REF] Pulz | Photobioreactors: production systems for phototrophic microorganisms[END_REF][START_REF] Carvalho | Microalgal reactors: A review of enclosed system designs and performances[END_REF]. First, high biomass concentration leads to an increase in light absorption. A trade-off between high density cultures (where light is attenuated after a few centimetres) and a diluted culture where light is not completely absorbed must be achieved. Moreover, CO 2 must be supplied in a concentration that is optimal for biomass synthesis, without dropping the pH to deleterious levels. Finally, the production of oxygen and its storage in the medium may be highly inhibitory, and a low concentration should be maintained to achieve high productivities. Microalgae production is thus a complex process that should be strongly monitored and controlled to manage these tradoffs and guarantee an optimal working mode. Several photobioreactor models have been proposed, especially to deal with light transfer property in the culture medium [START_REF] Suh | A light distribution model for an internally radiating photobioreactor[END_REF][START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by chlamydomonas reinhardtii in a torus photobioreactor[END_REF][START_REF] Franco-Lara | Model-supported optimization of phototrophic growth in a stirred-tank photobioreactor[END_REF] In the specific case of biodiesel production, nitrogen limitation is known to increase the lipid content of phytoplankton. But, by changing the pigment composition and concentration [START_REF] Turpin | effects of inorganic n availability on algal photosynthesis and carbon metabolism[END_REF][START_REF] Geider | a dynamic model of photoadaptation in phytoplankton[END_REF][START_REF] Sciandra | Growth compensating phenomena in continuous cultures of Dunaliella tertiolecta limited simultaneously by light and nitrate[END_REF][START_REF] Stramski | Effects of temperature, nitrogen, and light limitation on the optical properties of the marine diatom thalassiosira pseudonana[END_REF][START_REF] Geider | a dynamic regulatory model of phytoplanktonic acclimation to light, nutrients, and temperature[END_REF], it also strongly affects the radiative transfer properties in the culture medium [START_REF] Stramski | Effects of temperature, nitrogen, and light limitation on the optical properties of the marine diatom thalassiosira pseudonana[END_REF]. The objective of our modelling approach is to propose a new model that can predict the behaviour of a photobioreactor in conditions of nitrogen limitation.

The first point that we have to solve in the development of a model is the level of complexity that must be chosen in order to find an optimal tradeoff between model accuracy and ability to properly analyse and calibrate the model. A too complicated model will have a complex behaviour that could not be analysed and lead to calibration procedures of poor quality, especially if the measurements of some of the model variables are not available at high frequency. Moreover, a high dimensional model is generally not mathematically handable, so that it is hardly possible to derive control and optimisation algorithms. On the contrary, a too simplistic model will not catch the main dynamics of the involved processes.

In order to propose models that keep a level of complexity compatible with the requested mathematical analyses, we will consider the simplest model that contains the elements to reproduce both the ability of microalgae to adapt their pigments to a given light level (photoadaptation) and the reduction of the pigments contents in case of nitrogen limitation. The basis of our development is the Droop model [START_REF] Droop | Vitamin B12 and marine ecology. IV. the kinetics of uptake growth and inhibition in Monochrysis lutheri[END_REF][START_REF] Droop | 25 years of algal growth kinetics, a personal view[END_REF] which has been deeply investigated [START_REF] Lange | The attractiveness of the Droop equations[END_REF][START_REF] Bernard | Transient Behavior of Biological Loop Models, with Application to the Droop Model[END_REF][START_REF] Vatcheva | Experiment selection for the discrimination of semi-quantitative models of dynamical systems[END_REF] and proved to accurately reproduce situations of nitrogen limitations [START_REF] Droop | 25 years of algal growth kinetics, a personal view[END_REF][START_REF] Sciandra | The limitations of continuous cultures with low rates of medium renewal per cell[END_REF][START_REF] Bernard | Nonlinear qualitative signal processing for biological systems: application to the algal growth in bioreactors[END_REF]. A link between cellular nitrogen and chlorophyll will then be introduced, so that a simplified light distribution model within the reactor can be proposed.

The paper is organized as follows. In a first part, we recall the Droop model. Then we introduce the light influence in this model. In a third part we propose a model to infer chlorophyll concentrations. Then we represent oxygen and CO 2 . Finally the radiative transfer is examined, and the photoadaptation equation is proposed. The model is then validated using data from Isochrysis galbana.

The Droop model

The Droop model, initially established to represent the effect of B 12 Vitamin internal quota on the growth rate of phytoplankton [START_REF] Droop | Vitamin B12 and marine ecology. IV. the kinetics of uptake growth and inhibition in Monochrysis lutheri[END_REF], has been shown appropriate to represent also the effect of macronutrients , such as nitrogen, on growth rate [START_REF] Droop | 25 years of algal growth kinetics, a personal view[END_REF]. Contrarily to Monod model in which the growth is related to the external concentration of inorganic nitrogen s, Droop model considers that growth is related to the nitrogen cell concentration, or quota q, implying that nutrient uptake and growth are uncoupled processes. The substrate uptake leads to an increase of this internal quota which is used to sustain growth.

(D) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ṡ = Ds in -ρ(s)x -Ds q = ρ(s) -μ(q)q ẋ = μ(q)x -Dx (1)
In this model the absorption rate ρ(s) and growth rate μ(q) are generally taken as Michaelis-Menten and Droop functions:

ρ(s) = ρ s s + K s μ(q) = μ(1 - Q 0 q ) (2)
where K s is the half saturation constant for substrate uptake and Q 0 the minimal cell quota: at this quota level, no algal growth can take place. This model is more accurate than Monod model for algal growth modelling [START_REF] Vatcheva | Experiment selection for the discrimination of semi-quantitative models of dynamical systems[END_REF], but it is more complex and has been less studied.

Property 1

The internal quota will stay between two bounds:

Q 0 ≤ q ≤ Q m (3)
Where Q m = Q 0 + ρ μ represents the maximum cell quota obtained in conditions of non limiting nutrient. The growth rate is also bounded :

0 ≤ μ(q) ≤ μ m = ρ m μ Q 0 μ + ρ m ) ( 4
)
where μ m is the maximum growth rate reached in non limiting conditions.

