Introduction on gene regulatory network modeling -4/42

Introduction on gene regulatory network modeling -5/42

Introduction on gene regulatory network modeling -6/42

Assumption 1

The regulation functions b l i (•) are multiaffine functions, that is, they are affine with respect to each s + (x j , θ k j ), for j ∈ {1, . . . n} and k ∈ {1, . . . , p j }. Assumption 1 can be shown to be generic for all regulation functions corresponding to Boolean functions written in minterm disjunctive normal form.

Assumption 2

Every step function s + (x j , θ k j ), with j ∈ {1, . . . n} and k ∈ {1, . . . , p j }, occurs in at most one b i (•), i ∈ {1, . . . , n}.

Assumption 2 is a rather weak modeling assumption, in the sense that there is usually no compelling biological reason for two genes to be regulated at exactly the same threshold.

Introduction on gene regulatory network modeling -9/42

Introduction on gene regulatory network modeling

A first simple network of two genes A first simple network of two genes PWL model corresponding to this network.

ẋ1 = -γ 1 x 1 + κ 1 s + (x 2 , θ 1 2 ) s -(x 1 , θ 2 1 ) ẋ2 = -γ 2 x 2 + κ 2 s + (x 1 , θ 1 1 ) s -(x 2 , θ 2 2 )
(1) where x 1 , x 2 are cellular protein or RNA concentrations κ 1 , κ 2 and γ 1 , γ 2 are positive synthesis and degradation constants, respectively,

θ 1 1 , θ 2 1 , θ 1 2 , θ 2
2 are constant strictly positive threshold concentrations of regulation s + and s -are step functions

s + (x j , θ k j ) = 1 if x j > θ k j 0 if x j < θ k j and s -(x j , θ k j ) = 0 if x j > θ k
Definition of Piece-Wise Linear (PWL) models

Definition of Piece-Wise Linear (PWL) models Notation x = (x 1 , . . . , xn) T ∈ Ω a vector of cellular protein or RNA concentrations, where Ω ⊂ R n + is a bounded n-dimensional hyperrectangular subspace of R n + For each concentration variable x i , i ∈ {1, . . . , n}, we distinguish a set of constant, strictly positive threshold concentrations {θ 1 i , . . . , θ p i i }, p i > 0. Θ = i∈{1,...,n},k∈{1,...,p i } {x ∈ Ω | x i = θ k i } the subspace of Ω defined by the threshold hyperplanes.

Introduction on gene regulatory network modeling Definition of Piece-Wise Linear (PWL) models Definition of Piece-Wise Linear (PWL) models Definition 1 (PWL model)

A PWL model of a gene regulatory network is defined by a set of coupled differential equations

ẋi = f i (x) = -γ i x i + b i (x) = -γ i x i + l∈L i κ l i b l i (x), i ∈ {1, . . . , n}, (3)
where κ l i and γ i are positive synthesis and degradation constants, respectively, L i ⊂ N are sets of indices of regulation terms, b l i : Ω \ Θ → {0, 1} are so-called Boolean regulation functions.

Boolean functions and step functions

Step functions can be associated with Boolean variables X k j such that

X k j (x) = (x j > θ k j ) = s + (x j , θ k j ) X k j (x) = (x j < θ k j ) = s -(x j , θ k j ), (4)
where X denotes the complemented variables of X .

Introduction on gene regulatory network modeling -7/42

Definition of Piece-Wise Linear (PWL) models

Generically, any Boolean function b l i (x) can be rewritten in minterm disjunctive normal form (DNF):

b l i (x) = 2 p -1 α=0 c l i,α mα(x), (5)
with c l i,α ∈ {0, 1}. For the set of variables X k j , j ∈ {1, . . . , n}, k ∈ {1, . . . , p j }, we have 2 p minterms, with p = j∈{1,...,n} p j ,

mα(x) = n j=1 p j k=1 X k j (x), α ∈ {0, . . . , 2 p -1}. ( 6
)
where X k j (x) is a literal defined either as the Boolean variable X k j or its negation X k j . Step functions s ± (x j , θ k j ) gives rise to mathematical complications, because the step functions are undefined and discontinuous at x j = θ k j . Reformulation as differential inclusions ẋ ∈ F (x) Numerous options for defining the set-valued function F . Filippov's definition of solutions as a absolutely-continuous function x(•) such that ẋ(t) ∈ F (x(t)) holds almost everywhere on [t 0 , T ] with x(t 0 ) = x 0 .

Filippov's solutions -11/42

Filippov's extension of PWL models

Discontinuous Dynamics in Ω \ Θ ẋi = f i (x) = -γ i x i + b i (x), i ∈ {1, . . . , n}. (3)
Definition 3 (F-extension of PWL models)

The F-extension of the PWL model ( 3) is defined by the differential inclusion

ẋ ∈ F (x), with F (x) = co { lim y →x, y ∈Θ f (y )} , x ∈ Ω, (7)
where co(P) denotes the closed convex hull of the set P, and {lim y →x, y ∈Θ f (y )} the set of all limit values of f (y ), for y ∈ Θ and y → x.

Filippov's solutions -12/42

Filippov's extension of PWL models

Filippov's extension of PWL models Properties

Classical definition of Filippov's extension

Existence of solutions under mild assumptions Uniqueness is not ensured.

Issues: Hardly tractable formulation for Numerics

ẋ ∈ F (x)
General time-discretization scheme. [START_REF] Dontchev | Difference methods for differential inclusions: a survey[END_REF])

x k+1 -x k h ∈ F (x k )
How to practically build F (x) ?

How to compute and to choose a selection σ ∈ F (x) ?

How to avoid numerical chattering ?

Filippov's solutions -13/42

Aizerman & Pyatnitskii's extension of PWL models

Discontinuous Dynamics in Ω \ Θ ẋi = f i (x) = -γ i x i + b i (x), i ∈ {1, . . . , n}. (3)
Introduction of selection σ = (σ 1 1 , . . . , σ p1 1 , . . . , σ 1 n , . . . , σ pn n ) T ∈ [0, 1] p .
Let us define the function g : R p → R n by

g i (σ) = l∈Ln κ l i bl i (σ), j ∈ {1, . . . , n}. (8)
where bl i (•) are obtained from b l i (•) by replacing every occurrence of s + (x j , θ k j ) and s -(x j , θ k j ) by σ k j and 1 -σ k j , respectively.

Reformulation of the PWL model ( 3)

ẋi = f i (x) = -γ i x i + g i (σ), i ∈ {1, . . . , n}, (9)
Filippov's solutions -14/42

Aizerman & Pyatnitskii's extension of PWL models Multi-valued step function

S + (x j , θ k j ) =      1 x j > θ k j [0,1] x j = θ k j 0 x j < θ k j and S -(x j , θ k j ) =      0 x j > θ k j [0,1] x j = θ k j 1 x j < θ k j . (10)
Interesting equivalence.

σ k j ∈ S + (x j , θ k j ) ⇐⇒ (x j -θ k j ) ∈ N [0,1] (σ k j ) (11)
Filippov's solutions -15/42

Aizerman & Pyatnitskii's extension of PWL models Definition 4 (AP-extension of PWL models)

The AP-extension of a PWL model ( 3) is defined by the following differential inclusion

ẋ ∈    G 1 (x) . . . Gn(x)    =         -γ 1 x 1 + g 1 (σ) . . . -γn xn + gn(σ)    σ k j ∈ S + (x j , θ k j ), j ∈ {1, . . . , n}, k ∈ {1, . . . , p j }      (12)

Properties

Definition related to the Utkin concept of equivalent control method Existence of solutions is not so generic. G is not convex ! Promising extension for Numerics. § Mixed complementarity systems or Differential Variational Inequalities Filippov's solutions -16/42 Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems Filippov's solutions

Relations between the extensions

Relations between the extensions Proposition 5 ((Machina and Ponosov, 2011))

Under Assumption 1 (multiaffine R.H.S.), F (x) = co(G (x)) for all x ∈ Ω.

Comments

Every Filippov solution with the AP-extension is a Filippov solution with the F-extension Not sufficient to ensure the existence of a solution.

Proposition 6

Under Assumptions 1 and 2, F (x) = G (x) for all x ∈ Ω.

Comments

We retrieve the existence of solutions.

