Pierre-Alain Toupance

Laurent Lefèvre

Bastien Chopard

Innfluence measurement in a complex dynamical model: an information theoretic approach

Keywords: reachability and observability analysis, Mutual and multi-information, voter model

We address the link between the controllability or observability of a stochastic complex system and concepts of information theory. We show that the most influential degrees of freedom can be detected without acting on the system, by measuring the time-delayed multi-information. Numerical and analytical results support this claim, which is developed in the case of a simple stochastic model on a graph, the so-called voter model. The importance of the noise when controlling the system is demonstrated, leading to the concept of control length. The link with classical control theory is given, as well as the interpretation of controllability in terms of the capacity of a communication canal.

Introduction

Causality is an important concept in many areas of science [START_REF] Pearl | Causality: Models, Reasoning and Inference[END_REF]. It helps to better understand the behavior of complex dynamical systems. In particular, it reveals how the different degrees of freedom of a system influence each other. In this paper we investigate how causality (considered in a pragmatic and intuitive way) can be used to discover efficient control strategies in a complex system.

The key ingredient of our approach is the concept of the most influential components in a complex system. This notion is defined here as the impact of controlling a given variable on the behaviour of the other variables. For instance, one can measure the change in the joint probability distribution (Kulback-Leibler divergence) when the value of a selected variable is imposed. Alternatively, we can measure the variation of an average quantity when a perturbation is applied. The variable for which this change is the most important is labelled as the most influential. Following this procedure we can rank the degrees of freedom of a system from the most to the less influential. Arguably the notion of influence depends on the quantity used to measure the effect of forcing the variable. Then, to control this quantity in the system, it will be more effective to act on the corresponding most influential nodes.

The aforementioned procedure is intrusive in the sense that it requires to act on the system to be able to determine the effects of a perturbation. Here we would like to consider a non-intrusive approach, essentially based on the observation of the system. The non-intrusive approach we proposed is based on a time delayed multi-information on the free system, measure that Schreiber [START_REF] Schreiber | Measuring information transfer[END_REF] has named transfer information. This procedure can be performed by simple sampling on the system variables, even if the underlying dynamics is unknown, like for instance in financial systems.

In order to show that the metrics of information theory are good tools to find the influencers of a dynamic system, we have chosen here as a dynamic system a voting model where the vote of an agent depends on the vote of its neighbours. Voting models have been studied in many articles. For example Castellano and all [START_REF] Castellano | Nonlinear q-voter model[END_REF] have defined a q-voter model in which an agent votes like its neighbours if the opinion is unanimous; otherwise the vote is random. This model has been used by Nycska [START_REF] Nyczka | Anticonformity or independence? insights from statistical physics[END_REF]. Our model is closer to those used by Mobilia [START_REF] Mobilia | On the role of zealotry in the voter model[END_REF] or Masuda [START_REF] Masuda | Opinion control in complex networks[END_REF]. In our case, we want to detect the influencers of the system. Morone [START_REF] Morone | Influence maximization in complex networks through optimal percolation[END_REF] has proposed a numerical method to find them by using the adjacency matrix of the graph. Here we want to study the system without necessarily knowing its topology, using information theory. To verify that the theory of information is a good way to evaluate the influence of an agent, we first make the agent play the role of zealots (an agent that does not change its vote, as defined by Mobilia [START_REF] Mobilia | Does a single zealot affect an infinite group of voters?[END_REF], or inflexible as defined by Galam [START_REF] Galam | The role of inflexible minorities in the breaking of democratic opinion dynamics[END_REF]). In the literature, there are many articles that deal with the effects of zealots on the voter model dynamic: for instance, in [START_REF] Mobilia | On the role of zealotry in the voter model[END_REF], Mobilia studied the role of zealots on the result of a vote and Masuda [START_REF] Masuda | Opinion control in complex networks[END_REF] showed the link between the role of zealots and their degree.

The most influential nodes can be determined by controlling successively each variables and measuring the impact on the average opinion of the entire group. We will show that the same ranking of influence can also be obtained by monitoring the time-delayed multi-infomation.

The determination of the most influential variables has a clear connection with the well developed theory of control, in which observability and controllability of a system are defined and explored. In section 4.7 we make the link between the standard concepts of control theory and our present approach. A important element of our discussion is related to the effect of noise on the possibility to control a system. The voter model shows that in presence of noise the influential nodes cannot force the opinion of the far enough agents, despite the existence of a connecting path. This result shows the limit of some previous approaches about the controllability of systems on a complex network [START_REF] Liu | Controllability of complex networks[END_REF].

The paper is organized as follows: section 2 introduces our voter model, then section 3 demonstrates the link between influence and time-delayed multi-information. Section 4 solves the 1D voter model analytically, in the meanfield regime and gives a formal link between influence and delayed multi-information. The link between the control length and the capacity of a communication channel is also given. Section 4.7 proposes a formulation of the 1D voter model in the usual framework of control theory. Loss of controllability is related to the noise intensity and the cost of controllability is expressed with a Gramian.

Voter Model

Simple models that abstracts the process of opinion formation have been proposed by many researchers [START_REF] Castellano | Statistical physics of social dynamics[END_REF][START_REF] Galam | Competing species dynamics: Qualitative advantage versus geography[END_REF]. The version we consider here is an agent-based model defined on a graph of arbitrary topology, whether directed or not.

A binary agent occupies each node of the network. The dynamics is specified by assuming that each agent i looks at every other agent in its neighborhood, and counts the percentage ρ i of those which are in the state +1 (in case an agent is linked to itself, it obviously belongs to its own neighborhood). A function f is specified such that 0 ≤ f (ρ i ) ≤ 1 gives the probability for agent i to be in state +1 at the next iteration. For instance, if f would be chosen as f (ρ) = ρ, an agent for which all neighbors are in state +1 will turn into state +1 with certainty. The update is performed synchronously over all n agents.

Formally, the dynamics of the voter model can be express as

s i (t + 1) = 1 with probability f (ρ i (t)) 0 with probability 1 -f (ρ i (t)) (2.1)
where s i (t) ∈ {0, 1} is the state of agent i at iteration t, and

ρ i (t) = 1 |N i | j∈Ni s j (t).
(2.2)

The set N i is the set of agents j that are neighbors of agent i, as specified by the network topology.

The global density of all n agents with opinion 1 is obviously obtained as

ρ(t) = 1 n n i=1 s i (t) (2.3)
According to the law of total probability, the probability p i that agent i votes +1 is

p i (t + 1) = (1 -)p Vi (t)) + (1 -p Vi (t)) = (1 -2 )p Vi (t) +
where is the probability to take a decision different from that of the neighborhood and p Vi (t) is the probability that the majority of neighbours of agent i vote 1 at time t.

In what follows, we will use a particular function f , (see Fig. 1)

f (ρ) = (1 -)ρ + (1 -ρ) = (1 -2 )ρ + (2.4)
From now on, the quantity is called the noise. We limit the noise in the range 0 ≤ ≤ 1/2. The upper value = 1/2 corresponds to a blind vote, with probability 1/2 for each outcome. To illustrate the behavior of this model, we consider a random scale-free [START_REF] Barabási | Scale-free characteristics of random networks: the topology of the world-wide web[END_REF] graph G, as simple instance of a social network, as proposed by Newman in the paper Random graph models of social networks [START_REF] Mark Ej Newman | Random graph models of social networks[END_REF]. In a scale-free network, a small number of nodes have many connections. These nodes are the leaders of the social network. And most nodes have very few connections. The majority of voters are represented by these nodes. The scale free graph structure is based on communities built around a leader (see the paper by Andrew Wu [START_REF] Andrew Y Wu | Mining scalefree networks using geodesic clustering[END_REF]). We use the algorithm of Bï¿oela Bollobï¿oes [START_REF] Bollobás | Handbook of graphs and networks: from the genome to the internet[END_REF] to generate a random graph scale free.

Figure 2 shows the corresponding density of agents with opinion 1, as a function of time. We can see that there is a lot of fluctuations due to the fact that states "all 0's" or "all 1's" are no longer absorbing states when = 0.

Characterisation of the influence of an agent

In this study we would like to characterize how the opinion of one agent influences that of its neighbors and that of the entire system. We will first propose an approach based on information theory, and then measure the influence directly by forcing (or controlling) the opinion of one agent. We will show that both characterizations are strongly correlated. The information theoretic quantities that will be considered are the time-delayed mutual information and the time-delayed multi-information. The purpose of considering a time delay is to capture the causal effect of one element on another.

Delayed mutual-and multi-information

Let us consider a set of random variables X i (t) associated with each agent i, taking their values in a set A. For instance, X i (t) = s i (t) would be the opinion of agent i at iteration t.

