domain: info.info-au math spi sdv

text:
Francis Mairet
email: francis.mairet@inria.fr

Olivier Bernard

Robustness of Closed-Loop Control to Biodiversity: a Didactic Example ⋆

Keywords: Biotechnology, stability analysis, nonlinear systems, bioreactor, multispecies, coexistence

Single-species models are generally used for the design of bioprocess control laws. Nonetheless, most of the bioprocesses, if not all, involve an important biodiversity (different species or mutants). Here we propose 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 illustrate 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.

INTRODUCTION

Most of the bioprocesses, if not all, involve an important biodiversity, even when a single microorganism is initially targeted. This diversity, desired or endured can involve different species (e.g. in anaerobic digestion) or, for one species, different strains initially present or resulting from natural mutation within the process (e.g. in pharmaceutical biotechnology). Nevertheless, theoretical developments in automatic control of bioprocesses are generally based on the single-species dogma (one macroscopic reaction involves one species). A few works have shed light on the importance of biodiversity for the modelling and control of bioprocesses. For example, the effect of multispecies has been shown in simulation for anaerobic digestion [START_REF] Ramirez | Modeling microbial diversity in anaerobic digestion through an extended adm1 model[END_REF]]. It has also been tackled in the development of optimal strategies (via simulation for the start-up of an anaerobic digester [START_REF] Sbarciog | Some considerations about control of multi-species anaerobic digestion systems[END_REF] and analytically for fed-batch operation [START_REF] Gajardo | Minimal time sequential batch reactors with bounded and impulse controls for one or more species[END_REF]).

Classically, robustness of control laws are considered with respect to parameter uncertainties or noise measurements, but to our knowledge, never to the presence of other species. Indeed, given a control law designed for one species, what happens when two or more species are actually present? To tackle this problem, we introduce the concept of multispecies robustness. A control law is said multispecies robust for a given set of species if the performance of the control law (in a sense that will be defined later) is not affected by the presence of any species from this set.

To define the multispecies robustness of a control law, one should study the asymptotic behavior of a system including several species competing for a substrate. This has been widely done in mathematical ecology in order to study the competition of species [Smith and 1995]. When a constant dilution rate is applied, the principle of competitive exclusion states that at most one species will survive. On the other hand, De Leenheer and [START_REF] De Leenheer | Feedback control for chemostat models[END_REF], [START_REF] Gouzé | Feedback control for nonmonotone competition models in the chemostat[END_REF], [START_REF] Mazenc | Further results on stabilization of periodic trajectories for a chemostat with two species[END_REF] have proposed control laws in order to obtain coexistence. Finally, recent developments have been proposed for the selection of species [Mairet et al. 2013, Bayen and[START_REF] Bayen | Optimization of the separation of two species in a chemostat[END_REF]]: here, the objective is to design a control strategy in order to select a species of interest. These papers propose some useful tools for studying multispecies robustness.

To illustrate our approach, we study the multispecies robustness of a control law proposed in [START_REF] Mailleret | Robust nonlinear adaptive control for bioreactors with unknown kinetics[END_REF] in order to regulate substrate concentration. 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.

FRAMEWORK AND DEFINITION

We consider a classical model of micro-organism growth limited by one substrate in a chemostat [START_REF] Dochain | Automatic control of bioprocesses[END_REF]]. Given x 1 and s the concentrations of biomass and substrate respectively, the model writes:

{ ṡ = u(t)(s in -s) -k 1 µ 1 (s)x 1 ẋ1 = (µ 1 (s) -u(t)) x 1 (1)
where u(t) is the dilution rate, s in is the input substrate concentration and k 1 is the pseudo yield coefficient. Hypothesis 1. The specific growth rate µ 1 (s) is a nonnegative C 1 function with µ 1 (0) = 0.

A species is defined by a parameter vector θ ∈ S ⊂ R n θ + , gathering stoechiometric and kinetic parameters (i.e. parameters of the growth rate µ ). We will refer to S as a set of species.

We consider a control law which stabilizes the substrate concentration towards a set-point s * . By definition, the multispecies robustness of this control law guarantees that, in presence of multispecies, the system (whose state vector is denoted ξ) will be stabilized at a point ξ with the same or better productivity P(ξ) than for the set-point in monoculture ξ * . Consider a control law u(ξ) which globally stabilizes System (1) towards the set-point s * . Definition 1. The control law u(ξ) is said to be (S, n, P) robust iff, for any n species x i , i = 2, ..., n + 1 such that θ i ∈ S, the control law globally stabilizes the system:

     ṡ = u(ξ)(s in -s) - n+1 ∑ i=1 k i µ i (s)x i ẋi = (µ i (s) -u(ξ)) x i , i = 1, 2, ..., n + 1.
(2) towards a point ξ such that P( ξ) ≥ P(ξ * ), where ξ * = (s * , (s in -s * )/k 1 , 0, ..., 0).

For example, for wastewater treatment, the productivity criterion is defined as the organic load removed: P(ξ) = u(ξ)(s in -s). This criterion will be considered in the following to illustrate our presentation.

APPLICATION

To highlight the concept that we introduce, we will consider the control law proposed in [START_REF] Mailleret | Robust nonlinear adaptive control for bioreactors with unknown kinetics[END_REF]. This must be seen as a simple didactic example to introduce the ideas and show their relevance for bioprocess management.

Control law design

In the following, we assume that the measurement of total growth in the bioreactor is available: Hypothesis 2. We consider that the following measurement is available:

y(ξ) = n+1 ∑ i=1 l i µ i (s)x i
where l i is the associated yield coefficient for species i.

In anaerobic digestion, the methane flow rate can be used given the low solubility of methane (see e.g. [START_REF] Mailleret | Robust nonlinear adaptive control for bioreactors with unknown kinetics[END_REF], [START_REF] Karafyllis | A vector lyapunov function characterization of input-to-state stability with application to robust global stabilization of the chemostat[END_REF]). For other bioprocesses, the total growth can be estimated using observer-based estimator [START_REF] Bastin | On-line estimation and adaptive control of bioreactors[END_REF]Dochain 1990, Perrier et al. 2000] based on measurement of O 2 or CO 2 for example.

Given a set-point s * ∈ (0, s in ), we consider the feedback law proposed by [START_REF] Mailleret | Robust nonlinear adaptive control for bioreactors with unknown kinetics[END_REF]: Theorem 3. Under Hypotheses 1 and 2, the feedback control law

u(ξ) = γy(ξ) (3)
with γ = k1 l1(sin-s * ) globally stabilizes System (1) towards the positive set point (s * , x * 1 ), where

x * 1 = (s in -s * )/k 1 .
Proof. See [START_REF] Mailleret | Robust nonlinear adaptive control for bioreactors with unknown kinetics[END_REF].

Robustness of this control law with respect to fluctuations in s in has been considered in [START_REF] Karafyllis | A vector lyapunov function characterization of input-to-state stability with application to robust global stabilization of the chemostat[END_REF].

Here, we will study its multispecies robustness. For sake of simplicity, we restrict our analyses to the presence of only one additional species with initial conditions (s,

x 1 , x 2 ) in the attractive invariant manifold {(s, x 1 , x 2 ) | s + k 1 x 1 + k 2 x 2 = s in }. Under this hypothesis, System (2) with Control law (3) becomes: ẋi =   µ i (s) -γ 2 ∑ j=1 l j µ j (s)x j   x i , i = 1, 2. ( 4
) with s = s in - ∑ 2 i=1 k i x i .