σ ∈ S + (x, θ), (13)
into inclusion into normal cones, Complementarity Problems (CP) and finite-dimensional Variational Inequalities (VI)

2. An implicit event-capturing time-stepping scheme, mainly based on the backward Euler scheme which allows to deal with the switch-like behaviour and the sliding motion

Reformulation as MCS/DVI

The relation

σ ∈ S + (x, θ), (14)
can equivalently reformulate in the form of an inclusion as

(x -θ) ∈ N [0,1] (σ). (15)
In turn, the relation ( 15) are equivalent to the complementarity conditions

0 1 -σ ⊥ (x -θ) + 0 0 σ ⊥ (x -θ) -0, ( 16
)
where the symbol x ⊥ y means x T y = 0 and y + , y -respectively stand for the positive and negative parts of y . Finally, an equivalent formulation of ( 15) is given by the following VI : find σ ∈ [0, 1] such that

(θ -x) T (σ -σ ) 0 for all σ ∈ [0, 1]. ( 17
)
Numerical Methods for the AP-extension -20/42

Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems Numerical Methods for the AP-extension Reformulation as MCS/DVI

Reformulation as MCS/DVI

Let us now define the affine function y : R n → R p such that

y (x) = Cx -θ =               x 1 -θ 1 1 . . . x 1 -θ p 1 1 . . . xn -θ n 1 . . . xn -θ pn 1               T ∈ R p (18)
where

C ∈ R p×n with C ij ∈ {0, 1} and θ = [θ 1 1 , . . . , θ p 1 1 , . . . , θ 1 n , . . . , θ pn n ] T .
Numerical Methods for the AP-extension -21/42

Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems Numerical Methods for the AP-extension Reformulation as MCS/DVI

Reformulation as MCS/DVI

The AP-extension of the PWL system in Definition 4 can be written compactly as

ẋ = -diag(γ)x + g (σ) y (x) = Cx -θ ∈ N [0,1] p (σ) ( 19
)
where diag(γ) ∈ R n×n is the diagonal matrix made of the components γ i , i = 1 . . . n. § We get

Mixed Complementarity Systems (MCS) or Differential Variational Inequalities (DVI)

Numerical Methods for the AP-extension -22/42

Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems Numerical Methods for the AP-extension

The general time-discretization framework.

An event-capturing scheme. The one-step problem

The proposed time discretization of ( 19) over a time-interval [t k , t k+1 ] of length h:

   x k+1 = x k -h diag(γ)x k+τ + h g (σ k+1 ), y k+1 = Cx k+1 -θ, y k+1 ∈ N [0,1] p (σ k+1 ). ( 20
)
with the initial condition x 0 = x(t 0 ).

x k+τ = τ x k+1 + (1 -τ )x k for τ ∈ [0, 1]

Formulation into MCP/VI

Let us define the vector

z k+1 = x k+1 σ k+1 ∈ R n+m , (21)
and the function H : R n+m → R n+m as

H(z k+1 ) = x k+1 -x k + h diag(γ)x k+τ -h g (σ k+1 ) θ -Cx k+1 . ( 22
)
Then the problem (20) can be recast into the following inclusion

-H(z k+1 ) ∈ N R n ×[0,1] p (z k+1 ). ( 23
)
Numerical Methods for the AP-extension -23/42

Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems Numerical Methods for the AP-extension

The general time-discretization framework.

Existence of solutions Proposition 7

Let H : R n+m → R n+m be the function defined in ( 22). Under Assumption 1, the problem to find z

∈ R n × [0, 1] p such that -H(z) ∈ N R n ×[0,1] p (z), (24)
has a nonempty and compact solution set.

Sketch of the proof

Substitute x k+1 in the inclusion to get a reduced inclusion

-h(σ) ∈ N [0,1] p (σ) (25)
with

h(σ) = θ -Cx = θ -C diag(1/(1 + hτ γ)) [(In -h(1 -τ ) diag(γ))x k + h g (σ)] (26)
h is continuous and [0, 1] p is compact convex Apply Corollary 2.2.5 (Facchinei and Pang, 2003, page 148) Numerical Methods for the AP-extension -24/42

Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems Numerical Methods for the AP-extension Solution methods for MCP

Solution methods

Definition 8 (Mixed complementarity Problem(MCP) [START_REF] Dirkse | The path solver: A non-monotone stabilization scheme for mixed complementarity problems[END_REF])

Given a function H : R n+m → R n+m and lower and upper bounds l, u ∈ R n+m , the

Mixed complementarity Problem (MCP) is to find z ∈ R n+m and w , v ∈ R n+m + such that (MCP)        H(z) = w -v l z u (u -z) T v = 0 (z -l) T w = 0 . (27)

Numerical algorithms

MILES [START_REF] Rutherford | Miles: A mixed inequality and nonlinear equation solver[END_REF] classical Newton-Josephy method, PATH [START_REF] Ralph | Global convergence of damped Newton's method for nonsmooth equations, via the path search[END_REF][START_REF] Dirkse | The path solver: A non-monotone stabilization scheme for mixed complementarity problems[END_REF] NE/SQP [START_REF] Gabriel | An inexact NE/SQPmethod for solving the nonlinear complementarity problem[END_REF][START_REF] Pang | NE/SQP: A robust algorithm for the nonlinear complementarity problem[END_REF]) generalized Newton's method based on the minimum function QPCOMP [START_REF] Billups | QPCOMP: A quadratic programming based solver for mixed complementarity problems[END_REF] NE/SQP SMOOTH [START_REF] Chen | A class of smoothing functions for nonlinear and mixed complementarity problems[END_REF]

Enumerative solution methods

With the classical Newton-Josephy method linearization, we get a MLCP

y α+1 = W α+1 σ α+1 + q α+1 y α+1 ∈ N [0,1] p (σ α+1 ) , (28)
where

W α+1 = hC M -1 B(σ α ), q α+1 = CM -1 [(In -h(1 -τ )diag(γ))x k + hg (σ α ) + hB(σ α )σ α ] -θ. M = In + hτ diag(γ) B(σ) = ∇σg (σ)
Efficient enumerative solvers (see for instance [START_REF] Faiz | An implicit enumeration procedure for the general linear complementarity problem[END_REF][START_REF] Sherali | Enumeration approach for linear complementarity problems based on a reformulation-linearization technique[END_REF][START_REF] Júdice | On the solution of np-hard linear complementarity problems[END_REF] ) Enumerating several solutions corresponding to various modes Qualitative insight on the nature of solutions Numerical Methods for the AP-extension -26/42

Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems Numerical Methods for the AP-extension Solution methods for MCP

Stationary Points

Finding stationary points of the AP-extension of PWL systems is equivalent to solve the following MCP 0 = -diag(γ)x + g (σ)

y (x) = Cx -θ ∈ N [0,1] p (σ) (29)
or more compactly

C diag(1/(1 + γ))g (σ) -θ ∈ N [0,1] p (σ) (30)
With the same reasoning as in the proof of Proposition 7, the VI/CP (30) has a nonempty compact set of solutions.

We have just to checked that some solutions belong to Ω

3 .

 3 The use of efficient numerical solvers for the one-step problem which results from the time-discretization of the CP/VI formulation of the problem Numerical Methods for the AP-extension -19/42 Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems Numerical Methods for the AP-extension Reformulation as MCS/DVI

  smooth approximations of the NCP, SEMISMOOTH (DeLuca et al., 1996) semismooth Newton with Fischer-Burmeister function, ... Numerical Methods for the AP-extension -25/42 Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems Numerical Methods for the AP-extension Solution methods for MCP

