Vincent Calvez
email: vincent.calvez@ens-lyon.fr

Pierre Gabriel
email: pierre.gabriel@uvsq.fr

Stéphane Gaubert
email: stephane.gaubert@inria.fr

Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems

We study a growth maximization problem for a continuous time positive linear system with switches. This is motivated by a problem of mathematical biology (modeling growth-fragmentation processes and the PMCA protocol). We show that the growth rate is determined by the nonlinear eigenvalue of a max-plus analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the solutions or subsolutions of which yield Barabanov and extremal norms, respectively. We exploit contraction properties of order preserving flows, with respect to Hilbert's projective metric, to show that the non-linear eigenvector of the operator, or the "weak KAM" solution of the HJ equation, does exist. Low dimensional examples are presented, showing that the optimal control can lead to a limit cycle.

I. INTRODUCTION

We investigate in this note the optimal control of time continuous positive linear dynamical systems in infinite horizon. We wish to compute the maximal growth rate that can be obtained from infinitesimal combinations of a set of nonnegative matrices.

More precisely, we consider a compact set M ⊂ M n (R) of irreducible Metzler matrices. That is to say, we assume that for all m ∈ M and for all i = j, m ij ≥ 0. In addition for every partition of indices {1 . . . n} = I ∪ • J one can pick i ∈ I and j ∈ J such that m ij > 0. A direct consequence of compactness is uniform irreducibility: there exists a constant ν > 0 such that for all m ∈ M, and every partition of indices one can pick i ∈ I and j ∈ J such that m ij ≥ ν.

Let K be the nonnegative orthant in R n , K + the positive orthant, and K 0 = K \ {0}. For t > 0, x ∈ K and a measurable control function M : [0, t] → M, we define x M ∈ W 1,∞ ([0, t], R n ) as the solution of the following linear problem with control M :

ẋM (s) = M (s)x M (s), x M (0) = x . (1)
We also denote x M (s) = R(s, M )x, where R is the resolvent. Finally we denote in short L ∞ (0, t) the set of measurable (bounded by assumption) control functions M : [0, t] → M. We are interested in control functions maximizing the growth rate lim sup

t→∞ 1 t log x M (t) . (2)
We assume w.l.o.g. that M is convex. The results presented here are still valid for nonconvex sets M, provided the controls are replaced by relaxed controls which take values in the closed convex hull co (M).

For a constant control M (s) ≡ m we have R(t, m) = e tm . It is an immediate consequence of the Perron-Frobenius theorem that, being φ m ∈ K + a left Perron-Frobenius (PF) eigenvector of m, the linear function v(x) = φ m , x satisfies the following identity,

(∀t ∈ R + ) (∀x ∈ K) e λ(m)t v(x) = v(x m (t)) ,
where λ(m) ∈ R is the dominant eigenvalue of m. The following result can be thought of as a non-linear extension of the Perron-Frobenius theorem.

Theorem 1: Under previous assumptions there exist a real λ(M) and a function v : K → R + , homogeneous of degree 1, positive on K 0 , globally Lipschitz continuous, which satisfy the following identity

(∀t ∈ R + ) (∀x ∈ K) e λ(M)t v(x) = sup M ∈L ∞ (0,t) v(x M (t)) , (3)
The scalar λ(M) is unique as soon as v belongs to the class of homogeneous functions of degree 1 which are locally bounded on K, and it determines the optimal growth rate (2). Moreover, u = log v is characterized as a viscosity solution of an ergodic Hamilton-Jacobi PDE:

-λ(M) + H(D y u(y), y) = 0 , y ∈ S ,

where S is the standard simplex.

The Hamiltonian H will be given in Section II. Corollary 2 (Ergodicity): Let v 0 : K → R + be a continuous function, homogeneous of degree 1, positive on K 0 . Define v(t, x) = sup M ∈L ∞ (0,t) v 0 (x M (t)). Then we have the following ergodicity result,

(∀x ∈ K 0 ) lim t→+∞ 1 t log(v(t, x)) = λ(M) .
Moreover the convergence is locally uniform on K 0 . Theorem 1 is closely related to results belonging to the theory of stability of linear inclusions. There, matrices are not necessarily assumed to be Metzler matrices. The nonlinear eigenvalue λ(M) coincides with the joint spectral radius [START_REF] Rota | A note on the joint spectral radius[END_REF]. In his seminal paper [START_REF] Barabanov | An absolute characteristic exponent of a class of linear nonstationary systems of differential equations[END_REF], Barabanov proved the existence of extremal norms in R n which saturates (3), under a different irreducibility condition. Later the same author investigated the behaviour of extremal trajectories in the three-dimensional case n = 3, first when M has the specific structure of a segment with a rank one matrice for direction [START_REF] Barabanov | On the aizerman problem for third-order timedependent systems[END_REF], secondly under a uniqueness condition for extremal trajectories verifying the Pontryagin Maximum Principle (PMP) [START_REF] Barabanov | Asymptotic behavior of extremal solutions and structure of extremal norms of linear differential inclusions of order three[END_REF] (see also the recent improvement by Gaye et al [START_REF] Gaye | Properties of barabanov norms and extremal trajectories associated with continuous-time linear switched systems[END_REF]). We also refer to [START_REF] Wirth | The generalized spectral radius and extremal norms[END_REF] for an alternative proof of the existence of Barabanov extremal norms, and to the work of Chitour, Mason and Sigalotti [START_REF] Chitour | On the marginal instability of linear switched systems[END_REF] for the analysis of situations in which there are obstructions to the existence of such norms.

Several authors have analyzed specially the stability of positive linear systems. Very recently, Mason and Wirth [START_REF] Mason | Extremal norms for positive linear inclusions[END_REF] have established the existence of an extremal norm, that is, a viscosity subsolution of the spectral problem (3) (the equality relation being replaced by ≥), corresponding to a critical subsolution of the ergodic Hamilton-Jacobi equation. They use an irreducibility condition which is milder than our, but which does not guarantee the existence of a viscosity solution. Conditions for the existence of subsolutions are typically less restrictive. It is an interesting issue to see whether the assumptions of Theorem 1 could be relaxed.

We emphasize that we take advantage of an illuminating connection between problem (3) and the weak KAM theory in Lagrangian dynamics [START_REF] Fathi | The Weak KAM Theorem in Lagrangian Dynamics[END_REF]. In particular long-time dynamics of optimal trajectories appear to be encoded in the so-called Aubry sets. Such eigenproblems have been widely studied in ergodic control, and also by dynamicians in the setting of the weak KAM theory, where the eigenfunction is known as a weak KAM solution. However, basic existence results for eigenvectors rely on controllability conditions which are not satisfied in our setting.

We exploit tools from the theory of Hamilton-Jacobi PDE to prove Theorem 1, combined with techniques from Perron-Frobenius theory. In particular, we use the Birkhoff-Hopf theorem in a crucial way. The latter states that a linear map leaving invariant the interior of a closed, convex and pointed cone is a strict contraction in Hilbert's projective metric. The contraction of the controlled flow turns out to entail the existence of the eigenvector. We note that tools from Lagrangian dynamics (Mather sets) have been recently applied by Morris to study joint spectral radii [START_REF] Morris | Mather sets for sequences of matrices and applications to the study of joint spectral radii[END_REF]. This deserves to be further studied in the present setting.

The same type of equations has been studied in the context of infinite dimensional max-plus spectral theory. In particular, the existence of continuous eigenfunctions for max-plus operators with a continuous kernel is established in [START_REF] Kolokoltsov | Idempotent analysis and applications[END_REF]. More general conditions, exploiting quasi-compactness techniques, can be found in [START_REF] Mallet-Paret | Eigenvalues for a class of homogeneous cone maps arising from max-plus operators[END_REF]. It would be interesting to see whether such techniques to the present problems.

A natural question that arises in the literature is whether the knowledge of {λ(m)} m∈M , say (∀m ∈ M) λ(m) < 0 guarantees the stability of the differential inclusion [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]. A positive answer has been given in [START_REF] Gurvits | On the stability of switched positive linear systems[END_REF] in dimension n = 2. A negative answer has been given in (possibly) high dimension in the same work. Soon after, Fainshil et al give a counterexample in dimension n = 3 [START_REF] Fainshil | On the stability of positive linear switched systems under arbitrary switching laws[END_REF]. It is a pair of matrices such that every convex combination has a negative spectral radius but the associated joint spectral radius is positive.

We address similar questions in the present note, namely whether λ(M) = max m λ(m) or λ(M) > max m λ(m). We give a new and self-contained proof of the positive answer in dimension n = 2. We also give three dimensional numerical examples with positive and negative answers. The case where λ(M) > max m λ(m) is of particular interest. To find such a numerical example we restrict to the case where M is a segment, and the maximum of λ(m) is attained at an interior point. We investigate periodic perturbations of the optimal constant control in the spirit of [START_REF] Clairambault | An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations[END_REF], [START_REF] Clairambault | Comparison of Perron and Floquet eigenvalues in age structured cell division cycle models[END_REF]. More precisely we compute the second order directional derivative of the Floquet eigenvalue. We derive a criterion about the local optimality of the constant control with respect to periodic perturbations. We exhibit a numerical example for which this condition is satisfied. Numerical simulations of the full optimal control problem clearly shows the convergence of the optimal trajectory towards a limit cycle, suggesting that the optimal control in infinite horizon is indeed a BANG-BANG periodic control. It is worth noticing that the criterion that we derive is the exact opposite of a so-called Legendre condition in geometric optimal control theory [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF], [START_REF] Bonnard | Second order optimality conditions in the smooth case and applications in optimal control[END_REF]. The latter condition ensures the local optimality of the extremal trajectory (here the trajectory corresponding to the maximal Perron eigenvalue) for short times.

