J Harmand
email: jerome.harmand@inrae.fr

A Rapaport
email: alain.rapaport@inrae.fr

D Dochain
email: denis.dochain@uclouvain.be

Increasing removal rate can globally stabilize two-step biological systems

Keywords: Anaerobic digestion, nitrification, chemostat, multi-stability, stabilization

We revisit two-step mass-balance models of biological processes as met to describe numerous biological systems including the anaerobic digestion or the nitrification process in view of its global stabilization. We show that when a bi-stability occurs, it can be possible to globally stabilize the dynamics toward an unique positive equilibrium by increasing the removal rate. We give sufficient conditions on the growth functions of the model for this situation to appear. This illustrates that for biological multi-step reactional systems, increasing the residence time (e.g. decreasing the input flow rate) may not be the only way to stabilize the dynamics.

Introduction

In most of continuous cultures, it is well known that increasing the removal rate (or equivalently reducing the residence time inside the reactors) can destabilize the dynamics, in the sense that it enlarges the attraction basin of the wash-out equilibrium. This can be easily shown on the classical mathematical model of the chemostat, whatever the kinetics includes inhibition or not (see [START_REF] Harmand | The Chemostat: Mathematical Theory of Microorganism Cultures[END_REF]). For growth inhibited by the substrate, bi-stability systematically occurs for large values of the input concentration of substrate. This feature has practical impacts on positive equilibrium (when it exists) because it cannot be globally stable, and the dynamics can conduct the system to the wash-out of the biomass, when the state belongs to the attraction basin of the washout equilibrium. Ways to guarantee a global stability is either to fix a lower removal rate, which is penalizing for the performance of the process, or to control the removal rate with a feedback loop, which temporarily diminishes the removal rate when the state is far from the positive steady state [START_REF] Bastin | On-line Estimation and Adaptive Control of Bioreactors[END_REF][START_REF] Rapaport | Robust regulation of a class of partially observed nonlinear continuous bioreactors[END_REF][START_REF] Schaum | Saturated PI control of continuous bioreactors with Haldane kinetics[END_REF][START_REF] Satishkumar | Control of unstable bioreactor using fuzzy-tuned PI controller[END_REF]. In any case, the removal has to be reduced at a certain stage.

Here we consider a more complex reaction scheme that are two-step systems, as met for instance in many models representing the anaerobic digestion process [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF] or the nitrification process [START_REF] Volcke | Steady-state multiplicity of two-step biological conversion systems with general kinetics[END_REF]. For these systems we show that there exist situations presenting a bi-stability for which increasing (and not decreasing) the removal rate conducts the system to a globally asymptotically stable steady state, in opposition to classical stabilizing practices.

In the paper, we denote by R + the set of non-negative numbers and by R +, the set of positive numbers.

Let us consider the general mathematical model of a two-step mass-balance biological process, given by the following equations:

                   ẋ1 = µ 1 (s 1 )x 1 -αDx 1 ṡ1 = -µ 1 (s 1 )x 1 + D(s in 1 -s 1 ) ẋ2 = µ 2 (s 2 )x 2 -αDx 2 ṡ2 = -µ 2 (s 2 )x 2 + µ 1 (s 1 )x 1 + D(s in 2 -s 2 ) (1)
where the parameter D denotes the removal rate.

This model is presented under the original form proposed in [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]. The first reaction involves a microbial species of concentration x 1 which grows on a substrate of concentration s 1 with a monotonic specific rate µ 1 . The incoming flow fed the culture with substrate of concentration s in 1 . The second reaction involves a second microbial species of concentration x 2 which grows on another substrate of concentration s 2 , with a specific growth rate denotes µ 2 . This reaction is also fed by the first one which produces the second substrate. In addition, the incoming flow rate may contain (or not) substrate of concentration s in 2 . The parameter α ∈ (0, 1) reflects the fact that the effective removal rate of the biomass is impacted by a retention inside the tank, differently to the abiotic resource. Here, the yield coefficients of the transformations of substrate s i into biomass x i (i = 1, 2), and of the production of substrate for the second reaction by the first one, have been all kept equal to 1 (this is always possible without any loss of generality, by a right choice of the concentration units).

In many biological systems, such as the anaerobic digestion or the nitrification processes, it is often reported in the literature that the second reaction is inhibited by large values of s 2 , which amounts to consider the following hypotheses.

Assumption 1 The functions µ 1 , µ 2 belong to C 1 (R + , R + ) and fulfill the following properties.

(i) µ 1 is increasing on R + with µ 1 (0) = 0.

(i) There exists ŝ2 > such that µ 2 is increasing on [0, ŝ2 ) and decreasing on (ŝ 2 , +∞), with µ 2 (0) = 0 and µ 2 (+∞) = 0.

The model (1) has a cascade structure: the first reaction is independent of the second one and the (x 1 , s 1 ) sub-system follows the classical (mono-specific) chemostat model. However, the (x 2 , s 2 ) sub-system is more complex to study as it receives substrate from the first reaction and µ 2 is non-monotonic. This model and some of its variants has been already well studied in the literature [START_REF] Benyahia | Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes[END_REF][START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF][START_REF] Sbarciog | Determination of appropriate operating strategies for anaerobic digestion systems[END_REF], depending on the operating parameters (s in 1 , s in 2 , D). In particular, it has been shown that the dynamics may exhibit a multiple-stability, and the complete operating diagram has been established in [START_REF] Sari | The operating diagram for a two-step anaerobic digestion model[END_REF][START_REF] Volcke | Steady-state multiplicity of two-step biological conversion systems with general kinetics[END_REF]. The purpose of the present work is to complement those studies, investigating how to adapt the value of the removal rate D to ensure a global stability of the dynamics. For sake of completeness, we first recall in the next section the set of possible asymptotic behaviors of the model.

Stability analysis

Let us first denote, for convenience,

µ m 1 := sup{µ 1 (s 1 ) ; s 1 ∈ R + )} , µ m 2 := sup{µ 2 (s 2 ) ; s 2 ∈ R + )} = µ 2 (ŝ 2 )
(where µ m 1 could be equal to +∞).

We define the break-even concentration λ 1 associated to the first reaction as the function

λ 1 (D) := µ -1 1 (αD), αD < µ m 1 . (2)
Then, we define the following quantity

s in 2 (D) := s in 2 +      s in 1 -λ 1 (D) , αD < µ 1 (s in 1 ) 0 , αD ≥ µ 1 (s in 1 ) (3)
that is playing an important role in the analysis of the equilibria, as an "effective" input concentration for the second reaction.

We define also the break-even concentrations λ - 2 , λ + 2 associated to the second reaction as functions such that

[λ - 2 (D), λ + 2 (D)] := {s ∈ R + ; µ 2 (s) ≥ αD}, αD ≤ µ m 2 . (4)
One has the following result about equilibria of system (1) and their stability.

Proposition 1

The asymptotic behavior of the solutions of system (1) with initial condition in (R + * × R + ) 2 is given by one of the following cases.

(1) When αD > max(µ 1 (s in 1 ), µ m 2 ), any solution converges to the "double wash-out" steady-state E 0,0 := (0, s in 1 , 0, s in 1 ).

(

) When max(µ 1 (s in 1 ), µ 2 (s in 2 )) ≤ αD ≤ µ m 2 2
, the solution converges either to E 0,0 or to the equilibrium E 0, := (0,

s in 1 , (s in 2 -λ - 2 (D))/α, λ - 2 (D)), except for initial conditions on a set of null measure (3) When µ 1 (s in 1 ) < αD ≤ µ m 2 and s in 2 ≤ λ + 2 (D), any solution converges to E 0, . (4) When µ 1 (s in 1 ) > αD > µ 2 (s 2 ) for any s 2 < s in 2 (D), any solution converges to E ,0 := ((s in 1 -λ 1 (D))/α, λ 1 (D), 0, s in 2 (D)).
(5) When αD < µ 1 (s in 1 ), αD ≤ µ m 2 and s in 2 (D)) > λ + 2 (D), the solution converges either to E ,0 or to the positive equilibrium

E , := ((s in 1 - λ 1 (D))/α, λ 1 (D), (s in 2 (D) -λ - 2 (D))/α, λ - 2 (D)), except for initial condi- tions on a set of null measure. (6) When αD < µ 1 (s in 1 ), αD ≤ µ m 2 and λ - 2 (D) < s in 2 (D) ≤ λ + 2 (D), any solution converges to E , .
This result has already been proved in [START_REF] Benyahia | Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes[END_REF][START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF][START_REF] Sbarciog | Determination of appropriate operating strategies for anaerobic digestion systems[END_REF], and we recall quickly here the arguments of the proof based on the study of the single chemostat model, for which the classical results are recalled in the appendix. Let us stress that the statement of Proposition 1 does not distinguish the number of unstable equilibriums nor the hyperbolic characteristics of the equilibriums, differently to [START_REF] Benyahia | Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes[END_REF]. Our goal here is simply to characterize the possible asymptotic behaviors towards a stable equilibrium, up to a set of initial conditions of null measure.