  Reduced study around the stationary point x 1 = θ 2 1 and x 2 = θ 2

	Repressilator Repressilator
	Introduction on gene regulatory network modeling Repressilator consisting of three genes (Elowitz and Leibler, 2000) 2 Repressilator consisting of three genes (Elowitz and Leibler, 2000) Introduction on gene regulatory network modeling
	A first simple network of two genes Definition of Piece-Wise Linear (PWL) models 10 θ2 1 8 8.2 θ2 1 A first simple network of two genes Definition of Piece-Wise Linear (PWL) models		θ2 2		8.01			θ2 1
	Filippov's solutions Notions of Filippov's solutions Filippov's extension of PWL models Aizerman & Pyatnitskii's extension of PWL models Relations between the extensions Numerical Methods for the AP-extension Principles Reformulation as MCS/DVI The general time-discretization framework. Solution methods for MCP Illustrations The first simple network of two genes Synthetic oscillator with positive feedback Repressilator Conclusions & Perspectives 5 6 7 8 9 5 6 7 8 9 10 x2 x1 θ2 2 (a) Explicit scheme 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 7.9 8 8.1 8.2 8.3 8.4 8.5 x2 x1 (b) Explicit scheme (zoom 1) 6 7 8 9 10 x2 θ2 2 θ2 1 6.6 6.8 7 7.2 7.4 7.6 7.8 8 8.2 x2 θ2 2 θ2 1 0 0.5 1 1.5 2 2.5 x1 Filippov's solutions Notions of Filippov's solutions (LacI) 0 0.5 1 1.5 2.5 Filippov's extension of PWL models 2 Aizerman & Pyatnitskii's extension of PWL models 1.8 Relations between the extensions 1.6 2 x2 (tetR) 0 1.5 2 2.5 0.4 0.8 1.4 Numerical Methods for the AP-extension 1.2 Principles 1 x3 (cI) Reformulation as MCS/DVI x3 (cI) The general time-discretization framework. 0.6 Solution methods for MCP (cI) 1 x3 0.2 Illustrations 0.5 0 20 40 60 t (time) 1.2 1.4 1.6 1.8 2 0 x2 (TetR) The first simple network of two genes Synthetic oscillator with positive feedback Repressilator 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Conclusions & Perspectives	1	7.95 7.96 7.97 7.98 7.99 8 7.985 (c) Explicit scheme (zoom 2) 7.99 7.995 8 8.005 8.01 8.015 8.02 8.025 x1 7.96 7.97 7.98 7.99 8 8.01 θ2 1 θ1 θ2 θ3 80 100 0 0.2 0.4 0.6 0.8 x2 x2 x1 (LacI)	θ2 2 θ2 2
	5	5	6	7	8	9	10	6.4	7.9	8	8.1	8.2	8.3	8.4	8.5		7.95 7.985	7.99	7.995	8	8.005	8.01	8.015	8.02	8.025
				x1								x1									x1
			(d) Implicit scheme											(f) Implicit scheme (zoom 2)
																	Numerical Methods for the AP-extension -27/42 Illustrations -33/42 Illustrations -38/42 Illustrations -38/42

Numerical Methods for the AP-extension -28/42 (e) Implicit scheme (zoom 1)

Conclusions & Perspectives -41/42

Conclusions & Perspectives -39/42

Conclusions & Perspectives -40/42

References -41/42

Conclusions & Perspectives -41/42

The first simple network of two genes PWL model corresponding to this network.

(31)

The parameters are:

Illustrations -29/42

Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems

Illustrations

The first simple network of two genes Different trajectories of system (1) depicting the nature of the three equilibria. Illustrations -30/42

Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems

Illustrations

The first simple network of two genes

Reduced study around the stationary point

Illustrations -31/42

Reduced study around the stationary point

(34) is strongly monotone and has an unique solution (see (Facchinei and Pang, 2003, Theorem 2.3.3))

2. No numerical chattering in comparison with explicit schemes [START_REF] Dontchev | Difference methods for differential inclusions: a survey[END_REF] Application of Lemma 3 in [START_REF] Acary | General stability theory when only attractive surfaces are involved in a subset of interest[END_REF].

Illustrations -32/42

The first simple network of two genes

Reduced study around the stationary point

Lemma 9

The equilibrium point θ = (θ 2 1 , θ 2 2 ) is finite-time stable in Ω.

Sketch of the proof

Recast into monotone DI, ż ∈ T (z)

Use Lyapunov function V (z) = 1 2 z T z Since 0 ∈ T (0) and Br (0) ∈ T (0), r > 0, then we have

Synthetic oscillator with positive feedback

The variables x 1 and x 2 represent the concentrations of the proteins LacI and GlnG, respectively.

The following parameter values have been used in the simulations: γ 1 = γ 2 = 0.032, κ 1 = 0.08, κ 2 = 0.16, and θ 1 = 1, θ 1 2 = 1, and θ Repressilator consisting of three genes [START_REF] Elowitz | A synthetic oscillatory network of transcriptional regulators[END_REF])

The variables x 1 , x 2 , and x 3 represent the concentrations of the proteins LacI, TetR, and CI, respectively.

The following parameter values have been used in the simulations: γ 1 = γ 2 = γ 3 = 0.2, κ 1 1 = κ 1 2 = κ 1 3 = 4.8 10 -4 , κ 2 1 = κ 2 2 = κ 2 3 = 4.8 10 -1 , and θ 1 = θ 2 = θ 3 = 1.

Illustrations -37/42

General Piecewise Linear models Extension to general Piecewise Linear models -42/42


Vincent Ej Jassey
email: vincent.jassey@epfl.ch

Geneviève Chiapuso

Philippe Binet

Alexandre Buttler

Fatima Laggoun-Défarge

Frédéric Delarue

Nadine Bernard

Edward P Mitchell

Marie-Laure Toussaint

Andre-Jean Francez

Geneviève Chiapusio

Edward Ad Mitchell

Marie- Laure Toussaint

André-Jean Francez

Daniel Gilbert

Above-and belowground linkages in Sphagnum peatland: climate warming affects plant-microbial interactions

Keywords: aboveground, belowground, climate warming, microbial communities, peatlands, polyphenols, testate amoebae, plant-soil-microorganism interactions, SEM

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

Introduction

Ongoing global warming is causing ecological communities to rapidly change, resulting in modifications to their interactions, and to ecosystem functioning and services. Climate warming phenomena can directly affect aboveground communities, by changing plant community composition, carbon allocation patterns, or the quality of plant-derived organic matter, which indirectly affect soil biota [START_REF] Wardle | Linking between aboveground and belowground biota[END_REF][START_REF] Veteli | Do elevated temperature and CO 2 generally have counteracting effects on phenolic phytochemistry of boreal trees?[END_REF]De Dyn et al., 2008;[START_REF] Wardle | Linking vegetation change, carbon sequestration and biodiversity: insights from island ecosystems in a long-term natural experiment[END_REF]. The microbial subsystem in turn supports a wide range of key ecosystem processes by breaking down plant material, and therefore determines the nature and the extent of plant community and functioning [START_REF] Wardle | Linking between aboveground and belowground biota[END_REF][START_REF] Bardgett | Microbial contributions to climate change through carbon cycle feedbacks[END_REF][START_REF] Singh | Microorganisms and climate change: terrestrial feedbacks and mitigation options[END_REF]. Aboveground-belowground linkages and feedbacks may respond strongly to climate change yet such integrated effects remain poorly quantified [START_REF] Singh | Microorganisms and climate change: terrestrial feedbacks and mitigation options[END_REF]Eiseinhauer et al., 2012). Understanding warming-induced changes on plants and soil organisms is highly relevant to the growing interest that these components have in driving terrestrial ecosystem processes, both above-and belowground [START_REF] Wardle | Communities and Ecosystems: Linking the Aboveground and Belowground components[END_REF][START_REF] Bardgett | Aboveground -Belowground Linkages, Biotic interactions, Ecosystem processes, and Global Change[END_REF].

Peatlands dominated by Sphagnum mosses store more C than any other terrestrial ecosystem owing to imbalance between litter inputs and C outputs from soil respiration [START_REF] Davidson | Temperature sensitivity of soil carbon decomposition and feedbacks to climate change[END_REF][START_REF] Dise | Peatland response to global change[END_REF]. Complex linkages between above-and belowground communities regulate this C-sequestration, which may decrease or even reverse in response to warming, leading to a positive feedback to global warming [START_REF] Bardgett | Microbial contributions to climate change through carbon cycle feedbacks[END_REF][START_REF] Dorrepaal | Carbon respiration from subsurface peat accelerated by climate warming in the subarctic[END_REF][START_REF] Fenner | Drought-induced carbon loss in peatlands[END_REF]. However, despite recent interest in linkages between above-and belowground communities as powerful mutual drivers of ecosystem processes [START_REF] Wardle | The influence of biotic interactions on soil biodiversity[END_REF][START_REF] Ward | Long-term consequences of grazing and burning on northern peatland carbon dynamics[END_REF][START_REF] Kardol | How understanding aboveground-belowground linkages can assist restoration ecology[END_REF][START_REF] Wardle | Linking vegetation change, carbon sequestration and biodiversity: insights from island ecosystems in a long-term natural experiment[END_REF], only a limited number of field experiments considering response to ecosystem change have been conducted in peatlands [START_REF] Fenner | Elevated CO2 effects on peatland plant community carbon dynamics and DOC production[END_REF][START_REF] Kim | Funtional and structural responses of bacterial and methanogen communities to 3-year warming incubation in different depths of peat mire[END_REF][START_REF] Weddon | Summer warming accelerates sub-artic peatland nitrogen cycling without changing enzyme pools or microbial community structure[END_REF]. Knowledge of how climate warming impacts on the mechanisms behind above-and belowground linkages in peatlands is currently insufficient. Indeed, changes in linkages at one level of organization (e.g. species characteristics and interactions, community composition or carbon storage) can affect above-and belowground biota, and their linkages, at other levels of organization, thus destabilizing peatland functioning.

