Tino Laidin
email: tino.laidin@univ-lille.fr

Lorenzo Pareschi
email: l.pareschi@hw.ac.uk

CONSERVATIVE POLYNOMIAL APPROXIMATIONS AND APPLICATIONS TO FOKKER-PLANCK EQUATIONS

Keywords: Fokker-Planck equation, Galerkin spectral method, conservative methods, spectral accuracy. 2020 Mathematics Subject Classification: 65N35 (Primary), 82C40 (Secondary)

We address the problem of constructing approximations based on orthogonal polynomials that preserve an arbitrary set of moments of a given function without loosing the spectral convergence property. To this aim, we compute the constrained polynomial of best approximation for a generic basis of orthogonal polynomials. The construction is entirely general and allows us to derive structure preserving numerical methods for partial differential equations that require the conservation of some moments of the solution, typically representing relevant physical quantities of the problem. These properties are essential to capture with high accuracy the long-time behavior of the solution. We illustrate with the aid of several numerical applications to Fokker-Planck equations the generality and the performances of the present approach.

1. Introduction. Computational techniques that maintain certain fundamental properties or structures of the underlying mathematical problem in their discrete approximations play a crucial role in the study and analysis of ODEs and PDEs, offering insights into complex physical phenomena that are often inaccessible through analytical means alone [START_REF] Gosse | Computing Qualitatively Correct Approximations of Balance Laws. Exponential-Fit, Well-Balanced and Asymptotic-Preserving[END_REF][START_REF] Hairer | Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations[END_REF][START_REF] Jin | Asymptotic-preserving schemes for multiscale physical problems[END_REF]. These structure-preserving properties could include symmetries, conservation laws, or other structural characteristics that are crucial for accurately representing the behavior of the system being modeled.

Capturing the long-time behavior of solutions to PDEs is also intricately linked to the structure-preserving properties of numerical methods. In fact, preserving key structural properties at a discrete level, such as conservation laws and invariant quantities, ensures that the numerical approximation retains the essential characteristics of the original problem over extended time intervals.

In this regard, the preservation of moments of the solution stands out as a fundamental aspect, especially when considering kinetic and mean-field equations, where the moments represent the macroscopic observable physical quantities [START_REF] Bessemoulin-Chatard | On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system[END_REF][START_REF] Dimarco | Numerical methods for plasma physics in collisional regimes[END_REF][START_REF] Kraus | Metriplectic integrators for the Landau collision operator[END_REF][START_REF] Pareschi | Structure preserving schemes for nonlinear Fokker-Planck equations and applications[END_REF]. In kinetic equations, for example, the long time behavior of the systems is fully determined by the knowledge of some invariant moments. Achieving such properties, however, is particularly challenging due to the inherent complexity of many physical systems and the necessity to accurately capture these dynamics.

A substantial body of literature has been devoted to the development of structurepreserving methods for various types of PDEs, with a focus on preserving specific physical properties. For instance, in the context of Fokker-Planck equations, numerous studies have explored different numerical strategies aimed at maintaining conservation of mass, momentum and energy with the aim to describe accurately the steady state solution of the problem [START_REF] Bailo | Fully discrete positivity-preserving and energydissipating schemes for aggregation-diffusion equations with a gradient-flow structure[END_REF][START_REF] Buet | On the Chang and Cooper scheme applied to a linear Fokker-Planck equation[END_REF][START_REF] Chang | A practical difference scheme for Fokker-Planck equations[END_REF][START_REF] Larsen | Discretization methods for one-dimensional Fokker-Planck operators[END_REF][START_REF] Mohammadi | Analysis of the Chang-Cooper discretization scheme for a class of Fokker-Planck equations[END_REF][START_REF] Pareschi | Structure preserving schemes for nonlinear Fokker-Planck equations and applications[END_REF].

Spectral methods, which rely on expansions in terms of orthogonal polynomials, have garnered significant attention for their effectiveness in solving collisional kinetic equations of Boltzmann-type. The spectral accuracy and efficiency of these methods make them particularly well-suited for capturing the non equilibrium dynamics of such systems [START_REF] Hu | A fast spectral method for the inelastic Boltzmann collision operator and application to heated granular gases[END_REF][START_REF] Mouhot | Fast algorithms for computing the Boltzmann collision operator[END_REF][START_REF] Mouhot | Convolutive Decomposition and Fast Summation Methods for Discrete-Velocity Approximations of the Boltzmann Equation[END_REF][START_REF] Pareschi | Numerical Solution of the Boltzmann Equation I : Spectrally Accurate Approximation of the Collision Operator[END_REF][START_REF] Pareschi | On the stability of spectral methods for the homogeneous Boltzmann equation[END_REF][START_REF] Pareschi | Fast spectral methods for the Fokker-Planck-Landau collision operator[END_REF]. However, due to the lack of conservations, the long time behavior of such methods may lead to accumulation of errors and their extension to a conservative setting has represented a challenging task in numerical simulations. Recent advancements in conservative spectral methods based on L 2 -minimization frameworks have shown promise in addressing these challenges [START_REF] Alonso | Convergence and error estimates for the Lagrangian-based conservative spectral method for Boltzmann equations[END_REF][START_REF] Gamba | A fast spectral method for the Boltzmann collision operator with general collision kernels[END_REF][START_REF] Gamba | Shock and Boundary Structure formation by Spectral-Lagrangian methods for the Inhomogeneous Boltzmann Transport Equation[END_REF][START_REF] Pareschi | Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation[END_REF].

Building upon these developments, our main goal in this manuscript is to extend the L 2 -minimization setting to derive families of orthogonal polynomials capable of preserving the moments of the solution, thereby enhancing the accuracy and efficiency of structure-preserving numerical methods. To this end, we draw upon the well established theory of orthogonal polynomials, which provides a powerful framework for constructing accurate approximations of PDEs [START_REF] Canuto | Spectral Methods in Fluid Dynamics[END_REF][START_REF] Funaro | Polynomial approximations of differential equations[END_REF][START_REF] Gautschi | Orthogonal Polynomials: Computation and Approximation[END_REF][START_REF] Gottlieb | Numerical Analysis of Spectral Methods: Theory and Applications[END_REF][START_REF] Koekoek | Hypergeometric Orthogonal Polynomials and their q-Analogues[END_REF]. In particular, we have developed a general framework for constructing constrained orthogonal polynomial approximations which maintain the accuracy properties of the unconstrained polynomials for smooth solutions. This framework enabled us to construct spectrally accurate moment-preserving Galerkin-type approximations for several Fokker-Planck equations originating from various fields, ranging from classical physics to social sciences and operations research. Consequently, these equations are defined in different domains, both bounded and unbounded, and possess different steady states, requiring the adoption of suitable orthogonal polynomial bases.

Let us also remark, that the connection between orthogonal polynomials and probability theory to design uncertainty quantification methods further underscores the importance of our approach, as classical families of continuous and discrete orthogonal polynomials are intimately linked to probability distributions and our considerations naturally generalize to such contexts [START_REF] Xiu | Numerical Methods for Stochastic Computations: A Spectral Method Approach[END_REF].

In the following sections, we provide a detailed exposition of our methodology and present numerical results demonstrating the efficacy of our approach. The rest of the manuscript is organized as follows. In Section 2, we introduce some notations and present our moment-preserving approach based on a constrained L 2 -minimization setting. We also study the convergence properties and prove a general result on spectral accuracy. Section 3 is then devoted to testing the novel polynomial approximation, first by approximating probability densities with given moments and subsequently by considering different Fokker-Planck equations on bounded and unbounded domains. Some concluding remarks are provided in the last section.

Conservative approximations by orthogonal polynomials.