PROOF. Let us first note that one has ẋ1 = 0 when x 1 = 0, and ẋ2 = 0 when x 2 = 0. By uniqueness of the solutions of the system of differential equations (1), we deduce that the solutions verify x 1 (t) > 0, x 2 (t) > 0 for any t > 0. At s 1 = 0, one has ṡ1 = Ds in 1 > 0 and ṡ2 > Ds in 2 > 0, which shows that the hyperplanes s 1 = 0, s 2 = 0 are repulsive. We deduce that the solutions verify s 1 (t) > 0, s 2 (t) > 0 for any t > 0.

Consider the variables z 1 = x 1 + s 1 and z 2 = s 1 + x 2 + s 2 . From equations (1), one obtains ż1 ≤ D(s in 1 -αz 1 ), ż2 ≤ D(s in 1 + s in 2 -αz 2 ) from which one deduces that z 1 and z 2 are bounded. This shows that the non-negative solutions of (1) are bounded.

The (x 1 , s 1 ) dynamics is independent of the variables x 2 , s 2 and follows the the classical chemostat model with monotonic growth function, whose steady state analysis is recalled in the Appendix (Proposition 4). Two cases are distinguished:

• case A: αD ≥ µ 1 (s in 1 )
: any solution of the (x 1 , s 1 ) sub-system converges to the wash-out state E 0 1 := (0, s in 1 ).

• case B: αD < µ 1 (s in 1 ): any solution of the (x 1 , s 1 ) sub-system with x 1 (0) > 0 converges to the positive state

E 1 := ((s in 1 -λ 1 (D))/α, λ 1 (D)).
Then, the cascade structure of the dynamics (1), along with the boundedness of its solutions and the asymptotic behavior of the (x 1 , s 1 ) sub-system allows to proceed with the stability analysis of the system on the reduced dynamics of the (x 2 , s 2 ) subsystem:

     ẋ2 = µ 2 (s 2 )x 2 -αDx 2 ṡ2 = -µ 2 (s 2 )x 2 + µ 1 (s eq 1 )x eq 1 + D(s in 2 -s 2 ) (5)
where (x eq 1 , s eq 1 ) is the steady state of the (x 1 , s 1 ) sub-system (i.e. E 0 1 or E 1 when it exists, according to Proposition 4).

In case A, the subsystem [START_REF] Mairet | Three-reaction model for the anaerobic digestion of microalgae[END_REF] is the classical chemostat model with nonmonotonic growth, whose steady state analysis is recalled in Proposition 5 (see Appendix). Three cases are then possible depending on the value of αD with respect to µ m 2 and µ 2 (s in 2 ), which are exactly the cases (1), ( 2), (3) given of the Proposition statement.

In case B, notice that one has µ 1 (s eq 1 )x eq 1 = D(s in 1 -λ 1 (D)) which allows to rewrite the reduced (x 2 , s 2 ) dynamics as

     ẋ2 = µ 2 (s 2 )x 2 -αDx 2 ṡ2 = -µ 2 (s 2 )x 2 + D(s in 2 (D) -s 2 ) ( 6
)
where s in 2 (D) is defined in (3). This is again the classical chemostat model but with non-monotonic growth and the effective input concentration s in 2 (D), for which the steady state analysis given in Proposition 5 of the Appendix applies. This gives straightforwardly the cases (4), ( 5), ( 6) of the statement. 2

In practice, only case ( 6) is desirable because it guarantees that in any situation the wash-out of both species is avoided. Usually, the removal rate D is the operating parameter that can be easily manipulated. In the next section, we study how to change the value of D to be in case [START_REF] Harmand | The Chemostat: Mathematical Theory of Microorganism Cultures[END_REF] when the original operating conditions are not in this case.

Wash-out avoidance

In this section, we consider situations for which the attraction basin of equilibria with wash-out of biomass 1 or 2 or both is non empty. According to Proposition 1, this happens in cases (1) to [START_REF] Mairet | Three-reaction model for the anaerobic digestion of microalgae[END_REF]. We study now how to play only with the value of the removal rate D to move to case [START_REF] Harmand | The Chemostat: Mathematical Theory of Microorganism Cultures[END_REF].

Consider the domains

D 5 := {(s in 1 , s in 2 , D) ∈ R 3 + ; αD < µ 1 (s in 1 ), αD ≤ µ m 2 , s in 2 (D) > λ + 2 (D)} D 6 := {(s in 1 , s in 2 , D) ∈ R 3 + ; αD < µ 1 (s in 1 ), αD ≤ µ m 2 , λ - 2 (D) < s in 2 (D) ≤ λ + 2 (D)}
which are the sets of operating parameters (s in 1 , s in 2 , D) that correspond to cases (5) and ( 6) of Proposition 1.

Consider the interval

I := (0, min(µ m 1 , µ m 2 )/α) and introduce the functions defined on I:

ν -(D) := λ 1 (D) + λ - 2 (D), ν + (D) := λ 1 (D) + λ + 2 (D), D ∈ I
(that are such that ν -< ν + on I), which allow to describe the domains D 5 , D 6 as follows, using the expression (3) of s in 2 (D) when αD < µ 1 (s in 1 ).

D 5 = {(s in 1 , s in 2 , D) ∈ R 2 + × [0, µ m 2 /α] ; αD < µ 1 (s in 1 ); ν + (D) < s in 1 + s in 2 } D 6 = {(s in 1 , s in 2 , D) ∈ R 2 + × [0, µ m 2 /α] ; αD < µ 1 (s in 1 ); ν -(D) < s in 1 + s in 2 ≤ ν + (D)}
Note from expressions (2) and (4) that the functions λ 1 and λ - 2 are increasing, while λ + 2 is decreasing. The function ν -is thus increasing. One has also Note that ν -< ν + on I implies that one has η -(s) > η + (s) for any s > s m .

Our main result is the following.

Proposition 2 Consider a triplet (s in 1 , s in 2 , D) with s in 1 > 0, s in 2 ≥ 0 and D > 0, that do not belong to D 6 .

(i) For D ∈ (0, D) sufficiently small, (s in 1 , s in 2 , D) belongs to D 6 .

(ii) If (s in 1 , s in 2 , D) belongs to D 5 with the condition

α η + (s in 1 + s in 2 ) < µ 1 (s in 1 ) (7)
fulfilled, then any D ∈ (η

+ (s in 1 + s in 2 ), min(µ 1 (s in 1 )/α, η -(s in 1 + s in 2
)) is such that D > D and (s in 1 , s in 2 , D) belongs to D 6 .

PROOF. Fix s in 1 > 0 and s in 2 ≥ 0.

Note first that ν -(0) = 0 and ν + (0+) = +∞ imply that one has ν -(D) < s in 1 + s in 2 ≤ ν + (D) for D > 0 small enough. Therefore, for any D > 0, there exists D ∈ (0, D) such that (s in 1 , s in 2 , d) ∈ D 6 for any d ≤ D. This shows (i).

Let us now study if it possible to have (

s in 1 , s in 2 , D) ∈ D 6 with D > D when D > 0 is such that (s in 1 , s in 2 , D) / ∈ D 6 .
According to Proposition 1, in cases ( 1), ( 2) or ( 3), one has αD ≥ µ 1 (s in 1 ) and D has then to be reduced to fulfill the condition αD < µ 1 (s in 1 ) required in case [START_REF] Harmand | The Chemostat: Mathematical Theory of Microorganism Cultures[END_REF].

In case (4), either one has αD > µ m 2 , and D has to be reduced to fulfill the condition αD ≤ µ m 2 of case [START_REF] Harmand | The Chemostat: Mathematical Theory of Microorganism Cultures[END_REF], or one has s in 2 (D) ≤ λ - 2 (D). This latter situation amounts to write s in 1 + s in 2 ≤ ν -(D) and as the function ν -is increasing, D has again to be reduced to obtain the condition s in 1 + s in 2 > ν -(D) of case [START_REF] Harmand | The Chemostat: Mathematical Theory of Microorganism Cultures[END_REF].

