Andre Cardoso Barato

Raphaël Chetrite
email: raphael.chetrite@unice.fr

Haye Hinrichsen
email: hinrichsen@physik.uni-wuerzburg.de

David Mukamel
email: david.mukamel@weizmann.ac.il

Gallavotti-Cohen-Evans-Morriss Like

A C Barato

We investigate a new symmetry of the large deviation function of certain time-integrated currents in non-equilibrium systems. The symmetry is similar to the well-known Gallavotti-Cohen-Evans-Morriss-symmetry for the entropy production, but it concerns a different functional of the stochatic trajectory. The symmetry can be found in a restricted class of Markov jump processes, where the network of microscopic transitions has a particular structure and the transition rates satisfy certain constraints. We provide three physical examples, where time-integrated observables display such a symmetry. Moreover, we argue that the origin of the symmetry can be traced back to time-reversal if stochastic trajectories are grouped appropriately.

Introduction

In Nature most processes are not in thermodynamic equilibrium. For example, whenever a system is exposed to a flux of matter or energy in the stationary state, then it is generally not possible to describe it with standard methods of equilibrium statistical mechanics. For systems out of equilibrium the probability distribution of configurations is normally not known and a theory which allows one to calculate macroscopic quantities is not available. Therefore, general results valid for nonequilibrium processes are of great theoretical interest. This applies in particular to fluctuation relations [1][2][3][4][5][6][START_REF] Jiang | Mathematical Theory of Nonequilibrium Steady States[END_REF][START_REF] Andrieux | [END_REF][9][10], which are fairly general statements valid for any system out of equilibrium.

A large variety of nonequilibrium systems can be modelled as continuous-time Markov jump processes, meaning that the system jumps spontaneously from one classical configuration to the other with certain rates. If these rates obey detailed balance, the system relaxes into a equilibrium state. However, if detailed balance is broken, the system will relax into a nonequilibrium stationary state where the probability current between microstates is non-zero [11]. Maintaining these non-vanishing probability currents requires an external drive which continuously produces entropy in the environment. Remarkably, this entropy production can be quantified without knowing the explicit structure of the environment [12]. It turns out that the average entropy production is zero if and only if detailed balance is fulfilled, meaning that entropy production can be used as an indicator of nonequilibrium.

Since the entropy production depends on the specific sequence of microscopic transitions (the so-called stochastic path), it is a fluctuating quantity with a certain probability distribution [13]. A fluctuation relation is an equation that restricts the functional form of this distribution. There are two different kinds of fluctuation relations, namely, finite time fluctuation relations and infinite time fluctuation relations (see [9]). Finite time fluctuation relations describe the distribution of the total entropy (system + environment) and hold exactly for any time interval [13]. Relations of this kind include the Jarzynski equality [14] and Crooks relation [15]. On the other hand, infinite time fluctuation relations are asymptotically valid for the entropy produced in the environment (which is also known as action functional) [5]. Here we will deal with this infinite time relation, and we refer to it as the fluctuation theorem or the Gallavotti-Cohen-Evans-Morriss (GCEM) symmetry.

Large deviation theory [START_REF] Ellis | Entropy, Large Deviations, and Statistical Mechanics[END_REF][START_REF] Hollander | Large Deviations[END_REF][START_REF] Touchette | [END_REF][19] is the appropriate mathematical framework to investigate the fluctuation theorem. Using these methods the GCEM symmetry can be recast as a symmetry of the large deviation function for the probability distribution of the entropy. This symmetry does not yet allow us to calculate macroscopic observables, but at present it is the most general result for systems out of equilibrium.

Even though the fluctuation theorem is very general, one might argue that it is also very specific in the sense that it is valid only for one particular functional of the stochastic path, namely, the entropy produced in the environment. It is therefore interesting to find out if there are other physically relevant functionals with a similar symmetry. As a first step in this direction, we recently demonstrated that the height of an interface in a certain growth model defines a physically relevant time-integrated current different from the entropy with a symmetric large deviation function [20]. Interestingly, this symmetry can only be observed for a particular system size of the model because only then the network of microscopic transitions acquires a particular form.

The objective of the present paper is to generalize the new symmetry found in [20]. We prove that for a class of jump processes with a particular network of states we can find time-integrated currents different from entropy displaying a symmetric large deviation function. This symmetry is similar to the GCEM symmetry because it restricts the form of a large deviation function related to a time-integrated current. However, it is important to note that our symmetry is different from the GCEM symmetry because it refers to a different time-integrated current and has a slightly different physical origin.

It is known that the GCEM symmetry is a direct consequence of the fact that the entropy is given by the weight of a stochastic path divided by the weight of the time-reversed path. In this sense, the origin of the GCEM symmetry is related to time-reversal. An interesting question, which we address here for a particular case, would be to explain the origin of the symmetry. We show that it is also associated with time-reversal, but in a more hidden way: it comes to light only when we perform an appropriate grouping of stochastic trajectories.

The organization of the paper is as follows. In Sec. 2 we define time-integrated currents and briefly review the fluctuation theorem. In Sec. 3 we prove the new symmetry relation. Some physical examples where our symmetry appears in physically meaningful time-integrated currents are presented in Sec. 4. Before concluding we discuss the origin of the symmetry for a simple four states system in Sec. 5.

2 Time-integrated currents and the fluctuation theorem Continuous-time Markov jump processes are defined by a space of microscopic configurations c ∈ Ω in which the system evolves by spontaneous transitions c → c ′ at rate w c→c ′ . The probability P (c, t) of finding such a system at time t in the configuration c evolves according to the master equation

d dt P (c, t) = c ′ =c P (c ′ , t)w c ′ →c -P (c, t)w c→c ′ . (1)
For simplicity we assume that stationary probability distribution exists P (c) = lim t→∞ P (c, t). If this distribution obeys the condition of detailed balance P (c)w c→c ′ = P (c ′ )w c ′ →c the stationary state is an equilibrium state, otherwise the system is out of equilibrium.

A stochastic trajectory during the time interval [t 0 , t f ] is a sequence of M jumps

-→ C M,t : c(t 0 ) → c(t 1 ) → c(t 2 ) → . . . → c(t M ) (2)
taking place at times t 1 , t 2 , . . . , t M ∈ [t 0 , t f ]. Note that, the length of the time interval T = t f -t 0 is given while the number of jumps M is a random variable that may assume different values for different trajectories. In what follows we assume that all microscopic transitions are such that, i.e. w c→c ′ = 0 ⇔ w c ′ →c = 0, meaning that any stochastic path can be reversed.

Time-integrated currents

A time-integrated current is a functional of the stochastic trajectory

J[ -→ C M,t ] = M-1 i=0 θ c(ti)→c(ti+1) , (3)
which changes its value by θ c→c ′ whenever a jump from c → c ′ occurs. The increments are assumed to be antisymmetric, i.e. θ c→c ′ = -θ c ′ →c . Using the master equation the expectation value is

J = t f t0 dt c,c ′ θ c→c ′ P (c, t)w c→c ′ . (4)
In the stationary state this expression reduces to J = T c,c ′ θ c→c ′ P (c)w c→c ′ , i.e. the current increases on average linearly with T . Since the system relaxes towards a stationary state, in the limit of T → ∞ the quotient J/T tend to a constant, which is lim

T →∞ J T → c,c ′ θ c→c ′ P (c)w c→c ′ . (5)
More specifically, one expects that the corresponding probability distribution P J T = x becomes more and more peaked around this value as T → ∞. Assuming that the large deviation principle holds, the large deviation function of this probability distribution is defined by [START_REF] Ellis | Entropy, Large Deviations, and Statistical Mechanics[END_REF][START_REF] Hollander | Large Deviations[END_REF][START_REF] Touchette | [END_REF], lim

T →∞ P J T = x = exp[-T I(x)]. (6)
The function I(x) measures the rate at which the current deviates from its average value.

Entropy production and fluctuation theorem

A very prominent time-integrated current is the entropy. If we have a jump process describing a physical system in contact with external reservoirs, this quantity describes the amount of entropy which is generated by the external driving in a fictitious external environment [12,13,21]. More specifically, each transition c → c ′ changes the entropy in the environment by ln

w c→c ′ w c ′ →c
. Therefore, the accumulated entropy production along a stochastic path -→ C M,t is a time-integrated current of the form

J s [ -→ C M,t ] = M-1 i=0 ln w c(ti)→c(ti+1) w c(ti+1)→c(ti) . (7)
The fluctuation theorem reads [5]

I s (x) -I s (-x) = -x, (8)
where I s (x) is the large deviation function associated with lim T →∞ P ( Js T = x). This property of the entropic current is also referred to as the GCEM symmetry.

Determining the large deviation function

The master equation ( 1) can be rewritten in the form

d dt P (c, t) = - c ′ Lcc ′ P (c ′ , t), ( 9
)
where L is the Markov generator with elements

Lcc ′ = -w c ′ →c if c = c ′ λ(c) if c = c ′ . (10)
Here

λ(c) = c ′ =c w c→c ′ (11)
denotes the escape rate from configuration c. For each time-integrated current of the form (3), one can now define a modified generator L(z) by

L(z) cc ′ = -w c ′ →c exp(-z θ c ′ →c ) if c = c ′ λ(c) if c = c ′ . (12)
The scaled cumulant generating function Î(z), of the respective time-integrated current J, is defined by lim

T →∞ exp(-zJ) = exp(-T Î(z)). ( 13
)
It can be shown that Î(z) is given by the minimum eigenvalue of the modified generator (12) [5]. Moreover, the Grätner-Ellis theorem [START_REF] Ellis | Entropy, Large Deviations, and Statistical Mechanics[END_REF][START_REF] Hollander | Large Deviations[END_REF][START_REF] Touchette | [END_REF] states that I(x) is given by the Legendre-Fenchel transform of Î(z), i.e.,

I(x) = max z Î(z) -xz , (14)
with z real. Note that Î(0) = 0 because in this case L(z) reduces to the Markov generator L with the minimum eigenvalue 0. The GCEM symmetry [START_REF] Andrieux | [END_REF] in terms of the scaled cumulant generating function of the entropy Îs (z) is

Îs (z) = Îs (1 -z). (15)
The advantage of dealing with the scaled cumulant generating function is that it is easier to calculate in several situations. In the present case it corresponds to determine the minimum eigenvalue of the Perron-Frobenius matrix (12). Note that, the curve Îs (z) has a convex shape, vanishes at z = 0 and z = 1, and reaches its minimum at z = 1/2.

Non-entropic time-integrated currents

Is it possible to find other time-integrated currents with a GCEM-like symmetry? Previous works (see e.g. [9,22]) have shown that such currents do exist. However, these examples are unsatisfactory in so far as the proposed currents differ from the entropy only initially while they become proportional to the entropy in the long time limit. If we call such a current J r , this means that Îr (z) = Îr (E -z). Moreover, since the current J r becomes proportional to entropy in the long time limit, Îr (zE) = Îs (z), i.e., the rescaled scaled cumulant generating functions of J r and J s have the same functional form.