In order to get conservative approximations, we construct a conservative projection on the space of orthogonal polynomials by generalizing the constrained formulation approach introduced in [START_REF] Pareschi | Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation[END_REF] for trigonometric polynomials. In particular, we will give an explicit formulation of the orthogonal polynomial of best approximation in the weighted least square sense, constrained by preservation of moments, and show that it preserves spectral accuracy for smooth solutions like the classical orthogonal polynomial approximation.

Orthogonal polynomials.

Let us first set up the mathematical framework of our analysis. To simplify our treatment we will consider the one-dimensional case. Given a real function f (x) ∈ L 2 ω (Ω), with Ω ⊆ R and ω(x) > 0 a positive function, we define

(2.1) f L 2 ω = Ω |f (x)| 2 ω(x) dx 1/2
, which has the associated inner product

(2.2) f, g ω = Ω f (x)g(x)ω(x) dx.
We assume that ω(x) > 0 is such that

(2.3) Ω |x| q ω(x) dx < ∞, q = 0, 1, 2, . . .
We consider the problem of approximating functions in the weighted L 2 ω (Ω) space defined by the norm (2.1). In the case ω(x) = 1 we will use standard notations • L 2 and •, • to denote the L 2 norm and the associated inner product.

A polynomial basis {p k }, k = 0, 1, . . . for L 2 ω (Ω) is a set of functions such that any f in the space can be expressed uniquely as

(2.4) f (x) = ∞ k=0 fk p k (x).
A set of polynomials {p k } is called orthogonal if p h , p k ω = 0 for h = k. We refer to Table 1 for some examples of standard polynomial families Orthogonal bases are particularly nice, both for theory and numerical approximation. In fact, from

f, p k ω = ∞ h=0 fh p h , p k ω = fk p k , p k ω ,
the coefficients in (2.4) are easily represented as

(2.5) fk = f, p k ω p k , p k ω = f, p k ω p k 2 L 2 ω .
A first result that we recall is that the truncated approximation

(2.6) f N (x) = N k=0 fk p k (x)
is the best approximation of f in the subspace

S N = span {p k | k = 0, 1, . . . , N }. More precisely, let P ω,N : L 2 ω (Ω) → S N be the orthogonal projection upon S N in the inner product of L 2 ω (Ω) (2.7) f -P ω,N f, p k ω = 0, ∀ p k ∈ S N .
With these definitions P ω,N f = f N , is the solution of the following minimization problem

f N = argmin g N -f L 2 ω : g N ∈ S N .
By orthogonality, one also obtains the Parseval's identities

f 2 L 2 ω = ∞ k=0 | fk | 2 p k 2 L 2 ω , f N 2 L 2 ω = N k=0 | fk | 2 p k 2 L 2 ω ,
which gives the error estimate

f -f N 2 L 2 ω = ∞ k=N +1 | fk | 2 p k 2 L 2 ω .
An important feature of the orthogonal polynomial approximations on S N is related to their spectral convergence properties for smooth solutions. We report here a result for Jacobi type approximations [START_REF] Funaro | Polynomial approximations of differential equations[END_REF] Theorem 2.1. If f ∈ H r ω (Ω), where r ≥ 0 is an integer, then there exists a constant C > 0 dependent on α, β and r such that

(2.8) f -f N L 2 ω ≤ C N r (1 -x 2 ) r/2 d r f dx r L 2 ω , N > r.
Name ω(x) Ω Jacobi (1 -x) α (1 + x) β , α, β > -1 [-1, 1] -Legendre 1, α = β = 0 [-1, 1] -Chebyshev 1st kind 1 √ 1 -x 2 , α = β = - 1 2 [-1, 1] -Chebyshev 2nd kind 1 -x 2 , α = β = 1 2 [-1, 1] Laguerre x α e -x , α > -1 R + Hermite |x| 2α e -x 2 , α > -1 R 2.
2. Approximation of moments. Next, we discuss some properties of the projection operator P ω,N , in particular those concerning approximation of moments of nonnegative functions. We will denote for q = 0, 1, 2, . . .

(2.9) m q = Ω f (x)x q dx = f, x q , m q,N = Ω f N (x)x q dx = f N , x q .
Note that

m q = ∞ k=0 fk µ q,k , m q,N = N k=0 fk µ q,k ,
where for k, q ≥ 0 (2.10)

µ q,k = p k , x q ,
is the q-th moment of the k-th order polynomial. It is well-known that moments of the weight function ω(x) permits to generate explicitly the orthogonal polynomials [START_REF] Koekoek | Hypergeometric Orthogonal Polynomials and their q-Analogues[END_REF]. In our case, however, we are interested in the behavior of the moments of the orthogonal polynomials themselves defined in (2.10). An important characterization of monic orthogonal polynomials is the classical three-term recurrence relation [START_REF] Funaro | Polynomial approximations of differential equations[END_REF][START_REF] Gautschi | Orthogonal Polynomials: Computation and Approximation[END_REF][START_REF] Gottlieb | Numerical Analysis of Spectral Methods: Theory and Applications[END_REF][START_REF] Koekoek | Hypergeometric Orthogonal Polynomials and their q-Analogues[END_REF]

p k (x) = (a k x + b k )p k-1 (x) + c k p k-2 (x), k ≥ 1, (2.11)
where we assumed p 0 (x) = 1, p -1 (x) = 0 and the sequences {a k } k≥1 , {b k } k≥1 and {c k } k≥1 depend on the particular polynomial family under consideration.

Equation (2.11) permits to recursively compute the value of p k (x) starting from the values of p 0 (x) and p 1 (x).

From (2.11) one gets the following relations for the moments coefficients

µ q,k =a k µ q+1,k-1 + b k µ q,k-1 + c k µ q,k-2 , k ≥ 1, q ≥ 0, (2.12)
with µ q,0 =

x q+1 q + 1 Ω , µ q,-1 = 0, q ≥ 0, which, therefore, can also be computed recursively from (2.12) up to the moment q provided we know q + h moments of p k-h (x) for h = 1, . . . , k.

Unbounded domains.

Note, however, that for unbounded domains Ω, like the case of Laguerre or Hermite polynomials (see Table 1), the moments µ q,0 , q ≥ 0, are not finite. In such cases, one resorts on suitable orthogonal function families with respect to the inner product •, • defined in the symmetrically weighted case [START_REF] Bessemoulin-Chatard | On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system[END_REF][START_REF] Holloway | Spectral velocity discretizations for the Vlasov-Maxwell equations[END_REF][START_REF] Manzini | Convergence of spectral discretizations of the vlasov-poisson system[END_REF] as

P k (x) = p k (x)w 1/2 (x), so that P h , P k = p h , p k ω , ∀ h, k ≥ 0.
For functions that we write in the form f (x) = h(x)w 1/2 (x) we have

(2.13) f (x) = ∞ k=0 ĥk P k (x), with (2.14) ĥk = f, P k P k , P k = h, p k ω p k 2 L 2 ω .
Clearly, the orthogonal functions P k satisfy an analogous three term relation as (2.11) with P 0 (x) = w 1/2 (x), and similarly their moments satisfy (2.12) with

P 0 , x q = x q+1 q + 1 w 1/2 (x) Ω , µ q,-1 = 0, q ≥ 0.
The above quantity is bounded for Laguerre and Hermite polynomials thanks to (2.3). We can then define f N (x) as the truncated approximation to the first N + 1 terms of (2.13). We have

(2.15) f -f N , P k = 0, ∀ P k = p k w 1/2 , p k ∈ S N .
In the weighted function family representation Parseval identity reads

f 2 = ∞ k=0 | ĥk | 2 P k 2 2 .
We report here a result for Hermite approximations [START_REF] Fok | Combined Hermite spectral-finite difference method for the Fokker-Planck equation[END_REF][START_REF] Funaro | Polynomial approximations of differential equations[END_REF] Theorem 2.2. If f ∈ H r ω (Ω), where r ≥ 0 is an integer, then there exist a constant C > 0 and r such that