In case ( 5), one has

s in 1 + s in 2 > ν + (D) which implies D < η + (s in 1 + s in 2 ) by definition of η + . If α η + (s in 1 + s in 2 ) < µ 1 (s in 1 ), then any D in the interval (η + (s in 1 + s in 2 ), µ 1 (s in 1 )/α) verifies s in 1 + s in 2 ≤ ν + ( D). If moreover one has D < η -(s in 1 + s in 2 ) (recall that η + < η -), one guarantees the inequality s in 1 + s in 2 < η -(

D). This proves the point (ii). 2

The surprising fact, when compared to the usual chemostat model, is that increasing the removal rate D can bring stability in certain situations given in (ii). Indeed, increasing the removal rate amounts to reduce the residence time, which is usually a factor of instability. Here, the key point for such a phenomenon to occur relies on the possible non monotonicity of the function ν + , which implies that the function η + is non identically equal to min(µ m 1 , µ m 2 )/α. Let us show that the conditions of case (ii) of Proposition 2 can be really met.

Proposition 3 Assume one has µ m 2 ≥ µ m 1 . Let D m := max{D ∈ I ; ν + (D) = s m }. (i) Any triplet (s in 1 , s in 2 , D) such that s in 1 + s in 2 > s m and D > D m with ν + (D) < s in 1 + s in 2 belongs to D 5
and verifies the inequality (7), provided that s in 2 is sufficiently small.

(ii) Any triplet (s in 1 , 0, D) that belongs to D 5 satisfies the inequality [START_REF] Khedim | Effect of control parameters on biogas production during the anaerobic digestion of protein-rich substrates[END_REF].

PROOF.

Notice first that when µ m 2 ≥ µ m 1 , µ m 1 is necessarily finite and one has I = (0, µ m 1 /α).

Take any s > s m and denote D = η + (s). For any D > D m in I such that ν + (D) < s, one has λ 1 (D) < s -λ + 2 (D). As D < D and λ + 2 is decreasing, one has also

λ 1 (D) < s -λ + 2 ( D). (8)
Consider now any s in 2 < min(s, λ + 2 ( D)) and take s in

1 = s -s in 2 .
For this choice of s in 1 and s in 2 , inequality [START_REF] Rapaport | Robust regulation of a class of partially observed nonlinear continuous bioreactors[END_REF] gives

λ 1 (D) < s in 1 (along with ν + (D) < s in 1 + s in 2 ).
As the interval I is equal to (0, µ m 1 /α) and the function µ 1 is increasing, having λ 1 (D) < s in 1 is equivalent to have αD < µ 1 (s in 1 ). This shows that the triplet (s in 1 , s in 2 , D) belongs to D 5 (notice that one has necessarily αD < µ m 2 as µ m 1 ≤ µ m 2 ).

For D = D, one has ν + ( D) = s. In the same manner, one obtains λ 1 ( D) < s in 1 for the former choice of s in 1 and s in 2 , or equivalently α D < µ 1 (s in 1 ). This shows that the inequality αη + (s in 1 + s in 2 ) < µ 1 (s in 1 ) is necessarily fulfilled.

For the particular case s in 2 = 0, note that ν + (D) < s in 1 implies λ 1 (D) < s in 1 which, in turns, implies αD < µ 1 (s in 1 ). From the definition of ν + , this shows that the inequality αη + (s in 1 ) < µ 1 (s in 1 ) is verified. 2

Numerical illustrations

Typical instances of functions that fulfill Assumption 1 are given by the Monod expression for the first reaction

µ 1 (s 1 ) = µ m 1 s 1 K 1 + s 1 . (9)
and the Haldane one for the second

µ 2 (s 2 ) = μ2 s 2 K 2 + s 2 + s 2 2 K i (10)
for which one has ŝ2 = K 2 K i .

Then, the break even concentrations defined in Section 2 have the expressions

λ 1 (D) = αDK 1 µ m 1 -αD , αD < µ m 1
for the Monod function [START_REF] Schaum | Saturated PI control of continuous bioreactors with Haldane kinetics[END_REF], and for the Haldane function ( 10)

λ ± 2 (D) = μ2 -αD ± (μ 2 -αD) 2 -4(αD) 2 K 2 K i 2 αD K i , αD ≤ µ m 2 Figure 1
gives the values of the parameters chosen for the numerical computation, along with the graphs of the corresponding functions µ 1 , µ 2 . The numerical values of the parameters were chosen such that the graphical representations be very clear and the phenomenon very well put in evidence. We will come back on this point in the discussion at the end of this section but the phenomenon appears for several parameter sets reported in the literature even if it may be less obvious and more difficult to represent graphically (cf. for instance [START_REF] Sbarciog | Determination of appropriate operating strategies for anaerobic digestion systems[END_REF]) as well as for more complicated models, (cf. [START_REF] Khedim | Effect of control parameters on biogas production during the anaerobic digestion of protein-rich substrates[END_REF][START_REF] Volcke | Steady-state multiplicity of two-step biological conversion systems with general kinetics[END_REF] for example). For these values, one has ŝ2 2.646 and µ m 2 0.626. So, we are in the case µ m 2 > µ m 1 of Proposition 3.

µ m 1 K 1 μ2 K 2 K i α 0.6 0.4 1.1 1 7 0.7 (
Figure 2 gives the graphs of the associated break-even concentrations λ 1 , λ ± 2 and the functions ν ± . Remind that λ 1 and λ - 2 are increasing while λ + 2 is decreasing. The function ν + is always above ν -(which is increasing) and ν + tends to +∞ on the boundary of the interval

I = (0, µ m 1 /α) when µ m 1 ≤ µ m 2 .
As it is not always easy to grasp domains in R 3 , we fix values of s in 2 and depict model of the anaerobic digestion of microalgae, this property was highlighted. In particular, when comparing the two models of the anaerobic digestion initially proposed in [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF] and [START_REF] Mairet | Three-reaction model for the anaerobic digestion of microalgae[END_REF], it was first noticed that the inhibition coefficient identified in the first model, ( [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]) was too large to play a role in the system dynamics (recall that having a large inhibition coefficient in the Haldane function is equivalent to considering a Monod kinetics). Then, in modifying this coefficient (increasing the inhibition of the second step kinetics in decreasing the value of the inhibition coefficient in the Haldane kinetics), it was possible to show that the region in which the biogas is maximum exhibits a bistability. Comparing these results with those obtained when studying the second model (from [START_REF] Mairet | Three-reaction model for the anaerobic digestion of microalgae[END_REF]) yielded to plot the Figure 11 of [START_REF] Khedim | Effect of control parameters on biogas production during the anaerobic digestion of protein-rich substrates[END_REF] where this property is visible (but not further discussed). Finally, this property has been described in [START_REF] Hanaki | Mathematical study of a two-stage anaerobic model when the hydrolysis is the limiting step[END_REF] for two-step biological systems with density-dependent kinetics in the first step, but again without giving any condition to test it nor proposing a control strategy to take advantage of it. Here we give sufficient conditions that can easily be tested for the phenomenon to appear.

Notice that the proposed strategy consists in decreasing the residence time and not increasing it as usually done to stabilize bio-processes. This has some practical advantages: it allows to treat a larger quantity of matter and for industrial plants which impose a minimal flow rate to be processed, it does not require any upstream storage. Of course, increasing the residence time increases the output concentration at steady-state, but once about the steadystate one may reduce the flow rate to its nominal value.

Conclusion

This study reveals the role played by the sum of the break-even concentrations, the function D → λ 1 (D) + λ + 2 (D), in the counter-intuitive phenomenon of increasing the removal rate to stabilize a two-steps bioprocess model. More precisely, we show that when this function is non-monotonic on its domain, this phenomenon occurs provided that the input concentration s in 2 of substrate of the second reaction is null of not too large. This result provides a new way to stabilize such processes in certain situations, without increasing the residence time, as it is often done which may be penalizing in an industrial context.

ν

  -(0) = 0 and ν -(D) → +∞ when D → min(µ m 1 , µ m 2 )/α. One can then define the inverse function η -(s) := (ν -) -1 (s), s > 0 The function ν + is not necessarily monotonic but one has ν + (D) → +∞ when D → 0 + . Denote s m := min D∈ Ī ν + (D) and one can define the function η + (s) := max{D ∈ I ; ν + (D) ≤ s}, s > s m .

  a) Parameters values. Graphs of µ 1 and µ 2 .

Fig. 1 .

 1 Fig. 1.

Fig. 2 .

 2 Fig. 2. Graphs of the functions λ ± i (left) and ν ± = λ 1 + λ ± 2 (right).

Fig. 3 .

 3 Fig. 3. Cross-sections of the domains D 5 and D 6 for s in 2 = 0. From any operating point (s in 1 , D) ∈ C 5 it is possible to reach C 6 by simply increasing D.

Fig. 4 .

 4 Fig. 4. Cross-sections of the domains D 5 and D 6 for s in 2 = 2 and s in 2 = 5. From an operating point (s in 1 , D) ∈ C 5 with s in 1 not too far from s m -s in 2 , it is possible to reach C 6 by simply increasing D.

