Francis Filbet

AN ASYMPTOTICALLY STABLE SCHEME FOR DIFFUSIVE COAGULATION-FRAGMENTATION MODELS

Keywords: coagulation-fragmentation equation, finite volume method AMS subject classifications. 65R20, 82C05

This paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation with diffusion in space. A finite volume scheme is developed, based on a conservative formulation of the space nonhomogeneous coagulation-fragmentation model, it is shown that the scheme preserves positivity, total volume and global steady states. Finally, several numerical simulations are performed to investigate the long time behavior of the solution.

Introduction

Coagulation and fragmentation processes occur in the dynamics of cluster growth and describe the mechanisms by which clusters can coalesce to form larger clusters or fragment into smaller ones. Models of this type have found important applications in areas ranging over polymer kinetics [START_REF] Ziff | The kinetics of cluster fragmentation and depolymerisation[END_REF], aerosols [START_REF] Friedlander | Smoke, Dust and Haze, Fundamentals of aerosol dynamics[END_REF], cluster formation in astrophysics [START_REF] Makino | On the mass distribution of planetesimals in the early runaway stage[END_REF] to the animal grouping in biology [27].

This paper is devoted to the numerical simulation of the dynamics of the density function f = f (t, x, y) ≥ 0 of particles (polymers, clusters) with position x ∈ Ω ⊂ R N (N ≥ 1), volume y ∈ R + := (0, ∞), and time t ≥ 0. The distribution function f is subject to coagulation and fragmentation phenomena with respect to the volume variable y and diffusion in space x ∈ Ω and is governed by the following equation [START_REF] Aizenman | Convergence to equilibrium in a system of reacting polymers[END_REF] ∂f ∂t -d(y)

∆ x f = Q(f ),
where Q(f ) is the coagulation-fragmentation operator acting on the volume variable y ∈ R + and these particles diffuse in the environment Ω, which is assumed to be a smooth bounded domain with normalized volume |Ω| = 1. Moreover, the model ( 1) is supplemented with an initial datum

(2) f (0, x, y) := f in (x, y), and in the following analysis we will consider homogeneous Neumann boundary conditions on ∂Ω

(3)

∇ x f (t, x, y) • ν(x) = 0, (t, x, y) ∈ R + × ∂Ω × R + ,
with ν the outward unit normal to Ω. We also assume the diffusion coefficient d(y) to be non degenerate in the sense that there exists d * , d * ∈ R + such that (4)

d * ≥ d(y) ≥ d * > 0, y ∈ R + .
Let us first describe precisely the coagulation and fragmentation process. In the simplest coagulationfragmentation models the clusters are usually assumed to be fully identified by their size (or volume, or number of particles). The coagulation-fragmentation models we consider in this paper describe the time evolution of the cluster size distribution as the system of clusters undergoes binary coagulation and binary fragmentation events. More precisely, denoting by C y the clusters of size y with y ∈ R + , the basic reactions taken into account herein are [START_REF] Carrillo | Exponential Decay Towards Equilibrium for the Inhomogeneous Aizenman-Bak Model[END_REF] C y + C y ′ a(y, y ′ ) -→ C y+y ′ , (binary coagulation) and ( 6)

C y b(y -y ′ , y ′ ) -→ C y-y ′ + C y ′ , (binary fragmentation),
where a and b denote the coagulation and fragmentation rates respectively, and are assumed to depend only on the size of the clusters involved in these reactions. Thus, the coagulation-fragmentation operator is defined by ( 7)

Q(f ) = Q c (f ) -Q f (f ),
where the coagulation operator is

(8) Q c (f )(x, y) = 1 2 y 0 a(y ′ , y -y ′ ) f (x, y ′ ) f (x, y -y ′ ) dy ′ - ∞ 0 a(y, y ′ ) f (x, y) f (x, y ′ ) dy ′
whereas fragmentation mechanism by which a single particle splits into two pieces is given by ( 9)

Q f (f )(x, y) = 1 2 y 0 b(y ′ , y -y ′ ) dy ′ f (x, y) - ∞ 0 b(y, y ′ ) f (x, y + y ′ ) dy ′ .
The coagulation coefficient, a = a(y, y ′ ), characterizes the rate at which the coalescence of two particles with respective volumes y and y ′ produces a particle of volume y + y ′ , whereas the fragmentation coefficient, b = b(y, y ′ ), represents the rate at which the fragmentation of one particle with volume y + y ′ produces two particles of volume y and y ′ . Both coefficients a and b are nonnegative functions and [START_REF] Escobedo | Gelation and mass conservation in coagulation-fragmentation models[END_REF]    a(y, y ′ ) = a(y ′ , y), b(y, y

′ ) = b(y ′ , y), a, b ∈ L ∞ loc R + × R + .
For symmetric kernels, we observe that during the microscopic coagulation and fragmentation processes, as depicted in equations ( 5)-( 6), the number of particles varies with time while the total volume of particles is conserved.

In terms of f , the total number of particles and the total volume of particles at time t ≥ 0 are respectively given by

M 0 (t, x) := R + f (t, x, y) dy, M 1 (t, x) := R + y f (t, x, y) dy.
Besides existence and uniqueness results, few is known on the qualitative behavior of solutions of coagulation-fragmentation equations except when coagulation and fragmentation coefficients are linked by the so called "detailed balanced condition" : there exists a nonnegative function

M ∈ L 1 (Ω × R + , (1 + y)dxdy) such that (11) a(y, y ′ ) M (y) M (y ′ ) = b(y, y ′ ) M (y + y ′ ), (x, y, y ′ ) ∈ Ω × R + × R + .
This condition implies that M is a stationary solution to the coagulation-fragmentation equation

d(y) ∆ x f + Q(f ) = 0.
We further assume that, for each R ∈ R + , the equilibrium M satisfies the positivity condition [START_REF] Filbet | Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation[END_REF] inf

y∈[0,R] M (y) > 0.
An additional and interesting feature of the detailed balanced condition is that there exists an entropy H given by

H(f |M ) = Ω×R + f (x, y) ln f (x, y) M (y) -1 + M (y) dy dx,
which satisfies the following H theorem

dH dt = - 1 2 D(f ) - Ω×R + (∇ x f ) 2 f (t, x, y) dy dx ≤ 0,
where the entropy dissipation of the coagulation-fragmentation operator is defined by

D(f ) = Ω R + ×R + a f f ′ -b f ′′ ln(a f f ′ ) -ln(b f ′′ ) dy dy ′ dx,
with the shorthand notation f := f (t, x, y), f ′ := f (t, x, y ′ ) and f ′′ := f (t, x, y + y ′ ). Since D(f ) only vanishes when f is an equilibrium, it naturally means that

f (t) -→ M, when t → ∞.
The main purpose of this work is to present a numerical scheme to solve (1) built upon a finite volume discretization with respect to the space variable x ∈ Ω and volume variable y ∈ R + . The analysis of the so-obtained scheme would allow to prove the convergence of the discretized particle density towards a solution to the continuous problem on a fixed time interval [0, T ] (T > 0), see for instance [START_REF] Bourgade | Convergene of a Finite Volume Scheme for the Coagulation-Fragmentation Models[END_REF], but here our aim is different. It will consist in the study of the long time behavior of the numerical solution on a fixed mesh in space and volume variables. Indeed, often in applications we are interested in steady states or in the long time behavior of the solution, it is then important to design a numerical scheme, which has good stability properties uniformly in time and remains consistent with respect to the exact solution for long time. Thus, the scheme proposed in this paper is designed such that it preserves qualitative properties of the exact solution as steady states of the coagulation-fragmentation operators (8)- [START_REF] Erasmus | Numerical treatment of the population balance equation using a Spline-Galerkin method[END_REF]. Moreover, an estimate of the entropy dissipation due to an appropriate finite volume approximation with respect to space and volume variables is given and the scheme is shown to give a consistent approximation of the continuous problem in the long time asymptotic limit t → ∞.

Before describing more precisely our results, let us recall that the coagulation and fragmentation equations ( 1) with ( 8)-( 9) have been the object of several studies recently.

On the one hand, among the various approaches for the approximation of coagulation and fragmentation models, we may distinguish between deterministic and Monte Carlo methods. We refer for instance to [START_REF] Erasmus | Numerical treatment of the population balance equation using a Spline-Galerkin method[END_REF][START_REF] Filbet | Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation[END_REF][START_REF] Lee | On the validity of the coagulation equation and the nature of runaway growth[END_REF] for deterministic methods, [START_REF] Babovsky | On a Monte Carlo scheme for Smoluchowski's coagulation equation[END_REF][START_REF] Guias | A stochastic approach for simulating spatially inhomogeneous coagulation dynamics in the gelation regime[END_REF] for stochastic methods, and the references therein. Concerning the convergence analysis of numerical methods for coagulation and fragmentation models, we refer to [START_REF] Lécot | A quasi-Monte Carlo scheme for Smoluchowski's coagulation equation[END_REF] for a rigorous study of quasi Monte-Carlo methods. For deterministic approximations, the situation is different since the relationship between discrete and continuous models has been considered by some authors, see the survey paper [START_REF] Drake | A general mathematical survey of the coagulation equation[END_REF] and [START_REF] Aizenman | Convergence to equilibrium in a system of reacting polymers[END_REF]. A rigorous setting for the formal analysis performed in [START_REF] Aizenman | Convergence to equilibrium in a system of reacting polymers[END_REF] under general assumptions on the coagulation and fragmentation coefficients has been given in [START_REF] Laurenc | From the discrete to the continuous coagulation-fragmentation equations[END_REF]. Then, similar techniques are used to prove convergence of discrete schemes to the exact solution in [START_REF] Bourgade | Convergene of a Finite Volume Scheme for the Coagulation-Fragmentation Models[END_REF]. However, few is known concerning the stability and the analysis of the numerical solution in the long time asymptotic limit. For the continuous model, in reference [START_REF] Laurenc | The continuous coagulation-fragmentation equations with diffusion[END_REF], the authors are able to apply the techniques of weak compactness, as in the paper by R.J. DiPerna and P.-L. Lions [START_REF] Diperna | Global solutions of Boltzmann's equation and the entropy inequality[END_REF] about the Boltzmann equation, to prove the weak stability of weak solutions. In the case when an H-theorem holds, they can obtain some partial information about the large-time behavior of solutions. More recently, J. A. Carrillo et al. prove exponential decay of the solution towards equilibrium for the inhomogeneous Aizenman-Bak model [START_REF] Carrillo | Exponential Decay Towards Equilibrium for the Inhomogeneous Aizenman-Bak Model[END_REF] using entropy dissipation methods [START_REF] Desvillettes | On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation[END_REF]. Our analysis for discrete models will be inspired by these works.

We now briefly outline the contents of the paper. In the next section, we introduce the numerical approximation of (1) and state the stability result which we prove in Section 3. In the final section (Section 5), some numerical simulations are performed with the numerical scheme presented in Section 2 and the long time behavior of the solution is investigated.