Equilibria

System (4) admits the following equilibria:

• Substrate depletion (s = 0):

E 0 = (x 0 1 , x 0 2 ) such that k 1 x 0 1 + k 2 x 0 2 = s in . This corresponds to an equilibrium line. • Washout (s = s in ): E w = (0, 0). • Monospecies 1(s = s * ): E 1 = (x * 1 , 0), where x * 1 = 1 γl1 • Monospecies 2 : E 2 = (0, x * 2 ), where x * 2 = 1 γl2 iff k 2 l 2 < γs in (C1)
The substrate concentration at equilibrium is s = s * 2 := s in -k2 l2γ . Note that E 2 exists iff we have s * 2 > 0. • Coexistence : this equilibrium point is possible if the two growth functions have an intersection point:

∃s c ∈ (0, s in ) | µ 1 (s c ) = µ 2 (s c ). (C2) E c = (x 1 , x2 ) where { γ(l 1 x1 + l 2 x2 ) = 1 (k 1 x1 + k 2 x2 ) = s in -s c .
E c exists iff x1 and x2 are positive, i.e. iff:

x1 = l 2 (s c -s * 2 ) k 2 l 1 -k 1 l 2 > 0 and x2 = l 1 (s c -s * ) k 1 l 2 -k 2 l 1 > 0.
(C3) These conditions impose that the substrate concentrations of monospecies equilibrium are located on either side of the intersection point of the growth functions:

• if l2 k2 > l1 k1 , then s * < s c < s * 2 . • if l1 k1 > l2 k2 , then s * 2 < s c < s * (recalling that s * 2 ≤ 0 if E 2 does not exist).
For each non-trivial equilibrium E • , we denote P • the corresponding productivity:

P 1 = µ 1 (s * )(s in -s * ) P 2 = µ 2 (s * 2 )(s in -s * 2 ) P c = µ 1 (s c )(s in -s c )

Local stability analysis

The Jacobian matrix J of System (4) is:

J = [ B 1 -γ(y(ξ) + x 1 A 1 ) -k 2 µ ′ 1 (s)x 1 -γx 1 A 2 -k 1 µ ′ 2 (s)x 2 -γx 2 A 1 B 2 -γ(y(ξ) + x 2 A 2 ) ] (5)
where

A i = l i µ i (s) -k i (l 1 µ ′ 1 (s)x 1 + l 2 µ ′ 2 (s)x 2 ), B i = µ i (s) -k i µ ′ i (s)x i .
Local stability of E 0 . The eigenvalues of the Jacobian matrix J evaluated at E 0 are:

λ 1 = 0 λ 2 = µ ′ 1 (0)x 0 1 (γs in l 1 -k 1 ) + µ ′ 2 (0)x 0 2 (γs in l 2 -k 2 ). (6) Given that γs in l 1 > k 1 , E 0 is unstable if γs in l 2 > k 2 , i.e. if E 2 exists (Condition (C1)).
Otherwise, further analysis allows us to conclude that a subspace of E 0 is locally stable.

Stability of E w . At E w , the Jacobian matrix J becomes:

J = [ µ 1 (s in ) 0 0 µ 2 (s in ) ] (7)
so E w is unstable.

Local stability of E 1 and E 2 .

The eigenvalues of the Jacobian matrix J evaluated at E 1 are:

λ 1 = -µ 1 (s * ) and λ 2 = µ 2 (s * ) -µ 1 (s * ) (8) Thus, E 1 is locally stable iff µ 1 (s * ) > µ 2 (s * ). (C4) Similarly, E 2 is locally stable iff µ 2 (s * 2 ) > µ 1 (s * 2 ). ( C5
)
Local stability of E c . At the equilibrium point E c , the trace and determinant of J writes:

tr(J) = -µ 1 (s c ) -γ x1 x2 [µ ′ 2 (s c ) -µ ′ 1 (s c )] (k 2 l 1 -k 1 l 2 ) det(J) = µ 1 (s c )γ x1 x2 [µ ′ 2 (s c ) -µ ′ 1 (s c )] (k 2 l 1 -k 1 l 2 ) (9) Thus, E c is locally stable iff [µ ′ 2 (s c ) -µ ′ 1 (s c )] (k 2 l 1 -k 1 l 2 ) > 0. (C6)

Multispecies robustness

We now consider the following assumption on the growth functions: Hypothesis 4. The specific growth rates µ i (s) are assumed to be of Michaelis-Menten type:

µ i (s) = μi s s + K i . ( 10
)
where μi and K i are respectively the maximum growth rate and the half-saturation constant.

Remark that only the product/substrate yield coefficient α i := l i /k i matters to compute local stability and productivity. As a consequence, a species can be defined by the set of three parameters:

θ i = (μ i , K i , α i ) (11)
and we consider

S := [μ -, μ+ ] × [K -, K + ] × [α -, α + ]. (12)
Now, we can try to define the largest set1 S R ⊂ S such that Control law ( 3) is (S R , 1, P) robust.

Proposition 5. Given initial conditions (x 0 1 , x 0 2 ) ∈ R * + 2 such that k 1 x 0 1 + k 2 x 0 2 < s in (i.
e. s 0 > 0), the asymptotic behavior of System ( 4) is given in Table 1.

Proof. First, note that System (4) is dissipative: all positive trajectories lie in the bounded set

{(x 1 , x 2 ) ∈ R 2 + | k 1 x 1 + k 2 x 2 ≤ s in }.
If (C1) does not hold, a subspace of E 0 is locally stable (so the control law is not robust). Now, we restrict our analysis to the subset {θ ∈ S | γs in α 2 > 1}.

Let z = l 1 x 1 + l 2 x 2 . In the coordinates (z, x 1 ), System (4) becomes:

{ ż = y(ξ)(1 -γz) ẋ1 = (µ 1 (s) -γy(ξ)) x 1 . ( 13
)
Given positive initial conditions such that k 1 x 0 1 + k 2 x 0 2 < s in , we have y(t) > 0, ∀t ≥ 0 (since E 0 and E w are repulsive). Thus, z tends to γ -1 , which shows the absence of periodic solutions or cycles. Therefore, any trajectory converges to an equilibrium point. The previous studies of equilibria and local stability allow us to conclude for the different cases.

The asymptotic behavior of System (4) is illustrated on Figures 1 and2 for the different cases. The control law is robust whenever all the stable equilibrium lead to a productivity greater or equal to P 1 (see Table 1).

Restriction to more realistic cases. In order to restrict our analysis to a more realistic case, we consider a subset of S more relevant from an ecological point of view. We consider that each parameter represents a trait that is involved in the fitness of the species. A species with the best values for each trait will outcompete all the other species in any environmental conditions, and such super mutant should not exist. Actually, one may assume tradeoffs between the different traits. For m traits, we consider that the m archetypes (or specialist, i.e. a species with the best value for one trait and the worst values for the others) define a Pareto front where all the species lie [START_REF] Shoval | Evolutionary trade-offs, pareto optimality, and the geometry of phenotype space[END_REF]. In our example, we consider three archetypes x μ, x K , x α , with:

θ μ = (μ + , K + , α -) θ K = (μ -, K -, α -) θ α = (μ -, K + , α + )
Note that the best value for K is K -. These archetypes define the subset S: 4), Case A-C (see Proposition 5 and Table 1). Top: Specific growth rates as a function of substrate concentration. Bottom: Phase portraits with some trajectories (in purple). Filled circle: stable equilibrium, blank circles: unstable equilibrium. Blue: species x 1 , green: species x 2 , red: coexistence.

S := {θ

= aθ μ +bθ K +cθ α , ∀(a, b, c) ∈ [0, 1] 3 | a+b+c = 1} (14) Thus, for any triplet (a i , b i , c i ) ∈ [0, 1] 3 | a i + b i + c i = 1, the parameter vector of the species x i (a i , b i , c i ) is given by: θ i =   a i μ+ + (1 -a i )μ - b i K -+ (1 -b i )K + c i α + + (1 -c i )α -   (15)
The morphospace, i.e. the space of trait values, is represented on a ternary plot in Figure 3. Each vertex of the triangle represents an archetype. We consider Control law (3) designed for a species x 1 (0.4, 0.3, 0.3) located near the center of the morphospace. After discretizing the morphospace, we test for each additional species which conditions (C1, C4, C5) hold in order to determine the asymptotic behavior of the system (see Proposition 5 and Table 1). This allows us to define S R (represented in color on Figure 3), a subset of S, such that Control law ( 3) is (S R , 1, P) robust.

First, one can see that the presence of an additional species can increase the productivity. On the other hand, the control law is not robust for a large subsets of S, corresponding mainly to an additional species with a smaller yield coefficient than species x 1 (α 2 < α 1 ). Two situations may occur: the productivity at a stable equilibrium point (E 2 or E c ) is smaller than the productivity P 1 , or there is a reactor shutdown (s and y converge towards zero). A small decrease in productivity can actually be tolerated (although it is not considered in the present definition of multispecies robustness). On the other hand, the reactor shutdown is much more problematic and represents a real drawback of Control law (3). Anyway, parameter γ can be tuned in order to avoid as far as possible such situation, i.e. to increase the robustness to biodiversity of the control law.

The study of robustness for n species is obviously more delicate and is now under investigation. 4), Case D-E (see Proposition 5 and Table 1). Same legend as Figure 1.

Fig. 3. Robustness of Control law (3) on a ternary plot of the morphospace S. Each vertex represents an archetype: x μ(1, 0, 0), x K (0, 1, 0), x α (0, 0, 1). The diamond represents species x 1 (0.4, 0.3, 0.3). The set of species S R , such that Control law (3) is (S R , 1, P) robust, is represented in color. The color map represents the relative increase of productivity (with respect to P 1 ).