Fig. 5 .

 5 Fig. 5. Cross-sections of the domains 5 and D 6 for s in 2 = 7 and s in 2 = 9. From any operating point (s in 1 , D) ∈ C 5 it it not possible to reach C 6 by simply increasing D.

Acknowledgments

The work has been achieved in the framework of the PHC program 'TOUR-NESOL' 2018-19 between France and Belgium-Wallonia. The authors are very grateful to Professor Tewfik Sari for fruitful discussions.

the cross-sections of the domains D 5 , D 6 in the (s in 1 , D) plane. For a fixed value of s in 2 , these cross-sections are

It can be simply interpreted in terms of intersections of epigraphs or hypographs of the functions λ 1 , ν + -s in 2 and ν --s in 2 in the (D, s in 1 ) symmetric plane:

Accordingly to Proposition 2, the complementary of

s} is covered by cases (1), ( 2) and (3).

In Figure 3, one can see that for s in 2 = 0, C 5 and C 6 cover almost but not all the domain {αD < µ 1 (s in 1 )} (which is below the red curve), and that it is always possible to reach the domain C 6 (in green) from C 5 (in pink) by increasing D (for fixed s in 1 ), in accordance with Proposition 3. The red curve, which is the graph of the function µ 1 /α, is above the domain C 5 .

For s in 2 = 2 or s in 2 = 5, C 5 and C 6 cover all the domain {D < µ 1 (s in 1 )/α} and the graph of the red curve touches the boundary C 5 (see Figure 4). However, it still possible to reach C 6 from C 5 by increasing D only if s in 1 is close enough to the value s m -s in 2 (remind that s m is the minimum of the function ν + ).

Increasing the values of s in 2 slides further the cross-section C 5 to the left on the (s in 1 , D) plane and it is no longer possible to reach C 6 from C 5 by simply increasing D (see Figure 5), exactly when the extreme point (s m -s in 2 , D m ) of C 5 does no longer lie in the domain {αD < µ 1 (s in 1 )} (i.e. when it is no longer located below the red curve). Then, the only way to stabilize the system playing only with the removal rate D is to decrease its value such that (s in 1 , D) is below the green curve (which is the graph of the function ν + -s in 2 ).

Notice that this interesting property of two-step processes was already visible in certain figures of some papers of the literature, notably in relation to the analysis of the anaerobic digestion process (cf. for instance the Figure 4 of the paper by Sbarciog et al., 2010, [13]). However, the property was not underlined in the paper as a way to stabilize the system. It should be noticed that this property obviously appears in more complex models. In [START_REF] Khedim | Effect of control parameters on biogas production during the anaerobic digestion of protein-rich substrates[END_REF], who studied a

Appendix

We recall here the classical results about the chemostat model

when µ is a monotonic or non-monotonic function.

Proposition 4 Assume that the function µ is increasing on (0, s in ) and define the break-even concentration λ(D) such that µ(λ(D)) = αD, αD ∈ (0, µ(s in ))

• When αD ≥ µ(s in ), the system (11) has an unique equilibrium E 0 := (0, s in ) ("wash-out"), which is moreover globally asymptotically stable on R 2 + . • When αD < µ(s in ), the system (11) admits an unique positive equilibrium E + := ((s in -λ(D))/α, λ(D)) (in addition to the equilibrium E 0 ), which is moreover globally asymptotically stable on the domain R + × R + .

Proposition 5 Assume that there exists ŝ ∈ (0, s in ) such that the function µ is increasing on (0, ŝ) and decreasing on (ŝ, s in ). Define the break-even concentrations λ -(D), λ + (D) as follows

• If αD > µ(ŝ), the system (11) has the unique equilibrium E 0 := (0, s in ), which is globally asymptotically stable on R 2 + . • If αD < µ(s in ), the system (11) admits an unique positive equilibrium E -:= ((s in -λ -(D))/α, λ -(D)) which is globally asymptotically stable on R + ×R + . • If αD ∈ [µ(s in ), µ(s )], the system (11) presents a bi-stability between E - and E 0 . From any initial condition in R + × R + excepted on a set of null measure, the solution converges asymptotically to E -or E 0 .


Developing yeast-based Biosensors

The mystery of Ciguatera

History :

 First description : Chen Tsang Chi, chinese doctor of the T'ang dynasty (~650).

 From the XVth century, Colomb, Magellan, Cortes, Vasco de Gama describe the disease in the Caribbeans ; Fernandes de Quiros, Cook, Bligh,… in Pacific.

 First precise description of a ciguatoxin intoxication : 7th September 1774 Nouvelles-Hébrides :

 The term ciguatera : attributed to cuban Felipe Poey, 1866 for a neurodigestive intoxication due to a marine gasteropode (Livona pica) whose cuban name is « cigua ».

October 1, 2020 5

Cet après-midi, un des naturels, ayant harponné un poisson, mon secrétaire l' acheta et me l' envoya après mon retour. Il était d' une nouvelle espèce, un peu comme un poisson-soleil, avec une grosse tête longue et hideuse. Ne nous doutant pas qu' il pouvait nous empoisonner, nous donnâmes l' ordre de l' apprêter pour le souper. Mais par bonheur il fallut si longtemps pour le dessiner et le décrire qu' il n' était plus temps de le faire cuire, de sorte qu' on n' apprêta que le foie et les rognons auxquels monsieur Forster et moi goûtâmes tout juste. Vers trois heures du matin, nous nous trouvâmes atteints d' une extraordinaire faiblesse et d' un engourdissement de tous les membres. J' avais presque perdu le sentiment du toucher et je ne pouvais distinguer, entre ceux que j' avais la force de soulever, les corps lourds des légers. Un quart d' eau et une plume avaient le même poids pour ma main. Nous prîmes tous les deux de l' émétique et après cela nous fîmes une suée qui nous apporta beaucoup de soulagement. Le matin, un des cochons qui avait mangé les entrailles fut trouvé mort.

THE CAUSAL AGENT : CIGUATOXINES (CTXS) :

Gambierdiscus toxicus, (B Anderson, MBL.)

Ciguatoxines Molecular Structures

Pacific Ciguatoxines Caribbean Ciguatoxines (Nicholson and Lewis, Marin Drugs, 2006).

October 1, 2020 10

Ciguatera Food Poisoning

 Considered as the number one Food poisoning of marine origine in the world (~200 000 cases /year).

 Body sensitization effect : increase of symptoms severity upon next exposure, cross effects (ex : alcohol).

SYMPTOMES

The Targets : Voltage Gated Sodium Channels (Nicholson and Lewis, Marin Drugs, 2006).

 Current assays : Mouse bioassay, RBA, CBA, HPLC.

URGENT NEED FOR A SIMPLE DETECTION TOOL !!!

 Based on these data, can we design a biosensor for Ciguatoxins ?

 Yeast based bio sensors : interesting for several reasons

Live Cell Assays

 Wide range of application domains: pharmacy industry cosmetics, chemical hazard evaluation (REACH), alternatives to animal testing, diagnostic, management of environnemental issues and bioterrorism related threats.



Relevant regarding curent regulations and industrial needs: Market estimated over 4 billions dollars !

State of the Art at project start :

No direct cytotoxic effect of Ciguatoxines on yeast cells.

-Express a mammalian VGSC in S. cerevisiae, or its fixation domain fused to an endogenous yeast channel Need for modelization -The closest homolog of mammalian VGSCs in yeast is Cch1, the calcium channel, which activates the Calcineurine pathway.

Is this pathway activated by CTXs ?

Biosensor strains design

October 1, 2020 18 -71 permanents + 30 à 60 étudiants.

Ciguatoxines

Cna

Cmd

Expression protéine rapporteur

Crz

Signal colorimétrique ou fluorescent

Voie de signalisation cellulaire

Levure modèle Saccharomyces cerevisiae

Cch1 Ca 2+

Response to Algal extracts

-Quatre tutelles : -Université de Polynésie française -IRD -Ifremer -Institut Louis Malardé .

-Structurée en trois thèmes : Canaux VGSC comme cibles therapeutiques

  Context: personal scientific background Chargée de Recherche INRA (MICA) 2 themes : -Cell wall Integrity and stress response in yeasts and fungi, Role of KNR4 protein (since 2001) -Biosensors Development (2016-) Competences : -Fungal cell walls, -Cellular signalisation pathways -Microbial adhesion In LISBP (Laboratoire d'Ingénierie des Système Biologiques et des Procédés). Team : Physiology and Functional Genomic of Eukaryotes.