The main objective of the present paper is to show that it is possible to find symmetric currents which differ from the entropy even in the limit T → ∞ so that their rescaled scaled cumulant generating functions differ from Îs (z), even tough they are both symmetric and touch the horizontal axis at the same points (see below in Figs. 4, 6, and8).

Counting the degrees of freedom

Before proceeding let us point out that there are certain restrictions that reduce the degrees of freedom in the space of time-integrated currents. As such currents are specified by an antisymmetric matrix θ cc ′ = -θ c ′ c there are in principle N (N -1)/2 degrees of freedom, where N is the number of states. However, not all of them are independent. To see this let us consider the elementary time-integrated currents

J c→c ′ [ -→ C M,t ] = M-1 i=0 δ c,c(ti) δ c ′ ,c(ti+1) -δ c ′ ,c(ti) δ c,c(ti+1) (16)
from which all other currents can be constructed by linear combination. The current

J c→c ′ [ -→ C M,t
] is simply the number of transitions c → c ′ minus the number of reverse transitions c ′ → c along the stochastic path -→ C M,t . The sum over all destinations

N c = c ′ J c→c ′ (17)
is just the number how often the system reaches the configuration c minus the number how often this configuration is left, hence N c can only take the values 0 and ±1. This implies that

c ′ J c→c ′ T → 0. ( 18
)
in the limit T → ∞, meaning that these particular linear combinations of elementary currents do not contribute to the large-deviation function. This reduces the degrees of freedom by the number of independent relations [START_REF] Touchette | [END_REF], which is maximally N .

A symmetric time-integrated current different from entropy

In this section we prove that for jump processes with a particular network of states and suitably chosen transition rates one can define a current with a symmetric large deviation function which differs from the one for the entropy. The structure of this network is shown in Fig. 1. It consist of configurations c i k organized in columns labeled by a lower index k = 0, . . . , Q -1, each of them including n k different configurations labeled by an upper index i = 1 . . . n k . Spontaneous jumps are allowed only between configurations in neighboring columns with periodic boundary conditions, as indicated by straight lines in the figure. Moreover, we assume that the number of columns Q is even and that even columns carry only a single configuration, i.e. n 0 = n 2 = n 4 = . . . = 1. This forces the system to go through periodically arranged bottlenecks of single configurations.

On this network of configurations we consider the current

J r = Q-1 k=0 θ k n k i k =1 n k+1 i k+1 =1 J c i k k →c i k+1 k+1 , (19)
where θ k are numbers and J c→c ′ are the elementary currents defined in Eq. ( 16). This current increases by θ k if the process process jumps from column k to k + 1 and decreases by the same amount if the system jumps in the opposite direction, no matter which of the configurations within a column is selected.

Counting cycles

Assume that the process starts at a particular configuration, say c 0 . Depending on the yet unspecified transition rates, the process will perform a random walk from column to column. Whenever it returns to its starting point c 0 , it is easy to see that the current defined above will take the value J r = mΘ, where Θ = Q-1 k=0 θ k and m ∈ Z. The number m tells us how often the system completed a cycle through all columns 0 → 1 → 2 → . . . → Q -1 → 0. Therefore, if the rates are chosen in such a way that the random walk through the columns is biased to the right, m will be on average positive. With this picture in mind it is intuitively clear that the expectation value of m in the long-time limit is related to the average current through each of the bottlenecks.

To prove this intuitive argument, we apply the restriction [START_REF] Touchette | [END_REF] to each of the configurations in the network. For single but multiply connected configurations at even columns this restriction tells us that the incoming current is equal to the outgoing current in the long time limit, i.e. for even k we have

n k-1 i=1 J c i k-1 →c k T = n k+1 i=1 J c k →c i k+1 T . (20)
For the simply connected configurations in columns with odd k, we have instead

J c i k →c k+1 T = J c k-1 →c i k T , (21)
where i = 1, . . . , n k . From the above relations it is easy to show that the current in the large deviation regime is given by

J r T = Θ n1 i=1 J c0→c i 1 T . ( 22
)
This result proves that the large deviation properties of the current do not depend individually on the contributions θ k but only on their sum Θ = k θ k . This means that all currents of the form (19) are proportional in the long-time limit and therefore characterized (up to rescaling) by the same large deviation function.

Structure of the characteristic polynomial

We now prove that for suitably chosen transition rates the scaled cumulant generating function of J r exhibits the symmetry (15). Following Ref. [START_REF] Andrieux | [END_REF] we first consider the characteristic polynomial

P (z, x) = det xI -L(z) = π sgn(π) c xδ c,πc -L(z) c,πc (23)
where the sum runs over all permutations π of the available configurations. Here L(z) denotes the modified generator (12) with the matrix elements

L(z) c ′ ,c = -w c→c ′ exp(-z θ c→c ′ ) + δ c,c ′ λ(c) (24)
where λ(c) is again the escape rate (11). Since the off-diagonal entries represent the possible microscopic transitions only those permutations will contribute to the determinant which correspond to a set of nonintersecting transition cycles. In the network of configurations shown in Fig. 1 there are three types of closed transition cycles, namely local cycles which do not change the current J r , and cycles extending over the whole system in positive or negative direction, changing the current by ±Θ (see Fig. 2).

In the determinant this means that permutations corresponding to local cycles are z-independent since the exponential factors drop out. Conversely, complete cycles extending over the whole system contribute to the sum with terms which are proportional to e ±zΘ , respectively. We can therefore split the characteristic polynomial (23) into three parts

P (z, x) = f (x)e -zΘ + f (x)e zΘ + g(x) . (25)
Labelling each complete cycle

c 0 → c i1 1 → c 2 → c i3 3 . . . → c iQ-1 Q-1 → c 0 by a multiindex i := (i 1 , i 3 , . . . , i Q-1
) the first two functions can be expressed as

f (x) = i T i R i (x) , f (x) = i T i R i (x) (26)
where T i and T i are the products of all rates along the cycle in forward and backward direction, respectively. Explicitly,

T i = w c0→c i 1 1 w c i 1 1 →c2 . . . w cQ-2→c i Q-1 Q-1 w c i Q-1 Q-1 →c0 , T i = w c0→c i Q-1 Q-1 w c i Q-1 Q-1 →cQ-2 . . . w c2→c i 1 1 w c i 1 1 →c0 . (27)
Moreover, R comprises all diagonal entries coming from the configurations that are not involved in the cycle:

R i (x) = Q/2 k=1 n 2k-1 j 2k-1 =1,j 2k-1 =i 2k-1 -λ(c j 2k-1 2k-1 ) + x (28)
There is also a complicated expression for g(x) but, as we will see below, this function is not needed for finding the symmetry.

Symmetry condition and derivation of the constraints

Clearly, a symmetry of the characteristic polynomial P (z, x) = P (E -z, x) is a sufficient condition for the symmetry Îr (z) = Îr (E -z) of the minimal eigenvalue. Because of Eq. ( 25) this means that the condition

f (x) = exp(EΘ)f (x) (29)
implies a GCEM symmetry of the current defined in (19). Since f (x) and f (x) are polynomials in x of order N -Q, where N = k n k is the total number of configurations, we can compare the coefficients on both sides. Equating the leading order x N -Q we obtain

exp(EΘ) = i T i i T i . ( 30
)
The above formula says that exp(EΘ) is given by the sum of the product of the transition rates of each possible forward cycle that goes trough all the columns (increasing the current by Θ) divided by the sum of the product of the transition rates of each possible backward cycle.

The comparison of the N -Q remaining orders yields a set of constraints for the transition rates. To this end we rewrite Eq. ( 28) as a power series

R i (x) = N -Q n=0 (-1) n x N -Q-n σ n (Λ i ) , (31)
where

σ n (Y 1 , Y 2 , . . . , Y N -Q ) = 1≤l1<l2<...<ln≤N -Q Y l1 Y l2 .
. . Y ln is the elementary symmetric polynomial and where

Λ i = λ(c j k k ) k = 1, 3 . . . , Q -1; j k = 1, . . . , n k ; j k = i k (32)
is a set of N -Q arguments, consisting of all escape rates which are not part of the cycle labeled by i.

Inserting this power series into Eq. ( 26) and comparing the coefficients in Eq. ( 29) one is led to N -Q constraints for the transition rates of the form

i T i i T i = i T i σ n (Λ i ) i T i σ n (Λ i ) n = 1, 2, . . . , N -Q (33)

Simple solutions of the constraints

Even though these constraints appear to be complicated, they are fulfilled trivially by setting

λ(c i k ) := λ k (34)
for all k = 0, . . . , Q -1 and i = 1, . . . , n k meaning that all configurations in the same column have the same escape rate. In this case the function R i (x) = R(x) does no longer depend on the specific choice of the cycle labeled by the multiindex i, meaning that the polynomials f (x) and f (x) reduce to

f (x) = R(x) i T i , f (x) = R(x) i T i . ( 35
)
Clearly these functions satisfy equation (29), which implies in a symmetric characteristic polynomial and therewith a GCEM-like symmetry of the current (19).

Another trivial solution that fulfills the constraints is the following. Let us take a column k that has more than one state and define the quantity

F i k k = w c k-1 →c i k k w c i k k →c k+1 w c k+1 →c i k k w c i k k →c k-1 , (36)
where i k = 1, 2, . . . , n k . If this quantity is constant in all columns with more than one site, i.e.

F k = F i k k for i k = 1, . . . , n k , (37)
then the ratio

T i T i = Q/2 k=1 F 2k-1 (38)
is independent on the cycle labeled by the multiindex i because the reversal any complete cycle changes the product of the rates always by the same factor, hence we arrive at the symmetry condition

f (x) =   Q/2 k=1 F 2k-1   f (x) . (39)
Finally, we get a larger class of solutions by mixing the conditions (34) and (37), i.e., some of the Q columns with more than one state have constant escape rates while the remaining columns have constant F k . Using similar arguments one can again show that the ratio f /f is constant and, therefore, the constraints (33) are fulfilled. Nevertheless, we note that not all the solutions of the constraints equations are of this type.

The relation between J r and J s

Depending on the transition rates the current J r may be proportional to entropy. If this is the case, then J r displays the GCEM symmetry. However, when J r is not proportional to entropy in the large deviation regime and still with a symmetric large deviation function, then we have a symmetry different from the GCEM symmetry. In the following we derive the condition on the transition rates such that J s is proportional to J r .