To measure the influence between agents i and j, we define the τ -delayed mutual information w i,j as

w i,j (t, τ ) = I(X i (t), X j (t + τ )) (3.1) = (x,y)∈A 2 p xy log p xy p x p y (3.2)
with p xy = P(X i (t) = x, X j (t + τ ) = y) p x = P(X i (t) = x) and p y = P(X j (t + τ ) = y)

We also define the τ -delayed multi-information w i to measure the influence of one agent i on all the others

w i (t, τ ) = I(X i (t), Y i (t + τ )) (3.3)
Y i (t + τ ) = k =i X k (t + τ ) (3.4)
These information metrics can be computed by the method of sampling. We consider N = 10 5 instances of the system in order to perform an ensemble average. According to the central limit theorem, we know that, with this number of instances, we obtain a precision of 3 × 10 -2 with a risk of 5% for the approximate values of the probabilities that we compute (see Appendix B for details).

Non-intrusive characterisation of the nodes influences:

delayed multi-information The τ -delayed multi-information can be used as a measure of the influence of opinion of each node i on the vote of the other agents. For instance, Fig. 3 shows w i (τ = 2) in a steady state, where the origin of time is arbitrary. We observe that some agents i exhibit a more pronounced peak of multi-information towards the rest of the system, suggesting that the opinion of these agents may affect the global opinion of all agents. Note that this results is obtained only by probing the systems, without modifying any of its components. For this reason, we describe this approach as "non-intrusive". The algorithms used throughout this paper to numerically evaluate the delayed mutualand multi-informations in the voter model example are described in Appendix C.

As we can see in Fig. 4, the influence of each agent decreases strongly with noise. We deduce that with a high noise it will be difficult to control the system. Brede [START_REF] Brede | Transmission errors and influence maximization in the voter model[END_REF] obtained similar results about the effect of the noise on the control.

Intrusive characterization: forcing

In this section, we consider another way to measure the influence of an agent on the system. We call this approach "intrusive" as it implies a perturbation, and no longer just an observation.

To measure the influence of agent i, its opinion is forced to a chosen value, for instance the value 1. As a result the density (2.3) of opinions 1 in the system

ρ(t) = 1 n n j=1 s j (t) (3.5)
can be averaged over a large number N of independent realizations, to give a quantity ρ(t) i , where the subscript i indicate which agent has been forced to 1. If t is large enough, ρ i no longer depends on t.

The influence can be measured in a steady state, or from the initial state where all agents are initialized uniformly to 0 or 1 with probability 1/2, respectively.

The color representations of the graphs (Figures 5 and6) show that the multi-information give some information about the controllability and the observability of the system. In the case the multi-information is calculated from the initial state, these figures emphasize the link between the multi-information and the influence of an agent. We can then identify the agents that allow the best control of the system when their vote is forced.

With the multi-information, we get the same results as Masuda [START_REF] Masuda | Opinion control in complex networks[END_REF]. Choosing hubs as zealots is a good strategy to control the system and the effect of the control is strongly correlated with the hub's degree.

The measurement obtained in the steady state for the delayed multi-information is different from that observed in the transient regime. Low-impact agents can get a high multi-information by being a proxi of an influential neighbor. In this case, the multi-information rather evaluates the observability than the controlability.

The 1D Voter model

The previous section gave an illustration of the link between influence defined by intrusive forcing and the influence measured by observing the time-delayed multi-information. In this section, we propose an analytical meanfield solution of the voter model, in a one-dimensional topology. This solution will formally specify the proposed links. In particular we will introduce a characteristic control length.

Presentation

We consider the case of n voters organized along a line so that voter i looks at voter i -1 and itself to take its decision.

Chain network have been already studied. Brede [START_REF] Brede | Resisting influence: how the strength of predispositions to resist control can change strategies for optimal opinion control in the voter model[END_REF] has demontred that the influence of indirect control on an agent decreases with the distance from the controlled agent when there is noise.

Agent i = 0 has no left neighbor and will have a controlled dynamics. For instance its opinion will be always 1. The other agents are initialized randomly in {0, 1}.

Since agent 1 is looking at agent 0, its next state will likely to be 1. And so on for agent 2, 3, . . . , n. Intuitively, we could expect that the entire system will become 1, due to the control imposed by agent 1. But noise is changing this conclusion.

If p i (t) is the probability that agent i is 1 at time t, we can write the equation

p i (t + 1) = p i (t)W 1→1 (t) + (1 -p i (t))W 0→1 (t) (4.1)
where W a→b is the probability that the state evolves from a to b. In a meanfield approximation, we can write,

W 0→1 = p i-1 (t)f (1/2) + (1 -p i-1 (t))f (0) W 1→1 = p i-1 (t)f (1) + (1 -p i-1 (t))f (1/2) (4.2)
Before attempting to solve the above system analytically, we can observe its behavior numerically. We can see in Fig. 7 that if the noise is absent ( = 0), the entire system is indeed controlled by the left-most agent whose state is always 1. But, as sson as the noise is increased ( = 0.01) the control is not effective anymore. There is a critical noise = c (n) below which a system of size n can be controlled by the first node, and above which the influence of the driving node is diluted by the noise. Figure 8 shows the density of agents with opinion 1, as a function of time, for different intensities of noise, . We observe in this figure the effect of the system size. For smaller systems, the effect of controlling agent i = 1 is more effective than for larger n.

Probability distribution of opinions

We can determine the probability distribution in the case of the linear voter model. We have With

p i (t + 1) = p i (t)W 1→1 (t) + (1 -p i (t))W 0→1 (t) = p i (t)(W 1→1 (t) -W 0→1 (t)) + W 0→1 (t)
W 1→1 (t) = p i-1 (t)f (1) + (1 -p i-1 (t))f (1/2) = p i-1 (t)(1 -) + (1 -p i-1 (t)) 1 2
and

W 0→1 (t) = p i-1 (t)f (1/2) + (1 -p i-1 (t))f (0) = p i-1 (t) 1 2 + (1 -p i-1 (t))
we obtain

p i (t + 1) = 1 2 - p i (t) + 1 2 - p i-1 (t) + (4.3)
As p 0 (t) = 1, we obtain

p 1 (t + 1) = 1 2 - p 1 (t) + 1 2 (4.4)
Let P (t) be the vector of probability defined by

P (t) =      p 1 (t) p 2 (t) . . . p n (t)     
With this notation, the system can be expressed in a matrix form

P (t + 1) = AP (t) + B (4.5) with A = ( 1 2 -)          1 0 . . . . . . 0 1 1 0 . . . . . . 0 1 1 . . . 0 . . . . . . . . . 0 0 . . . 0 1 1          and B =      1/2 . . .     
This equation can be solved recursively and gives

P (t) = A t P (0) + t-1 j=0 A j B (4.6)
The explicit forms for power matrices A j are given in Appendix A.

Stationnary system

Let us write

Π =      π 1 π 2 . . . π n     
the stationary distribution. From relation (4.5), it obeys It is an arithmetico-geometric sequence which can be solved for all agents i as

AΠ+B = Π ⇔      ( 1 2 + )π 1 = 1 2 ∀i ∈ 2; n , π i = 1 -2 1 + 2 π i-1 + 2 1 + 2 (4.7)
     π 1 = 1 1 + 2 π i = 1 2 + 1 -2 1 + 2 i-1 (π 1 - 1 2 ) (4.8)
with

π 1 - 1 2 = 1 2 1 -2 1 + 2
Further, we can write eq.( 4.8) as

π i = 1 2 1 + 1 -2 1 + 2 i = 1 2 + 1 2 exp -i ln 1 + 2 1 -2 = 1 2 + 1 2 exp - i c (4.9)
where c is defined as

c = 1 ln 1+2 1-2 (4.10)
and referred to as the control length as it gives a value for i above which the exponential falls quickly to zero. It is a characteristic distance from the controlled agent where its inluence starts to fade. We see that, when approaches 1/2, the length of control c converges to 0, which corresponds to a total loss of the controlability of the system. Figure 9 shows that c decreases very quickly to 0 when increases to 1/2.

Average vote of the system

In the case of a stationnary system, we can calculate the average density of agents with vote 1.

S = 1 n n i=1 π i
with n is the number of free agents.

According to (4.9), we have

S = 1 n n 2 + 1 2 n i=1 1 -2 1 + 2 i (4.11)
When = 0, we have and we obtain

n i=1 1 -2 1 + 2 i = 1 -2 1 + 2 n-1 i=0 1 -2 1 + 2 i (4.12) = 1 -2 1 + 2 1 -1-2 1+2 n 1 -1-2 1+2 ( 4
S = 1 2 + 1 2n   1 -2 1 + 2 1 -1-2 1+2 n 1 -1-2 1+2   = 1 2 + 1 2n 1 -2 4 1 - 1 -2 1 + 2 n (4.14)
In Figure 8, we see that the simulations are in agreement with this theoretical result. Indeed, the density of agents who votes 1 oscillates around the mean value S represented by the dashed lines in the figure.