-

  Among the most powerful marine biotoxines (1 microgramme can kill a man) No simple detection test available !!! -Produced by the dinoflagellate micro-algae Gambierdiscus toxicus, upon corral reefs damage -Polycyclic polyethers, liposoluble, MW~1100 Da -3 famillies : Pacific = P-CTXs, Caribbean = C-CTXs, Indian Ocean = I-CTXs.

  Thanks for your attention !Interested in Microbial Adhesion ? Come to BioPhysAdh meeting, 10-11 septembre, Toulouse. website : https://biophysadh.sciencesconf.org  2017: 2 Communications,  2018 : 1 Publication (Env. Res.) i.t. injections (lamotrigine) TTX R , NaV1.7

(Thewes, 2014)

October 1, 2020

The Targets : Voltage Gated Sodium Channels Scientific Questions :

 Cell wall permeability/reactivity towards toxins?


Hélène Martin-Yken

OcéaSafe Project : Developing yeast-based biosensors for Marine Toxins

October 1, 2020 4

The Mystery of Ciguatera

History :

 First description : Chen Tsang Chi, chinese doctor of the T'ang dynasty (~650).

 From the XVth century, Colomb, Magellan, Cortes, Vasco de Gama describe the disease in the Caribbeans ; Fernandes de Quiros, Cook, Bligh,… in Pacific.

 First precise description of a ciguatoxin intoxication : James Cook, Sept. 7th 1774 Nouvelles-Hébrides :

 The term ciguatera : attributed to cuban Felipe Poey, 1866 for a neurodigestive intoxication due to a marine gasteropode (Livona pica) whose cuban name is « cigua ».

October 1, 2020 5

Cet après-midi, un des naturels, ayant harponné un poisson, mon secrétaire l' acheta et me l' envoya après mon retour. Il était d' une nouvelle espèce, un peu comme un poisson-soleil, avec une grosse tête longue et hideuse. Ne nous doutant pas qu' il pouvait nous empoisonner, nous donnâmes l' ordre de l' apprêter pour le souper. Mais par bonheur il fallut si longtemps pour le dessiner et le décrire qu' il n' était plus temps de le faire cuire, de sorte qu' on n' apprêta que le foie et les rognons auxquels monsieur Forster et moi goûtâmes tout juste. Vers trois heures du matin, nous nous trouvâmes atteints d' une extraordinaire faiblesse et d' un engourdissement de tous les membres. J' avais presque perdu le sentiment du toucher et je ne pouvais distinguer, entre ceux que j' avais la force de soulever, les corps lourds des légers. Un quart d' eau et une plume avaient le même poids pour ma main. Nous prîmes tous les deux de l' émétique et après cela nous fîmes une suée qui nous apporta beaucoup de soulagement. Le matin, un des cochons qui avait mangé les entrailles fut trouvé mort.

No simple detection test available !!! THE CAUSAL AGENT : CIGUATOXINES (CTXS)

Gambierdiscus toxicus, (B Anderson, MBL.)

Ciguatoxines Molecular Structures

Pacific Ciguatoxines Caribbean Ciguatoxines (Nicholson and Lewis, Marin Drugs, 2006).

October 1, 2020 10

Ciguatera Food Poisoning

 Considered as the number one Food poisoning of marine origine in the world (~200 000 cases /year).

 Body sensitization effect : increase of symptoms severity upon next exposure, cross effects (ex : alcohol).

SYMPTOMES

CTXs Targets : Voltage Gated Sodium Channels -VGSC play key roles in electric signal propagation in excitable tissues (nerves, heart, muscles).

-Mechanism : activation of VGScs provokes a transiant depolarization, responsible of the ascendant phase of the action potential.

-CTXs stay fixed and prevent repolarization of VGSCs. -71 permanents + 30 à 60 étudiants.

La famille des canaux VGSCs de mammifères

-Quatre tutelles : -Université de Polynésie française -IRD -Ifremer -Institut Louis Malardé .

-Structurée en trois thèmes : October 1, 2020 31

Developing yeast-based Biosensors

  Integrity and stress response in yeasts and fungi, Role of KNR4 protein (since 2001) -Biosensors Development (2016-…) Competences : -Fungal cell walls, -Cellular signalisation pathways -Microbial adhesion In LISBP (Laboratoire d'Ingénierie des Système Biologiques et des Procédés), Team : Physiology and Functional Genomic of Eukaryotes.

-

  Among the most powerful marine biotoxines (1 microgramme can kill a man) -Produced by the dinoflagellate micro-algae Gambierdiscus toxicus, upon corral reefs damage -Polycyclic polyethers, liposoluble, MW~1100 Da -3 famillies : Pacific = P-CTXs, Caribbean = C-CTXs, Indian Ocean = I-CTXs.

  i.t. injections (lamotrigine) TTX R , NaV1.7 Canaux VGSC comme cibles therapeutiques CTXs Targets : Voltage Gated Sodium Channels (Nicholson and Lewis, Marin Drugs, 2006). CTXs Detection : problematic -Current assays : heavy, difficult, long, expensive, and/or unethical (Mouse bioassay, RBA, CBA, HPLC…) URGENT NEED FOR A SIMPLE DETECTION TOOL !!! -Can we design a biosensor for Ciguatoxins ? -Yeast based bio sensors : interesting for several reasons State of the Art at project start : No direct cytotoxic effect of Ciguatoxins on yeast cells. -Express a mammalian VGSC in S. cerevisiae, or its fixation domain fused to an endogenous yeast channel Need for modelization -The closest homolog of mammalian VGSCs in yeast is Cch1, the calcium channel, which activates the Calcineurine pathway. Is this pathway activated by CTXs ? Adhesion ? Come to BioPhysAdh meeting, 10-11 septembre, Toulouse. website : https://biophysadh.sciencesconf.org



  2018 : 1 Publication (Env. Res.)  5 Projects submitted  2016: 1 funded (PEPs CNRS)  2017: 3 non funded, 1 submitted (decembre ANSES).

(Thewes, 2014)

October 1, 2020

CTXs Targets : Voltage Gated Sodium Channels

Scientific Questions :

 Cell wall permeability/reactivity towards toxins?

Investigation tools : cell wall mutants, tagged toxins.

 Membrane target (receptor)? Binding?

Investigation tools : Cloning, SMFS with Atomic Force Microscopy.

 Cellular signalisation pathways?

Investigation tools : reporters, Fluorescent proteins, western blotting, RT-PCR.

Perspectives :

Understanding yeast cellular response to CTXs -Cell wall transfert -Competition CTXs -Rosmarinic acid and tetrodotoxin.

-Inhibitory Effect of HIT0 extracts ???? -Activators / repressors / blokers of VGSC and Cch1 -Hyper-active point mutants of Cch1 and Crz1, Chimera Cch1-VGSC

Response Optimisation

Developp a generic approach, apply to other toxins and drugs binding VGSCs.

Future applications : build a bio-sensor : huge interest for Pharmacology and Environmental domains


Philippe Charlier

Isabelle Huynh-Charlier

Olivia Munoz

Michel Billard

Luc Brun

Geoffroy Lorin

La Grandmaison

G Lorin De La Grandmaison

The microscopic (optical and SEM) examination of dental calculus deposits (DCD). Potential interest in forensic anthropology of a bio-archaeological method

Keywords: Dental calculus, Forensic anthropology, Identification Circumstances of death, Pathological status, Optical microscopy, SEM, Bio-archaeology

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The first method of dental calculus microscopic examination was developed in 1987 by Dobney [START_REF] Dobney | A method for evaluating the amount of dental calculus on teeth from archaeological sites[END_REF]. After being charted and measured on the whole of the teeth, the layers of dental calculus deposits (DCD) were sampled and examined according to various processes after partial decalcification: binocular magnifying glass, optical and electronic microscope (SEM or TEM). It was then possible to identify food remains trapped in the dental plaque and thus preserved: fragments of cereals, vegetable fibers, phytolithes, pollens, seeds, animal hairs, parasites, insects included accidentally, etc. All these elements may bring extra information to the botanical, zoological and parasitological data of the individual or a whole population when all results are compared [START_REF] Juan-Tresserras | Identification of phytoliths from prehistoric dental remains from the Iberian Peninsula and the Balearic Islands Madrid: Centro de Ciencas Medioambientales[END_REF]. The study of dental calculus takes part directly in the appreciation of the framework, alimentation or environment of a subject, but also in the evaluation of its health condition [START_REF] Charlier | Les dents d'Agnès Sorel[END_REF].

The electronic microscope allows a much more fine and precise observation thanks to a very large enlargement (bacteria, fungal filaments and viruses may thus be visualized); it may however give only descriptive information not allowing a precise diagnosis for the species present. Nevertheless the methods of molecular biology could be exploited in this case to typify, thanks to the characterization and comparison of their DNA or RNA.