As a corollary of this property, most of Droop model parameters can be straightforwardly identified from the measurements of the internal quota during exponential growth and at the end of a batch phase, when growth rate is zero. In this situation q = Q 0 . The internal quota in nonlimited growth conditions (when q = Q m ) together with the maximum growth rate provides then the values of μ and ρ. Parameter K s can be deduced (together with ρ) from a batch experiment where both s and x are measured.

The Droop model has been widely studied [START_REF] Lange | The attractiveness of the Droop equations[END_REF][START_REF] Bernard | Transient Behavior of Biological Loop Models, with Application to the Droop Model[END_REF][START_REF] Vatcheva | Experiment selection for the discrimination of semi-quantitative models of dynamical systems[END_REF] and validated [START_REF] Droop | 25 years of algal growth kinetics, a personal view[END_REF][START_REF] Sciandra | The limitations of continuous cultures with low rates of medium renewal per cell[END_REF][START_REF] Bernard | Nonlinear qualitative signal processing for biological systems: application to the algal growth in bioreactors[END_REF][START_REF] Vatcheva | Experiment selection for the discrimination of semi-quantitative models of dynamical systems[END_REF]. However, it cannot directly be used in the case of photobioreactors for two main reasons:

• In its rough form it does not include the effect of light intensity

• It does not account for light attenuation due to the cell density

Improvement of Droop model to deal with light limitation

The Droop model does not take light into account. Adding light is however not straightforward. Indeed, an idea would be to introduce the light effect into the maximum growth rate μ = μ(I):

μ(q, I) = μ(I)(1 - Q 0 q ) (5)
where

μ(I) = μ I I + K sI + I 2 K iI (6)
With such a model, the maximum cell quota would be:

Q m (I) = Q 0 + ρ μ(I) (7)
During night period, μ(0) = 0 and thus equation ( 7) leads to an infinite maximal quota. Indeed, with such a formulation no growth occurs at night, so that the substrate can be indefinitely taken up into the quota without being consumed for growth. If an increase of the maximum cell quota in the absence of light is acceptable [START_REF] Laws | nutrient-and light-limited growth of thalassiosira fluviatilis in continuous culture, with implications for phytoplankton growth in the ocean[END_REF], obviously it cannot become infinite.

We impose therefore that the uptake rate stops as cells become replete [START_REF] Geider | a dynamic regulatory model of phytoplanktonic acclimation to light, nutrients, and temperature[END_REF]:

ρ(s, q) = ρ s s + K s (1 -q/Q l ) (8)
with

Q l > Q 0 .
It leads to the following model including light effect, where we have also added a term dealing with respiration. As in [START_REF] Geider | a dynamic regulatory model of phytoplanktonic acclimation to light, nutrients, and temperature[END_REF], we assume that the rate of nitrogen loss (due both to mortality and remineralization) is the same than the respiration rate.

(D L ) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ṡ = Ds in -ρ s s+K s (1 -q/Q l )x -Ds q = ρ s s+K s (1 -q/Q l ) -μ I I+K I (q -Q 0 ) ẋ = μ I I+K sI +I 2 /K iI (1 -Q 0 q )x -Dx -Rx (9)
Now, with uptake rate (8), it is straightforward to show that the absence of light prevents growth, but that the maximum cell quota is lower than parameter Q l :

Property 2 The internal quota stays between two bounds:

Q 0 ≤ q ≤ Q m (I) ≤ Q l ( 10
)
Where Q m (I) = ρ+ μ(I)Q 0 ρ+ μ(I)Q l Q l represents the maximum cell quota obtained in replete nutrient conditions. In these conditions, the associated maximum growth rate, at irradiance I is now given by :

0 ≤ μ(q, I) ≤ μ M (I) = μ(I)(1 - Q 0 Q m (I) ) (11)
Proof: The equation of the quota, in conditions of abundance of nutrients in the environment ( ρ s s+K s ρ), becomes:

q = ρ(1 -q/Q l ) -μ(I)(q -Q 0 ) = (μ(I) + ρ Q l )(Q m (I) -q) ( 12
)
where

Q m (I) = ρ + μ(I)Q 0 ρ + μ(I)Q l Q l ( 13
)
It is thus clear from (12) that q tends towards Q m (I) at a rate (μ(I) + ρ

Q l

).

Relationship between chlorophyll and particulate nitrogen

With the same spirit of keeping the model very simple, we propose a direct relationship to compute chlorophyll from particulate nitrogen xq, following the work of [START_REF] Pawlowski | Modélisation de l'incorporation du carbone photosynthétique en environnement marin contrôlé par ordinateur[END_REF], for a culture photoacclimated at a light intensity I * , we have:

Chl = γ(I )xq (14)
where γ(I ) = γ max K I I + K I [START_REF] Lange | The attractiveness of the Droop equations[END_REF] This expression results from experimental observations obtained at steady state. Figure 1 presents data obtained at various light/dilution rate experiments. Finally, since photoadaptation is not instantaneous after a light change, we represent the light adaptation dynamics, as follows:

İ = μ(q, I)(I -I ) (16)
where we assume that the chlorophyll adaptation rate is the growth rate.

Prediction of pH, O 2 and CO 2 transfer

CO 2 availability is a key issue to guarantee that phytoplanktonic growth is not limited by the inorganic carbon resource. At the industrial scale, most of the efficient production systems inject CO 2 in the medium, and the risk of medium acidification is high if the influent gas flow rate is not closely regulated. For an efficient control it is therefore necessary to have a predictive model both for dissolved CO 2 and pH. In the same spirit, high concentrations of O 2 can lead to growth inhibition and it is thus also important to monitor and control oxygen concentration.

The gaseous flow rates are computed as follows:

Q CO 2 = K L a CO 2 (CO 2 -CO sat 2 ) and Q O 2 = K L a O 2 (O 2 -O sat 2 ) ( 17
)
where CO sat 2 and O sat 2 , respectively the saturating concentration of CO 2 and O 2 , are linearly related to the partial pressure in the gas phase. The transfer coefficient K L a for carbon dioxide is assumed to be proportional to that of oxygen (K L a CO 2 = 0.93 K L a O 2 ), due to the low difference in aqueous diffusivity of the two gases [START_REF] Rubio | Prediction of dissolved oxygen and carbon dioxide concentration profiles in tubular photobioreactors for microalgal culture[END_REF].

