Ying Hu
email: ying.hu@univ-rennes1.fr.

Zhongmin Qian
email: qianz@maths.ox.ac.uk.

BMO Martingales and Positive Solutions of Heat Equations

In this paper, we develop a new approach to establish gradient estimates for positive solutions to the heat equation of elliptic or subelliptic operators on Euclidean spaces or on Riemannian manifolds. More precisely, we give some estimates of the gradient of logarithm of a positive solution via the uniform bound of the logarithm of the solution. Moreover, we give a generalized version of Li-Yau's estimate. Our proof is based on the link between PDE and quadratic BSDE. Our method might be useful to study some (nonlinear) PDEs.

Introduction

In this article, we study positive solutions u of a linear parabolic equation

L - ∂ ∂t u = 0 in (0, ∞) × M , (1.1)
where M is either the Euclidean space R n and L is an elliptic or sub-elliptic operator of secondorder L = 1 2 m α=1 A 2 α + A 0 , {A 0 , • • • , A m } is a family of vector fields on R n , or M is a complete manifold of dimension n with Riemannian metric (g ij ), and 2L is the Laplace-Beltrami operator

∆ = 1 √ g n i,j=1 ∂ ∂x i √ gg ij ∂ ∂x j ,
where g denotes the determinate of (g ij ) and (g ij ) is the inverse of the matrix (g ij ).

The well-posedness and the regularity theory for (1.1) are parts of the classical theory in partial differential equations, see [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF], [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF] and [START_REF] Lieberman | Second order parabolic differential equations[END_REF] for details. On the other hand, it remains an interesting question to devise precise estimates of a solution u in terms of the (geometric) structures of (1.1). There is already a large number of papers devoted to this question. Among many interesting results, let us cite two of them which are most relevant to the present paper. The first result is a classical result under the name of semigroup domination, first discovered by Donnelly and Li [START_REF] Donnelly | Lower bounds for the eigenvalues of Riemannian manifolds[END_REF], which says that if the Ricci curvature is bounded from below by C, then

|∇P t u 0 | ≤ e -Ct P t |∇u 0 |, for u 0 ∈ C 1 b (M ), (1.2)
where (P t ) t≥0 is the heat semigroup on M , so that the left-hand side is the norm of the gradient of u(t, •) = P t u 0 a solution to the heat equation

1 2 ∆ - ∂ ∂t u = 0 in (0, ∞) × M (1.3)
with initial data u(0, •) = u 0 , while the right-hand side P t |∇u 0 | is a solution of (1.3) with initial data |∇u 0 |. The second result is Li-Yau's estimate first established in [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF]. If the Ricci curvature is non-negative, and if u is a positive solution of (1.3) then

|∇ log u| 2 -2 ∂ ∂t log u ≤ n t for t > 0 .
(1.4)

In fact, in the same paper [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF], Li and Yau also obtained a gradient estimate for positive solutions in terms of the dimension and a lower bound (which may be negative) of the Ricci curvature, though less precise. Their estimates in negative case have been improved over the years, see for example [START_REF] Yau | On the Harnack inequalities of partial differential equations[END_REF], [START_REF] Yau | Harnack inequality for non-self-adjoint evolution equations[END_REF], [START_REF] Bakry | Harnack inequalities on a manifold with positive or negative Ricci curvature[END_REF] and [START_REF] Bakry | A logarithmic Sobolev form of the Li-Yau parabolic inequality[END_REF].

In this paper we prove several gradient estimates for the positive solutions of (1.1). Let us first mention some simple ones for illustration.

Theorem 1.1 Let M be a complete manifold with non-negative Ricci curvature. Suppose u is a positive solution of (1.3) with initial data u 0 > 0, then

|∇ log u(t, •)| 2 ≤ 4 t || log u 0 || ∞ , for t > 0, (1.5)
where || • || ∞ denotes the L ∞ norm on M .

Remark 1.2 As pointed out by the referee, (1.5) can be derived from the reverse logarithmic Sobolev inequality due to Bakry and Ledoux [START_REF] Bakry | A logarithmic Sobolev form of the Li-Yau parabolic inequality[END_REF]. In fact, from the reverse logarithmic Sobolev inequality, tP t (u 0 )(x)| log P t (u 0 )| 2 (x) ≤ 2[P t (u 0 log u 0 )(x) -P t (u 0 )(x) log P t (u 0 )(x)],

from which we derive (1.5). However, our method is useful to study estimates for other (nonlinear) PDEs with subelliptic operators, see Theorems 3.8 and 3.9 in Section 3.

Remark 1.3 Theorem 1.1 is very closed to Harnack estimate for the heat equation which is dimension free, see R. Hamilton [START_REF] Hamilton | A matrix Harnack estimate for the heat equation[END_REF]. The relation between the Bakry-Ledoux reverse logarithmic Sobolev inequality and a slight improvement of Hamilton's Harnack inequality was discussed in a very interesting paper by X. D. Li [START_REF] Li | Hamilton's Harnack inequality and the W-entropy formula on complete Riemannian manifolds[END_REF].

Indeed we will establish a similar estimate for the heat equation with a sub-elliptic operator, under similar curvature conditions, and indeed we will establish a gradient estimate for a complete manifold whose Ricci curvature is bounded from below.

Theorem 1.4 Let M be a complete manifold of dimension n with non-negative Ricci curvature. Suppose u is non-negative solution to the heat equation of (1.3)

with initial data u 0 > 0. If C ∈ [0, ∞] such that -∆ log u 0 ≤ C, then |∇ log u| 2 -2 ∂ ∂t log u ≤ C t n C + 1 for t ≥ 0.
By setting C = ∞ we recover Li-Yau's estimate (1.4).

The novelty of the present paper is not so much about the gradient estimates in Theorem 1.1 and Theorem 1.4, what is interesting of the present work is the approach we are going to develop in order to discover and prove these gradient estimates. Our approach brings together with the martingale analysis to the study of a class of non-linear PDEs with quadratic growth. Of course the connection between the harmonic analysis, potential theory and martingales is not new, which indeed has a long tradition, standard books may be mentioned in this aspect, such as [START_REF] Doob | Stochastic processes[END_REF], [START_REF] Doob | Classical potential theory and its probabilistic counterpart[END_REF], [START_REF] Durrett | Brownian motion and martingales in analysis[END_REF] and etc., what is new in our study is an interesting connection between the BMO martingales and positive solutions of the heat equation (1.3).

To take into account of the positivity, it is better to consider the Hopf transformation of a positive solution u to (1.3), i.e. f = log u, then f itself solves a parabolic equation with quadratic non-linear term, namely

1 2 ∆ - ∂ ∂t f = - 1 2 |∇f | 2 in [0, ∞) × M . (1.6)
The preceding equation (1.6) is an archetypical example of a kind of semi-linear parabolic equations with quadratic growth which has attracted much attention recently associated with backward stochastic differential equations, for example Kobylanski [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF], Briand-Hu [START_REF] Briand | BSDE with quadratic growth and unbounded terminal value[END_REF], Delbaen et al. [START_REF] Delbaen | Backward SDEs with superquadratic growth[END_REF] and etc.

The main idea may be described as the following. Suppose f is a smooth solution of the nonlinear equation (1.6), and X t = B t + x where B is a standard Brownian motion on a complete probability space. Let

Y t = f (T -t, X t ) and Z t = (Z i t ) where Z i t = ∇ i f (T -t, X t
), ∇ i is the covariant derivative written in a local orthonormal coordinate system. Then, Itô's lemma applying to f and X may be written as

Y T -Y t = n i=1 T t Z i s dB i s - 1 2 T t |Z s | 2 ds. (1.7)
On the other hand, it was a remarkable discovery by Bismut [START_REF] Bismut | Théorie probabiliste du contrôle des diffusions[END_REF] (for a special linear case) and Pardoux-Peng [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF] that given the terminal random variable Y T ∈ L 2 (Ω, F T , P), there is actually a unique pair (Y, Z) where Y is a continuous semimartingale and Z is a predictable process which satisfies (1.7). The actual knowledge that Z is the gradient of Y may be restored if Y T = f 0 (X T ). The backward stochastic differential equation (1.7) with a bounded random terminal Y T , which has a non-linear term of quadratic growth and thus is not covered by Pardoux-Peng [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF], was resolved by Kobylanski [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF]. Observe that the martingale part of Y is the Itô integral of Z against Brownian motion B (which is denoted by Z.B). It can be shown that, if Y is bounded, then Z.B is a BMO martingale up to time T , so that the exponential martingale

E(h(Z).B) t = exp n i=1 t 0 h i (Z s )dB i s - 1
E Q T t |Z s | 2 ds F t ≤ 4||Y || ∞ .
Finally the sub-martingale property of |Z t | 2 allows to move |Z s | 2 (for s ∈ (t, T )) out from the time integral on the left-hand side of the previous inequality, which in turn yields the gradient estimate.

Let us now give a heuristic probabilistic proof to Theorem 1.4 to explain from where such estimates come from. Let f = log u, and G = -∆f . Then one can show that

G = |∇f | 2 -2f t
(where f t stands for the time derivative ∂ ∂t f for simplicity). Moreover, G satisfies

(L - ∂ ∂t )G = 1 n G 2 + H, where H = (|∇∇f | 2 - 1 n G 2 ) + 2Ric(∇f, ∇f ),
and H ≥ 0. We suppose here G > 0. Consider the BSDE:

dY t = Z t dB t + 1 n Y 2 t dt, Y T = G(0, x + B T ). Then Y t ≥ G(T -t, x + B t ).
Setting

U t = 1 Y t , V t = - Z t Y 2 t ,
then (U, V ) satisfies the following quadratic BSDE:

dU t = - 1 n dt + V t dB t + |V t | 2 U t dt.
Using BMO martingale techniques, one can prove that there exists a new probability measure Q under which Bt = B t + t 0

Vs

Us ds is a Brownian motion. Hence

dU t = - 1 n dt + V t d Bt ,
from which we deduce that

U 0 = T n + E Q [U T ],
and

Y 0 = 1 T n + E Q 1 Y T
which yields the estimate in Theorem 1.4.

Even though the above heuristic proof is probabilistic (which can be made rigorous), we prefer to give a pure analytic proof in the last section.

The paper is organized as follows. Next section is devoted to some basic facts about quadratic BSDEs including BMO martingales. Section 3 establishes the gradient estimates for some linear parabolic PDEs on Euclidean space, while Section 4 establishes these estimates on complete manifold. Last section is devoted to establish a generalized Li-Yau estimate via analytic tool.