Monitoring study had shown how peat mosses (Sphagnum fuscum) stabilised vegetation composition in response to climate warming by encroaching on vascular plant space [START_REF] Keuper | A Race for Space? How Sphagnum fuscum stabilizes vegetation composition during long-term climate manipulations[END_REF], underlying the key role of Sphagnum in peatlands. By efficiently accumulating nutrients, Sphagnum mosses act as ecosystem engineers, creating unfavourable conditions to vascular plant life [START_REF] Van Breemen | How Sphagnum bogs down other plants[END_REF][START_REF] Turetsky | The role of bryophytes in carbon and nitrogen cycling[END_REF][START_REF] Gunnarsson | Global patterns of Sphagnum productivity[END_REF].

These mosses are also tightly linked to microbial communities through a variety of both direct and indirect mechanisms that ultimately exert control on peatland ecosystem C dynamics [START_REF] Lindo | The Bryosphere: an integral and influential component of the earth's biosphere[END_REF]. Microbial communities living in Sphagnum mosses constitute a crucial detrital network for nutrient and C cycling, where protozoans (and especially testate amoebae) play a central role [START_REF] Gilbert | The microbial loop at the surface of a peatland: Structure, function, and impact of nutrient input[END_REF][START_REF] Mitchell | Structure of microbial communities in Sphagnum peatlands and effect of atmospheric carbon dioxide enrichment[END_REF][START_REF] Lindo | The Bryosphere: an integral and influential component of the earth's biosphere[END_REF]. Sphagnum also produces organochemical compounds such as polyphenols, which are known to have a strong inhibitory effect on microbial breakdown of organic matter, therefore favouring peat accumulation [START_REF] Verhoeven | Decomposition of carex and sphagnum litter in fens -effect of litter quality and inhibition by living tissue-homogenates[END_REF]. For instance, a reduction in polyphenol content may stimulate bacterial and microbial enzymatic activity [START_REF] Fenner | Drought-induced carbon loss in peatlands[END_REF]Jassey et al., 2011a). However, despite the global significance of microbial communities in SOC dynamics of peatlands, their sensitivity to climate change has so far received little attention (Jassey et al., 2011b;[START_REF] Kim | Funtional and structural responses of bacterial and methanogen communities to 3-year warming incubation in different depths of peat mire[END_REF][START_REF] Tsyganov | Sphagnum-dwelling Testate Amoebae in Subarctic Bogs are More Sensitive to Soil Warming in the Growing Season than in Winter: the Results of Eight-year Field Climate Manipulations[END_REF][START_REF] Weddon | Summer warming accelerates sub-artic peatland nitrogen cycling without changing enzyme pools or microbial community structure[END_REF] and considerable gaps remain in our understanding of the impacts of warming on aboveground-belowground relationships in peatlands following changes in microbial components. Long-term global warming is expected to considerably alter peat-forming areas, potentially involving substantial C loss due to modifications in microbial processes linked to Sphagnum mosses [START_REF] Dorrepaal | Carbon respiration from subsurface peat accelerated by climate warming in the subarctic[END_REF][START_REF] Delarue | Effects of short-term ecosystem experimental warming on water-extractable organic matter in an ombrotrophic Sphagnum peatland (Le Forbonnet, France)[END_REF][START_REF] Fenner | Drought-induced carbon loss in peatlands[END_REF]Jassey et al., 2011a,b). Understanding key mechanisms behind the changing Sphagnummicrobialvascular plant interactions in responses to climate change, is obviously needed to better understand ongoing processes and to predict more accurately future changes in the functioning of peatlands.

In this paper, we focus on the ways in which warming affects above-and belowground subsystems as key components of C fluxes in Sphagnum-peatlands. Specifically, we analyse the effect of climate warming on key above-and belowground components in a Sphagnum peatland in the Jura Mountains (France) to explore whether changes in peatland properties will be brought by plant communities, microbial communities, or by interactions between the two. Our aims were to (1) quantify over the course of two years how the soil microbial food web (biomass of bacteria, fungi, protozoans, and metazoans), plants communities (species composition and diversity), Sphagnum-polyphenols and soil water chemistry composition changed in response to warming, and (2) to clarify whether climate warming destabilized peatland functioning by changing plant-soil-microbial interactions, including polyphenol phytochemical interactions. Linkages amongst these variables were investigated using a pathrelation network and structural equation models (SEM) to gain a mechanistic understanding of how warming effects materialized on above-and belowground linkages.

Materials and methods

Field site and experimental design

The experimental site is an undisturbed Sphagnum-dominated peatland situated in the Jura Mountains (The Forbonnet peatland,46°49'35''N,6°10'20''E). Above-(mosses, vascular plants, polyphenols produced by Sphagnum) and belowground (soil water chemistry, microbial communities living in Sphagnum carpet) components were measured across a transitional fen-bog area in June 2008[START_REF] Dise | Peatland response to global change[END_REF][START_REF] Andersen | Peat, water and plant tissue chemistry monitoring : a seven-year case-study in a restored peatland[END_REF](begin of the warming experiment April 2008). The experimental site has an area of 30m x 30m and the vegetation consists of a mosaic of hummocks and lawns (Jassey et al. 2011a). The moss layer is dominated by S. fallax in lawns and by a mixture of Sphagnum fallax, S. magellanicum and S. palustris in hummocks whereas the evergreen dwarf shrubs Eriophorum vaginatum and Andromeda polifolia, the deciduous dwarf shrubs Vaccinum oxycoccus and the graminoids Calluna vulgaris and Carex rostrata characterize the field layer. We chose 12 sampling plots (six ambient and six warmed plots) across the area placed so as to cover all the possible microforms along the gradient. Six plots were placed in lawns and six plots in hummocks, and in each area three plots were assigned as ambient conditions and three as warmed conditions.

We used open-top chambers (OTC) to simulate the regional climate expected for the coming decades (IPCC, 2007;Jassey et al., 2011a). The OTCs significantly increased mean air temperature in spring and summer by up to about 1.3°C, and decreased the water content of Sphagnum in summer by up to about 30% [START_REF] Delarue | Effects of short-term ecosystem experimental warming on water-extractable organic matter in an ombrotrophic Sphagnum peatland (Le Forbonnet, France)[END_REF], Jassey et al., 2011a). The water table depth did not change with warming and was ranged from 13.4 cm to 20.8 cm between 2008 and 2010.

Belowground measurements

We analyzed the entire microbial community using a morpho-type functional group approach rather than focusing on a given group, such as bacteria or fungi. Samples of S. fallax were collected for the study of microbial communities from ten permanent markers in each plot and the average altitude (microtopography) was measured. This sampling design allowed for multiple sampling over time and collection of a composite sample from each plot, avoiding any bias due to spatial heterogeneity (Mitchell et al., 2000a). Microbial communities were analysed from surface (0-3 cm) and litter (3-6 cm) layers of Sphagnum shoots. Microbial samples were fixed in glutaraldehyde (2%) and extracted following the standard method described in Jassey et al. (2011b). We used direct observation by inverted microscopy (x200 and x400 magnification) to determine precisely the abundance and biomass of individual microbial functional groups (fungi, microalgae, ciliates, testate amoebae, rotifers and nematodes). Flow cytometry (FAC-SCalibur flow cytometer, Becton Dickinson) was used for bacterial counts. Fluorescent microbeads (Molecular probes) of diameter 1µm were added to each sample as an internal standard. Bacterial samples were stained with SYBR Green I (1/10,000 final conc.) for 15 min in the dark and run at medium speed (ca 40 µL min -1 ). For each specimen, the average biovolume (µm 3 ) was estimated by assuming geometrical shapes using image analysis and converted to biomass using conversion factors [START_REF] Gilbert | The microbial loop at the surface of a peatland: Structure, function, and impact of nutrient input[END_REF].

Data were expressed as micrograms of carbon per gram of Sphagnum dry mass (µgC.g -1 DM).

The trophic position of testate amoebae was also estimated by measuring their body size and their shell-aperture size using image analysis. We chose these two specific morphological traits because they are linked to their feeding habit. The species with a low ratio shell aperture size over body size (0 < shell-aperture size/body size < 0.18) were considered as having a low trophic position (i.e. primarily bacterivors and algivors), and species with a high one (0.20 < shell-aperture size/body size) as having a high trophic position (i.e. primarily predators of other protists and micro-metazoan) in the microbial food web [START_REF] Yeates | Testate amoebae as predators of nematodes[END_REF][START_REF] Mitchell | Structure of microbial communities in Sphagnum peatlands and effect of atmospheric carbon dioxide enrichment[END_REF]Jassey et al., 2012).

Water chemistry (20 mL) was analysed in each sampling plot at 10 cm depth from piezometers during each sampling campaign. Total dissolved nitrogen (DN) and dissolved organic carbon (DOC) of peatland water were determined with a SHIMADZU SSM-5000A total C and N analyser. Ammonium (NH 4 + ), nitrates (NO 3 -) and phosphates (PO 4 3-and total phosphorus) were analysed colorimetrically using a continuous flow analyser (FLOWSYS, Systea, Roma) after filtering the bog water at 0.45 µm.