With such observations made on anthropological samples, we would like to show that the microscopic examination of DCD may be particularly helpful for providing decisive data for the identification of a body and the attribution of an identity in a forensic purpose.

Materials and methods

We submitted several archaeological samples of DCD to a classical optical microscope (OM) and scanning electronic microscope (SEM) analysis in order to try to identify any tissue or, at least, cellular fragments still present in this substance that could be helpful for the determination of the individual habits (with a purpose of identification) and/or pathologies (with a purpose of determination of a manner of death). It consisted in a direct application of microscopic examination coupled with an elemental surface analysis.

The material included nine archaeological samples, four analyzed with an OM and five with a SEM.

Almost all of the skeletons from the Etruscan-Celtic necropolis of Monterenzio Vecchia (=MV; made up of 40 subjects, close to Bologna, Italy; archaeological researches under the authority of Prof. Daniele Vitali and Thierry Lejars) showed variable quantity of dental calculus deposits (Fig. 1a), but, due to conservative reasons, only five of them were sampled (tombs MV 7, MV 15, MV 20, MV 21 and MV 31). These 3-5-mm-long fragments of DCD were sampled directly from the teeth (lingual position of mandibular premolar or molar) during the bioarchaeological survey performed by one of us (P.C.). Evaluation of age at death for these individuals was carried out according to tooth attrition, closing of skull sutures and, when present, importance of osteo-arthritis lesions [START_REF] Charlier | Ostéo-archéologie de deux necropolis étrusco-celtiques: Monte Bibele et Monterenzio Vecchia (Bologne, Italie). Reconstitution d'une pathocénose à l'échelle de la vallée de l'Idice[END_REF].

They were then cleaned by a fine scraping with non-contaminating plastic instruments and a delicate vaporization of pressured air on all the faces during 10 min. No gold covering was realized in order to be able to perform elemental analysis on the surface. Environmental SEM examination of each sample was directly performed on a Philips XL30 CP with X-ray microanalysis (energy-dispersive X-ray spectroscopy). The resolution of the microscope was 3.5 nm at 30 kV using the secondary electron detector. All images were stored using a standard resolution digital frame store.

A study with optical microscope was also carried out on other samples of DCD taken from:

 A 20 to 25-years-old female skeleton (grave The methodology employed for the sampling and preparation of DCD was in all points comparable with that of putrefaction fluid deposits (PFD), previously published by our research team [START_REF] Charlier | The microscopic (optical and SEM) examination of putrefaction fluid deposits (PFD) potential interest in forensic anthropology[END_REF]:

 The first step consisted in a very fine fragmentation of the sample. This was justified by the fact that DCD has no architecture of its own and that this fragmentation does not consist in a loss of information (since we worked in a microscopic scale). In addition, this reduction allowed for a better penetration of the fixative substances.  In order to rehydrate and fix the samples, they were immersed in 20% diluted acetic acid for 48 h, in order to slowly decalcify the fragments of DCD.  Our solution was obtained using the following two phases: a liquid phase made of the DCD in suspension and a solid-phase slope. These two phases have been studied separately.  The cytological analysis of the DCD in suspension began with the sampling of 200 lL from the supernatant. This liquid was then centrifuged (800 turns per minute for 10 min) in order to obtain two spots per slide. Four slides were produced by sample: two slides were colored by the technique of Papanicolaou after a fixing of the spots with a lacquer; the two other blades were colored by the technique of the May-Grünwald-Giemsa (MGG) after fixing of the spots to the air.

 The study of the remaining solid phase (base of centrifugation) was carried out after a new centrifugation (3500 turns per minute for 10 min). The supernatant was kept in reserve (for a later possible cytological study) while the base was recuperated then fixed 24 h in the AFA (acetic acid, formaldehyde and alcohol). Soon after fixation and decalcification, the sample was put in cassette on a foam, followed by the traditional circuit of inclusion (dehydratation in xylene and increasing alcohol baths, then inclusion in liquid paraffin, cooling, section with the microtome (from 6 to 10 lm), deposit on an albumenized slide, air-drying free, dewaxing and rehydratation by immersion in xylene then in increasing alcohol baths to distilled water). Four colorations were carried out for each sample: periodic acid Schiff (PAS), Gram, toluidine blue and hematoxylineosin-saffron (HES).

Results

KHB-1 (grave 1, US 169)

A total of three fragments of 1-mm length from the lingual face of a mandibular molar were studied one by one with OM. In all cases, numerous vegetal structures have been observed, associated with degraded Malpighian epithelial cells (Fig. 1b andc). Some fungal filaments and rare crystals were also seen.

RH-5 (grave 405, individual B)

Were sampled two fragments of 1-and 3-mm length from the lingual face of a mandibular molar, secondary studied one by one with OM. In both samples, many crystals were observed (Fig. 1d) accompanied by particularly numerous vegetable fibers (Fig. 1e).

Monterenzio Vecchia 7 (MV 7)

Two fragments of 2-and 3-mm length were studied successively with the SEM. The elemental analysis of the matrix showed in both samples a composition rich in phosphorus, calcium and oxygen (calcium phosphate). Within this matrix, crystals made up of silicon and oxygen was observed in great number, with a hexagonal characteristic form (quartz crystals: Fig. 1f). Lastly, some fungal filaments and spores were highlighted (Fig. 1g).

Monterenzio Vecchia 15 (MV 15)

An important fragment of 4-mm length was examined with the SEM. The elemental analysis of the matrix, which appeared as a fibrillar accumulation (Fig. 2a) sometimes grinding the surface of dental enamel (Fig. 2b), highlighted a phospho-calcium nature (Fig. 3) with some crystals of quartz and alumino-silicate. More in-depth, towards dental enamel, many filamentous elements of vegetable origin broad from 5 to 7 lm, enchased in the anfractuosities dug in the matrix have been observed (Fig. 2c-e). Elemental analyses of these formations (vegetable hairs rather than fungal filaments?) showed a complete phospho-calcium mineralization.

Monterenzio Vecchia 20 (MV 20)

A total of three samples of 2-mm length were observed with the SEM. The elemental analysis of the matrix showed in all samples a composition rich in phosphorus, calcium and oxygen comparable that of the individuals MV 7 and MV 15 (calcium phosphate). Within this matrix, quartz crystals were observed whose composition was confirmed by the elemental examination.

In some places, irregular formations from 50 to 70 lm length (Fig. 2f) have been seen, whose elemental analysis showed a very great high content in manganese.

Monterenzio Vecchia 21 (MV 21)

The study with the SEM of two samples of 3-mm length showed in both cases an aluminosilicate matrix comparable with that of the individual MV 31. Within this matrix, some crystals were observed, of comparable composition (Fig. 4a). Lastly, many fungal filaments were present on the sectional surface, in close contact with enamel or in the thickness of the dental layers of sediment, carrying out true tunnels within the calcified matrix (Fig. 4b-e).

Monterenzio Vecchia 31 (MV 31)

The study with the SEM of four samples of 1-mm length showed in all cases the existence of several foreign bodies wedged in the matrix: vegetable fibers, red blood cells, grains of starch. On the surface, fungal filaments were observed. An elemental analysis of the matrix showed that prevalence of silicon clearly predominated with relative high levels of calcium and magnesium (Fig. 5). Taking into account of the strong oxygen and aluminum rates, matrix seemed to be composed of alumino-silicate. Carbonaceous, magnesium and sulfur impurities were finally found, but also some quartz crystals, whose nature was confirmed by the elemental study (Fig. 4f).

SEM control sample

A 1-mm length fragment of spongy bone (diploe from the skull of the individual MV 21) was examined with the SEM in order to check elemental proportioning. The Haversian structure of the bone was confirmed with the microscopic examination associated with the elemental analysis, as well as the hydroxy-apatite composition of the osseous spans (high content in calcium, phosphorus and oxygen: Ca10(PO4)6(OH)2 in chemical formula).

Agnès Sorel

Many structures were highlighted, in particular food remains such as vegetal fiber with full refringence conservation under polarized light (Fig. 6a). Some Malpighian cells were also observed, coming from the mouth mucosa (Fig. 6b) and one bacteria of spirochete type particularly visible after a coloration of Gram (Fig. 6c) whose precise characterization was impossible to give (very likely Treponema denticola, a cultivable anaerobia saprophyte bacteria of dental calculus).

Pierre Hazard

Vegetal fibers (comparable to those of KHB-1) were seen (Fig. 6d), accompanied by crystals and Malpighian epithelial cells (Fig. 6e).