Numerical scheme and main results

In order to compute an approximation of this model using a finite volume method in space variable x ∈ Ω and volume variable y ∈ R + , we observe that the coagulation-fragmentation operator can be written in a conservative form. Indeed, writing equation (1) in a "conservative" form, as proposed in [START_REF] Makino | On the mass distribution of planetesimals in the early runaway stage[END_REF][START_REF] Tanaka | Steady-state size distribution for the self-similar collision cascade[END_REF], enables to describe precisely the time evolution of the total volume. Also, this formulation is particularly well adapted to a finite volume discretization which, in turn, is expected to give a precise account of volume conservation. Precisely, the coagulation and fragmentation terms can be written in divergence form:

         y Q c (f )(x, y) = - ∂C(f ) ∂y (x, y), y Q f (f )(x, y) = - ∂F(f ) ∂y (x, y),
where the operator C(f ) is given by

C(f )(x, y) := y 0 ∞ y-u u a(u, v) f (x, u) f (x, v) dvdu , (x, y) ∈ Ω × R + , (13)
and F(f ) is

F(f )(x, y) := y 0 ∞ y-u u b(u, v) f (x, u + v) dvdu , (x, y) ∈ Ω × R + . (14)
Finally, the coagulation-fragmentation equation reads

       y ∂f ∂t -d(y) y ∆ x f = - ∂ C(f ) ∂y + ∂ F(f ) ∂y , f (0, x, y) = f in (x, y), (x, y) ∈ Ω, ×R + (15)
and we assume that the initial datum f in is a nonnegative function which satisfies:

(16) f in ∈ L 1 (Ω × R + ) ∩ L 1 (Ω × R + , ydxdy).
Here and below, the notation L 1 (Ω × R + , ydxdy) stands for the space of the Lebesgue measurable real-valued functions on Ω × R + which are integrable with respect to the measure ydxdy.

When designing the volume discretization of the coagulation and fragmentation terms, it is necessary to truncate the infinite integrals in formulae ( 13)- [START_REF] Filbet | Solving the Boltzmann equation in N log 2 N[END_REF]. But this means restricting the domain of action of kernels a and b to a bounded set of volumes y, that is, preventing coagulation to occur among particles with volume exceeding a fixed value. The discretization we propose is based on the conservative truncation method for the coagulation and fragmentation terms. Given a positive real R, let (x, y) ∈ Ω × (0, R)

C R (f )(x, y) := y 0 R-u x-u u a(u, v) f (x, u) f (x, v) dvdu , = y 0 R y u a(u, w -u) f (x, u) f (x, w -u) dw du . (17) In that case, C R (f )(x, 0) = C R (f )(x, R) = 0 foreach x ∈ Ω
so that the total volume of the solution is now nonincreasing with respect to time.

As regards the fragmentation term, the truncation is also a conservative truncation on the fragmentation term. Using the same idea, we introduce for (x, y) ∈ Ω × (0, R)

F R (f )(x, y) := y 0 R-u y-u u b(u, v) f (x, u + v) dv du. = y 0 R y u b(u, w -u) f (x, w) dv du. (18)
Then, the conservative coagulation-fragmentation operator satisfies exactly the conservation of total volume, so that the following equation is indeed a truncated conservative coagulation and fragmentation equation:

       y ∂f R ∂t -d(y) y ∆ x f R = - ∂C R (f R ) ∂y (x, y) + ∂F R (f R ) ∂y (x, y) f (0, x, y) = f in (x, y), (x, y) ∈ Ω × (0, R), (19)
since d dt R 0 Ω y f R (t, x, y) dx dy = 0.
Under the detailed balance condition [START_REF] Eymard | Finite volume methods[END_REF], model [START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF] has also a steady state M R , only depending on y and such that

M R (y) = M (y), y ∈ [0, R]. and 1 |Ω| R 0 Ω f in (x, y) dx dy = R 0 M R (y) dy.
Convergence for large values of R has been thoroughly studied in the recent past. We briefly mention some results for the coagulation equation (that is, with b = 0). These results adapt easily to the coagulation-fragmentation equation but under different assumptions on the kernels. Let us mention that when a(y, y ′ )/(y y ′ ) → 0 as y + y ′ → +∞, convergence as R → +∞ of the solutions to [START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF] toward a solution of ( 13)-( 15) can be proven by using the approach developed in [START_REF] Laurenc | From the discrete to the continuous coagulation-fragmentation equations[END_REF]. Thus, since the convergence of solutions to [START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF] towards solutions of ( 13)-( 15) is well established in rather general situations, this paper will only focus on the convergence of a sequence built on a numerical scheme towards a solution to the equation ( 19) when the truncature R is fixed. In the remainder of the paper, for the sake of clarity, we drop the subscript R and write f instead of f R for a solution of equation [START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF]. Parameter R being fixed, this should raise no confusion. Now, we turn to the discretization of equation [START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF]. Having reduced the computation to a bounded interval, the second step is to introduce the space and volume discretizations. To this end, we set ∆y ∈ (0, 1) and N y a large integer, and denote by (y i-1/2 ) i∈{0,...,Ny} a mesh of (0, R), where y i-1/2 = i ∆y, and

Λ i = [y i-1/2 , y i+1/2 ) for i ≥ 0.
Concerning the space variable, we choose an admissible mesh of Ω given by a family T of control volumes (open and convex polygons in 2-D), a family E of edges and a family of points (x K ) K∈T which satisfy Definition 5.1 in [START_REF] Eymard | Finite volume methods[END_REF]. It implies that the straight line between two neighboring centers of cells (x K , x L ) is orthogonal to the edge σ = K|L. In the set of edges E, we distinguish the interior edges σ ∈ E int and the boundary edges σ ∈ E ext . For a control volume K ∈ T , we denote by E K the set of its edges, E int,K the set of its interior edges, E ext,K the set of edges of K included in Γ = ∂Ω.

In the sequel, we denote by d the distance in R N , m the measure in R N . We assume that the family of mesh considered satisfies the following regularity constraint : there exists ξ > 0 such that

(20) d(x K , σ) ≥ ξ d(x K , x L ), for K ∈ T , for σ ∈ E int,K , σ = K|L.
The size of the mesh is defined by

(21) δ = max K∈T (diam(K)) .
For all σ ∈ E, we define the transmissibility coefficient:

τ σ =          m(σ) d(x K , x L ) , for σ ∈ E int , σ = K|L, m(σ) d(x K , σ) , for σ ∈ E ext,K .
Next, X(T ) will be the set of functions from Ω to R which are constant over each control volume K ∈ T . We define the approximation f h (0) of the initial datum f in as usual by

(22) f h (0) = Ny i=0 K∈T f in K,i 1 K×Λ i , with f in K,i = 1 m(K) ∆y Λ i K f in (x, y) dx dy ,
where 1 E denotes the characteristic function of the subset E of Ω and converges strongly to f 0 in L 1 (Ω × (0, R)) as h = (∆y, δ) goes to 0.

Let us now introduce the numerical scheme itself. For each integer i ∈ {0, • • • , N y } and each K ∈ T , we define the approximation of f (t, x, y) for t ∈ R + and (x, y) ∈ K × Λ i as f K,i (t). The sequence (f K,i ) K,i is defined by the following discretization of the coagulation-fragmentation equation : for K ∈ T , i ∈ {0, . . . , N y }, we solve the ordinary differential system (23)

         m(K) df K,i dt -d(y i-1/2 ) σ∈E K τ σ D K,σ f K,i = m(K) Q K,i , f K,i (0) := f in K,i . We have set (24) Q K,i = - C K,i+1/2 -C K,i-1/2 y i-1/2 ∆y + F K,i+1/2 -F K,i-1/2 y i-1/2 ∆y
where the flux C K,i+1/2 in ( 24) represents an approximation of the coagulation operator [START_REF] Guias | A stochastic approach for simulating spatially inhomogeneous coagulation dynamics in the gelation regime[END_REF], F K,i+1/2 is the approximation of the fragmentation part [START_REF] Jeon | Existence of gelling solutions for coagulation-fragmentation equations[END_REF] and both are defined by

C K,i+1/2 = i j=0 Ny-1 l=i+1 ∆y 2 y j-1/2 a j,l-j f K,j f K,l-j , (25)
where a i,j := a(y i , y j ) with y i = (i + 1/2) ∆y and

F K,i+1/2 = i j=0 Ny-1 l=i+1 ∆y 2 y j-1/2 b j,l-j f K,l , (26)
where b i,j := b(y i , y j ) whereas the fluxes at the boundary are

C K,-1/2 = F K,-1/2 = C K,Ny+1/2 = F K,Ny+1/2 = 0, K ∈ T . ( 27
)
Concerning the approximation of the diffusion in space, we set ν K,σ the unit normal to σ outward from K and define an approximation of ∇ x f • ν Kσ on σ by ( 28)

D K,σ f K,i = f L,i -f K,i , if σ = K|L ∈ E int,K , 0, if σ ∈ E ext,K ,
for all K ∈ T . This discretization obviously relies on a simple finite volume approach for the space and volume variables (see, e.g. [START_REF] Bourgade | Convergene of a Finite Volume Scheme for the Coagulation-Fragmentation Models[END_REF]).

The following function f h defined on R + × Ω × [0, R] will be useful in the sequel.

(29)

f h (t, x, y) = K∈T Ny i=0 f K,i (t) 1 K×Λ i .
We may now state our main result.

Theorem 2.1. Assume that the coagulation and fragmentation kernels satisfy (4), ( 10) and ( 12) and f in satisfies [START_REF] Friedlander | Smoke, Dust and Haze, Fundamentals of aerosol dynamics[END_REF]. We consider a uniform volume mesh in y and require the mesh T in space to satisfy condition ( 20), [START_REF] Laurenc | From the discrete to the continuous coagulation-fragmentation equations[END_REF].

Then, there exists a unique solution f h to the finite volume scheme ( 23)- [START_REF] Tanaka | Steady-state size distribution for the self-similar collision cascade[END_REF], which satisfies

(i) nonnegativity f K,i (t) ≥ 0, (K, i) ∈ T × {0, . . . , N y }, t ∈ R + , (ii) conservation of total volume d dt   K,i m(K) ∆y y i-1/2 f K,i (t)   = 0, ∀ t ∈ R + , (iii) entropy dissipation dH(f h |M ) dt + ∆y K,i σ∈E K d(y i-1/2 )τ σ D K,σ f K,i ln f K,i M i = -D(f h ),
where

H(f h |M ) = K,i m(K) ∆y f K,i ln f K,i M i -1 + M i and D(f h ) = 1 2 K,i i l=0 ∆y 2 m(K) (a l,i-l f K,l f K,i-l -b l,i-l f K,i ) ln a l,i-l f K,l f K,i-l b l,i-l f K,i .
(iv) Moreover, the finite volume scheme is asymptotic preserving i.e.

f h (t) -→ M h , as t -→ ∞.
where

M h (x, y) = M i := exp(-α y i ), ∀(x, y) ∈ K × [y i-1/2 , y i+1/2 ) and α ∈ R + is such that K∈T Ny i=0 m(K)y i-1/2 M i = K∈T Ny i=0 m(K)y i-1/2 f in K,i .

A priori estimates

As we mentioned in the introduction, our goal is not to prove that the sequence of functions (f h ) h converges in some sense to a function f as h = (∆y, δ) goes to 0, but to study the long time behavior of the numerical solution and to prove its uniform stability with respect to time. First, we prove that the solution f h to the scheme ( 23)-( 27) enjoys properties similar to those of function f given by ( 19) which we gather in Proposition 3.2 below.