For the network of states shown in Fig. 1 the entropy current is given by

J s T = 1 T Q-1 k=1,3,5,... n k i=1 J c k-1 →c i k ln w c k-1 →c i k w c i k →c k-1 + n k i=1 J c i k →c k+1 ln w c i k →c k+1 w c k+1 →c i k . ( 40
)
Using relation (21), in the long time limit the above term divided by T becomes

J s T = 1 T Q-1 k=1,3,5,... n k i=1 J c k-1 →c i k ln w c k-1 →c i k w c i k →c k+1 w c k+1 →c i k w c i k →c k-1 . (41)
Let us first consider the second solution of the constraints (37). In this case the quotient of rates appearing in the logarithm does not depend on i, therefore, using relation ( 20) we obtain

J s T = 1 T   Q-1 k=1,3,5,... ln F k   n1 i1=1 J c0→c i 1 1 . (42)
In this case J r is proportional to the entropy and thus exhibits a GCEM symmetry. However, we can also conclude in opposite direction that for systems obeying the constraint equations (33) in such a way that (37) is not satisfied for all k, then the current J r is not proportional to J s . In this case we expect to have a new type of symmetry different from the GCEM symmetry.

We point out that there is also a multi-dimensional version of the fluctuation theorem [5,[START_REF] Andrieux | [END_REF]23], where the joint probability distribution of a set of currents (which when summed give the entropy) displays a symmetric large deviation function. It is important to note that our symmetry is also different from this multi-dimensional case. This becomes clear if the fluctuation theorem obtained in [START_REF] Andrieux | [END_REF] using the cycle decomposition approach [12] is considered. Following [START_REF] Andrieux | [END_REF], the fact that the product of the transition rates associated with a fundamental cycle in a given direction divided by the product of the transition rates in the opposite direction is independent of the states within the cycle, is analogous to our condition (37) holding for all columns. Transitions network with odd Q So far we have considered transition networks shown in Fig. 1 with an even number of columns Q. The above results can be easily generalized to the case where Q is odd. In this case there are at least two adjacent columns with only one configuration.

In order to do this generalization we can consider an index k that runs only over columns with more than one state. Note that we might also have odd columns with one state. Moreover we take Q as the number of columns with more than one state, such that k = 1, . . . , Q. For the case where we have an even number of columns and all odd columns have more than one state we have Q = Q/2 and k = 2 k -1.

Note that the number of possible complete forward (and backward) cycles is given by Q k=1 n k and only the columns with more than one state are relevant in determining a complete cycle. Now if we consider the multiindex i = {i k} and change definition (32) to

Λ i = λ(c jk k ) k = 1, 2 . . . , Q; j k = 1, . . . , n k; j k = i k (43)
the formulas (30), ( 31) and (33) take the same form if Q is odd. Therefore, the proof also works for the Q odd case.

Examples

In order to illustrate how this new symmetry can be established we consider some examples of physical systems described by jump processes with a network of states of the type shown in Fig. 1, where the current J r has a clear physical meaning. The examples include a molecular motor in a particular network of states, the so-called restricted solid-on-solid (RSOS) model with four sites, and a growth model where the nucleation of the first particle and the completion of the layer take place in the time-scales of the same order.

Molecular motor with four different states

A molecular motor is a biological protein that converts chemical energy into mechanical work by hydrolysis of adenosinetriphosphate (ATP) to adenosinediphosphate (ADP) and phosphate (P) [START_REF] Howard | Mechanics of Motor Proteins and the Cytoskeleton[END_REF] (see also [START_REF] Lacoste | Fluctuation relations for molecular motors[END_REF] for a review in fluctuation relation for molecular motors). Unlike macroscopic motors, which move unidirectionally in a well-defined cycle, a molecular motor performs a random walk driven by the chemical potential difference ∆µ = µ ATP -µ ADP -µ P , moving preferentially forward if ATP is in excess. This allows one to model molecular motors by stochastic jump processes [26]. Assuming that the motor is thermally equilibrated with its local environment the jump rate is proportional to exp(β(∆µ -W )), where W = F l is the mechanical work to transport the cargo with force F over the distance l.

The simplest model of a molecular motor would require a cycle of two states. In such a model the consumption of ATP would be proportional to the mechanical work. However, realistic molecular motors are often characterized by several possible transition cycles, i.e. the motor protein can advance by one step through different paths of intermediate configurations with different energy consumption. In what follows we consider a hypothetical network of four configurations with two transition paths denoted by A and B, as shown in Fig. 3. One interesting question is whether such nano-machine leads to a better efficiency at maximum power than a simple linear chain [27]. Here we are interested in providing a simple example where our theory can be applied. Clearly, this network belongs to the class of networks studied in the previous section with Q = 3 columns.

If both paths are characterized by different chemical potential differences ∆µ A and ∆µ B , the jump rates in positive direction will be proportional to X A = exp [β (∆µ A -F l)] and X B = exp [β (∆µ B -F l)], respectively, while the rates for jump in opposite direction do not depend on the chemical potential. Moreover, we included a constant factor Y in path B, accounting for a possibly different attempt frequency. Finally we assume that the transition rates between c 0 and c 2 are symmetric and equal to 1, as shown in Fig. 3. The corresponding time evolution operator in a canonical basis {c 0 , c A , c B , c 2 } reads

L =    (1 + X A + Y X B ) -1 -Y -1 -X A + X A ) 0 -1 -Y X B 0 Y (1 + X B ) -Y -1 -X A -Y X B 2 + Y    . (44)
We are now going to show that the propagation of the molecular motor defines a time-integrated current with a clear physical meaning that exhibits a symmetry in the large deviation regime that differs from the one for the entropy. To this end, we note that there are two possible complete cycles one going trough c A and the other trough c B . The product of the rates along these cycles in forward and backward direction are given by

T A = w c0→cA w cA→c2 w c2→c0 = X 2 A T A = w c0→c2 w c2→cA w cA→c0 = 1 T B = w c0→cB w cB →c2 w c2→c0 = Y 2 X 2 B T B = w c0→c2 w c2→cB w cB →c0 = Y 2 (45)
Following the procedure described in the previous section we consider the time-integrated current (19). Note that the polynomials ( 26) are of order one for the present model. Comparing the leading order terms in Eq. ( 29) we obtain the condition

exp(EΘ) = T A + T B T A + T B . ( 46
)
Comparing the terms of order x 0 we obtain a constraint on the transition rates of the form

T A + T B T A + T B = T A λ B + T B λ A T A λ B + T B λ A , (47)
where λ A = X A + 1 and λ B = Y (X B + 1) are the escape rate in the configurations C A and C B , respectively. This equation has two solutions, namely T A /T A = T B /T B , for which (37) holds so that the current is proportional to entropy in the large deviation regime, and λ A = λ B , which is the one that gives symmetry different from GCEM. Therefore, we need λ A = λ B , which implies

Y = (1 + X A )/(1 + X B ). ( 48
)
Without loss of generality we choose ∆µ A > ∆µ B , which gives Y > 1. In this case, the above restriction on Y means that the cycles that go trough configurations with higher chemical potential evolve at a slower time-scale.

Assuming this condition to hold, we consider a time-integrated current of the form (19), namely the mechanical work

J m = F l J cA→c2 + J c0→cA + J cB →c2 + J c0→cB (49)
for which the factor (30) is given by In order to demonstrate that this quantity exhibits a non-GCEM symmetry, we compute the smallest eigenvalues Îs (z) and Îm (z) of the modified time evolution operators (12) for both the entropy

E = 1 2F l ln T A + T B T A + T B = 1 2F l ln X 2 A + X 2 B (1 + X B ) 2 (1 + X A ) 2 + (1 + X B ) 2 . ( 50
Ls (z) =     (1 + X A + Y X B ) -X z A -Y X z B -1 -X (1-z) A (1 + X A ) 0 -X z A -Y X (1-z) B 0 Y (1 + X B ) -Y X z B -1 -X (1-z) A -Y X (1-z) B 2 + Y     (51)
and for the mechanical work

Lm (z) =    (1 + X A + Y X B ) -1 exp(F lz) -Y exp(F lz) -1 -X A exp(-F lz) (1 + X A ) 0 -1 exp(F lz) -Y X B exp(-F lz) 0 Y (1 + X B ) -Y exp(F lz) -1 -X A exp(-F lz) -Y X B exp(-F lz) 2 + Y    . ( 52
)
As expected, the large deviation function for the entropy obeys the GCEM symmetry Îs (z) = Îs (1 -z) while the mechanical work fulfills the symmetry Îm (z) = Îm (E -z) with E given in (50). Moreover, plotting I s (z) with I m (Ez) in Fig. 4 we see that the rescaled Legendre transforms of the large deviation functions are different. Therefore, the mechanical work is a time-integrated current with a clear physical meaning that exhibits a new type of symmetry that differs from the GCEM symmetry of the entropy.

Restricted solid on solid growth model in a four sites lattice

The second example is a restricted solid-on-solid (RSOS) model for interface growth [20]. In this model the interface configuration is described by height variables h i ∈ Z residing on the sites i of a one-dimensional lattice. Particles are deposited everywhere with rate q while they evaporate with rate 1 from the edges and with rate p from the interior of plateaus, provided that the restriction h i -h i±1 is not violated (see left panel of Fig. 5). On an infinite lattice one observes that the interface roughens according to the predictions of the Kardar-Parisi-Zhang universality class [28].

Let us now consider a small system with L = 4 sites with periodic boundary conditions. Moreover, let us identify all configurations which differ only by translation either in space or in height direction so that the interface can be in six possible configurations. By doing that, it can be demonstrated that the network of states takes the from shown in the right panel Fig. 5 (see [20] for details). This network belongs to the class of networks studied in the previous section. Moreover, one can easily show that the escape rates λ(c 3 B ) = λ(c 3 A ) = 2q + 2 in column 3 are constant and that the rates across column 1 obey the condition (37),

w c0→c A 1 w c A 1 →c2 w c2→c A 1 w c A 1 →c0 = w c0→c B 1 w c B 1 →c2 w c2→c B 1 w c B 1 →c0 = q 2 p . (53)
Hence, as explained in the previous section, the process fulfills the constraint (33).

The most important time-integrated current of the form ( 19) with a clear physical meaning is the total interface height which increases (decreases) by one whenever a deposition (evaporation) happens. The corresponding modified generator is given by

Lh (z) =        2q + p + 2 -4pe z -e z 0 -2qe -z -2qe -z -qe -z 4p + 4q 0 -e z 0 0 -qe -z 0 q + 1 -pe z 0 0 0 -4qe -z -qe -z 1 + p + 3q -2e z -2e z -2e z 0 0 -2qe -z +2 + 2q 0 -pe z 0 0 -qe -z 0 2 + 2q        (54)
with the factor E = 1 4 ln 3q 4 2p+p 2 , given by relation (30). Again its lowest eigenvalue Îh (z) has to be compared with the lowest eigenvalue Îs (z) of the corresponding modified generator for the entropy current, which reads

Ls (z) =        2q + p + 2 -q z (4p) 1-z -q z 0 -2q 1-z -(2q) 1-z p z -q 1-z (4p) z 4p + 4q 0 -(4q) z 0 0 -q 1-z 0 q + 1 -q z p 1-z 0 0 0 -(4q) 1-z -q 1-z p z 1 + p + 3q -2q z -2 1-z q z -2q z 0 0 -2q 1-z 2 + 2q 0 -(2q) z p 1-z 0 0 -2 z q 1-z 0 2 + 2q        . (55)
Plotting the scaled cumulant generating function in Fig. 6 we can see that both are symmetric and not proportional. Unfortunately, for L > 4 sites the height current is no longer symmetric because in this case the network of transitions is not of the form shown in Fig. 1. In the next example, we introduce a model where the space of states is of the form displayed in Fig. 1 for any system size.