Delayed mutual information

In this section we will compute the influence of an agent based on the τ -delayed mutual information, w i,j (τ ), between agents i and j, as defined in eq. (3.2). These values are obtained by a sampling of the simulation of the 1D voter model, with n = 50 agents. Measurements are performed when the system has reached a stationnary state, that is after t iterations such that all the probabilities A t P (0) are smaller than a certain threshold. In our case, we take the threshold at 10 -4 .

In Fig. 10, we notice that the mutual information w i,j (τ ) is zero if j < i, has a plateau for j < i + τ , shows a peak for j = i + τ , and decreases for j > i + τ . This observation reflects the fact that agent i can only influence agents on its right as the voting decision of an agent is based on the state of its left neighbor. plateau shows the influence of the past j -i iterations. The influence of i over j is maximum for j = i + τ as it takes τ iterations for the vote of i to travel from i to j. For j > i + τ the influence is due to the steady state regime.

The results that we have are consistent with those that Brede [START_REF] Brede | Resisting influence: how the strength of predispositions to resist control can change strategies for optimal opinion control in the voter model[END_REF] obtained recently when he studied the dependence between the vote of the first agent and that of the others. He computed analytically the dependance of the average stationary vote of a node on a distance i to the node controlled, and he obtained that this dependence decreases exponentially with i, as obtained from the mutual information.

In Fig. 11 we consider the behavior of w i,j (j -i). It suggests the following relationship

∀j > i, w i,j (j -i) = α i exp [-λ i (j -i)] (4.15)
where α i and λ i depend on the noise level, .

The coefficients of correlation between ln(w i,j (j -i)) and j, for different values of the noise are found to be between -1 and -0.99, thus confirming the relation proposed in eq. (4.15). The value of α i and λ i can be determined with a least squares method.

Consequently, the value of the delayed mutual information w i,j (j -i) decreases quickly as j departs from i. This reflects the difficulty to control agent j from agent i.

This interpretation is confirmed by Fig. 12 which shows the relation between the values of λ ≡ λ 1 and the control with a = 0.973 and b = -0.003, independent of the value of . The coefficient of correlation is 0.997, in agreement with the proposed linear link between λ and 1/ c .

Control of the density of vote 1

The previous section suggests that the influence of an agent decreases exponentially with the distance to others, with a characteristic length which decreases as the noise increases. This result follows both from studying an intrusive action on the system, or by simply observing it. In this section we exploit this result to find a strategy to control the full system by acting on more than one agent. In practice we consider the situation where agents i = kd are forced to vote 1, where k ∈ {0, 1, 2, . . .} and d is given by the control length c or 1 λ1 . Fig. 13 shows the simulation results for d chosen as 1 λ1 . The density of agents voting 1 increases significantly. The quantity nλ 1 is the number of controlled agents and is a good indicator to evaluate the cost to control the system.

Noise and information capacity

In the previous section, by evaluating the mutual information, we found that the cost of control increased greatly when the noise increases. This result can be related to the notion of capacity, as defined in the standard theory of information. In the linear voter model, agent i + 1 can be considered as a channel of communication where the input message is the vote of agent i at time t and the output message is the vote of agent i + 2 at time t + 2.

The information channel capacity C 2 is defined as (see [8])

C 2 = sup P X i I(X i (t), X i+2 (t+2)) = sup β∈[0;1] I(X i (t), X i+2 (t+2)) (4.17) With P(X i (t) = 1) P(X i (t) = 0) = β 1 -β , we obtain P(X i+1 (t + 1) = 1) = (1 -)P(X i (t) = 1) + P(X i (t) = 0) P(X i+1 (t + 1) = 0) = P(X i (t) = 1) + (1 -)P(X i (t) = 0)
which we write as

P(X i+1 (t + 1) = 1) P(X i+1 (t + 1) = 0) = A β 1 -β oï¿oe A = 1 - 1 - Therefore, P(X i+2 (t + 1) = 1) P(X i+2 (t + 1) = 0) = A 2 β 1 -β where A 2 = (1 -) 2 + 2 2 (1 -) 2 (1 -) (1 -) 2 + 2 (4.18) = 1 -2 (1 -) 2 (1 -) 2 (1 -) 1 -2 (1 -) (4.19) (4.20)
corresponds to a binary symmetric channel, with a probability of error

p e = 2 (1 -)
. We know that the capacity of a binary symmetric channel with a probabiliy of error p e is

C = 1 -H (p e ) (4.21) with H 2 (p e ) = -p e log (p e ) -(1 -p e ) log 2 (1 -p e ) Therefore C 2 = 1 -H 2 (2 (1 -)) (4.22)
Now, we consider all agents from i + 1 to i + m -1 as a channel of communication between agents i et i + m.

We denote C m the capacity of this channel (it depends only on m, the length of the channel). Following the same derivation as before, we obtain

P(X i+m (t + m) = 1) P(X i+m (t + m) = 0) = A m β 1 -β
Since A is a symmetric matrix it can be cast in a diagonal form with an orthonormal basis. The eigenvalues are λ 1 = 1 and λ 2 = 1 -2 . Thus, A m can be expressed as

A m = P T 1 0 0 (1 -2 ) m P with P = 1 √ 2 1 1 1 -1 Thus A m = 1 2 1 + (1 -2 ) m 1 -(1 -2 ) m 1 -(1 -2 ) m 1 + (1 -2 ) m
and we obtain a symmetric binary channel of length m with a probabily of error

m = 1 2 (1 -(1 -2 ) m )
Therefore, the capacity of this channel is

C m = 1 + m log 2 ( m ) + (1 -m ) log 2 (1 -m )
We know that the capacity is an upper bound of the mutual information for each value of . In Fig. 14, the capacity C m is shown as a function of its length m, for different values of the noise. The fact that the capacity C m decreases with m and with the noise, gives another confirmation of the increasing difficulty to control agent i + m by forcing the vote of agent i.

Observability analysis in the case of a voter model with linear topology

The linear voter model analysis given above may be interpreted in terms of reachability or observability using classical tools from system (control) theory (see [START_REF] Athanasios | Approximation of large-scale dynamical systems[END_REF], chapter 4 for an introduction to the control notions used hereafter). Roughly speaking, reachability denotes the posssibility to drive the dynamics of the whole network to a desired value for the state of all nodes by choosing the values of some of them (the inputs) at every time instant. Observability is the dual problem (see e.g. [START_REF] Athanasios | Approximation of large-scale dynamical systems[END_REF]) and denotes the ability to recover the initial state value of all agents from the observation of some of nodes (the outputs). Many results exist related to the reachability analysis of dynamical network systems. Structural analysis aims at analysing the reachability based on the local or global interaction topology within the network [START_REF] Rahmani | Controllability of multi-agent systems from a graphtheoretic perspective[END_REF][START_REF] Liu | Controllability of complex networks[END_REF][START_REF] Liu | Observability of complex systems[END_REF]. The drawbacks of such analyses is that they do not provide any quantitative information about the amount of energy which may be observed in the ouputs or which is required from the inputs. Such a quantitative information is requested when trying to understand the effects of the distance or noise on the reachability and observability of network systems. Therefore, numerous metrics have been defined related to these energies [START_REF] Pasqualetti | Controllability metrics, limitations and algorithms for complex networks[END_REF], including the reachability and observability Gramians which are precisely measures for the control or observation energies and wich are analyzed hereafter in this paper. The computation of the reachability Gramians in the case of target control problems -where inputs are designed to manipulate a group of target nodes rather than the whole network -is the subject of many recent papers [START_REF] Vosughi | Target control and source estimation metrics for dynamical networks[END_REF][START_REF] Henk | A distance-based approach to strong target control of dynamical networks[END_REF]. Qualitative properties and bounds for their eigenvalues have been established [START_REF] Roy | Controllability-gramian submatrices for a network consensus model[END_REF]. These results apply for general interconnexion topologies and many network dynamical processes with a diffusive structure, hence also for most voter models. However, in the case of very simple topologies and dynamics -such as in the case of the linear voter model investigated in this section -it is possible to compute analytically the observability matrix and Gramian and deduce anlytical results on the decay of the output signal energy with the control length and with the noise. It is proved hereafter that one recovers then exactly the behaviour exhibited by the delayed mutual information.