II. TECHNIQUES OF PROOF OF THEOREM 1

We present in this section the main elements of the proof of Theorem 1. In this Section we write in short λ = λ(M).

Step #1. Homogeneity and projection of the dynamics onto the simplex. The infinitesimal version of (3) writes as a Hamilton-Jacobi equation in the viscosity sense,

λv(x) = max m∈M D x v(x), mx . (5)
Using the homogeneity of the function v we can project (5) onto the simplex S = {x ∈ K : 1, x = 1}. We write

v(x) = 1, x v x 1, x ,
where v is defined on S. Then problem ( 5) is equivalent to finding (λ,v) such that

λv(y) = max m∈M (L(y, m)v(y) + D y v(y), b(y, m) ) , (6)
where the pay-off L and the vector fields b are given by

L(y, m) = 1, my , b(y, m) = my -L(y, m)y .
Note that each vector field b(•, m) is tangent to the simplex S. It gives indeed the projected dynamics on the simplex: if

x M is solution to (1) then y M = x M 1,x M is solution to the non-linear ODE ẏM (s) = b(y M (s), M (s)) . (7)
Step #2. Computation of a Lipschitz constant with respect to Hilbert's projective metric Recall that Hilbert's (projective) metric is defined, for all x, y ∈ K + , by

d(x, y) = log max 1≤i,j≤n x i y j x j y i . (8)
It is a metric in the set of half-lines included in the interior of K. In particular, d(x, y) = 0 iff x and y are proportional.

It is known to be a weak Finsler structure [START_REF] Nussbaum | Finsler structures for the part metric and Hilbert's projective metric and applications to ordinary differential equations[END_REF], obtained by thinking of K + as a manifold with the seminorm

h x = max i h i x -1 i -min j h j x -1 j in the tangent space at point x. Then, d(x, y) = inf 1 0 γ(s) γ(s) ds
where the infimum is taken over all differentiable paths γ contained in the interior of K, such that γ(0) = x and γ(1) = y.

Let q ∈ K + and m ∈ P. We aim to compute the Lipschitz constant of the function l(x) = q,mx q,x , when the source set is endowed with the Hilbert metric. For a given matrix m = (m ij ) we denote |m|= (|m ij |) the matrix obtained by taking absolute values of the coefficient pointwise.

The following lemma is established by exploiting the Finsler's nature of Hilbert projective metric, along the lines of [START_REF] Nussbaum | Finsler structures for the part metric and Hilbert's projective metric and applications to ordinary differential equations[END_REF]. See also [START_REF] Gaubert | Dobrushin ergodicity coefficient for markov operators on cones, and beyond[END_REF].

Lemma 3: Let l : (K, d) → (R, |•|) defined as l(x) = q,mx q,x .
It is Lipschitz continuous with the following bound on the Lipschitz constant,

Lip l ≤ sup x∈K0 inf a∈R q, |m -a.id| x q, x .
Step #3. Exponential contraction of the flow (after some time). A key technical ingredient is the following lemma, which shows that for a fixed time τ > 0, the flow maps the closed cone to its interior. Lemma 4: Let τ > 0. Define the cone K τ ⊂ K as the convex closure of images of K by the flow after a time step τ > 0,

K τ = co   M ∈C(τ ) R(τ, M )K   .

It satisfies the following properties,

• K τ is stable with respect to every flow R(s, M ), s ≥ 0, M ∈ L ∞ (0, s). • K τ is included in the interior of the cone, closed, and bounded in Hilbert's projective metric. A classical result of Birkhoff and Hopf shows that a linear map sending a (closed, convex, and pointed) cone to its interior is a strict contraction in Hilbert's projective metric, see for instance [START_REF] Lemmens | Non-linear Perron-Frobenius theory[END_REF] for more information. We deduce from the Birkhoff-Hopf theorem and from Lemma 4 the following contraction result for the flow.

Lemma 5: There exist a time T > 0 and a positive rate µ > 0 such that the flow R(t, M ) is uniformly exponentially contractive for t ≥ T :

(∀t ≥ T ) (∀M ∈ L ∞ (0, t)) (∀(x, y) ∈ K + × K + ) d(R(t, M )x, R(t, M )y) ≤ e -µt d(x, y) . ( 9
)
Remark 6: If inf m∈M min i =j m ij > 0, one can choose T = 0 in the Lemma 5, and accordingly,

µ = inf m∈M min i =j 2(m ij m ji ) 1/2 > 0 . (10)
See also [START_REF] Gaubert | Dobrushin ergodicity coefficient for markov operators on cones, and beyond[END_REF].

Step #4. Weak KAM Theorem. As suggested by the expected exponential growth, we make a logarithmic transformation. Let introduce ů = logv. The original problem (3) writes equivalently: find a real λ and a function ů, defined on the simplex S, such that

λt+ů(y) = sup M ∈L ∞ (0,t) t 0 L(y M (s), M (s)) ds + ů(y M (t)) , (11)
for all t ≥ 0, or in its infinitesimal setting: find a real λ and a function ů such that ů is the viscosity solution of the stationary Hamilton-Jacobi equation

-λ + H(D y ů(y), y) = 0 , y ∈ S , (12)
where the Hamiltonian is defined as

H(p, y) = max m (L(y, m) + p, b(y, m) ).
The existence of a solution (λ, u) is known as a weak KAM Theorem in the context of dynamical systems, see the work by Fathi [START_REF] Fathi | Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens[END_REF], [START_REF] Fathi | The Weak KAM Theorem in Lagrangian Dynamics[END_REF]. Here, we follow the now classical argument of Lions-Papanicolaou-Varadhan to prove the existence of such a pair (λ, ů), the vector ů being obtained as a rescaled limit of the solution u ǫ of a Hamilton-Jacobi PDE with discount rate ǫ > 0. In doing so, we make use of the contraction property of Lemma 5 with respect to Hilbert's projective metric.

Step #5. Calibrated trajectories. Before we proceed with the end of the proof (boundedness of ů and uniqueness of λ), we recall some definitions from [START_REF] Fathi | The Weak KAM Theorem in Lagrangian Dynamics[END_REF] adapted to our context.

Definition 7 (Calibrated trajectories): A Lipschitz curve γ : I → S + defined on the interval

I ⊂ R, associated to some control M ∈ L ∞ (I), γ = y M , is calibrated if for every t ≤ t ′ ∈ I, we have ů(γ(t ′ )) -ů(γ(t)) = t ′ t (L(y M (s), M (s)) -λ) ds
Along the lines of [START_REF] Fathi | The Weak KAM Theorem in Lagrangian Dynamics[END_REF], we show that calibrated trajectories do exist.

Step #6. Regularity of ů up to the boundary ∂S and uniqueness of λ. First of all we deduce from the fixed point formulation [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi equations with state constraints[END_REF] that ů is Lipschitz continuous on the whole S with the respect to the ℓ 1 norm |•| 1 . Notice that the previous argument only yields local Lipschitz continuity due to the singularity of the Hilbert metric at the boundary ∂S. From the fixed point formulation [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi equations with state constraints[END_REF] we have in particular,

λ+ů(y) = sup M ∈L ∞ (0,1) 1 0 L(y M (s), M (s)) ds + ů(y M (1)) .
(13) It suffices to observe that for all M ∈ L ∞ (0, 1), y M (1) = R(1, M )y ∈ K 1 which is a compact subset of S with respect to the Hilbert metric. Thus K 1 is at uniform positive distance from the boundary ∂S and there exists a constant C(K 1 ) such that for all (x, y)

∈ K 1 ×K 1 , d(x, y) ≤ C(K 1 )|x-y| 1 .
Finally we observe that ( 13) is a supremum of Lipschitz functions as it is the case for ů(y M (1)):

|ů(y M (1)) -ů(x M (1))| ≤ (Lip ů| K1 ) d(R(1, M )y, R(1, M )x) ≤ (Lip ů| K1 ) C(K 1 )|R(1, M )y -R(1, M )x| 1 ≤ (Lip ů| K1 ) C(K 1 ) sup M ∈L ∞ (0,1) R(1, M ) 1 |y -x| 1 .
Therefore ů is globally Lipschitz on S with respect to the ℓ

1 norm |•| 1 .
As a consequence we can uniquely extend ů to a continuous function defined on S.

The uniqueness of λ is then deduced from a classical argument, that we skip, as well as the proof of Corollary 2.