Aboveground measurements

We performed vegetation surveys by the point-intercept method [START_REF] Buttler | Permanent plot research in wet meadows and cutting experiment[END_REF][START_REF] Keuper | A Race for Space? How Sphagnum fuscum stabilizes vegetation composition during long-term climate manipulations[END_REF]. We used a 50x50 cm Plexiglas frame placed above a permanently marked quadrat by means of 4 adjustable poles. A ruler with 20 holes was moved along 20 different positions to obtain 400 measuring points. A metal pin with a 1 mm diameter tip was lowered through each hole in the ruler and each contact of the pin with green living vegetation was recorded by species until the pin reached the moss substrate. Moss and vascular plant abundances were expressed as percentage of mean number of hits (%). Moreover, we measured the average microtropography of the marked vegetation quadrat.

Total water-soluble Sphagnum-polyphenols were quantified in living segments of Sphagnum shoots (0-6 cm) during each sampling campaign using Folin methods and gallic acid as standard (see Jassey et al., 2011b for details). These samples were collected around the same ten permanent markers used for microbial communities for each plot.

Numerical analyses

We carried out linear mixed effects models and ANOVAs to test for differences between treatments and years in the following variables: (i) biomass of individual microbial groups;

(ii) abundance of individual testate amoeba species; (iii) individual water chemical components; (iv) total abundance of plants (mosses and vascular plants) and abundance of individual moss and vascular plant; and (v) polyphenol content in Sphagnum. Linear mixed effects were used to assess the effects of warming (difference of temperature between ambient and warmed plots, delta TC), time and microtopography (fixed effects) on measured of biotic and abiotic variables while accounting for the temporal repeated measurements in each plot on three dates. We fitted all models including plot as a random effect on the intercept, i.e. we corrected for the inflation of the residual degrees of freedom that would occur if we were using repeated measurements as true replicates [START_REF] Pinheiro | Mixed-Effects Models in S and S-PLUS. Springer. Pomeroy L.R Food web connections: links and sinks[END_REF]. Then ANOVAs were applied on the different models to test the fixed effects, and differences among the levels of the fixed effects in the final model were determined using multiple comparison post-hoc analyses (general linear hypothesis test). Finally, we used the F-values from the ANOVAs to quantify the impact of warming on the abundance of individual testate amoeba species.

Higher F-values indicated stronger warming effect on the individual testate amoeba species.

We evaluated the warming effect on vegetation and microbial community structure, and water chemistry composition using the principal response curve (PRC) method. PRC was used to focus on the time-dependent treatment effect and applied on the Hellinger- effect on the individual species/group. Permutation tests with 1,000 permutations, stratified by year, were performed for every canonical axis. Additionally, variance partitioning using redundancy analyses (RDA) and adjusted R 2 were applied to compare the respective effect of microtopography, time and warming on each biotic and abiotic matrices. The significance of the each explanatory variables included in RDA were tested using 1,000 permutations.

Above-and belowground data was organized into a path-relation network and subjected to structural equation modelling (SEM) (Grace & Keeley, 2006;[START_REF] Grace | Does species diversity limit productivity in natural grassland communities[END_REF][START_REF] Grace | On the specification of structural equation models for ecological systems[END_REF]. This enabled us to explore simultaneous influences of several potentially important drivers of peatland functioning that may be affected by warming, and thereby identify their relative importance to obtain a better understanding of peatland response to warming. SEM was based on the overall dataset and predicted causal relationships between variables were based upon prior knowledge, theory and past experience on the role of above-and belowground factors in peatland functioning. ) retained by stepwise selection from the water chemistry matrix. We used RV-coefficients from multiple factor analysis (MFA) to construct our correlation matrices used in SEM [START_REF] Legendre | Numerical Ecology 2 nd Edition[END_REF][START_REF] Grace | On the specification of structural equation models for ecological systems[END_REF]. MFA symmetrically linked the six groups of descriptors described above. MFA was selected because it allowed the simultaneous coupling of several groups or subsets of variables defined on the same objects and assess the general structure of the data (Escofier & Pagès, 1994). RVcoefficients (Pearson correlation coefficient) were used to measure the similarities between two data matrices [START_REF] Robert | Unifying tool for linear multivariate statistical-methods -rvcoefficient[END_REF][START_REF] Josse | Testing the significance of the RV coefficient[END_REF]. SEM was performed using Amos 5 (Amos Development Corporation, Crawfordville, FL, USA). For each analysis, R 2 values were obtained for each dependent matrix, showing the amount of the variance explained by the model (Grace & Keely, 2006;[START_REF] Grace | On the specification of structural equation models for ecological systems[END_REF][START_REF] Jonsson | Structural equation modelling reveals plant-community drivers of carbon storage in boreal forest ecosystems[END_REF].

All statistical analyses were performed with R 2.10.1 using the lme, vegan and FactoMineR packages [START_REF] Husson | FactoMineR: Factor Analysis and Data Mining with R[END_REF][START_REF] Oksanen | Community Ecology Package. R package version 1[END_REF]R Development Core Team, 2010).

Results

Belowground subsystem

Microbial network structure

The structure of microbial communities clearly responded to warming as showed by PRCs (Fig. 1a,b). Microbial community structure in warmed plots significantly deviated from ambient plots at the surface layers (F = 4.6, P = 0.02) whereas the same but weaker warming effect was recorded at the litter layers (F = 3.1, P = 0.05). The response of warmed plots clearly appeared in 2009 in both sampling depth, one year after the beginning of the warming experiment (Fig. 1a,b). The respective effects of microtopography, time and temperature increase on the variation of microbial communities living at the surface layers were of 13.7%, 6.6% and 10.5%, respectively (Table 1), while their respective effects at the litter layers were of 0.6%, 24.7% and 14.3%, respectively. Only some microbial groups were affected by microtopography variations (e.g. microalgae and testate amoebae) along Sphagnum shoots (Table S1). Furthermore, RDA and variance partitioning revealed that the interaction between microtopography and warming were not significant along Sphagnum shoots (Table 1), highlighting a similar warming effect amongst lawns and hummocks.

The detailed examination of the time-dependent warming effect on individual microbial groups revealed a strong negative effect on testate amoebae, especially at the surface layers (Fig. 1a; Table S1). This effect was largely due to a decline of the biomass of the testate amoebae with high trophic level (F = 5.6, P = 0.002) of 40% and 75%, after one and two warming years respectively. The total biomass of this category of species decreased from 440 µgC g -1 DM to 131 µgC g -1 DM in warmed conditions over two years (Table S1).

Although the decrease of the biomass of testate amoebae with high trophic level was not significant at the litter layers (F = 3.9, P = 0.063), the trend of their response followed that of testate amoebae in surface layers with a decrease of 28% and 41% of their biomass after one and two warming years respectively (Table S1). Moreover, we recorded that the warming effect on the abundance of main testate amoeba species was positively correlated to the ratio shell aperture size over body size at the surface layers (r = 0.68, F = 4.25, P < 0.04; Fig. 1c).

Here again, such trend was non-significant in the litter (r = 0.36, F = 2.3, P = 0.17). We further observed that the microbial network from surface and litter became increasingly enriched in bacteria and depleted in top-predators with warming. The relative biomass of bacteria in the surface layers significantly increased from 31% in ambient plots to 45% in warmed plots, and from 29% to 36% in the litter layers (Fig. 2, Table S1). In parallel, the relative biomass of testate amoebae with high trophic level decreased from 31% in ambient plots to 12% in warmed plots in the surface layers and from 21% to 11% in the litter layers respectively. Consequently, the decrease in testate amoebae with high trophic level explained 10% (F = 4.86, P = 0.03) and 16% (F = 7.91, P < 0.01) of the rise of the relative bacterial biomass in warmed plots from surface and litter, respectively.

Water chemistry composition

PRC analysis showed that water chemistry composition was not affected by warming (F = 2.1, P = 0.11). Time mainly explained the variance in water chemistry composition with 17% of the total variation. However, variance partitioning revealed a significant interaction between time and temperature increase on water chemistry composition with 21.9% of the variance (Table 1). Warming slightly affected water chemistry composition over the course of the two years. Elevated temperatures modified the annual dynamic of some chemical components such as PO 4 3-, DOC, and NO 3 -(Table 2), even if these variations remained very low. Most importantly, we found in warmed plots that nitrates (r = -0.64, F = 8.4, P < 0.01)

and DOC (r = -0.66, F = 3.8, P = 0.05) were negatively correlated to the decrease of the biomass of the testate amoebae with high trophic levels, whereas no such relationships were found in ambient plots (Fig. 3a,b,c). In the same way, no such relationships were found with the biomass of the testate amoebae with low trophic levels, both in ambient and warmed plots.