BSDE and BMO martingales

Let us begin with an interesting result about BSDEs with quadratic growth. The kind of BSDEs we will deal with in this paper has the following form

dY = m j=1 Z j F j (Y, Z)dt + m j=1 Z j dB j , Y T = ξ, (2.1)
with terminal value ξ ∈ L ∞ (Ω, F T , P) which is given, where B = (B 1 , • • • , B m ) is a standard Brownian motion, (F t ) t≥0
is the Brownian filtration associated with B, and F j are continuous function on R × R m with at most linear growth: there is a constant

C 1 ≥ 0 such that |F (y, z)| ≤ C 1 (1 + |y| + |z|) ∀(y, z) ∈ R × R m .
According to Peng [START_REF] Peng | Probabilistic interpretation for systems of quasilinear parabolic partial differential equations[END_REF] and as we have seen in the Introduction, if u is a bounded smooth solution to the following non-linear parabolic equation

∂ ∂t u + d j=1 F j (u, ∇u) ∂u ∂x j = 1 2 ∆u in [0, ∞) × R m , (2.2)
with initial data u 0 , then

Y t = u(T -t, B t + •) and Z t = ∇u(T -t, B t + •) is a solution pair of (2.1) with terminal value Y T = u 0 (B T + •).
The special feature of (2.2) is that the maximum principle applies, which implies that global solutions (here global means for large t) exist for the initial value problem of the system as long as the initial data is bounded (though, this constraint can be relaxed a bit, but for the simplicity we content ourself to the bounded initial data problem). The maximum principle implies that as long as u is a solution to (2.2) then |u(x, t)| ≤ ||u 0 || ∞ . Therefore, if the initial data u 0 is bounded, and F j are global Lipschitz, then, according to Theorem 6.1 on page 592, [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF], u exists for all time, and both u and ∇u are bounded on

R m × [0, T ].
The maximum principle for (2.1) however remains true even for a bounded random terminal value (so called non Markovian case), which in turn yields that the martingale part of Y is a BMO martingale. This is the context of the following Proposition 2.1 Suppose that ξ ∈ L ∞ (Ω, F T , P). There exists a unique solution (Y, Z) to (2.1) such that Y is bounded and M = Z.B is a square integrable martingale. Moreover M = Z.B is a BMO martingale up to time T , and

||Y (t)|| ∞ ≤ ||ξ|| ∞ ∀t ∈ [0, T ].
Proof. The existence and uniqueness is already given in [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF]. The fact that M = Z.B is a BMO martingale up to time T is proved in [START_REF] Morlais | Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem[END_REF]. Then there exists a constant C 2 > 0 such that

E T t |Z s | 2 ds F t ≤ C 2 . Let N t = d j=1 t 0 F j (Y s , Z s )dB j s . Since N, N T -N, N t = T t j |F j (Y s , Z s )| 2 ds ≤ T t C 2 1 (1 + |Y s | + |Z s |) 2 ds, so there exists a constant C 3 > 0 such that E { N, N T -N, N t | F t } ≤ C 3 .
Therefore N is a BMO martingale. Hence the stochastic exponential E(-N ) is a martingale up to T . Define a probability measure Q on (Ω, F T ) by dQ/dP = E(-N ) T . Then, according to Girsanov's theorem Bt = B t + N, B t is a Brownian motion up to time T under Q, and (Y, Z) is a solution to the simple BSDE dY t = Z t .d Bt under the probability Q, whose solution is given by

Y t = E Q {ξ|F t } = E{E(-N ) T E(-N ) -1 t ξ|F t } for t ≤ T . (2.
3)

It particularly implies that ||Y t || ∞ ≤ ||ξ|| ∞ .

Stochastic flows and gradient estimates

Let A 0 , A 1 , • • • , A m be m+1 smooth vector fields on Euclidean space R n , where n is a non-negative integer. Then, we may form a sub-elliptic differential operator of second order in R n :

L = 1 2 m α=1 A 2 α + A 0 , (3.1)
here we add a factor 1 2 in order to save the constant √ 2 in front of Brownian motion which will appear frequently in computations in the remaining of the paper. Our goal is to devise an explicit gradient estimate for a (smooth) positive solution u of the heat equation

L - ∂ ∂t u = 0, on (0, ∞) × R n , (3.2)
by utilizing the BSDE associated with the Hopf transformation f = log u, which satisfies the semi-linear parabolic equation

L - ∂ ∂t f = - 1 2 m α=1 |A α f | 2 , on (0, ∞) × R n . (3.3)

Stochastic flow

The first ingredient in our approach is the theory of stochastic flows defined by the following stochastic differential equation

dϕ = A 0 (ϕ)dt + m α=1 A α (ϕ) • dw α , ϕ(0, •) = x, (3.4)
where •d denotes the Stratonovich differential, developed by Baxendale [START_REF] Baxendale | Brownian motions in the diffeomorphism group[END_REF], Bismut [START_REF] Bismut | Mécanique aléatoire[END_REF], Eells and Elworthy [START_REF] Eells | Stochastic dynamical systems. Control theory and topics in functional analysis[END_REF], Malliavin [START_REF] Malliavin | Géométrie différentielle stochastique[END_REF], Kunita [START_REF] Kunita | On the representation of solutions of stochastic differential equations[END_REF] and etc. The reader may refer to Ikeda and Watanabe [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF] for a definite account. To ensure the global existence of a stochastic flow, we require the following condition to be satisfied.

Condition 3.1 Let A α = n j=1 A j α ∂ ∂x j . Assume that A j α have bounded derivatives.
By writing (3.4) in terms of Itô's stochastic integrals, namely

dϕ j = A j 0 + 1 2 m α=1 A i α ∂A j α ∂x i (ϕ)dt + m α=1 A j α (ϕ)dw α , ϕ(0, •) = x, (3.5)
where (and thereafter) Einstein's summation convention has been used: repeated indices such as l is summed up from 1 up to n. The existence and uniqueness of a strong solution follow directly from the standard result in Itô's theory, which in turn determines a diffusion process in R n with the infinitesimal generator L.

In fact, more can be said about the unique strong solution, and important consequences are collected here which will be used later on. Suppose w = (w t ) is a standard Brownian motion (started at 0) with its Brownian filtration (F t ) t≥0 on the classical Wiener space (Ω, F, P) of dimension m, so that w = (w t ) t≥0 is the coordinate process on the space Ω of continuous paths in R m with initial zero. Then, there is a measurable mapping ϕ : R + × Ω × R n -→ R n and a probability null set N , which possess the following properties.

1. w → ϕ(t, w, x) is F t -measurable for t ≥ 0 and x ∈ R n , and ϕ(0, w, x) = x for every w ∈ Ω \ N and x ∈ R n .

2. t → ϕ(t, w, x) is continuous, that is ϕ(•, w, x) ∈ C(R + , R n ), for w ∈ Ω \ N and x ∈ R n . t → ϕ(t, •, x) is a continuous semimartingale for any x ∈ R n . 3. x → ϕ(t, w, x) is a diffeomorphism of R n for each w ∈ Ω \ N and t ≥ 0. That is x → ϕ(t, w, x)
is smooth and its inverse exists, and the inverse is also smooth.

4. The family {ϕ(t, •, x) : t ≥ 0, x ∈ R n } is a stochastic flow:

ϕ(t + s, w, x) = ϕ(t, θ s w, ϕ(s, w, x))
for all t, s ≥ 0, x ∈ R n and w ∈ Ω \ N , where θ s : Ω → Ω is the shift operator sending a path w to a path θ s w(t) = w(t + s) for t ≥ 0.

5. For each x ∈ R n , ϕ(t) = ϕ(t, •, x) (or denoted by ϕ(t, x)) is the unique strong solution of (3.4).

6. Let J i j (t, w, x) = ∂ϕ i (t,w,x) ∂x j for i, j ≤ n. Then J i j (0, w, x) = δ i j and J solves the following SDE

dJ i j = ∂A i 0 ∂x l (ϕ)J l j dt + m α=1 ∂A i α ∂x l (ϕ)J l j • dw α , J i j (0) = δ i j , (3.6)
and its inverse matrix K = J -1 = (K i j ) solves

dK i j = -K i l ∂A l 0 ∂x j (ϕ)dt - m α=1 K i l ∂A l α ∂x j (ϕ) • dw α , K i j (0) = δ i j . (3.7)
In our computations below, we have to use Itô's integrals rather than Stratonovich's ones. Therefore we would like to rewrite (3.6, 3.7) in terms of Itô's differential, so

dJ i j = m α=1 ∂A i α ∂x l (ϕ)J l j dw α + ∂A i 0 ∂x l + 1 2 m α=1 A k α ∂ 2 A i α ∂x l ∂x k + ∂A k α ∂x l ∂A i α ∂x k (ϕ)J l j dt, (3.8)
and

dK i j = - m α=1 K i l ∂A l α ∂x j (ϕ)dw α -K i l ∂A l 0 ∂x j + 1 2 m α=1 A k α ∂ 2 A l α ∂x j ∂x k - ∂A k α ∂x j ∂A l α ∂x k (ϕ)dt.
(3.9)

Structure assumptions

We introduce some technical assumptions on the structure of the Lie algebra generated by the family of vector fields {A 0 , A 1 , • • • , A m }, in addition to Condition 3.1. Recall that A α = A j α ∂ ∂x j , and A j α,β , A j α,β,γ etc. are the corresponding coefficients in Lie brackets

[A α , A β ] = A j α,β ∂ ∂x j , [A α , [A β , A γ ]] = A j α,β,γ ∂ ∂x j etc.,
where Let us suppose the following Frobenius integrability condition: there exist some bounded smooth coefficients c l β,α (x), such that

A j α,β = A i α ∂A j β ∂x i -A i β ∂A j α ∂x i , A k β,β,α = A j β A i β ∂ 2 A k α ∂x i ∂x j -A j β A i α ∂ 2 A k β ∂x i ∂x j + A i β ∂A k α ∂x j ∂A j β ∂x i -2A j β ∂A i α ∂x j ∂A k β ∂x i + A i α ∂A j β ∂x i ∂A k β ∂x j . (3.10) etc. Let R k α = m β=1 A k β,β,α = m β=1 [A β , [A β , A α ]] k . (3.11) Condition 3.2 There is a constant C 1 ≥ 0 such that for any ξ = (ξ i ) i≤n , θ β = (θ i,β ) i≤n ∈ R n (β = 1, • • • , m), it holds that m α,β=1 n k=1 A k α θ k,β 2 + 2 m α,β=1 n k=1 ξ k A k β,α n i=1 A i α θ i,β +2 m α,β=1 n k=1 ξ k A k α n i=1 A i β,α θ i,β ≥ -C 1 m α=1 n k=1 A k α ξ k 2 i.e. m α,β=1 A α , θ β 2 + 2 A α , ξ A β,α , θ β + 2 A α , θ β A β,α , ξ ≥ -C 1 m α=1 A α , ξ 2 . Condition 3.3 There is a constant C 2 ≥ 0 such that for any ξ = (ξ i ) i≤n ∈ R n n i,k=1 ξ i   m α=1 (A i α R k α + 2A i α A k 0,α ) + m α,β=1 A i β,α A k β,α   ξ k ≥ -C 2 m α=1 n k=1 A k α ξ k
A β,α = m l=1 c l β,α A l , β = 0, 1, . . . , m, α = 1, . . . , m.
In other words, the Lie brackets A β,α , β = 0, 1, . . . , m, α = 1, . . . , m, must lie in the linear span of A 1 , . . . , A m . Then the conditions (3.2) and (3.3) are satisfied. Indeed,

[A β , [A β , A α ]] j = A i β ∂ ∂x i (c l β,α A j l ) -c l β,α A i l ∂A j β ∂x i = c l β,α A j β,l + ∂c l β,α ∂x i A i β A j l = (c l β,α c k β,l + ∂c k β,α ∂x i A i β )A j k . This means that [A β , [A β , A α ]]
also lies in the linear span of A 1 , . . . , A m , and the conditions (3.2) and (3.3) are easily checked.