III. QUALITATIVE PROPERTIES OF THE OPTIMAL EXPONENT λ. A. Optimality of stationnary controls in dimension 2

Proposition 8 (Optimality and relaxed control): The optimal growth rate λ(M) is greater or equal than any Perron eigenvalue λ(m) for m ∈ M.

Proof: An immediate proof of this statement is obtained by choosing a constant control M ≡ m in [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi equations with state constraints[END_REF]. We denote by z m ∈ S the corresponding eigenvector. Since z m is a stationary point for the dynamics, we have

λ(M)t + ů(z m ) ≥ t 0 L(z m , m) ds + ů(z m ) ≥ λ(m)t + ů(z m ) .
Therefore λ(M) ≥ λ(m). A similar proof is obtained by noticing that u m (y) = log φ m , y is a supersolution of [START_REF] Castilla | In vitro generation of infectious scrapie prions[END_REF].

Our next result shows that in dimension 2, the optimal growth is achieved by constant controls.

Theorem 9: Assume that n = 2. Then

λ(M) = max m∈M λ(m) .
We skip the proof of this result, which exploits the Pontryagin maximum principle, but rather give an heuristic argument. The weak KAM statement, i.e. the existence of a pair (λ, ů) solution of the stationary Hamilton-Jacobi equation, generates an optimal vector field b * . It is determined by the rule b * (y) = b(y, m * ) where m * ∈ M realizes the maximum of the Hamiltonian H in [START_REF] Castilla | In vitro generation of infectious scrapie prions[END_REF]. Since Equation ( 12) is stationary, the vector field b * is autonomous. However it is not defined everywhere on the simplex. For instance it cannot be defined on the points where ů is not differentiable nor on the points where the maximum value of H is attained for several m * ∈ M. Anyway, up to this regularity issue, an autonomous vector field on the one-dimensional simplex is expected to exhibit fairly simple dynamics, e.g. convergence towards an equilibrium point. Simple arguments show that equilibria are in fact Perron eigenvectors. By optimality they have to be associated with the maximal possible eigenvalue for m ∈ M.

A stronger result (where the unique optimal control is exhibited) can be found in [START_REF] Coron | Optimization of an amplification protocol for misfolded proteins by using relaxed control[END_REF] in a particular case coming from the modelling of the PMCA.

B. Floquet perturbations of the maximal Perron eigenvalue.

In this subsection, we give a few insights why we cannot hope for λ(M) = max m∈M λ(m) in dimension n ≥ 3. We shall focus on the possible existence of limit cycles on the simplex which have a better reward than the maximal Perron eigenvalue.

The arguments used to justify Theorem 9 cannot be transposed to a higher dimension. Another way to attack the problem is to test the optimal Perron eigenvalue against periodic perturbations. The question goes as follows: is it possible to find a larger Floquet eigenvalue in the neighbourhood of the maximal Perron eigenvalue? To address this issue we consider a simplified framework where M is a segment. We denote M = {G + αF , α ∈ [a, A]}, and λ(α) = λ(G+αF ). We assume that there exists α * ∈ (a, A) such that λ(α * ) is a local maximum of λ(α).

We assume for the sake of simplicity that the matrix G + α * F is diagonalizable. We denote by (e The following Proposition gives the second order condition for λ(α * ) being a local maximum relatively to periodic perturbations of the control. We denote by f T the time average over one period,

f T = 1 T T 0 f (t) dt .
Proposition 10: The directional derivative of the Floquet eigenvalue vanishes at ǫ = 0:

dλ F (α * + ǫγ) dǫ ǫ=0 = 0 . (14)
Hence, α * is also a critical point in the class of periodic controls. The second directional derivative of the Floquet eigenvalue writes at ǫ = 0:

d 2 λ F (α * + ǫγ) dǫ 2 ǫ=0 = 2 n i=2 γ 2 i T (φ * 1 F e * i )(φ * i F e * 1 ) λ * 1 -λ * i , (15)
where γ i (t) is the unique T -periodic solution of the relaxation ODE γi (t)

λ * 1 -λ * i + γ i (t) = γ(t) .
The idea of computing directional derivatives has been used in a similar context in [START_REF] Michel | Optimal proliferation rate in a cell division model[END_REF] for optimizing the Perron eigenvalue in a continuous model for cell division. See also [START_REF] Clairambault | An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations[END_REF] for a more general discussion on the comparison between Perron and Floquet eigenvalues.

Taking γ ≡ 1 in Equation ( 15), we get the second derivative of the Perron eigenvalue at α * ,

d 2 λ dα 2 (α * ) = 2 n i=2 (φ * 1 F e * i )(φ * i F e * 1 ) λ * 1 -λ * i , (16)
which is nonpositive since α * is a maximum point. Therefore we are led to the following question: is it possible to construct counter-examples such that the sum ( 15) is positive for some periodic control γ, whereas the sum ( 16) is nonpositive? This is clearly not possible in dimension n = 2 because the sum in ( 15) is reduced to a single nonpositive term by [START_REF] Clairambault | An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations[END_REF]. For n ≥ 3, considering periodic perturbations γ(t) = cos(ωt), we get the formula

d 2 λ F (α * + ǫγ) dǫ 2 ǫ=0 = n i=2 λ * 1 -λ * i ω 2 + (λ * 1 -λ * i ) 2 (φ * i F e * 1 )(φ * 1 F e * i ) .
An asymptotic expansion when ω → +∞ indicates that if the condition

n i=2 (λ * 1 -λ * i )(φ * i F e * 1 )(φ * 1 F e * i ) > 0 ( 17
)
is satisfied, then ( 15) is positive for some ω large enough.

C. Legendre type condition for local optimality on short times

Within the framework described in the previous section, we introduce the endpoint mapping

F T : L ∞ (0, T ) → K, α(•) → x(T ),
which maps a control α ∈ L ∞ (0, T ) to the terminal value x(T ) of the corresponding trajectory, i.e. the solution of the ODE ẋ(s) = (G + α(s)F )x(s) .

We analyse in the following the behaviour of this mapping in the neighbourhood of the best constant control α * and its associated trajectory x(t) = e λ * 1 t e * 1 ,. Moreover we make the link with the computations on the Floquet eigenvalue in the previous section. Consider a variation α(•) = α * + ǫ γ(•) ∈ L ∞ (0, T ) (not necessarily periodic) and define the quadratic form

Q(γ) := φ * 1 (D 2 α * F T )(γ, γ).
A straightforward computation gives the expression

Q(γ) = 2e λ * 1 T T 0 t 0 γ(t)γ(s) φ * 1 Fe (G+α * F -λ * 1 I)(t-s) Fe * 1 dsdt
Looking at the leading terms in Q when T is small, we get a sufficient condition for the control α * to be locally optimal for small times on the hyperplane

L ∞ 0 := γ ∈ L ∞ (0, T ), T 0 γ(t) dt = 0 . Proposition 11: If the condition φ * 1 F (G + α * F -λ * 1 I)F e * 1 > 0. ( 18
)
is satisfied, then the quadratic form Q restricted to L ∞ 0 is negative definite for short times with respect to the negative Sobolev space H -1 (0, T ), that is to say

∃δ > 0, ∃ǫ > 0, ∀T ∈ (0, ǫ), Q| L ∞ 0 (γ) ≤ -δ γ 2 H -1 .
We skip the proof of this result.

Condition ( 18) is the so-called generalized Legendre condition of our problem. The generalized Legendre condition appears in the study of optimality for totally singular extremals, i.e. when the second derivative of the Hamiltonian is identically zero along the trajectory. A typical example is provided by the single-input affine control systems, namely, ẋ(t) = f 0 (x(t)) + α(t)f 1 (x(t)), where α(t) ∈ R, and f 0 , f 1 are smooth vector fields. In this case the generalized Legendre condition writes

p(•)[f 1 , [f 0 , f 1 ]]x(•) > 0 where [•,
•] is the Lie bracket of vector fields. Our linear control system belongs to this class of problems, and straightforward computations show that for x(t) = e λ * 1 t e * 1 and

p(t) = e -λ * 1 t φ * 1 , we have p(•)[F, [G, F ]]x(•) = φ * 1 F (G + α * F -λ * 1 I)F e * 1 .
The generalized Legendre condition ensures that the quadratic form Q| KerD α * F T is definite negative. Then it allows to deduce that the trajectory x(•) is locally optimal for short final times T in the C 0 topology (we refer to [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF] and [START_REF] Bonnard | Second order optimality conditions in the smooth case and applications in optimal control[END_REF] for details).

Here we proved the negativity of Q on L ∞ 0 instead of KerD α * F T under the condition [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi equations with state constraints[END_REF]. Our aim is to make clearer the link with the computation of the second derivative of the Floquet eigenvalue. It follows from the previous section that the second derivative of λ F is positive for periodic controls cos(ωt) when ω is large if condition ( 17) is satisfied. Considering T small and ω = k 2π

T with k ≥ 1, we have that cos(ωt) ∈ L ∞ 0 and ω → +∞ when k → +∞. The following proposition points out the consistency between conditions ( 18) and [START_REF] Carvalho | On Poincaré-Bendixson theorem and non-trivial minimal sets in planar nonsmooth vector fields[END_REF].