DISCUSSION

The robustness with respect to parametric uncertainty is classically studied for bioprocess control laws. Here, we have defined the multispecies robustness. These two approaches are clearly not equivalent. Actually, the n species can be seen as one species with time-varying kinetic and stoechiometric parameters. Such time-varying aspect is generally not considered in parametric uncertainty. Moreover, the multispecies robustness also involves species competition/selection/coexistence. These phenomena, which affect bioprocess productivity, are not considered in the robustness with respect to parametric uncertainty.

Link with species competition

The classical theory of species competition [START_REF] Smith | The theory of the chemostat: dynamics of microbial competition[END_REF] does not apply when the chemostat is operated with a closed loop control, and some surprises can arise. For example, if we consider a slow growing species x 2 (with µ 2 (s) < µ 1 (s), ∀s ∈ (0, s in )), we expect that this species will rapidly be outcompeted (as it happens in open loop), so it should not affect the asymptotic behavior of the system. Actually, with Control law (3), the substrate depletion equilibrium E 0 can become locally stable in presence of x 2 (if α 2 γs in < 1), so this species can cause reactor shutdown (see Figure 1, Case A).

Coexistence of two species

The stable coexistence of two species is possible whenever their specific growth rates intersect. In open loop, this is not possible in practice since the dilution should be chosen exactly equal to the rate at which growth curves intersect. On the other hand, De Leenheer and [START_REF] De Leenheer | Feedback control for chemostat models[END_REF] have proposed a feedback control for the stable coexistence of two species. Assuming increasing growth rates and measurement of both biomass concentrations x 1 and x 2 , the feedback u = c 1 x 1 + c 2 x 2 + ϵ (where c 1 , c 2 , and ϵ are constant which should be chosen such that some conditions hold) globally stabilizes the system towards a coexistence point. [START_REF] Gouzé | Feedback control for nonmonotone competition models in the chemostat[END_REF] have extended this approach for non-monotone growth rates with only the measurement of the sum of biomass concentrations x 1 +x 2 . In both studies, the original system is gathered in a planar competitive system. This allows to exclude the existence of periodic solution using Poincaré-Bendixson Theorem. Here, we have shown that the feedback law (3) can also lead to stable coexistence. In this case, we obtain a planar system which is not competitive (and neither cooperative). A change of coordinate (System (13)) is used to show the absence of periodic solution.

To guarantee stable coexistence for two species with this control law, one must choose the set-point s * accordingly.

Assuming without loss of generality µ ′ 1 (s c ) > µ ′ 2 (s c ), the coexistence point E c is globally asymptotically stable iff α 1 > α 2 and 0 < s * 2 < s c < s * < s in . This last condition is fulfilled whenever on choose s * ∈ (s * , s * ), where s * = max {s c , s in (1 -α 2 /α 1 )} , s * = min {s in , s in (1 -α 2 /α 1 ) + s c } . If α 1 < α 2 , Control law (3) does not allow stable coexistence.

Challenges

Maintaining a high bioreactor productivity despite invasive species (or new individuals resulting from natural mutations) turns out to be a challenging problem to ensure process robustness. New questions arise now, such as detecting and preventing the apparition of "bad species". Observers could be set-up to early identify new competing species with reduced productivity capability. Control strategies, such as extremum-seeking [START_REF] Guay | Adaptive extremum seeking control of continuous stirred tank bioreactors with unknown growth kinetics[END_REF], could be adapted in order to consider multispecies and maintain conditions favorable for the settlement of mote productive species. As demonstrated in this paper, such control strategies could use this opportunity to continuously improve the reactor performance.

CONCLUSION

We have introduced the concept of multispecies robustness for bioprocess control laws. We have illustrated our approach with a control law proposed in [START_REF] Mailleret | Robust nonlinear adaptive control for bioreactors with unknown kinetics[END_REF]. Depending on the characteristics of the additional species, some counter-intuitive results may appear such as coexistence, or even reactor shutdown when a slowgrowing species is introduced. This framework should be used to design multispecies robust control laws and therefore better tame biodiversity within a biotechnological process.

  Waltman ⋆ This work benefited from the support of the Facteur 4 research project founded by the French National Research Agency (ANR).

Fig. 1 .

 1 Fig. 1. Asymptotic behavior of System (4), Case A-C (see Proposition 5 and Table1). Top: Specific growth rates as a function of substrate concentration. Bottom: Phase portraits with some trajectories (in purple). Filled circle: stable equilibrium, blank circles: unstable equilibrium. Blue: species x 1 , green: species x 2 , red: coexistence.

Fig. 2 .

 2 Fig. 2. Asymptotic behavior of System (4), Case D-E (see Proposition 5 and Table1). Same legend as Figure1.

Table 1 .

 1 Equilibrium and robustness for System (4) (see Proposition 5)

	Case	Conditions	E 0	Ew	E 1	E 2	Ec	Robustness
	A	¬C1	LS	uns	-	no	-	F
	B	C1.C4.(¬C5)	uns uns	GS	uns	no	T
	C D	C1.(¬C4).C5 C1.(¬C4).(¬C5) a	uns uns uns uns uns uns uns GS	no GS	T iff P 2 = µ 2 (s * 2 )(s in -s * 2 ) ≥ P 1 T iff Pc = µ 1 (sc)(s in -sc) ≥ P 1
	E	C1.C4.C5 b	uns uns	LS	LS	uns T iff P 2 = µ 2 (s * 2 )(s in -s * 2 ) ≥ P 1
	LS: locally stable, GS:globally stable, uns: unstable, -:whatever, T/F: True/False.
	a: or equivalently C1.C2.C3.C6.
	b: or equivalently C1.C2.C3.(¬C6).

note that we ignore subsets of null measure corresponding to nongeneric cases, since they will never appear in practice (for example the cases s * = sc, or s *

2 = sc).


text:
Ghjuvan Micaelu Grimaud
email: ghjuvan.grimaud@inria.fr

Francis Mairet
email: fran-cis.mairet@inria.fr

Olivier Bernard

Modelling thermal adaptation in microalgae : an adaptive dynamics point of view

Keywords: temperature, evolution, adaptive dynamics theory, microalgae

Controlling microalgae outdoor cultures is a growing challenge as these organisms could be used at the industrial scale to produce biofuels. In this context, understanding how temperature affects microalgae is a major field. Because of their high division rate, microalgae rapidly adapt to their environment through a process of selection -mutation so that it is not possible to neglect the evolution part. We present here a simple Monod-like model to account for temperature effect on microalgae. We then use it in an evolutionary perspective thanks to the Adaptive Dynamics theory to understand how temperature drives evolution. We analyze the model for a constant temperature, showing that the optimal temperature trait tends to equal the environment temperature. We then study the case where the temperature is periodically fluctuating, and we find a stable equilibrium. Finally, we simulate the model to show that evolutionary branching in two distinct morphs can occur under this condition, which could be a first step to find a criterion for strain selection.

INTRODUCTION

Autotrophic micro-organisms called 'microalgae' brought together a variety of unicellular photosynthetic eukaryotes. Some of them are able to accumulate an important amount of neutral lipids [START_REF] Mata | Microalgae for biodiesel production and other applications: a review[END_REF], which can be turned into biofuel, others can store important quantities of carbohydrates, used for methane production. Thus, they are a promising biotechnology resource for the future [START_REF] Wijffels | Potential of industrial biotechnology with cyanobacteria and eukaryotic microalgae[END_REF]. Their growth is mostly influenced by light, but temperature is the second determining factor because microalgae are ectotherms (i.e they can't regulate their internal temperature). The temperature effect on growth is often neglected in the modelling of microalgae. In regard to the increasing interest for outdoor production of microalgae, temperature effect on growth has to be investigated [START_REF] Ras | Temperature effect on microalgae: a crucial factor for outdoor production[END_REF].

Because of their high division rate, microalgae are able to rapidly adapt to their environment through a process of selection -mutation. In outdoor cultures, the daily range of temperature fluctuation can be more than 20 • C [START_REF] Bechet | Mechanistic modeling of broth temperature in outdoor photobioreactors[END_REF]. It is thus important to understand how temperature drives evolution. To date, there is only one model which predicts microalgae adaptation to temperature [START_REF] Thomas | A global pattern of thermal adaptation in marine phytoplankton[END_REF], by simulations based on the Adaptive Dynamics theory. However, this model implies several assumptions that have been put in doubt [START_REF] Boyd | Marine phytoplankton temperature versus growth responses from polar to tropical waters outcome of a scientific community-wide study[END_REF].

We propose here a simple Monod-like model that represents the effect of a constant temperature on growth. We employ it to model microalgae adaptation to temperature. In line with [START_REF] Thomas | A global pattern of thermal adaptation in marine phytoplankton[END_REF], we use the Adaptive Dynamics theory. We keep the model as simple as possible to study it analytically. In a second part, we use the model under periodic temperature to account for more realistic conditions. Finally, we simulate strain separation through evolutionary branching under fluctuating temperature.