Discussion

Background

The whole of the oral cavity is covered by a translucent film composed of salivary glycoproteins, being able to reach several millimeters thickness. During the intra-uterine life, the oral cavity is sterile, but at the time of the passage of the pelvi-genital area, a bacterial colonization occurs, amplified by the first times of milk alimentation (contamination by the skin surface of the breast). This layer also sticks to the surface of the teeth, especially on the irregularities of the surface. An accumulation of Gram-negative and anaerobic bacteria takes place, of yellowish color, extremely sticky, that is called ''bacterial plaque" (or ''dental plaque"). Its development, supra-or sub-gingival, arises in three phases: on the physiological preexistent film, some bacteria adhere then proliferate and form colonies, i.e. aggregation of filamentous organizations and spirochetes [START_REF] Bernimoulin | Recent concepts in plaque formation[END_REF]. Also amalgamate with this substance some fragmentary food remains and cells from the mouth mucosa that are incompletely altered by digestive enzymes and/or bacterial proteases; these elements calcify progressively in the dental plaque, this calcification permitting thus differential diagnosis with external contamination after death or sampling [START_REF] Theilade | A transmission electron microscopic study of 7-day old bacterial plaque in human tooth fissures[END_REF].

Composed of around 100 billion bacteria per gram, the dental plaque may calcify and form the ''dental calculus" or ''dental tartar". This calcification is not exactly complete, because some sectors remain non-mineralized in the form of channels or gaps still comprising in their centre an important aerobic and anaerobic bacterial colony [START_REF] Tan | Study of bacterial viability within human supragingival dental calculus[END_REF]. Another type of formation of dental calculus is a spontaneous calcium salt precipitation on the surface of the teeth; in this case, deposits are present especially at the openings of the excretion channels of salivary glands (lingual faces of the mandible incisors and vestibular faces of the maxillary molars). Many microbial and fungal species were found in this kind of deposit [START_REF] Howell | Cultivable bacteria in developing and mature human dental calculus[END_REF]: Actinomycetes (Actinomyces israeli and A. naeslundii mainly), Veillonellae, bacteroides, fusobacteries, not identified Gram-positive bacilli, etc.

The elemental analysis of dental calculus also showed its extreme diversity of composition [START_REF] Mcdougall | Analytical transmission electron microscopy of the distribution of elements in human supragingival dental calculus[END_REF]. Four major elements were identified: calcium and phosphorus (mainly), magnesium (up to 3% of the mass), zinc (up to 1% of the mass). Other minor elements were only found as traces: titanium, nickel, cobalt, arsenic, strontium, zirconium, cadmium, selenium and lead. The following inorganic molecules were also identified [START_REF] Jin | Supragingival calculus: formation and control[END_REF]: brushite, di-hydrate phosphate dicalcium, phosphate octo-calcic, hydroxyapatite, whitlockite. The variation of composition of the dental calculus is extremely important between individuals [START_REF] Galgut | A transmission electron microscopy study of supragingival calculus[END_REF]. Factors that are potentially involved in this important variability are [START_REF] Roberts-Harry | Morphology and elemental composition of subgingival calculus in two ethnic groups[END_REF]: food, salivary pH, oral hygiene, the immunizing state of the subject, environment, oral anatomy of the individual, salivary flow, masticatory disorders, ethnic origin, age of the subject, etc. Lastly, dental calculus has been accused in the development of periodontal diseases, i.e. the irritating and inflammatory diseases concerning the support tissue of the tooth [START_REF] White | Dental calculus: recent insights into occurrence, formation, prevention, removal and oral health effects of supragingival and subgingival deposits[END_REF][START_REF] White | Recent advances in methods for the assessment of dental calculus research and clinical implications[END_REF].

Precedent studies on dental calculus from skeletal remains

Very early, the anthropological interest of dental calculus has been recognized. A simple methodology was developed, initially allowing the extraction of phytolithes from dental calculus deposits of prehistoric herbivores. Its preparation from a skeletal human sample is particularly easy [START_REF] Middleton | Extraction of oral phytoliths from herbivore dental calculus[END_REF]:  Two washings of the taking away with distilled water allowing the elimination of a possible postmortem contamination;  A final superficial washing with diluted hydrochloric acid;  A direct examination with optical microscopy.

Dental calculus was described on bones of Portuguese Middle Pleistocene [START_REF] Trinkaus | Later Middle Pleistocene human remains from the Almonda Karstic system, Torres Novas, Portugal[END_REF], but Vandermeersch was the first, in 1994, to use the electron microscope (SEM) to identify micro-organisms directly involved in the development of the dental plaque [START_REF] Vandermeersch | Middle Paleolithic dental bacteria from Kebara, Israël[END_REF] on an individual of the Middle Paleolithic discovered on the site of Kebara, Israel.

A microbiological study carried out on skeletons dated between 500,000 to 12,000 years B.C. from Chile, Spain and Near East showed the rather frequent presence of Streptococcus mutans in dental calculus.

Nevertheless, it seems that the distribution of this bacterial species was related neither to the food mode, nor to the conditions of life [START_REF] Linossier | Paleomicrobiological study in dental calculus: Streptococcus mutans[END_REF]; this observation is opposed to that of a Mesolithic bone collection from Yugoslavia that showed sex differences of calculus and dental wear distribution revealing different food modes [START_REF] Edynak | Yugoslav mesolithic dental reduction[END_REF]. Dental calculus was found particularly frequent in a population of the Bronze Age (2000 years B.C.) in Bahrain [START_REF] Hojgaard | Dentition on Bahrain 2000 BC[END_REF] and in a population of breeder-farmer-fishermen of Umm an-Nar (Abu-Dhabi) [START_REF] Hojgaard | Dentition on Umm an-Nar (Trucial Oman), 2500 BC[END_REF]; within a collection of almost 11,000 teeth coming from Harappa (Pakistan), a contemporary site, 6.8% of the teeth were decayed with almost generalized layers of calculus deposits [START_REF] Lukacs | Dental paleopathology and agricultural intensification in South Asia: new evidence from Bronze Age Harappa[END_REF]. The systematic odontological study of 12 skeletal collections dated from the Bronze Age (3000 years B.C.) to the 16 th century A.D. coming from the Arabian Gulf [START_REF] Littleton | Fish-eaters and farmers: dental pathology in the Arabic Gulf[END_REF], was the occasion to work out a classification of pathological odontological characteristics:

 Marine mode (severe dental wear, weak frequency of decays, dental abscesses caused by dental wear, weak ante-mortem dental loss);  Pastoral, marine and agrarian mode mixed (moderate dental wear, moderated dental calculus, low level of decays, low level of dental abscesses caused by dental wear, weak rate of antemortem dental losses);  Agro-pastoral mixed mode (weak and moderate dental wear, important frequency of decays, dental calculus, abscesses caused by dental wear, severe dental ante-mortem losses);  Mode of intensive agriculture (severe dental wear).

The anthropological study of a dental sample from Monte Bibele (Etruscan-Celtic necropolis of the north of Italy, close to Bologna and Monterenzio Vecchia) showed the frequent presence of dental calculus [START_REF] Gualandi | Food habits and dental disease in an iron-age population[END_REF] that indicated, for the authors, a large extent of proteins in the alimentation of the population. The great frequency of ante-mortem dental losses and dental abscesses was in agreement with the importance of decays and calculus. The microscopic observation of enamel surfaces showed the presence of an important share of plants in the food. The analysis of the whole of the dental characteristics made it possible to conclude that this population had an economy with agrarian prevalence.

The examination of two skeletal collections coming from late medieval Serb cemeteries showed many dental anomalies: important dental wear, severe alveolar attrition, frequent decays and generalized dental calculus. Such anomalies were reported as secondary to an abrasive food mode and a deplorable oral hygiene [START_REF] Srejic | Dental paleopathology in a Serbian medieval population[END_REF]. Lastly, dental calculus of 20 Hungarian mummies dated between the 18th and 19th century A.D. were studied with optical and electronic microscope, and under polarized light. Positive Gram bacteria were identified within all the preparations, while negative Gram bacteria were present in only 12 subjects, and fungal filaments in only three individuals. Flesh-colored food remains were visible in five individuals, and vegetable fragments in the totality of the individuals. Non-organic elements and cellular remains could be identified within all the preparations [START_REF] Torok | Microscopic study of dental calculus in cadavers from the 18th-19th centuries[END_REF].

Comparative analysis of our results

KHB-1 (grave 1, US 169) and RH-5 (grave 405, individual B)

In these populations of fish-eaters, the numerous vegetal structures observed may be of diverse origins: food (herbs, vegetables), fibers from the fishnet when restored, or from the basket and rope work, using the mouth as a third hand (such activities are attested at RH-5 by mineralized mat and rope remains). The extreme wear on the lower central incisors of the subject from RH-5, associated with the ante-mortem loss of both superior first premolars as well as interproximal grooves occurring as symmetrical transverse furrows worn into the mesial or distal surfaces of the teeth, could be the consequences of such a use of the mouth (Fig. 6f). Degraded Malpighian epithelial cells very likely originate from the mouth mucosa.