Proposition 12: We have

φ * 1 F (G+α * F -λ * 1 I)F e * 1 = - n i=2 (λ * 1 -λ * i )(φ * 1 F e * i )(φ * i F e * 1 )
.

This relation is instructive since it emphazises the relation between a condition for optimality for small times and a condition for optimality with high frequences.

D. Lack of controllability/coercivity and the ergodic set

Classical arguments for proving ergodicity results such as Theorem 1 or Corollary 2 rely on short time dynamics of the system. This is the case for instance of the ergodicity result in Capuzzo-Dolcetta and Lions [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi equations with state constraints[END_REF], and of the Weak KAM Theorem of Fathi [START_REF] Fathi | Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens[END_REF]. In the former the authors assume a uniform controllability condition,

(∃r > 0) (∀y ∈ S) B(0, r) ⊂ co{b(y, m) | m ∈ M} ,
The latter relies on a regularizing property of the Lax-Oleinik semi-group which holds true for Tonelli Lagrangians. Both cases imply that the Hamiltonian is coercive, i.e. lim |p|→+∞ H(p, y) = +∞, a property which is not satisfied in our case. One noticeable exception can be found in [7, Section VII.1.2], where the controllability condition is replaced by a dissipativity condition which is somehow similar to our uniform contraction estimate in Lemma 5.

The lack of controllability is clear from Lemma 4, where some strict subsets of the simplex are positively invariant by all the flows. In a couple of papers, Arisawa made clear the equivalence between ergodicity (in the sense of Corollary 2) and the existence of a so-called ergodic set when controllability is lacking. The ergodic set satisfies the following properties: it is non empty, closed and positively invariant by the flows; it is attractant; it is controllable. We refer to [START_REF] Arisawa | Ergodic problem for the Hamilton-Jacobi-Bellman equation. I. Existence of the ergodic attractor[END_REF], [START_REF] Arisawa | Ergodic problem for the Hamilton-Jacobi-Bellman equation[END_REF] for the precise meaning of this statement, and to Section V for illustrations of ergodic sets in lowdimensional examples.

IV. ILLUSTRATION IN DIMENSION 2

We first illustrate our results on a simple two-dimensional example. Let a ∈ (0, 1 2 ). We consider the one-parameter family of matrices

M = 0 1 -α α 0 , α ∈ [a, 1 -a] .
The dominant eigenvalue is α(1α), with a maximum attained at α = 1 2 . We identify [0, 1] and S under the parameterization

S = 1 -θ θ , θ ∈ [0, 1] .
We have L(θ, α) = α(1-θ)+(1-α)θ, and b(θ, α) = α(1θ) 2 -(1-α)θ 2 . We look for a solution (λ, ů) to the Hamilton-Jacobi equation [START_REF] Castilla | In vitro generation of infectious scrapie prions[END_REF]. For computing ů we have to combine two branches, corresponding to the choice α = a or α = 1a. We realize that the first branch has a vertical asymptote at θ = z a since b(•, a) vanishes at this point, whereas the second branch has a vertical asymptote at θ = z 1-a . Therefore there is only one possible combination which gives a bounded viscosity solution on S.

In this example the ergodic set is the segment [z a , z 1-a ].

V. APPLICATION TO THE OPTIMIZATION OF GROWTH-FRAGMENTATION PROCESSES IN DIMENSION 3

We consider a toy model for a stage-structured linear polymerization-fragmentation process. It is inspired from a nonlinear discrete polymerization-fragmentation introduced in [START_REF] Masel | Quantifying the kinetic parameters of prion replication[END_REF] for the dynamics of prion proliferation. We do not take into account nonlinear saturation effects, and we further reduce the size of the system to n = 3. Polymers can have three states relative to their lengths: small (monomers), medium (oligomers), large (polymers). We denote by x i , i = 1, 2, 3 the density of polymers in each compartment. Transition rates due to growth in size of polymers from smaller to larger compartments (polymerization) are denoted by τ i , i = 1, 2. Transition rates due to fragmentation from larger to smaller compartments are denoted by β i , i = 2, 3. The corresponding matrices are

G =   -τ 1 0 0 τ 1 -τ 2 0 0 τ 2 0   and F =   0 2β 2 β 3 0 -β 2 β 3 0 0 -β 3   . (19)
Denoting by q = (1 2 3) T the vector of relative sizes of polymers, we have the following properties: 1 T G = 0 (conservation of the number of polymers by growth) and q T F = 0 (conservation of the total size of polymers by fragmentation). Optimal control issues come up in the development of efficient diagnosis tools for early detection of prion diseases from blood samples. The protocol PMCA (Protein Misfolding Cyclic Amplification) has been introduced by Soto and co-authors [START_REF] Soto | Cyclic amplification of protein misfolding: application to prion-related disorders and beyond[END_REF], [START_REF] Castilla | In vitro generation of infectious scrapie prions[END_REF] as very powerful method to achieve this goal. It aims at quickly generating in vitro detectable quantities of PrPsc being given minute quantities of it. PMCA consists in successive switching between incubation phases (where aggregates are expected to grow following a seeding-nucleation scenario alimented by purified PrPc) and sonication phases (where breaking of polymers is expected to increase the number of nucleation sites). This clear distinction between two phases with a control parameter which is the intensity of sonication makes the framework of (1) very weel adapted to model PMCA.

The minimal model for PMCA goes as follows: introduce α : [0, t] → [a, A] the intensity of sonication (i.e. fragmentation). The goal is to maximize the total size of polymers q, x α (t) following the linear growth-fragmentation process: Fig. 2. Illustration of convergence of optimal trajectories towards the optimal Perron eigenvector for the three-dimensional growth-fragmentation process [START_REF] Fathi | Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens[END_REF] with typical parameters. We have plotted the curve of Perron eigenvectors (black line), the boundary of the ergodic set (blue line), and an arbitrary optimal trajectory (red line). The optimal trajectory enters the ergodic set and then converges towards a limit cycle. Right picture is a zoom of the left one. The numerical grid is plotted on the axes: the space step is ∆x = 5E-4 to fall much below the width of the ergodic set. Parameters are τ 1 = 2E -2, τ 2 = 1, β = 4E -2, and a = 2, A = 8.

ẋα (s) = (G + α(s)F )x α (s), x α (0) = x . (20)
A generalization of this model, which includes an incidence of the sonication on the growth process, is investigated in [START_REF] Coron | Optimization of an amplification protocol for misfolded proteins by using relaxed control[END_REF]. For problem [START_REF] Fathi | The Weak KAM Theorem in Lagrangian Dynamics[END_REF], corollary 2 implies that q, x α (t) has an exponential growth with exponent λ(M). When τ 1 = τ 2 it is clearly better to sonicate as much as possible (α(s) ≡ A) because smaller monomers are equally efficient at growing in size than intermediate oligomers, but they are more numerous for a given size. However there are some biological evidence that polymerization rate is size-dependent: polymerization of intermediate aggregates have been postulated to be the most efficient [START_REF] Silveira | The most infectious prion protein particles[END_REF] (see also [START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF] for a continuous PDE model and a discussion of this phenomenon). Mathematically speaking we have a precise description of the variations of λ(α) as stated in the following Proposition.

Proposition 13: The Perron eigenvalue λ(α) of G + αF reaches a maximum value for some α * ∈ (0, +∞) if and only if τ 2 > 2τ 1 . Furthermore we face the following alternative:

• either τ 2 ≤ 2τ 1 and λ(α) increases from 0 to τ 1 , • or τ 2 > 2τ 1 and λ(α) increases from 0 to λ(α * ) and then decreases from λ(α * ) to τ 1 . We refer to [START_REF] Calvez | Optimal growth for linear processes with affine control[END_REF] for the details of the proof of Proposition 13.

Qualitative analysis of the optimal control and associated optimal trajectories rely on the description of relevant sets in the simplex. The ergodic set introduced by Arisawa [2], [START_REF] Arisawa | Ergodic problem for the Hamilton-Jacobi-Bellman equation[END_REF] can be characterized as the set enclosed by two remarkable trajectories: each starting from one of the two extremal Perron eigenvectors (resp. e 1 (a) and e 1 (A)) and evolving with constant control (resp. A and a). This set is of particular interest since it attracts all trajectories, not necessarily optimal ones. However it does not give any insight about the fate of optimal trajectories inside the ergodic set.