MODELLING TEMPERATURE EFFECT ON MICROALGAE GROWTH

The influence of temperature on all ectotherms growth is a well-known asymetric curve, called thermal reaction norm (Fig. 1 A). In microalgae, two models have been used to describe temperature effect on growth [START_REF] Norberg | Biodiversity and ecosystem functioning: A complex adaptive systems approach[END_REF][START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF]. We use the model from [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF] based on the Cardinal Temperature Model with Inflection (CTMI) of [START_REF] Rosso | An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model[END_REF] because its parameters can be directly estimated on the available data:

µ(T ) =              0 for T < T min µ opt λ(T ) β(T ) for T min < T < T max 0 for T > T max (1)
with

λ(T ) = (T -T max )(T -T min ) 2 β(T ) = (T opt -T min ) [ (T opt -T min )(T -T opt ) -(T opt -T max )(T opt + T min -2T ) ] (2)
and

T opt > T min + T max 2 (3)
T corresponds to the temperature of the environment, µ(T ) is the growth rate of a given species, T min and T max are the minimum and maximum temperatures that sustain growth, T opt is the optimal temperature for growth and µ opt is the theoretical maximum growth rate. Contrary to [START_REF] Norberg | Biodiversity and ecosystem functioning: A complex adaptive systems approach[END_REF] used in [START_REF] Thomas | A global pattern of thermal adaptation in marine phytoplankton[END_REF], this model does not imply any assumption concerning the structural link between the cardinal temperatures and the maximal growth rate (e.g. if T opt is hotter, µ opt will be higher).

In the following, we use a Monod-type model including the temperature effect on microalgae growth in a chemostat:

M : { Ṡ = D(S in -S) -µ(T )ρ(S)X Ẋ = µ(T )ρ(S)X -DX (4)
where S is the nutrient concentration in the chemostat, X is the algal biomass concentration, D is the dilution rate and with:

ρ(S) = S K + S (5) K is a half-saturation coefficient.
It is possible to calculate the non zero equilibrium (S * , X * ) of system (M ):

S * = KD µ(T ) -D X * = (S in -S * ) (6)
with the hypothesis µ(T )ρ(S in ) -D > 0.

In the following, we use a Lyapunov function taken from [START_REF] Harisson | Global stability of predator-prey interactions[END_REF] to prove that (6) is globally asymptotically stable. Lemma 1. The Lyapunov candidate function is given:

V (S, X) = ∫ S S * µ(T )ρ(w) -µ(T )ρ(S * ) µ(T )ρ(w) dw + ∫ X X * w -X * w dw (7) with V : B → R 2 where B is an open containing (S * , X * ). V (S, X) is zero at (S * , X *
), positive at all other points (S, X), defined and monotone increasing when |X -

X * | or |S -S * | increases. The time derivative of V is: V (S, X) = 1 µ(T )ρ(S) (D -µ(T )ρ(S)) [ µ(T )ρ(S) (S in -S * ) -D(S in -S) ] (8)
Proof: It is obvious that V (S * , X * ) = 0. Moreover, since the integrands are of the same sign as X -X * and S -S * , the integrals are positive and increasing as |X -X * | and |S -S * | increase. Proposition 2. If D < µ(T )ρ(S in ), System (M ) admits one non zero equilibrium which is globally asymptotically stable.

Proof: It is sufficient to prove that (8)< 0 ∀(S, X) ∈ B, (S, X) ̸ = (S * , X * ).

If S > S * (resp. S < S * ), then D -µ(T )ρ(S) < 0 (resp. > 0) because µ(T )ρ(S * ) = D, and

[ µ(T )ρ(S)(S in -S * ) - D(S in -S)
] > 0 (resp. < 0) because S in -S < S in -S * (resp. S in -S > S in -S * ). Thus, (8)< 0 is verified, and (S * , X * ) is globally asympotitcally stable.

EVOLUTIONARY MODEL

General case

We now study the system (M ) in the context of adaptive dynamics, introducing a mutant X mut with an adaptive trait a mut (i.e. a quantifiable trait that is likely to evolve) different from the resident trait a, with µ mut (a mut , T ) and µ(a, T ) [START_REF] Dieckmann | The dynamical theory of coevolution: A derivation from stochastic ecological processes[END_REF]. For further introduction to adaptive dynamics, see [START_REF] Dieckmann | A beginners guide to adaptive dynamics[END_REF]. To define the canonical equation of adaptive dynamics, i.e. the equation of the evolution of trait a [START_REF] Dieckmann | The dynamical theory of coevolution: A derivation from stochastic ecological processes[END_REF], we need to find the mutant growth rate (per capita) in the resident population at equilibrium, called invasion fitness f (a mut , a). Proposition 3. The invasion fitness for System (4) is given by:

f (a mut , a) = D ( λ mut λ β β mut -1 ) (9)
Proof: This results from the steady-state condition of the resident system, and from the hypothesis that the mutant is initially rare. We can thus replace S by its equilibrium value S * :

f (a mut , a) = µ mut (a mut , T )ρ(S * ) -D (10)

Modelling the evolution of the optimal temperature trait

We choose to study the adaptive trait a = T opt , assuming that temperature will mainly affect the optimal conditions for growth. Because of the constraint (3) on T opt , we choose to study the case:

T > T min + T max 2 (11)
We calculate the selection gradient, which gives the direction of the selection, using (9):

∂f (a mut , a) ∂a mut amut=a = - β ′ (a) β(a) D (12) with β ′ (a) = -6a 2 + (6T + 2T max + 4T min )a +(-2T max -4T min )T (13)
We can deduce the canonical equation of the adaptive trait, that describes the evolution of T opt at the evolutionary time scale θ:

da dθ = -M p σ 2 X * D β ′ (a) β(a) (14)
where M p is the probability to be a mutant at each apparition, and σ is the mutation step.

We then search for the evolutionary singular strategy. We find that da/dθ = 0 for a * = T , which means that the optimal temperature trait tends to equal the environment temperature (Fig. 1 B). Note that the same results can be obtained with the Droop model (not presented here for sake of brevity).

We investigate the stability of the singular strategy examining the sign of the fitness invasion second order derivative.

Proof:

The following conditions are respected:

H1. ∂ 2 f (a mut , a) ∂a 2 mut amut=a=a * < ∂ 2 f (a mut , a) ∂a 2 amut=a=a * H2. ∂ 2 f (a mut , a) ∂a 2 mut amut=a=a * < 0
Indeed, we have:

∂ 2 f (a mut , a) ∂a 2 mut amut=a=a * = -D β ′′ β -2β ′2 β 2 ∂ 2 f (a mut , a) ∂a 2 amut=a=a * = D β ′′ β (15)
β ′ (a * ) = 0, and so it is sufficient to prove:

-D β ′′ (a * ) β(a * ) < 0 (16) Yet, we have β(a * ) = -(T -T min )(T -T max )(T min - T ) < 0 because T min < T < T max . Also, β ′′ (a * ) = 2T max + 4T min -6T . β ′′ (a * ) < 0 is equivalent to T > (2T min + T max )/3
which is true because of (11). Thus, ( 16) is true. H1 and H2 are verified.

Structural link between adaptive traits

In a real case of evolution, it is possible that several adaptive traits evolve concurrently, e.g. T max and T opt . In bacteria, [START_REF] Rosso | An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model[END_REF] point out that there exists a structural link between T max and T opt . That is, T max can be written as a linear function of T opt . We compiled data from [START_REF] Miller | Evolution of thermotolerance in hot spring cyanobacteria of the genus Synechococcus[END_REF] for cyanobacteria and from [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF] for microalgae to verify whether this relation is true for these organisms (Fig. 2).

The data reveal a good correlation:

T max = mT opt + p ( 17
)
with m = 0.93 and p = 9.83, r = 0.941 (Fig. 2). This property must depend on the genetic interactions between traits, but the exact reason for such a link is still to understand.

In expression (9), we replace T max by (17). By taking the adaptive trait a = T opt , the selection gradient becomes:

∂f (a mut , a) ∂a mut amut=a = D λ ′ (a)β(a) -β ′ (a)λ(a) λ(a)β(a) (18) with λ ′ (a) = -m(T -T min ) 2 (19)
We search for the evolutionary singular strategy by setting Eq.( 18) equal to zero. As for ( 14), we find that a * = T , and thus T * max = mT + p (Fig. 1 C). This is an important result in the context of co-evolving species, because whether T max evolve or not, the thermal reaction norm will differently affect the species fitness in its environment, and so its competition with other species or stains.