The density processes Z α

Let us consider a smooth solution f to the following non-linear parabolic equation

L - ∂ ∂t f = h(f, A α f ), on R + × R n , (3.12)
where h is a C 1 -function on R × R m , though our archetypical example is f = log u and u is a positive solution to equation (3.3). By Itô's formula,

Y t = Y T - m α=1 T t Z α dw α - T t h(Y, Z)ds, (3.13)
where

Y (t, w, x) = f (T -t, ϕ(t, w, x)), Z α (t, w, x) = (A α f ) (T -t, ϕ(t, w, x))
for α = 1, • • • , m, and Z = (Z α ). The arguments w and / or x will be suppressed if no confusion may arise. Equivalently

dY = m α=1 Z α dw α + h(Y, Z)dt. (3.14)
Our aim in this part is to show that Z is an Itô process, and derives stochastic differential equations for Z (which in turn gives its Doob-Meyer's decomposition).

It is clear that both Y and Z α (α = 1, • • • , m) are continuous semimartingales. Taking derivatives with respect to x i (i = 1, • • • , n) in the equation (3.14) one obtains

dY i = m α=1 Z α i dw α + h y (Y, Z)Y i + m α=1 h zα (Y, Z)Z α i dt, (3.15)
where

Y i = ∂ ∂x i Y and Z α i = ∂ ∂x i Z α , i = 1, • • • , n, and
h y = ∂ ∂y h(y, z) , h zα = ∂ ∂z α h(y, z), α = 1, • • • , m.
On the other hand, by definition,

Y i (t, •, x) = ∂ ∂x i Y (t, •, x) = ∂f ∂ϕ j (t, ϕ(t, •, x))J j i (t, •, x), so ∂f ∂ϕ k (t, ϕ(t, •, x)) = K l k (t, •, x)Y l (t, •, x).
It follows that

Z α (t, •, x) = A k α (ϕ(t, •, x)) ∂f ∂ϕ k (ϕ(t, •, x)) = A k α (ϕ(t, •, x))K l k (t, •, x)Y l (t, •, x), (3.16)
which implies that Z is a continuous semimartingale. The equation (3.16) is not new, and has been used by many authors in different contexts.

We next would like to write down the stochastic differential equations that Z α must satisfy by using the relation (3.16), which is however an easy exercise on integration by parts. Indeed, we have

dZ α = Y l K l k ∂A k α ∂x j (ϕ)dϕ j + Y l A k α (ϕ)dK l k + A k α (ϕ)K l k dY l +Y l ∂A k α ∂x j (ϕ)d ϕ j , K l k + K l k ∂A k α ∂x j (ϕ)d ϕ j , Y l +A k α (ϕ)d K l k , Y l + 1 2 Y l K l k ∂ 2 A k α ∂x i ∂x j (ϕ)d ϕ i , ϕ j . (3.17)
Using the SDEs (3.4, 3.9) and the BSDE (3.15), through a lengthy but completely elementary computation, we establish the following Doob-Meyer's decomposition for Z

dZ α = m β=1 U α,β dw β + h z β (Y, Z)dt + K l k m β=1 A k β,α Z β l dt +K l k Y l   A k 0,α + 1 2 m β=1 A k β,β,α + A k α h y (Y, Z) - m β=1 A k β,α h z β (Y, Z)   dt, (3.18)
where repeated indices are added up from 1 to n,

U α,β = K l k A k β,α Y l + A k α Z β l (3.19) and A k β,α = [A β , A α ] k , A k β,β,α = [A β , [A β , A α ]] k .
We are now in a position to work out the Doob-Meyer's decomposition for

|Z| 2 = m α=1 |Z α | 2
which simply follows from Itô's formula and (3.18). In order to simplify our displayed formula, we introduce the following notations:

ξ i = n j=1 K j i Y j , θ k,β = n j=1 K j k Z β j , for i, k = 1, • • • , n and β = 1, • • • , m, so that Z α = n j=1 A j α ξ j and |Z| 2 = m α=1   n j=1 A j α ξ j   2 . (3.20) Let d wβ = dw β + h z β (Y, Z)dt,
which is a Brownian motion under probability Q with the Cameron-Martin density

dQ dP Ft = exp   - m β=1 t 0 h z β (Y, Z)dw β - 1 2 t 0 m β=1 |h z β (Y, Z)| 2 ds   . (3.21)
Then, by an elementary computation,

d|Z| 2 = 2 m α,β=1 Z α ξ k A k β,α + ξ i A i α A k α θ k,β d wβ +   m α,β=1 A k α θ k,β A i α θ i,β + 2ξ k m α,β=1 A k β,α A i α + A i β,α A k α θ i,β   dt +2   h y (Y, Z) m α=1 A i α A k α - m α,β=1 A i α A k β,α h z β (Y, Z)   ξ i ξ k dt +   m α=1 A i α R k α + 2A k 0,α + m α,β=1 A i β,α A k β,α   ξ i ξ k dt. (3.22) Lemma 3.5 If h(Y, Z) = h(Y, |Z| 2 ), then d|Z| 2 = 2 m α,β=1 Z α ξ k A k β,α + ξ i A i α A k α θ k,β d wβ +   m α,β=1 A k α θ k,β A i α θ i,β + 2ξ k m α,β=1 A k β,α A i α + A i β,α A k α θ i,β   dt +   m α=1 A i α R k α + 2A k 0,α + 2A k α h y (Y, |Z| 2 ) + m α,β=1 A i β,α A k β,α   ξ i ξ k dt. (3.23) Proof. In this case i,k,α,β ξ i A i α A k β,α h z β (Y, Z)ξ k = 2 i,k,α,β h ′ ξ i A i α A k β,α A l β ξ l ξ k = -2 i,k,α,β h ′ ξ l A l β A k β,α A i α ξ i ξ k , so i,k,α,β ξ i A i α A k β,α h z β (Y, Z)ξ k = 0,
and thus (3.23) follows directly from (3.22).

Gradient estimates

Recall that f is a smooth solution to the non-linear parabolic equation (3.12), where the nonlinear term h(Y, Z) has at most quadratic growth. In order to devise explicit estimate for A α f (α = 1, • • • , m), we assume the following condition to be satisfied.

Condition 3.6 h(y, z) depends only on (y, |z| 2 ), i.e. there is a continuously differentiable function denoted again by h so that h(y, z) = h(y, |z| 2 ), and we assume that

∂ ∂y h(y, z) ≥ 0.
Then, under Conditions (3.1, 3.2, 3.3, 3.6), according to (3.23), we have

d|Z| 2 ≥ -K|Z| 2 dt + 2 Z α ξ k A k β,α + ξ i A i α A k α θ k,β d wβ , (3.24)
where

K = C 1 + C 2 .
Lemma

E Q {M t |F s } ≥ M s , ∀0 ≤ s < t ≤ T . (3.25)
Proof. By Itô's formula dM = KM dt + e Kt d|Z| 2 , hence, for any 0 ≤ s ≤ t ≤ T , we have

M t -M s = K t s M r dr + t s e Kr d|Z| 2 ≥ 2 t s e Ks m α,β=1 Z α ξ k A k β,α + ξ i A i α A k α θ k,β d wβ ,
which yields (3.25).

We are now in a position to prove the following gradient estimate.

Theorem 3.8 Assume that Conditions (3.1, 3.2, 3.3) are satisfied. Then

m α=1 |A α log u(t, x)| 2 ≤ 4K 1 -e -K(T -t) || log u 0 || ∞ , (3.26)
for any positive solution u of (3.2).

Proof. Apply the computations in the preceding sub-section to f = log u, and h(y, z) = -1 2 |z| 2 . Then, under the probability Q (defined by (3.21))

Y t = Y T - m α=1 T t Z α d wα + T t m α=1 Z α h zα (Y, Z) -h(Y, Z) ds, thus Y t = Y T - m α=1 T t Z α d wα - 1 2 T t |Z s | 2 ds,
and therefore

E Q 1 2 T t |Z s | 2 ds F t = E Q { Y T -Y t | F t } ≤ 2||Y T || ∞ ≤ 2|| log u 0 || ∞ . (3.27)
On the other hand, M t = e Kt |Z t | 2 is a submartingale, thus one has

E Q |Z s | 2 F t ≥ e K(t-s) |Z t | 2 , ∀s ∈ [t, T ], so E Q 1 2 T t |Z s | 2 ds F t ≥ 1 2 T t e K(t-s) |Z t | 2 ds = 1 -e -K(T -t) 2K |Z t | 2 . ( 3.28) Putting (3.27, 3.28)
together, we obtain

|Z t | 2 ≤ 4K 1 -e -K(T -t) || log u 0 || ∞ ,
which yields (3.26).