So far we have only access to local second-order conditions [START_REF] Coron | Optimization of an amplification protocol for misfolded proteins by using relaxed control[END_REF] to test the optimality of the best constant control. Numerical tests suggest that we always have λ(M) = max m λ(m) in the case of [START_REF] Fathi | Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens[END_REF], that is to say the optimal trajectory converges towards the optimal Perron eigenvector 3. Illustration of the optimal limit cycle for the numerical example [START_REF] Gaubert | Dobrushin ergodicity coefficient for markov operators on cones, and beyond[END_REF]. We have plotted the curve of Perron eigenvectors (black line), the boundary of the ergodic set (blue line), and an arbitrary optimal trajectory (red line). The optimal trajectory enters the ergodic set and then converges towards a limit cycle. Right picture is a zoom of the left one. The numerical grid is plotted on the axes: the space step is ∆x = 1E-3 to fall much below the width of the limit cycle. in the simplex. This is confirmed by finite-volume numerical simulations of the ergodic Hamilton-Jacobi equation (4), see Figure 2. We computed the optimal control α * as a function of the position in the simplex. We observed a clear separation between two connected regions of the simplex (result not shown), corresponding to the extremal choices α = a or α = A. This gives an optimal vector field which drives optimal trajectories. Outcomes of the numerical simulations are consistent with the presumable stability of the best constant control against periodic perturbations. A three-dimensional example with an optimal limit cycle Proposition 9 rules out the existence of optimal limit cycles in dimension n = 2. Although the previous example of the three-dimensional growth-fragmention process did not exhibit limit cycles apparently, we were able to find another three-dimensional example by testing random choices of matrices with respect to the stability criterion ( 16

We assume as in the previous example that the control α takes values in [a, A]. The maximal Perron eigenvalue is obtained for α * ≈ 0.415. The stability criterion [START_REF] Coron | Optimization of an amplification protocol for misfolded proteins by using relaxed control[END_REF] has been checked numerically: the optimal constant control is not stable with respect to periodic perturbations at high frequency.

Therefore we expect limit cycles to attract optimal trajectories in the simplex in the spirit of the Poincaré-Bendixson theory. This has been checked using finite-volume numerical simulations of the ergodic Hamilton-Jacobi equation (4), see Figure 3. It is worth mentioning that a similar counterexample has been proposed in [START_REF] Fainshil | On the stability of positive linear switched systems under arbitrary switching laws[END_REF] to answer a question raised in [START_REF] Gurvits | On the stability of switched positive linear systems[END_REF]. Our quantitative approach based on secondorder conditions provides another example. Furthermore it illustrates the rich possible dynamics of optimal trajectories. The connexion with the Poincaré-Bendixson theory seems appealing. However, proving that limit cycles are the only alternative to pointwise convergence seems out of reach at the moment, due to the complexity of the Poincaré-Bendixson theory in the case of discontinuous vector fields [START_REF] Carvalho | On Poincaré-Bendixson theorem and non-trivial minimal sets in planar nonsmooth vector fields[END_REF].

Fig. 1 .

 1 Fig.1. Solution of the Hamilton-Jacobi equation[START_REF] Castilla | In vitro generation of infectious scrapie prions[END_REF] for the critical value λ =1 2 . The two branches (dotted lines, red and blue) correspond to the two possible choices α = a, resp. α = 1 -a. The bold line corresponds to the only possible combination of the two branches which gives a bounded viscosity solution.

  Fig.3. Illustration of the optimal limit cycle for the numerical example[START_REF] Gaubert | Dobrushin ergodicity coefficient for markov operators on cones, and beyond[END_REF]. We have plotted the curve of Perron eigenvectors (black line), the boundary of the ergodic set (blue line), and an arbitrary optimal trajectory (red line). The optimal trajectory enters the ergodic set and then converges towards a limit cycle. Right picture is a zoom of the left one. The numerical grid is plotted on the axes: the space step is ∆x = 1E-3 to fall much below the width of the limit cycle.


Muhammad Shahid

Camille Dumat
email: camille.dumat@ensat.fr

Jérôme Silvestre

Eric Pinelli

Effect of fulvic acids on lead-induced oxidative stress to metal sensitive Vicia faba L. plant

Keywords: Speciation, Pigment contents, Metal toxicity, Reactive oxygen species

is a ubiquitous environmental pollutant capable to induce various morphological, physiological, and biochemical functions in plants. Only few publications focus on the influence of Pb speciation both on its phytoavailability and phytotoxicity. Therefore, Pb toxicity (in terms of lipid peroxidation, hydrogen peroxide induction, and photosynthetic pigments contents) was studied in Vicia faba plants in relation with Pb uptake and speciation. V. faba seedlings were exposed to Pb supplied as Pb(NO 3 ) 2 or complexed by two fulvic acids (FAs), i.e. Suwannee River fulvic acid (SRFA) and Elliott Soil fulvic acid (ESFA), for 1, 12, and 24 h under controlled hydroponic conditions. For both FAs, Pb uptake and translocation by Vicia faba increased at low level (5 mg l -1 ), whereas decreased at high level of application (25 mg l -1 ). Despite the increased Pb uptake with FAs at low concentrations, there was no influence on the Pb toxicity to the plants. However, at high concentrations, FAs reduced Pb toxicity by reducing its uptake. These results highlighted the role of the dilution factor for FAs reactivity in relation with structure; SRFA was more effective than ESFA in reducing Pb uptake and alleviating Pb toxicity to V. faba due to comparatively strong binding affinity for the heavy metal.

Introduction

Lead is widely used in many industrial processes and can reach in soil [START_REF] Uzu | Foliar lead uptake by lettuce exposed to atmospheric fallouts[END_REF]Shahid et al. 2011a). This metal is known to interfere with morphological, physiological, and biochemical functioning of exposed plants and can induce a broad range of noxious effects [START_REF] Islam | Effect of Pb toxicity on leaf growth, physiology and ultrastructure in the two ecotypes of Elsholtzia argyi[END_REF]Shahid et al. 2011b) such as, decrease in seed germination, root elongation, and plant biomass and inhibition of chlorophyll biosynthesis (Shahid et al. 2011c). Inside the cell, Pb can affect respiration and enzyme reactions (Shahid et al. 2011c). The presence of Pb in plants can also induce the production of reactive oxygen species (ROS) such as hydrogen peroxide (H 2 O 2 ), which can disrupt the redox status of cells and cause lipid peroxidation [START_REF] Shahid | Review of Pb availability and toxicity to plants in relation with metal speciation; role of synthetic and natural organic ligands[END_REF]. According to Shahid et al. (2011c), Pb-induced toxicity to plant is closely related to its form in the soil solution.

Humic substances (HSs) influence the speciation, the bioavailability, and the acute toxicity of metals in soils and natural waters [START_REF] Dumat | Reduced adsorption of caesium on clay minerals caused by various humic substances[END_REF][START_REF] Staunton | Possible role of organic matter in radiocaesium adsorption in soils[END_REF][START_REF] Dumat | Study of the trace metal ion influence on the turnover of soil organic matter in cultivated contaminated soils[END_REF]). These substances are a complex mixture of partially "decomposed" and otherwise transformed organic materials and constitute a major sorbent phase for contaminants in soils [START_REF] Essington | Soil and water chemistry: an integrative approach, 1st edn[END_REF]). The properties of HSs are due to their physico-chemical characteristics such as molecular weight, molecular size, and substructures, and their functionality depends on the nature and source of organic matter and on their functional groups. The carboxylic and phenolic hydroxyl groups of HSs are the main binding sites for metals [START_REF] Zeng | Effect of dissolved humic substances on the photochemical degradation rate of 1-aminopyrene and atrazine[END_REF]. Most of the data on HSs refer to average properties and structure of a large assembly of components with diverse structure and molecular weight.

Over the last decades, more than 200 short-term bioassays utilizing plants, microorganisms, or insects have been developed and used to evaluate the environmental risk [START_REF] Debenest | A new cell primo-culture method for freshwater benthic diatom communities[END_REF]2009;Marcato-Romain et al. 2009a). Plant assays are highly sensitive, easy to be used in experiment, inexpensive, and good predictors of carcinogenicity [START_REF] Ennever | The ability of plant genotoxicity assays to predict carcinogenicity[END_REF]).

In the present study, the effect of Pb speciation on its uptake by Vicia faba and toxicity to the plant was assessed by measuring different stress biomarker parameters, such as lipid peroxidation, ROS, and pigment contents. These parameters were selected due to their high sensitivity and common use to evaluate plant toxicity to environmental pollutants, such as heavy metal [START_REF] Mishra | Lead detoxification by coontail (Ceratophyllum demersum L.) involves induction of phytochelatins and antioxidant system in response to its accumulation[END_REF]Cenkci et al. 2010;[START_REF] Singh | Lead bioaccumulation potential of an aquatic macrophyte Najas indica are related to antioxidant system[END_REF]. Lead was supplied as salt or complexed with FAs.

Materials and methods

Plant materials and growth conditions

The V. faba (common name, broad beans; cultivar, Primabel; type, aguadulce; family, Fabaceae) seedlings were cultured according to El [START_REF] Hajjouji | Assessment of the genotoxicity of olive mill waste water (OMWW) with the Vicia faba micronucleus test[END_REF]Marcato-Romain et al. (2009a). The V. faba seeds were germinated under optimal germination conditions, i.e., under the darkness at 22°C and 100% of moisture. When the primary roots were about 2-3-cm long, the seedlings were transplanted and cultured in modified Hoagland solution (Sigma) with the following macro elements: 5 mM KNO 3 , 5 mM Ca(NO 3 ) 2 , 2 mM KH 2 PO 4 , and 1.5 mM MgSO 4 ; the micro elements were: 9.11 μM MnSO 4 , 1.53 μM ZnSO 4 , 0.235 μM CuSO 4 , 24.05 μM H 3 BO 3 , 0.1 μM Na 2 MoO 4 , and 268 μM Fe/ ethylenediaminetetraacetic acid (EDTA). Nutrient solution was renewed on alternate days to keep the nutrient composition and pH constant. Plants were grown under controlled conditions in a phytotron with 16-h photoperiod at 70% relative humidity and day/night temperatures of 24/22°C. Light was supplied by 600 W Osram Nav-T super high pressure sodium lamp providing a minimum photosynthetical photon flux density of 500 μmol m -2 s -1 at the top of the plant [START_REF] Pourrut | Potential role of NADPH-oxidase in early steps of leadinduced oxidative burst in Vicia faba roots[END_REF].