Growth process with instantaneous monolayer completion

As a third example, let us define a growth process with deposition and evaporation which has the special property that after a nucleation the actual monolayer is completed on a very short time scale. The model is defined on a one-dimensional lattice with particles of species α and β, assuming that only α particles can be deposited on top of β particles and vice versa (see Fig. 7). Once a particle is deposited with rate d α,β i at site i on a flat surface, the subsequent event is either the completion of the layer or the evaporation of the particle with rate e α,β i . This describes a limit in which a monolayer is completed almost instantaneously after the first deposition. The transition rates for the completion of a layer is 1 and the transition rate for the reversed event, that is, the evaporation of L -1 particles of a monolayer, is ǫ. The network of configurations of this model is of the form shown in Fig. 1 with Q = 4, where c 0 and c 2 correspond to flat interface configurations while the columns c 1 and c 3 have L states, each corresponding to a single particle nucleated at one of the L sites. In this model the physically relevant time-integrated current of the form (19) is the interface height, which increases by 1/L when a particle is deposited on a flat interface, and by (L -1)/L whenever a monolayer is completed. The product of rates on a forward and backward cycle through c i 1 and c j 3 are T ij = d α i d β j and T ij = ǫ 2 e α i e β j while the escape rates are λ(c i 1 ) = λ α i = 1 + e α i and λ(c i 3 ) = λ β i = 1 + e β i . Hence, the and for the factor (30) we get

E = ln L i,j=1 d α i d β j ǫ 2 L i,j=1 e α i e β j . (58)
The constraints (33) take the form

L i,j=1 d α i d β j ǫ 2 L i,j=1 e α i e β j = L i,j=1 d α i d β j σ k λ α 1 , . . . , λ α i-1 , λ α i+1 , . . . , λ α L , λ β 1 , . . . , λ β j-1 , λ β j+1 , . . . , λ β L ǫ 2 L i,j=1 e α i e β j σ k λ α 1 , . . . , λ α i-1 , λ α i+1 , . . . , λ α L , λ β 1 , . . . , λ β j-1 , λ β j+1 , . . . , λ β L , (59)
where k = 1, 2, . . . , 2L -2. Whenever the deposition and evaporation rates fulfill the above equation, the probability distribution of velocity will symmetric with respect to (58). Again, this symmetry is generally different from the GCEM symmetry. As an example in Fig. 8 we plot the scaled cumulant generating function for the entropy and height for a system with L = 3 sites, where the transition rates are chosen in such a way that the escape rate is constant in columns c 1 and c 3 so that the relation (59) is satisfied.

On the origin of the symmetry

In the following we demonstrate that the new symmetry we proved here also comes from time-reversal. However, in order to see the we have to consider a functional of a group of trajectories instead of a single trajectory. We will restrict our analysis to the four-state system shown in Fig. 3, but we expect the same kind of proof to be valid for general networks of the form shown in Fig . 1. Before going into detail, we provide a simple demonstration of the GCEM symmetry, where it becomes clear that its physical origin is related to time-reversal.

A simple demonstration of the fluctuation theorem

As in Sect. 2 we consider stochastic trajectory with M jumps in the time interval [t 0 , t f ] and jumps from c(t i ) to c(t i+1 ) take place at times t i . We denote this trajectory by -→ C M,t and its weight is given by

W [ -→ C M,t ] = exp[-λ(c(t M ))(t f -t M )] M-1 i=0 w c(ti)→c(ti+1) exp[-λ(c(t i ))(t i+1 -t i )], (60)
where λ(c) is the scape rate from state c. We note that we should also multiply the weight by the probability distribution of the initial state, but we are considering a uniform initial distribution of states. The reversed trajectory, where the system starts at state c(t M ) at time t f (again with the uniform distribution) and jumps from state c(t i+1 ) to state c(t i ) at time t f + t 0 -t i+1 , is denoted by ← -C M,t f +t0-t . The weight of this trajectory is written as

W [ ← - C M,t f +t0-t ] = exp[-λ(c(t M ))(t f -t M )] M-1 i=0 w c(ti+1)→c(ti) exp[-λ(c(t i ))(t i+1 -t i )]. (61)
The entropy current ( 7) is related to the weight of a trajectory divided by the weight of the timereversed trajectory by exp

(-J s [ -→ C M,t ]]) = W [ ← - C M,t f +t0-t ] W [ -→ C M,t ] , (62)
From (62) it follows that

W [ -→ C M,t ] exp(-X)δ(J s [ -→ C M,t ] -X) = W [ ← - C M,t f +t0-t ]δ(J s [ -→ C M,t ] -X) . (63)
Summing over all possible trajectories and using

J s [ -→ C M,t ] = -J s [ ← - C M,t f +t0-t ]
we obtain the fluctuation theorem (8), i.e., P (J s = -X) P

(J s = X) = exp(-X). ( 64
)
This demonstrates that the GCEM symmetry is a direct consequence of the fact that the entropy is the weight of a trajectory divided by the weight of the time-reversed trajectory.

Time-reversal of a group of trajectories

We now consider Markov jump process with the network of states given in Fig. 3. In the following we will denote the states c A and c B by c A 1 and c B 1 , respectively. Here, instead of considering one stochastic trajectory, we consider a certain group of trajectories (or class of trajectories), which can be defined as follows. Two trajectories belong to the same class if they have the same number of jumps taking place at the same times t i , following of the same sequence of columns. For example, the trajectories

c 0 → c A 1 → c 2 → c 0 and c 0 → c B 1 → c 2 → c 0 belong to the same class.
More generally, a trajectory that goes trough column 1 for K different times pertains to a class with 2 K trajectories.

We denote this group of trajectories by { -→ C M,t } and its weight, which is the sum of the weights of each trajectory in the group, by R

[{ -→ C M,t }]. We define a quantity J such that exp(-J[{ -→ C M,t }]) = R[{ ← - C M,t f +t0-t }] R[{ -→ C M,t }] , (65)
where R[{ ← -C M,t f +t0-t }] is the sum of the weights of the reversed trajectories in the group and are using a tilde to denote functionals of the group of trajectories. Note that, unlike the entropy, the current J is a functional of a group of trajectories. What we show next is that in the case λ 1 = λ(c A 1 ) = λ(c B 1 ) (which is the sufficient condition (34) for the symmetry), then the the functional Jr , which is generated by the current J r in the way explained below, is equal to J, provided the increments θ k defined in (19) are chosen appropriately.

The difference between a current that is a functional of the stochastic path -→ C M,t and a current that is a functional of the group of paths { -→ C M,t } is in the increment when there is a jump to an state at column 1. For the second kind of currents the increment is as follows. Let us consider a trajectory which at time t i-1 hops to a state c(t i ) that pertains to the column c 1 , and stays in that state during the time interval t i -t i-1 . The contribution to the current J

[{ -→ C M,t }] is given by ln w c(ti-1)→c A 1 w c A 1 →c(ti+1) exp[-λ(c A 1 )(t i -t i-1 )] + w c(ti-1)→c B 1 w c B 1 →c(ti+1) exp[-λ(c B 1 )(t i -t i-1 )] w c(ti+1)→c A 1 w c A 1 →c(ti-1) exp[-λ(c A 1 )(t i -t i-1 )] + w c(ti+1)→c B 1 w c B 1 →c(ti-1) exp[-λ(c B 1 )(t i -t i-1 )]
.

(66) If the escape rate is constant at column 1 the above term becomes

ln w c(ti-1)→c A 1 w c A 1 →c(ti+1) + w c(ti-1)→c B 1 w c B 1 →c(ti+1) w c(ti+1)→c A 1 w c A 1 →c(ti-1) + w c(ti+1)→c B 1 w c B 1 →c(ti-1)
.

Note that, the same relation holds if the relation ( 37) is valid at column 1. The fact that the scape rates are constant simplify the situation considerably: in this case the dependence on the time interval t i -t i-1 disappears. From now on we assume that the escape rates are constant at column 1.

The current J r is a functional that is invariant within the class of trajectories, i.e., all the trajectories in the same group have the same value of J r . This important property comes from the fact the the current ( 19) is defined in a way such that it does not discriminate between different states in the same column. Hence, J r induces a current Jr defined on a class of trajectories by the relation

Jr [{ -→ C M,t }] = J r [ -→ C M,t ] for -→ C M,t ∈ { -→ C M,t }. ( 68
)
If we consider the current J r with the increments

θ 0 = θ 1 = 1 2 ln w c0→c A 1 w c A 1 →c2 + w c0→c B 1 w c B 1 →c2 w c2→c A 1 w c A 1 →c0 + w c2→c B 1 w c B 1 →c0 , θ 2 = ln w c2→c0 w c0→c2 , ( 69
)
then it is clear that Jr [{ -→ C M,t }] is equal to the current given in (65). It then follows that

R[{ -→ C M,t }] exp(-X)δ( Jr [{ -→ C M,t }] -X) = R[{ ← - C M,t f +t0-t }]δ( Jr [{ -→ C M,t }] -X) . (70)
Now summing over all possible group of trajectories we get P ( Jr =-X) P ( Jr =X) = exp(-X). Since relation (68) gives P ( Jr = X) = P (J r = X) we finally obtain P (J r = -X) P (J r = X) = exp(-X).

(71)

This last relation implies the symmetry for the current J r with the choice of increments (69). However, as we proved in (22), at large time all the currents J r are proportional for any choice of increments θ k . Therefore, this argument shows that the symmetry in the current J r comes also from time-reversal, but at a more coarse grained level. We point out that functionals of a group of paths have been considered also in [29], however in a rather different context.

We believe that the same demonstration can be extended for the more general network of Fig. 1. However, formalizing the proof for this more general case leads to much more cumbersome formulas. We pretend to address this problem in future work.

Conclusion

In this paper we have investigated a new symmetry of time-integrated currents which is generally different from the GCEM-symmetry of the entropy current in the long time limit. It is valid for a very restricted class of Markov processes in the sense that they have the very peculiar network of configurations shown in Fig. 1. Moreover, the symmetry appears only when the transition rates fulfill a certain set of constraints. Nevertheless, to our knowledge, this is the only case where a current not proportional to the entropy in the large deviation regime displays a symmetric large deviation function.