In order to compute the carbon dioxide concentration, we assume that, for non-calcifying species, the water alkalinity (Alk) is hardly affected by microalgal growth and is thus constant [START_REF] Zeebe | CO 2 in seawater: equilibrium, kinetics, isotopes[END_REF]. As a consequence the bicarbonate concentration is kept to constant level: [HCO - 3 ] = Alk. Note that this hypothesis has to be revisited if ammonium is used as a nitrogen source: it should be replaced by Alk = Alk 0 + [NH + 4 ] (where Alk 0 is constant), inducing thus a decrease in alkalinity.

It follows that CO 2 c -Alk

Where c is the inorganic carbon concentration. We finally propose a pH computation formula from the model state variables [START_REF] Bernard | Multimodel analysis of the response of the coccolithophore Emiliania huxleyi to an elevation of pCO 2 under nitrate limitation[END_REF]. We use the affinity constant of the acid base couple CO 2 /HCO - 3 :

K a = [HCO - 3 ]h [CO 2 ]
where h = [H + ] is the proton concentration.

h = ( c Alk -1)K a
pH is thus computed by pH =log 10 (h).

The evolution of dissolved inorganic carbon and oxygen are then directly obtained:

ċ = D(c in -c) -μ(q, I)x + R r x -K L a CO 2 (c -Alk -CO sat 2 ) ȯ = D(o in -o) + R q μ(q, I)x -R r x -K L a O 2 (o -O sat 2 ) (18)
Where R q is the photosynthetic quotient, c in and o in the influent inorganic carbon and oxygen, respectively.

Modelling light gradient in the photobioreactor

We investigate here a simple representation of light attenuation inside a photobioreactor of thickness L, due to high biomass. As a result, the light decrease in the photobioreactor can be represented by a Beer-Lambert exponential decrease with a rate proportional to qx. When I 0 is the irradiance at the surface, we have thus:

E(z) = I 0 exp(-(aγ(I )q.x + b)z) (19)
The average light received by the cell culture between 0 and L is therefore:

Ī = I 0 L 0 exp(-(aγ(I )q.x + b)z)dz L = I 0 (aγ(I )q.x + b)L [1 -exp(-(aγ(I )q.x + b)L)] ( 20
)
Note that this approximation could be improved by using more accurate models of the radiative transfer that would take the pigment composition into account [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by chlamydomonas reinhardtii in a torus photobioreactor[END_REF][START_REF] Pruvost | Numerical investigation of hydrodynamic and mixing conditions in a torus photobioreactor[END_REF]. In order to get a more workable equation, we may replace (20) by a rational expression with equivalent behaviour:

Ī I 0 K g (aγ(I )qx + b)L + K g ( 21
)
with K g = 1.25. This expression is a good approximation of (20) since 1-exp(-x)

x K g K g +x . As a consequence, the influence of light should not be represented from the intensity of the light source, as was the case in the term I 0 I 0 +K I ; it should be replaced by a similar term that takes the average light intensity reaching the microorganisms:

μ(I 0 , qx) = μ I 0 I 0 +K sI + I 2 0 K iI = μ I 0 K g ((aγ(I )qx+b)L+K g ) I 0 K g ((aγ(I )qx+b)L+K g )+K sI ((aγ(I )qx+b)L+K g ) 2 +I 2 0 K 2 g /K iI
However, [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by chlamydomonas reinhardtii in a torus photobioreactor[END_REF] showed that a more accurate way of estimating the growth rate consists in computing the average value of μ(I 0 , qx). The function μ(I 0 ) is increasing up to a light intensity Ī0 = K iI K sI exp(γ(I )xqL), and is then decreasing

It can be approximated with an excellent accuracy by an Haldane function.

μ(I 0 , I , x, q) = μ Ī Ī + K sI + Ī2 K iI exp(γ(I )xqL/2) (23)
Proof: The first point is straightforward computation. The second point results from extensive numeric tests.

The nitrogen limited photobioreactor model

This section synthesises the results of the previous sections, and gathers the various model elements together.

The simplified reduced model writes now, for an incident light intensity I 0 :

(PDM) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ṡ = Ds in -ρ s s+K s (1 -q/Q l )x -Ds q = ρ s s+K s (1 -q/Q l ) -μ(I 0 , I , x, q)(q -Q 0 ) ẋ = μ(I 0 , I , x, q)(1 -Q 0 q )x -Dx -Rx İ = μ(I 0 , I , x, q)(1 -Q 0 q )( Ī -I ) (24)
with the average light intensity Ī = I 0 K g (aγ(I )qx+b)L+K g . The average growth function is given by equation [START_REF] Sciandra | Growth compensating phenomena in continuous cultures of Dunaliella tertiolecta limited simultaneously by light and nitrate[END_REF].

The model has been validated using the experimental data deriving from [START_REF] Flynn | carbon-nitrogen relations at whole-cell and free amino-acid levels, during batch growth of isochrysis galbanna (prymnesiophyceae) under conditions of alternating light and dark[END_REF], as it is shown on figure 2. Parameter values are given in table 1.

Conclusion

We have developed a dynamical model that accurately represents the dependence of microalgae growth on nitrogen and light. This model takes into account the light gradient into the photobioreactor and predicts both carbon and oxygen fluxes. It strongly represents the coupling between biology (microalgae growth) and physics (radiative transfer properties).

Light heterogeneity in the medium induces a complex process of photoadaptation resulting from two opposite feedbacks. After an incident light shift, the biomass concentration increases and so do the pigments in the medium. It results in reducing the range of average light increase compared to the incident light shift. This negative feedback is completed by a positive feedback, since photoadaptation to a higher light intensity leads to a reduction of the pigment content and thus a higher light penetration. This shows that modelling is the only possibility to qualitatively describe what happens in a photobioreactor. Such a model can be the base for the optimisation of biomass or subproducts yield.

Figure 1 :

 1 Figure1: Representation of the ratio Chlorophyll a over particulate nitrogen for various light conditions and dilution rates (data from[START_REF] Pawlowski | Modélisation de l'incorporation du carbone photosynthétique en environnement marin contrôlé par ordinateur[END_REF]).