In general, we may proceed with f = ψ(u) where ψ is a concave function, and u is a positive solution to (1.1), thus f solves (3.3) with

h(y, z) = 1 2 ψ ′′ (ψ -1 (y)) |ψ ′-1 (y))| 2 |z| 2 .
We can proceed as above. Under the probability Q

Y t = Y T - m α=1 T t Z α d wα + T t m α=1 Z α h zα (Y, Z) -h(Y, Z) ds, so Y t = Y T - m α=1 T t Z α d wα + 1 2 T t ψ ′′ (ψ -1 (Y s )) |ψ ′-1 (Y s ))| 2 |Z s | 2 ds.
It is important to note that if

ψ (3) ψ ′ ≤ 2|ψ ′′ | 2 ,
then h y (y, z) ≥ 0, thus from Lemma 2.2 in [START_REF] Delbaen | Backward SDEs with superquadratic growth[END_REF],

E Q T t |Z s | 2 ds F t ≤ 4||Y T || 2 ∞ ≤ 4|| log u 0 || 2 ∞ . (3.29)
On the other hand M t = e Kt |Z t | 2 is a submartingale, thus one has

E Q |Z s | 2 F t ≥ e K(t-s) |Z t | 2 , ∀s ∈ [t, T ],
and therefore

E Q T t |Z s | 2 ds F t ≥ T t e K(t-s) |Z t | 2 ds = 1 -e -K(T -t) K |Z t | 2 . ( 3.30)
Putting (3.29, 3.30) together, we can obtain

|Z t | 2 ≤ 4K 1 -e -K(T -t) || log u 0 || 2 ∞ ,
which yields the following estimate.

Theorem 3.9 Assume that Conditions (3.1, 3.2, 3.3) are satisfied. Moreover, ψ is concave and satisfies:

ψ (3) ψ ′ ≤ 2|ψ ′′ | 2 . Then m α=1 |A α ψ(u(t, x))| 2 ≤ 4K 1 -e -K(T -t) ||ψ(u 0 )|| 2 ∞ , (3.31)
for any positive solution u of (3.2).

Heat equation on complete manifold

In this section, we study positive solutions of the heat equation

1 2 ∆ - ∂ ∂t u = 0, in [0, ∞) × M, (4.1)
where M is a complete manifold of dimension n, ∆ is the Beltrami-Laplace operator. In a local coordinate system so that the Riemann metric ds 2 = g ij dx i dx j and

∆ = 1 √ g n i,j=1 ∂ ∂x i g ij √ g ∂ ∂x j ,
where g = det(g ij ) and (g ij ) is the inverse matrix of (g ij ). We prove the following Theorem 4.1 Suppose the Ricci curvature Ric ≥ -K for some K ≥ 0, and suppose u is a positive solution of (4.1) with initial data u 0 > 0. Then

|∇ log u| 2 (t, x) ≤ 2K 1 -e -Kt 2 || log u 0 || ∞ . (4.2)
Remark 4.2 This estimate can also be derived from the reverse logarithmic Sobolev inequality, as in Remark 1.2.

The preceding theorem is proved by using similar computations as in the proof of Theorem 3.7 but working on the orthonormal frame bundle O(M ) over M .

Recall that a point γ = (x, e) ∈ O(M ), where (e 1 , • • • , e n ) is an orthonormal basis of the tangent space T x M at x ∈ M . Let π : γ = (x, e) → x be the natural projection from O(M ) to M . O(M ) is a principal fibre bundle with its structure group O(n). For the general facts on differential geometry, we refer to Kobayashi and Nomizu [START_REF] Kobayashi | Foundations of differential geometry[END_REF].

Suppose The following relations will be used in what follows.

x = (x 1 , • • • , x n ) is
Lα f (γ) = Lα f • π(γ) = e k α ∂f ∂x k , for γ = (x i , e k j ) ∈ O(M ) . (4.3)
We need the following geometric facts, whose proofs are elementary. where Ric is the Ricci curvature.

3) We also have

∆ O(M ) Lα f -Lα ∆ O(M ) f = 1 2 n β=1 [ Lβ , Lα ] Lβ f . (4.7)
Proof. The first identity (4.5) follows from the torsion-free condition. To prove (4.6) we choose a local coordinate which is orthonormal and dg ij = 0 (so that Γ k ij = 0) at the point we evaluate tensors, thus

Lα Lβ Lβ f = e j β e i β e q α ∂ 3 f ∂x q ∂x i ∂x j - ∂Γ k ij ∂x q e q α e i β e j β ∂f ∂x k , Lβ Lα Lβ f = e j α e i β e q β ∂ 3 f ∂x q ∂x i ∂x j -e q β e i α e j β ∂Γ k ij ∂x q ∂f ∂x k , and therefore [ Lβ , Lα ] Lβ f = ∂Γ k ij
∂x q e q α e i β e j βe i α e q β e j

M f t = f (t, ϕ(t)) -f (0, ϕ(0)) - t 0 1 2 ∆ O(M ) f (s, ϕ(s))ds is a local martingale for any f ∈ C 1,2 (R + × O(M )), and
M f t = n α=1 t 0 (L α f )(s, ϕ(s))dw α s . ( 4.9)
We may express ϕ(t, w, γ) = (X(t, w, γ), E(t, w, γ)),

where X(t, w, γ) = π(ϕ(t, w, γ)), then {X(t, •, γ) : t ≥ 0} is a diffusion process on M starting from x = π(γ) with infinitesimal generator 1 2 ∆. {X(t, •, γ) : t ≥ 0} is a Brownian motion on M starting from x = π(γ).

In a local coordinate system (x k , e i j ), write

ϕ(t, •, γ) = (X k (t, •, γ); E i j (t, •, γ)) so that E α (t) = E i α (t) ∂ ∂x i .
Then the SDE (4.8) may be written as

dX k t = n α=1 E k α (t) • dw α t , dE i j (t) = -n α,β,k=1 Γ(X t ) i βk E k j (t)E β α (t) • dw α t . (4.10) Let F (t) = (F (t) i j ) = E(t) -1 . Then F (t)E(t) = I so that dF i j (t) = n α,β,l=1 Γ(X t ) l βj F i l (t)E β α (t) • dw α t . (4.11) If f ∈ C 1,2 (R + × M ), then f (T, X T ) = f (t, X t ) + T t n α,k=1 E k α (s) ∂f ∂x k (s, X s )dw α s + T t ∂ ∂s + 1 2 ∆ f (s, X s )ds, (4.12

Gradient estimate

Suppose now f satisfies the non-linear heat equation

1 2 ∆ - ∂ ∂t f = - 1 2 |∇f | 2 . ( 4.14)
Let f be the horizontal lifting of f i.e. f = f • π, so that f satisfies the parabolic equation on

O(M ): 1 2 ∆ O(M ) - ∂ ∂t f = - 1 2 n β=1 | Lβ f | 2 . (4.15) Let T > 0. Let Y t = f (T -t, X t ) and Z α t = n k=1 E k α (t) ∂f ∂x k (t, X t ) = ( Lα f )(t, (X t , E(t))), (4.16
)

for α = 1, • • • , n. Then |∇f (T -t, •)| 2 (X t ) = n α=1 |Z α t | 2 , (4.17)
therefore, according to (4.12),

Y T = Y t + T t Z α t dw α s - 1

Li-Yau's estimate

In this section we prove Theorem 1.4. Thus, u is a positive solution to the heat equation

1 2 ∆ - ∂ ∂t u = 0, on [0, ∞) × M, (5.1)
where M is a complete Riemannian manifold of dimension n, with non-negative Ricci curvature. Then f = log u is a solution to the semi-linear heat equation

1 2 ∆ - ∂ ∂t f = - 1 2 |∇f | 2 , f (0, •) = f 0 . ( 5
Let G = -∆f = |∇f | 2 -2f t .
By combining (5.3) and (5.5) together, we obtain

∂ ∂t G ≤ LG - 1 n G 2 .
(5.6)

Suppose that C > 0 and -∇ log u 0 ≤ C. Set

F = t n + 1 C (-∆f ) . Since ∂ ∂t -L F = 1 n G + t n + 1 C ∂ ∂t -L G ≤ 1 n G - t n + 1 C 1 n G 2 = 1 n G (1 -F ) ,
and therefore

L - ∂ ∂t (F -1) ≥ 1 n G (F -1) .
As -∆ log u 0 ≤ C, (F -1)(0, •) ≤ 0. Applying the maximum principle, F -1 ≤ 0, from which we obtain the estimate in Theorem 1.4.

Corollary 5.1 Suppose u is a positive solution to (5.1)

2 . 3 . 4

 234 Remark Condition (3.1) is standard in the literature, while Conditions (3.2) and (3.3) are satisfied if A is elliptic or A satisfies the Frobenius integrability condition.

. 2 α

 2 a local coordinate system on M , then it induces a local coordinate system γ = (x k , e i j ) on O(M ) so that e j = e i j ∂∂x i . If L is a vector field, then L denotes the horizontal lifting of L to O(M ): L(x, e) = L i (x) ∂ ∂x i -Γ(x) k ij L i (x)e j l ∂ ∂e k l in a local coordinate system, where Γ k ij are the Christoffel symbols associated with the Levi-Civita connection, and L = L i ∂ ∂x i . For α = 1, • • • , n and γ = (x, e), then Lα denotes the horizontal lifting of e α , that is Lα (x, e) = e The system { L1 , • • • , Ln } is called the system of canonical horizontal vector fields. The mapping L : R n → Γ(T O(M )) where L ξ = ξ α Lα , is defined globally, and is independent of the choice of a local coordinate system. Therefore ∆ O(M ) = n α=1 L is well defined sub-elliptic operator of second order on the frame bundle O(M ), called the horizontal Laplacian. If f ∈ C 2 then ∆ O(M ) f • π = ∆f . For simplicity, any function f on M is lifted to a function f on O(M ) defined by f = f • π which is invariant under the group action by O(d).