Aboveground subsystem

Mosses and vascular plants abundances

Plant community structure was stable across years and did not respond to warming over the course of the two years, as showed by the PRC (F = 0.61, P = 0.96). Our results revealed that microtopography highly affected the composition of vegetation communities across the vegetation patterns (F = 12.8, P = 0.001; Table 1). The detailed examination of the warming effect on vegetation in lawns and hummocks showed that the total abundance of vascular plants significantly increased with warming in both microtypes (F = 5.1, P = 0.032, ANOVA)

and that of mosses decreased in lawns (F = 7.15, P = 0.019, ANOVA). Total vascular plant abundance increased from 29% of mean number of hits to 31% in lawns and from 37% of mean number of hits to 41% in patterns of hummocks and hollows (Fig. 4a). Such increase was due to a rise in the abundance of the evergreen low shrub Andromeda polifolia (F = 8.37, P < 0.01; Fig. 4c). The abundance of other evergreen vascular plants (Eriophorum vaginatum, Vaccinium oxycoccus), graminoids (Carex limosa, C. rostrata, C. pauciflora) and forbs (Drosera rotundifolia) did not respond to climate warming. Total moss abundance decreased in lawns from 55% to 45% after two warming years (Fig. 4b), especially Sphagnum fallax which significantly decreased from 49.0% to 42.9% (F = 4.1, P = 0.03; Fig. 4d). Other moss species such as S. magellanicum and S. palustris did not respond to warming.

Sphagnum-polyphenols. Total water-soluble polyphenol content was comprised between 0.5 mg.g -1 DM and 2.5 mg.g -1 DM (Table 2). Phenolic content was not affected by warming over the course of the two warming years, but a significant negative link between phenolics and air temperatures was found. Furthermore, we found a negative relationship between watersoluble polyphenol and the biomass of testate amoebae with high trophic levels in warmed plots (r = -0.6, F = 9.9, P < 0.01) whereas the same relationship in ambient plots was not significant (r = -0.2, F = 1.7, P = 0.21) (Fig. 5). Moreover, variance partitioning revealed that water-soluble polyphenols and temperature increase explained 13% (F = 6.2, P = 0.018) and 19% (F = 8.9, P = 0.005) of the decrease of the biomass of testate amoebae with high trophic levels at the surface layers, respectively. Here again, such relationship was not found with testate amoebae with low trophic levels, as well as in the litter layers.

Effect of warming on above-and belowground linkages

The use of a path-relation network showed that warming affected above-and belowground interactions (Fig. 6). The final models adequately fitted the data under ambient (χ 2 7 = 6.51, P = 0.51; AIC = 34.19; GFI = 0.88; RMSEA < 0.001; RMR = 0.14) and warmed conditions (χ 2 7 = 1.00, P = 0.99; AIC = 29.03; GFI = 0.98; RMSEA < 0.001; RMR = 0.034). Under ambient conditions, the model showed that microbial community structure in surface layers had significant links to microbial community structure in litter (P < 0.01), and that Sphagnumpolyphenols were negatively linked to microbial community structure in surface and litter (P = 0.09) (Fig. 6a). Significant associations were further found between microbial communities living in litter and water chemistry composition (P < 0.01; Fig. 6a). The model also showed a strong negative association between moss community and vascular plant community (P < 0.01). The ambient model explained a low proportion of the variance in mosses (8%) and microbial community structure at the surface (2%), whereas it explained 17.5%, 25.5% and 46.4% of microbial community structure in litter, water chemistry, and vascular plant community, respectively (Fig. 6a).

In warmed plots, treatment effect on testate amoebae with high trophic level significantly increased the link of microbial communities living in surface with moss community structure (P < 0.01; Fig. 6b). The link between Sphagnum-polyphenols and microbial communities also increased and shifted from a negative association in ambient plots to a positive association in warmed plots, especially on microorganisms living in surface layers. This analysis also revealed a decrease of the negative association between mosses on vascular plants community structure in warmed conditions. Moreover, the model showed that the relationships between water chemistry and microbial communities increased with warming, especially in litter layers (P < 0.01). The SEM model on warmed plots explained a larger proportion of the variance in moss community structure (19.4%), water chemistry (49.7%), microorganisms in surface (13%) and litter (27.3%) layers, whereas that of vascular plants diminished (19.5%) (Fig. 6b).

Discussion

Our results from a field manipulation experiment in a Sphagnum peatland show that warming affects total vascular plant and moss abundances as well as the structure and functioning of soil microbial communities so resulting in a modification of linkages between above-and belowground subsystems.

Warming effect on microbial communities resulted to rapid trophic alteration in the microbial food web. The impact of warming on microbial structure decreased with depth; the response of microbial communities was highly significant in the surface layers, similar but weaker in the litter layers. Such result is in line with previous observations of a weak warming effect on Sphagnum-polyphenol, enzymatic activities and water-extractable organic matter at this depth [START_REF] Delarue | Effects of short-term ecosystem experimental warming on water-extractable organic matter in an ombrotrophic Sphagnum peatland (Le Forbonnet, France)[END_REF]Jassey et al., 2011a;Jassey et al., 2012b). Warmed communities disproportionately lost top-predators and omnivores, and became increasingly dominated by decomposers and autotrophs (Fig. 2), showing that the stability of the microbial food web was affected by warming [START_REF] Petchey | Environmental warming alters food-web structure and ecosystem function[END_REF][START_REF] Dossena | Warming alters community size structure and ecosystem functioning[END_REF][START_REF] Heckmann | Interactive effects of body-size structure and adaptive foraging on food-web stability[END_REF]. More specifically, warming impacted testate amoebae with a large body size and shell aperture size, key organisms of the structure and functioning of microbial network due to their position at the top of the food web [START_REF] Mitchell | Structure of microbial communities in Sphagnum peatlands and effect of atmospheric carbon dioxide enrichment[END_REF][START_REF] Wilkinson | Testate Amoebae and Nutrient Cycling with Particular Reference to Soils[END_REF]Jassey et al., 2012a). This decrease was linked and coincided with an increase in the relative biomass of bacteria, both in the surface and litter layers. Such changes in the relative distribution of microbial group among trophically defined levels potentially lead to differences in ecosystem functioning beyond those expected from ambient conditions [START_REF] Petchey | Environmental warming alters food-web structure and ecosystem function[END_REF][START_REF] Petchey | Size, foraging, and food web structure[END_REF][START_REF] Dossena | Warming alters community size structure and ecosystem functioning[END_REF]. Although it was not explore here, warming has been recognized to alter the structure of bacterial communities by favouring bacterial communities over methanogens, which may result changes in balance between CO 2 and CH 4 fluxes from peatlands [START_REF] Kim | Funtional and structural responses of bacterial and methanogen communities to 3-year warming incubation in different depths of peat mire[END_REF]. Furthermore, the strong impact of climate warming on testate amoebae is of particular interest because they link mosses and microbial heterotrophic producers (i.e. fungi and bacteria) via the microbial loop. This loop describes a trophic pathway in microbial food web where dead organic matter is returned to higher trophic levels via its incorporation into bacterial biomass coupled with the microbial food chains (e.g. bacteria  ciliate  testate amoebae; [START_REF] Gilbert | The microbial loop at the surface of a peatland: Structure, function, and impact of nutrient input[END_REF]. This recycles C and nutrients from dead organic matter to plant communities [START_REF] Ledeganck | Plant functional group diversity promotes soil protist diversity[END_REF]Mitchell et al., 2003). The decline of testate amoebae with high trophic levels probably caused a shortening of microbial food chains that sould greatly affect the microbial food web stability and structure, which further could destabilize ecosystem processes they control [START_REF] Petchey | Environmental warming alters food-web structure and ecosystem function[END_REF][START_REF] Mitchell | Structure of microbial communities in Sphagnum peatlands and effect of atmospheric carbon dioxide enrichment[END_REF][START_REF] Wilkinson | Testate Amoebae and Nutrient Cycling with Particular Reference to Soils[END_REF][START_REF] Heckmann | Interactive effects of body-size structure and adaptive foraging on food-web stability[END_REF]. Indeed, testate amoebae with large shell aperture consume a wide range of prey such as small testate amoebae, ciliates, rotifers and small nematodes, typically bacterivores and algivores species [START_REF] Wilkinson | Testate Amoebae and Nutrient Cycling with Particular Reference to Soils[END_REF]Jassey et al., 2012a). Such decrease could accelerate the turnover of microbial biomass due to the diminution of their top-down control (predation), speeding up decomposition and nutrient recycling via the microbial loop in long-term [START_REF] Petchey | Environmental warming alters food-web structure and ecosystem function[END_REF][START_REF] Mitchell | Structure of microbial communities in Sphagnum peatlands and effect of atmospheric carbon dioxide enrichment[END_REF][START_REF] Dossena | Warming alters community size structure and ecosystem functioning[END_REF]. Thus these changes in microbial community structure and stability suggest that climate warming will impact belowground subsystem functioning (Allison & Martiny, 2008).