We showed three physical examples where this current is a relevant physical observable. We considered a toy model for a molecular model, where the mechanical work has a symmetric large deviation function different from the one for the entropy. Moreover, we analyzed two growth models where the height displays such a symmetry.

As is the case of the GCEM symmetry, the origin of our symmetry seems to be related to timereversal. However, as we showed in Sec 5 for a 4 states system, the symmetry becomes clear only when we consider a functional of a group of stochastic trajectories. That is, the symmetric current as a functional of the group of trajectories is given by the weight of the trajectories divided by the weight of the reversed trajectories. We demonstrated the symmetry with this grouping of trajectories argument only for the specific 4 states system, but we expect the same kind of proof also to be valid for the whole class of networks considered here.

A natural extension of the this work is to look for other currents, in a more general space of states, with a symmetric large deviation function. This can be done by looking for the conditions that the transition rates have to fulfill in order for the characteristic polynomial of the modified generator to be symmetric. We expect that whenever this characteristic polynomial is symmetric, the origin of the symmetry should be related to time-reversal (of a trajectory or a group of trajectories). One might speculate about a situation where the characteristic polynomial of the modified generator is not symmetric, however its minimum eigenvalue is still symmetric. In this case the origin of the symmetry might be also related to time-reversal, but of some most probable stochastic trajectory dominating a sum over different trajectories. Ultimately, it would be of great theoretical interest for nonequilibrium statistical physics to find out which are the time-integrated currents with a symmetric large deviation function and what might be the origin of these symmetries.

Fig. 1

 1 Fig. 1 Artificial network of configurations c i k with periodic boundary conditions. The process can jump between configurations that are connected by a line.

Fig. 2

 2 Fig. 2 Local transition cycle (left) and complete transition cycle through all columns (right).

Fig. 3

 3 Fig. 3 Configuration network of a molecular motor with two possible transition cycles A and B (see text).

)Fig. 4

 4 Fig.4Scaled cumulant generating functions for F = β = l = 1, ∆µA = 3, and ∆µB = 2. The black line corresponds to the entropy and the red line corresponds to mechanical work (with z → z/E). They are obtained from the minimum eigenvalue of (51) and (52), respectively. They are both symmetric and not proportional.

Fig. 5

 5 Fig. 5 Restricted solid on solid model. Left: Dynamic rules for (a) deposition and (b) evaporation. Right: Transition network of the model with four sites and periodic boundary conditions (see text).

Fig. 6

 6 Fig.6Scaled cumulant generating functions for the RSOS model for p = 0.02 and q = 10. The red line corresponds to height (with z → z/E) and the black line to the entropy, they are obtained from the minimum eigenvalue of (54) and (55), respectively.

Fig. 7

 7 Fig. 7 Possible cycle of transitions for the case L = 4. The α particles are blue and the β particles red. The left-most configuration is a flat interface with β particles.

Fig. 8

 8 Fig. 8 Scaled cumulant generating functions for the growth model with fast monolayer completion. We used L = 3, d α 1 = 1, d α 2 = 2, d α 3 = 3, d β 1 = 2, d β 2 = 2, d β 3 = 5, e α 1 = e α 2 = e α 3 = 2, e β 1 = e β 2 = e β 3 = 3, and ǫ = 0.05. The black line is related to entropy and the red line to height.

polynomials ( 26 )

 26 with degree 2L -2 are given by f (x) =

Acknowledgements The support of the Israel Science Foundation (ISF) is gratefully acknowledged. ACB and RC thank the Weizmann Institute of Science for hospitality.


Lise C Fechne
email: lise.fechner@cemagref.fr

Catherine Gourlay-Francé

Adeline Bourgeault

Marie-Hélène Tusseau-Vuillemin

Diffuse urban pollution increases metal tolerance of natural heterotrophic biofilms

Keywords: metal tolerance, beta-glucosidase, freshwater biofilm, ARISA, environmental quality standards, PICT, heterotrophic communities

This study is a first attempt to investigate the impact of urban contamination on metal tolerance of heterotrophic river biofilms using a short-term test based on β-glucosidase activity. Tolerance levels to Cu, Cd, Zn, Ni and Pb were evaluated for biofilms collected at three sites along an urban gradient in the Seine river (France). Metallic pollution increased along the river, but concentrations remained low compared to environmental quality standards. Biofilm metal tolerance increased downstream from the urban area.

Multivariate analysis confirmed the correlation between tolerance and contamination and between multimetallic and physico-chemical gradients. Therefore, tolerance levels have to be interpreted in relation to the whole chemical and physical characteristics and not solely metal exposure. We conclude that community tolerance is a sensitive biological response to urban pressure and that mixtures of contaminants at levels lower than quality standards might have a significant impact on periphytic communities.

Introduction

Over the past decades, human activities in urban areas have contributed to the increase of chemical contamination of aquatic ecosystems [START_REF] Meybeck | Global analysis of river systems: From Earth system controls to Anthropocene syndromes[END_REF]. Typical urban pollution can be described as a mixture of contaminants at low, sublethal concentrations [START_REF] Tusseau-Vuillemin | Dissolved and bioavailable contaminants in the Seine river basin[END_REF]. It is difficult to evaluate the long-term biological consequences of such diffuse, chronic contamination as it does not engender easily-detectable biological responses in situ [START_REF] Beasley | Assessment of heavy metal and PAH contamination of urban streambed sediments on macroinvertebrates[END_REF][START_REF] Bourgeault | Lessons from a transplantation of zebra mussels into a small urban river: An integrated ecotoxicological assessment[END_REF].

The distinction between chemical-induced changes and changes due to other causes is not always clear as endpoints used to evaluate toxicity may vary due to environmental and/or biological variability [START_REF] Blanck | Pollution-Induced Community Tolerance -a new ecotoxicological tool[END_REF][START_REF] Sabater | Monitoring the effect of chemicals on biological communities. The biofilm as an interface[END_REF][START_REF] Soldo | Long-term effects of copper on the structure of freshwater periphyton communities and their tolerance to copper, zinc, nickel and silver[END_REF]. Assessment of the impacts of urban contaminants in the field therefore requires the use of efficient tools to link biological impairment to chemical contamination.

The PICT (Pollution-Induced Community Tolerance) approach was proposed as an efficient ecotoxicological tool to relate tolerance acquisition at the community level to exposure to toxicants [START_REF] Blanck | A Critical Review of Procedures and Approaches Used for Assessing Pollution-Induced Community Tolerance (PICT) in Biotic Communities[END_REF][START_REF] Blanck | Pollution-Induced Community Tolerance -a new ecotoxicological tool[END_REF]. It relies on the assumption that sensitive components of the exposed community (species, genotypes or phenotypes) will be replaced by more tolerant ones during exposure, thus leading to an increase of community tolerance. Tolerance development can be measured as a shift in the Effect Concentration (usually EC50) that is obtained with a short-term toxicity test based on a physiological endpoint. In particular, the PICT approach was proposed as a tool to demonstrate causative links between toxicants and their adverse biological effects (tolerance levels), due to its ability to discriminate between selection pressures [START_REF] Blanck | A retrospective analysis of contamination and periphyton PICT patterns for the antifoulant irgarol 1051, around a small marina on the Swedish west coast[END_REF], although co-tolerance phenomenons might confound interpretation of the results [START_REF] Blanck | A Critical Review of Procedures and Approaches Used for Assessing Pollution-Induced Community Tolerance (PICT) in Biotic Communities[END_REF].

Biofilms (or periphyton) are complex, natural assemblages of micro-organisms that develop on submerged substrata. They typically consist of bacteria, micro-algae, protozoa and fungi embedded in a polymeric matrix. Because of their short-generation time, these micro-organismes respond quickly to disturbances and are thus regarded as early-warning indicators to detect the effects of toxicants [START_REF] Sabater | Monitoring the effect of chemicals on biological communities. The biofilm as an interface[END_REF].

Moreover, biofilms provide a community-level perspective on biological responses, which is more ecologically relevant to assess the impacts of contaminants than data obtained from single-species tests [START_REF] Clements | Community responses to contaminants: using basic ecological principles to predict ecotoxicological effects[END_REF][START_REF] Lehmann | Induced Metal Tolerance in Microbenthic Communities from Three Lowland Rivers with Different Metal Loads[END_REF][START_REF] Mcclellan | Pollution-induced community tolerance as a measure of species interaction in toxicity assessment[END_REF]. The PICT methodology has been applied using natural biofilms to link tolerance levels to river contamination for instance with herbicides [START_REF] Pesce | In situ relationships between spatio-temporal variations in diuron concentrations and phototrophic biofilm tolerance in a contaminated river[END_REF], using photosynthetic activity as endpoint in toxicity tests, or with Zn, with both photosynthetic and bacterial activity (measured as thymidine incorporation), as endpoints [START_REF] Blanck | Variability in Zinc Tolerance, Measured as Incorporation of Radio-Labeled Carbon Dioxide and Thymidine, in Periphyton Communities Sampled from 15 European River Stretches[END_REF]. Yet ecotoxicological investigations attempting to link the biological responses of collected organisms to the presence of stressors in the field remain scarce [START_REF] Clements | Integrating observational and experimental approaches to demonstrate causation in stream biomonitoring studies[END_REF].

Most PICT studies have focused on tolerance measurements of phototrophic algal communities (periphyton or phytoplankton) exposed to herbicides or Cu (for instance [START_REF] Blanck | A retrospective analysis of contamination and periphyton PICT patterns for the antifoulant irgarol 1051, around a small marina on the Swedish west coast[END_REF]Guasch et al., 1998;[START_REF] Mcclellan | Pollution-induced community tolerance as a measure of species interaction in toxicity assessment[END_REF][START_REF] Serra | Influence of phosphorus on copper sensitivity of fluvial periphyton: The role of chemical, physiological and communityrelated factors[END_REF]. However, in urban fluvial ecosystems, heterotrophic communities play an important ecological role, in particular in bio-mineralization processes. Hence, exoenzymes, which control essential metabolic pathways, are interesting to study the effects of environmental perturbation on the functioning of aquatic ecosystems [START_REF] Admiraal | Extracellular enzyme activity associated with suspended matter in the River Rhine[END_REF]. In particular, βglucosidase activity represents the activity of heterotrophic micro-organisms [START_REF] Chrost | Microbial Enzymes in Aquatic Environments[END_REF]. It is therefore an interesting endpoint for short-term tests designed to measure the tolerance of heterotrophic communities. Moreover, several toxicicy tests used in PICT studies are performed using radioactive reagents such as labeled-thymidine for measurements of bacterial growth or labeled-CO2 for photosynthesis measurements (see for instance [START_REF] Blanck | Variability in Zinc Tolerance, Measured as Incorporation of Radio-Labeled Carbon Dioxide and Thymidine, in Periphyton Communities Sampled from 15 European River Stretches[END_REF]. β-glucosidase activity, in contrast, can be measured using non-toxic fluorescent susbtrates. Recently, it was successfully used as the endpoint in toxicity tests measuring the tolerance to metals of natural freshwater biofilms (Fechner et al., 2010a). Additionally, this short-term test was shown to be a fast and sensitive means to measure biological effects at the community level in microcosms' experiments in which biofilms were exposed to metals (Cd, Ni and Zn) at environmental levels (Fechner et al., 2011a). Since sewers and road and roof run-offs are major metal contamination sources [START_REF] Thévenot | Critical budget of metal sources and pathways in the Seine River basin (1994-2003) for Cd, Cr, Cu, Hg, Ni, Pb and Zn[END_REF], metals are important to consider when trying to assess the impact of urban contaminations.