(2.16) f -f N L 2 ω ≤ C N r/2 f H r ω , N > r.
Let us remark that, in general, moments are not preserved by the projection. Neither in standard nor in the weighted case. In fact, we have (2.17)

m q -m q,N = f -f N , x q = f -f N , x q ω(x) ω
which, by (2.7), is equal to zero only if x q /ω(x) belongs to S N . A special case is represented by the Legendre polynomials which by construction preserve all moments of the approximated function f (x) up to the order q ≤ N . Finally, we have for each ϕ ∈ L 2 ω (Ω) by Hölder's inequality

(2.18) | f, ϕ ω -f N , ϕ ω | ≤ ϕ L 2 ω f -f N L 2 ω .
Therefore, the projection error on the moments decays faster than algebraically, for the weighted norms, when the solution is infinitely smooth. Although this is in general a guarantee of the accuracy of the methods, in practice the loss of conservations can lead to accumulations of error in the approximation of PDEs over long time scales, resulting in the determination of inaccurate steady states [START_REF] Gosse | Computing Qualitatively Correct Approximations of Balance Laws. Exponential-Fit, Well-Balanced and Asymptotic-Preserving[END_REF][START_REF] Pareschi | Structure preserving schemes for nonlinear Fokker-Planck equations and applications[END_REF].

Moment preserving orthogonal polynomials.

Although the error on moments is spectrally small for smooth functions, in many applications moments represent structural properties of the PDE and are related to conservation of physical properties such as mass, momentum and energy. In such cases, the design of a numerical method that is structure preserving requires the construction of a projection operator that does not affect the relevant moments. To this end, we will construct a different projection operator on the space of polynomials, P c ω,N : L 2 ω (Ω) → S N such that it preserves a finite number of moments. More precisely, we require the projection to satisfy for q = 0, 1, . . . , M

P c ω,N f, x q = f, x q ,
but maintaining the convergence properties of the original orthogonal projection.

To this aim, it is natural to consider the following constrained best approximation problem

(2.19) f c N = argmin g N -f 2 L 2 ω : g N ∈ S N , g N , x q = f, x q , q = 0, 1, . . . , M . Now, since g N ∈ S N we can represent it in the form g N (x) = N k=0 ĝk p k (x)
and then by Parseval's identity

g N -f 2 L 2 ω = ∞ k=0 |ĝ k -fk | 2 p k 2 L 2 ω ,
where we assumed ĝk = 0, k > N . We require that (2.20)

g N , x q = N k=0 ĝk µ q,k = f, x q , q = 0, 1, . . . , M
or, after introducing the vector of moments Φ = (1, x, x 2 , . . . , x M ) T , in vector form

g N , Φ = N k=0 ĝk Φk = f, Φ , where Φk = (µ 0,k , µ 1,k , . . . , µ M,k ) T .
Let us now solve the minimization problem (2.19) using the Lagrange multiplier method. Let λ ∈ R M +1 be the vector of Lagrange multipliers, we consider the objective function

L(ĝ, λ) = ∞ k=0 |ĝ k -fk | 2 p k 2 L 2 ω + λ T N k=0 ĝk Φk -f, Φ = 0, with ĝ ∈ R N +1 the vector of coefficients ĝk , k = 0, . . . , N .
Stationary points are found by imposing

∂L(ĝ, λ) ∂ĝ k = 0, k = 0, 1, . . . , N ; ∂L(ĝ, λ) ∂λ q = 0, q = 0, 1, . . . , M.
From the first condition, one gets

(2.21) 2(ĝ k -fk ) p k 2 L 2 ω + λ T Φk = 0,
whereas the second condition corresponds to (2.20).

Multiplying the above equation by Φk

/ p k 2 L 2 ω
and summing up over k we can write

2 N k=0 (ĝ k -fk ) Φk + N k=0 1 p k 2 L 2 ω Φk ΦT k λ = 0
and, using (2.20) as well as the fact that Φk ΦT k are symmetric and positive definite matrices of size M + 1 one obtains

(2.22) λ = -2 N k=0 1 p k 2 L 2 ω Φk ΦT k -1 ( f, Φ -f N , Φ ).
Now, rewriting the first condition (2.21) as

ĝk = fk - 1 2 p k 2 L 2 ω ΦT k λ,
and plugging the expression (2.22) of λ in it, one obtains that the minimum is achieved for ĝk = f c k , given by the following definition.

Definition 2.3. For f ∈ L 2 ω (Ω), we define the conservative orthogonal projec- tion P c N,ω f = f c N in S N , where f c N is given by (2.23) f c N (x) = N k=0 f c k p k (x),
and we define the moment constrained coefficients as (2.24)

f c k = fk + ĈT k ( f, Φ -f N , Φ ), ĈT k = 1 p k 2 L 2 ω ΦT k N h=0 1 p h 2 L 2 ω Φh ΦT h -1
.

The following result states the spectral accuracy of the conservative best approximation in least square sense (2.23), and generalizes Theorem 2.1. In order to prove this result, one needs the following hypothesis: Hypothesis 2.4. The moments of the approximated solution are spectraly accurate in the classical L 2 -norm. Namely, there exists C > 0 such that

(2.25) U -U N ≤ C N r f H r ω Φ 2,L 2 ω
This assumption is different from the spectral accuracy on the moment, in weighted norm, one could obtain from (2.18). This property depends heavily on the function under consideration and we will see in numerical simulations that it may only be a technical detail.

Theorem 2.5. If f ∈ H r ω (Ω)
, where r ≥ 0 is an integer, we have

(2.26) f -f c N L 2 ω ≤ C Φ N r f H r ω
where the constant C Φ depends on the spectral radius of the matrix

M = N h=0 1 p h 2 L 2 ω Φh ΦT h ,
and on Φ 2 2,L 2 ω = M q=0 Φ q 2 L 2 ω
where Φ q , q = 0, 1, . . . , M are the components of the vector Φ.

Proof. We can split

f -f c N 2 L 2 ω ≤ f -f N 2 L 2 ω + f N -f c N 2 L 2 ω .
The first term is bounded by the spectral estimate of truncated approximation (2.8), whereas for the second term, by Parseval's identity, we have

f c N -f N 2 L 2 p = N k=0 | f c k -fk | 2 p k 2 L 2 ω .
Now, using the definition (2.24) we get

N k=0 | f c k -fk | 2 p k 2 L 2 ω = N k=0 | ĈT k (U -U N )| 2 p k 2 L 2 ω ≤ U -U N 2 2 N k=0 ĈT k 2 2 p k 2 L 2 ω .
where • 2 denotes the euclidean norm of the vector. The assumption 2.4 of spectral accuracy, in classical L 2 norm, on the moments implies

U -U N 2 ≤ C N r f H r ω M q=0 Φ q 2 L 2 ω 1/2 = C N r f H r ω Φ 2,L 2 ω where for Φ = (Φ 0 , . . . , Φ M ) T we defined Φ 2 2,L 2 ω = M q=0 Φ q 2 L 2 ω
. Finally, by defi-

nition of ĈT k = 1 p k 2 L 2 ω ΦT k N h=0 1 p h 2 L 2 ω Φh ΦT h -1
, one has

N k=0 ĈT k 2 2 p k 2 L 2 ω ≤ N k=0 Φk 2 2 N h=0 1 p h 2 L 2 ω Φh ΦT h -1 2 2 . Now let M = N h=0 Φh ΦT h / p h 2 L 2 ω . It follows that (2.27) N k=0 ĈT k 2 2 p k 2 L 2 ω ≤ Φ 2 2,L 2 ω M -1 2 2 .
As a sum of positive definite symmetric matrices, M enjoys the same properties. Consequently,

(2.28)

N k=0 ĈT k 2 2 p k 2 L 2 ω ≤ Φ 2 2,L 2 ω ρ 2 (M -1 ),
where ρ(M -1 ) denotes the spectral radius of the matrix M -1 . Combining everything, one finally obtains

f c N -f N L 2 ω ≤ C N r/2 f H r ω Φ 2,L 2 ω ρ(M -1 )
which proves (2.26).