Interestingly, in other Sphagnum-dominated peatlands, summer warming accelerates nitrogen cycling due to changes in belowground subsystem [START_REF] Weddon | Summer warming accelerates sub-artic peatland nitrogen cycling without changing enzyme pools or microbial community structure[END_REF]. In our study, the largest differences in water chemistry composition were found amongst sampling years. Annual variations in evapotranspiration and oxido-reduction processes associated with the water table position probably explained these variations [START_REF] Proctor | Seasonal and shorter-term changes in surface-water chemistry on four English ombrogenous bogs[END_REF][START_REF] Andersen | Peat, water and plant tissue chemistry monitoring : a seven-year case-study in a restored peatland[END_REF]. However, our results highlighted that water chemistry composition responded to warming in interaction with annual variations by slightly increasing nutrient (as NO 3 -) and C-exports (as DOC) levels in the soil solution after two years. Reasons of such slight effect may be explained by the fact that microorganisms living in surface layers are exposed to the chemistry of capillary water rather than water deeper in the peat (i.e. water table) [START_REF] Lamentowicz | Contrasting Species-Environment Relationships in Communities of Testate Amoebae, Bryophytes and Vascular Plants Along the Fen-Bog Gradient[END_REF]. As a consequence of this vertical gradient, the relationships between bryophyte and water deeper in the peat are not optimal (Mitchell et al., 2000b;Hajkova et al., 2004). Warming effects on water chemistry would have been probably higher on water extracted from mosses themselves but this was not analysed here. Nevertheless, increasing amount of DOC and NO 3 -in warmed plots, even varying annually, may be interpreted as an indication of a slight destabilization of the belowground functioning of peatland [START_REF] Evans | Long-term increases in surface water dissolved organic carbon: Observations, possible causes and environmental impacts[END_REF][START_REF] Carrera | Soil invertebrates control peatland C fluxes in response to warming[END_REF][START_REF] Weddon | Summer warming accelerates sub-artic peatland nitrogen cycling without changing enzyme pools or microbial community structure[END_REF]. Moreover, the actual amount of C and nutrient fluxes from the microbial loop is dependent of the length of food chains in microbial food webs (Pomeroy, 2006). Thus, the negative correlations between NO 3 -/DOC and the biomass of testate amoebae with high trophic levels suggest that the trophic alteration of the microbial structure due to potential shortening of food chains affected these biogeochemical fluxes.

We clearly showed that warming affected belowground subsystem, however, these phenomena are often indirectly linked to changes in aboveground communities [START_REF] Wardle | Linking between aboveground and belowground biota[END_REF]. A growing number of studies have showed how changes in plant communities affect soil communities [START_REF] Wardle | Plant removals in perennial grassland: Vegetation dynamics, decomposers, soil biodiversity, and ecosystem properties[END_REF][START_REF] Kardol | How understanding aboveground-belowground linkages can assist restoration ecology[END_REF] and how the soil biota in turn affects the plant community, leading to feedbacks between the plant and soil subsystems (De Dyn et al., 2003;[START_REF] Kardol | Temporal variation in plant-soil feedback controls succession[END_REF][START_REF] Wardle | Linking vegetation change, carbon sequestration and biodiversity: insights from island ecosystems in a long-term natural experiment[END_REF]. The increase of vascular plants and the decrease of Sphagnum in response to warming probably modified the quantity and/or quality of plant-derived organic matter, which in turn affected microbial communities [START_REF] Delarue | Effects of short-term ecosystem experimental warming on water-extractable organic matter in an ombrotrophic Sphagnum peatland (Le Forbonnet, France)[END_REF]. Although it was not explored here, we cannot exclude an increase of root exudates from vascular plants associated to their increase in response to warming, which could stimulate decomposers such as bacteria [START_REF] Delarue | Effects of short-term ecosystem experimental warming on water-extractable organic matter in an ombrotrophic Sphagnum peatland (Le Forbonnet, France)[END_REF]. In addition, we revealed significant relationships between Sphagnum-polyphenols and microbial communities in warmed plots, suggesting these aboveground factors interact in driving belowground subsystem. Sphagnum-polyphenols play a fundamental ecological role in the regulation of soil microbiota and hence of microbial processes by way of chemical interactions (Hättenschwiler & Vitousek, 2000;[START_REF] Inderjit | The ecosystem and evolutionary contexts of allelopathy[END_REF]Jassey et al., 2011a). For the first time, we found that warming greatly impacted polyphenol-microbial community interactions by a shift from a negative to a positive effect in warmed conditions (Fig. 6), and that the amount of Sphagnumpolyphenols have a similar influence than warming on key microorganisms such as testate amoebae. The strong negative correlation between polyphenols and testate amoebae with high trophic levels in warmed conditions revealed a potential inhibitory effect of phenols on this category of microorganisms in context of warming (Jassey et al., 2011b). Even if warming did not directly impact the total amount of Sphagnum-polyphenols, qualitative changes induced by climate change are quite possible, as previously showed in dwarf shrubs ecosystems [START_REF] Hansen | Long-term experimental warming, shading and nutrient addition affect the concentration of phenolic compounds in artic-alpine deciduous and evergreen dwarf shrubs[END_REF]. Such findings strongly suggest that polyphenols produced by Sphagnum play a key role in the regulation of testate amoebae community structure, as well as in the microbial network stability in ways that can positively affect decomposition (Jassey et al., 2011b;[START_REF] Inderjit | The ecosystem and evolutionary contexts of allelopathy[END_REF]. Indeed, by their inhibitory effect on top-predators, polyphenols may indirectly enhance C and nutrient mineralization. Bacterial growth and microbial OM breakdown could be stimulated by such inhibition of top-predators, leading to an increase of nutrient levels and C release in peatlands, which further would destabilize the productivity of plant communities and peatland carbon stock [START_REF] Wardle | Linking between aboveground and belowground biota[END_REF][START_REF] Fenner | Drought-induced carbon loss in peatlands[END_REF]Jassey et al., 2011a).

Reciprocally, we also assume that direct effects of climate warming on the microbial food web modify the linkages between above-and belowground communities. Warming effect on microbial food webs could greatly influence aboveground productivity and plant community structure through the stimulation of nutrient dynamics, which indirectly influence nutrient uptake of aboveground consumers such as vascular plants [START_REF] Mitchell | Structure of microbial communities in Sphagnum peatlands and effect of atmospheric carbon dioxide enrichment[END_REF][START_REF] Carlson | Community development along a proglacial chronosequence: are above-ground and below-ground community structure controlled more by biotic than abiotic factor?[END_REF][START_REF] Kardol | How understanding aboveground-belowground linkages can assist restoration ecology[END_REF]. The rise of temperature involved a decrease of the negative link between mosses and vascular plants, highlighting a reduction of Sphagnum mosses repression on vascular plant life (Fig. 6), as shown in response to other perturbations such as fertilization and drainage [START_REF] Berendse | Raised atmospheric CO 2 levels and increased N deposition cause shifts in plant species composition and production in Sphagnum bogs[END_REF][START_REF] Lang | Determinants of cryptogam composition and diversity in Sphagnum-dominated peatlands: the importance of temporal, spatial and functional scales[END_REF]. Even if vegetation composition did not change overall with warming, we recorded significant changes in total vegetation cover in course of two years of warming with a decrease of Sphagnum in lawns and an increase of vascular plants in lawns and hummocks. The observed slight increases of nutrient and C levels in soil solution are likely to alter the competition between mosses and vascular plants [START_REF] Berendse | Raised atmospheric CO 2 levels and increased N deposition cause shifts in plant species composition and production in Sphagnum bogs[END_REF][START_REF] Keuper | A Race for Space? How Sphagnum fuscum stabilizes vegetation composition during long-term climate manipulations[END_REF]. Sphagnum mosses are known to be efficient at absorbing nitrogen, preventing deep-rooted vascular plants growing in a dense Sphagnum carpet [START_REF] Van Breemen | How Sphagnum bogs down other plants[END_REF][START_REF] Turetsky | The role of bryophytes in carbon and nitrogen cycling[END_REF][START_REF] Gunnarsson | Global patterns of Sphagnum productivity[END_REF], but excessive tissue nitrogen accumulation due to warming effect on the microbial food web should exacerbate the decline of Sphagnum and the increase of vascular plants [START_REF] Breeuwer | The effect of temperature on growth and competition between Sphagnum species[END_REF][START_REF] Breeuwer | Response of Sphagnum species mixtures to increased temperature and nitrogen availability[END_REF][START_REF] Limpens | How litter quality affects mass loss and N loss from decomposing Sphagnum[END_REF]. Such a warming-induced increase of vascular plant cover would further contribute to diminishing carbon sequestration in Sphagnum-peatlands due to a plant litter more easily degradable [START_REF] Verhoeven | Decomposition of carex and sphagnum litter in fens -effect of litter quality and inhibition by living tissue-homogenates[END_REF].