On the one hand, we first re-write the discrete coagulation-fragmentation obtained form the conservative form [START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF] in a new form which is consistent with the classical formulation ( 7)-( 9). On the other hand, we find a discrete version of the weak formulation from which we prove a priori estimates. Proposition 3.1. Assume the approximation of the coagulation-fragmentation model ( 17)-( 19) is given by the finite volume scheme ( 23)- [START_REF] Tanaka | Steady-state size distribution for the self-similar collision cascade[END_REF]. Then, the discrete operator satisfies

Q K,i = - C K,i+1/2 -C K,i-1/2 y i-1/2 ∆y + F K,i+1/2 -F K,i-1/2 y i-1/2 ∆y with Q K,i = 1 2 i j=0 ∆y (a j,i-j f K,j f K,i-j -b j,i-j f K,i ) (30) - Ny-1 j=i ∆y (a i,j-i f K,i f K,j-i -b i,j-i f K,j )
Proof: Starting from the definition of the fluxes ( 25) and ( 26), we set

Q K,i = - C K,i+1/2 -C K,i-1/2 y i-1/2 ∆y + F K,i+1/2 -F K,i-1/2 y i-1/2 ∆y
and by construction

y i-1/2 Q K,i = - i j=0 Ny-1 l=i+1 ∆y y j-1/2 (a j,l-j f K,j f K,l-j -b j,l-j f K,l ) + i-1 j=0 Ny-1 l=i ∆y y j-1/2 (a j,l-j f K,j f K,l-j -b j,l-j f K,l ) ,
which can be easily simplified into

y i-1/2 Q K,i = - Ny-1 l=i+1 ∆y y i-1/2 (a i,l-i f K,i f K,l-i -b i,l-i f K,l ) + i-1 l=0 ∆y y l-1/2 (a l,i-l f K,l f K,i-l -b l,i-l f K,i ) .
Then, adding and subtracting the term y i-1/2 (a i,0 f K,0 f K,0 -b i,0 f K,i ), we get

y i-1/2 Q K,i = i l=0 ∆y y l-1/2 (a l,i-l f K,l f K,i-l -b l,i-l f K,i ) - Ny-1 l=i ∆y y i-1/2 (a i,l-i f K,i f K,l-i -b i,l-i f K,l ) .
Hence, it only remains to treat the first term of the right hand side: we compute for a symmetric sequence g i,j such that g i,j = g j,i for each (i,

j) ∈ {0, • • • , N y } 2 i l=0 l g l,i-l = i l=0 i g l,i-l - i l=0 (i -l) g l,i-l = i l=0 i g l,i-l - i l=0 l g i-l,l .
Thus, using the symmetry g i-l,l = g l,i-l we finally get

1 2 i l=0 i g l,i-l = i l=0 l g l,i-l .
Therefore, with y i-1/2 = i ∆y and taking successively g i,j = a i,j f K,i f K,j and g i,j = b i,j , it yields the result

Q K,i = 1 2 i l=0 ∆y (a l,i-l f K,l f K,i-l -b l,i-l f K,i ) - Ny-1 l=i ∆y (a i,l-i f K,i f K,l-i -b i,l-i f K,l ) .
Next, we prove the following estimates Proposition 3.2. Assume the approximation of the coagulation-fragmentation operator ( 17)-( 18) is given by the finite volume scheme ( 25)- [START_REF] Makino | On the mass distribution of planetesimals in the early runaway stage[END_REF]. Then, for each K ∈ T the discrete operator (Q K,i ) 0≤i≤Ny satisfies a discrete weak formulation : for any sequence (ϕ i ) 0≤i≤Ny and for each K ∈ T (31)

Ny-1 i=0 ∆y Q K,i ϕ i = - 1 2 Ny-1 i=0 i l=0 ∆y 2 (a l,i-l f K,l f K,i-l -b l,i-l f K,i ) (ϕ l + ϕ i-l -ϕ i )
Proof: We multiply (30) by ∆y ϕ i and sum over

i ∈ {0, • • • , N y -1} to get Ny-1 i=0 ∆y Q K,i ϕ i = ∆y 2 (I 1 -I 2 ) , (32)
with

I 1 = 1 2 Ny-1 i=0 i j=0 (a j,i-j f K,j f K,i-j -b j,i-j f K,i ) ϕ i I 2 = Ny-1 i=0 Ny-1 j=i (a i,j-i f K,i f K,j-i -b i,j-i f K,j ) ϕ i .
We keep the first term I 1 as it is and only treat the term I 2 . First, we commute indexes (i, j) and then perform a change of index on I 2 which yields

I 2 = Ny-1 j=0 j i=0 (a i,j-i f K,i f K,j-i -b i,j-i f K,j ) ϕ i = Ny-1 i=0 i l=0 (a i-l,l f K,i-l f K,l -b i-l,l f K,i ) ϕ i-l . ( 33
)
Using the symmetry a i,j = a j,i and b i,j = b j,i for each (i, j) ∈ {0, • • • , N y -1}, we also have that (34)

I 2 = Ny-1 i=0 i l=0 (a l,i-l f K,l f K,i-l -b l,i-l f K,i ) ϕ l .
Finally, gathering (32)-(34), we get the result

Ny-1 i=0 ∆y Q K,i ϕ i = - 1 2 Ny-1 i=0 i l=0 ∆y 2 (a l,i-l f K,l f K,i-l -b l,i-l f K,i ) (ϕ l + ϕ i-l -ϕ i ) .
Now we are ready to prove Theorem 2.1.

Proof of Theorem 2.1

We assume that the kinetic and diffusion coefficients fulfil ( 4), ( 10) and ( 12), respectively, and that a and b are positive a.e. in R + × R + . We are also given an initial datum f in satisfying ( 16) and denote by f h the solution to [START_REF] Lee | On the validity of the coagulation equation and the nature of runaway growth[END_REF] constructed in ( 24)-( 26).

On the one hand, existence and uniqueness of a solution to

(35) df K,i dt -d(y i-1/2 ) σ∈E K τ σ D K,σ f K,i = m(K) Q K,i ,
directly follows by applying the classical Cauchy-Lipschitz theorem for a finite set of ordinary differential equations. Existence of a global solution will be a consequence of the following uniform estimates with respect to time.

On the other hand, since the discrete distribution function is solution to (35), the nonnegativity of f h (t) follows from the monotonicity of the discrete operator σ∈E K τ σ D K,σ f K,i and the structure of the discrete coagulation-fragmentation operator (30) written as the sum of a gain operator and a local loss term.

Next, we prove the total volume conservation (36) d dt

Ny-1 i=0 K∈T m(K) ∆y y i-1/2 f K,i (t) = 0.

Indeed; we multiply (35) by ∆y y i-1/2 and sum over (K, i) ∈ T × {0, . . . , N y -1}

d dt K,i m(K) ∆y y i-1/2 f K,i (t) = - 1 2 K,i ∆y d(y i-1/2 ) σ∈E K τ σ D K,σ f K,i D K,σ (y i-1/2 ) + K,i m(K) ∆y Q K,i y i-1/2 .
Thus using a discrete integration by part, we prove that the first term of the right hand side is zero, whereas applying Proposition 3.2 with ϕ i = y i-1/2 , it also gives that

K,i m(K) ∆y Q K,i y i-1/2 = 0,
which concludes the proof of (ii). Now, let us prove the stabilization towards an equilibrium in the long time when the kinetic coefficients a and b satisfy the detailed balance condition [START_REF] Eymard | Finite volume methods[END_REF]. As mentioned in the introduction the detailed balance condition [START_REF] Eymard | Finite volume methods[END_REF] ensures an analogue of the Boltzmann H-theorem for the coagulation-fragmentation equations which we derive now for the discrete solution to [START_REF] Lee | On the validity of the coagulation equation and the nature of runaway growth[END_REF]. We set

H(f h |M ) = K,i m(K) ∆y f K,i ln f K,i M i -1 + M i
and take in the discrete weak formulation (31), the test function ϕ such as

ϕ i = ln(f K,i /M i ) with M i = M (y i-1/2
) and noticing that

dH dt (f h ) = K,i m(K) ∆y df K,i dt ln f K,i M i ,
and recalling (31) with

ϕ i = ln(f K,i /M i ), we obtain dH dt (f h ) + ∆y K,i σ∈E K d(y i-1/2 )τ σ D K,σ f K,i ln f K,i M i = -∆y K,i m(K) Q K,i ln f K,i M i , which yields (37) dH dt (f h ) + ∆y K,i σ∈E K d(y i-1/2 ) τ σ (D K,σ f K,i ) 2 f K|L,i = -D(f h ), with f K|L,i = (f K,i -f L,i )/(ln(f K,i /f L,i )) > 0.
Then, we treat the right hand side using the detailed balance condition [START_REF] Eymard | Finite volume methods[END_REF] and get

D(f h ) = 1 2 K,i i l=0 ∆y 2 (a l,i-l f K,l f K,i-l -b l,i-l f K,i ) ln a l,i-l f K,l f K,i-l b l,i-l f K,i ,
which proves (iii). Now, we study the long time asymptotic behavior of the numerical solution f h . To this aim we establish the following estimates Lemma 4.1. The numerical solution f h satisfies the following uniform estimates with respect to time t ∈ R + : there exists κ 0 > 0 such that

K,i m(K) ∆y (1 + y i-1/2 ) f K,i (t) + H(f h (t)|M ) ≤ κ 0 .
Moreover, for all t ∈ R + , we have

(38) ∆y t 0   Ny i=0 σ∈E int d(y i-1/2 ) 1/2 m(σ) |Df K,i,σ (s)|   2 ds ≤ κ 0 . and (39) ∆y 2 t 0 K,i i l=0 (a l,i-l f K,l f K,i-l -b l,i-l f K,i ) ln a l,i-l f K,l f K,i-l b l,i-l f K,i ds ≤ κ 0 ,
Proof: Using the total volume conservation and the entropy dissipation, we have already proven that

K,i m(K) ∆y y i-1/2 f K,i (t) + H(f h (t)|M ) ≤ C 0 .
It remains to show that there exists a constant C 0 > 0, only depending on the initial datum f in and M , such that (40)

K,i m(K) ∆y f K,i (t) ≤ C 0 .
Indeed, on the one hand we notice that

K,i m(K) ∆y f K,i (t) ≤ e 2 K,i m(K) ∆y 1 {f K,i (t)≤e 2 M i } M i + 1 2 K,i m(K) ∆y 1 {f K,i (t)>e 2 M i } f K,i ln f K,i (t) M i , ≤ e 2 K,i m(K) ∆y M i + 1 2 K,i m(K) ∆y f K,i ln f K,i (t) M i .
On the other hand, observing that r ln(r) ≥ r | ln(r)| -2/e for r > 0, we have

f K,i ln f K,i M i ≥ f K,i ln f K,i M i - 2 M i e ,
hence it gives

K,i m(K) ∆y f K,i ln f K,i (t) M i ≤ K,i m(K) ∆y f K,i ln f K,i (t) M i + 2 e K,i m(K) ∆y M i , ≤ H(f h |M ) + K,i m(K) ∆y f K,i + 2 e M i .
Combining the two inequalities yields

1 2 K,i m(K) ∆y f K,i (t) ≤ e 2 + 1 e K,i m(K) ∆y M i + 1 2 H(f h (t)|M ), ≤ e 2 + 1 e K,i m(K) ∆y M i + 1 2 H(f h (0)|M ) =: C 0 .
Next, let us prove (39). We start with the entropy estimates (37), which we integrate with respect to time t ∈ R + , it gives

H(f h (t)|M ) + t 0 ∆y K,i σ∈E K d(y i-1/2 ) τ σ (D K,σ f K,i ) 2 f K|L,i ds + t 0 D(f h (s))ds ≤ H(f h (0)|M ).
Then, since H is a convex function, we know that H(f h (t)|M ) is bounded from below and we get the result (39).