Let us consider as previously a linear topology with n + 1 voting agents. In section 4, we mostly considered the case where agent was forced to vote 1. Here we consider a more general case. For l, m ∈ {0, . . . , n}, forcing the vote of agent l may be considered as a control action, while observing the vote of agent m may be considered as an output measurement. Since we are interested in the deviation from 1/2 of the probability to vote 1 (thus measuring the influence of a forcing action, for instance), we define these deviations as state space variables pi (t) := p i (t) -

1 2 ∈ - 1 2 ; 1 2 (5.1)
for all t ≥ 0 and i ∈ {0, . . . , n}. We will consider in the sequel, with no loss of generality, a forcing of agent 0 vote and an observation of agent n vote, since the influence in the considered linear voter is unidirectional (from left to right). Therefore, the input variable, ũ, and output variable, ỹ, will be defined as

ũ(t) := p 0 (t) - 1 2 ; ỹ(t) := p n (t) - 1 2 (5.2)
Using these state space, input and output variables, the dynamical voter model (4.5) transforms into the state space system

p(t + 1) = Ap(t) + bũ(t) (5.3)
ỹ(t) = c T p(t)
with the state vector p(t) := [p 1 (t), . . . , pn (t)] T ∈ R n and the internal dynamics matrix (generator) A defined as For any time t ≥ 0, any initial probability distribution p(0) := p0 ∈ -1 2 ; 1 2 n and any control (forcing) signal values ũ(t) ∈ -1 2 ; 1 2 , the solution φ (ũ; p0 ; t) of the state space equations (5.3) may be written φ (ũ; p0 ; t) = A t p0 + t-1 j=0 A (t-1)-j bũ(j)

A := 1 2 -         1
(5.6)

Note that the matrix A has a unique eigenvalue λ(A) = ( 1 2 -), with multiplicity n and such that |λ(A)| < 1 (since the noise satisfies 0 ≤ < 1 2 ). Therefore the trajectory (5.6) is bounded when t → ∞ and the dynamical system (5.3) is said stable.

A state p ∈ -1 2 ; 1 2 n is said unobservable if the corresponding output can not be distinguished from the output associated with the zero state, that is if

y(t) = c T φ (0; p; t) = 0 (5.7)
for all t ≥ 0 (in the observability analysis, only the free response dynamics is analyzed and ũ is set to zero). The whole state space system (5.3) is said observable if the set of unobservable states reduces to {0}. With the solution (5.6) and Cayley theorem, it is easy to prove that this is the case if and only if the observability matrix

O n = c ; A T c ; . . . ; (A n-1 ) T c T (5.8)
is full rank or when the infinite observability Gramian

W o := lim n→∞ O T n O n = ∞ k=0 (A k ) T cc T A k (5.9)
is strictly positive definite. The infinite observability Gramian gives additionnal quantitative information about how much the system or a particular state is observable. Indeed, the largest observation energy (i.e. the maximum energy for the output signal) is reached when t → ∞ and equals for any given state space trajectory φ (0; p; t). Therefore, with the appropriate change of state space coordinates, the components of the initial condition (or subspaces) may be re-ordered, from the less to the most observable ones. If some of the infinite horizon observability Gramian eigenvalue are zero, then the corresponding vector spaces are unobservable. If some of these eigenvalues are small, then initial conditions variations in the corresponding subspaces will cause low energy variations in the output signal.

In the linear voter model example, rather than measuring the influence of forcing permanently the agent 0 to vote 1 (with a constant input signal ũ(t) = 1 2 , ∀t ≥ 0) on the vote of agent n, we could instead analyze to effect of considering the initial probability distribution p := [1, 0, . . . , 0]

T ∈ R n+1 (5.11) on agent n+1, by measuring the corresponding observation energy. We will consider a long range time horizon k > n for which the influence of the initial state of agent 1 has reached agent n + 1 in the line. The last row of matrix A k may be written (see Appendix A):

A k (n+1,•) :=        ( 1 2 -) k k n k n-1 . . . k 2 k 1 when k ≥ n ( 1 2 -) k 0 . . . 0 k k . . . k 2 k 1 when k < n
(5.12) According to definition (5.9), since we are measuring the vote of agent n+1, we get for the components of the infinite observability Gramian

W o i,j := ∞ k=0 A k (n+1,i) A k (n+1,j) (5.13)
for all i, j ∈ {1, . . . , n}. Measuring the influence of the initial vote of agent 1, we start with the initial probability distribution (5.11) and get, for the agent n + 1, the observation energy

ỹ 2 2 := ∞ k=0 A k (n+1,1) 2 = ∞ k=0 ( 1 2 -) 2k A k (n+1 ,1) 2
(5.14) With equation (5.12), one gets

ỹ 2 2 = ∞ k=n ( 1 2 -) 2k k n 2 (5.15) = 1 (n!) 2 ( 1 2 -) 2n ∞ p=0 ( 1 2
-) 2p (p + n)! p! 2
Using the lower bound

(p + 1) n < (p + n)! p! (5.16) one gets 1 ((n -1)!) 2 ( 1 2 -) 2n 4(3 -2 ) (1 + 2 ) 3 < ỹ 2 2
(5.17)

On the other hand, since

∞ p=0 ( 1 2 -) 2p (p + n)! p! 2 ≤ ∞ p=0 ( 1 2
-) p (p + n)! p! 2 =   k≥n ( 1 2 -) k-n k(k -1) . . . (k -(n -1))   2 = n!2 n+1 (1 + 2 ) n+1 2 (5.18)
we get the following upper bound for the observation energy

ỹ 2 2 < 4(1 -2 ) 2n (1 + 2 ) 2n+2 (5.19)
It is worthwile to notice how this upper bound behaves with the number of agents along the line and with the noise . For instance, the upper bound (5.19) decreases with the number of agents and the corresponding observation energy is divided by two when k supplementary agents are added in the line, with The lower bound (5.17) decreases similarly, with the same order, when the noise decreases. However it decreases much faster with the number of agents in the voter line since this lower bound for the observation energy is divided by 1-2 2n 2 when only one agent is added to the n previous ones. Note that we performed the observability analysis on the linear voter model. We could as well develop the dual reachability analysis for the same example. In this analysis, the initial condition is assumed to be zero and one analyzes the forced solution of the state space model (5.3). More specifically, one could be interested in its reachability property. A state p is said reachable when there is a an input signal ũ(t) such that lim t→∞ φ (ũ, 0, t) = p

k ≥ 1 2 1 log 2 1+2 1-2 = c 2
It may be proved (see, e.g. [START_REF] Athanasios | Approximation of large-scale dynamical systems[END_REF]) that, among those input signals which can reach the state p from a zero intial condition, the one with minimum energy may be written as

ũ * 2 2 = pT (W c ) -1 p
where the infinite reachability Gramian W c is defined as

W c := ∞ k=0 A k bb T (A T ) k (5.22)
Therefore, a reachability Gramian analysis may be used to compute the forcing of agent 1 with minimal energy requested to reach a state p where all agents in the line vote 1, that is such that pi = 1, for all i ∈ 1, . . . , n + 1. However, in this case, it would be necessary to compute the sum of all the elements in (W c ) -1 , which is a much more involved computation than the one performed for the observability analysis. Besides, the duality between reachability and observability for linear systems [START_REF] Athanasios | Approximation of large-scale dynamical systems[END_REF] and the particular topology of the linear voter model lead us to the conjecture that the reachability analysis would not bring any new result fundamentally different from the ones obtained through the observability analysis.

Conclusions

In this paper we show that time delayed mutual-and multi-informations are promising tools to better grasp the behavior of a dynamical system on complex networks. In particular it can be used to determine the most influential degrees of freedom and the most observable variables. This knowledge can be obtained without perturbing the system, by just probing its behavior.

We claim that influential nodes are those that are the most interesting to control or monitor to (i) force a system to reach a given target, or (ii) to have a proxy giving an information on the state of the entire system.

We illustrated our approach in a simple stochastic dynamical model on a graph, a so-called voter model, where agents iteratively adapt their opinion to that of the majority of their neighbors, with however a given noise level. We first discussed the case of a general scale-free topology, where only numerical results can be obtained. Then we consider a 1D topology for which analytical results can be obtained. There, we rigorously showed that the influence of an agent on the entire system can be equivalently measured by actually forcing its behavior, or, in a non-intrusive way, by measuring the time delayed multi-information of this agent with respect to the rest of the system. In particular, we proposed the concept of a control length, which indicates a characteristic distance above which the influence of a controlled agent fades exponentially.

The link with classical control theory has been proposed and the control length has been related to the reachability Gramian, thus indicating that the cost of control becomes intractable at large distance. The importance of the noise is clearly shown as being a central element in the possibility of observing or controlling a system, as opposed to previous literature that claimed that a causality path was sufficient to achieve control [START_REF] Liu | Controllability of complex networks[END_REF].