The objective of our study is to detect pollution-induced effects on fluvial biofilms collected in a typical urban river (the Seine river, France), under chronic, multi-metallic exposure at low, environmental concentrations. For that purpose, we used the recently-developed short-term toxicity test based on βglucosidase activity to detect differences in Cu, Cd, Ni, Pb and Zn tolerance levels of the heterotrophic component of fluvial biofilms collected in situ.

Material and Methods

Biofilms were collected along the Seine river (France) upstream (site 1) and downstream (sites 2 and 3) from the Paris urban area after several weeks of in-situ colonization. Biofilms were analyzed using general descriptors such as dry weight (DW), ash-free dry weight (AFDW) and chlorophyll a (Chl-a) concentrations and biofilm tolerance to metals (Cu, Cd, Ni, Pb and Zn) was evaluated using short-term toxicity tests based on β-glucosidase activity. Modifications of both bacterial and eukaryotic community structures were also assessed with ARISA (Automated-Ribosomal Intergenic Spacer Analysis) fingerprints. Multivariate analysis was then used to study the correlations between tolerance and metal contamination.

Study sites

Biofilms were collected on three sites located along the Seine river (North of France). Site 1 (Marnay-sur-Seine) was located in a non-urbanized area, approximately 200 km upstream from Paris (Strahler: 6, median annual flow 50 m3/s), and sites 2 (Bougival, Strahler: 7, mean annual flow 184 m3/s) and 3 (Triel-sur-Seine, Strahler: 8, mean annual flow 340 m3/s) were both downstream from Paris. In particular, site 3 was located about 20 km downstream the discharge of the sewer treatment plant Seine-Aval (nominal capacity 8 10 6 inhabitants, flow 24 m3/s). Total, dissolved and labile metal concentrations, as well as 15 physico-chemical parameters were monitored monthly at the three sites over one year [START_REF] Priadi | Urbanization impact of the Greater Paris Region on metal mobility in suspended sediments in the Seine River, France: Role of metal oxides[END_REF]. Only the data collected over the period corresponding to this study (3 samplings from May to July 2009, with one sample for total and dissolved metal concentrations and for physico-chemical parameters at each sampling date) was used. P-PO4 and SiO2 concentrations were measured once (July 2009) with the Ascorbic Acid Method and the Silicomolybdate Method respectively. Each labile metal concentration corresponds to three Diffusive Gradients in Thin films (DGTs) [START_REF] Zhang | Performance Characteristics of Diffusion Gradients in Thin Films for the in Situ Measurement of Trace Metals in Aqueous Solution[END_REF] immersed in the river during one month [START_REF] Priadi | Urbanization impact of the Greater Paris Region on metal mobility in suspended sediments in the Seine River, France: Role of metal oxides[END_REF]. DGTs provide an estimate of time-weighted average concentrations of inorganic and weakly-complexed dissolved metals [START_REF] Tusseau-Vuillemin | Dissolved and bioavailable contaminants in the Seine river basin[END_REF].

Collection of periphytic communities

Biofilms were grown on Low Density PolyEthylene membranes (approximately 15 * 30 cm 2 ) vertically attached to plastic crates as in Fechner et al. (2010a). On day 0, 10 membranes were immersed in the river at each site. After 28, 35 and 42 days of colonization, 2 to 3 colonized membranes were collected from each sampling site and carried back to the laboratory in 250 mL glass-bottles placed in a cooler. Membranes were then hand-scraped to collect periphyton and make biofilm suspensions in mineral water (Grand Barbier, Mont-Dore, France) as in Fechner et al. (2010a). Biofilms scraped from the membranes collected at one site were pooled together to provide a unique biofilm suspension for each date and each site. Aliquots of the biofilm suspensions were then assigned to various analyses for periphyton characterization or tolerance measurements. To obtain bacterial and eukaryotic Automated Ribosomal Intergenic Spacer Analysis (ARISA) fingerprints of each biofilm sample, aliquots of biofilm suspensions were centrifuged (15 minutes, 10 000 g, 4°C). Supernatants were discarded and pellets were stored at -80°C for further use.

Biofilm chemical characterization

At each sampling date, biofilm DWs, AFDWs and Chl-a concentrations were measured in triplicate as in Fechner et al. (2010b). C/N ratios were calculated from the particulate organic carbon and nitrogen (POC and PON) concentrations in the biofilm suspensions. POC and PON were determined from biofilm pellets (centrifugation: 2000g, 15 minutes), after acidic digestion (HCl, 10%), with a carbon analyzer (VarioELIII, Elementar, Germany) in triplicate. Concentrations of total accumumated metals were measured in the biofilms collected on day 35 as in (Fechner et al., 2011a). Acumulated metal concentrations were analyzed with an Inductively-Coupled Plasma-Mass Spectrophotometer (ThermoFisher Scientific). The accuracy and the precision of the measuring device were regularly controlled with certified reference natural water samples (NIST 1640).

ARISA fingerprinting of biofilms

ARISA fingerprints were obtained at each sampling date as in Fechner et al. (2010b). Briefly, DNA was extracted from biofilm pellets using the Power Soil DNA Isolation Kit (Mobio Laboratories, Inc., Carlsbad, US). PCR amplification of bacterial 16S-23S ITS was performed using primers ITSF/ITSReub [START_REF] Cardinale | Comparison of Different Primer Sets for Use in Automated Ribosomal Intergenic Spacer Analysis of Complex Bacterial Communities[END_REF] and amplification of eukaryotic ITS1-5.8S-ITS2 regions was performed using primers 2234C/3126T (set euk, appropriate to study eukaryotes: algae, ciliates, etc.) (Ranjard et al., 2001) and ARAlg18S/ITS4 (Fechner et al., 2010b) (set diat, more specific to diatoms). Amplicons were separated on an electrophoresis Bioanalyzer (2100 Electrophoresis Bioanalyzer, Agilent Technologies) and fluorescence data was converted into electrophoregrams using 2100 Expert software (Agilent Technologies).

Electrophoregrams were processed using the StatFingerprints R package [START_REF] Michelland | StatFingerprints: a friendly graphical interface program for processing and analysis of microbial fingerprint profiles[END_REF]. To compare efficiently fingerprint profiles, the area under each curve was normalized to 1 and proximities between pairs of fingerprinting profiles were calculated using the Manhattan distance [START_REF] Lear | Impact of catchment land use on bacterial communities within stream biofilms[END_REF]. The distance matrix thus constructed was analyzed using nMDS -non metric multidimensional scaling -in order to visualize similarities in community structure in 2D. NMDS constructs a twodimensional map showing each fingerprint (ARISA profile) as one plot so that highly similar fingerprints are plotted together. A stress value is used to compute a goodness of fit analysis between the reproduced (in the map) and the actual distances. A stress value < 0.1 corresponds to a good ordination [START_REF] Clarke | Non-parametric multivariate analyses of changes in community structure[END_REF].

Tolerance measurements

The tolerance to metals (Cd, Cu, Ni, Pb, Zn) of heterotrophic biofilm micro-organisms was assessed using the normalized EC50 values obtained with the toxicity test based on β-glucosidase activity developed by Fechner et al. (2011a;2010a). Briefly, biofilms were exposed for one hour to acute concentrations of metals (at least six concentrations of metal varying between 0.001 and 10 mM were tested in triplicate for each Fluorescence measurements were converted into MUF concentrations by calibrating the spectrofluorimeter with a range of MUF solutions prepared in demineralised water. % inhibition of β-glucosidase activity was then calculated using MUF concentrations measured for each metal concentration tested. 50% effect concentrations (EC50) and maximum inhibition levels were determined from the dose/effect curves, which were obtained by plotting the % inhibition of β-glucosidase activity as a function of metal concentration, and by fitting the data to the Hill model [START_REF] Hill | The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves[END_REF][START_REF] Vindimian | REGTOX: The solution for dose-response modeling with Exel[END_REF]. The general form of the curve fitted on the data is:

H H H EC x x x F 50 max % ) ( + × =
where x is the metal concentration, %max is the predicted effect when the toxicant concentration is infinite, and H is the Hill number (slope of the toxicity curve). Each parameter (%max, EC50 and H) was adjusted for the calculation by non-linear regression. 95% confidence intervals around the fitted parameters were estimated using a Bootstrap method. Parameters (EC50 values and maximum % inhibition) were considered significantly different when 95% confidence intervals were not overlapping.

Community tolerance to metals was assessed by calculation of a normalized EC50 value by dividing the EC50 value by the Total Suspended Solids (TSS) concentration of the periphyton suspension used for the short-term test (Fechner et al., 2010a). TSS-normalized EC50 values provide a reliable and robust estimation of metal tolerance and do not depend on experimental conditions.

Cu tolerance levels were measured on the biofilms collected on day 28 at the three sites, Cd and Zn tolerance levels on the biofilms collected on day 35, and Ni and Pb tolerance levels on the biofilms collected on day 42.

Statistical analyses

The influence of the environmental parameters on biofilm tolerance levels and biofilm parameters was investigated using Principal component analysis (PCA). The data was log-transformed and normalized before analysis. Statistical analyses were performed using XLStat 2010 (Addinsoft, France).

Results

Environmental data upstream to downstream the river An increase in metallic concentrations was revealed between site 1 (upstream) on the one hand and sites 2 and 3 (downstream from Paris) on the other hand. Mean metal concentrations (Figure 1) were always higher at the downstream sites, by a factor ranging from 1.3 (ratio of total Ni concentrations between site 1 and site 2) to 10.2 (ratio of total Pb concentrations between site 1 and site 2). Mean total Cu, Cd, Pb, Zn and Cr as well as dissolved and labile Pb concentrations were higher at site 2 compared to site 3, while the other mean metal concentrations were higher at site 3, even though the differences between both downstream sites were not statistically significant. Dissolved metal concentrations remained much lower than Environmental Quality Standards (EQSs) from the European Waterframework Directive for Cd, Ni and Pb. For Cu and Zn, dissolved metal concentrations were in the range of EQSs at both downstream sites (Table 1).