Remark 2.6. The conservative best approximation in least square (2.24) can be represented in term of the standard projection as (2.29)

P c N f = P N f + N k=0 ĈT k f -P N f, Φ .

Numerical examples and applications.

In this section, we conduct numerical experiments to illustrate the properties of the proposed method. The initial test focuses on evaluating the accuracy of the moment-preserving approach. Subsequently, in Tests 2 to 4, we employ the conservative method within a PDE framework, considering scenarios involving both bounded and unbounded domains. The ensuing analysis includes the assessment of the numerical scheme's accuracy, its conservation properties, and an exploration of the long-time behavior of solutions.

Test 1: Approximation of functions.

Bounded domain: [-1, 1]. Let us consider the approximation of the oscillating and non symetric function

(3.1) f (x) = sin(2πx) + x 2 cos(2πx), x ∈ [-1, 1]
We first consider a standard approximation using Legendre polynomials. Figure 1 illustrates the approximate solutions and Table 2 shows that the standard method is indeed spectrally accurate. In addition, we observe that moments are indeed preserved without constraining the projection as predicted by (2.17). Note that errors are sometimes extremely close to 0 but not exactly 0 because the Gaussian quadrature used may introduces small machine epsilons.

Let us now compare the standard and constrained approximations for three polynomial families: Chebyshev 1 st kind, Chebyshev 2 nd kind, and general Jacobi with parameters α = 1 and β = -1/2. The associated outcomes are illustrated in Figure 2 and errors are detailed in Tables 3, 4, and 5. Within the Jacobi family, the deliberate choice of α and β serves the purpose of disrupting the basis symmetry to evaluate the method's robustness in this context. Across all three cases, it is clear that standard approximations exhibit spectral accuracy, with moments converging in a spectral manner. Consequently, our analysis falls within the range of Theorem 2.5, affirming spectral accuracy for constrained approximations and exact conservation of the first M = 4 moments. This property shows clearly in the Tables mentioned earlier. It is noteworthy that the constraining matrix Ĉk may exhibit pronounced ill-conditioning for large values of M , potentially introducing numerical artifacts. Preconditioners may therefore be required to improve numerical stability.

N f -f N 2 |m 0 -m 0,N | |m 0 -m 1,N | |m 0 -m 2,N | |m 0 -m 3,N | 8
Unbounded domain : R. Continuing, we extend our approximation to R utilizing Hermite functions. Let H k denote the k th Hermite polynomial. We define the k th symmetrically weighted Hermite functions as

(3.2) ψ k = H k e -x 2 /2 .
Specifically, we want to approximate the difference between two Gaussian functions deliberately crafting a non-symmetric outcome:

(3.3) f (x) = 3 √ 2π e (x+3)/2 - 1 √ π e (x-2) , x ∈ R
We illustrate the results in Figure 3 and, again, the results (See Table 6) on the standard approximation ensures that we are within the scope of Theorem 2.5. As predicted, the conservative method exhibits spectral accuracy, along with conservation of moments.

Unbounded domain : R + . We now look at the approximation on R + using symmetric Laguerre functions. We define for a Laguerre polynomial L k the k th Laguerre function as (3.4) Let us consider the function

ξ k = L k e -x/2
Standard N f -f N 2 |m 0 -m 0,N | |m 0 -m 1,N | |m 0 -m 2,N | |m 0 -m 3,N | 8
N f -f c N 2 |m 0 -m c 0,N | |m 0 -m c 1,N | |m 0 -m c 2,N | |m 0 -m c 3,N | 8
Standard N f -f N 2 |m 0 -m 0,N | |m 0 -m 1,N | |m 0 -m 2,N | |m 0 -m 3,N | 8
N f -f c N 2 |m 0 -m c 0,N | |m 0 -m c 1,N | |m 0 -m c 2,N | |m 0 -m c 3,N | 8
Standard N f -f N 2 |m 0 -m 0,N | |m 0 -m 1,N | |m 0 -m 2,N | |m 0 -m 3,
N f -f c N 2 |m 0 -m c 0,N | |m 0 -m c 1,N | |m 0 -m c 2,N | |m 0 -m c 3,N | 8
(3.5) f (x) = (x 3 -2x + sin(x))e -x , x ∈ R +
The approximation is illustrated in Figure 4 and, consistent with prior cases, the observations in Table 7 shows the spectral accuracy of the conservative method. Although errors on the moments exhibit a slight increase compared to previous cases, it is noteworthy that this discrepancy can be attributed to numerical artifacts arising from the ill-conditioning of the constraining matrix Ĉk and numerical integration intricacies.

To conclude this first test let us again consider Laguerre functions but for the approximation of

(3.6) f (x) = 1 √ 2πσv exp - (ln(v) -µ) 2 2σ , x ∈ R +, * .
where σ = 0.2 and µ = ln(40) -0.2. This function is badly approximated by negative exponentials at infinity, yet it holds significance for Test 4. As depicted in Figure 5, spectral accuracy is not attained in the approximation of (3.6). Consequently, this experiment falls outside the assumptions of Theorem 2.5. Nevertheless, it is noteworthy that the resultant conservative approximation successfully preserves the 0 th order moment and achieves fifth-order accuracy. In the next tests, we apply the conservative method in a PDE setting to various Fokker-Planck equations.

Standard N f -f N 2 |m 0 -m 0,N | |m 0 -m 1,N | |m 0 -m 2,N | |m 0 -m 3,N | 8
N f -f c N 2 |m 0 -m c 0,N | |m 0 -m c 1,N | |m 0 -m c 2,N | |m 0 -m c 3,N | 8

Test 2: Kinetic Fokker-Planck equation.

Let us now consider the following kinetic Fokker-Planck equation on R:

(3.7)    ∂ ∂t f = ∂ v ((v -µ)f + T ∂ v f ) := L F P (f ), f (0, v) = f 0 (v).
In this case, the unknown f (t, v) describes the amount of particles moving with velocity v at time t. In this setting, the particle will be forced towards a mean velocity µ and an average temperature T . It is well known that this equation admits a steady state f ∞ given by a Maxwellian distribution:

(3.8) f ∞ = ρ √ 2πT exp -(v -µ) 2 2T ,
where ρ is the total mass of particles. In the following, let us consider the initial data (3.9) This distribution admits a mass ρ 0 , mean velocity µ 0 and temperature T 0 given by (3.10)

f 0 (v) = 1 √ π e -(v+2) 2 + e -(v-2) 2 Standard N f -f N 2 |m 0 -m 0,N | |m 0 -m 1,N | |m 0 -m 2,N | |m 0 -m 3,N | 8
N f -f c N 2 |m 0 -m c 0,N | |m 0 -m c 1,N | |m 0 -m c 2,N | |m 0 -m c 3,N | 8
ρ 0 = R f 0 (v) dv, µ 0 = 1 ρ 0 R vf 0 (v) dv, T 0 = R (v 2 -µ 0 )f 0 (v) dv.
In addition, we set the parameters of L F P in (3.7) as

(3.11) µ = µ 0 and T = T 0 .
With these choice of parameters, one can in particular expect preservation of mass, mean velocity and temperature. We now consider the symmetrically weighted Hermite functions:

ψ k = H k e -x 2 /2
. By definition of the Hermite polynomials, we have the relations:

(3.12)

H k = 2vH k-1 -2(k -1)H k-2 H k = 2kH k-1 .
Therefore, the Hermite functions ψ k satisfy:

(3.13) Using these relations, one can explicitly compute (3.14)