The presence of crystals may lastly be explicated by the desert and/or beach sand transported by the wind to the aliments, the grit involuntarily assimilated with dried fish [START_REF] Macchiarelli | Prehistoric ''fish-eaters" along the eastern Arabian coasts: dental variation, morphology and oral health in the Ra's al-Hamra community (Qurum, Sultanate of Oman, 5 Millennia B[END_REF], or by the fragmentation of the grinding stone with pulverization of crystal fragments into the food [START_REF] Cavulli | Stone vessels from KHB-1, Ja'lan region, Sultanate of Oman[END_REF]. These particles could explain the important and premature abrasion of dental enamel surface for the young age of one of the subjects (KHB-1, grave 1, US 169). Vegetal fragments were found in all the samples, confirming the importance of this alimentation for this population. Unfortunately, their extreme fragmentation did not allow us a precise identification of the species present. They were accompanied by some crystals (quartz or alumino-silicate) originating from the fragmentation of grinding stone, due to the geographical localization of the site (Apennines, central part of north Italy, close to Firenze and Bologna), responsible for an important and premature dental wear in this population.

For the individual MV 20, the elemental analysis of some crystals showed a very high content in manganese, an element frequently found in ancient ceramics paintings: a food (vessel) or work-related (ceramist or painter) origin of this compound was thus very probable. The nature of this kind of crystal gave then pertinent information about the habits of this individual (type of alimentation vessel, work exposure to pollutant, etc.).

The matrix composition was variable between the five individuals sampled of Monterenzio Vecchia: alumino-silicate at some, phospho-calcium at others (Table 1). More, it is important to note that the opposition between MV 7, MV 15 and MV 20 on one hand, and MV 21 and MV 31 on the other hand, confirmed the existence of two different groups in the necropolis according to the topographical repartition of anthropological variations [START_REF] Charlier | Ostéo-archéologie de deux necropolis étrusco-celtiques: Monte Bibele et Monterenzio Vecchia (Bologne, Italie). Reconstitution d'une pathocénose à l'échelle de la vallée de l'Idice[END_REF] (olecranial perforation, Patella emarginata, supernumerary bones on the skull vault, enamel hypoplasia). These groups may have been founded on familial, ethnic or status differences, with variations, for example, in the alimentation process and/or composition of saliva.

Conclusion

The whole of the analyses provided important information on the food, environmental habits, work-related intoxication of the skeletonized individuals with possibilities of familial or social group differentiation.

As for the microscopic examination of PFD [START_REF] Charlier | The microscopic (optical and SEM) examination of putrefaction fluid deposits (PFD) potential interest in forensic anthropology[END_REF], some interesting data may be obtained pending some precautions, particularly by carrying out an examination and elemental analysis of the adjacent sediment in order to check if the recorded values testify to an external contamination or one from the original source.

4. 3 . 2 .

 32 Monterenzio Vecchia (MV 7, MV 15, MV 20, MV 21, MV 31)

Figure 1 :

 1 Figure 1: (a) Dental calculus on a molar of the subject MV15 (Etruscan-Celtic necropolis of Monterenzio Vecchia, Bologna, Italy); (b and c) vegetal structures and Malpighian epithelial cells from the DCD of the subject KHB-1 (grave 1, US 169) after cytocentrifugation (HES coloration); (d) crystal structures from the DCD of the subject RH-5 (grave 405, individual B) after cytocentrifugation (HES coloration); (e) Vegetal fibers and crystal structure from the DCD of the subject RH-5 (grave 405, individual B) after cytocentrifugation (HES coloration) and (f) quartz crystal present in the DCD of the subject MV 7 (SEM: enlargement x 1600) and (g) fungal filaments with spores present in the DCD of the subject MV 7 (MEB, enlargement x 3200).

Figure 2 :

 2 Figure 2: (a) Fibrillary aspect of the surface of the DCD of the subject MV 15 (MEB, enlargement x 160); (b) inner surface of the DCD carrying out a moulding of enamel at the subject MV 15 (MEB, enlargement x 1300); (c-e) details of the vegetable structures enchased in the DCD of the subject MV 15 (MEB, enlargement from _ 80 to _ 2700) and (f) formation rich in manganese (of food origin?) in the DCD of the subject MV 20 (MEB, enlargement x 520).

Figure 3 :

 3 Figure 3: Elemental spectrum of the DCD matrix of the subject MV 15.

Figure 4 :

 4 Figure 4: (a) Crystal enchased in the DCD of the subject MV 21 (MEB, enlargement x 630); (b-e) prints of the fungal filaments in the DCD of the subject MV 21 (MEB, enlargement from x 300 to x 1250) and (f) quartz crystal enchased in the DCD of the subject MV 31 (MEB, enlargement x 1240).

Figure 5 :

 5 Figure 5: Elemental spectrum of the DCD matrix of the subject MV 31.

Figure 6 :

 6 Figure 6: (a) Vegetable fiber from the DCD of a canine of Agnes Sorel observed in OM under polarized light, with a characteristic refringence; (b) group of malpighian epithelial cells from the DCD of a canine of Agnès Sorel after cytocentrifugation (HES coloration); (c) a bacteria of spirochete type from the DCD of a canine of Agnès Sorel after cytocentrifugation (Gram coloration); (d) vegetal fibers from the DCD of Pierre Hazard after cytocentrifugation (HES coloration); (e) crystals and Malpighian epithelial cells from the DCD of Pierre Hazard after cytocentrifugation (HES coloration) and (f) interproximal notch in the right second upper molar of the subject RH-5 (grave 405, individual B).

  1, US 169) from the Middle Holocene settlement of KHB-1 at Ra's al-Khabbah (5th to 4th millennium B.C.) in the Sultanate of Oman, excavated by Fabio Cavulli (University of Trento, Italy) in the frame of the Joint Hadd Project, an international archaeological research program directed by Prof. Serge Cleuziou (University of Paris I, France) and Maurizio Tosi (University of Bologna, Italy). Concerning the dental pathologies, we have noticed several lesions of enamel hypoplasia, attesting successive stresses during childhood.

	The
	degree of attrition was moderate and calculus present but slight. No carious lesions neither
	abscesses were recorded [5].
	 A 20 to 49-years-old female skeleton (grave 405, individual B) from the Neolithic graveyard of
	RH-5 at Ra's al-Hamra (4th millennium B.C.) located in the Capital area of the Sultanate of
	Oman [6], currently excavated by one of us (O.M.), under the authority of the Ministry of
	Heritage and Culture of Oman and Prof. Maurizio Tosi, and previously excavated by Sandro
	Salvatori (respectively University of Naples and Centro di Studi e Ricerche Ligabue of Venice,
	Italy).
	 A mandible canine of Agnès Sorel, favorite mistress of the French King Charles VII, buried in the
	Saint-Ours collegial of Loches (Indre-and-Loire, France) in 1450. Excavation and
	bioarchaeological analyses were realized by one of us (P.C.). Cause of death was an acute
	mercury intoxication following the beginning of an anti-helminthic cure. The body was
	embalmed in Normandy, and then transported to Loches, some 500 km far away. The dental
	status was very good [3], with only post-mortem teeth loss due to a recuperation by the local
	population in order to use them as historical souvenirs or prostheses.
	 A maxillary molar of Pierre Hazard (royal notary buried in Maubec, Isère, France) in the 15 th
	century. Macroscopic bio-archaeological analyses were realized by one of us (M.B.). His body
	was embalmed with lime after an opening of the body and a complete extraction of all organs.
	Dental calculus and moderate alveolar attrition were observed.

Acknowledgments

The authors would like to thank warmly V. Mazel and P. Richardin for the MEB analyses carried out in their laboratory (C2RMF, Louvre Museum), and the anonymous reviewers for their important comments and modifications proposed for our text.

SEM examination may be made without any preparation and authorizes a complete preservation of the sample; observation of biological and mineral structures is then possible (red blood cells, vegetal fibers, crystals, etc.) with the possibility of elemental surface analysis. Optical microscopy examination of DCD is destructive but may supports direct diagnoses and complementary analyses as such as immuno-cyto-chemistry or DNA extraction; more the examination of red blood cells offers the opportunity of intra-cytoplasmic elements diagnosis such as parasites (Plasmodium sp., for example, in a context of malaria [START_REF] Charlier | A medieval endemic parasitosis[END_REF]) or metals (gold, for example, in a context of chronic poisoning [START_REF] Charlier | Fatal alchemy a gold poisonous elixir in the French 16th court[END_REF]).

Conflict of interest

The authors have declared no conflict of interest.

Tables and figures