Finally, we show there exists C 0 >, only depending on f in and M , such that (41)

t 0    Ny i=0 σ∈E int σ=K|L m(σ) d(y i-1/2 ) 1/2 |Df K,i,σ (s)|    2 ds ≤ C 0 , ∀ t ∈ R + .
To this aim, we start with the Cauchy-Schwarz inequality

Ny i=0 σ∈E int σ=K|L m(σ) d(y i-1/2 ) 1/2 |Df K,i,σ |, ≤    Ny i=0 σ∈E int σ=K|L τ σ d(y i-1/2 ) |Df K,i,σ | 2 f K|L,i    1/2    Ny i=0 σ∈E int σ=K|L m(σ)d(x K , x L ) f K|L,i    1/2 .
Applying ( 20) and ( 36), (37), it follows that

   Ny i=0 σ∈E int σ=K|L ∆y m(σ) d(y i-1/2 ) 1/2 |Df K,i,σ |    2 ≤ 2 ξ    Ny i=0 σ∈E int σ=K|L ∆y τ σ d(y i-1/2 ) |Df K,i,σ | 2 f K|L,i      Ny i=0 K∈T ∆y m(K) f K,i   , ≤ 2 C 0 ξ    Ny i=0 σ∈E int σ=K|L ∆y τ σ d(y i-1/2 ) |Df K,i,σ | 2 f K|L,i    .
Then, for each t ∈ R + we integrate the latter inequality with respect to time over the interval [0, t], and use the entropy dissipation (37) which guarantees that the right hand side is uniformly bounded with respect to time. Now, we prove that the numerical solution converges to a steady state. Let (t n ) n≥0 be a sequence of positive real numbers such that t n → +∞ and set f n (t) = f h (t n + t) for n ≥ 1 and t ∈ R + . Owing to the construction of f n , it is easily seen that f n is a weak solution to ( 23)- (27) on [0, +∞) with initial datum f h (t n ). We fix T ∈ R + and infer from above that sup

t∈[0,T ] K,i m(K) ∆y (1 + y i ) f n K,i (t) + H(f n (t)|M ) ≤ κ 0 .
Moreover, by construction of f n and applying Lemma 4.1, it implies that ∆y

T 0 Ny i=0 σ∈E int d(y i-1/2 ) 1/2 m(σ) |Df n K,i,σ (t)| dt (42) ≤ ∆y t n +T t n Ny i=0 σ∈E int d(y i-1/2 ) 1/2 m(σ) |Df K,i,σ (t)| dt -→ n → ∞ 0.
and

∆y 2 T 0 K,i i l=0 a l,i-l f n K,l f n K,i-l -b l,i-l f n K,i ln a l,i-l f n K,l f n K,i-l b l,i-l f n K,i dt (43) ≤ ∆y 2 t n +T t n K,i i l=0 (a l,i-l f K,l f K,i-l -b l,i-l f K,i ) ln a l,i-l f K,l f K,i-l b l,i-l f K,i dt -→ 0, as n → ∞.
Thanks to theses estimates, we deduce that there are a subsequence of (f n ) n∈N (not relabeled) and a weak solution f to ( 23)-( 27) such that (44)

f n → f , in C([0, T ]), as n → +∞.
Moreover, passing to the limit in (30), it easily follows that (45)

Q K,i (f n ) -→ Q K,i ( f )
, in L 1 (0, T ), as n → +∞.

and passing to the limit in the discrete diffusion operator, we get

(46) d(y i-1/2 ) σ∈E K τ σ Df n K,i,σ -→ d(y i-1/2 ) σ∈E K τ σ D fK,i,σ in L 1 (0, T ), as n → +∞.
On the one hand, we use (43) and (44) to conclude that D( f ) = 0, a.e. in (0, T ).

Consequently, f satisfies a i,j fK,i (t) fK,j (t) = b i,j fK,i+j (t), for almost every t ∈ (0, T ) and each (K, i, j) ∈ T × {0, . . . , N y } 2 . In particular, it implies that

Q K,i ( f (t)) = 0, (K, i, j) ∈ T × {0, . . . , N y } 2 .
On the other hand, (42) and (46) ensure that (47)

D K,σ fK,i = 0, (K, i, σ) ∈ T × {0, . . . , N y } × E
for almost every t ∈ (0, T ). Therefore, f (t) does not depend on time and satisfies the steady states equation [START_REF] Lee | On the validity of the coagulation equation and the nature of runaway growth[END_REF]. Moreover, using the zero flux boundary conditions (3) and (47), f does not depend on K ∈ T . Recalling that the differential equation ( 23) conserves global volume, we conclude there is α ∈ R + such that fK,i := M i , a.e. in (0, T ),

where

Ny-1 i=0 K∈T ∆y m(K) y i-1/2 M i = Ny-1 i=0 K∈T ∆y m(K) y i-1/2 f in K,i .

Numerical simulations

This section is devoted to the numerical study of the convergence to equilibrium under the detailed balance condition and when this condition is not satisfied. We also investigate numerically the case with non homogeneous Dirichlet boundary conditions. 5.1. Detailed balance kernels and convergence to equilibrium. We assume that the coagulation and fragmentation coefficients fulfil the detailed balance condition: there exists a nonnegative function

M ∈ L 1 1 (R + × R + ), such that (48) a(y, y ′ ) M (y) M (y ′ ) = b(y, y ′ ) M (y + y ′ ), (y, y ′ ) ∈ R + × R + .
Since in that case there exists a Lyapunov H at the discrete level, we have proven that the numerical solution f h converges to a discrete equilibrium M h . Here, we choose kernels a and b as follows: a(y, y ′ ) = b(y, y ′ ) = 1, which yields M (y) = exp(-y/ √ M 1 ). It is the Aizenman-Bak model for reacting polymers which diffuse in space with a non degenerate size-dependent coefficient d(y) = α > 0. In [START_REF] Carrillo | Exponential Decay Towards Equilibrium for the Inhomogeneous Aizenman-Bak Model[END_REF], the authors demonstrate that the entropy-entropy dissipation methods developed by Desvillettes and Villani for the Boltzmann equation [START_REF] Desvillettes | On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation[END_REF] applies directly and gives the exponential convergence with explicit rates towards global equilibrium for constant diffusion coefficient in any spatial dimension or for the non degenerate diffusion in dimension one. Thuerefore, the global equilibrium is given by M (y) = exp(-y/ M 1 ), y ∈ R + , where the number M 1 is given by Let us first mention that in [START_REF] Filbet | Solving the Boltzmann equation in N log 2 N[END_REF][START_REF] Filbet | High order numerical methods for the space non-homogeneous Boltzmann equation[END_REF] a similar problem i.e.; trend to equilibrium of the solution to the nonhomogeneous Boltzmann equation, is investigated numerically. It is shown that the relative entropy with respect to the local equilibrium oscillates with time when the solution becomes close to the equilibrium. Here, we will show that the situation is completely different and much simpler.

M 1 := 1 |Ω| Ω R + y f in (x,
In Figure 1, we report the evolution of the total number of particles M 0 , the high order moments in y of f h with respect to time and the Lyapunov functional H(f |M ). As expected, the total volume M 1 (t) remains constant throughout time evolution and the moments stabilize to a fixed value. As regards the asymptotic profile, our numerical results are in fair agreement with the equilibrium M (x) = exp(-x). Moreover, we observe that the scheme is able to give the correct behavior of the numerical entropy H(f |M ), which converges exponentially fast to zero [START_REF] Carrillo | Exponential Decay Towards Equilibrium for the Inhomogeneous Aizenman-Bak Model[END_REF]. However, we observe that trend to equilibrium for the coagulation-fragmentation with diffusion is much more simpler than trend to equilibrium for the Boltzmann equation [START_REF] Filbet | Solving the Boltzmann equation in N log 2 N[END_REF] since no oscillation occurs for the relative entropy H(f |M loc ). Indeed, after a short transient regime, the solution f converges to equilibrium as an exponential with respect to time. dissipation. Indeed, we choose kernels a and b as follows:

a(y, y ′ ) = (y y ′ ) 1/2 , b(y, y ′ ) = 1.
The situation becomes much more complicated since we do not know neither the expression of the steady state nor the expression of entropy. As an initial datum, we take f in (x, y) = exp(-α(x) y), with α(x) = 1+0.5 cos(4 π x 1 ) cos(4 π x 2 ) and x = (x 1 , x 2 ) ∈ [-1/2, 1/2] 2 . Moreover, the diffusion is taken to be d(y) = 0.1/(1 + y), which is degenerated for large y. We choose the truncation such that R = 20, N y = 64 and next 128 and finally ∆t = 0.002. In Figure 2, we still report the evolution of the total number of particles M 0 and other high order moments in y of f h , that is M 1 , M 2 and M 3 . As expected, the total volume M 1 (t) remains constant throughout time evolution, whereas high order moments vary and next stabilize to a fixed value, which means that the solution converges to a steady state. In the same figure (right hand side), we also report the distribution function with respect to y in log scale for large time, which indicates that the tail with respect to y of the steady state f is still exponentially decreasing and f is of course constant in x ∈ Ω. 5.3. Convergence to equilibrium for non homogeneous boundary conditions. In this last section, we study the coagulation-fragmentation operator with diffusion in space and mixed boundary conditions in x ∈ Ω. More precisely, we aim to approximate the solution in Ω = (0, 1/8) × (0, 1) to

                     ∂f ∂t -d(y) ∆ x f = Q(f ), , f (t = 0, x, y) = f in (x, y), (x, y) ∈ Ω × R + , f (t, x 1 = 0, x 2 , y) = exp (-y/α(x 2 )) , (x 2 , y) ∈ ∂Ω × R + , ∇ x f • ν(x) = 0, x 1 = 1/8, or x 2 = 0, or x 2 = 1, with α(x 2 ) = (1 + cos(4πx 2
))/2 and ν the external unit normal to ∂Ω.

We consider an initial condition, which is at equilibrium for the coagulation fragmentation operator f in (x, y) = exp(-y/α(x)), (x, y) ∈ Ω × R + , with α(x) = (1 + cos(32 π x 1 ) cos(4 π x 2 ))/2.

We perform numerical simulations using the finite volume scheme with diffusion coefficient d(y) = 0.01/(1 + y) and a = b ≡ 1. The following figures (Fig 3, 4 and5) are illustrations of the profile of the number of particles M 0 (t, x) and total volume M 1 (t, x) for t ∈ R + and x ∈ Ω and also the projections P (t, x 2 , y) = 1/8 0 y f (t, x, y)dx 1 , x 2 ∈ (0, 1), y ∈ R + .

It allows us to observe the qualitative behavior of the system. The profiles are in that case smooth and also converges to an equilibrium. For such a computations we have considered N y = 64 and 128 × 128 grid points for the space variable x ∈ Ω. In this situation, we cannot characterize explicitly the equilibrium in space since the diffusion coefficient is not constant and H is not valid here due to the Dirichlet boundary conditions for x 1 = 0 and then the total volume is not preserved at all. However, we still observe that the numerical solution converges to a discrete equilibrium (see Fig. 345).

Conclusion

In this paper, we make use of different principles (volume conservation, entropy dissipation, existence of steady states) which allow to build a stable and accurate numerical scheme for the nonlinear dynamics of coagulation, fragmentation and diffusion equations. Such an algorithm is able to recover the main properties of the exact solution and in some particular cases, it is proven that the method is asymptotically stable, in the sense that the numerical solution converges to an equilibrium which is consistent with the exact equilibrium of the system. Numerical simulations illustrate the efficiency of the algorithm even when we cannot prove convergence to equilibrium.