Property 3

 3 Assuming that K sI > 4K sI , there exists two positive constants I a and I b (with I b > I a ), such that: (I + I a )(I + I b ) = I 2 + K iI I + K iI K sI . The average value of the growth rate μ(I) can then be computed as follows: μ(I 0 ) = μ I b -I a 1 γ(I )xqL ln I 0 + I a I 0 + I b I 0 exp(-γ(I )xqL) + I b I 0 exp(-γ(I )xqL) + I a (22)

3 )Figure 2 :

 32 Figure 2: Simulation of the photobioreactor Droop model and comparison with experimental data from [10].

Table 1 :

 1 Parameter values used for the simulation of the PDM model

	Parameter Value Unit
	μ	1.8	day -1
	Q 0 K sI K iI ρ	0.053 mgN. mgC -1 50 μE.m -2 s -1 1000 μE.m -2 s -1 1.8 mmolN.mgC -1 .day -1
	Q l K sI γ max K I	0.11 50 0.12 50	mg N.mgC -1 μE.m -2 s -1 mgChl.mgN -1 μE.m -2 s -1

Proceedings MATHMOD 09 Vienna -Full Papers CD Volume

Acknowledgements: This paper presents research results supported by the ANR-06-BIOE-014 Shamash project.


Context

The herd spends 5 months of the year in a building (early November  early April) in free stalling

Conclusion and implications of this study

Many advantages of this natural resource Provided that the reeds are 'managed' sustainably:

Respecting the biological cycle of the plant by harvesting late in the season enables the reed beds to retain their role as a habitat for many animal species

Reed is a good alternative to cereal straw Its cost can be competitive compared to straw

Reeds could be given a place on organic livestock farms that wish to make better use of natural resources, while also preserving them and biodiversity!

Reed can also constitute a good additional fodder in summer as this plant has interesting nutritive value

The trials, visited by farmers (December 2019)

Many thanks to:

•


L Bekere

V Lokshin

M Sigalov

R Gvishi

P Zhao

D J Hagan

E W Van Stryland

V Khodorkovsky

Chromenes involving a two--photon absorbing moiety: photochromism via intramolecular resonance energy transfer †

New

derivatives involving the photochromic 2H-benzo[h]chromene moieties covalently linked to the 2,7-bis(carbazolyl)fluorene--derived two--photon absorbing moiety were designed to enable the possibility of resonance energy transfer from the fluorene donor to the photochromic acceptor. Rapid coloration of colorless solutions was observed upon one-and two--photon absorption.

Photochromism, defined as a reversible transformation of chemical species, induced in one or both directions by electromagnetic radiation 1 attracts considerable interest owing to the potential use of this phenomenon in various applications. Among such applications, the development of optical data storage materials, An alternative approach involves resonance energy transfer (RET) from a 2PA fluorophore as a donor to a photochrome as an acceptor. Previously, we demonstrated that a mixture of a fluorophore and a spiroxazine--derived photochrome absorbs two photons and exhibits RET.

A two-fold enhancement in the rate of the photochromic conversion of a diarylethene--derived photochrome in the presence of 2-photon absorbing fluorenes was also demonstrated. [START_REF] Belfield | [END_REF] Because of the strong dependence of RET efficiency on donor--acceptor separation, effective RET between separate molecules requires high concentrations of the components and is best suitable for aggregates or polymers. 14 This concentration requirement can be removed when the acceptor and donor are chemically linked. Several derivatives involving both a photochromic and a fluorescent moiety have been reported, targeting fluorescence switching using the RET phenomenon. In this case, however, the colored (open) forms of the photochromes played the role of energy donors, see for instance. 15 Diarylethenes were demonstrated to be the most suitable candidates as 2PA switching molecules so far, although their direct conjugation with the 2PA moiety can lead to relatively low 2PA cross section values. Please do not adjust margins Please do not adjust margins detailed account of DFT calculations on derivatives 1 -3 will be published elsewhere). At such distance between the donor and the acceptor moieties, both Förster--and Dexter--types of RET may be enabled. The structure variation 2 vs. 1 provides a shorter distance between the two moieties, whereas the variation 3 vs. 2 should give rise to a red shift in absorption of the respective colored form owing to the presence of the amino group in the para--position of one of the phenyl groups.

Fluorene derivative 7a Derivative 1a crystallizes out of acetonitrile solution as colorless crystals of two different shapes. The X--ray analysis ‡ showed that the rectangular prisms involve one molecule of acetonitrile per molecule of 1a (Fig. 2), while the hexagonal prisms are solvent free. The molecular geometry of 1a in the crystals of both types is very similar and is characterized by the 'unfolded' conformation with the distance between the photochromic and 2PA moiety of about 7 Å, in contrast to the 'folded' configuration produced by the geometry optimization. There The longest wavelength absorption bands of derivatives 1 -3, a --series in methylene chloride are observed as shoulders between 370 (2, 3) and 380 nm (1), (ε about 10,000) on the strong absorption bands of the fluorene moiety (340 nm, ε about 80,000), overlapping well with the fluorescence band of 7a (Fig. 3). The absorption spectrum of 1a represents an exact sum of the spectra of 1b and 7a indicating the absence of electronic interaction between the photochromic and 2PA moieties. Minor deviations in the absorption band positions and shapes in the spectra of 2a and 3a may indicate the presence of weak through--space interaction between the two functional moieties. No concentration dependence of the absorption spectra within the range of 10 4), yellow--orange for 1 and 2 and violet for 3. A hypsochromic shift of the colored forms generated at 330 nm from 3a is observed (Fig. 5a) during thermal discoloration owing to the different stabilities of the colored form isomers. Only derivatives 1a -3a undergo coloration upon laser irradiation at 620 nm (2PA)(Fig. 5b).