Mean physico-chemical parameters of the river water at the three sites (Table S1) during biofilm colonization in situ correspond to annual mean values reported elsewhere [START_REF] Priadi | Characterization of Bacterial and Fungal Soil Communities by Automated Ribosomal Intergenic Spacer Analysis Fingerprints: Biological and Methodological Variability[END_REF]. Except for calcium which remained stable and pH which tended to decrease upstream to downstream, all parameters tended to increase from site 1 to both downstream sites. It was not always possible to distinguish both downstream sites according to the values of the physico-chemical parameters. The impact of urbanization was reflected by high TSS, DOC, POC, nitrate and phosphate concentrations at sites 2 and 3.

Biofilm characterization

MDS analysis of ARISA fingerprints clearly groups biofilms according to the sampling sites (Figure 2). Both eukaryotic (Figure 2b) and bacterial (Figure 2a) communities reflect differences along the river, as ARISA fingerprints from both downstream sites are close together in the 2D plots. No clear pattern can be observed on eukaryotic fingerprints obtained using set diat for DNA amplification (Figure 2c), which is probably due to the high specificity of this primer set, resulting in a low number of fluorescence peaks on each fingerprint (Fechner et al., 2010b).

Biofilms could not be distinguished between site 1 on the one hand and both downstream sites on the other hand according to classical biofilm descriptors (Table 2). Biomass parameters were highest for the site 2 biofilms (DW, AFDW and Chl-a), but lowest for the site 3 biofilms at each sampling date.

Concentrations of metals accumulated within the biofilms reflected the increase of the multi-metallic exposure along the river as they were always lower for the site 1 biofilm, except in the case of Ni and Cr (Figure 3).

Metal tolerance and maximum inhibition levels

Dose/effect curves obtained from the toxicity tests are shown in Figure 4 (corresponding normalized EC50 values and 95% confidence intervals are given in Table 3). Normalized EC50 values increased along the river and remained similar for both downstream sites for Cu, Cd and Ni as confidence intervals overlap. Metal tolerance levels of the three biofilms generally correlated well with in situ metal concentrations in the water (Pearson correlation coefficients appear in Table S2), except in the case of total Zn and total and labile Pb.

Maximum inhibition levels of β-glucosidase activity are also shown in Table 3. A maximum inhibition level below 100% means that β-glucosidase activity cannot be fully inhibited during metal exposure even at high metal concentrations, which was already identified as a biological response to metal exposure in a previous microcosm study using the same toxicity test (Fechner et al., 2011a). In the present study, maximum inhibition levels tend to decrease upstream to downstream as tolerance increases (Table 3).

PCA of biofilm parameters and metal concentrations

The PCA plot is shown in Figure 5a. The first two components accounted for respectively 85.9 % and 14.1 % of the total variance. Axis 1 reflected the urban gradient as it was positively correlated to metal concentrations (mostly the labile and dissolved metal concentrations) and physico-chemical parameters (only pH was negatively correlated to axis 1). Axis 2 reflected mostly the total metal concentrations (Pb, Cr, Cd, Zn Cu) and the labile Pb concentration. Sites 1, 2 and 3 were displayed along the first axis, according to the urban exposure gradient (Figure 5b). Site 2 appeared in the upper part of the plot as the total metal and CemOA : archive ouverte d'Irstea / Cemagref labile lead concentrations were higher at this site. Figure 5a also shows that all tolerance values were positively correlated and that all the environmental parameters (metal concentrations and physico-chemical parameters) were positively correlated except for pH. It also indicates a closer relationship between tolerance and dissolved and labile concentrations of Cd, Cu and Zn. PCA thus suggested that labile and dissolved metal concentrations provide a better estimation of the effective metal fraction inducing community tolerance to Cd, Cu and Zn than total metal concentrations. Moreover, tolerance levels were not correlated to biofilm gross parameters (DW, AFDW, Chl-a).

Discussion

Identifying the causes of tolerance acquisition in situ

Identifying the biological impacts of toxic pressures is difficult in the field as toxicant-induced effects have to be discriminated from effects induced by other biotic or abiotic factors. Hence, identifying stressors and demonstrating a causative link between stressors and ecologically-relevant biological responses in the field remains a major scientific challenge for ecotoxicologists [START_REF] Clements | Integrating observational and experimental approaches to demonstrate causation in stream biomonitoring studies[END_REF]. The PICT approach proposes to evaluate community tolerance as a biological response to exposure to a toxicant [START_REF] Blanck | Pollution-Induced Community Tolerance -a new ecotoxicological tool[END_REF]. Increase in tolerance is therefore considered to be directly linked to exposure to a specific stressor.

However, proper identification of the factors that are responsible for community tolerance acquisition in situ remains difficult [START_REF] Blanck | Variability in Zinc Tolerance, Measured as Incorporation of Radio-Labeled Carbon Dioxide and Thymidine, in Periphyton Communities Sampled from 15 European River Stretches[END_REF][START_REF] Pesce | In situ relationships between spatio-temporal variations in diuron concentrations and phototrophic biofilm tolerance in a contaminated river[END_REF], and in particular in the present study where biofilms were collected in an area of diffuse, low-level multi-contamination. Indeed co-tolerance phenomenons and variations of both habitat characteristics (for instance light or water flow) and of physico-chemical parameters during exposure have to be taken into account to correctly interpret tolerance measurements.

PCA showed that there was a positive correlation between biofilm tolerance and the majority of environmental parameters, including both metals and physico-chemical parameters. This is not surprising as many parameters are inter-dependent, since urbanization induces a rise in both metal and nutrient concentrations. Variations of environmental parameters like light (Guasch et al., 1998;Guasch and Sabater, 1998), pH [START_REF] Guasch | Effect of copper on algal communities from oligotrophic calcareous streams[END_REF] and nutrients like phosphorus [START_REF] Serra | Influence of phosphorus on copper sensitivity of fluvial periphyton: The role of chemical, physiological and communityrelated factors[END_REF] have been shown to modify periphyton tolerance levels. Therefore, the increase of tolerance measured along the Seine river has to be related to the global urban gradient (both metal concentrations and physico-chemical parameters).

Co-tolerance to metals is a well-known phenomenon in algal and bacterial species and has been observed for indoor-exposed periphyton (Fechner et al., 2011a;[START_REF] Soldo | Long-term effects of copper on the structure of freshwater periphyton communities and their tolerance to copper, zinc, nickel and silver[END_REF]. It is therefore likely to occur also for in-situ exposed communities collected in an urban contamination gradient. In the present study, a significant increase of tolerance to Cu was observed from site 2 to site 3 although mean Cu concentrations in the river and biofilm-accumulated Cu concentrations were similar at both sites. Therefore, it is possible that exposure to metals other than Cu conferred tolerance to Cu for the site 3 biofilm. As regards Ni, biofilm tolerance levels were similar for both downstream sites although the lowest biofilm-accumulated Ni level was observed for the site 2 biofilm. Therefore, the high Ni tolerance of the site 2 biofilm might also be related to co-tolerance to other metals: for instance mean total concentrations of Cu, Pb, Zn and Cd were higher at site 2 (compared to site 3). Therefore, it is not easy to link tolerance and exposure levels in a context of diffuse multi-contamination. Although tolerance acquisition was observed for several metals using the same methodlogy in controlled microcosm experiments (Fechner et al., 2011a), it is thus probable that, for in situ experiments, tolerance acquisition should be used to identify the global impact of urban pressure on communities and not to identify the impact of a specific toxic agent.

Relevance of tolerance measurements using β-glucosidase activity to assess multi-metallic exposure β-glucosidase activity used as endpoint to measure community tolerance appeared to be a particularly sensitive biological variable to detect the impact of multi-metallic exposure. Indeed, no clear relationship was revealed between contamination levels and the other biofilm descriptors, whereas periphyton tolerance to Cd, Cu, Ni, Pb and Zn increased along the river as already observed in a study focusing on biofilm tolerance to Cu (Fechner et al., 2011b). PCA confirmed the positive correlation between metal tolerance of natural biofilms and the metallic contamination. Moreover, the increase of tolerance to metals along the river is concomitant with changes in both eukaryotic and bacterial community structure as revealed by ARISA fingerprints. This observation showed that community tolerance was probably related to species composition. Indeed, disappearance of sensitive species during exposure is one of the phenomenons which leads to tolerance acquisition at the community level [START_REF] Blanck | A Critical Review of Procedures and Approaches Used for Assessing Pollution-Induced Community Tolerance (PICT) in Biotic Communities[END_REF]. Although higher tolerance levels could also be related to acclimatization processes, the similar community structures observed for both downstream sites suggested that heterotrophic communities collected downstream from the urban area were different from the heterotrophic communities collected upstream. Increases of tolerance to metals (Cd and Zn) had already been described in other field studies for natural bacterial communities using thymidine incorporation as the endpoint for tolerance measurements [START_REF] Admiraal | Short-term toxicity of zinc to microbenthic algae and bacteria in a metal polluted stream[END_REF][START_REF] Lehmann | Induced Metal Tolerance in Microbenthic Communities from Three Lowland Rivers with Different Metal Loads[END_REF]).

Yet, biofilms were collected in highly-impacted areas, due to mining activity: even the upstream site was highly contaminated when compared to sites 2 and 3 of the present study. Indeed, in the present study, individual metal contamination levels remained very low. For instance, total Zn concentrations measured at sites 2 and 3 were lower than the range of concentrations monitored in the field study of [START_REF] Blanck | Variability in Zinc Tolerance, Measured as Incorporation of Radio-Labeled Carbon Dioxide and Thymidine, in Periphyton Communities Sampled from 15 European River Stretches[END_REF]. Community tolerance measurements using β-glucosidase activity as endpoint thus appear as good indicators of the biological impacts of a typical diffuse urban metallic exposure.