Here, we have only considered the time evolution of a distribution function, but the method can be easily coupled with Euler or Navier-Stokes equations after some adaptations. Typical applications are transport problems (including linear and nonlinear diffusion or fluid dynamics), which are coupled with population balance dynamics represented by a distribution function f depending on space x ∈ Ω and "size" variable y ∈ R + .

On the other hand, for some coagulation and fragmentation kernels [START_REF] Filbet | Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation[END_REF][START_REF] Bourgade | Convergene of a Finite Volume Scheme for the Coagulation-Fragmentation Models[END_REF], total volume is not conserved at all and then a non conservative formulation can be used to dicretize the coagulation and fragmentation operators using the same kind of finite volume scheme. Then, the finite volume

Figure 1 .

 1 Figure 1. Evolution of (a) the total number of particles M 0 , the total volume M 1 , M 2 and M 3 (b) the functional H(f |M ), H(f |M loc ) and H(M loc |M ) in log scale.

5. 2 .Figure 2 .

 22 Figure 2. Evolution of (a) the total number of particles M 0 , the total volume M 1 , M 2 and M 3 (b) the functional H(f |M ), H(f |M loc ) and f h (t ≃ ∞, y) in log scale.

Figure 4 .

 4 Figure 4. Time evolution of the volume density M 1 (x) at time t = 0; 0.33; 0.66 and 4

Figure 5 .

 5 Figure 5. 2D projection (x 1 , y) of the density y f (t, x, y) at time t = 0; 0.33; 0.66 and 4

  Figure 3. Time evolution of the density M 0 (x) at time t = 0; 0.33; 0.66 and 4 approach allows to use non uniform meshes for the volume variable y ∈ R + , which is particularly well suited in this case[START_REF] Filbet | Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation[END_REF].

								0								0
								0.02								0.02
								0.04								0.04
								0.06								0.06
								1								1
								0.08								0.08
								0.8								0.8
	0						0.6	0.1	0						0.6	0.1
	0.02	0.04					0.4	0.12	0.02	0.04					0.4	0.12
	0	0.2	0.06	0.08 0.4	0.6	0.2	0.8	1	0	0.2	0.06	0.08 0.4	0.6	0.2	0.8	1
				0.1								0.1
					0.12 0								0.12 0
								0								0
								0.02								0.02
								0.04								0.04
								0.06								0.06
								1								1
								0.08								0.08
								0.8								0.8
	0						0.6	0.1	0						0.6	0.1
	0.02	0.04					0.4	0.12	0.02	0.04					0.4	0.12
	0	0.2	0.06	0.08 0.4	0.6	0.2	0.8	1	0	0.2	0.06	0.08 0.4	0.6	0.2	0.8	1
				0.1								0.1
					0.12 0								0.12 0

Acknowledgement. This work is partially supported by the programm "ANR Jeunes Chercheurs" JCJC-0136 (projet MNEC) from the french Ministery of Research.


Audrey Bellier

Myriam Gominet

Philippe Mazodier
email: mazodier@pasteur.fr

Post-translational control of the Streptomyces lividans ClgR regulon by ClpP

Keywords: Clp and Lon proteases, degradation, ClgR activator, Streptomyces, differentiation

est destinée au dépôt

Introduction

Proteolysis of intracellular proteins is crucial for all living organisms : protein quality control is required to quickly remove misfolded or damaged proteins ; and regulation of cellular processes is accomplished by degrading unstable key regulators of stress response, cell cycle or differentiation. In eukaryotic cells, these functions are served by the energy-dependent 26S proteasome complex [START_REF] Baumeister | The proteasome: paradigm of a self-compartmentalizing protease[END_REF]. In bacteria, different ATP-dependent proteases contribute to intracellular proteolysis : Clp (ClpAP, ClpCP, ClpXP), HslUV (or ClpYQ), Lon and FtsH [START_REF] Gottesman | Proteases and their targets in Escherichia coli[END_REF]. They all contain 2 distinct domains : an ATPase chaperone domain responsible for substrate recognition and unfolding, and a proteolytic domain. These domains can either be on the same polypeptide chain (Lon and FtsH) or on 2 different subunits : the ATPase subunits (ClpX, ClpA, ClpC or HslU) and the proteolytic subunits (ClpP or HslV).

Different substrates of Clp proteases in Escherichia coli have been identified. For many of them, Clp ATPases recognize the NH2-or COOH-terminal region of the substrate. The MuA transposase and the SsrA-tagged polypeptides, which are degraded by ClpXP, have a substrate recognition motif located at their C-terminal sequence [START_REF] Gottesman | Protein quality control: triage by chaperones and proteases[END_REF][START_REF] Levchenko | Disassembly of the Mu transposase tetramer by the ClpX chaperone[END_REF]. For other substrates, like RepA, the P1 plasmid replication protein [START_REF] Hoskins | Protein binding and unfolding by the chaperone ClpA and degradation by the protease ClpAP[END_REF], or the O phage replication protein [START_REF] Gonciarz-Swiatek | Recognition, targeting, and hydrolysis of the lambda O replication protein by the ClpP/ClpX protease[END_REF], recognition sites are near the NH2 terminus. Recently, ClpXP substrates have been trapped in vivo using an E. coli inactive variant of ClpP. Analysis of more than 50 trapped proteins revealed five distinct classes of ClpX-recognition motifs : 3 N-terminal and 2 C-terminal motifs [START_REF] Flynn | Proteomic discovery of cellular substrates of the ClpXP protease reveals five classes of ClpX-recognition signals[END_REF]. One of these classes includes the C-terminal motif of SsrA-tagged proteins, ending with 2 alanine residues.

Identifying the proteolytic targets of specific proteases is critical for understanding their diverse physiological roles. In Bacillus subtilis, the ComK competence transcriptional factor is specifically degraded by ClpCP until the competence is induced [START_REF] Turgay | Competence in Bacillus subtilis is controlled by regulated proteolysis of a transcription factor[END_REF] ; the anti-sporulation  F factor is degraded by ClpCP during sporulation [START_REF] Pan | Self-reinforcing activation of a cell-specific transcription factor by proteolysis of an anti-sigma factor in B. subtilis[END_REF] and the CtsR class III heat shock gene repressor is degraded by ClpXP at 37°C [START_REF] Derre | The CtsR regulator of stress response is active as a dimer and specifically degraded in vivo at 37 degrees C[END_REF] and by ClpCP under stress conditions [START_REF] Kruger | Clp-mediated proteolysis in Gram-positive bacteria is autoregulated by the stability of a repressor[END_REF]. Clp proteases thus play crucial roles in competence, sporulation and heat shock response in B. subtilis. In Caulobacter crescentus, the CtrA response regulator protein that directly controls more than 50 genes during the cell cycle is eliminated by ClpXP before cells enter the DNA replication phase [START_REF] Jenal | An essential protease involved in bacterial cell-cycle control[END_REF] and the flagellar and chemotaxis machineries are regulated by ClpXP proteolysis [START_REF] Alley | Polar localization of a bacterial chemoreceptor[END_REF][START_REF] Grunenfelder | Identification of the protease and the turnover signal responsible for cell cycledependent degradation of the Caulobacter FliF motor protein[END_REF][START_REF] Potocka | Degradation of a Caulobacter soluble cytoplasmic chemoreceptor is ClpX dependent[END_REF].

Clp proteases also play an important role in biological functions of the Gram-positive soil bacterium Streptomyces, a model for bacterial differentiation with regard to its complex life cycle. On solid media, spore germination leads to growth of a basal mycelium which then differentiates into an aerial mycelium, and finally septates and differentiates into spores. Two classes of mutants have been characterized : the bld (bald) mutants that fail to produce aerial hyphae and the whi (white) mutants whose aerial hyphae fail to complete the production of normal, grey-pigmented spores [START_REF] Chater | Regulation of sporulation in Streptomyces coelicolor A3(2): a checkpoint multiplex?[END_REF]. The S. lividans clpP1clpP2 mutant has a bld phenotype and is therefore unable to complete the differentiation cycle [START_REF] Crecy-Lagard | Alteration of the synthesis of the Clp ATP-dependent protease affects morphological and physiological differentiation in Streptomyces[END_REF]. This suggests that one or several ClpP1 targets need to be degraded for normal aerial mycelium formation.

However, only one S. lividans ClpP1 target has been identified to date : PopR, the transcriptional activator of the clpP3clpP4 operon [START_REF] Viala | The clpP multigenic family in Streptomyces lividans: conditional expression of the clpP3 clpP4 operon is controlled by PopR, a novel transcriptional activator[END_REF]. PopR is primarly degraded by ClpP1 and the carboxy-terminal residues Ala-Ala play an essential part in the degradation process [START_REF] Viala | ClpP-dependent degradation of PopR allows tightly regulated expression of the clpP3 clpP4 operon in Streptomyces lividans[END_REF]. Recently, we have shown that ClgR, which is encoded by a gene paralogous to popR, activates expression of clpP1, clpC1, lon and clgR [START_REF] Bellier | ClgR, a novel regulator of clp and lon expression in Streptomyces[END_REF]. ClgR and PopR DNA-binding domain regions share over 50 % amino-acid sequence identity, and like PopR, ClgR has 2 alanine residues at its C-terminus.

Therefore, ClgR seemed a good candidate as a substrate for ClpP1. Moreover, we had observed that clgR overexpression led to a strong differentiation delay, suggesting that ClgR or members of its regulon could be involved in aerial mycelium formation.

We investigated ClgR stability in wild-type and clpP1 mutant strains. We found that ClpP1 degrades ClgR and that the 2 C-terminal alanines are required for degradation. We also found that Lon, whose synthesis is activated by ClgR, is also degraded by ClpP1 and that its degradation also involves the 2 C-terminal alanine residues. ClgR thus activates clpP1 gene expression, and ClpP1 degrades several proteins encoded by genes of the ClgR regulon. Via specific proteolysis, ClpP1 ensures a negative post-translational control of the ClgR regulon.

Results

ClgR a new target for ClpP1-dependent degradation

Since the ClgR primary sequence ends, like PopR, with 2 alanine, we assumed that it might be a new target for ClpP1 proteolysis. We used pAB54 and pAB55 plasmids, respectively overexpressing the native clgR, clgR-AA, or clgR-DD, where the two carboxy-terminal alanine residues were replaced by aspartate residues, from the strong erm*p promoter. Levels of ClgR were examined by Western blotting in the wild-type strain or in the clpP1 mutant harbouring these plasmids (Fig. 1). The level of ClgR-AA is significantly increased in the clpP1 mutant compared to the wild-type strain. Moreover, ClgR-DD accumulates to a very high level in the wild-type strain, to a level that is quite similar to that of ClgR-AA in the clpP1 mutant.

Taken together, these results indicate that ClpP1 is required to degrade the activator of its own synthesis, ClgR, and that the 2 C-terminal alanine residues are essential for degradation of ClgR.

Presence of ClgR during S. lividans cell cycle

In order to determine if clgR is differentially expressed during the cell cycle, protein extracts were prepared from plate cultures at the 3 different stages of differentiation : basal mycelium, aerial mycelium and sporulation ; and analyzed by Western blot experiments.

We first used pAB50 and pAB51 strains, overexpressing the native clgR, clgR-AA, or clgR-DD, under the control of its own promoter on a multicopy plasmid. However, no obvious effects on expression of the ClgR regulon were observed and the ClgR protein was only weakly detectable at the basal mycelium stage in these strains (data not shown).