Lemma 4 . 3 Lemma 4 . 4 (

 4344 For α = 1, • • • , n we have ∆ O(M ) Lα -Lα ∆ O(M ) Suppose f is a smooth function on M and f = f • π is the horizontal lifting of f to O(M ). 1) For α, β = 1, • • • , n [ Lα , Lβ ] f = 0 . Lα f )([ Lβ , Lα ] Lβ f ) = Ric(∇f, ∇f ),(4.6)

  t)) • dw α t , ϕ(0) = γ ∈ O(M ),(4.8)on the classical Wiener space (Ω, F, P) of dimension n. The stochastic flow associated with (4.8) is denoted by {ϕ(t, •, γ) : t ≥ 0}, which is a diffusion process in O(M ) with the infinitesimal generator

  not only Z. B is again a BMO martingale (where B is the martingale part of B under the new probability Q), but also t → |Z t | 2 is a non-negative submartingale. Next by utilizing the BSDE (1.7), we can see the BMO norm of Z. B under Q is dominated at most 2 ||Y || ∞ , that is

	then, under Q,
	2	0	t	|h(Z s )| 2 ds
	is a uniformly integrable martingale (up to time T ), as long as h is global Lipschitz continuous. The
	main technical step in our approach is that, due to the special feature of our non-linear term in (1.6),
	we can choose h i (z) = z i (one has to go through the detailed computations below to see why this
	choice of h i is a good one), and making change of probability measure to Q by dQ dP = E(h(Z).B) T ,

  3.7 Assume that Conditions (3.1, 3.2, 3.3, 3.6) are satisfied. Then M t = e Kt |Z t | 2 is submartingale under the probability Q (up to terminal time T ):

  2 |Z s | 2 ds F t ≤ 4||f 0 || ∞ . (4.22) But on the other hand e Kt 2 |Z t | 2 is submartingale under Q, thus

	Define a probability Q by dQ dP F T	= R T where
			R t = exp	0	t	Z β dw β -	1 2	0	t	|Z| 2 ds .
	T T We will consider (4.18) as a backward stochastic differential equation. t n α=1 |Z α t | 2 ds. Then Y T = Y t + T t Z α t d wα s + 1 2 n t α=1 |Z α t | 2 ds, Applying Itô's formula to Lα f and the stochastic flow ϕ(t, •, γ) to obtain dZ α t = n β=1 hence T E Q Lβ ( Lα f )(t, ϕ(t, •, γ))dw β t t	(4.18)
	so	E Q	t	T	+ |Z s | 2 ds F t 1 2 ∆ O(M ) + =	∂ ∂t	t	Lα f (t, ϕ(t, •, γ))dt, T E Q |Z s | 2 F t ds	(4.19)
		d|Z| 2 = 2 = 2	n α=1 n α,β=1 Z α t dZ α t + Z α t ( Lβ Lα f )dw β n ≥ e Kt 2 |Z t | 2 α,β=1 t + n α,β=1 T t | Lβ Lα f | 2 dt e -Ks 2 ds = 1 -e -K 2 (T -t) K 2 |Z t | 2 ,
	However, by Lemma 4.3, which yields that and hence (4.2).			+2 |Z t | 2 ≤ n α=1 Z α t 1 -e -K 1 2 ∆ O(M ) + 2K 2 (T -t)	∂ ∂t ||f 0 || ∞ , Lα f dt .	(4.20)
	1 2	∆ O(M ) +	∂ ∂t			Lα f = Lα	1 2	∆ O(M ) +	∂ ∂t	f +	1 4	n β=1 [ Lβ , Lα ] Lβ	f
											= -	n β=1 ( Lβ f )( Lα Lβ f ) +	1 4	n β=1 [ Lβ , Lα ] Lβ f ,
	we therefore have
						n						n
		d|Z| 2 = 2	α=1	Z α t	β=1	( Lβ Lα f ) dw β t -Z β t dt
			+	n α,β=1	| Lβ Lα f | 2 dt +	1 2	n α,β=1	( Lα f )([ Lβ , Lα ] Lβ f )dt.
	By using Lemma 4.3 we obtain
						n						n
		d|Z| 2 = 2	α=1	Z α t	β=1 ( Lβ Lα f ) dw β t -Z β t dt
			+	n α,β=1	| Lβ Lα f | 2 dt +	1 2	Ric(∇f, ∇f )(T -t, X t )dt.	(4.21)

| Lβ Lα f | 2 (t, ϕ(t, •, γ))dt

  By using the Bochner identity one can verify that∂ ∂t |∇f | 2 = L|∇f | 2 -|∇∇f | 2 -2Ric(∇f, ∇f ) . (5.4)Since ∆f is the trace of the Hessian ∇∇f so that |∇∇f | 2 ≥ 1 n (∆f ) 2 , thus, since the Ricci curvature is non-negative, then

	∂ ∂t	|∇f | 2 ≤ L|∇f | 2 -	1 n	(∆f ) 2 .	(5.5)
						.2)
	Taking derivative in the parabolic equation (5.2) with respect to the time parameter t one
	obtains	∂ ∂t	f t = Lf t ,	(5.3)
	where	L =	1 2	∆ + ∇f.∇ .

Acknowledgement The authors would like to thank the referee for the helpful comments.

Curie ITN Grant, "Controlled Systems", GA no.213841/2008.

This follows by integrating the gradient estimates in Theorem 1.4 along geodesics, see [START_REF] Bakry | Harnack inequalities on a manifold with positive or negative Ricci curvature[END_REF] for details.


Joan Artigas

N Pascault

Agnes Bouchez

J Chastaine

D Debroas

J F Humbert

J Leloup

R D Tadonleke

A Ter Halle

S Pesce

Comparative sensitivity to the fungicide tebuconazole of biofilm and plankton microbial communities in freshwater ecosystems

Keywords: Algae, Bacteria, Fungicide, Photosynthetic efficiency, Adaptation, Substrate-induced respiration

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Comparative sensitivity to the fungicide tebuconazole of biofilm and plankton microbial communities in freshwater ecosystems

Introduction

Freshwater ecosystems in agricultural watersheds are widely exposed to multiple pesticidecontamination events, including fungicides which pose major ecotoxicological threats to aquatic microbial community biodiversity and function (e.g. [START_REF] Montuelle | The periphyton as a multimetric bioindicator for assessing the impact of land use on rivers: an overview of the Ardières-Morcille experimental watershed (France)[END_REF]Rasmussen et al., 2012a). The intensive use of modern azole fungicides like fluconazole, propiconazole and tebuconazole (TBZ) [START_REF] Richardson | Water analysis: emerging contaminants and current issues[END_REF] has prompted studies aiming to understand their effects on aquatic detritivores and microbial decomposers [START_REF] Bundschuh | Fungal composition on leaves explaINS pollutant-mediated indirect effects on amphipod feeding[END_REF]Rasmussen et al., 2012b). TBZ has been found in numerous freshwater ecosystems in Europe, including rivers and lakes, and is reported as relatively persistent even after wastewater treatment [START_REF] Berenzen | A comparison of predicted and measured levels of runoff-related pesticide concentrations in small lowland streams on a landscape level[END_REF][START_REF] Kahle | Azole fungicides: occurrence and fate in wastewater and surface waters[END_REF].

The main aquatic microbial communities potentially threatened by TBZ are fungi. TBZ affects fungal activity-related processes such as litter decomposition (Rasmussen et al., 2012b and[START_REF] Bundschuh | Fungal composition on leaves explaINS pollutant-mediated indirect effects on amphipod feeding[END_REF] and inhibits ergosterol biosynthesis, which hampers fungal mycelium growth [START_REF] Copping | Chemistry and mode of action of crop protection agents[END_REF]. Exposing submerged leaves to TBZ led to lower fungal biomass accumulation and reduced the ability of microorganisms to decompose cellulose and hemicellulose compounds [START_REF] Artigas | Effects of the fungicide tebuconazole on microbial capacities for litter breakdown in streams[END_REF]. Important bacterial groups such as nitrogen-fixing and nitrifying bacteria can also be directly or indirectly affected by fungicides [START_REF] Johnsen | Pesticide effects on bacterial diversity in agricultural soils -a review[END_REF]. Specifically, TBZ has been reported to stimulate ammonifying bacteria populations at early exposure in soils and nitrifying bacteria in later exposure phases [START_REF] Chen | Effects of the fungicides benomyl, captan and chlorothalonil on soil microbial activity and nitrogen dynamics in laboratory incubations[END_REF][START_REF] Cycoń | Microbiological characteristics of a sandy loam soil exposed to tebuconazole and λ-cyhalothrin under laboratory conditions[END_REF][START_REF] Muñoz-Leoz | Tebuconazole application decreases soil microbial biomass and activity[END_REF]. This stimulating effect on bacterial groups could be associated with increased levels of nutrients and energy sources released from dead fungal hyphae killed by TBZ [START_REF] Cycoń | Microbiological characteristics of a sandy loam soil exposed to tebuconazole and λ-cyhalothrin under laboratory conditions[END_REF] or by chemicals present in the fungicide formulation becoming available to bacteria [START_REF] Muñoz-Leoz | Tebuconazole application decreases soil microbial biomass and activity[END_REF]. Although such results suggest that similar effects will be found in aquatic ecosystems, with consequences on microbial processes, only a few studies have explored the effects of TBZ on aquatic microbial communities other than fungi in biofilms and plankton (Tlili et al., 2011a and[START_REF] Artigas | Effects of the fungicide tebuconazole on microbial capacities for litter breakdown in streams[END_REF].

Microbial community exposure to TBZ varies between stream and lake ecosystems. Pesticide mobilization in agricultural watersheds mostly depends on drift, run-off and drainage processes from the watershed (Kolpin et al., 1996), and often coincides with fungicide applications in conjunction with cultural practices [START_REF] Rabiet | Assessing pesticide concentrations and fluxes in the stream of a small vineyard catchment: effect of sampling frequency[END_REF]. However, the arrival of fungicides into surface water depends on watershed topography (especially field slope), soil characteristics (i.e. organic carbonwater binding coefficient), and the half-life of the molecule (49-610 days for TBZ in aerobic soils; [START_REF] Strickland | Tebuconazole dissipation and metabolism in Tifton loamy sand during laboratory incubation[END_REF]. Once in the aquatic ecosystem, hydrological conditions in lotic ecosystems mostly generate peak exposure scenarios (lasting a few hours to days) at high contaminant concentrations, whereas lentic ecosystems establish a more chronic exposure scenario (several months) at lower contaminant concentration [START_REF] Simmons | A critical reviewof the biogeochemistry and ecotoxicology of selenium in lotic and lentic environments[END_REF]. Consistently with this pattern, TBZ concentrations recorded in rivers (often at µg L-1) exceed those found in lakes (often at ng L-1; [START_REF] Richardson | Water analysis: emerging contaminants and current issues[END_REF][START_REF] Rabiet | Assessing pesticide concentrations and fluxes in the stream of a small vineyard catchment: effect of sampling frequency[END_REF][START_REF] Kahle | Azole fungicides: occurrence and fate in wastewater and surface waters[END_REF]. However, TBZ concentrations in rivers are highly variable over time [START_REF] Rabiet | Assessing pesticide concentrations and fluxes in the stream of a small vineyard catchment: effect of sampling frequency[END_REF] and chronic exposure scenarios to low fungicide concentrations can also be detected (i.e. [START_REF] Reilly | Occurrence of boscalid and other selected fungicides in surface water and groundwater in three targeted use areas in the United States[END_REF]. Knowing that level and duration of exposure of microbial communities to the pesticide are important factors determining the adaptation processes of microorganisms ( Le [START_REF] Jeune | Planktonic microbial community responses to added copper[END_REF][START_REF] Pesce | Response of spring and summer riverine microbial communities following glyphosate exposure[END_REF], microbial communities from stream biofilms and lakes are likely to display contrasted susceptibility to this fungicide.