We conclude that climate warming greatly destabilises peatland functioning by changing the interactions between plant and microbial communities. These results imply that warming changes peatlands directly by modifying the structure of microbial communities with the loss of top-predators and the increase of bacteria and indirectly by changing ecosystem functions via modifications of plant-soil-microbial interactions through the increase of nutrients cycling and the positive and/or negative effect of Sphagnum-polyphenol on the microbial food web. Our observations support the hypothesis that microbial food web associated to mosses positively contribute to global warming by controlling ecosystem feedbacks [START_REF] Lindo | The Bryosphere: an integral and influential component of the earth's biosphere[END_REF][START_REF] Singh | Microorganisms and climate change: terrestrial feedbacks and mitigation options[END_REF]Jassey et al., 2011b). Consequently the moss-microbiota system exerts a strong control over the structure and dynamics of peatland ecosystems. Furthermore, these key findings add another crucial new contributor to the list of mechanisms by which mosses, as ecosystem engineers, tightly regulate biogeochemical cycling and climate feedback in peatlands [START_REF] Van Breemen | How Sphagnum bogs down other plants[END_REF][START_REF] Cornelissen | Comparative cryptogam ecology: a review of bryophyte and lichen traits that drive biogeochemistry[END_REF][START_REF] Gornall | Artic mosses govern belowground environment and ecosystem processes[END_REF]. Warming effect on testate amoeba species in relation to the size of their shell-aperture over their body size. The warming effect was determined using the F-value from ANOVA tests.

Testate amoebae that contributed to less than 3% of maximum abundance in all samples were removed from the data set to reduce the influence of rare taxa in this analysis. Lines are regression line, asterisks indicate significant regression at P < 0.05 level (ANOVA test).

  transformed vegetation and microbial assemblages and the standardized water chemistry matrix, including the response of individual plant species, microbial groups or chemical components. PRC analysis is a special case of partial RDA (pRDA) for multivariate responses in repeated observation design. It is based on a pRDA adjusted for overall changes in community structure response over time, as compared to control plots (van den[START_REF] Van Den | Principal response curves: analysis of timedependent multivariate responses of a biological community to stress[END_REF]. In diagrams, the curves represent the time trajectory for the controls as a horizontal line maintained to 0 (dashed line) and deviation of community structure in warmed plots in course of two years (solid line). This is achieved by taking the control treatment as the reference against which the other treatments are contrasted and by defining -time‖ as the horizontal axis of the diagram (van den[START_REF] Van Den | Principal response curves: analysis of timedependent multivariate responses of a biological community to stress[END_REF]. With the help of the species weights on the right Y-axis, the PRC can be used to infer about the response of individual species to warming. The species scores on the right Y-axis allow an interpretation at the species level, i.e. values indicate the contribution of individual species/group/component to the deviation of the community structure observed in warmed plots. Individual group with positive scores on right Y-axis are inferred to show a negative effect of warming, whereas groups with negative scores show a positive response to warming. A group near zero scores either show no response or a response unrelated to the pattern shown in PRC. Higher values (positive or negative) on the right Y-axis indicate stronger warming

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: Barplot of the relative contribution of microbial groups to the total microbial

Figure 3 :

 3 Figure 3: (a) Nitrates (NO 3 -) and (b) labile C (DOC) concentrations plotted against the

Figure 4 :

 4 Figure 4: Total abundance of (a) vascular plants and (b) mosses (expressed as % of mean

Figure 5 :

 5 Figure 5: Total water-soluble polyphenol content (mg.g -1 DM) plotted against the biomass of

Figure 6 :

 6 Figure 6: Structural equation models of above-and belowground components in a

  The adequacy of the model was determined via several tests, i.e. χ 2 tests, goodness fit index (GFI), Akaike value (AIC), root square mean error of approximation (RMSEA) and root mean square residual (RMR)(Grace & Keeley, 2006;[START_REF] Jonsson | Structural equation modelling reveals plant-community drivers of carbon storage in boreal forest ecosystems[END_REF][START_REF] Eisenhauer | Global change belowground: impacts of elevated CO2, nitrogen, and summer drought on soil food webs and biodiversity[END_REF]. Adequate model fits are indicated by

	nonsignificant χ 2 tests (P > 0.05), high GFI (0.8 < GFI < 1), low AIC, low RMSEA (< 0.05)
	and low RMR (RMR < 0.05) (Grace et al., 2010). Models were built and separately tested for
	ambient and warmed plots based on the biomass of individual microbial groups in surface and
	litter, abundances of mosses and vascular plants, polyphenol content and a subset of four
	variables (DOC, NH 4	+ , NO 3	-and PO 4	3-

Table 2 :

 2 Mean (± SE) of water chemical variables measured in a Sphagnum-dominated peatland subjected to experimental warming (WAR) or ambient conditions (AMB) in 2008, 2009 and 2010. Significant warming effects are shown in bold (ANOVA tests). W = warming effect.

			2008	2009	2010		F-value P -value
			AMB WAR AMB WAR AMB WAR	warming	time	microtopography
	pH	Mean SE	3.88 0.03	3.41 0.17	4.54 0.30	4.23 0.26	3.96 0.05	3.91 0.07	1.97 P = 0.17	0.76 P = 0.39	4.2 P = 0.07
	DOC (mg L -1 )	Mean SE	42.0 4.3	40.2 0.8	40.7 1.1	40.5 0.7	47.3 1.0	48.5 0.8	4.2 P = 0.001	87.2 P = 0.001	3.2 P = 0.11
	DN (mg L -1 )	Mean SE	1.00 0.20	0.70 0.03	0.74 0.11	0.63 0.01	0.70 0.01	0.71 0.02	0.88 P = 0.35	1.8 P = 0.19	0.2 P = 0.90
	PO4 3-(mg L -1 )	Mean SE	1.9 0.2	2.3 0.7	11.5 2.3	10.5 0.2	10.1 0.3	9.7 0.3	8.93 P = 0.001	28.2 P = 0.001	0.17 P = 0.69
	Ptot (mg L -1 )	Mean SE	12.4 0.4	13.4 0.8	20.3 2.2	19.6 2.2	11.7 0.5	13.5 2.0	0.24 P = 0.78	0.25 P = 0.61	0.55 P = 0.47
	NH4 + (mg L -1 )	Mean SE	55.0 6.2	50.2 1.7	43.4 2.6	51.5 2.8	43.2 2.2	40.2 9.3	1.1 P = 0.31	5.23 P = 0.03	4.32 P = 0.07
	NO3 -(mg L -1 )	Mean SE	8.7 2.0	7.7 1.5	10.3 4.4	4.4 2.6	25.7 1.6	28.0 1.8	6.23 P = 0.02	21.2 P = 0.001	0.29 P = 0.59
	Phenol (mg.g -1 DM)	Mean SE	1.20 0.06	1.11 0.07	1.00 0.09	0.95 0.13	1.49 0.20	2.31 0.22	0.03 P = 0.87	5.49 P = 0.03	10.4 P = 0.001

Acknowledgements

This research is a contribution of the ANR PEATWARM project (Effect of moderate warming on the functioning of Sphagnum peatlands and their function as carbon sink). PEATWARM is supported by the French National Agency for Research under the ‗‗Vulnerability: Environment-Climate'' Program (ANR-07-VUL-010). Further funding was provided to VEJ.J by the Franche-Comté Region, to A.B by EPFL and WSL, and to EAD.M by WSL, EPFL and University of Neuchâtel. The authors would like to thank T. Sime-Ngando and J. Colombet (University of Clermont-Ferrand, FR) for their help with the flow cytometry, and G. Bernard for field assistance. R.D. Bardgett (University of Lancaster, UK) R. Payne (Manchester Metropolitan University, UK) and R.T.E. Mills (Swiss Federal Research Institute WSL, CH) are kindly acknowledged for their fruitful comments on the manuscript and English edits.