We then used pAB54M2 and pAB55M2 strains, overexpressing clgR-AA or clgR-DD, cloned in N-terminal translational fusion with two M2 Flag epitopes and from the strong erm*p promoter. Using polyclonal anti-S. lividans ClgR antibodies or anti-M2 monoclonal antibodies to analyze the crude extracts, signals were detected only at the basal mycelium stage in both strains (Fig. 2), with a stronger intensity in the pAB55M2 strain (clgR-DD), in agreement with an increased stability of the ClgR-DD protein. We then decided to use these strains for the rest of the study presented here.

Expression of clgR during the cell cycle

The erm*p promoter had been described as a constitutive promoter (Bibb, 1994). Therefore, the results described above showing that the ClgR protein is only present at the basal mycelium stage were unexpected, especially in the pAB55 strain (expressing clgR-DD),

where ClgR proteolysis by ClpP1 is, if not abolished, at least strongly decreased. This led us to check clgR expression from the plasmid erm*p promoter or from the chromosomal clgR native promoter in strains harbouring pAB54 and pAB55. RNA was extracted from plate cultures at the 3 different stages of differentiation and clgR expression was determined by RT-PCR experiments.

In pAB54 and pAB55 strains, the erm*p promoter expression is stronger during the basal mycelium stage and then decreases throughout the cell cycle (Fig. 3a). The endogenous clgR promoter activity is different (Fig. 3b). Indeed, in the pAB54 strain, clgR expression from its native promoter remains low during the cell cycle. In the pAB55 strain, expression from the clgR native promoter is much more important at the basal mycelium stage than in the pAB54 strain (increasing more than ten times). Its expression remains strong at the aerial mycelium stage, and finally decreases during sporulation.

We finally tested the global clgR expression in the cell, reflecting clgR expression from both the erm*p promoter and its native promoter (Fig. 3c). While global clgR expression remains low and constant in the pAB54 strain, its expression is strongly increased at the basal and aerial mycelium stages in the pAB55 strain (about three-fold) and finally decreases at the sporulation stage to reach a level of expression that is comparable to that in the pAB54 strain.

Correlation between clgR expression and activation of clpP1 and clpC1 expression

In order to confirm the activation of the ClgR regulon at a specific time within the cell cycle, as observed at the protein level [START_REF] Bellier | ClgR, a novel regulator of clp and lon expression in Streptomyces[END_REF], clpP1 and clpC1 expression were measured by RT-PCR using RNAs extracted from plate cultures (Fig. 4). In the pAB54 strain (clgR-AA), expression of clpP1 and clpC1 was higher at the basal mycelium stage and then decreased throughout the cell cycle. In the ClgR-DD strain, clpP1 and clpC1 were strongly expressed at the basal and aerial mycelium stages, and reached a low level of expression during sporulation. Therefore, activation of clpP1 and clpC1 expression is directly correlated to global clgR expression in the cell.

Moreover the RNA expression profile of clpP1 and clpC1 is consistent with the protein levels shown previously [START_REF] Bellier | ClgR, a novel regulator of clp and lon expression in Streptomyces[END_REF]. However, in the Western blot experiments in the pAB54 strain, ClpP1 levels were constant, while ClpC1 protein levels actually decreased, in agreement with the RNA expression profile. In order to explain this difference between the RNA and protein levels, the stability of ClpP1 and ClpC1 was tested. For this purpose, cultures were grown on filters on rich medium until they formed the basal mycelium, a developmental stage where they express both clpP1 and clpC1. The culture disks were then transferred to rifampicin plates in order to block transcription, or to chloramphenicol plates in order to block translation. Proteins were extracted after 2 or 6 hours incubation on these plates and detected by Western blotting (Fig. 5). While ClpC1 rapidly disappears on rifampicin and chloramphenicol plates, ClpP1 is still detected even after 6 hours in both conditions.

These results indicate that ClpP1 is quite stable, whereas ClpC1 appears to be labile.

Presence of the Lon protease during the cell cycle

Since we had previously shown that the lon gene also belongs to the ClgR regulon, Lon protein levels were compared between the wild-type strain and the pAB54 (clgR-AA) or pAB55 (clgR-DD) strains by Western blotting of crude extracts from plate cultures (Fig. 6). In the wild-type strain, Lon is present at a high level in the basal mycelium stage and then disappears. However in the pAB54 or in the pAB55 strains, no obvious effects of clgR overexpression on Lon protein levels could be detected.

Since the Lon peptide sequence also ends by 2 alanines, Lon could also be a target for ClpP1dependent proteolysis. In this case, the effects of clgR overexpression on lon expression would be missed due to clpP1 overexpression, and correspondingly increased Lon degradation.

Lon is a new target for ClpP1 proteolysis

In order to distinguish between transcriptional and post-translational ClpP1-dependent control of Lon levels, the lon gene was cloned under the strong erm*p promoter. Moreover, in order to be able to specifically detect the Lon protein, whose expression is controlled by erm*p, but not the endogenous Lon protein, 2 M2 Flag epitopes were added to the N-terminal of the Lon protein. Finally, to test if the 2 alanines could be the substrate degradation motif for Lon, both lon-AA (native gene) and lon-DD genes were cloned.

Protein levels from crude extracts from liquid cultures were tested by Western blot experiments with monoclonal anti-M2 antibodies (Fig. 7). The level of M2-Lon-AA is higher in the clpP1 mutant than in the wild-type strain. The M2-Lon-DD levels are higher than the M2-Lon-AA levels, and they are about the same in the wild-type strain and in the clpP1 mutant.

The Lon protease is thus a new target for ClpP1-dependent proteolysis and the 2 C-terminal amino residues are essential for degradation by the Clp protease.

Effects of ClgR on the stabilized Lon-DD protein

In order to discrimate the post-translational effects (ie degradation of Lon by ClpP1) from the transcriptional effects (ie activation of lon promoter by ClgR) of clgR overexpression on lon expression, lon-DD gene was cloned under its own promoter in an integrative vector to give pAB63. Lon protein levels were then tested by Western blotting on plate cultures in wild-type or in pAB55 (clgR-DD) strains harbouring pAB63 (Fig. 8). As observed in the wild-type strain, Lon-DD is only present at the beginning of the cell cycle. However, in the pAB55 strain, the Lon-DD protein is now detected throughout the cell cycle, suggesting that the Lon disappearance observed previously in the ClgR-DD strain (Fig. 7) was due to its degradation ; and that the absence of Lon persistence in the wild-type strain expressing lon-DD is due to a shut-off of ClgR-dependent transcription after the basal mycelium stage. and in each case, the 2 alanine residues have been shown to be crucial for their degradation by ClpP1. Indeed, replacement of these 2 carboxy-terminal residues by 2 aspartates greatly increased the stability of these proteins. Several previously described Clp targets also end by 2 alanines : the SsrA tag that is added to the C-terminus of truncated proteins to target them for ClpXP degradation [START_REF] Gottesman | The ClpXP and ClpAP proteases degrade proteins with carboxy-terminal peptide tails added by the SsrA-tagging system[END_REF] ; the LexA autocleavage C-terminal fragment which is degraded by ClpXP in E. coli [START_REF] Neher | Latent ClpX-recognition signals ensure LexA destruction after DNA damage[END_REF] and the essential response regulator CtrA in C. crescentus, whose proteolysis is prevented when the last 2 alanines are replaced by 2 aspartates [START_REF] Domian | Cell type-specific phosphorylation and proteolysis of a transcriptional regulator controls the G1-to-S transition in a bacterial cell cycle[END_REF].

It was then tempting to speculate that in Streptomyces, the 2 alanine residues at the carboxyterminus could be sufficient to target proteins for ClpP1 and ClpP2-dependent degradation.

However, we have shown that the protein encoded by the S. coelicolor SCO3558 gene, homologous to cicA, which encodes a phosphotransferase in C. crescentus, and which ends with 2 alanines preceded by a hydrophobic residue (proline) in Streptomyces does not accumulate in a clpP1 mutant (data not shown). This result was unexpected since the recent analysis of the sequences of 50 trapped proteins using an inactive variant of ClpP in E. coli has revealed 5 classes of recognition signals, one of which is a C-terminal motif similar to the SsrA terminus (Leu-Ala-Ala) [START_REF] Flynn | Proteomic discovery of cellular substrates of the ClpXP protease reveals five classes of ClpX-recognition signals[END_REF]. Therefore, in Streptomyces, a hydrophobic residue followed by 2 alanines at the C-terminus is not sufficient to target proteins for Clp degradation. Therefore, we cannot exclude that there may be other features in the sequences, besides these 3 carboxy-residues, that play an important role for degradation by ClpP1 and ClpP2 in Streptomyces.

Degradation of Lon by the Clp protease is shown here for the first time. However, ATPdependent protease degradation by other ATP-dependent proteases has already been shown. Indeed, in E. coli, the ClpA subunit strongly accumulates in a clpP mutant [START_REF] Gottesman | The ATP-dependent Clp protease of Escherichia coli. Sequence of clpA and identification of a Clp-specific substrate[END_REF] and in B. subtilis, ClpE and ClpX are rapidly degraded in wild-type cells during permanent heat stress but remained almost stable in a clpP mutant, suggesting ClpPdependent degradation [START_REF] Gerth | Fine-tuning in regulation of Clp protein content in Bacillus subtilis[END_REF]. Therefore, amounts of ATP-dependent proteases in the cell seem to require fine tunings, combining transcriptional and post-translational regulation.

Clp-dependent degradation of ClgR is probably conserved in all Actinomycetes. Indeed, in C. glutamicum, ClgR is only present in clpC but not in wild-type cells, whereas the levels of clgR mRNAs are comparable in both strains [START_REF] Engels | clpC and clpP1P2 gene expression in Corynebacterium glutamicum is controlled by a regulatory network involving the transcriptional regulators ClgR and HspR as well as the ECF sigma factor sigmaH[END_REF]. However, the motif responsible for its degradation is probably different since C. glutamicum ClgR does not end with 2 alanines. Moreover, the motif is probably not located at the C-terminus since the addition of 10 amino-acid residues does not interfere with its degradation [START_REF] Engels | clpC and clpP1P2 gene expression in Corynebacterium glutamicum is controlled by a regulatory network involving the transcriptional regulators ClgR and HspR as well as the ECF sigma factor sigmaH[END_REF]. ClgR is not the only example of regulators that activate expression of proteases responsible for their own degradation. In S. lividans, PopR degradation is primarily dependent on ClpP1 and ClpP2, but can also be achieved by ClpP3 ClpP4, whose expression is activated by PopR [START_REF] Viala | ClpP-dependent degradation of PopR allows tightly regulated expression of the clpP3 clpP4 operon in Streptomyces lividans[END_REF]. In E. coli, the  32 heat shock transcriptional sigma factor controls ftsH expression and is itself degraded by FtsH [START_REF] Blaszczak | On the mechanism of FtsHdependent degradation of the sigma 32 transcriptional regulator of Escherichia coli and the role of the Dnak chaperone machine[END_REF]. More generally, degradation of transcriptional activators by ATP-dependent proteases appears to be quite common. In E. coli, the RcsA capsular biosynthesis transcriptional activator is degraded by Lon (Torres-Cabassa and [START_REF] Torres-Cabassa | Capsule synthesis in Escherichia coli K-12 is regulated by proteolysis[END_REF] and the  S stress sigma factor is degraded by ClpXP [START_REF] Schweder | Regulation of Escherichia coli starvation sigma factor ( s ) by ClpXP protease[END_REF], and in C. crescentus the CtrA essential regulator is degraded by ClpXP [START_REF] Jenal | An essential protease involved in bacterial cell-cycle control[END_REF]. For regulatory proteins, control by proteolysis can allow a rapid reduction of their cellular levels, in response to a specific signal, triggering the corresponding response. For example, the  32 and  S sigma factors levels rapidly increase after a stress, and the CtrA regulator accumulates and then drastically decreases at specific stages of the C. crescentus cell cycle. We could then assume that ClpP-dependent proteolysis of ClgR is achieved until some stress or specific environmental conditions occur.