As an additional link of our approach to existing concepts, we showed that controllability can also be considered in the framework of the capacity of communication channel, as defined in information theory by Shannon. We showed that this capacity drops as agent are separated by a distance above the control length.

In a forthcoming paper we will apply our approach to other complex systems, in particular those for which the underlying dynamics and topology of interaction are not known. We already obtained (not shown here) that the time delayed multi-information can be used to infer the topology of the graph of Fig. 5. Further, we want to use the concept of observability as a way to detect early warning signal of possible tipping points in a complex dynamical system. In simple words, we want to analyze the idea that the most influential degree of freedom is the best variable to observe to know in advance if a given system is likely to move to another regime. These nodes being the most influential ones, we can argue that their evolution will dictate the evolution of the other variables.

A p = ( 1 2 -) p          1

Figure 1 :

 1 Figure 1: The probablility f (ρ) used in this study. The noise is visible as the values f (0) and 1 -f (1).

Figure 2 :

 2 Figure 2: Graph of the time evolution of the density of opinion 1 with noise = 0.001 and n = 200 agents connected through a scale-free network.

Figure 3 :

 3 Figure 3: τ -delayed multi-information w i (τ ) (τ = 3) as a function of i, for graph G with n = 50 agents and noise level = 0.001.

Figure 4 :

 4 Figure 4: τ -delayed multi-information w i (τ ) (τ = 3) as a function of i, for graph G with n = 50 agents and with different noise level = 0.0001 (black curve) , = 0.001 (blue curve), = 0.01 (green curve) and = 0.05 (red curve).

Figure 5 :

 5 Figure 5: Scale free graph colored as a function of the values of the influence (left) and the τ -delay multi-information (right), for τ = 4. In this case, the multi-information is computed from the initial state.

Figure 6 :

 6 Figure 6: Scale free graph colored as a function of the values of the influence (left) and the τ -delay multi-information (right), for τ = 4.In this case, the multi-information is computed when the system is in a steady state regime.

Figure 7 :

 7 Figure 7: Space-time diagram of the evolution of the states of a n = 500 agents of a voter model organized in a line. Line t of the figure depicts the configuration of the n voter at iteration t. We can see the first 500 iterations. Left: = 0. Right: = 0.01 .

Figure 8 :

 8 Figure 8: Density of agents with value 1 as a function of time, for different noise levels, and two different system sizes, n = 500 and n = 50. The dashed lines are the predictions of the meanfield analytical approach, see eq. (4.14).

Figure 9 :

 9 Figure 9: Control length c as a function of the noise , according to eq. (4.10).

Figure 10 :

 10 Figure 10: Delayed mutual information, w i,j (τ ), as a function of agent j, for different values of i. The different curves correspond to i = 1, 2, 5, 10, 20 et 40, from left to right, respectively. The vote of agent i = 0 is forced to 1 and the noise is = 0.01. The delay is τ = 1 (left panel), τ = 3 (middle panel) and τ = 5 (right panel).

Figure 11 :

 11 Figure11: Delayed mutual information w i,j (j -i) as a function of j, for agents i = 1, 2 5, 10, 20 and 30 (curves from left to right, respectively). The vote of agent i = 0 is forced to 1 and the noise is = 0.001 (left panel), = 0.01 (middle panel) and = 0.05 (right panel).

Figure 12 :

 12 Figure 12: λ as a function of 1/ c

Figure 13 :

 13 Figure 13: Evolution of the density of vote 1 during the time when only the vote of the agent 1 is forced to 1 (blues curves) and when every vote of the agents j ∝ 1 λ is forced to 1 (greens curves). left = 0.001. middle : = 0.01, right: = 0.05.

Figure 14 :

 14 Figure 14: Capacity Cm of the channel between agent i and i + m, as a function of m, for different noise levels .

4 )

 4 The control column matrix b and observation row matrix c T are defined respectively as

  ỹ

(5. 20 ) 2 2 = O ( 1 - 2 ) 2 -

 202122 When the noise increases, the observation energy upper bound decreases ỹ 2n → 0 as → 1

C

  Appendix A. Explicit evaluation of A pIn 4.6, we need to calculate A p with is a nilpotent matrix and ∀k ∈ N, A k = k p=0 k p C p . Therefore, for p n -1:

  0 . . . . . . 0

	p 1 p 2 . . .	. . . . . . . . .	. . . . . . . . .	. . . . . . . . . 0 . . .
	p
	n-1

Acknowledgment

We thank Gregor Chliamovitch and Alex Dupuis for initiating several of the ideas developed in this paper, during the FP7 project SOPHOCLES (2012-2015), and for the reading of the manuscript.

Appendix B. Accuracy and confidence for the numerical evaluation of probability distributions

Let us consider an attribute of the members of a population which appears with probability p. For a sample of size n drawn in this population, let F n be the random variable equal to the proportion of those elements having this attribute. According to the Moivre-Laplace theorem, the quantity

converges in distribution to a Gaussian distribution

where t 1-α/2 is the real number defined by P(X

n . For N = 10 5 and α = 0, 05, we have t 1-α/2 = 1.96, we obtain an approximation value of p with a precision of 0.03, with a risk of 5%.

Appendix C. Algorithms for the computations of mutual and multi-information

We consider a scale free graph G, with n agents. To compute the τ -delayed mutual information between 2 agents i and j, we generate N runs. For every run, we have a matrix S defined by: for i ∈ 1, n , and for j ∈ 1, t + τ , such that S[i][j] is the state of the agent i at the moment j.

We use 4 n × n matrix (N00, N01, N10 and N11), initialized to zeros. For every run, we compare the vote of the agent i at the time t and the vote of the agent j at the time t + τ

We then compute w i,j (t, τ ), the τ -delayed mutual information at time t between agents i and j, according to definition (3.2). We obtain

. Multi-information

To compute the delayed multi-information, as for the delayed mutual information, we execute N runs, and for every run, we compute the state matrix S. We use two n × n matrix, N 0 and N 1 (n is the number of agents) defined by : ∀(i, j), N 0[i][j] is equal to the number of runs where the vote of the agent i is 0 and the number of agents who voted 1 is j -1 at time t + τ . ∀(i, j), N 1[i][j] is equal to the number of runs where the vote of the agent i is 1 and the number of agents (without the agent i) who voted 1 is j -1. These matrices give us the probability distribution of the couple of random variable (X i (t), Y i (t + τ )) that we have in the definition of the delayed multi-information, eq. (3.3).


Tobias Müller

Benjamin Neuhäuser

Uwe Ludewig

Gabriella Houdinet

Sabine D Zimmermann

Pierre-Emmanuel Emmanuel Courty

Daniel Wipf
email: daniel.wipf@inrae.fr

Orcid U Ludewig

Orcid S Zimmermann

Orcid Pe

Orcid D Wipf

New insights into HcPTR2A and HcPTR2B, two high-affinity peptide transporters from the ectomycorrhizal model fungus Hebeloma cylindrosporum

Keywords: ectomycorrhiza, peptide transporter, gene expression, heterologous expression

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

Introduction

Nitrogen (N) is one of the most limiting factors for plant growth even though it is among the most abundant elements on Earth [START_REF] Vance | Symbiotic nitrogen fixation and phosphorus acquisition. Plant nutrition in a world of declining renewable resources[END_REF]. The efficiency and ability to use such limiting nutrients is an important component of plant productivity [START_REF] Mckane | Resource-based niches provide a basis for plant species diversity and dominance in arctic tundra[END_REF]. N forms available in the soil are largely dependent on different environmental factors like the climate, soil type, acidification and nitrification processes, but also on vegetation and management, such as fertilization. When the soil is relatively acidic, cold or poorly aerated, ammonium (NH4 + ) will remain the predominant N-form of plants, but under conditions favouring nitrification, the conversion to nitrate (NO3 -) will follow ammonification [START_REF] Marschner | Mineral nutrition of higher plants[END_REF]. In forest soils, N is present either in its main inorganic forms NH4 + and NO3 -, or in organic compounds like amino acids, peptides and proteins [START_REF] Marschner | Mineral nutrition of higher plants[END_REF][START_REF] Chalot | Physiology of organic nitrogen acquisition by ectomycorrhizal fungi and ectomycorrhizas[END_REF].

Plants generally acquire N in its inorganic forms, followed by an intracellular reduction into NH4 + , assimilation into glutamine and subsequently other amino acids and organic compounds. Since reduction of NO3 -to NH4 + is associated with a high energetic demand, the direct uptake and usage of NH4 + is often favourable. Nevertheless, adaptation of plants under conditions of high N competition or low inorganic N availability has selected plants with diverse mechanisms of acquiring N from their surroundings [START_REF] Zak | The vernal dam: Plantmicrobe competition for nitrogen in Northern hardwood forests[END_REF][START_REF] Kaye | Competition for nitrogen between plants and soil microorganisms[END_REF][START_REF] Aerts | The mineral nutrition of wild plants revisited: a re-evaluation of processes and patterns[END_REF]. In addition to using the inorganic N sources, some plants cooperate with ectomycorrhizal fungi to indirectly access the large pool of organic N in the soil. These fungi use high affinity peptide transporters to absorb amino acids and peptides from the soil.