ψ k = 2vψ k-1 -2(k -1)ψ k-2 , ψ k = kψ k-1 - 1 2 ψ k+1 , ψ k = k(k -1)ψ k-2 -k + 1 2 ψ k + 1 4 ψ k+2 .
L F P (ψ k ) = ψ k + (v -µ)ψ k + T ψ k = k(k -1)(1 + T )ψ k-2 - µ 2 ψ k-1 + -kT - (T -1) 2 ψ k + µ 2 ψ k+1 + 1 4 (T -1) ψ k+2 .
We can now proceed to use a Galerkin method to solve (3.7). We denote by P N and P c N the standard and constrained Hermite function approximation. We want to find

f N ∈ S N solution to (3.15)    ∂ ∂t f N = P N (L F P (f N )) , f N (0, v) = P N (f 0 )(v).
The moment constrained problem writes

(3.16)    ∂ ∂t f c N = L c F P,N (f c N ), f N (0, v) = P c N (f 0 )(v)
, where L c F P,N is solution to (3.17)

L c F P,N (f ) = argmin g N -L F P (f ) 2 L 2
ω : g N ∈ S N , g N , x q = 0, q = 0, 1, 2, 3 . Let us emphasize that (3.17) corresponds to (2.19) where we ensure the first three moments of L F P to be zero. For the time stepping method, we utilize a classical forth order Runge-Kutta (RK4) method with fixed time step ∆t = 10 -4 . In Figure 6 one can observe a very good agreement between the two solutions as well as a trend towards the steady state f ∞ . This observation suggests that the constrained approximation may be able to finely capture long-time properties of the solution.

Spectral accuracy and conservations. Let us now investigate the accuracy of the method. In Figure 7 we set a final time T f = 0.1 and compare the solution to a reference one computed with N = 32 modes. We do the same comparison for the first three moments. A first observation is that the standard method does exhibit spectral accuracy both on the solution and on the first three moments. This therefore falls within the scope of Theorem (2.5) and the constrained method is indeed spectrally accurate. Note that in this particular case, the mean velocity is 0, hence the machine accuracy in the bottom left panel of Figure 7. Long-time behaviour. Let us conclude this test by investigating the long time behaviour of the solution observed in Figure 6. In Figure 8 The top left panel shows the evolution of the norms f N (t n ) -f ∞ 2 and f c N (t n ) -f ∞ 2 while the remaining panels show the variation of the moments |ρ n -ρ 0 |, |µ n -µ 0 | and |T n -T 0 |. We observe that the constrained method indeed preserves the first three moments up to machine accuracy. As a consequence, the long-time behaviour should be better approximated. The standard approximation does seem to converge towards the steady state but ultimately, accumulation of errors on the moments overcomes the dynamic of the solution, pushing it away from the equilibrium. On the contrary, our proposed method does not exhibit this re-bounce and converges to a plateau that decreases as one increases the number of modes. One can mention that equilibrium preserving techniques [START_REF] Filbet | On steady-state preserving spectral methods for homogeneous Boltzmann equations[END_REF][START_REF] Pareschi | On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation[END_REF] have been developed to overcome this saturation but these are beyond the scope of this article.

Test 3: A model of Opinion formation.

For this second test, we turn our attention to the following opinion model [START_REF] Furioli | Fokker-planck equations in the modeling of socio-economic phenomena[END_REF] where v ∈ [-1, 1]:

(3.18)    ∂ ∂t g = λ 2 ∂ vv (1 -v 2 )g + ∂ v ((v -m)g) := L Op g, g(0, v) = g 0 (v).
In this case, the unknown g(t, v) describes the distribution of individuals at time t with an opinion trait v. The modelling parameters are chosen so that |m| < 1 and λ < 1 + |m| to prevent blow-ups as v → ±1. In addition, we supplement this equation . with a no flux boundary condition to ensure the conservation of mass:

(3.19) λ 2 ∂ v (1 -v 2 )g + (v -m)g v=±1 = 0.
In particular, one can show that this condition in fact boils down to the homogeneous Dirichlet boundary condition g(t, ±1) = 0. This equation also admits a steady state given by

(3.20) g ∞ (v) = c m,λ (1 + v) 1+m λ -1 (1 -v) 1-m λ -1
where the constant c m,λ is for normalization. In order to approximate solutions to (3.18) we want to consider general Jacobi polynomials p α,β k k=0,...,N associated to

the weight ω(v) = (1 -v) α (1 + v) β .
However, this polynomial basis does not satisfy the homogeneous Dirichlet boundary conditions. To solve this issue, we construct a new polynomial basis obtained from the original Jacobi polynomials (see e.g. [START_REF] Shen | Spectral Methods: Algorithms, Analysis and Applications[END_REF]). For α, β > -1, let

(3.21) ζ k = p α,β k + a k p α,β k+1 + b k p α,β
k+2 , k = 0, . . . , N -2 where the constants a k and b k are solution to the linear system:

(3.22) ζ k (-1) = 0 ζ k (1) = 0.
This new polynomial family now satisfies the boundary conditions but it is, by construction, no longer orthogonal. Since orthogonality is a key ingredient of the conservative spectral method, we need to apply a Gram-Schmidt algorithm to obtain an orthonormal basis. Note that in practice the process of orthonormalization may induce an accumulation of machine errors that can lead to a significant loss in orthogonality for large number of modes. We can now expand g in the basis of polynomials (ζ k ) k=0,...,N -2 :

(3.23)

g(t, v) = N -2 k=0 ĝk (t)ζ k (v),
and then proceed as before to solve (3.18) using a Galerkin method. We define the standard and constrained problems as

(3.24)    ∂ ∂t g N = P N (L Op (g N )) , g N (0, v) = P N (g 0 )(v), and
(3.25)    ∂ ∂t g c N = L c Op,N (g c N ), g N (0, v) = P c
N (g 0 )(v). Since (3.18) preserves only the first moment, we define L c Op,N as the solution to

(3.26) L c Op,N (g) = argmin g N -L Op (g) 2 L 2 ω : g N ∈ S N , g N , 1 = 0 .
The time steeping is again achieved using an RK4 time integrator with time step ∆t = 10 -4 . Before presenting numerical results, let us mention that the space spanned by the newly constructed basis does not contain the monomials v q , q = 0, 1, 2, . . . anymore. As a consequence, in regards of (2.17), even if one considers Legendre polynomials (ω = 1) to construct the new basis, one does not expect exact conservation of moments. However, as will be shown through numerical experiments, one still recovers spectral accuracy on the moments for the standard method which ensures spectral accuracy on the constrained solution. In regards of this discussion, we now present only the choice of Legendre polynomials for the construction of the new basis (ζ k ) k=0,...,N -2 . Note however that classical families such as Chebyshev first and second kind provide similar results. As an initial data, we take (3.27) g 0 (v) = c 0 (1 + v) 12 (1 -v) 6 + (1 + v) 13 (1 -v) 25 , where c 0 is for normalizing the first moment. Then we set m = 0 and λ = 0.1 in the definition of L Op in (3.18), therefore expecting an equilibrium of the form

g ∞ (v) = c ∞ (1 -v 2 ) 9 .
Figure 9 shows the evolution of the approximate distributions g N and g c N . As in the previous case, we observe a very good agreement between the standard and constrained method. Moreover, the solutions also appear to converge towards the steady state.

Spectral accuracy and conservations. Let us now investigate the accuracy of the method and its mass conservation properties. Indeed, at the continuous level, equation (3.18) along with no flux boundary conditions preserves only the 0th order moment. In the next study, we set a final time T f = 0.1 and compare the approximation error between the standard an constrained method. We observe in Figure 10 that we again fall into the framework of the convergence theorem 2.5 and the constrained method behaves as predicted. Long-time behaviour. Let us conclude this test by looking at the long time behaviour of the approximated solution to (3.18). In order to quantify the observation made on Figure 9, we show in Figure 11 the evolution of the norms g N -g ∞ 2 and g c N -g ∞ 2 as well as the evolution of the mass variation |m 0 0 -m 0 (t n )|. Similarly as before, we observe that the conservation of moments allows to stabilize the long time dynamic of the solution.