Treatments

After preculturing of 15 days (fifth to sixth foliar stage), the plants were exposed to 5 μM Pb in the presence or absence of two FAs, i.e., Suwannee River fulvic acid (SRFA) and Elliott Soil fulvic acid (ESFA), from the International Humic Substances Society (IHSS, Colorado School of Mines, Golden, CO, USA), used as model of natural dissolved organic matter. The chemical properties of these FAs are given in Table 1. These substances were applied at two levels, i.e., 5 and 25 mg l -1 . The HSs were also applied alone as control at the highest level (25 mg l -1 ). The concentration of KH 2 PO 4 in the nutrient solution was reduced, also in the control, to 0.2 mM in order to prevent phosphate precipitation (Shahid et al. 2011c). All Pb treatments were applied for 1, 12, and 24 h. Since organic ligands have no effects on the studied parameters of V. faba up to 24 h (Shahid 2010), FAs alone was only applied for 24 h.

Calculation of Pb speciation in nutrient solution

The speciation of Pb in nutrient solution (free Pb 2+ ions and complexed by FAs, Table 2) was calculated using Windermere 1).

Lead content analysis

Before all experiments, Pb content was analyzed in the nutrient solution so as to determine the possible precipitation of Pb. All solutions (Table 2) were prepared from solutions of Pb [50 μM Pb as Pb(NO 3 ) 2 , Sigma] and FAs (100 mg l -1 SRFA and ESFA) in milli-Q water. The pH values of all solutions were adjusted to 5±0.1 using distilled HNO 3 (15 M, suprapur 99.9%, Sigma). Inductively coupled plasma-atomic emission spectrometry (ICP-AES, Jobin Yvon) with an IRIS Intrepid II XDL/ Thermo Electron Corporation was used to determine Pb contents after filtration (0.22 μm). Both the pH and total Pb concentration remained constant in all treatments after 24 h (Table 2). Lead cell uptake by V. faba seedlings was determined according to [START_REF] Pourrut | Potential role of NADPH-oxidase in early steps of leadinduced oxidative burst in Vicia faba roots[END_REF]. After harvest, the seedlings were separated into root stem and leaves. Lead bound to the rhizoderm was removed by HCl as described by [START_REF] Uzu | Study of lead phytoavailability for atmospheric industrial micronic and sub-micronic particles in relation with lead speciation[END_REF]. Plants tissues were dried at 80°C for 48 h and digested by a 1:1 mixture of 65% HNO 3 (Sigma) and 30% H 2 O 2 (Sigma) at 80°C over 6 h using DigiPrep Jr (SCP Sciences). After dilution with milli-Q water, the metal contents were analyzed using ICP-AES. Virginia tobacco leaves (CTA-VTL-2, polish certified reference material; ICHTJ) were used as a reference material for verifying the accuracy of the analytical procedure. Certified values for Pb in tobacco leaves were 22.1±1.2 mg Pb kg -1 dry weights. Measured values of the three replicates were 22.1±0.9, 22.3±1.0, and 22.0±0.7 mg Pb kg -1 dry weight.

Determination of lipid peroxidation

In order to evaluate the Pb-induced oxidative damages on cell membrane, thiobarbituric acid reactive substances (TBARS, Sigma) were determined as reported by [START_REF] Hodges | Improving the thiobarbituric acid-reactive-substances assay for estimating lipid peroxidation in plant tissues containing anthocyanin and other interfering compounds[END_REF]. Plant samples were homogenized in hydroalcoholic solution (80/20: v/v, Sigma) under liquid nitrogen at 4°C under darkness followed by incubation at 95°C with thiobarbituric acid. After centrifugation (3,000g for 10 min), the absorbance of the supernatant was recorded at 532 nm. The value for nonspecific absorption at 600 nm was subtracted. The amount of TBARS was calculated using the extinction coefficient (155 mM -1 cm -1 ) and the equations reported by [START_REF] Hodges | Improving the thiobarbituric acid-reactive-substances assay for estimating lipid peroxidation in plant tissues containing anthocyanin and other interfering compounds[END_REF].

Evaluation of H 2 O 2 contents

The H 2 O 2 content was measured according to [START_REF] Islam | Effect of Pb toxicity on leaf growth, physiology and ultrastructure in the two ecotypes of Elsholtzia argyi[END_REF]. Each sample (500 mg) was homogenized with 5 mL 0.1% (w/v) trichloroacetic acid (Sigma) under liquid nitrogen followed by centrifugation at 12,000g for 20 min. The mixture assay contained 0.5 mL of the supernatant added to 0.5 mL of 10-mM potassium phosphate buffer (pH 7.0, Sigma) and 1 mL of 1 M KI (Sigma). Absorbance was determined at 390 nm and the content of H 2 O 2 was calculated using a standard curve under the same conditions.

Pigment content assay

The frozen leaves samples (600 mg) of V. faba were ground with liquid nitrogen. Pigments were extracted by incubating leaves in 10-mL acetone 80% (v/v, Sigma) for 24 h at 4°C under darkness. After centrifugation at 1,000g for 10 min, (1987).

Statistical analysis

The variables were tested for differences between treatments using an analysis of variance (one-way) followed by LSD Fisher's test. This statistical analysis was performed using the software Statistica, edition'98 (StatSoft Inc., Tulsa, OK, USA). In each bioassay, the mean values are average of 12 replicates and one separate plant was used for each replicate. These 12 replicates were run in two separate experiments (six replicates in each experiment). An asterisk or a solid dot indicates significant differences at P<0.05 as measured by LSD Fisher's test.

Results and discussions

Effect of FAs on Pb uptake by V. faba

Table 3 presents Pb contents in V. faba roots, stem, and leaves in function of time. V. faba roots were exposed to Pb in the presence or absence of FAs. In roots, without FAs, there was a rapid uptake and accumulation of 27±8, 59±8, and 95±9 mg g -1 of Pb after 1, 12, and 24 h, respectively. In contrast to Pb uptake, its translocation towards shoot tissues was very slow and only 3.6±1.1 and 3.2±0.5 mg g -1 of Pb were transferred, respectively, into stem and leaves after 24 h. This limited translocation of Pb from roots to shoot tissues has been already reported (Shahid et al. 2011b). Indeed, in plant roots, Pb precipitates as insoluble salts or is immobilized by molecule such as sugar, pectins, celluloses, and hemicelluloses (Shahid et al. 2011b). This metal sequestration in roots is not common to all heavy metals but its intensity is very specific to Pb. Several studies reported that more than 90% of absorbed Pb is stored in the plant roots [START_REF] Uzu | Study of lead phytoavailability for atmospheric industrial micronic and sub-micronic particles in relation with lead speciation[END_REF]Shahid et al. 2011b). In our case, 93% of absorbed Pb was sequestered in V. faba roots. Application of both FAs at low levels (Pb-SRFA-5 and Pb-ESFA-5) increased Pb uptake and Pb translocation to shoot tissues. The Pb accumulation in roots, stem, and leaves increased, respectively, by 16%, 42%, and 22% for Pb-ESFA-5 and by 12, 25% and 13% for Pb-SRFA-5 after 24 h. In contrast, at higher levels of FAs (Pb-SRFA-25 and Pb-ESFA-25), a significant (P<0.05) decrease in Pb uptake and translocation was observed except for Pb-ESFA-25 (Table 3). The decrease in Pb accumulation by Pb-ESFA-25 in roots, stem, and leaves was, respectively, 22%, 22%, and 19% after 24 h. In the case of Pb-SRFA-25, this decrease was 35%, 28%, and 25%, respectively, in roots, stem, and leaves after 24 h.

Several previous studies described the interaction between Pb and HSs and their results are contradictory. Some studies observed a significant increase in Pb uptake under HSs influence [START_REF] Halim | Potential availability of heavy metals to phytoextraction from contaminated soils induced by exogenous humic substances[END_REF][START_REF] Karaca | Effects of a humic deposit (gyttja) on soil chemical and microbiological properties and heavy metal availability[END_REF][START_REF] Salati | Fresh organic matter of municipal solid waste enhances phytoextraction of heavy metals from contaminated soil[END_REF], whereas others observed a decrease [START_REF] Worms | Uptake of Cd(II) and Pb(II) by microalgae in presence of colloidal organic matter from wastewater treatment plant effluents[END_REF]). In our experiment, the contrasting effects of FAs towards Pb uptake, at low and high concentrations could be explained by FA reactivity due to its structure [START_REF] Dumat | Adsorption of cesium by synthetic clay-organic matter complexes: effect of the nature of organic polymers[END_REF][START_REF] Staunton | Possible role of organic matter in radiocaesium adsorption in soils[END_REF][START_REF] Kleber | Advances in understanding the molecular structure of soil organic matter: implications for interactions in the environment[END_REF]. When added at high concentration, FA adopts a ball structure, which can strongly complex Pb and reduce its plant uptake. Under diluted conditions, FA adopts a more hydrophilic fibrous structure that forms soluble and mobile complexes with Pb, which can enter the plants more easily due to small size and this increases Pb plant uptake [START_REF] Evangelou | The influence of humic acids on the phytoextraction of cadmium from soil[END_REF][START_REF] El-Ghamry | Amino and humic acids promote growth, yield and disease resistance of faba bean cultivated in clayey soil[END_REF].