Peptide transporters can be categorized into three super-families: (i) the ATP-binding cassette (ABC family); (ii) the oligopeptide (OPT) transporter family; (iii) and the peptide transporter (PTR/POT) family [START_REF] Steiner | The PTR family: a new group of peptide transporters[END_REF][START_REF] Reddy | The major facilitator superfamily (MFS) revisited[END_REF]. OPT and PTR/POT families are mainly involved in the transport of peptides in fungi [START_REF] Dunkel | Roles of different peptide transporters in nutrient acquisition in Candida albicans[END_REF]. Some PTRs are capable of transporting other nitrogen-containing substrates like nitrate [START_REF] Chiang | Mechanisms and functional properties of two peptide transporters, AtPTR2 and fPTR2[END_REF]). In the ectomycorrhizal fungus (EMf) Hebeloma cylindrosporum, two putative di-and tripeptide transporters, HcPTR2A and HcPTR2B were previously isolated [START_REF] Benjdia | Peptide uptake in the ectomycorrhizal fungus Hebeloma cylindrosporum: Characterization of two di-and tri-peptide transporters (HcPTR2A and B)[END_REF]. The functionality of both transporters was shown by complementation of yeast mutants. Closer characterization by uptake studies with 3 [H] -labelled dipeptides could only be performed for HcPTR2A, while a more detailed functional analysis of HcPTR2B was still lacking. It is worthwhile to note that a dipeptide transporter was characterized in another mycorrhizal association type, namely in an arbuscular mycorrhizal fungus [START_REF] Belmondo | A dipeptide transporter from the arbuscular mycorrhizal fungus Rhizophagus irregularis is upregulated in the intraradical phase[END_REF].

In this study, we investigated and compared the transport characteristics of HcPTR2A and HcPTR2B in detail by functional expression in Xenopus laevis oocytes and confirmed high affinity peptide transport for both transporters. We identified a low affinity amino acid transport activity of HcPTR2A as well as the voltage-dependent substrate affinities and transport kinetics for several peptides for both transporters. Moreover, differential expression patterns of these two transporters in the presence of the host plant argue for different specific roles in plant nutrition via the ectomycorrhizal association.

Material and methods

Cloning of HcPTR2A and HcPTR2B into pOO2 and synthesis of cRNA

For expression in X. laevis oocytes, the genes coding the HcPTR transporters were transferred from the yeast expression vector into the X. laevis expression vector pOO2 [START_REF] Ludewig | Uniport of NH4 + by the root hair plasma membrane ammonium transporter LeAMT1;1[END_REF]. Coding regions of the genes came from the cDNA library in Montpellier (HcPTR2A and HcPTR2B) and were inserted into the yeast expression vector pFL61 [START_REF] Lambilliotte | Large-scale identification of genes in the fungus Hebeloma cylindrosporum paves the way to molecular analyses of ectomycorrhizal symbiosis[END_REF]). For isolation of the genes, the vector was restricted with NotI and then ends blunted with T4-polymerase (Fermentas, Vilnius, Lithuania) following the manufacturer's instructions. pOO2 was blunted with SmaI and dephosphorylated to prevent re-ligation. Ligation was done as described by Sambrook et al. (1998) and the orientation was checked by PCR using a gene and vector specific primer pair.

For the synthesis of cRNA, 10 µg of the cDNA containing pOO2 vectors were linearized by digestion with MluI and linearization was verified by gel electrophoresis. The linearized and phenol chloroform cleaned plasmid was used as a template for cRNA synthesis using the "SP6 mMESSAGE mMACHINE Kit" (Life Technologies GmbH, Darmstadt; Germany) following the manufacturer's instructions. Quality of the cRNA was checked by RNA gel electrophoresis and cRNA concentrations were determined photometrically.

Preparation of Xenopus laevis oocytes and microinjection of cRNA

Oocytes were surgically removed from anesthetized X. laevis. Oocytes were defolliculated by collagenase treatment (Ca 2+ -free ND96 with 0,2% collagenase; ND96: 96 mM NaCl, 2 mM KCl, 1 mM MgCl2, 1.8 mM CaCl2, 2.5 mM sodium pyruvate, 5 mM HEPES adjusted to pH of 7.4 by NaOH). After five washing steps with Ca 2+ -free ND96 supplemented with 1 mg ml -1 BSA solution, oocytes were manually separated for development stages V and VI [START_REF] Dumont | Oogenesis in Xenopus laevis (Daudin). 1. Stages of oocyte development in laboratory maintained animals[END_REF]. Before microinjection of cRNA, the freshly prepared oocytes were stored overnight at 4 °C.

Oocytes were microinjected with 50 nl of 1 µg µl -1 cRNA solutions with the help of a glass micropipette and a micromanipulator (Drummond Scientific, Broomall, PA). Control oocytes were injected with 50 nl H2O. Before recordings, the injected oocytes were kept at 16 °C for three to five days in ND96 solution. The buffer was changed every day depending on the oocytes and solution conditions. Pipettes for injections and recordings were pulled using a Narishige PP-830 puller (Tokyo, Japan) or L/M-3P-A puller (List Medical Electronic, Darmstadt, Germany), respectively.

Measurements of steady-state inward currents across the oocyte plasma membrane

Micropipettes pulled from 1.5 mm thin wall borosilicate glass GB150TF10 (Science Products GmbH, Hofheim, Germany) were used for recordings. Pipettes were filled with 3 M KCl and connected to the headstage using a chloride silver wire. Pipette resistance was 1.5 -3.5 MΩ in ND96 solution.

Two-electrode voltage-clamp (TEVC) was done using a Dagan TEVC-200A amplifier (Dagan Corp, Minneapolis, USA). Currents were recorded via a LM12 interface (Dagan Corp.) using Clampex 5.5.1 (Axon Instruments Inc., Union City, CA).

For measuring the steady-state inward currents, the oocyte plasma membrane was held at -30 mV, and membrane currents were measured after stepping from the holding potential (Vh) to test potentials (Vm) between -140 mV and +40 mV in 20 mV increments. Each voltage pulse was applied for 250 ms.

The currents were filtered at 50 Hz. Peptides and amino acid substrates were supplied by perfusion of the bath in various concentrations indicated in the figures.

RNA extraction and quantitative reverse transcription-PCR

Growth conditions of H. cylindrosporum mycelium alone and in indirect contact with young seedling roots of Pinus pinaster mimicking the symbiotic interface for 48 h were performed as described [START_REF] Becquer | Establishing a symbiotic interface between cultured ectomycorrhizal fungi and plants to follow fungal phosphate metabolism[END_REF]Torres-Aqino et al. 2017). Briefly, fungal mycelia, grown for three weeks in liquid cultures (N6), and pine seedlings, grown from seeds, cultivated separately in liquid nutritive medium N1 for two months and finally washed (CaSO4 0,2 mM, pH 5.5), were incubated together for 48 h in an interaction medium (MgSO4 0,2 mM, CaCl2 0,5 mM, MES 5 mM, TRIS 5 mM; pH 5.5). One fungal mycelium was incubated with three pine seedling roots (indirect contact) or in the same conditions alone (three repetitions). RNA extraction from these fungal mycelia was performed using the Spectrum™ Plant Total RNA Kit (Sigma), RNA was treated by DNAse (Deoxyribonuclease I, Amplification Grade, Invitrogen), and cDNA synthesis was performed by retrotranscription (SuperScript™ III Reverse Transcriptase, Invitrogen).

RNA from P. pinaster-Laccaria bicolor ectomycorrhiza, harvested three and 28 days post-infection, were gently provided by Pr F. Canavos (University of Malaga, Spain). Synthesis of these ectomycorrhizas and RNA extraction are described in Flores-Monterroso et al. (2013). PCR reactions and data analysis were performed as described in [START_REF] Courty | Phylogenetic analysis, genomic organization and expression analysis of multicopper oxidases in the ectomycorrhizal basidiomycete Laccaria bicolor[END_REF] with 3 biological replicates.