Test 4:

A Call center service time model. This final test deals with a call center service time model introduced in [START_REF] Gualandi | Call center service times are lognormal: A Fokker-Planck description[END_REF]:

(3.28)      ∂ ∂t h = λ 2 ∂ vv v 2 h + γ 2 ∂ v v ln v v L h := L CC h, h(0, v) = h 0 (v).
In this context, the distribution density h(t, v) describes the service time of agents in a call center. The constants 0 < γ < 1, λ > 0 and v L > 0 are modelling parameters.

In particular, the latter corresponds to an ideal time for agents to complete their task.

To supplement this equation, we consider a no-flux boundary condition It is clear that this condition is automatically satisfied by (3.28). It can also be shown that (3.28) admits an equilibrium distribution of the form:

(3.29) λ 2 ∂ v v 2 h + γ 2 ln v v L h v=0 = 0.
(3.30) h ∞ (v) = 1 √ 2πσv exp - (ln(v) -µ) 2 2σ
where σ = λ γ and µ = ln(v L ) -σ. As in Test 1, in order to deal with integrability at infinity, we want to consider the symmetrically weighted Laguerre functions :

ξ k = L k ω 1 2 .
In addition, to deal with the poor approximation of the steady state observed in Figure 5 we consider a micro-macro approach. Since the equilibrium is known, we can decompose the unknown as

(3.31) h = h ∞ + h,
where h is the perturbation that can take negative values. Since (3.28) is linear in h, and by definition of the steady state, the perturbation also solves (3.28). The only difference is the initial data.

     ∂ ∂t h = λ 2 ∂ vv v 2 h + γ 2 ∂ v v ln v v L h , h(0, v) = h 0 (v) -h ∞ (v).
From this procedure, we expect to observe that h converges towards 0 for long time.

Consequently, the long time behaviour of the solution should be better approximated since 0 now belongs to the approximation space. However, one cannot say anything about the short time behaviour as the initial data could still be poorly approximated.

As previously, we can then define the standard and constrained problems on the perturbation as The time stepping is yet again achieved through a RK4 method with time step ∆t = 10 -4 . Let us now fix the parameters λ = 0.5, γ = 0.9 and v L = 40 in (3.28). As an initial data, we consider

(3.32)    ∂ ∂t hN = P N L CC ( hN ) , hN (0, v) = P N ( h0 -h ∞ )(v), and
(3.33)    ∂ ∂t hc N = L c CC,N ( hc N ), hN (0, v) = P c N ( h0 -h ∞ )(v).
(3.35) h 0 (v) = (x 3 -2x + sin(x))e -x .
This function is actually well approximated by Laguerre functions. We show on Figure 12 the evolution of the standard and constrained distributions. A first observation is that there is again a good agreement between the two mathod. In addition, as expected from the micro-macro approach, the steady state is well approximated. Accuracy and conservation. To assess the spectral accuracy of the method, we consider a final time T f = 0.1 along with a time step ∆t = 10 -4 for the time integration. We present on Figure 13 the study of the convergence of the method. As expected, in short times, the distribution is not very well approximated in our chosen basis. As a consequence, we are only able to recover a convergence that is quite slow. However, the up-side is that the mass is still exactly preserved by the constrained approximation. Now, contrary to previous tests, it is important to note that the standard method is not spectrally accurate on the mass. Long-time behaviour. To conclude this section, and as in previous tests, we investigate the longtime behaviour of the solution through the norms h n -h ∞ 2 and h c n -h ∞ 2 . We observe in Figure 14 that the micro-macro approach allows to approximate really well the steady state with both methods. In addition, the mass variation induced by the standard approach decays exponentially fast towards 0 therefore ensuring that it also converges towards h ∞ albeit more slowly than the conservative approach.

Conclusions.

We extended and analyzed a class Galerkin spectral methods for PDEs based on general families of orthogonal functions that can preserve the moments of the solution. Leading examples of such problems arise in kinetic and mean-field theory where preservation of the moments of the distribution function is of paramount importance to describe correctly the long-time behavior. The methods were designed using an L 2 best constrained approximation formalism in the space of orthogonal polynomials. Thanks to this setting, the resulting methods allow the spectral accuracy property of the solution to be maintained for the conservative polynomial series. The new approximation was then used to derive moment preserving methods for several Fokker-Planck equations in distinct physical domains and by adopting different standard Galerkin projections, including the use of micro-macro decompositions.

Given its generality, the method admits numerous extensions. Among the most interesting are certainly the construction of spectrally accurate and conservative methods for the Vlasov equation. Another interesting direction is the application to kinetic models in the socio-economic domain where equilibrium states are often unknown and thanks to the present approach can be computed with spectral accuracy.

Figure 1 .

 1 Figure 1. Constrained Legendre approximation for (3.1)

Figure 2 .

 2 Figure 2. Constrained approximations for Chebyshev 1 st kind (Top Left), Chebyshev 2 nd kind (Top Right) and Jacobi α = 1, β = -1/2 (Bottom) for (3.1)

Figure 3 .

 3 Figure 3. Constrained Hermite approximation for (3.3)

Figure 4 .

 4 Figure 4. Constrained Laguerre approximation for (3.5)

Figure 5 .

 5 Figure 5. Error Analysis: Solution and first Moment using Laguerre functions for (3.6)

Figure 6 .

 6 Figure 6. Fokker-Planck model: Snapshots of the distribution at t = 0, 0.1 and 5, N = 32.

10 1 2 × 10 1 3 × 10 1 4 × 10 1 6 × 10 1 NFigure 7 .

 1013117 Figure 7. Fokker-Planck model: Approximation error on the solution (Top Left) and absolute error on the mass (Top Right), mean velocity (Bottom Left) and temperature (Bottom Right) at T f = 0.1.

Figure 8 .

 8 Figure 8. Fokker Planck model: Convergence towards the steady state (Top Left) and absolute error on the mass (Top Right), mean velocity (Bottom Left) and temperature (Bottom Right)

Figure 9 .

 9 Figure 9. Opinion model: Snapshots of the distribution at times t = 0, 0.4 and 5, N = 24.

Figure 10 .

 10 Figure 10. Opinion model: Approximation error on the solution (Left) and on the 0 th order moment (Right) at T f = 0.1

Figure 11 .

 11 Figure 11. Opinion model: Convergence towards the steady state (Left) and mass variation (Right)

Figure 12 .

 12 Figure 12. Call Center model: Snapshots of the distributions at times t = 0, 2 and 5, N = 32.

Figure 13 . 2 L 2 ω

 1322 Figure 13. Call Center model: Approximation error on the solution (Left) and on the 0 th order moment (Right) at T f = 0.1.

Figure 14 .