Increasing polarity of the solvent (methylene chloride vs. toluene) brings about small shifts in the position of the absorption band maxima of the colored forms. The species generated by irradiation of 1a and 2a at 315, 330, 350 (1PA) and 620 (2PA) nm show similar absorption spectra (Fig. 5b). The exception is 3a: irradiation at 315 - 350 nm gives rise to a very broad absorption band covering all visible range red shifted by 16 nm in methylene chloride compared to toluene. The spectrum of femtosecond pulse at irradiation 620 nm overlaps with the absorption of the colored isomers and gives rise to the band shape distortion owing to photo--discoloration of the longer wavelength absorbing isomers (Fig. 5b). The color fading observed for this derivative is very rapid: the half--life time t 1/2 in methylene chloride is just 2 seconds.

The half--life times of other derivatives vary in a non-systematic way and the variations can stem from steric rather than from the electronic factors. This phenomenon was observed also in the case of other chromene derivatives. [START_REF] Hepworth | Photochromic naphthopyrans[END_REF] Derivatives 1b -3b do not exhibit coloration under 620 nm laser irradiation. A more detailed quantitative optical characterization of these compounds, including nonlinear properties, is currently under way.

Conclusions

The non--conjugative tethering of the relatively bulky 2PA moiety to the photochromic 2H--benzo[h]chromenes (the derivatives of a--series) does not negatively affect their performance as photochromic materials as compared to the derivatives of b--series. All derivatives of the a--series undergo coloration under femtosecond laser irradiation at 620 nm and quenching of fluorescence from the 2PA moiety corroborates the occurrence of the efficient RET process, whose exact nature still has to be established.

Notes and references ‡ Single crystals of 1a were grown from acetonitrile. X--ray crystallography data were collected on a Bruker--Nonius KappaCCD diffractometer with CCD detector using MoKα radiation (λ = 0.71073 Å). Crystallographic data for the structures reported in this paper have been deposited with the Cambridge Crystallographic Data Centre: 1a involving one acetonitrile molecule: CCDC 1033558; solvent--free crystals: CCDC 1033559.

7 . 8 .

 78 two--photon absorption (2PA) instead of one--photon absorption for inducing photochromic transformations offers further advantages and opens new possibilities in data recording and biomedical applications.5 Most of the known organic photochromes possess very small 2PA cross section values and the multiphoton--induced photochromic phenomena can be observed in solid state 6 or in polymer matrix containing 1.3 M of a photochromic material in the presence of gold nanoparticles Rendering large 2PA cross sections by chemical modification of the photochrome structure, such as attaching a molecule with the large 2PA cross section, is difficult to achieve, as the resulting molecule may exhibit weaker photochromic response or even lose the photochromic properties Porphyrin-perinaphthothioindigo and azo conjugates can be mentioned as a successful realization of this approach, 9,10 for a recent review see 11 .

16

  Here we report on the synthesis of a series of new bifunctional molecules involving the 2,7-bis(carbazolyl)fluorene--derived 2PA moiety covalently linked by the (CH 2 ) 3 bridge to the chromene--derived photochromic moiety (1a -3a) along with the model derivatives lacking the 2PA fragment (1b -3b) and give an account of their spectroscopic behavior in solution. The structures of 2H--benzo[h]chromene (2H--naphtho[1,2-b]pyran) derivatives 1 -3 are shown in Fig. 1. These derivatives were designed so that the longest wavelength absorption of the photochromic moiety overlaps with the fluorescence band of 9,9'--(9--(3--hydroxypropyl)--9--methyl--9H--fluorene--2,7--diyl)bis--9H--carbazole, and the distance between the photochromic and the 2PA moieties is about 5 - 6 Å to provide efficient RET and to exclude the possibility of through--bond and through--space intermolecular electronic interaction (This distance is calculated from the B3LYP/6--31G(d) geometry optimization. A This journal is © The Royal Society of Chemistry 20xx

17

 17

Fig. 1 .

 1 Fig. 1. Derivatives 1 -3 with the 2PA moiety (a--series) and model compound (b--series).

19 , 5 20and 6

 1956 18 and the respective bromide 8a and iodide 9a (Scheme 2) were used for the modifications of the photochromic units 4 (prepared by analogy to 20 ) owing to the high fluorescence quantum yield and relatively large 2PA cross section of the respective 2,7-bis(amino)fluorene derivatives 21 . The synthetic routes toward 1 - 3 are shown in Scheme 2. These compounds were purified by column chromatography and characterized by the NMR and HRM spectra. The best yields of derivatives 2 and 3 were achieved using iodides 9a,b.

Scheme. 1 .

 1 Scheme. 1. Synthesis of chromenes 1 -3 and precursors 7 --9.

Fig. 2 .

 2 Fig. 2. ORTEP representation of the molecular structure of 1a with one molecule of acetonitrile (hydrogen atoms are omitted, thermal ellipsoids are presented at 50% of probability).

  Please do not adjust margins Please do not adjust margins are several short intermolecular distances in the crystal lattice in both types of crystals that may constrain the molecule in this conformation.

Fig. 3 .

 3 Fig. 3. Normalized absorption spectra in methylene chloride at 2x10 --5 M: (a) 1; (b) 2; (c) 3. Red: a - series; dashed: b - series; dotted: 7a; blue: normalized fluorescence of 7a (λex = 330 nm).

6 M

 6 was observed. Whereas the dicarbazolyl fluorene precursor 7a is strongly fluorescent (the quantum yield ca. 80%), derivative 1a exhibits only weak fluorescence (the quantum yield ca. 1%) at about 380 nm and derivatives 2a and 3a are practically non-fluorescent. Such strong fluorescence quenching may indicate the occurrence of the efficient RET process. Irradiation of solutions of all six derivatives at 315, 330 or 350 nm (1PA) brings about rapid coloration owing to the formation of the colored open forms of the photochromes as a mixture of isomers (Scheme 4, Fig.

Scheme 2 .

 2 Scheme 2. Photochromism of derivatives 1 -3.

Fig. 4 .

 4 Fig.4. Irradiation in toluene at 330 nm; (a) 3a (1.2x10 --5 M); (b) 3b (10 --4 M).

Fig. 5 . 4 |

 54 Fig. 5. (a) Thermal discoloration of 3a irradiated at 330 nm in methylene chloride; (b) UV--Vis absorption spectra of 1a (black), 2a (blue) and 3a (red) irradiated at 620 nm (solid lines) and 330 nm (dashed lines) in methylene chloride.

Table 1 .