Comparative studies assessing pollutant toxicity between lotic (stream) and lentic (lake) ecosystems are rare in the literature [START_REF] Munawar | A comparative bioassessment of sediment toxicity in lentic and lotic ecosystems of the North American Great Lakes[END_REF][START_REF] Simmons | A critical reviewof the biogeochemistry and ecotoxicology of selenium in lotic and lentic environments[END_REF][START_REF] Simmons | A critical reviewof the biogeochemistry and ecotoxicology of selenium in lotic and lentic environments[END_REF], despite their potential utility for better understanding how the ecological, hydrological and biogeochemical characteristics of each type of ecosystem can modulate pollutant ecotoxicity. In this global framework, we adopted an experimental approach to investigate the effects of TBZ on non-targeted microalgal and bacterial communities living in stream biofilms and lake plankton samples collected in natural ecosystems. For each ecosystem (stream and lake), samples were collected in sites qualified as "less polluted" and "polluted" based on previous in situ measurements in order to assess the influence of pre-exposure to TBZ on their susceptibility to the fungicide. The responses of all these communities to TBZ (at two doses, 2 and 20 µg TBZ L-1) were examined using a range of structural and functional descriptors. This experimental scheme was thus designed to test the respective effects of community lifestyle (benthic versus planktonic) and pre-exposure to the pollutant on the responses of non-targeted aquatic microbial communities to TBZ.

Materials and methods

Biological communities

Biofilm and plankton microbial communities from less-polluted (LP) and polluted (P) sites were obtained from stream and lake ecosystems in eastern France. Biofilms were obtained from two sites (upstream in Saint Joseph and downstream in Saint Ennemond) of Morcille River (Beaujolais vineyard region, 46.15°N 4.60°E), a siliceous stream composed by three nested sub-watersheds that experiences an upstream-to-downstream gradient of agricultural contamination [START_REF] Rabiet | Assessing pesticide concentrations and fluxes in the stream of a small vineyard catchment: effect of sampling frequency[END_REF][START_REF] Montuelle | The periphyton as a multimetric bioindicator for assessing the impact of land use on rivers: an overview of the Ardières-Morcille experimental watershed (France)[END_REF]. This pollution gradient is characterized by increases in dissolved organic carbon, phosphate (125% to 300% increase between upstream and downstream sites, respectively) and copper, arsenic, herbicide (diuron and its breakdown products) and fungicide (azoxystrobin, carbendazim, tebuconazole, procymidone and dimetomorph) concentrations [START_REF] Montuelle | The periphyton as a multimetric bioindicator for assessing the impact of land use on rivers: an overview of the Ardières-Morcille experimental watershed (France)[END_REF]. Previous surveys have shown that TBZ concentrations ranged from 0.001-0.1 µg L-1 at the upstream LP site of Saint Joseph to 0.01-4 µg L-1 at the downstream P site of Saint Ennemond (Margoum C., pers. comm.). Artificial glass substrata (20 cm2 tiles) were left to colonize for one month at both study sites. Biofilm colonization was conducted during late summer, when the highest TBZ contamination peaks had been detected at the P site [START_REF] Montuelle | The periphyton as a multimetric bioindicator for assessing the impact of land use on rivers: an overview of the Ardières-Morcille experimental watershed (France)[END_REF]. The resulting biofilms from LP and P sites were collected, transported cold (4 °C) to the laboratory, and placed in indoor stream channels for TBZ experiments.

Plankton communities were obtained from two different peri-alpine lakes experiencing contrasting degrees of agricultural contamination -one less polluted (Lake Aiguebelette, mean TBZ concentration < 0.001 µg L-1) and one polluted (Lake Geneva, mean TBZ concentration = 0.001 µg L-1). Lake Aiguebelette (LP site) is situated in a small forested watershed (70 km2) bearing low anthropogenic pressure, whereas Lake Geneva (P site) has a much larger watershed (7395 km2) bearing high pressure due to urbanization and agriculture. Samples (400 L) from each lake were collected from the 0-50 cm layer and filtered through a 50 µm pore-size filter to remove zooplankton. Samples were collected at a central part in each lake (Lake Aiguebelette: 5°47′56″, 45°33′02″, Lake Geneva: 6°35′34″, 46°27′22″). Samples with plankton communities from LP and P sites were transported in plastic containers, stored overnight at 4 °C, and used for subsequent TBZ experiments in indoor microcosms.

Laboratory experiments

The experiments were performed in glass indoor stream channels (L × W × D = 63 cm × 11 cm × 4 cm) for biofilms and in transparent polycarbonate bottles (20 L) for plankton. Each type of microbial community (LP-site biofilm, P-site biofilm, LP-site plankton, P-site plankton) was subjected to three different treatments consisting of control (non-contaminated), low-TBZ dose (2 µg L-1) and high-TBZ dose (20 µg L-1) exposure.

For biofilms, each microbial community (LP and P sites) and treatment was tested in three independent channel replicates (18 channels in total), each connected to a separate 5 L reservoir (Pyrex, USA) by means of an aquarium pump (NEWA MJ 750). Water temperature was set to 20 °C and flow conditions were set to 0.19 L s-1, all under a 12 h/12 h light/dark cycle. Before TBZ contamination, stream biofilms were allowed to adapt for one week to the indoor channels by recirculating tap water (previously dechlorinated using an active carbon filter). After TBZ contamination, the water was not replaced during the 22-day experiment but N and P were added to prevent nutrient limitation in the biofilms (weekly added quantities: 225 µg N L-1 and 65 µg P L-1). For plankton, LP and P-site communities were exposed to the three different treatments in triplicate. Bottles containing plankton communities (18 in total) were incubated at 20 °C under a 12 h/12 h light/dark cycle and without nutrient addition. Air circulation was supplied to prevent anoxia in the plankton communities.

Solutions of TBZ (C16H22CIN3O, Sigma Aldrich; 10 mg L-1 in deionized water) were added to the biofilm and plankton communities, and biological community and water samples were taken on days 0, 3, 7, 10, 14, 17, 22 post addition. For biofilms, samples were scraped from glass tiles and suspended in the water of the corresponding channel for subsequent biological analyses. Plankton samples were taken with a pipette directly from bottles after stirring. Assays to determine substrate-induced respiration, bacterial production, chlorophyll concentration and photosynthetic activity were processed immediately after sampling, while samples taken to determine bacterial density were fixed in 2% formaldehyde until analysis. Water samples to determine nutrient concentrations were taken into glass flasks (100 mL) and analyzed immediately. Further water samples taken to analyze TBZ concentration (100 mL) and TBZ metabolites (500 mL) were stored frozen (-20 °C) until analysis.

Chemical analyses

Water temperature and pH were measured at each sampling time using portable meters (WTW 3110, Germany). Filtered (0.45 µm) water samples were analyzed to determine the concentration of inorganic nutrients [soluble reactive phosphorus (SRP), dissolved inorganic nitrogen (DIN = ammonia + nitrite + nitrate) and dissolved organic carbon (DOC)] following French standard operating procedures.

TBZ quantification was determined by UPLC (ACQUITY, Waters, Manchester, UK) using a Hypersil Gold PFP stationary phase (100 m × 2.1 mm × 1.9 µm) with detection at 220 nm. The mobile phase consists of 50% acetonitrile/50% H2O at 0.5 mL/min. Analysis was performed at 30 °C with an injection volume of 6 µL. TBZ degradation metabolites were determined by LC/Q-TOF (Waters/Micromass, Manchester, UK). For metabolite analyses, 25 µL were injected onto a reversephase column (X terra, Waters, C18, 3.5 µm, 100 mm × 2.1 mm) at a flow rate of 0.3 mL min-1 with a mobile phase composed of acetonitrile (solvent A) and acidified water (solvent B: formic acid 0.3% v/v; pH 2.6). The gradient was 5% A from 0-5 min, 5-95% A (linear) from 5-15 min, and 95% A from 15-25 min. The desolvation chamber and ion source block temperatures were set at 350 °C and 100 °C, respectively. The voltages for the probe and ion source components were set to 2.1 kV for the stainless steel capillary, 35 V for the sample cone, and 2 V for the extractor cone. A collision energy gradient (15-35 eV) was used for preliminary fingerprints, and specific energies of 15-45 eV were used to unambiguously assign the MS/MS fragments.

Biological analyses

Bacterial density

Bacterial densities were quantified by DAPI staining for biofilms [START_REF] Porter | The use of DAPI for identifying and counting aquatic microflora[END_REF] and by flow cytometry for plankton communities [START_REF] Tadonléké | Microbial food webs in boreal humic lakes and reservoirs: ciliates as a major factor related to the dynamics of the most active bacteria[END_REF]. For biofilms, suspensions corresponding to 1 cm2 of scraped biofilm were sonicated for 2 min to facilitate disaggregation, diluted 1:2, and stained for 5 min with 4,6-diamidino-2-phenylindole (DAPI, Sigma-Aldrich) at a final concentration of 2 µg mL-1. Suspensions were then filtered through 0.2 µm irgalan black-stained polycarbonate membranes (Nuclepore, Whatman, England) and mounted on microscope slides. Fifteen fields were counted per filter, accounting for a total of 400-800 cells, using a fluorescence microscope (Nikon Labophot-2, Japan) at 1000 × magnification. Bacterial densities were calculated as cells per unit of biofilm surface area in cm2.

For biofilms, additional bacterial suspensions (1 cm2 of biofilm surface area) were diluted 1:2 and stained with propidium iodide (PI) and SYTO 9™ (BacLight) for 15 min at room temperature in the dark. Samples were subsequently filtered through 0.2 µm irgalan black-stained polycarbonate membranes and mounted on microscope slides. Counting of dead (red-stained) and live (greenstained) bacterial cells was performed on 20 random microscope fields using an epifluorescence microscope (Nikon Labophot-2, Japan) at 1000 × magnification [START_REF] Freese | Bacterial abundance, activity, and viability in the eutrophic riverWarnow, Northeast Germany[END_REF]. The ratio between live and dead bacterial cells (L/D bacterial ratio) was calculated.

For plankton, total bacterial abundance was determined using a flow cytometer (FacsCalibur, Becton-Dickinson) equipped with a laser emitting at 488 nm. Samples were stained with SYBR Green II (Invitrogen), and an internal standard (yellow-green 0.92 µm fluorescent latex beads) was included. Samples were run at low speed (10-12 µL min-1), and bacterial abundances were calculated as cells mL-1.