In the wild-type strain, the ClgR protein is almost not detected. In order to get a strong protein signal and to have effects on the ClgR regulon members expression, we had to use a strain overexpressing clgR under a strong promoter, and to modify the protein in order to increase its stability. However, the fact that we have not been able to inactivate clgR, presumably because it is required for clpC1 expression, which we think is essential, suggests that control by ClgR is crucial for the cell, probably under specific conditions.

The stress or input signal has not been identified yet. However, the cinA gene (coding for a homolog of Streptococcus pneumoniae competence induced protein) located upstream from clgR, belongs to the disulphide stress  R regulon [START_REF] Paget | Defining the disulphide stress response in Streptomyces coelicolor A3(2): identification of the sigmaR regulon[END_REF]. The intergenic region is 120 bp long and there is no obvious transcription terminator between the two genes, so its conceivable that they form an operon. clpP1 expression is not induced by heat shock [START_REF] Viala | The clpP multigenic family in Streptomyces lividans: conditional expression of the clpP3 clpP4 operon is controlled by PopR, a novel transcriptional activator[END_REF], but the ClpP1 complex is likely to be involved in misfolded protein degradation under other stress conditions, which could lead to an increase of ClgR stability, and consequently clgR expression. Analogous mechanisms have been described, such as the HspR regulon, where the DnaK chaperone acts as a transcriptional co-repressor by binding to the HspR repressor. In heat stress conditions, DnaK is recruited by misfolded proteins and the HspR regulon is therefore not induced [START_REF] Bucca | The HspR regulon of Streptomyces coelicolor: a role for the DnaK chaperone as a transcriptional corepressor[END_REF].

It is interesting to note that lon expression is controlled by both ClgR and HspR. This suggests that ATP-dependent proteases are tightly regulated in Streptomyces, allowing a flexible response to a variety of signals.

According to Western blot experiments, it seems that ClgR is specifically present during the basal mycelium stage, suggesting that ClgR could be degraded during the aerial mycelium and spores stages, and that proteolysis would stop once the spore germinates. Arrest of degradation combined with clgR positive autoregulation leads to a very rapid increase of ClgR level. Similarly, CtrA is degraded by ClpXP and one of the ctrA promoters is under positive feedback control [START_REF] Domian | Feedback control of a master bacterial cell-cycle regulator[END_REF]. Together with CtrA phosphorylation, the coupling of these mecanisms results in the specific pattern of CtrA activity required for the temporal and spatial control of the Caulobacter cell cycle.

The fact that in the wild-type strain, ClgR, ClpC1 and Lon are specifically present at the basal mycelium stage of differentiation, while the ClpP1 protein is present throughout the cell cycle led us to suggest a negative feedback control of the ClgR regulon by ClpP1. ClgR expression is specifically activated during the basal mycelium stage, leading to activation of clgR itself and of the ClgR regulon genes, including those encoding the ClpP1, ClpP2, ClpC1 subunits and the Lon protease. ClpP1, in association with an ATPase that is still unidentified, specifically degrades ClgR and Lon. ClgR degradation will consequently end ClgR regulon activation and Lon degradation will allow a rapid elimination of Lon from the cell. Therefore, ClpP1 is responsible for a negative feedback on ClgR activation. ClpP1, which is a very stable protein, is the only protein so far encoded by the ClgR regulon that is present throughout the cell cycle.

The clpP1P2 mutant fails to form aerial mycelium, suggesting that Clp protease targets must be degraded for normal differentiation. In other words, Clp target protein accumulation will block the cell cycle at the basal mycelium stage. We have shown that ClgR, ClpC1 and Lon are only present at this stage, suggesting that their functions are required before aerial mycelium formation. Therefore, we could assume that overproduction of these 3 proteins in the clpP1P2 mutant could be responsible for the bld phenotype. This is not confirmed by the pAB55 strain (overexpressing clgR-DD) phenotype, which is not bald but only strongly delayed in its cell cycle. However, the ClgR protein level in the pAB55 strain might be lower than the level in the clpP1 mutant, since we could assume that the -DD motif is not sufficient to completely stabilize the ClgR protein. Indeed, it has been shown that even if the global regulator CtrA is greatly stabilized when the 2 alanines are modified to 2 aspartates, the 56 Namino-terminal residues are also part of the degradation signal [START_REF] Ryan | The CtrA response regulator essential for Caulobacter crescentus cell-cycle progression requires a bipartite degradation signal for temporally controlled proteolysis[END_REF]. Moreover, the strong differentiation delay observed in the pAB55 strain is in agreement with the necessity for clearance from the cell of the ClgR protein and/or proteins encoded by genes that belong to the ClgR regulon, at the end of the basal mycelium stage, in order to successfully complete the differentiation cycle.

Experimental procedures

Bacterial strains and media S. lividans strain 1326 was obtained from the John Innes Culture Collection, and S. lividans 1326 clpP1::Am R was constructed in our laboratory [START_REF] Crecy-Lagard | Alteration of the synthesis of the Clp ATP-dependent protease affects morphological and physiological differentiation in Streptomyces[END_REF]. YEME medium was used for liquid growth [START_REF] Hopwood | Genetic manipulation of Streptomyces. A laboratory manual[END_REF]. NE medium [START_REF] Murakami | Thiostrepton-induced gene expression in Streptomyces lividans[END_REF] and R5 [START_REF] Hopwood | Genetic manipulation of Streptomyces. A laboratory manual[END_REF] were used for Streptomyces growth on plates.

Antibiotic apramycin, hygromycin and thiostrepton were added to final concentrations of 25, 100 and 25 g ml -1 to solid medium, and to 20, 50 and 10 g ml -1 to liquid medium respectively ; and antibiotic rifampicin and chloramphenicol were added to final concentrations of 100 and 50 g ml -1 to solid medium Gibson, 1984) was used as the general cloning host and E. coli BL21DE3, containing the pREP4 plasmid (Amrein et al., 1989) for protein production and purification.

E. coli TG1 (
E. coli strains were grown in LB medium. Antibiotics ampicillin and hygromycin were added to final concentrations of 100 g ml -1 .

DNA manipulation and transformation procedures

Plasmid DNA extracted was from E. coli using a Qiagen Kit. DNA fragments were purified from agarose gels with Ultrafree-DA (Amicon-Millipore). Restriction enzymes were used as recomended by the manufacturers. DNA fragments were amplified by the PCR technique [START_REF] Mullis | Specific synthesis of DNA in vitro via a polymerasecatalyzed chain reaction[END_REF][START_REF] Saiki | Primer-directed enzymatic amplification of DNA with a thermostable DNA polymerase[END_REF]. Standard electroporation procedures were used for E. coli transformation.

Cells were resuspended in 0.5 ml of cold deionized water and added to 0.5 g of glass beads

(106-m diameter ; Sigma), 0.4 ml of 4 % Bentone (Rheox) and 0.5 ml of phenol-chloroformisoamyl alcohol, pH8.0 (Amresco). The cells were disrupted in a Fastprep desintegrator (Bio101, Inc.) for 30 s 3 times at 4°C. After centrifugation for 2 min at 4°C and 20,800 g, the supernatants were collected, treated with phenol-chloroform (1:1, v/v) twice and then with chloroform-isoamyl alcohol (24:1, v/v) twice. 250 l of ethanol was added and the sample was applied to an RNeasy mini column (QIAGEN) following the RNA cleanup protocol recomended by the manufacturers. RNA sample was eluted in 35 l of deionized water. RNA concentrations were determined by measuring absorbance at 260 nm.

Real-time quantitative PCR

Primers (Table 1) were designed with BEACON Designer software. RNA (10 g) was treated twice with 30 U of RNase-free DNase I (Roche) for 30 min at 37°C. Dnase was removed by applying the sample to an RNeasy mini column (QIAGEN) following the RNA cleanup protocol. RNA sample was eluted in 30 l of deionized water. cDNA synthesis was performed with random hexamers (Roche) using SuperScript II RT (Invitrogen) according to the protocol recommended by the manufacturers. Real-time quantitative PCR was performed in a 25 l reaction volume containing cDNA, 12.5 l of SYBR PCR master mix (Applied Biosystems) and 1 l of gene-specific primers (10 M). Amplification and detection of specific products were performed with the iCyclerIQ Multi-Color real-time PCR detection system (Biorad) with the following cycle profile : one cycle at 95°C for 3 min, 40 cycles at 95°C for 15 s, 55°C for 15 s and 72°C for 15 s. The specificity of the amplified product is verified by generating a melting curve with a final step of 80 cycles of 10 s at an intitial temperature of 55°C, then increased of 0.5°C each cycle until 95°C. Loss of fluorescence is observed at the denaturing/melting temperature of the product [START_REF] Ririe | Product differentiation by analysis of DNA melting curves during the polymerase chain reaction[END_REF]. To check whether contaminating chromosomal DNA was present, each sample was tested in control reactions that did not contain reverse transcriptase. For each condition, quadriplate assays were done. The analysis gave a threshold cycle (CT) value for each sample, which is defined as the cycle at which a significant increase in amplification product occcurs. The CT value was calculated for each quadriplate reactions. A CT value was then calculated for each sample by substracting the mean CT value of the target gene from the mean CT value of the hrdB reference gene (hrdB is encoding for an essential and constitutively expressed  factor).

The data was transformed from a logarithmic to linear scale by using the formula x=2-CT [START_REF] Livak | Analysis of relative gene expression data using realtime quantitative PCR and the 2(-Delta Delta C(T)) Method[END_REF].

Protein extraction and Western blotting experiments

Cultures of S. lividans 1326 carrying pHM11a (control), pAB54, pAB55, pAB70M2, pAB71M2 or pAB63 were grown on cellophane discs laid down on the surface of solid NE plates. Proteins were prepared from mycelia of the different strains at different stages of growth (basal mycelium, aerial mycelium, sporulation). Mycelium was resuspended in the sonication buffer (20 mM Tris, 5 mM EDTA, 1 mM -mercaptoethanol, 0,5 mM PMSF) and lysed by sonication. The resulting suspension was centrifuged for 15 min at 4°C and 20,800 g, and the supernatant was treated with 0,3% SDS 5 min at 85°C. The sample was centrifuged for 15 min at 4°C and the protein concentration of the supernatant was determined by the method of Bradford [START_REF] Bradford | A rapid and sensitive method for the quantitation of microgram quantities of protein utilizing the principle of protein-dye binding[END_REF]. Ten g of protein extract were subjected to SDS-PAGE as described by Laemmli [START_REF] Laemmli | Cleavage of structural proteins during the assembly of the head of bacteriophage T4[END_REF]. The proteins were transferred to a nitrocellulose membrane (Hybond C), which was then probed with rabbit polyclonal anti-Streptomyces

ClgR (1:500) (Agro-Bio), anti-Streptomyces Lon (1 :5,000) or anti-Flag M2 monoclonal antibodies (Sigma). Signals were detected with the ECL Western blotting Detection Kit (Amersham Biosciences).