FLUCTUATING TEMPERATURE

Ecological timescale

We now study the system (M ) in the context of fluctuating temperature:

{ T = T inf if t mod τ ∈ [ϵ; τ 1 -ϵ[ T = T supp if t mod τ ∈ [τ 1 + ϵ; τ -ϵ] ( 20
)
with T inf < T supp . By applying the conservation principle, we have the equality X + S = S in at equilibrium, that is to say S = S in -X. Thus, the system (M ) can be reduced to a one dimension differential equation: [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF] are represented in blue, data from [START_REF] Miller | Evolution of thermotolerance in hot spring cyanobacteria of the genus Synechococcus[END_REF] are in red. From Miller and Castenholz (2000), we took data for the Synechococcus strains C9, SH94, OH2,20,30,29,28,4. Vertical and horizontal bars errors are represented, computed with the same method as [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF].

Ẋ = [µ(T (t))ρ(S in -X) -D]X (21)
where T (t) is given by Eq. ( 20). We define g(t, X)

def = µ(T (t))ρ(S in -X) -D.
We follow the same reasoning as [START_REF] Butler | A mathematical model of the chemostat with periodic washout rate[END_REF] and [START_REF] Butler | Predator-prey system with periodic coefficients[END_REF] who study a similar Monod-type model, but with a time varying dilution rate D(t) instead, and a predator-prey system with periodic coefficients, respectively.

Theorem 1: For D < min(µ(T)ρ(S in ), equation ( 21) has a unique nontrivial positive periodic solution ψ(t) which is globally orbitally asymptotically stable. Moreover, we have min s∈[0,τ ] A(s) ≤ ψ(t) ≤ max s∈[0,τ ] A(s) with

A(t) := [D(S in + K) -µ(T (t))S in ]/(D -µ(T (t))
(which corresponds to the steady state biomass concentration for constant T ).

Proof: ∂g/∂X exists and is continuous for (t,

X) ∈ R 1 × R 1 + with R 1 + = {X ∈ R 1 : X ≥ 0}. Moreover, ∃A(t) > 0 ∋ [X -A(t)]g(t, X) < 0 ∀X > 0, X ̸ = A(t). Indeed, given D < min(µ(T)ρ(S in ), we have: [X -A(t)] g(t, X) < 0 ∀X > 0, X ̸ = A(t) (22)
Thus, Massera's theorem [START_REF] Massera | The existence of periodic solutions of systems of differential equations[END_REF] easily implies the existence of a periodic solution ψ(t) of ( 21

) satisfying min s∈[0,τ ] A(s) ≤ ψ(t) ≤ max s∈[0,τ ] A(s).
ψ(t) is the unique solution of ( 21), given that X∂g(t, X)/∂X < 0 for all (t, X) ∈ R 1 × R 1 + . Indeed, we have:

X ∂g(t, X) ∂X = µ(T )X -K (S in -X + K) 2 (23)
Because X > 0, K > 0, X∂g(t, X)/∂X < 0 is always verified. Following the same reasoning as [START_REF] Butler | Predator-prey system with periodic coefficients[END_REF], X∂g(t, X)/∂X < 0 implies that g(t, X) is strictly decreasing as a function of X, for X > 0, for all t. So, if two solutions ψ(t) and ψ 2 (t) exist, with ψ(t) < ψ 2 (t), it implies that ψ ′ (t)/ψ(t) = g(t, ψ(t)) > g(t, ψ 2 (t)) = ψ ′ 2 (t)/ψ 2 (t) for all t. Integrating this inequality over [0, τ ] leads to a contradiction which prove the uniqueness of ψ(t).

Using theorem 1, we have the following inequality with the periodic temperature (20):

D(S in + K) -µ(T inf )S in D -µ(T inf ) ≤ X(t) ≤ D(S in + K) -µ(T supp )S in D -µ(T supp ) (24)
which means that

DK µ(T inf ) -D ≤ S(t) ≤ DK µ(T supp ) -D (25)
If the time spent at each temperature is sufficiently long, then, when T = T inf (resp. T = T supp ), the substrate concentration converges towards its equilibrium S * inf = DK/(µ(T inf )-D) (resp. S * supp = DK/(µ(T supp )-D)). We assume that the state transition between S * inf and S * supp is negligible. From a biological point of view, this assumption implies that the growth rates at both temperatures are quite similar.

Evolutionary timescale

As explained previously, we assume that the resident population reaches rapidly its equilibrium for each temperature applied. In the case with the adaptive trait a = T opt and considering a constant T max , we obtain:

           f (a mut , a) = D ( β(T inf ) β mut (T inf ) -1 ) if T = T inf f (a mut , a) = D ( β(T supp ) β mut (T supp ) -1 ) if T = T supp (26)
It is possible to use the average mutant growth rate in the resident population at steady-state [START_REF] Ripa | Mutant invasions and adaptive dynamics in variable environments[END_REF]:

f (a mut , a) = 1 τ ∫ τ 0 f (a mut , a; t) dt which is equivalent to: f (a mut , a) = D [ β(a, T inf ) β(a mut , T inf ) τ 1 τ + β(a, T supp ) β(a mut , T supp ) τ -τ 1 τ -1 ] (27)
We thus deduce the selection gradient: and so the canonical equation of the adaptive trait with fluctuating temperature is:

∂ f (a mut , a) ∂a mut amut=a = D [ - τ 1 τ β ′ (T inf ) β(T inf ) - τ -τ 1 τ β ′ (T supp ) β(T supp ) ] (28)
da dθ = M p σ X * ∂ f (a mut , a) ∂a mut amut=a (29)
Equation ( 29) is not analytically tractable. We perform simulations of (29) to search for the evolutionary singular strategy. Result (Fig. 3 C) shows that, at steady, T opt does not converge to the average temperature T mean . This result is still available if we replace T max by a linear function of T opt (Fig. 3 B), even if a * is different. The asymmetric property of the thermal reaction norm is probably the reason for such outcome.

Evolutionary Branching conditions

Under particular conditions, Adaptive Dynamics predict that evolution can converge to a specific singular strategy called 'branching point' where selection becomes disruptive, so that two strains move apart [START_REF] Metz | How should we define fitness for general ecological scenarios ?[END_REF].

The mutant can invade the resident and reciprocally, and both stably coexist.

We investigate the evolutionary dynamics of (3) by using a pairwise invasibility plot (PIP) (Fig. 4) [START_REF] Geritz | Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree[END_REF]. This allow us to determine the stability of the singular strategy, studying graphically the sign of the mutant invasion fitness. For temperatures comprised between T inf = 20 can mutually invade in such a way that two strains separate. For a range of values around the previous ones, we observe that the singular strategy bifurcates in three singular strategies (Fig. 4 B,C,D). In Fig. 4 D, there is still a branching point, which seems to allow a strong separation between strains because of the large area of positive invasion fitness.

To confirm the results found previously, we perform a simulation of an evolutionary branching. In line with [START_REF] Mirrahimi | Evolution of species trait through resource competition[END_REF], we consider a model where the adaptive trait a becomes a continuous trait: Fig. 5. Evolutionary branching of trait a for the parameters of Table 1. The trait a is expressed in • C, t is arbitrarily expressed in hours. The colored lines correspond to a population size for each trait value.

     ∂ t X(a, t) = X(a, t)[µ(a, T )S(t) -D] + ϵ∆X(a, t) S(t) = DS in D + ∫ µ(a, T )X(a, t) da (30)
where X(a, t) is the species density with trait a = T opt , S(t) is the quasi-static approximation of resource dynamics, ϵ is the mutation rate. We use the parameters listed in Table 1. Fig. 5 shows that an evolutionary branching really occurs. Two general morphs with two distinct T opt appear and stabilize.

CONCLUSION

We have proposed a simple model of temperature effect on microalgae growth based on the model developped by [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF]. We used it in an evolutionary perspective thanks to the adaptive dynamics. We found that, under constant temperature, the optimal temperature tends to equal the environment temperature.

In a second part, we showed that there exists a structural link between T max and T opt . In the evolutionary sense, the optimal temperature also tends to the environment temperature, but the thermal reaction norm is different. Thus, the outcome of evolution if several microalgae species coexist is mostly influenced by this structural link, which may differ between species depending upon the genetics and physiological constraints.

Finally, we studied our model under a simple fluctuating temperature signal. We showed that a stable periodic solution exists. At evolutionary time scale, the fluctuating temperature allows strains to separate, and evolutionary branching occurs. We simulated the strain separation and found results consistent with our theoretical approach. This may be a first step to understand how species coexist under fluctuating temperature. It could serve to find a criterion for selecting species with the highest growth rate under particular temperature conditions, which is of key interest for microalgae outdoor production.

Fig. 1 .