Bacterial activity

Bacterial activity in biofilms was estimated using substrate-induced respiration assays [START_REF] Campbell | A rapid microtiter plate method tomeasure carbon dioxide evolved fromcarbon substrate amendments so as to determine the physiological profiles of soil microbial communities by using whole soil[END_REF] as modified by Tlili et al. (2011b). Bacterial activity in plankton was estimated by means of bacterial production using 3H-leucine incorporation assays [START_REF] Tadonléké | Microbial food webs in boreal humic lakes and reservoirs: ciliates as a major factor related to the dynamics of the most active bacteria[END_REF]. For biofilms, deep-well microtiter plates were filled with biofilm suspensions (500 µL) of the corresponding treatment, and 30 µL of glucose (6.2 mg C per well) was added. Controls without glucose were also included to determine basal respiration. Immediately after inoculation, the detection microplate (containing cresol red dye) was sealed to the deep-well plate with a silicone seal. The assembled MicroResp system was incubated in the dark at room temperature for 15 h. After incubation, detection microplate absorbance was measured at 590 nm (Biotek Synergy HT spectrophotometer), amounts of released CO2 were calculated, and the results were expressed in µg CO2 mgAFDW-1 h-1.

For plankton, bacterial production was estimated from 3H-leucine (Leu) incorporation by bacteria after prior determination of saturating concentration [START_REF] Tadonléké | Microbial food webs in boreal humic lakes and reservoirs: ciliates as a major factor related to the dynamics of the most active bacteria[END_REF][START_REF] Tadonléké | Responses of lake bacterioplankton activities and composition to the herbicide diuron[END_REF]. The samples were counted in a scintillation counter (2100-TR, Packard Instruments). The results were expressed in pmol [3H]leucine L-1 h-1 incorporated.

Chlorophyll-a concentration and photosynthetic activity

"In vivo" measurements of chlorophyll-a fluorescence in biofilms and plankton communities were performed using the Phyto-PAM fluorometer (Heinz Walz, Germany). In dark-adapted communities (ca. 30 min), Phyto-PAM excites chlorophyll fluorescence at different measuring lights (470, 520, 645 and 665 nm), and the reading at 665 nm (F0) is proportional to chlorophyll-a concentration [START_REF] Rysgaard | Biomass, production and horizontal patchiness of sea ice algae in a high-Arctic fjord (Young Sound, NE Greenland)[END_REF]. Photosynthetic efficiency measurements were based on the maximal quantum yield (YII) of algae which reflects number of functional photosystem II [START_REF] Schreiber | New type of dual-channel PAM chlorophyll fluorometer to highly sensitive water toxicity biotests[END_REF]. A single saturation pulse was applied to calculate maximal quantum yield, which was later computed considering F0 (steady-state fluorescence) and Fm (maximum fluorescence after saturation pulse) parameters.

Statistical analyses

Three-way ANOVA tests were applied on biological descriptors and on water chemical characteristics to test differences between control and TBZ-contaminated treatments in LP and P-site communities over time. Biological descriptors were classified as functional (substrate-induced respiration, bacterial production, L/D cell ratio, photosynthetic efficiency) and structural (bacterial density and chlorophylla concentration). The sources of variation were calculated for Time (T), Treatment (control, 2 and 20 µg L-1) and Site (clean vs. polluted) and their corresponding interactions. Multiple comparison tests (Tukey test) were also applied on the three-way ANOVA to determine differences between treatments. Prior to all ANOVA analyses, biological and chemical variables were log-transformed and resultant probability values adjusted by the Dunn-Sidak correction. SPSS Statistics 17.0 was used for analyses.

The percentage of difference (D) of biological descriptors in TBZ treatments (Dtreatment) relative to controls (Dcontrol) was calculated using the following equation:

Results

Physical and chemical characteristics of the water

Water temperature and pH conditions were similar between biofilm (Temperature = 20.4-20.6 and pH = 8.5-8.6) and plankton (Temperature = 20.5-20.9 and pH = 8.1-8.3) TBZ experiments, whereas water nutrient concentrations (DOC, DIN and SRP) varied between samples from LP and P sites (biofilms and plankton) and between TBZ treatments (only biofilms) (Fig. 1, Table 1).

Water nutrient concentration comparisons between LP and P sites revealed that DOC concentration was greater in LP-site communities whereas SRP concentration was greater in P-site communities during the 22-day experiment (Fig. 1). Conversely, water DIN concentration was similar between LP and P-site biofilms but not LP and P-site plankton (average DIN in LP 313.4 ± 7.3 µg L-1 and DIN in P 259 ± 13 µg L-1, Table 1). Water nutrient concentrations (especially SRP) tended to increase during experiments, except in P-site plankton communities where decreases of DOC (-62% between Time 0 and Time 3) and DIN (-26% between Time 3 and Time 22) concentrations were observed (Fig. 1).

Water DOC, DIN and SRP concentrations varied between control and TBZ treatments in biofilm experiments but not in plankton experiments (Table 1). In biofilms, TBZ treatments (at 2 and 20 µg L-1) had greater DOC, DIN and SRP concentrations in comparison to controls (except for SRP in LP, Tukey test, Fig. 1). Chronologically, the differences in nutrient concentrations between control and TBZ treatments for biofilms were observed first for DIN (Time 3) and later for DOC and SRP (Time 14) concentrations.

TBZ concentrations

TBZ concentration analyses in water (Fig. 2) confirmed the exposure of biofilm and plankton communities to low and high fungicide doses. TBZ concentrations at the beginning of experiments (Time 0) were 4.6 ± 0.7 µg L-1 (low dose) and 27.3 ± 1.9 µg L-1 (high dose) for biofilms, and 1.8 ± 0.2 µg L-1 (low dose) and 21 ± 1.3 µg L-1 (high dose) for plankton. TBZ concentrations decreased over the three-week experiment, revealing an attenuation of fungicide concentration in the water phase. However, this decrease was greater in biofilms (range 60-75% between Time 0 and Time 22) than in plankton communities (range 15-18%). Secondary metabolites of TBZ, such as hydroxyltebuconazole and desbutyl-tebuconazole, were detected in water at the end of biofilm experiments (Time 22, data not shown).

Bacterial communities

Bacterial densities

Bacterial densities at the beginning of the experiment (Time 0) were similar between LP-site (3.2 × 107 cells cm-2) and P-site (3 × 107 cells cm-2) biofilms but different between LP-site (3.4 × 106 cells L-1) and P-site (6.3 × 106 cells L-1) plankton communities. Cell densities showed significant temporal variation in biofilm and plankton microcosm experiments, but only plankton communities were affected by TBZ (Table 1).

TBZ increased bacterial densities in plankton communities from the two studied sites (Fig. 3). The increases were globally stronger at the high TBZ dose (+ 21% in LP-site plankton and + 48% in P-site plankton; Fig. 3, Tukey P < 0.05) but appeared relatively transient over time (weak differences at Time 22). TBZ effects on biofilm bacteria were also significant but were site-and time-dependent ( Table 1). While TBZ treatment resulted in increases in bacterial densities in LP-site biofilms at Time 3 and Time 6, it tended to decrease bacterial densities in P-site biofilms between Time 6 to Time 10. L/D cell ratios (only available for biofilms) in the controls averaged 0.23 for the LP site and 0.31 for the P site between Time 0 and Time 22 (Fig. 4). The presence of TBZ led to different patterns of decrease in L/D cell ratios in communities from the LP and P sites (average -48% and -35%, respectively) (Site × TBZ effect, Table 1). These effects on biofilm L/D cell ratios were fairly persistent over time and were constant between TBZ doses

Bacterial activity

SIR rates measured in control biofilms from the two sites remained fairly constant over the experiment (1145 ± 117 mgCO2 mgAFDW-1 h-1, average Time 0-Time 22), except for a sharp decrease at Time 3, independently of TBZ exposure (Fig. 5). TBZ decreased SIR in LP-site biofilms (range: -20 to -28%) compared to controls (Site × TBZ effect, Table 1). negative effect was more intense between Time 10 and Time 17 and was independent of TBZ dose (Tukey test, P < 0.05, Fig. 5). Conversely, TBZ tended to stimulate SIR in P-site biofilms (range: + 23 to + 48%), but this effect was not statistically significant.

Bacterial production in plankton increased during the experiments but without an observable effect of TBZ (Table 1). Bacterial production in controls (including LP and P sites) ranged from 6-9 pmol [3H]leucine L-1 h-1at Time 0 to 42-68 pmol [3H]leucine L-1 h-1 at Time 22. It was only at Time 22 (Fig. 5) when slight TBZ effects were observed in LP-site plankton communities (range: + 40 to + 50% compared to controls).

Algal communities

Chlorophyll-a concentration

Chlorophyll-a measurements in control treatments evidenced differences between biofilm and plankton LP and P-site communities ( Table 1). For biofilms, average chlorophyll-a concentration in controls was greater in the LP site (82.7 ± 5.6 µg chl cm-2) than the P site (39.2 ± 3.9 µg chl cm-2), whereas for plankton communities average chlorophyll-a concentration was greater in the P site (35 ± 4.8 µg chl L-1) than the LP site (14.5 ± 0.5 µg chl L-1). Chlorophyll-a values were not affected by TBZ in either biofilms or plankton ( Table 1), but showed significant temporal variations in the microcosms (Time × Site effect; Table 1 and Fig. 6).

Photosynthetic efficiency

Photosynthetic efficiency (YII) values in control treatments stayed constant for biofilms (0.58 ± 0.01 LP site, 0.56 ± 0.008 P site) and plankton communities (0.42 ± 0.03 LP site, 0.52 ± 0.01 P site) over the 22 days of the experiment. TBZ only had a detectable effect on YII in LP-site biofilms (Site × TBZ effect; Table 1, Fig. 7), which experienced a stronger decrease in YII at the higher TBZ dose treatment (-11% on average) than at the lower TBZ dose (-5.3%) (Tukey test, P < 0.05). These decreases were especially marked at the end of experiment between Time 17 and Time 22 ( Fig. 7). Conversely, TBZ had no effect on YII in P-site biofilms or LP-and P-site plankton communities.

Discussion

This research has shown that biofilm and plankton microbial communities exposed to TBZ displayed different responses in some of the biological variables measured, suggesting that the effects of TBZ are modulated by lifestyle (benthic versus planktonic) and pre-exposure to pollutants (less polluted versus polluted).

Biofilm communities experienced functional alterations in presence of TBZ, as shown by a significant decrease in SIR and photosynthetic activity, especially in communities from the LP site. SIR inhibition might be partly explained by a direct TBZ effect on fungal activity in the biofilm (Table 2), as has been reported for soil microbial communities [START_REF] Muñoz-Leoz | Tebuconazole application decreases soil microbial biomass and activity[END_REF]. The observed decrease in SIR may be due to the death of fungi suggested by the increase in water DIN concentrations [START_REF] Cycoń | Microbiological characteristics of a sandy loam soil exposed to tebuconazole and λ-cyhalothrin under laboratory conditions[END_REF]. Unfortunately, fungal biomass was not measured in this experiment, so we are unable to assess the proportion of dead fungi affecting SIR in biofilms. However, other studies support this view. For example, significant decreases in fungal biomass have been observed in benthic assemblages exposed for several weeks to TBZ concentrations [START_REF] Bundschuh | Fungal composition on leaves explaINS pollutant-mediated indirect effects on amphipod feeding[END_REF][START_REF] Artigas | Effects of the fungicide tebuconazole on microbial capacities for litter breakdown in streams[END_REF].