 14 Figure 14. Call Center model: Convergence towards the steady state (Left), and mass variation (Right)

Table 1

 1 Families of orthogonal polynomials

Table 2

 2 Error Analysis: Legendre approximation for(3.1)

	9.004e-03	2.429e-15	2.387e-15	2.387e-15	2.331e-15
	16 2.197e-05	1.388e-17	0	0	0
	32 1.050e-13	8.327e-17	1.110e-16	0	0

Table 3

 3 Error Analysis: Standard vs. Conservative with Chebyshev 1 st kind approximation of(3.1)

Table 4

 4 Error Analysis: Standard vs. Conservative with Chebyshev 2 nd kind approximation of(3.1)

Table 5

 5 Error Analysis: Standard vs. Conservative with Jacobi (α = 1, β = -

	1.787e-01	6.939e-17	0	2.776e-17	5.551e-17
	16 6.695e-05	1.110e-16	0	1.110e-16	5.551e-17
	32 1.868e-12	1.388e-16	5.551e-17	5.551e-17	0

Table 6

 6 Error Analysis: Standard vs. Conservative Hermite function approximation for(3.3)

	4.324e-01	0	1.776e-15	3.553e-15	2.842e-14
	16 2.879e-03	2.220e-16	1.776e-15	0	0
	32 5.055e-09	2.220e-16	1.776e-15	0	0

Table 7

 7 Error Analysis: Standard vs. Conservative using Laguerre Function approximation for(3.5)

Acknowledgment. This work has been written within the activities of GNCS and GNFM groups of INdAM (Italian National Institute of High Mathematics). LP has been supported by the Royal Society under the Wolfson Fellowship "Uncertainty quantification, data-driven simulations and learning of multiscale complex systems governed by PDEs". The partial support by ICSC -Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing, funded by European Union -NextGenerationEU and by MUR PRIN 2022 Project No. 2022KKJP4X "Advanced numerical methods for time dependent parametric partial differential equations with applications" is also acknowledged. T.L. would like to thanks Thomas Rey for fruitful discussions on the implementation of the method. T.L. acknowledges support from Labex CEMPI (ANR-11-LABX-0007-01) and the MSCA DN-2022 program


Mauricio Lopez-Obando

Ruan De Villiers

Beate Hoffmann

Linnan Ma

Alexandre A De Saint

Jens Kossmann

Yoan Coudert

C Jill Harrison

Catherine Rameau

Paul Hills
email: phills@sun.ac.za

Sandrine Bonhomme
email: sandrine.bonhomme@inra.fr

Physcomitrella patens MAX2 characterization suggests an ancient role for this F-box protein in photomorphogenesis rather than strigolactone signalling

Keywords: Bryophyte, Moss, Hormone signaling, Strigolactone, Photomorphogenesis, F-box protein

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

Introduction

Strigolactones (SLs) are plant hormones that were first identified as root-exudate products, exogenously indicating the vicinity of a host plant to parasitic plants such as Striga [START_REF] Cook | Germination of Witchweed (Striga lutea Lour.): Isolation and Properties of a Potent Stimulant[END_REF] and Arbuscular Mycorrhizal (AM) fungi [START_REF] Akiyama | Plant sesquiterpenes induce hyphal branching in arbuscular mycorrhizal fungi[END_REF]. Roles for SLs in a range of endogenous developmental processes including shoot branching and root architecture were more recently described [START_REF] Waldie | Strigolactones and the control of plant development: lessons from shoot branching[END_REF][START_REF] Lopez-Obando | Strigolactone biosynthesis and signaling in plant development[END_REF]. SLs are present in most land plants [START_REF] Xie | The strigolactone story[END_REF]) and the charophyte algal sister lineage to land plants [START_REF] Delaux | Origin of strigolactones in the green lineage[END_REF], but signaling pathways are expanded in land plants relative to charophytes [START_REF] Bowman | Insights into Land Plant Evolution Garnered from the Marchantia polymorpha Genome[END_REF]. Therefore, SLs are key candidate facilitators for plant terrestrialization 480 million years (MY) ago [START_REF] Bowman | Insights into Land Plant Evolution Garnered from the Marchantia polymorpha Genome[END_REF]. SL synthesis and signaling pathways have well characterized roles in branching in seed plants such as pea, Arabidopsis, Petunia and rice [START_REF] Al-Babili | Strigolactones, a novel carotenoid-derived plant hormone[END_REF][START_REF] Waters | Strigolactone Signaling and Evolution[END_REF]. Genes cloned from SL-deficient mutants have identified synthesis steps requiring a carotenoid isomerase DWARF27 (D27), two CAROTENOID CLEAVAGE DIOXYGENASES (CCD7 and CCD8), at least one Cytochrome P450 MORE AXILLARY GROWTH 1 (MAX1) [START_REF] Al-Babili | Strigolactones, a novel carotenoid-derived plant hormone[END_REF], and the oxidoreductase-like enzyme LATERAL BRANCHING OXIDOREDUCTASE (LBO) [START_REF] Brewer | LATERAL BRANCHING OXIDOREDUCTASE acts in the final stages of strigolactone biosynthesis in Arabidopsis[END_REF]. In parallel, the study of SL-insensitive mutants has implicated several gene families in SL signaling. The first step of SL signaling is hormone perception, and this requires an α/β hydrolase enzyme, DECREASED APICAL DOMINANCE2/DWARF14/RAMOSUS3 (DAD2/D14/RMS3), that has been shown to interact with and cleave SL molecules in vitro [START_REF] Hamiaux | DAD2 is an alpha/beta hydrolase likely to be involved in the perception of the plant branching hormone, strigolactone[END_REF][START_REF] Nakamura | Molecular mechanism of strigolactone perception by DWARF14[END_REF][START_REF] De Saint Germain | An histidine covalent receptor and butenolide complex mediates strigolactone perception[END_REF][START_REF] Yao | DWARF14 is a non-canonical hormone receptor for strigolactone[END_REF].

Petunia DAD2 and rice D14 have been shown to interact in the presence of SLs with the Fbox proteins MORE AXILLARY GROWTH 2A (PhMAX2A) and DWARF3 (D3), which are orthologous to Arabidopsis MAX2 [START_REF] Hamiaux | DAD2 is an alpha/beta hydrolase likely to be involved in the perception of the plant branching hormone, strigolactone[END_REF][START_REF] Jiang | DWARF 53 acts as a repressor of strigolactone signalling in rice[END_REF][START_REF] Zhou | D14-SCF(D3)-dependent degradation of D53 regulates strigolactone signalling[END_REF][START_REF] Zhao | DWARF3 Participates In An SCF Complex And Associates With DWARF14 To Suppress Rice Shoot Branching[END_REF]). The current model for SL signaling mostly builds on studies of shoot branching in angiosperms, proposing that SL perception by D14/AtD14 induces the recognition of specific target proteins by an SCF D3/MAX2 complex. This process leads to ubiquitination and proteasome-mediated degradation of targets in a similar process to processes described for other plant hormones including gibberellins [START_REF] Lopez-Obando | Strigolactone biosynthesis and signaling in plant development[END_REF][START_REF] Waters | Strigolactone Signaling and Evolution[END_REF]. Whilst roles for MAX2 in SL signaling were first described around 15 years ago [START_REF] Stirnberg | MAX1 and MAX2 control shoot lateral branching in Arabidopsis[END_REF][START_REF] Johnson | Branching genes are conserved across species. Genes controlling a novel signal in pea are coregulated by other long-distance signals[END_REF][START_REF] Stirnberg | MAX2 participates in an SCF complex which acts locally at the node to suppress shoot branching[END_REF][START_REF] Shen | MAX2 Affects Multiple Hormones to Promote Photomorphogenesis[END_REF], the identification of DWARF53 (D53)/SUPPRESSOR OF MAX2-LIKE (SMXL) proteins as putative targets of the SCF D3/MAX2 complex is more recent [START_REF] Jiang | DWARF 53 acts as a repressor of strigolactone signalling in rice[END_REF][START_REF] Stanga | SUPPRESSOR OF MORE AXILLARY GROWTH2 1 controls seed germination and seedling development in Arabidopsis[END_REF][START_REF] Zhou | D14-SCF(D3)-dependent degradation of D53 regulates strigolactone signalling[END_REF][START_REF] Soundappan | SMAX1-LIKE/D53 Family Members Enable Distinct MAX2-Dependent Responses to Strigolactones and Karrikins in Arabidopsis[END_REF][START_REF] Wang | Strigolactone Signaling in Arabidopsis Regulates Shoot Development by Targeting D53-Like SMXL Repressor Proteins for Ubiquitination and Degradation[END_REF].