Tolerance acquisition at low exposure levels

It is noteworthy that an increase of biofilm tolerance to metals down the Seine river was actually detected despite low exposure levels. For instance, in a previous study using β-glucosidase activity measurements in microcosm experiments (Fechner et al., 2011a), no increase in tolerance acquisition was observed after a three-week exposure to Zn at 20 µg/L. In contrast to those results, in the present study, Zn concentrations at downstream sites 2 and 3 were lower in the Seine river, and yet, the increase in Zn tolerance was significant (considering 95% confidence intervals) after a similar exposure time. The higher sensitivity of natural periphyton (as opposed to laboratory-grown periphyton) could arise from a greater complexity of natural communities [START_REF] Serra | Copper accumulation and toxicity in fluvial periphyton: The influence of exposure history[END_REF]. There is also a possibility that metal-resistant bacteria present in biofilms collected at sites 2 and 3 could have been transported to the river by sewage or other contamination sources. This is especially true for site 3 which is located downstream from the effluent of a large sewage treatment plant. The difference in sensitivity could also be related to differences in metal speciation during exposure, as metal speciation might be very different from one study to another. As suggested by PCA, it is probable that labile and dissolved metals are better related to metal-induced tolerance than total metals, at least for some metals. DGT-labile metal concentrations have indeed been shown to provide good estimations of bioavailable metal concentrations to various aquatic organisms [START_REF] Ferreira | More than inorganic copper is bioavailable to aquatic mosses at environmentalaay relevant concentrations[END_REF][START_REF]Office of Water, Office of Science and Technology[END_REF][START_REF] Tusseau-Vuillemin | Performance of diffusion gradient in thin films to evaluate the toxic fraction of copper to Daphnia magna[END_REF]. More recently, [START_REF] Bradac | Cadmium speciation and accumulation in periphyton in a small stream with dynamic concentration variations[END_REF] showed that Cd-accumulation in periphyton may be controlled by either dissolved or DGT-labile Cd concentrations. But the relationship between DGT-labile metal concentrations and bioaccumulation or toxicity to periphyton remains to be investigated. [START_REF] Guasch | Use of multivariate analyses to investigate the contribution of metal pollution to diatom species composition: Search for the most appropriate cases and explanatory variables[END_REF] found that the metal content of periphyton collected in a field study accounted for a significant portion of diatom species variance in the biofilms. In the present study, although biofilm-accumulated metal concentrations are generally positively correlated with metal concentrations in the river water, complex mechanisms of metal accumulation in periphytic communities exposed in situ are brought to light. For instance, biofilm-accumulated Ni concentrations do not correspond to the gradient of Ni exposure measured in the field.

Another explanation to the difference of sensitivity between laboratory-or in situ-exposed biofilms could also be related to the mixture of micropollutants (both metallic and organic) in the Seine river. Mixture toxicity can indeed be observed even when the mixture components are present at low, non-significant effect concentrations [START_REF] Arrhenius | Predictability of the mixture toxicity of 12 similarly acting congeneric inhibitors of photosystem II in marine periphyton and epipsammon communities[END_REF]. Moreover, other typically urban micropolluants (for instance CemOA : archive ouverte d'Irstea / Cemagref PAHs, PCBs, etc.), some of which had already been detected in the Seine river [START_REF] Tusseau-Vuillemin | Dissolved and bioavailable contaminants in the Seine river basin[END_REF], might exert a selection pressure leading to adaptation of biofilms to urban contamination, and thus to metals.

Conclusions

This study was designed as a first attempt to relate heterotrophic biofilm metal tolerance to diffuse urban contamination at low exposure levels using a short-term test based on β-glucosidase activity. The results showed that periphyton tolerance to Cd, Cu, Zn, Pb and Ni increased upstream to downstream from the urban area of Paris (France) along the Seine river. Although the Seine river has long been subjected to metal contamination [START_REF] Meybeck | Historical perspective of heavy metals contamination (Cd, Cr, Cu, Hg, Pb, Zn) in the Seine River basin (France) following a DPSIR approach (1950-2005)[END_REF][START_REF] Thévenot | Critical budget of metal sources and pathways in the Seine River basin (1994-2003) for Cd, Cr, Cu, Hg, Ni, Pb and Zn[END_REF], current mean concentrations of dissolved metals in the water remained low when compared to either EQSs from the European Water Framework Directive or to National Recommended Water Quality Criteria from the US EPA (National Recommended Water Quality Criteria, 2009). Therefore, tolerance measurements at the community level are sensitive and environmentally relevant indicators of an urban disturbance, considered here both in terms of micropollutants and other physico-chemical parameters. Other authors have already pinpointed that, although they are based on ecotoxicological data, EQSs might not protect organisms at the community level [START_REF] Mcclellan | Pollution-induced community tolerance as a measure of species interaction in toxicity assessment[END_REF]. The study thus supports the potential role of biofilms as sensitive indicators of urban pollution in aquatic ecosystems. Moreover, the results suggest that exposure to a mixture of contaminants at levels lower than quality standards might have a significant impact on periphytic communities.

Acknowledgement

This study was supported by ONEMA and is part of the SISTEO project from the Ile-de-France Regional Council. The authors would like to thank Emmanuelle Uher for her great help with metal analysis and Sophie Ayrault for providing access to the ICP-MS platform at the LSCE. They also thank Aurélie Germain and François Versace for their valuable help both in the field and laboratory, Julien Guieu for editing the English version of the manuscript and Renata Behra (Eawag,) for her precious advice on discussing the results. 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2

  toxicity test). Metal exposure levels during the toxicity tests were checked by measuring metal concentrations in the stock solutions by flame AAS (Varian Inc., USA). After one hour of exposure, βglucosidase activity of the metal-exposed biofilms was measured spectrofluorometrically using Methylumbelliferyl) was measured using an LB 941 Tristar Ti fluorescence microplate reader (Berthold Technologies, Bad Wildbad, Germany) (excitation/emission filters: 355 and 460 nm).

Figure 1 .

 1 Figure 1. Histograms of mean total (a), dissolved (b) and labile (c) metal concentrations in the Seine river at sites 1, 2 and 3 over the period corresponding to the experiment (n = 3). Error bars show standard deviations (± σ). The labile metal concentrations were determined by the means of sets of 3 DGTs immersed at each site. Means and σ were calculated on three samplings performed between May and July 2009. For Cd, Zn, Mn and Ag, metal concentrations were multiplied by factors indicated in the histograms for more clarity. A * indicates a significant statistical difference between site 2 or 3 and site 1 (Kruskal Wallis: p < 0.1 and Dunn tests, p ≤ 0.05).

Figure 2 .

 2 Figure 2. PCA plot of environmental (metallic and physico-chemical parameters) and biofilm (tolerance values and biofilm gross descriptors) parameters. a: correlation circle. b: score plot showing the three sites. Abbreviations ared use for more clarity: lab for labile, dis for dissolved and tot for total metal concentrations. Biological parameters are shown in bold letters and physico-chemical parameters in italics.

Figure 3 .

 3 Figure3. Non-metric multidimensional scaling plots derived from the Euclidian distances between ARISA fingerprints obtained using StatFingerprints. 2D stress is respectively 0.10, 0.10 and 0.07 for respectively a) bacterial ARISA; b) eukaryotic ARISA performed using set euk for PCR amplification and c) eukaryotic ARISA performed using set diat for PCR amplification. Samples collected on day 28, 35 and 42 are referred to as "1", "2" and "3" respectively. Biofilms were collected in triplicates.

Figure 4 .

 4 Figure4. Concentrations of metals accumulated within periphyton from the three sites (each site is indicated by its number e.g. "1" for site 1). Concentrations are expressed in µg metal/g of biofilm dry weight and were multiplied by a factor indicated in the legend for Ag, Cd, Mn and Zn for more clarity.

Figure 5 .

 5 Figure 5. Dose/effect curves obtained with the β-glucosidase toxicity tests performed on the biofilms collected at the three sites with Cu, Cd, Ni, Pb and Zn. % inhibition values are shown as ▲ for the biofilms collected at site 1 (dotted line), as • for the biofilm collected at site 2 (dashed line) and as ■ for the biofilm collected at site 3 (full line). For each metal concentration tested, % inhibition of β-glucosidase activity are shown as triplicates. TSS-normalized metal concentrations are expressed as gmetal/gTSS.

  3

	Ag × 10 Ag × 10
	Cd × 10 Cd × 10
	Co Co
	Cr Cr
	Cu Cu
	Mn × 1/100 Mn × 1/100
	Ni Ni
	Pb Pb
	Sb Sb
	Zn × 1/10 Zn × 1/10
	Ag	Cd	Co	Cr	Cu	Mn	Ni	Pb	Sb	Zn
	×10	×10				× 1/100				× 1/10

Table 1 .

 1 Environmental Quality Standards from the European Water Framework Directive (for waters with CaCO3 > 200 mg/L), means and standard deviations of dissolved metal concentrations measured at the three sites during the experiment (means of three individual samplings between May and July 2009) for Cd, Ni, Pb, Zn and Cu. Concentrations are expressed in µg/L. Dissolved Cd concentrations measured at site 1 were either below the detection (first two samplings) or quantification (third sampling) limits.

	metal EQS	site 1	site 2	site 3
	Cd 0.25	0 *	0.008 (± 0.007) 0.015 (± 0.003)
	Ni	20 1.30 (± 0.60) 1.77 (± 0.49)	2.29 (± 0.11)
	Pb	7.2 0.02 (± 0.04) 0.24 (± 0.01)	0.23 (± 0.09)
	Zn	7.8 2.32 (± 0.43) 6.51 (± 3.11)	8.02 (± 1.37)
	Cu	1.4 0.58 (± 0.11) 1.38 (± 0.33)	1.50 (± 0.10)

*

Table 2 .

 2 Mean values and corresponding standard deviations (σ) of dry weights (DW), ash-free dry weights (AFDW), Chl-a concentrations and C/N ratios corresponding to the biofilms collected on the three sites (n=9 as each parameter was measured in triplicate at each sampling date). A * indicates a significant statistical difference between sampling sites (Kruskal Wallis and Dunn tests, p < 0.05).

			over the three sampling
	biofilm parameters	sites	dates
			mean	σ
		1	3.60	1.03
	DW (mg/cm 2 )	2	4.18	0.77
		3	1.23 *	0.31
		1	0.61	0.16
	AFDW (mg/cm 2 )	2	1.59 *	0.27
		3	0.63	0.18
		1	32.85	14.40
	Chl-a (mg/m 2 )	2	79.80 *	27.25
		3	21.15	9.30
		1	6.93	0.25
	C/N ratio	2	6.52	0.30
		3	5.76 *	0.18

Table 3 .

 3 Normalized EC50 values (mgmetal/gTSS) and maximum percentages of inhibition obtained with the dose effect curves. 95% confidence intervals are shown in brackets.

	metal	biological response	site 1	site 2	site 3
		EC50/TSS	0.54 (± 0.12)	2.16 (± 0.27)	3.65 (± 2.49)
	Cu
		% max	96.3 (95.8-96.9)	95.8 (94.4-97.6)	74.0 (67.8-80.4)
		EC50/TSS	0.47 (± 0.04)	2.13 (± 0.33)	1.84 (± 0.22)
	Cd
		% max	100.6 (99.7-101.5)	93.5 (91.7-95.8)	98.4 (97.6-99.2)
		EC50/TSS	3.17 (± 0.46)	14.63 (± 5.45)	16.69 (± 179.97)
	Ni
		% max	79.1 (77.2-81.1)	54.9 (50.0-60.0)	44.6 (38.5-68.2)
		EC50/TSS	27.12 (± 4.33)	80.36 (± 5.17)	170.85 (± 42.60)
	Pb
		% max	92.8 (91.0-94.5)	92.1 (91.7-92.4)	89.9 (88.4-91.7)
		EC50/TSS	1.12 (± 0.21)	2.10 (± 0.54)	4.11 (± 0.55)
	Zn
		% max	95.9 (94.6-97.4)	87.6 (82.9-90.9)	95.1 (91.9-98.6)

CemOA : archive ouverte d'Irstea / Cemagref