 1 Absorption maxima and the half--life times of the colored open isomers generated from 1 -3 at 330 nm (1PA) and 620 nm (2PA).

	Entry	λmax (nm)	a t1/2 (s)	a	λmax (nm)	b	t1/2 (s)	b
	1a	462	17		464 (465)	c	35
	1b	475	77		462			41
	2a	472	77		475 (475)	c	173
	2b	475	178		478			173
	3a	540	43		556 (480)	c	2
	3b	544	46		550			23
	a in toluene, b in methylene chloride, c irradiated at 620 nm.


Pierre-Emmanuel Morant

Quentin Thommen

François Lemaire

Constant Vandermoëre

Benjamin Parent

Marc Lefranc

Oscillations in the expression of a self-repressed gene induced by a slow transcriptional dynamics

Keywords: numbers: 87, 18, -h 87, 18, Vf 87, 16, Yc 82, 40, Bj

We revisit the dynamics of a gene repressed by its own protein in the case where the transcription rate does not adapt instantaneously to protein concentration but is a dynamical variable. We derive analytical criteria for the appearance of sustained oscillations and find that they require degradation mechanisms much less nonlinear than for infinitely fast regulation. Deterministic predictions are confirmed by stochastic simulations of this minimal genetic oscillator.

Networks of genes interacting via regulatory proteins modulating their activities are highly nonlinear systems which display a variety of dynamical behaviors [START_REF] Goldbeter | Biochemical Oscillations and Cellular Rhythms[END_REF][START_REF] Fall | Computational Cell Biology[END_REF][START_REF] Tiana | [END_REF]. Their modeling has generally assumed that gene activation is fast compared to other processes so that transcription rate reacts instantaneously to protein concentration. However, transcription is a complex process [4]. In the last years, it has been increasingly recognized that gene activity fluctuations can be slow and that this can affect the behavior of gene regulatory networks. In particular, slow transcriptional bursting and transcriptional memory have been observed experimentally [5,6,7]. Theoretically, it has been shown that slow activation dynamics can lead to bursts in expression [8,9], induce bistability [10] or modify the flipping rate of a genetic switch [START_REF] Walczak | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Morelli | [END_REF].

In this Letter, we show that slow promoter dynamics can also lead to oscillations by investigating how transcriptional dynamics modifies the behavior of a single gene repressed by its own protein [13,14,15,[START_REF] Goldbeter | Proc. R. Soc. Lond. B[END_REF][START_REF] Leloup | [END_REF]18,19,20]. This old problem of theoretical biology has been recently revived by the study of the Hes1 gene involved in the somite clock [21]. The usual view is that oscillations appear in this genetic circuit only when additional steps are inserted in the feedback loop [START_REF] Tiana | [END_REF]14]. In the Goodwin and Bliss oscillators [13,14,15], the gene protein catalyzes synthesis of the actual repressor. In early circadian models, transport of the repressor into the nucleus [START_REF] Goldbeter | Proc. R. Soc. Lond. B[END_REF][START_REF] Leloup | [END_REF] is a key oscillatory ingredient. In fact, the mere introduction of a time delay in the one-gene circuit model (accounting for protein transport or more generally a cascade of intermediate steps [22]) can destabilize it [START_REF] Tiana | [END_REF]18,19,20]. As will be of particular interest here, oscillations may also be induced by strongly nonlinear degradation mechanisms [23].

For simplicity, we study the case of an elementary kinetic equation describing regulation through protein-DNA binding [24], which we however prefer to view as a minimal description of transcriptional memory in more complex mechanisms. We derive an analytical expression of the oscillation threshold, and show that when the gene response time is appropriately tuned, the one-gene circuit can be destabilized (and oscillations induced) by degradation mechanisms much less nonlinear than for infinitely fast regulation. This result provides new insights into the interplay of nonlinearity and time delay. Stochastic simulations confirm that the main results of our analysis carry over to low copy number situations.

Our study is based on the following three-variable model describing the genetic circuit represented in Fig. 1:

Ġ = θ 0 (1 -G) -α 0 C(P )G (1a) Ṗ = n Ġ + β 0 M -δ P F (P ) (1b) Ṁ = µ 0 + λ 0 G -δ M H(M ) (1c)
where G, P and M represent gene activity, protein and RNA copy numbers. Eq. (1a) formally describes the kinetics of protein-DNA binding at rate α 0 and unbinding at rate θ 0 [24]. Possible cooperativity effects are taken into account via the function C(P ) and the number n of proteins binding DNA. Single-protein regulation corresponds to C(P ) = P , n = 1. More generally, Eq (1a) is a minimal model for the dynamics of an effective gene activity G slowly relaxing towards an equilibrium value given by the gene regulation function G = 1/[1 + C(P )/C(P 0 )], with P 0 the half-expression threshold. Such a model can be obtained as the leading approximation of a mechanistic model of transcription including all processes concurring to gene expression (e.g., protein-DNA interaction, formation of open complex, polymerase recruitment, chromatin remodeling,...) when there is a dominant limiting step. Eq. (1a) causes gene activity G to lag fast changes in protein level, and plays a dynamical role similar to explicit time delays [18,19,20] or to transport equations [START_REF] Leloup | [END_REF]. In Eq. (1b), the three terms correspond to binding/unbinding, translation and degradation. Eq. ( 1c) describes transcription at rate µ 0 + λ 0 G and RNA degradation. In order to understand how oscillations can be induced by tuning protein and RNA degradation, we derive the oscillation criterion for arbitrary degradation functions F (P ) and H(M ) with unit derivative at zero, δ P and δ M being the low-copynumber degradation rates. By renormalizing time, variables, parameters, cooperativity and degradation functions according to:

t = t ′ δ M , G = g, P = pP 0 , M = mM 0 , (2a)
P 0 = C -1 θ 0 α 0 , M 0 = δ P P 0 β 0 , θ = θ 0 δ M , (2b)
α = θ 0 P 0 δ M , δ = δ P δ M , λ = λ 0 M 0 δ M , µ = µ 0 M 0 δ M (2c) c(p) = C(P ) C(P 0 ) , f (p) = F (P ) P 0 , h(m) = H(M ) M 0 , (2d)
Eqs. ( 1) can be rewritten in dimensionless form