 1 Fig. 1. Thermal reaction norm for Nannochloropsis oceanica (A).

Fig. 2 .

 2 Fig. 2. Linear relationship between Tmax and Topt. Data from

Fig. 3 .

 3 Fig. 3. Evolution of the thermal reaction norm for T inf = 24 • C, Tsupp = 31 • C. (A) is the initial reaction norm, (B) is the evolutionary reaction norm for a = Topt and Tmax = mTopt +p at steady-state, and (C) is the evolutionary reaction norm for a = Topt and Tmax = 31.5 • C at steady state. The singular strategies are represented by black points.

Fig. 4 .

 4 Fig. 4. Pairwise Invasibility Plots across a range of values of T inf and Tsupp (expressed in• C). The other parameters are listed in Table 1. Grey and white represent positive and negative invasion fitness, respectively. Evolutionary singular strategy are represented by a black circle.

  Proposition 4. a * is Convergent Stable Strategy (CSS), which means that the singular strategy is attractive, and Evolutionary Stable Strategy (ESS), which means that the resident a * can't be invade by another mutant.

Table 1 .

 1 Model parameters for evolutionary branching.

	Parameters	Unit
	4 • C Tmax, maximall temperature for growth 40 • C T min , minimal temperature for growth
	T inf , inferior temperature applyed Tsupp, superior temperature applyed	20 • C 36 • C
	τ 1 , time during which T inf is applyed	12 h
	τ , period of temperature fluctuation	24 h

  • C). The other parameters are listed in Table 1. Grey and white represent positive and negative invasion fitness, respectively. Evolutionary singular strategy are represented by a black circle.

ACKNOWLEDGEMENTS

This work was supported by the ANR Facteur 4 project.


Marie Després-Lonnet
email: marie.despres-lonnet@univ-lille3.fr-dominique.cotte@univ-lille3.fr.

Dominique Cotte

Introduction

Ever since the invention of the web, it has been anticipated that the Internet would become the place where everyone could participate in building the global knowledge of tomorrow. However, the lack of technical ability has long prevented most people from producing online resources.

Things are quickly changing now, as new tools like blogs or wikis give anyone the opportunity to become both a writer and an online editor; without having to deal with complicated client/server transfer protocols or to take encoding and mark-up languages into account. The latent desire for such collaborative publication spaces explains the success of projects like Wikipedia® and the exponential growth of weblogs.

Nevertheless, merely providing adequate tools is not the main issue as far as online publishing is concerned. Recent hackers attacks against collaborative sites have shown that not all of us share the same views of a better tomorrow. Questions raised by Wikipedia® fans regarding regulation and means of preventing malevolent writers from interfering with their collaborative project, have shed light on an Internet specific practice: the massive use of nicknames behind which anyone can hide. To whom may one turn when authoring and editing responsibilities can no longer be clearly read from the traditional masthead but often have to be deciphered through anonymous web pages.

Our aim, in this paper, is not to address the numerous potential misuses of totally anonymous a media sphere, but to show that new authority markers emerge on the web. Markers that one must learn to look for, in order to measure the credibility of a web site.

Having put our focus on the editing world, it seemed logical for us to initiate our investigations with actors from the paper industry, having to adapt their practices to new contexts. We then chose to study two ambitious projects especially conceived for the web by newcomers but having strong links with emblematic realizations in terms of publication and knowledge diffusion: that is the encyclopedia and the newspapers.

Legitimizing fields and forms

Since the middle of the 15th century, when printing became an industry in Europe, professionals have invented multiple editorial processes to give texts a better chance to widely spread among the general public. These inventions related to work distribution, as well as tool improvement or page making up. This is not the place to develop the historical aspects of this evolution. We just want to stress the importance of those craftsmen and specialists in the process of selecting manuscripts, designing fonts, assembling pages in volumes, and, more generally, always innovating by introducing new tools and techniques. Even if printer and publisher became two different jobs later on, the interaction between what could be called the "editorial intention" and the printing means has always remained strong.

The very justification of the publisher1 is, of course, not only to be aware of technical modernization. He mostly plays a major part in the emergence of opinions, theories and knowledge. Apart from the technical processes that turn a manuscript into a printed book, the editorial corporation has created its own chain of selection, and validation in order to legitimate and bring texts to life. In this way, the editor takes a high responsibility. If we consider things from the reader's point of view, the very fact that a work or text has been published might legitimate its existence, if not its content. Someone (the company as a trademark or a famous name in the editorial field) has finally decided that, for some reason, such a text has or has not to be brought to the public. As a reader, I rely on the specialist and trust his choices. The brand name, the reputation of the company, the previous texts published in the same collection, give me clues about the sort of works and texts I am to find in a new publication.

Thus, the publisher has conceived a whole organization, built on a chain of responsibilities, to assert his legitimacy and professionalism. Every actor in the chain has an assigned role. The reader does not question this organization, whether he exactly knows how it functions or not. This is not a concern for him, except when the product is obviously corrupted.

In the same way, readers and consumers apprehend books, newspapers and magazines as almost natural "every day" items. Their structure and form have been integrated

for so long that they are now considered as obvious . Habits and acculturation forged from childhood and enforced both by school training and a frequent use lead us to forget that they are cultural forms invented and refined for years and years by highly qualified professionals and specialists, who have created complex editorial processes, starting from the basic invention of the printing press during the 15th century.

The way we comprehend a book or our favourite magazine also has a spatial and a corporal dimension. Varied tools, such as table of contents or page numbers, as well as "paratexts", guide us through the pages. We therefore find it easy to directly access the cultural or sport section when opening a newspaper. These tools and forms have been remarkably stable for years, even if, formally, the look of a modern newspaper, with colours, photographs, and a spaced out design has little to see with, let's say, the sixties' layout. The main editorial marks have not changed: a newspaper will assert the legitimacy of its content through elements such as its name ("The Times", "Le Monde"…), the reputation of its columnists, its editorial line, or its deontology codes or charters. They sometimes let the reader know what the rules are for printing such or such an article. For instance, one of French daily paper "Liberation" principle is never to publish the criticism of a book written by one of its employees.

The logic of legitimacy is highly developed in the case of scientific literature, with the principle of peer-review, double blind reading and so on. Thus, one usually accepts that information published by "Nature" or "New Scientist" is "true"2 , or at least validated by the scientific community.

However, these editorial marks are not the prerogative of scientific, intellectual or elitist contents. One may also identify multiple and more trivial genres, at first sight, from their particular layout and distinctive formal characteristics. Readers, and even non-readers, have integrated these codes and marks as cultural referents, which contribute to guide them and help them in their choices and activities. A novice reader will consider a thick book, printed in small fonts as "intellectual", while a university teacher will look at a highly illustrated publication, with contrasted colours and heavy printed titles, with some kind of despise.

Lastly, the very place where one can find a printed work says a lot about its content. Newspapers stands, bookshops or libraries, are also important actors in the assertion process. Selecting books, building a collection, putting certain publications in display and others in a stockroom, destroying unsold copies are decisive actions in the process of guiding readers.

The main issue for us is not to question the effective part played by these different factors in the process of reaching and convincing readers, but to analyze how habits, places, and actors have adapted to the digital age. How have readers and publishers dealt with this radical transfer from a support, with its own standards and forms, to another support that has nothing to do with the first one, except to mix lines of characters and pictures, so as to build a text and bring it to potential readers. More widely, how have the rules that governed the printing industry for centuries adapted to this new environment and how have readers built new strategies to know where they are in this new publication space?

Publishing on the Web

We chose to focus on newspapers online publications to get a better understanding of what had been the web sites policies of actors from the paper" world. We then analyzed their consequences in terms of relations to readers.

When they first came to the Internet, about ten years ago, newspapers firms did not fully invest it as a new medium or an additional publication space. In the nineties, the web was not the place where one had to be seen yet. Due to the lack of user-friendly tools and the absence of on-line publishing teams, site development was often delegated to computer specialists, not so well versed in the art of site design and publishing. If one adds that the first generation of HTML gave only few editing possibilities, the result was far from the paper products demand. The graphical quality of the first sites was quite poor and the editor's know how could hardly be seen from the average layout.

A visit to "the Internet Archive" gave us the opportunity to take interesting screen shots from old age newspapers sites. One can easily measure the gap between the first generation lay out and today's. The 1997 "Financial Times" site background was deep black; far from its ochre paper-tone and "Le Monde" graphical charter is only merely applied on the homepage of its 1996 site. Readers were thus often confronted with texts that could not, at first sight, by clearly see as complementary to their usual newspaper contents. The financial times site might as well have been a fake, as Domain Names had not been protected yet. Anyone could thus have easily bought this trademark dot com name and made a very different use of it.