Primers used for control (HcTub H. cylindrosporum α-tubulin, GenBank accession number CK993737 [START_REF] Tatry | Two differentially regulated phosphate transporters from the symbiotic fungus Hebeloma cylindrosporum and phosphorus acquisition by ectomycorrhizal Pinus pinaster[END_REF]) and LbEf-3 L. bicolor elongation factor (F: CTCATCCTTCCTGCTCTTCTTCACG and R: CGGTGGCTTTTGTAAGTGAATCACG)) or for analysis of HcPTR2 (HcPTR2A-F (TCGTTTCCCTCTCCACTGAC) and HcPTR2A-R (GCTCGTCCTCTTTGGCATC); HcPTR2B-F (GCGTTACTCAGCACGACCT) and HcPTR2B-R (ATCAGCGGCAATCACGAC)) and of LbPTR2 (LbPTR2A-F (GGCTTGCGGAGTGTTCTG) and LbPTR2A-R (TCTTCTTGAGGGCGGTGA); LbPTR2B-F (TCCTTGTCGCTGGTCTCG) and LbPTR2B-R (CGTTTTGGTGCCTTGGTG)) had an efficiency ranging between 90 and 110%.

Bioinformatic analysis

Transmembrane domain analysis was performed using the TMHMM server (http://www.cbs.dtu.dk/services/TMHMM/).

Results

PTR genes of Hebeloma cylindrosporum and predicted proteins

Initially, HcPTR2A and HcPTR2B had been identified by a functional yeast complementation screening of a cDNA library as peptide transporters [START_REF] Benjdia | Peptide uptake in the ectomycorrhizal fungus Hebeloma cylindrosporum: Characterization of two di-and tri-peptide transporters (HcPTR2A and B)[END_REF]). Sequencing of the H. cylindrosporum genome among other EMf [START_REF] Kohler | Convergent losses of decay mechanisms and rapid turnover of symbiosis genes in mycorrhizal mutualists[END_REF] allowed detailed analyses of these PTR genes and gene products. The sequences of HcPTR2A and HcPTR2B are randomly distributed in the genome (scaffold 3 and 22, respectively) whereas in other EMf, e.g. Suillus luteus, duplicated genes are on the same scaffold (Table S1). The length of the genomic nucleotide sequences for HcPTR2A and HcPTR2B are of 2,486 bp and 2,452 bp, respectively (Table S1). The coding exon sequences for these PTR genes were confirmed by EST alignment and cDNA sequencing; they are interrupted by short introns with a mean length of 55 and 58 bp, respectively. The actual number of introns is 13 for HcPTR2A and 11 for HcPTR2B (Table S1, Figure S1), which is similar to PTR transporters from other basidiomycetes, which normally show between 9 and 14 introns. The deduced protein sequences of HcPTR2A and HcPTR2B genes showed a length typical for fungal dipeptide transporters (591 aa and 603 aa, respectively) (Table S1). The THMM and Swissmodel predicted proteins with 10 and 11 transmembrane domains, for HcPTR2A and HcPTR2B, respectively (Figure S2). The sequence identity between the two proteins is 42.3%. A phylogenetic classification of the two HcPTR transporters is given in Figure 1.

Current/voltage (I/V) relationships at saturating substrate concentrations for HcPTR2A

To study the biochemical properties of HcPTR2A and HcPTR2B, four peptides exhibiting different properties were chosen. Two of them, LeuLeu and LeuLeuLeu, had been already tested in Saccharomyces cerevisiae [START_REF] Benjdia | Peptide uptake in the ectomycorrhizal fungus Hebeloma cylindrosporum: Characterization of two di-and tri-peptide transporters (HcPTR2A and B)[END_REF]. Previous uptake experiments with radiolabelled 3 [H]LeuLeu showed no significant uptake by HcPTR2B [START_REF] Benjdia | Peptide uptake in the ectomycorrhizal fungus Hebeloma cylindrosporum: Characterization of two di-and tri-peptide transporters (HcPTR2A and B)[END_REF]). Here, we expressed both transporters in X. laevis oocytes and measured inward currents induced by different peptides. At neutral pH, only negligible small ionic currents were elicited by the peptides, but substantial ionic inward currents of up to 0.5 µA were elicited at acidic external pH 4.5. I/V relationships at saturating substrate concentrations were normalized to the maximum induced steady-state current for each oocyte and are given in Figure 2 for both transporters, HcPTR2A and HcPTR2B, for different peptides. Only inward currents were observed, in accordance with the fact that the (uncharged)

substrates were applied externally and uptake was in form of peptide co-transport with H + , driven by the negative membrane potential. At positive potentials, the currents asymptotically approached zero, but no outward currents were observed. For comparison, water-injected oocytes were exposed to the same peptides, but no ionic currents were elicited, clearly showing that the recorded currents were carried by the respective heterologously expressed transporters.

Kinetics of HcPTR2A and HcPTR2B mediated peptide transport

Next, we measured the concentration-dependent transport rates. According to the screening results in yeast, several substrates were tested for their ability for transport competition [START_REF] Benjdia | Peptide uptake in the ectomycorrhizal fungus Hebeloma cylindrosporum: Characterization of two di-and tri-peptide transporters (HcPTR2A and B)[END_REF]).

HcPTR2A as well as HcPTR2B were both capable of transporting the LeuLeu, as well as the LeuLeuLeu peptide, over a broad range of concentrations (Figure 3). The Km of HcPTR2A and HcPTR2B for the uncharged di-and tripeptides LeuLeu and LeuLeuLeu, and for the zwitterionic peptide LysAsp were in a similar range, around 0.5-0.8 µM and 1.5-4 µM, respectively (Table 1,

Figure 3).

For the neutral dipeptide GlyAsn, containing the small glycine and polar asparagine, HcPTR2A and HcPTR2B had a much lower affinity with a Km in the range of 91 µM and 33.7 µM, respectively. For the dipeptides LysAsp and LeuLeu, and the tripeptide LeuLeuLeu, the transport saturation of both transporters was already reached at substrate concentrations below 30 µM. In contrast, for GlyAsn, significantly higher concentrations were needed to saturate the transport of HcPTR2A and HcPTR2B, in a range above 300 µM and 100 µM, respectively (Table 1, Figure 3).

In all cases, when the currents induced from -40 mV to -140 mV were sufficiently large, we quantified the affinity for each membrane potential. For HcPTR2A, these calculations could be done for all peptides, while for HcPTR2B, only the currents induced with LysAsp and LeuLeu were sufficiently large for this analysis. Interestingly, the concentration of peptides, at which the transport was halfmaximal saturated (Km) increased with more negative membrane potentials (Figure 4), suggesting that the Km was voltage-dependent. This was irrespective of the type of peptide tested and suggests that both transporters had higher affinity to their substrates at less negative voltages.

HcPTR2A mediates low affinity transport of large hydrophobic amino acids

Since all four tested peptides elicited large ionic currents in HcPTR2A, we further tested whether this transporter additionally mediated transport of individual amino acids. The large hydrophobic amino acids Leu, Phe and Met elicited the strongest currents and induced substantial inward currents at millimolar concentrations (Figure 5A). Transport rates were measured over a broad range of concentrations and revealed that the transport saturated with low affinities (Km = 3.37, 4.75 and 3.52 mM for Leu, Phe and Met, respectively) (Figure 5B). Non-injected control oocytes did not show amino acid-induced currents, confirming that the amino acid transport was carried by HcPTR2A.

However, only few of the 20 proteinogenic amino acids were transported by HcPTR2A at a substrate concentration of 1 mM. In addition to Leu, Phe and Met, the large neutral amino acid Tyr induced substantial currents (Figure 5C). Smaller currents were elicited by the small hydrophobic amino acids

Val and Ile, as well as the polar Asn and the charged Arg and Glu (Figure 5C).

HcPTR2B was down-regulated in the presence of a plant partner

Expression of both peptide transporters was quantified on the transcriptional level in fungal mycelia alone compared to mycelia incubated for 48 h in the presence of young pine seedling roots (Figure 6). The expression of HcPTR2A was similar in free-living mycelium and in mycelium in indirect contact with P. pinaster roots under same culture conditions. Interestingly, HcPTR2B was significantly down-regulated in presence of its plant host in the conditions of the device mimicking the interaction space already after 48 h arguing for a tight regulation at early stages of symbiotic signalling between ectomycorrhizal fungus and roots of the host plant. In addition, expression of both peptide transporters was quantified on the transcriptional level in P. pinaster-Laccaria bicolor ectomycorrhizal roots, harvested three and 28 days post-infection. LbPTR2A was only expressed in functional ectomycorrhizal roots (28 dpi). LbPTR2B was significantly down-regulated 8,5 times in 28 dpi ectomycorrhizal roots compared to 3 dpi ectomycorrhizal roots (normalized expressed of 0.68±0.16 versus 5.77±0.38).