g ′ = θ [1 -g(1 + c(p))] (3a) p ′ = nα [1 -g(1 + c(p))] + δ[m -f (p)] (3b) m ′ = µ + λg -h(m) (3c)
where

x ′ = dx/dt ′ . When f and g are monotonous and h[f (∞)] > µ, model (3)
has a single steady state (g * , p * , m * ) satisfying the fixed point equations:

g * = 1 1 + c(p * ) , m * = f (p * ), g * = h(m * ) -µ λ . (4)
Note that the steady state depends only on parameters λ and µ as well as on functions c, f and h, whereas parameters θ, α, δ control time scales. The behavior of the degradation and cooperativity functions in the neighborhood of the steady state is described by the slopes

s = df (p) dp p=p * , u = dh(m) dm m=m * , v = dc(p) dp p=p * In the case of linear degradation [f (p) = p, h(m) = m],
we have u = s = 1. Small or even negative values of the slopes s and u generally denote strongly nonlinear degradation mechanisms [START_REF] Fall | Computational Cell Biology[END_REF]23], including saturation.

To assess whether Eqs. (3) can display oscillations, we have searched for parameter values where the fixed point specified by (4) loses stability to a periodic solution via a Hopf bifurcation (i.e., a pair of conjugate eigenvalues of the linearized problem cross the imaginary axis). For simplicity, we assume perfect repression (µ = 0) and a large threshold P 0 (α ∼ 0). Under this approximation, the Routh-Hürwitz stability criterion [START_REF] Gradshteyn | Tables of Integrals, Series, and Products[END_REF] indicates that a Hopf bifurcation occurs when the quantity

H = σ + -δλvg 2 * + σ 2 τ + γστ 2 (5)
crosses zero to become negative, where τ = g * /θ is the gene response time, and the sum σ = δs + u and product γ = δsu are symmetric functions of degradation rates δs and u. At bifurcation, the oscillation period is τ osc = 2π τ /(σ + γτ ) where σ and γ satisfy H = 0. Cooperativity essentially changes feedback strength from δλ to δλv, thus we assume for simplicity single-protein regulation [c(p) = p, v = 1] thereafter. Eq. ( 5) shows that a strong feedback destabilizes the system while high degradation rates (large σ and γ) tend to stabilize it. In the single protein case, H > 0 when both protein and RNA are linearly degraded and no oscillations occur. Conversely, when protein and RNA degradations are completely saturated (s = u = σ = 0), H = -δλg 2 * τ < 0, indicating that oscillations then appear systematically. The behavior in intermediate cases depends on the value of the response time τ .

In the classical case τ = 0, H = σ and oscillations appear only for σ < 0. It is indeed known that negative effective degradation rates can lead to oscillations [START_REF] Fall | Computational Cell Biology[END_REF]23]. We thus restrict ourselves to showing that at finite τ , oscillations can occur for u, s > 0. More precisely, we want to understand how oscillations arise away from the saturated cases u = 0 or s = 0. To this end, we use the geometric slope average ν = √ us = γ/δ as an index (ν = 1 in the linear case), seeking to determine the maximum value of ν at which oscillations can be observed, and for which values of τ this extremum is achieved.

The quantities σ and γ play complementary roles. The total degradation rate σ controls instability onset for small to moderate τ . Moreover, Eq. ( 5) indicates that when σ, γ > 0, a necessary condition for oscillations is

σ < σ c = g * √ δλ = g * √ 2 t sw (6)
where t sw is the time during which a fully active gene synthesizes the amount of protein corresponding to halfrepression threshold. The degradation rate product γ is relevant only for large τ , blocking oscillations if it is too large. In particular, γ = 0 guarantees the onset of oscillations for sufficiently large τ whenever (6) holds. For u, s > 0 and a given value of σ, γ can take any value between σ 2 /4 and 0 depending on whether the two degradation rates are equal or completely unbalanced, one being equal to zero and the other to σ. We use below ǫ = 2 √ γ/σ ∈ [0, 1] as a balance indicator. Remarkably, we note that under the rescaling

σ = σ c Σ, γ = σ 2 c ǫΣ 2 2 , τ = T σ c , (7)
the oscillation condition can be rewritten without explicit parameter dependence:

H ǫ (Σ, T ) = Σ × ǫ 2 Σ 2 4 T 2 + Σ - 1 Σ T + 1 < 0 (8)
and defines a series of curves Σ ǫ (T ) such that given a balance index ǫ and a response time T , oscillations are found for Σ ≤ Σ ǫ (T ). Fig. 2 shows the limit curves Σ 1 (T ) and Σ 0 (T ) which are important to understand the bifurcation diagram: regardless of the value of ǫ, the circuit always (resp., never) oscillates when Σ < Σ 1 (T ) [resp., Σ > Σ 0 (T )]. To support our analysis, we have searched the parameter space of Eqs. (3) for oscillatory behavior for α, µ = 0, assuming for definiteness allosteric protein degradation and Michaelis-Menten RNA degradation:

f (p) = p × (a + p/κ) a + 2a(p/κ) + (p/κ) 2 , h(m) = χm χ + m (9)
Points in the (Σ, T ) plane associated with oscillating parameter sets are shown as black dots in Fig. 2. Agreement is excellent: all dots are below the Σ 0 (T ) curve and the few significantly above Σ 1 (T ) have one small degradation rate. We are thus confident that our analysis allows us to understand the behavior of Eqs. [START_REF] Tiana | [END_REF]. Two regions can be distinguished in Fig. 2. For T < 1, the instability threshold Σ ǫ (T ) is practically independent of ǫ and increases rapidly with T . For small T , Σ ǫ (T ) ∼ T (thus, the oscillation criterion is σ < σ 2 c τ ). In the T > 1 region, Σ ǫ (T ) reaches its maximum value

Σ m (ǫ) = 1/ √ ǫ + 1 at T = T m (ǫ) = 2 √ ǫ + 1/ǫ,
and then decreases as T -1 for T → ∞, except for ǫ = 0 where it monotonously increases towards Σ = 1. At fixed Σ, oscillations are thus found in a finite range of T , which widens gradually, and is eventually infinite, as ǫ → 0.