The other possible explanation of the reduced Pb uptake at high level of FAs could be the competition between Pb and other nutrient ions. For example, [START_REF] Nardi | Physiological effects of humic substances on higher plants[END_REF] reported that FAs increased plant growth by positively influencing the uptake of nutrients. This increased uptake of nutrients at high level of FAs might have competed with Pb uptake. This can explain the Pb reduced uptake during the first 24 h.

The kinetic of Pb uptake by V. faba roots and translocation to aerial parts is shown in Fig. 1. The rate of Pb uptake was maximum during first hour for root, whereas the maximum rate of translocation towards stem and leaves occurred between 2-12 h for all the treatments. This showed that Pb uptake by V. faba roots and translocation to shoots tissues was not linear during the first 24 h of incubation but occurred in phases. The application of fulvic acids increased the rate of Pb uptake and translocation at low level and decreased it at high level of application. However, the trends/phases of Pb uptake and translocation remained the same for all treatments. Data regarding short-term metal uptake kinetic are scarce however, [START_REF] Nedelkoska | Characteristics of heavy metal uptake by plant species with potential for phytoremediation and phytomining[END_REF] and [START_REF] Sgherri | Early production of activated oxygen species in root apoplast of wheat following copper excess[END_REF] also showed a triphasic kinetic profile for cadmium, nickel, and copper uptake during 24 h.

Moreover, the precise properties of a given HS sample depend on their source and the specific conditions of extraction. The different behavior of applied FAs with respect to Pb accumulation and kinetic of uptake can be due to their different sources since SRFA was extracted from water and ESFA from soil.

Effect of organic ligands on lipid peroxidation and H 2 O 2 induction by Pb

The effect of each FA alone was investigated on plant physiology. The high concentrations of both FAs (25 mg l -1 ), applied alone without Pb for 24 h, did not affect lipid peroxidation and H 2 O 2 production compared to the control (Figs. 2 and3).

The effect of FAs on Pb-induced generation of H 2 O 2 and lipid peroxidation is presented in Figs. 2 and3. Application of Pb alone caused overproduction of H 2 O 2 which resulted into lipid peroxidation in both roots and leaves. In roots, Pbinduced H 2 O 2 production and lipid peroxidation started immediately after Pb exposure at 1 h and continued up to 12 h. The increase in H 2 O 2 and TBARS contents by Pb was, respectively, 64% and 55% at 1 h and 39% and 52% at 12 h. In Values are means of 12 replicates. An asterisk indicates significant differences at P<0.05 for Pb-FA treatments compared to Pb alone whereas a solid dot indicates significant differences at P<0.05 for Pb and FAs alone compared to the control leaves, the Pb-induced increase in H 2 O 2 and TBARS contents started after 12 h but the effect was only significant (P<0.05) at 24 h (34% and 36%, respectively, for H 2 O 2 and TBARS compared to the control). The presence of Pb across the rootcell membrane, even in small amounts, is known to induce oxidative stress through overproduction of ROS [START_REF] Islam | Effect of Pb toxicity on leaf growth, physiology and ultrastructure in the two ecotypes of Elsholtzia argyi[END_REF][START_REF] Singh | Lead bioaccumulation potential of an aquatic macrophyte Najas indica are related to antioxidant system[END_REF]. This overproduction of ROS is one of the earliest responses of plant cells to heavy metal toxicity, and it is due to the imbalance between the generation and the neutralization of ROS by antioxidant mechanisms. These free radicals react (oxidize) with various cell components including DNA, proteins, and lipids/fatty acids and cause DNA damage, mitochondrial malfunction, and cell membrane damage [START_REF] Aarti | Effects of oxidative stress on chlorophyll biosynthesis in cucumber (Cucumis sativus) cotyledons[END_REF][START_REF] Mishra | Lead detoxification by coontail (Ceratophyllum demersum L.) involves induction of phytochelatins and antioxidant system in response to its accumulation[END_REF][START_REF] Pourrut | Potential role of NADPH-oxidase in early steps of leadinduced oxidative burst in Vicia faba roots[END_REF].

Lead ions are known to induce lipid peroxidation, decrease the level of saturated fatty acids, and increase the content of unsaturated fatty acids of membranes in several plant species [START_REF] Singh | Lead bioaccumulation potential of an aquatic macrophyte Najas indica are related to antioxidant system[END_REF].

In roots, Pb-ESFA-5 prolonged the Pb induction of H 2 O 2 to 24 h without any effect on lipid peroxidation (Figs. 2 and3). Application of Pb-SRFA-5 did not affect H 2 O 2 production but delayed the start of lipid peroxidation from 1 to 12 h which then continued up to 24 h. Lead-induced H 2 O 2 production and lipid peroxidation were reduced significantly (P<0.05) at 1 h by Pb-ESFA-25 and at 1 and 12 h by Pb-SRFA-25. In contrast, at 24 h, Pb-SRFA-25 increased H 2 O 2 contents and TBARS values. In leaves, addition of FAs in the presence of Pb at both levels of application did not sufficiently modify Pb-induced H 2 O 2 production and lipid peroxidation, except for Pb-SRFA-25, which significantly (P<0.05) reduced H 2 O 2 production induced by Pb at 24 h. This is the first time that the influence of FAs was studied on H 2 O 2 induction and lipid peroxidation. Marcato-Romain et al. (2009b) suggested a protecting effect of soil organic matter on Pb-induced genotoxicity to V. faba. On the basis of our results (Tables 2 and3), it is assumed that Pb-induced toxicity correlates with the concentration of free Pb 2+ ions in plants. Recently, Shahid et al. (2011c) calculated a linear relationship between free Pb 2+ ions concentration in nutrient solution and genotoxicity in the presence of EDTA. Application of FAs at high levels complexed Pb 2+ ions in the nutrient solution and reduced its uptake and consequently alleviated or delayed the toxicity. [START_REF] Pourrut | Potential role of NADPH-oxidase in early steps of leadinduced oxidative burst in Vicia faba roots[END_REF] previously demonstrated that Pb-induced toxicity is antagonized by the Ca 2+ ions due to the great inhibition of Pb entry into the roots, suggesting an important role of Pb level in the oxidative stress induction in plants.

The other possible hypothesis of reduced/delayed Pb toxicity when FAs were added at high level can be the minimum threshold level of Pb 2+ ions required inside the plant to induce the production of H 2 O 2 and lipid peroxidation. Moreover, the plant detoxification mechanism is proposed to be more efficient at lower than higher plant Pb level. [START_REF] Mishra | Lead detoxification by coontail (Ceratophyllum demersum L.) involves induction of phytochelatins and antioxidant system in response to its accumulation[END_REF] reported increased levels of protein accumulation involved in the maintenance of cell redox status or in the sequestration of the metal like phytochelatins under low Pb levels. This decreased uptake of Pb with the more efficient detoxification mechanism could be due to the reduced or delayed Pb-induced phytotoxic effects in the presence of higher than lower levels of FAs. At low level of FAs application, despite the increased Pb uptake, the Pb-induced toxicity remained slightly lower than that due to Pb alone. Under diluted conditions, a portion of Pb could be taken up by plants as Pb-FA complex due to fiber FA structure [START_REF] Kleber | Advances in understanding the molecular structure of soil organic matter: implications for interactions in the environment[END_REF]. Inside the plants, this complex either remains stable, and in this way slightly reduces Pb toxicity, or dissociates after some times, and thus, delays the toxicity. Shahid et al. (2011c) concluded that only stable organic-metal complexes could decrease Pb genotoxicity.

Effect of organic ligands on chlorophyll contents

The application of FAs alone has no effect on pigment contents compared to the control except for carotenoid contents, which decreased significantly (P<0.05) by 9% in the presence of Suwana river FA (Fig. 4). Humic substances are, generally, known to increase plant tolerance against metal stress and enhance pigment contents and plant growth [START_REF] Piccolo | Structural characteristics of humic substances as related to nitrate uptake and growth regulation in plant systems[END_REF][START_REF] El-Ghamry | Amino and humic acids promote growth, yield and disease resistance of faba bean cultivated in clayey soil[END_REF]. However, in the present study, the HSs did not enhance V. faba pigment contents, which can be due to the short duration of the experiment.