Discussion

Previously, two di-and tripeptide transporters (HcPTR2A and HcPTR2B) from H. cylindrosporum were isolated and partially characterized [START_REF] Benjdia | Peptide uptake in the ectomycorrhizal fungus Hebeloma cylindrosporum: Characterization of two di-and tri-peptide transporters (HcPTR2A and B)[END_REF]. For PTR2A, a Km value of 1.46 µM for LeuLeu was estimated from radiolabelled dipeptide uptake experiments in the S. cerevisiae mutant LR2 expressing HcPTR2A. The transport of other substrates was inferred from competition experiments [START_REF] Benjdia | Peptide uptake in the ectomycorrhizal fungus Hebeloma cylindrosporum: Characterization of two di-and tri-peptide transporters (HcPTR2A and B)[END_REF]). However, uptake experiments for PTR2B with 3 [H]-LeuLeu in S. cerevisiae previously showed a very high standard deviation so that no Km value could be estimated in yeast. To gain a better understanding regarding the transport properties and the substrate specificities of HcPTR2A and HcPTR2B, both genes were expressed in X. laevis oocytes. Both transporters elicited significant inward currents when oocytes were perfused with a peptidecontaining solution at acidic pH, in line with their function as proton co-transporters. Similarly to RiPTR2, a peptide transporter of the arbuscular mycorrhizal fungus Rhizophagus irregularis [START_REF] Belmondo | A dipeptide transporter from the arbuscular mycorrhizal fungus Rhizophagus irregularis is upregulated in the intraradical phase[END_REF]), HcPTR2A showed a wide substrate spectrum with very high affinities for LysAsp, LeuLeu and LeuLeuLeu. GlyAsn was transported with a significant lower affinity compared to the other peptides. Similar as in the heterologous host yeast, HcPTR2A had a lower affinity for peptides starting with a Gly. These results correspond to what has been observed in the uptake experiments of HcPTR2A in S. cerevisiae [START_REF] Benjdia | Peptide uptake in the ectomycorrhizal fungus Hebeloma cylindrosporum: Characterization of two di-and tri-peptide transporters (HcPTR2A and B)[END_REF]) and confirm the very high affinity of HcPTR2A (1.46 µM for 3 [H]-LeuLeu in S. cerevisiae uptakes and 0.73 µM for LeuLeu at -140 mV in oocytes). The slight difference can be due to the fact that in uptake experiments the membrane potential is not tightly controlled as in the voltage-clamp experiments and will be an average of all S. cerevisiae cells. Whereas the resting membrane potential of plants is between -120 mV and -180 mV [START_REF] Marschner | Mineral nutrition of higher plants[END_REF] the fungal membrane potential is expected even more negative, it can be down to -300 mV [START_REF] Slayman | Plasma-membrane proton pumps in plants and fungi[END_REF]. Interestingly, the Km value for HcPTR2A increased with more negative membrane potentials. A robust inward current was observed using GlyAsn as substrate. Further we observed solid inward currents when large, mainly hydrophobic, amino acids like Leu, Phe and Met were supplied in millimolar concentrations. The transport of these three amino acids was saturable as well but only reached half maximal transport rates at high concentrations of ~ 3 -5 mM. Therefore, it might be questioned whether this transport is physiologically relevant but affinity might also be voltage-dependent as shown for other substrates.

The second peptide transporter HcPTR2B as well showed a high affinity for LysAsp, LeuLeu and LeuLeuLeu. HcPTR2B showed a lower Km for GlyAsn (33.7 µM) when compared to HcPTR2A.

However, transport affinities were not as high as observed for HcPTR2A in response to LeuLeu and LeuLeuLeu. In contrast, the affinity to LysAsp and GlyAsn seemed to be higher.

These results indicate that both transporters are high affinity di-and tripeptide transporters, but have differences in their substrate specificity. Both peptide transporters show high affinity levels that have never been observed before for any plant or fungal peptide transporter from the PTR family. In X.

laevis experiments, a Km value of 33 µM for LeuLeu could be estimated for a fungal peptide transporter, originally identified from an Arabidopsis thaliana cDNA library [START_REF] Steiner | An Arabidopsis peptide transporter is a member of a new class of membrane transport proteins[END_REF]) but then turned out to be a fungal contamination [START_REF] Chiang | Mechanisms and functional properties of two peptide transporters, AtPTR2 and fPTR2[END_REF]). However, the Km value of fPTR2

does not increase with a more negative membrane potential as it could be shown for both H. cylindrosporum peptide transporters. Yeast uptake experiments with this transporter resulted in a Km value of 14 µM for 3 [H]-LeuLeu [START_REF] Steiner | An Arabidopsis peptide transporter is a member of a new class of membrane transport proteins[END_REF]) demonstrating the difference between both heterologous expression systems explained by a series of possible differences in e.g. cytosolic and/or membrane conditions. However, the results observed for H. cylindrosporum peptide transporters expressed in yeast and X. laevis show similar characteristics.

Other characterised peptide transporters derived from plants like Vicia faba PTR1 (Km 20 mM; show lower substrate affinities. If also not all possible substrates were tested, the broad substrate specificity of peptide transporters could be shown. This seems to be necessary as Dietrich et al. (2004b) demonstrated that from the 400 dipeptides from degraded Rubisco only seven were present more than six times in the intact protein, but most of them exist only one or two times (Dietrich et al. 2004a).

Keeping the high substrate affinity of both H. cylindrosporum peptide transporters in mind the question could be addressed if there is a reason for this. Therefore, one has to take a look at the original habitat of H. cylindrosporum and P. pinaster. It is divided into two habitats, the "dune" habitat where the percentage of organic matter ranges from 0.1% to 0.5% and the "forest" habitat with up to 3% of organic matter [START_REF] Guidot | Forest habitat characteristics affect balance between sexual reproduction and clonal propagation of the ectomycorrhizal mushroom Hebeloma cylindrosporum[END_REF]. This very low percentage of organic matter could be an explainable reason for the high affinities of the peptide transporters.

It is worth to note that the transport rate of HcPTR2B is much lower than that observed for HcPTR2A.

Our results on HcPTR2A and 2B expression are in concordance with [START_REF] Plett | The mutualist Laccaria bicolor expresses a core gene regulon during the colonization of diverse host plants and a variable regulon to counteract host-specific defenses[END_REF] who studied on arrays the expression level of Laccaria bicolor genes in the mycelium grown on medium containing agar and in ectomycorrhizal roots of Populus trichocarpa-L. bicolor grown for 2, 4, 6 and 12 weeks after contact. No expression was detected for protID668820 (homolog of HcPTR2A) whatever the condition was. For protein ID301981 (homologous of HcPTR2B) the expression was significantly reduced (from 6 to 30 times) in ectomycorrhizal roots whatever the time of contact between the fungus and the roots. In addition, we also that expression of LbPTR2 was significantly reduced in functional ectomycorrhizal roots of P. pinaster-L. bicolor. Here, we show even a strong reduction of expression after only two days of indirect contact of the fungus with roots from young pine seedlings arguing for an involvement in early signalling. Why should this peptide transporter be so down-regulated? Might it be that both transporters are differentially localized, HcPTR2A perhaps more in external hyphae exploring the soil, HcPTR2B more at the symbiotic interface, within the Hartig net? In this case, it would be evident that plant factors are down-regulating expression of HcPTR2B to avoid possible competition for peptides. One could suggest conserved roles for both PTRs during evolution in ectomycorrhizal fungi. Previous analysis on the expression of both HcPTRs in free living mycelium [START_REF] Avolio | Regulation of genes involved in nitrogen uptake and metabolism by different C/N ratios and nitrogen sources in the model ectomycorrhizal fungus Hebeloma cylindrosporum[END_REF]) highlighted the fact that HcPTR2B was expressed under all investigated conditions. So we could propose a role in a "basic" uptake of di-and tripeptides with a high affinity in free living mycelium. Whereas HcPTR2A, that has a higher affinity, would be

  [START_REF] Miranda | Peptide and amino acid transporters are differentially regulated during seed development and germination in faba bean[END_REF], A. thaliana PTR1 (Km 56 µM for AlaAla to 416 µM for AlaAsp;Dietrich et al. 2004a) or A. thaliana PTR2 (Km 52 µM for LeuLeu to 1,832 µM for AlaAsp;[START_REF] Chiang | Mechanisms and functional properties of two peptide transporters, AtPTR2 and fPTR2[END_REF]

Our work has partially been funded by grants from the Deutsche Forschungsgemeinschaft (Gottfried-Wilhelm-Leibniz; DFG WI1994/2-1 and 2-2) Conflicts of interest/Competing interests : not applicable Ethics approval (include appropriate approvals or waivers): not applicable Consent to participate (include appropriate statements): not

data and material (data transparency): all data and material are available Code availability (software application or custom code): Consent to participate