In the oscillation region, the index ν ∼ ǫΣ measuring distance from saturated degradation reaches its maximal value ν opt = g * λ/8 for T = T opt = 2 √ 2 and ǫ = 1, at the maximum of the Σ 1 (T ) curve. Our analysis thus unveils a resonance phenomenon in the dynamics of a self-regulated gene with dynamical transcription rate: this circuit bifurcates most easily to periodic behavior, or more generally is least stable, at a finite value of the gene relaxation time given by τ opt = 2 √ 2τ c where

τ c = 1 g * √ δλ = 1 σ c = t sw g * √ 2 = δ M × 1 g * P 0 λ 0 β 0 . (10)
The quantity τ c gives the time scale at which dynamical behavior departs from the fast regulation case. Because g * is determined by (4), computing precise lower bounds on τ c with (10) requires specifying the degradation mechanisms. Fixing λ 0 = β 0 = 10 mn -1 , P 0 = 100, and g * = 0.5, provides an estimate t c = τ c /δ M = 2 mn which is not unrealistically larger than typical gene induction times. At resonance, the oscillation period is τ osc = 4π 2/3 τ c (about 20 mn in the example above). Since a common interpretation of the diagram of Fig. 1 is that there are only two gene states (bound or unbound) [START_REF] Kepler | [END_REF], one may wonder whether our deterministic analysis is relevant. If g is viewed as a temporal average of gene activity, our results are valid when the response time τ is small compared to the oscillation period so that there are many binding/unbinding events by cycle [START_REF] Forger | Proc. Nat. Acad. Sci[END_REF]. Moreover, transcription is a complex process involving a number of distinct steps [4], and Eq. (1a) is the simplest way to model memory effects arising from cooperativity in the transcription machinery [6].

Anyhow, we now show that even when G is a stochastic variable jumping between 0 and 1, our main result still holds: there is a time scale near τ opt at which oscillations are enhanced. To this end we have carried out stochastic simulations of the reaction network of Fig. 1 using the Gillespie algorithm [START_REF] Gillespie | [END_REF], varying the response time τ at fixed P 0 . Instead of regular oscillations, a sequence of irregularly spaced peaks in protein concentration is observed. A natural question is then whether protein peaks occur more regularly at parameter values where the deterministic model oscillates, in particular when τ = τ opt . We define interpeak times as the time intervals ∆t between two crossings of P = 1.2P avg separated by at least one crossing of P = 0.8P avg , with P avg the mean protein level (thus imposing a minimum amplitude of 40%). Their distribution is characterized by the coefficient of variation η = σ∆t <∆t> . A typical variation of η with τ in our system is shown in Fig. 3. It definitely suggests that the deterministic analysis remains relevant in the stochastic regime, since there is clearly a time scale near τ opt where interpeak time fluctuations are minimal.

In conclusion, we have shown that a nontrivial transcriptional dynamics can destabilize a self-regulated gene. Although it is known that nonlinear degradation mechanisms can induce oscillations in this system, we observe a resonance-like effect such that a much weaker nonlinearity is required when the gene response time matches a characteristic time. Its expression can be computed analytically, which allows us to identify the parameter regions where this effect cannot be neglected. Stochastic simulations confirm the relevance of this time scale in the dynamics of the self-regulated gene. This shows that transcriptional dynamics is a possible source of oscillatory behavior besides other deterministic [START_REF] Goldbeter | Biochemical Oscillations and Cellular Rhythms[END_REF][START_REF] Fall | Computational Cell Biology[END_REF][START_REF] Tiana | [END_REF][START_REF] Goldbeter | Proc. R. Soc. Lond. B[END_REF][START_REF] Leloup | [END_REF]18,19,20,23] and stochastic [29,30] effects.

A natural question is whether our conclusions remain valid when more detailed transcriptional mechanisms or multiple sources of delay are taken into account. It can be shown that adding an explicit delay to Eqs. (3) only further destabilizes the circuit and thus does not essentially interfere with the effect of a gene response delay. Future studies should thus focus on cases where the transcriptional dynamics is more complex and features more than one limiting step.

FIG. 1 :

 1 FIG. 1: Reaction diagram of the self-regulated gene circuit.

FIG. 2 :

 2 FIG. 2: Bifurcation diagram of Eqs. (3) in the (Σ, T ) plane according to (8). A system with balance index ǫ oscillates for Σ < Σǫ(T ) (hatched areas). Black dots indicate oscillating parameter sets of (3), with θ, δ ∈ [10 -1 , 10], θ/α ∈ [10, 1000], λ ∈ [1, 10 3 ], λ/µ ∈ [10, 10 3 ], assuming the degradation mechanisms (9) with a ∈ [10 -6 , 1], κ, χ ∈ [1, 100]. Tc = 1 is the time scale at which transcriptional dynamics cannot be neglected, Topt = 2 √ 2 is the location of the maximum of Σ1(T ).

FIG. 3 :

 3 FIG. 3: Coefficient of variation of interpeak time intervals η vs. mean residence time τ for a stochastic simulation of (3)+(9) rescaled to original variables G, P , M at three values of threshold P0 (from top to bottom, P0 = 100, 500, 2500). Parameter values correspond to δ = 1, θ = 1, α = θ/P0, λ = 21.54, µ = 0.085, χ = 95.5, κ = 21.68, a = 10 -6 . τ1 and τ2 are the boundaries of the deterministic oscillation domain.


Toxicity of copper(I)-NHC complexes against human tumor cells: induction of cell cycle arrest, apoptosis, and

DNA cleavage.

  Marie-Laure Teyssot, Anne-Sophie Jarrousse, Aurélien Chevry, Angélique de Haze, Claude Beaudoin, Michèle Manin, Steven P. Nolan, Silvia Diez-Gonzalez, Laurent Morel, Arnaud Gautier To cite this version: Marie-Laure Teyssot, Anne-Sophie Jarrousse, Aurélien Chevry, Angélique de Haze, Claude Beaudoin, et al.. Toxicity of copper(I)-NHC complexes against human tumor cells: induction of cell cycle arrest, apoptosis, and DNA cleavage.. Chemistry -A European Journal, 2009, 15 (2), pp.314-8. 10.1002/chem.200801992. hal-00352346