Aknowledgments

We are grateful to T. Msadek for fruitful discussions and for critical reading

Figures

Table 1 :

  Therefore, as shown for clpC1 and clpP1, lon expression is activated throughout the cell cycle in the pAB55 strain.DiscussionPrevious studies have shown that the clpP1P2 operon, clpC1, clgR and lon genes are activated by ClgR. PopR, the activator of the clpP3 operon, is the only one identified target of the Clp protease in Streptomyces. In this study, we identify 2 new targets of ClpP1-dependent proteolysis : ClgR and the Lon protease, encoded by a gene that belongs to the ClgR regulon. These observations point to the existence of an interactive network within the ClgR regulon members : ClgR activates clpP1 expression and in response, ClpP1 degrades ClgR, allowing a negative feedback control of ClgR activation. The identification of these new targets of ClpP1 proteolysis, ClgR and Lon, which play important roles in the Streptomyces cell cycle, emphasize on the crucial role of ClpP1 in Streptomyces differentiation. The 3 identified targets of ClpP proteases in Streptomyces, PopR, ClgR and Lon, all end with 2 alanines preceded by a hydrophobic residue (leucine for PopR and valine for ClgR and Lon)

LEGENDSFig. 1 :

 1 Fig. 1 : Stabilization of ClgR-DD with respect to ClgR-AA in wild-type and clpP1. Crude

Fig. 2 :

 2 Fig. 2 : Detection of ClgR by Western blotting during S. lividans cell cycle. Crude extracts

Fig. 3 :

 3 Fig. 3 : Quantitative real-time RT-PCR analysis. Expression of clgR gene under Erm*p

Fig. 4 :

 4 Fig. 4 : Quantitative real-time RT-PCR analysis of clpP1 and clpC1 genes expression during

Fig. 5 :

 5 Fig. 5 : Stabilization of ClpP1 and ClpC1 proteins. Crude extracts from wild-type carrying

Fig. 6 :Fig. 7 :

 67 Fig. 6 : Detection of Lon by Western blotting during S. lividans cell cycle. Crude extracts

Fig. 8 :

 8 Fig. 8 : Detection of Lon by Western blotting in wild-type or in pAB55 (clgR-DD) strain

Streptomyces DNA and protoplasts were prepared and transformed as described by Hopwood et al. [START_REF] Hopwood | Genetic manipulation of Streptomyces. A laboratory manual[END_REF].

Plasmids and plasmid constructions

Plasmids pAB54 and pAB55 for clgR overexpression in Streptomyces have previously been described [START_REF] Bellier | ClgR, a novel regulator of clp and lon expression in Streptomyces[END_REF] RNA isolation from surface-grown cultures of S. lividans Mycelia of S. lividans 1326 with or without the pHM11a (control), pAB54 or pAB55 grown on cellophane discs were harvested with a spatula, dispersed in 2 ml of water and pelleted.


sur la route de mendeleïev : de la chimie à la physique des particules

Dès 1922, Bohr donne une interprétation quantique du tableau périodique, basée sur l'ancienne théorie des quanta, au moyen d'un remplissage d'orbites elliptiques de Bohr et Sommerfeld. À la fin des années 20, le modèle des couches de la physique atomique donnera une assise plus quantitative à cette description quantique du tableau. Mais la grande nouveauté viendra des années 70. Sur la base de modèles de théorie des groupes utilisés en physique des particules et dans la description de certains systèmes dynamiques, le groupe conforme SO(4,2) est proposé pour la classification des éléments chimiques. Il en résulte un nouveau tableau périodique (voire figure). Les grandes caractéristiques de ce nouveau tableau sont les suivantes : 1) l'hydrogène et l'hélium sont dans la famille des alcalins et des alcalino-terreux, réciproquement ;

2) les lanthanides et les terres rares ne sont pas relégués à la périphérie du tableau : ces éléments y occupent une place naturelle ; 3) chaque bloc de 6, 10, 14, … éléments est divisé en 2 sous-blocs ce qui correspond pour les terres rares à la subdivision en terres rares légères et terres rares lourdes ; les superactinides vont de Z = 139 à Z = 152 et non de Z = 122 à Z = 153 comme prédit par Seaborg ; 5) le nombre d'éléments dans le tableau est a priori infini. Pour finir, disons que cette approche présente de fortes potentialités pour une description quantitative (à la Gell-Mann/ Okubo, cf. la physique des particules) des éléments chimiques.

[Pour en savoir plus, voir M.R. Kibler, From the Mendeleev periodic Tableau périodique basé sur le groupe SO(4,2) x SU(2) Chaque élément est repéré par son numéro atomique Z et son symbole chimique. Par exemple Z = 111 correspond à Rg (roentgenium). Le symbole « X? » correspond à un élément observé qui n'a pas encore reçu de nom. Finalement, « no » correspond à un élément non observé.

  éléments vers 1870, on en connaîtra 86 vers 1940 et on en connaît actuellement 116. Mais et surtout l'après Mendeleïev se caractérise par le « démontage » de l'atome et le passage de la physique atomique à la physique nucléaire puis à la physique des particules. Tout commence avec la découverte de l'électron dans les rayons cathodiques par J.J. Thomson en 1897. En 1911, l'expérience de Rutherford de diffusion de particules α par une cible d'or va mettre en évidence le noyau atomique. (Cette expérience préfigure les expériences de diffusion inélastique profonde d'électrons sur protons réalisés en 1966 à Standford qui mettront en évidence les partons de Bjorken et Feynman ou quarks/aces de Gell-Mann/Zweig.) Ceci va conduire à abandonner le modèle du pain aux raisins de l'atome (une sphère positive piquée d'électrons) pour l'atome et conduira au modèle planétaire d'un noyau autour duquel se meuvent des électrons. Le passage à la physique nucléaire se fera en 1932 avec la découverte du neutron par Chadwick et le modèle de Heisenberg. Le monde des par-ticules est simple en 1932 : avec 4 particules (électron, proton, neutron et neutrino postulé par Pauli en 1930) et le photon on explique beaucoup de choses. Mais tout se complique avec l'arrivée du muon (en 1937), des pions (en 1947) et des particules dites étranges (dans les années 50). Il faut trouver un moyen de classer les nombreuses particules connues. Cela se fera en 1961 avec le modèle de la voie octuple de Gell-Mann et Ne'eman basé sur le groupe SU(3) : les hadrons (particules du type proton ou pion) sont classés dans de petits tableaux issus de la théorie des groupes et comportant 8 ou 10 cases, tableaux dans lesquels on range des particules ayant des propriétés voisines. Et c'est là que Gell-Mann va jouer à l'eka-particule : à l'époque on connaît 9 baryons ayant des propriétés voisines, que l'on peut donc disposer dans un tableau à 10 cases, et il postule une dixième particule (la particule Ω) pour remplir le tableau, particule qui sera observée deux ans plus tard en 1964. L'histoire va se continuer avec l'hypothèse des quarks (3 en 1964 et 6 en 2007), à partir desquels on construit tous les hadrons, et par la découverte de nouveaux électrons et neutrinos (ils sont maintenant 6, 3 électrons et 3 neutrinos). Ces 6+6 particules de matière peuvent être disposées dans un tableau périodique comportant 3 périodes ou générations de 2 quarks, un électron et un neutrino. Ceci nous amènera au modèle standard de la physique des particules et de leurs interactions ; mais ce sera pour une autre fois dans cette rubrique «Histoire des sciences». Pour cette gazette, nous allons revenir à Mendeleïev et voir comment les modèles basés sur la théorie des groupes (ou symétries) développés pour la physique des particules ont influencé la compréhension du tableau périodique.

hIstoIre

  On célèbre cette année le 100ème anniversaire de la mort de Dimitri Ivanovitch Mendeleïev. C'est dire que 2007 est riche en manifestations (congrès, expositions, etc.) et publications de livres et d'articles dédiés à Mendeleïev, un des pères de la classification périodique des éléments chimiques.

	Mendeleïev n'a pas été le seul à classer des
	éléments. Il a eu d'illustres précurseurs
	comme, par exemple, Lavoisier et
	ses 33 substances simples (dont 23
	seulement correspondent à des corps
	simples), Döbereiner et ses triades,
	Beguyer de Chancourtois et sa spi-
	rale sur laquelle sont rangés des élé-
	ments avec une périodicité d'ordre
	8, Newlands et ses octaves et Meyer
	et ses courbes. Alexandre-Émile Be-
	guyer de Chancourtois est souvent
	présenté comme le Mendeleïev fran-
	çais (voir l'article dans The New York
	Times du 21 novembre 2006 : « Did
	a Frenchman beat Mendeleev to the
	periodic table? »). Alors, pourquoi
	retient-on le nom de Mendeleïev
	? En effet, comme beaucoup de ses
	prédécesseurs, il classe selon les poids
	atomiques croissants (ce qui est une
	approximation : il faudra attendre
	Moseley et sa loi de 1913 pour re-
	connaître que la place d'un élément
	dans le tableau périodique est déter-
	miné par son numéro atomique Z
	plutôt que par son poids atomique).
	Mais, en 1869, il n'hésite pas à faire : 1)
	des inversions (comme Te/I et Ni/Co)
	qui vont à contrario des poids atomiques
	croissants, 2) laisser des cases vides, 3) pré-
	dire des nouveaux éléments (les fameux
	eka-éléments) et 4) donner les propriétés
	chimiques et physiques des éléments pré-
	dits. Cela n'est pas sans rappeler la prédic-
	tion, avec toutes les bonnes propriétés, de
	la particule étrange Ω par Gell-Mann en
	1962.
	L'après 69 (1869 !) va voir la découverte des
	eka-éléments (comme l'eka-aluminium ou
	gallium et l'eka-silicium ou germanium).
	La taille du tableau périodique ne va pas
	cesser d'augmenter : si on connaissait 63

  table to particle physics and back to the periodic table, Cs Ba Tl Pb Bi Po At Rn Lr Rf Db Sg Bh Hs Mt Ds Rg X? no no no no no no no no no no no no no no

					dans Foundations of Chemistry (2007),
					arXiv : quant-ph/0611287 et:
					http://spiral.univ-lyon1.fr/entree.
					asp?id=2770.]
	1 2
	H He
	3 4	5 6 7 8 9 10
	Li Be	B C N O F Ne
	11 12	13 14 15 16 17 18	21 22 23 24 25 26 27 28 29 30
	Na Mg	Al Si P S Cl Ar	Sc Ti V Cr Mn Fe Co Ni Cu Zn
	19 20	31 32 33 34 35 36	39 40 41 42 43 44 45 46 47 48	57 58 59 60 61 62 63 64 65 66 67 68 69 70
	K Ca	Ga Ge As Se Br Kr	Y Zr Nb Mo Tc Ru Rh Pd Ag Cd	La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb
	37 38	49 50 51 52 53 54	71 72 73 74 75 76 77 78 79 80	89 90 91 92 93 94 95 96 97 98 99 100 101 102 121 -138
	Rb Sr	In Sn Sb Te I	Xe	Lu Hf Ta W Re Os Ir Pt Au Hg	Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No	no
	55 56	81 82 83 84 85 86	103 104 105 106 107 108 109 110 111 112 139 140 141 142 143 144 145 146 147 148 149 150 151 152 …
					…
	87 88	113 114 115 116 117 118 153 154 155 156 157 158 159 160 161 162 …
	Fr Ra	X? X? X? X? no no	no no no no no no no no no no	…
	…
	…