Interestingly, the reduction of SIR in biofilms coincided with increases (day 3 to 10) in bacterial cell mortality (L/D cell ratio), which also suggests a possible direct toxic effect of TBZ on bacterial community. In the same way, TBZ also impaired the photosynthetic efficiency of biofilms from the less polluted site. This effect was probably indirect (given the mode of action of TBZ; [START_REF] Copping | Chemistry and mode of action of crop protection agents[END_REF] and could be the result of indirect effects of pesticides on the relationships between biofilm components [START_REF] Ricart | Effects of low concentrations of the phenylurea herbicide diuron on biofilm algae and bacteria[END_REF]. Previous studies have reported that the herbicide diuron affected both algal and bacterial activity in biofilms, suggesting that pesticides may cause a chain effect in biofilms [START_REF] Ricart | Effects of low concentrations of the phenylurea herbicide diuron on biofilm algae and bacteria[END_REF]. The effects on algae and bacteria observed here might result from direct effects of TBZ on fungal communities and possibly effects on protozoa (Table 2). Conversely, biofilm communities from the polluted site showed more variable responses to TBZ without any significant influence on the biological parameters measured. This finding suggests that these biofilms that were pre-exposed to TBZ and to other types of fungicides (see [START_REF] Montuelle | The periphyton as a multimetric bioindicator for assessing the impact of land use on rivers: an overview of the Ardières-Morcille experimental watershed (France)[END_REF] may have developed adaptational capacities leading to an enhanced tolerance to the TBZ molecule (or "pollution-induced community tolerance"; [START_REF] Blanck | A critical reviewof procedures and approaches used for assessing pollution induced community tolerance (PICT) in biotic communities[END_REF], as already observed in biofilms for other types of pesticides (i.e. the herbicide diuron; Tlili et al., 2011a).

Conversely to biofilms, plankton communities showed a stimulation of bacterial densities in the presence of TBZ. This response was similar between plankton communities from LP and P sites and could be explained by (i) increased levels of nutrients and energy sources released from dead fungi (mostly unicellular forms in plankton; [START_REF] Wurzbacher | Fungi in lake ecosystems[END_REF] and utilized by bacteria, ii) direct TBZ effects on the protozoan community [START_REF] Ekelund | The impact of the fungicide fenpropimorph (Corbel®) on bacterivorous and fungivorous protozoa in soil[END_REF] reducing grazing pressure on the bacterial community, and iii) possible use of TBZ as an extra carbon source by the bacteria. The first two hypotheses are linked to the interaction between plankton microorganisms. For instance, grazing by protozoan communities is one of the major drivers shaping bacterial community structure in plankton [START_REF] Güde | The role of grazing on bacteria in plankton succession[END_REF]. Studies using culture-dependent methods have shown direct reduction of protozoan populations by a variety of fungicides [START_REF] Ekelund | The impact of the fungicide fenpropimorph (Corbel®) on bacterivorous and fungivorous protozoa in soil[END_REF]. Therefore, it cannot be excluded that direct negative effects of fungicides on bacterial grazers occurred in our microcosms, indirectly resulting in increased bacterial densities in the plankton communities (Table 2). The third hypothesis, concerning the microbial decay of TBZ, can be ruled out, since the carbon supplied by TBZ (0.00125 mgC L-1 and 0.0125 mgC L-1 in the 2 and 20 µg L-1 treatments, respectively) was between two and three orders of magnitude below DOC levels in plankton microcosms (2.5-3.5 mg C L-1) and was thus relatively unimportant.

The different effects of TBZ between biofilm and plankton communities could be explained by difference in the nature of microbial communities, pre-exposure to TBZ, and possibly also the specific composition of microbial communities. The pyrosequencing approach employed here on 16S and 18S rDNA genes revealed that TBZ mostly affected the eukaryotic community and that this effect was different between biofilms and plankton (Pascault et al., in prep.). These pyrosequencing results supported our argument that the TBZ-induced reduction in protozoan populations might have led to increased bacterial densities in the plankton experiments. [START_REF] Bending | Fungicide impacts on microbial communities in soils with contrasting management histories[END_REF] also found that the effects of TBZ on fungal and protozoan community structure differed in soils with identical mineralogical composition but different microbial community composition. Apart from community composition, the presence of EPS matrix in biofilms could also have modulated the effects of TBZ compared to plankton (free-living organisms) which is left more exposed to TBZ. This hypothesis is supported by previous work performed with metal contaminants. Indeed, metal toxicity is usually found at higher concentrations in biofilms than free-living organisms, since the structure of biofilms reduces solute penetration into the cells [START_REF] Guasch | Metal ecotoxicology in fluvial biofilms: potential influence of water scarcity[END_REF]. However, the duration and frequency of contamination episodes play a major role, since toxicant diffusion within the biofilm is influenced by exposure time [START_REF] Guasch | Metal ecotoxicology in fluvial biofilms: potential influence of water scarcity[END_REF]. Given the exposure mode, it is reasonable to posit that biofilm (adapted to peak exposure scenarios, at higher TBZ concentrations in this study) and plankton (more adapted to chronic exposure, at lower TBZ concentrations in this study) microbial communities have developed different strategies to cope with contaminants in their ecosystems, but this remains a speculation that needs to be tested.

The attenuation kinetics of TBZ differed significantly between biofilm and plankton experiments, being four times greater in biofilms. Several factors can be put forward to explain the observed differences in TBZ attenuation, including adsorption processes and biotic degradation of the molecule. Previous studies indicate that TBZ is a relatively lipophilic molecule (log Kow = 3.7; [START_REF] Margni | Life cycle impact assessment of pesticides on human health and ecosystems[END_REF] that rapidly adsorbs onto organic matter [START_REF] Artigas | Effects of the fungicide tebuconazole on microbial capacities for litter breakdown in streams[END_REF]. Biofilms possess a matrix of extracellular polymeric substances (EPS) capable of rapidly adsorbing metal ions [START_REF] Guasch | Metal ecotoxicology in fluvial biofilms: potential influence of water scarcity[END_REF] and organic matter ( Freeman and Lock, 1995), including organic pesticides such as TBZ. Conversely, plankton cells are mostly free-living and so develop less EPS, which would mean that they have a lower pollutant adsorption capacity. Another possible explanation could be microbial decay of TBZ [START_REF] Obanda | Biotransformation of tebuconazole bymicroorganisms: evidence of a common mechanism[END_REF], since TBZ degradation metabolites (e.g. hydroxyltebuconazole and desbutyl-tebuconazole) were detected in water samples from biofilm experiments but not in samples from plankton experiments. However, the steep slopes observed in TBZ dissipation curves from biofilms suggested that an active sorption process took place between days 0 and 3 of experiment (~ 50% of total TBZ dissipated). This process may mirror that described for other organic contaminants in biofilms where two sorption phases are identified -a rapid initial process to an "equilibrium" concentration and a second phase of slower sorption [START_REF] Wicke | Experimental and modeling approach to study sorption of dissolved hydrophobic organic contaminants to microbial biofilms[END_REF]. Consequently, it can be argued that TBZ was mostly adsorbed by the EPS matrix rather than degraded by microorganisms, and that this potential to adsorb TBZ was greater in biofilms than in plankton communities.

The different sensitivity between communities from LP and P sites could be explained by the differences in pollution-induced community tolerances of biofilm and plankton microorganisms to TBZ. [START_REF] Blanck | A critical reviewof procedures and approaches used for assessing pollution induced community tolerance (PICT) in biotic communities[END_REF] postulated that toxicants exert a selection of sensitive species which get replaced by more tolerant species in biological communities. This principle concord with the contrasting responses to TBZ observed between biofilm communities from LP and P sites, but not for the plankton. The similar sensitivity to TBZ between P-site and LP-site plankton communities might be explained by the elevated TBZ concentrations used in our experiment, as the nominal concentrations of 2 and 20 µg TBZ L-1 can be encountered during contamination peaks in rivers but not in the lakes studied (mean TBZ concentration of 0.001 µg L-1 in the polluted lake). Therefore, the responses observed in our plankton experiments should be considered as extreme scenarios of TBZ contamination. Finally, increasing the TBZ dose (from 2 to 20 µg L-1) did not significantly enhance TBZ effects on biofilm and plankton community descriptors. Therefore, the toxicity of TBZ for the four types of aquatic microbial communities analyzed was detectable at a relatively low concentration (2 µg L-1 in this experiment). A dose-dependent response is usually expected in toxicity experiments, but many exceptions are observed, as in the case of indirect effects in complex biological communities like biofilms (bearing multiple microbial interactions) where the effects of a toxic compound do not necessarily increase linearly with pesticide dose [START_REF] Ricart | Effects of low concentrations of the phenylurea herbicide diuron on biofilm algae and bacteria[END_REF].

Conclusions

TBZ effects in biofilm and plankton microbial communities were moderate at the 2 and 20 µg L-1 TBZ doses tested, which suggests that the actual TBZ concentrations found in the ecosystems pose relatively little ecotoxicological risk to the ecology of aquatic microbial communities analyzed. However, we found different sensitivities to the TBZ molecule between biofilms and plankton, which can be explained by the specific structure and composition of the microbial community. TBZ effects on non-targeted microalgal and bacterial communities were mostly indirect (Table 2) and seemed to be related to the multiple interactions between biofilm components (i.e. fungal and protozoan communities), though these results need to be confirmed by further investigations specifically targeting those microbial groups ( Pascault et al., in prep.). TBZ toxicity assessment in aquatic ecosystems cannot be exclusively based on single-species responses but should also cover the responses of complex microbial communities (including targeted and non-targeted organisms) to the fungicide. Our study suggests that lifestyle (benthic versus pelagic), pre-exposure to TBZ (less-polluted versus polluted sites) and exposure mode (peak exposure versus chronic exposure scenarios) of microbial communities can accentuate the variability in microorganism responses to the TBZ fungicide. To improve TBZ risk assessment in aquatic microbial communities, further research should be performed by enlarging the spatial and temporal scale of microbial responses to TBZ.

Acknowledgments

This study was funded by the Sendefo project under the French ANR [National Research Agency] Contaminants-Ecosystems-Health (CES-2009) program. The authors thank the Irstea Water Chemistry Laboratory in Lyon for running the chemical analyses and ATT for editing the English-language version of the manuscript.

Conflict of interest

We state that this manuscript does not entail any conflict of interest.