The pigment composition remained unaffected after 1 h in all the treatments compared to the control (Fig. 4). This is due to slow Pb translocation which takes more than 1 h to reach the threshold level in leaves as indicated by Pb uptake kinetic (Fig. 1). The application of Pb alone reduced pigment contents at 12 and 24 h, but the effect was only significant (P<0.05) after 24 h, except for carotenoids content where the effect was also significant (P<0.05) at 12 h. Lead is known to stimulate the degradation of pigments [START_REF] Islam | Effect of Pb toxicity on leaf growth, physiology and ultrastructure in the two ecotypes of Elsholtzia argyi[END_REF]Shahid et al. 2011b).

Lead-induced degradation of pigment has been attributed to direct or indirect factors, such as distortion of chloroplast ultrastructure, inhibition of the synthesis of photosynthetic pigments and enzymes of Calvin cycle [START_REF] Mishra | Lead detoxification by coontail (Ceratophyllum demersum L.) involves induction of phytochelatins and antioxidant system in response to its accumulation[END_REF], impaired uptake of essential elements like Mn and Fe, damage of the photosynthetic apparatus or due to chlorophyll degradation by increased chlorophyllous activity (Shahid et al. 2011b). Recently, Cenkci et al. (2010) reported dose-dependent negative correlation between Pb concentration and pigment contents in Brassica rapa exposed to 0.5-5 mM Pb for 6 days. However, in our case, the rapidity (after 12 h) of the observed effect on the pigment content compared to other studies suggests the involvement of fast mechanisms, such as production of ROS [START_REF] Aarti | Effects of oxidative stress on chlorophyll biosynthesis in cucumber (Cucumis sativus) cotyledons[END_REF][START_REF] Singh | Lead bioaccumulation potential of an aquatic macrophyte Najas indica are related to antioxidant system[END_REF].

The low-level application of both FAs (5 mg l -1 ) on the pigment contents depended on the FA type. The effect of Pb-ESFA-5 was similar to that of Pb alone. In contrast, application of Pb-SRFA-5 reduced significantly (P<0.05) Pb-induced Chl-b degradation at 24 h. The application of FAs at high levels (25 mg l -1 ) inhibited the Pb-induced toxicity to pigment contents with Pb-SRFA-25 being more effective than Pb-ESFA-25. Addition of Pb-ESFA-25 significantly (P<0.05) reduced the degradation of Chlb induced by Pb. In the case of Chl-a and carotenoids contents, the values were always higher than those with Pb alone. Application of Pb-SRFA-25 alleviated the Pbinduced reduction of pigment contents throughout the experiment. The values of pigments remained close to control values in this treatment.

The results show that only high levels of FAs can effectively bind and reduce Pb toxicity to chlorophyll contents. This effect was due to the reduced Pb concentration in roots and leaves in the presence of high levels of FAs. [START_REF] Ruley | Effects of lead and chelators on growth, photosynthetic activity and Pb uptake in Sesbania drummondii grown in soil[END_REF] also reported similar effect for LMWOAs on Pbinduced degradation of chlorophyll contents in Sesbania drummondii. They stated that due to low binding affinity of citric acid, only high level of this acid can efficiently bind and reduce Pb 2+ toxicity. The mechanism of protection provided by HSs against Pb toxicity can be similar to that of the clay minerals, the adsorption of free Pb 2+ ions to fulvic acid, and can limit the uptake and toxicity of Pb to living organisms [START_REF] Babich | Abiotic factors affecting the toxicity of lead to fungi[END_REF]. [START_REF] Kruatrachue | Combined effects of lead and humic acid on growth and lead uptake of Duckweed, Lemna minor[END_REF] reported that the application of HSs inhibited the Pb-induced degradation of total chlorophyll contents and growth rate of Lemna minor.

Conclusions and perspectives

The present study highlights the important role of natural FAs to the fate of Pb in the soil-plant systems. The FAs are capable to alleviate Pb toxicity to plants by complexing highly toxic free Pb 2+ in solution and reducing Pb uptake. This protecting role of FAs against Pb uptake and toxicity depends on the applied concentration: only the high concentrations (25 mg l -1 ) FAs could effectively bind and reduce metal toxicity. Therefore, the behavior of pollutants in terms of bioavailability, uptake, and toxicity in soil can vary with concentration of HSs. Our results also proposed that Pb toxicity depends on metal binding capacity of FA. Due to the high binding constant, the SRFA bound more strongly to Pb compared ESFA and was more effective against Pb toxicity. Therefore, understanding metal speciation along with the type and level of complexing agents in the natural systems are important for the risk assessment and/or soil or water quality criteria.

Fig. 1 Fig. 2

 12 Fig. 1 Effect of SRFA and ESFA on rate of Pb uptake and translocation (in micrograms per gram per hour dry weight) in V. faba roots, stem, and leaves. Values are means of 12 replicates

Fig. 3

 3 Fig.3Effect of SRFA and ESFA on Pb-induced production of lipid peroxidation (TBARS in nanomoles per gram fresh weight) in V. faba roots and leaves. Values are means of 12 replicates. An asterisk indicates significant differences at P<0.05 for Pb-FA treatments compared to Pb alone whereas a solid dot indicates significant differences at P<0.05 for Pb and FAs alone compared to the control

Fig. 4

 4 Fig.4Effect of SRFA and ESFA on Pb-induced reduction of pigment contents (in micrograms per grams fresh weight) in V. faba leaves. Values are means of 12 replicates. An asterisk indicates significant differences at P<0.05 for Pb-FA treatments compared to Pb alone whereas a solid dot indicates significant differences at P<0.05 for Pb and FAs alone compared to the control

Table 1

 1 Properties of the two standard FAs from International Humic Substance Society

	Parameters	SRFA	ESFA
	Cat. No	2S101F	2S102F
	C (% w/w)	52.34	40.12
	H (% w/w)	4.36	4.28
	O (% w/w)	42.98	42.61
	N (% w/w)	0.67	3.75
	S (% w/w)	0.46	0.89
	P (% w/w)	0.004	0.12
	Carboxyl (mEq/g C)	11.17	13.24
	Phenolic (mEq/g C)	2.84	2.27
	Log K1 values are 3.76 and 3.67, and log K2 values are 9.84 and 9.53
	for SRFA and ESFA, respectively

Table 2

 2

	Experimental design
	and speciation of Pb in nutrient	Treatments	Composition	Pb-chelated	Pb-free	Measured Pb
	solution calculated by the			(%)	(%)	(μM)
	WHAM VI
		Control	Hoagland solution (HS)	-	-	0
		ESFA-25	HS + 25 mg l -1 ESFA	-	-	-
		SRFA-25	HS + 25 mg l -1 SRFA	-	-	-
		Pb	HS + 5 μM Pb	0	84	4.99±0.04
	Measured Pb (last column) indi-cates the Pb concentration (in micromolars) determined by	Pb-ESFA-5 Pb-ESFA-25 Pb-SRFA-5	HS + 5 μM Pb + 5 mg l -1 ESFA HS + 5 μM Pb + 25 mg l -1 ESFA HS + 5 μM Pb + 5 mg l -1 SRFA	16 36 12	70 57 74	5.01±0.03 5.01±0.02 4.98±0.03
	ICP-AES in the nutrient solution after 24 h without plants	Pb-SRFA-5	HS + 5 μM Pb + 5 mg l -1 SRFA	12	74	4.98±0.03

Table 3

 3 Effect of SRFA and ESFA on Pb concentration (in micrograms per gram dry weight) in V. faba roots, stem, and leaves

	Treatments	Time	Roots	Stem	Leaves
		(h)	Mean ± SD	Mean ± SD	Mean ± SD
	Pb	1	27±8	0.14±0.01	0.03±0.01
	Pb-SRFA-5		31±4	0.14±0.05	0.03±0.01
	Pb-SRFA-25		27±2	0.11±0.02	0.02±0.01
	Pb-ESFA-5		29±8	0.11±0.04	0.04±0.03
	Pb-ESFA-25		28±2	0.14±0.03	0.03±0.01
	Pb	12	59±8	2.9±0.7	2.2±0.4
	Pb-SRFA-5		60±13	3.1±0.4	2.3±0.3
	Pb-SRFA-25		47±12	2.4±0.6	2±0.3
	Pb-ESFA-5		69±5	3.5±0.5	2.7±0.5
	Pb-ESFA-25		50±6	2.5±0.3	2±0.5
	Pb	24	95±9	3.6±1.1	3.2±0.5
	Pb-SRFA-5		106±13	4.5±0.8	3.6±0.6
	Pb-SRFA-25		62±19*	2.6±0.6*	2.4±0.2*
	Pb-ESFA-5		110±15	5.1±0.3	3.9±1.1
	Pb-ESFA-25		74±8*	2.8±0.4	2.6±0.3
	Values are means of 12 replicates
	*P<0.05, significant

Acknowledgments The authors would like to thank the Higher Education Commission of Pakistan (www.hec.gov.pk) and the French Society for Export of Educative Resources (SFERE, http://www.sfere.fr/) for the scholarship granted to M. Shahid. The experimental work was funded by the "French Ministry of Higher Education and Research" (MESR, enseignementsup-recherche.gouv.fr) and a national project EC2CO-CYTRIX. We thank these organizations for their support which has enabled the realization of this work.