Nevertheless, regular readers are not so easily fooled. They have developed such closeness to their favourite publication that they have integrated its usual structure, organization and contents. Therefore, they know what to expect from the site and although the graphical arrangement has nothing to do with the paper copy they quickly make the link. For example, the Business Week oldest site only displayed a simplified table of contents, made of blue underlined titles that were nothing but the basic format for any hyperlink with no specific attributes. However, both the style and structure of this simple table of contents gave sufficient clues to make sure that it was the newspaper's very site.

Within a few years things changed dramatically. Both technical improvements and the Internet development brought companies to reconsider their strategy. By 2000, Web sites became a first range concern and financial means were affected to their development. Nevertheless, they were still seen as showcases, aiming at driving readers back to the paper edition. There were no Internet specific contents and, seemingly, no clear comprehension of the potential of this new medium.

Publishers tended to reproduce the schemes they traditionally relied on and screens reflected the paper pages layout. Headlines and sections were thought as electronic equivalents of those found on the paper editions. Though the scripting space had nothing in common with newspapers pages, sections were calibrated according to the old rules. This had heavy consequences on the way people comprehended the online contents versus the paper ones. For example, the graphical links created by placing items on the same page and under the same headline had not been rethought for screen display. Consequently, articles belonging to the same section were often placed one after the other, preventing readers from bringing together different points of view or seeing the complementarities between two articles. In the same way, contents were not adapted to their new editing context and one could often find markers such as "see above", "read next page" or "to be continued" that referred to nothing on the screen space.

Hyperlinks were merely used as page-turners and navigation was mostly considered, on a top down basis, from the home page to specific sections. The relationship with readers was still one of "emitting/receiving". The possibility of interacting with journalists had not been thought of at all and updating followed the paper edition periodicity.

In the meantime, the web has grown to a much wider extend. New editing tools, like blogs and wikis, made it quite easy for anyone to become both a writer and an online editor. Formerly confidential newsgroups can now easily be reached through the web and queried via search engines. The right to speak seems to be potentially offered to all of us. Emails, chats, and messengers have diversified the ways we can interact with others and the demand for interactivity has grown accordingly.

Newspapers could not stay behind. They now provide a wide range of tools thought to ease interaction between editorial offices and readers. Traditional "letters to the editor" often take the shape of emails, but new kinds of relations have also been initiated. Journalists have extended the range of their contributions and now often animate blogs on current affairs, giving readers the opportunity to react and give their opinions online. Yet, journalists keep the main part in the talk. Things are still initiated from their side as readers are only invited to comment. Comments are thus given a second range place as they are often presented on separate pages, only sorted according to their date of issue. Furthermore, there is a strong opposition between the journalist's name and the reader's nickname, leading to a symbolic hierarchy. The former builds his reputation on his name; the latter is only given the right to react. The discussion does not take place on equal terms. The figure of the journalist prevails against the public's opinion.

Regarding contents, things have also taken a different path. The web edition is now seen as necessary to offset the "freshness less" of the paper edition. Sites are updated every minute and dated agency dispatches scroll in continuously, attesting of the awareness given to the newest news. Sections are reordered along the day, according to events and scoops. The paper issue is still the support where events are analyzed and commented, while the site seems closer and closer to a press agency. Priority is given to reactivity and a greater proximity to readers, which leads to new offers, like instant opinion polls or online auctions Conclusively, we could roughly distinguish three main periods, as far as newspapers web publication is concerned:

-

The Pioneer stage, when both tools and experience were rather poor and led to quite tinkered sites.

-The Xerox age, with sites totally depending on the paper edition contents;

-The Agora era, the site now being the place where readers and journalists share opinions and get in closer touch, while the number of web sites grows on an exponential scale.

A world-wide publication space

Nowadays, with millions of sites potentially at hand and multiple services converging on portals, it has become harder and harder for a provider to take his place on the web. All the more since the screen is the one and only support, its very size commands any choice made as far as layout is concerned. Part of it is obviously taken by software frames, which diminishes the space devolved to contents. In the domain of making up, any parameter that has not been thought of by the conceiver will be overcome by "default" values. This goes from the choice of fonts to the place where images will appear on the screen. These different constraints lead to a sort of layout "normalization".

As sites are now countless, it becomes harder for users to find their way. Not only have specific markers faded out, but also a great number of signs that where external to the support itself, but of great importance to qualify and differentiate information sources, have disappeared. One now finds mail order catalogues, phone directories, accounting software, encyclopedias, mail, corporate information and so on and so forth, at the same place; this place being both the screen itself and the spot where the computer sits.

When confronted to more traditional media, readers were not left alone. They were guided both by the materiality of the item, should it be a book, a magazine or a newspaper, and by the spatial context where this item was to be found (the library, the bookshop or the newspapers stand). Thus, the web site plays multiple parts that were originally assumed by social actors, like librarians or booksellers. They also have to represent the "scene", that is to give users the necessary keys to envision the editing and publication context, when space has become text. This leads to a great semiotic complexity, as signs have to represent multiple types of information on an average 17' screen. A reader must then learn to decipher overlapping signs strata, and to relate them to the right context, so as to correctly interpret the situation he is confronted.

Therefore, information literacy at the Internet era would not only be the ability to find the appropriate information but above all to master the interpretation of the utterance "framework". That is to understand the situation in which the information is laid out, when every needed sign can only be found on the screen. Things are getting more and more challenging, as an always wider range of services are made available through the web, transforming navigators into mandatory gateways to any content, should it be on personal computers, local servers or the Internet.

That means hard work for providers; today's challenge being both to keep one's hand in and to remain visible among billions of potentially reachable documents. What we call "Editorial Utterance" is thus a first range concern: The screen has to be organized so as to give information and meta-information about the communication situation, as well as tools to interact with it. Signs must thus give multi-level indications, ending on a complex pluri-semiotic system. Our observations indeed show that sites only rarely give a comprehensive view of this intricacy. Furthermore, search engines make things notably more complex. On one hand, a site is built as a whole: Pages are logically linked to offer reading courses. For example, if we go back to newspapers sites, the front page is seen as the mandatory starting page to envision their entirety. On the other hand, pages are only seldom reached this way. Consultation mainly begins on a search engine site, which means that any page could become "front page". This parameter is only seldom taken into account by conceivers. People must thus develop strategies to validate information found on somehow unknown places. We have for example noted that students use the University Library OPAC as a validation tool: They start their book-search using Google® and when they find a seemingly interesting reference, but cannot clearly identify its source, they test it on the OPAC. If the book is referenced, it's certification enough for them, as they have a clear representation of the catalogue as being the very finality of the whole library validation cycle.

As we can see, to legitimate a text is a back and forth process. On one hand, providers might or might not give enough clues to assert their legitimacy. On the other hand, users might or might not be reassured by what they see on the screen. When things are not made clear enough for them, they find ways to complement their findings and cross their sources; exactly the way journalists do… Things then come full circle and seem to conform to the ideological project underlying the very project of designing a World Wide Web; that is to give anyone the opportunity to communicate directly, without the help of an intermediate.

The World Wide Web is indeed presented as the first place ever where everyone is given the opportunity to express whatever he wants to tell or demonstrate. The editorial or validation circuits that were presented earlier in this text are thus seen as constraints and limits that technology has fortunately permitted to overpass. Everyone might thus become a first range actor in the public space. If we follow this thesis, the only admitted mediation is a technical one. Difficulties and contradictions, that are bound to happen, are quite logically attributed to a lack of technical improvements.

For instance, it is clear that HTML 1 or 2 offered only few layout options. At this time, sites tended not to be so different one from the other. For the main part, pages appeared poorly organized, with a pre-eminence of "courier" or "times" fonts, white backgrounds and blue underlined hypertext links-as previously said about first generation newspapers sites. This in-differentiation often generated disorientation and reading difficulties. But, solutions to these problems were mostly envisioned from a technical point of view and "new versioning" was, and still is, the most frequently suggested answer to any handling problem, including those coming from a misinterpretation of the screen layout.

Web editors

Even more, technical evolution, as XML for example, is easily presented as a mean to automatically dominate forms and structure and to organize and to hierarchize information. Yet, material organization is not a by-product of support evolution. It clearly appears as a dramatic necessity. Even newcomers have to admit that "forms affect meanings" and that the structure of written material is more than helpful to build a general comprehension of the way the support is organized and has to be used.

The look and feel of today's sites have little to do with the appearance they had a few years ago. A study of this evolution shows that newspaper sites tend to mix forms that can best represent the paper media they are related to with internet specific ones. They will exhibit summaries and organize pages as sections, in order to imitate the paper structure. They seek marks that may help readers to quickly identify the kind of support they are consulting. For instance, a publication that is already well known outside the World Wide Web will insist on its reputation by obviously installing its logo or any other sign of identification. In the case of magazines, the paper copy is often quoted, with, for example, a picture of its cover [START_REF] Chartier | Forms and Meanings: Texts, Performances, and Audiences From Codex to Computer[END_REF] . These "tricks" can even be used to mislead the user. For instance, when comparing the layout of a site devoted to environment protection with the site of an international company, selling genetically modified seeds, we found the same graphical choices, including a green colour-based graphical charter, and the same type of discourse, offering, in both cases, all appearances of "neutral" points of view on the matter .

As far as newspapers are concerned, Google News® is an interesting case of mimesis. This service is presented as the most accomplished information tool. The main argument being that gathering news on a "computing" basis is the most "objective" way to be informed since selection does not depend on any editorial project, information hierarchy or human selection. At least this is the way the concept is being explained by Google itself:

"Google News gathers from more than 4,500

English-language news sources worldwide, and automatically arranges them to present the most relevant news first. Topics are updated every 15 minutes, so you're likely to see new stories each time you check the page. Pick the item that interests you and you'll go directly to the site which published that story".

Google News is indeed a highly unusual information service: News is solely compiled by computer algorithms. As a result, varied sources are gathered together, regardless of political viewpoints or ideologies, so as to enable readers to see how different currents of thought, report on the same story.

In the same time, in order to avoid intellectual property problems, Google stresses the fact that it does not deliver direct access to the content of the 4500 sources that are quoted, but only links to the site they belong to. This is a good way to transfer its editorial responsibility to others. Yet, the choice of the sources themselves is already a selection and a qualification of information. All the same, the "neutral" and "automatic" algorithms of Google gather articles in large categories, incredibly looking like headings and sections of classical newspapers.

As a matter of fact, "Google News" home page is designed to look like a newspaper site front page. Thus, even if "robots" are meant to avoid subjectivity in the "choice" of headings and titles, the very existence of a structure is a clear indication to readers. Main titles are reinforcing signs. News reported under them will anyway be considered as being more important or more relevant, should they have been chosen automatically or not.

The very fact that Google News tries and imitates the editorial format of a newspaper site confirms that, after ten years of trial and error, the online newspaper has found its own shape. Readers recognize this new generic model as easily as they identify their newspaper. For them, the site now belongs to the general category of "the press".

A way to abolish space and time?

The credibility of the project is thus based on a closeness/distance duality. The newspaper model is strongly sought-after, but online specificities are used to break away from it. The Wikipedia project [START_REF] Cotte | Le document comme Lego, ou La dialectique peut-elle casser des briques[END_REF] , founded a few years ago by Jimmy Wales, is quite different from Google News. However, it has similar characteristics, speaking of this "distinction/mimesis" dialectic between classic and new media.

Our investigations show that both space and time are frequently put forward to praise the qualities of online publication projects, in contrast with their paper counterparts. When Google News, insists on the importance of its reactivity [START_REF] Gerard | Paratexts : Thresholds of Interpretation[END_REF] as an attempt to almost abolish timeto add credit to its project, Wikipedia, uses exhaustivenessas an attempt to almost abolish spaceto differ its model from the classical, paper-based encyclopedia.

"The most obvious difference is that there are, in principle, no size limits in the Wikipedia universe. It is quite possible, for example, that when you finish typing in everything you want to say about poker, there might well be over 100 pages, and enough text for a full-length book by itself. This would certainly never be tolerated in a paper encyclopedia, which is why Encyclopaedia Britannica has such limited information on the topic (and on most other topics)" (Wiki is not paper, Wikipedia FAQ)"

The traditional paper-based encyclopedia is explicitly criticized; it's supposed lake of exhaustiveness and subjectivity being attributed to technical barriers and limitations. The choices made by an encyclopedia publisher are not considered as an editing policy. The intellectual decisions leading to stress specific aspects of a domain are merely seen as spatial constraints. Thus, once technology allows the publication of a boundary less text, a potentially "infinite" encyclopedia could be built. In its FAQ, Wikipedia gives the example of an article about the "Simpsons" TV series, and pleads for a general cover of the subject, comprising as many articles as the series has characters, or episodes and so on. This may appear as an opportunity to develop infinite knowledge, especially if we consider matters (TV series) that are not "noble" enough for a classical encyclopedia.

The concept which lies behind such a project is that any of us may be an expert in a specialized domain, should it be horror movies, gardening or Heidegger's philosophy. If errors are made, whether intentionally or not, the community will immediately react and correct the entry. The idea is that this cooperative process will lead to an almost perfect adjustment of the encyclopedia. Objectiveness, neutrality and "truth" of information are presented as the results of this collective, community-based auto-regulation. Articles are not signed and authorship is banned as anonymity is seen as the most accomplished form of democracy. The figure of the author is seen in his "authoritative" acceptation. To put contributors on an equal footing is synonymous with freedom of speech But things are not that simple. First, none of the classical encyclopaedias, starting from Diderot's have ever been "neutral" or objective even when they pretended to represent the whole of the knowledge of an era. Diderot and d'Alembert's encyclopaedia was an intellectual tool against the power of absolute monarchy in the 18th century. Ideological points of view, philosophically or politically oriented opinions, or, more simply, different cultural visions of the world around, weight upon any editorial project. Wikipedia will be no exception. Its conceivers have learnt that good will is not enough to avoid subjectivity. A recent New York Times article explains that some 82 entries are now "protected" by administrators, which means that no one else is allowed to modify their content, another 179 -including those about George W. Bush, Islam or Adolf Hitler -are "semi protected", that is only opened to people who have registered on the site for at least four days.

The fading figure of the editor comes back to light as selection becomes a major concern. The intervention of a human being is seen again as mandatory to authorize such or such contribution. Another attribution of the editor strongly reappears too, that is particularly interesting for us as information scientists: the "content organizer", a professional who treats information on a hierarchical basis, reshapes and structures information contents. Speaking of its huge, free content about "The Simpsons" and all its characters, Wikipedia FAQ admits that it may be useful to have :

"a table listing every episode, all neatly cross-linked and introduced by a shorter central page (our emphasis). Every episode name in the list could link to a separate page for each of those episodes, with links to reviews and trivia. Each of the 100+ poker games can have its own page with rules, history, and strategy."

This shows how important the question of structuring and organizing information is, whatever the media might be, paper or electronic. The technical possibilities of endlessly expanding the entries and contents of the encyclopaedia necessarily generate construction problems. As the archive expands, the need for mediation grows accordingly. An editorial know-how is more than ever required to help readers "navigate" through billions of information pages.

Conclusion

New user-friendly publication tools, such as groupware, blogs or wikis, make it easy for anyone to create a web site. Freedom of speech thus seems to be given to anyone and the web now is commonly seen as a "free", hierarchy less, "open" editing system. Online editing would then be the very democratic way of emitting and exchanging ideas, texts, opinions and information. Ambitious information projects, based on this vision of a potentially infinite writing space, freed from the technical and social6 constraints of the paper world, have thus been initiated during the past five years. However, the printer and publisher's know-how are now called back. The once criticized figure of the editor comes back in full light, as these projects expand and become more and more complex. On the Google site, the very form of the text on the screen seeks and imitates the structure of an online newspaper; as for Wikipedia, matters of selection, authority and information hierarchy reappear, when auto-regulation shows its limits.

Information literacy in the "Information age", greatly depends on the ability to decipher and interpret meta-information as new authority markers emerge, partly inherited from the "paper" world and strongly integrated both socially and culturally, partly created to take into account the specificity of online writing and publishing

French language does not have the distinction made in English between the publisher and the editor. Both roles are confounded under the single denomination of "editeur".

Though, from time to time a scholar tries and beats the validation system, whether to denounce its subjectivity or weakness (as Alan Sokal, whose hoax essay was published in "Social Text") or to gain a questionable renown (as Jacques Benveniste's essay about "the memory of water").

The hyperlinks associated with these images often directly open a PDF file, containing a picture of the front page as it appears on the paper support

Wikipedia is an attempt to make the most out of a traditionally high valued editorial shape: the encyclopedia, to promote a democratic ideal of sharing and building collective knocowledge, through the powerful means of collaborative tools. "Freewheeling", "volunteer administration", "online community" are keywords generally used to describe the project.

Google News site makes an extensive use of time related terms: the search bar is immediately followed by a red banner claiming « Search and browse 4,500 news sources updated continuously » and the freshness of each article is attested to the very minute « 15 minutes ago »...

This may be a typically French preoccupation, as the French public regularly manifests through opinion polls a huge distance and defiance towards its information supports. The annual inquiry directed by the French newspaper "La Croix", shows that only 48% of French population trusts printed press, and 44% television (La Croix, 23d February 2006).