Arabidopsis max2 mutants were also isolated in early genetic screens for delayed darkinduced senescence [START_REF] Woo | ORE9, an F-box protein that regulates leaf senescence in Arabidopsis[END_REF], and light hyposensitivity [START_REF] Shen | The F-box protein MAX2 functions as a positive regulator of photomorphogenesis in Arabidopsis[END_REF]). Whereas the involvement of SLs in leaf senescence has been confirmed [START_REF] Yamada | Strigolactone signaling regulates rice leaf senescence in response to a phosphate deficiency[END_REF][START_REF] Ueda | Strigolactone Regulates Leaf Senescence in Concert with Ethylene in Arabidopsis[END_REF], the photomorphogenesis phenotype of max2 mutants appears independent of the SL pathway [START_REF] Shen | MAX2 Affects Multiple Hormones to Promote Photomorphogenesis[END_REF]. Furthermore, a requirement for MAX2 in other butenolide signaling pathways was demonstrated by the isolation of max2 mutants in a genetic screen for insensitivity to smoke-derived karrikins [START_REF] Nelson | F-box protein MAX2 has dual roles in karrikin and strigolactone signaling in Arabidopsis thaliana[END_REF][START_REF] Waters | Specialisation within the DWARF14 protein family confers distinct responses to karrikins and strigolactones in Arabidopsis[END_REF]. Karrikins induce Arabidopsis seed germination and affect seedling photomorphogenesis through a similar but distinct signaling pathway to SLs [START_REF] Scaffidi | Strigolactone Hormones and Their Stereoisomers Signal through Two Related Receptor Proteins to Induce Different Physiological Responses in Arabidopsis[END_REF][START_REF] Waters | The karrikin response system of Arabidopsis[END_REF], and an α/β hydrolase protein closely related to D14, KARRIKIN INSENSITIVE 2 (KAI2), is required for the response to karrikins [START_REF] Waters | Specialisation within the DWARF14 protein family confers distinct responses to karrikins and strigolactones in Arabidopsis[END_REF]). Whilst karrikins have not been detected in plants, KAI2 is the presumed receptor of an unknown plant-produced KAI2-ligand (KL) [START_REF] Scaffidi | Carlactone-independent seedling morphogenesis in Arabidopsis[END_REF][START_REF] Waters | KAI2-and MAX2-mediated responses to karrikins and strigolactones are largely independent of HY5 in Arabidopsis seedlings[END_REF][START_REF] Conn | Evidence that KARRIKIN-INSENSITIVE2 (KAI2) Receptors may Perceive an Unknown Signal that is not Karrikin or Strigolactone[END_REF]. Thus, the MAX2 F-box protein is involved in several signaling pathways apart from strigolactone signaling.

Although several components of the strigolactone synthesis and signaling pathways are shared amongst land plants, their roles in early diverging land plant lineages and contribution to plant evolution are unknown [START_REF] Bowman | Insights into Land Plant Evolution Garnered from the Marchantia polymorpha Genome[END_REF]. We addressed this evolutionary question using the moss Physcomitrella patens (P. patens) as a model representing an ancient divergence in land plant evolution. Whilst CCD7 and CCD8 orthologues are found in the P. patens genome, a true orthologue of MAX1 is absent [START_REF] Delaux | Origin of strigolactones in the green lineage[END_REF]. We previously generated Ppccd8 SL-deficient mutants and demonstrated SL-functions in repressing radial plant growth and gametophore branching [START_REF] Proust | Strigolactones regulate protonema branching and act as a quorum sensing-like signal in the moss Physcomitrella patens[END_REF][START_REF] Hoffmann | Strigolactones inhibit caulonema elongation and cell division in the moss Physcomitrella patens[END_REF][START_REF] Coudert | Three ancient hormonal cues co-ordinate shoot branching in a moss[END_REF]. Consideration of signaling pathways has revealed no true orthologue of the D14 receptor gene, but there are 13 PpKAI2-LIKE genes that are closer to the KAI2 α/β hydrolase clade in P. patens [START_REF] Delaux | Origin of strigolactones in the green lineage[END_REF]Lopez-Obando et al. 2016a). Phylogenetic analyses have also identified a single putative homologue for the F-box protein gene MAX2 [START_REF] Delaux | Origin of strigolactones in the green lineage[END_REF]) and three to four PpSMXL genes [START_REF] Zhou | D14-SCF(D3)-dependent degradation of D53 regulates strigolactone signalling[END_REF]).

Here we wished to explore SL signaling pathway evolution, and we focused on the role of the P. patens MAX2 gene, testing whether PpMAX2 is involved in the SL response. We also generated PpMAX2 KO mutants and characterized their response to SL and red light at the phenotypic and molecular level. Our data indicate that similarly to MAX2 from Arabidopsis, PpMAX2 is involved in photomorphogenesis. However, PpMAX2 is probably not involved in generating a SL response.

Materials and Methods

P. patens growth conditions

The Gransden wild-type strain of P. patens was used and grown as previously described in a culture room at 24°C (day)/22°C (night) with a light regime of 16 h light/8 h darkness and a quantum irradiance of 80 µE m -2 s -1 [START_REF] Proust | Strigolactones regulate protonema branching and act as a quorum sensing-like signal in the moss Physcomitrella patens[END_REF]Lopez-Obando et al. 2016b). For phenotypic analysis, fragmented protonemal tissue was grown for 7 days on PP-NH 4 medium (=PP-NO 3 medium supplemented with 2.7 mM NH 4 -tartrate) then transferred to PP-NO 3 medium [START_REF] Ashton | Analysis of gametophytic development in the moss, Physcomitrella patens , using auxin and cytokinin resistant mutants[END_REF][START_REF] Hoffmann | Strigolactones inhibit caulonema elongation and cell division in the moss Physcomitrella patens[END_REF]. Sporogenesis was induced in Magenta vessels in which 21/28-day-old plants were grown on soil plugs (or PP-NO 3 medium) for 10 days as above and then transferred to a growth chamber at 15°C with 8 h of light per day and a quantum irradiance of 15 µE m -2 s -1 and rinsed once a week with sterile tap water till capsule maturity (after 2 to 3 months). For red light experiments, plants were grown on PP-NO 3 medium, at 24°C with a continuous red-light regime of 46 µE m -2 s -1 .

Generation of Ppmax 2-1, Ppmax2-2 and Ppccd8-Ppmax2 mutants

Moss protoplasts were obtained and transformed as described previously [START_REF] Trouiller | MSH2 is essential for the preservation of genome integrity and prevents homeologous recombination in the moss Physcomitrella patens[END_REF]. For the Ppmax2-1 mutant, a 735 bp PpMAX2 genomic 3' CDS flanking sequence fragment was cloned in the pBHRF vector [START_REF] Thelander | The moss genes PpSKI1 and PpSKI2 encode nuclear SnRK1 interacting proteins with homologues in vascular plants[END_REF], digested with NarI and HpaI. Next, an 886 bp PpMAX2 genomic 5' CDS flanking sequence fragment was inserted into AvrII/XhoI sites of the pBHRF vector carrying the 3' CDS flanking sequence (PpMAX2-KO1 construct). For the Ppmax2-2 mutant, a 1170 bp 5' CDS flanking sequence fragment and a 1170 bp 3' CDS flanking sequence fragment were amplified and subcloned into pJET1.2 vector (Fermentas) with a Geneticin/G418 resistance cassette from pMBL11a plasmid (Knight et al. 2002) (PpMAX2-KO2 construct). WT protoplasts were transformed with the PpMAX2-KO1 or the PpMAX2-KO2 constructs, and transformants were selected on 30 mg l - 1 Hygromycin B or 50 mg l -1 G418, respectively. For the Ppccd8-Ppmax2 double mutant, protoplasts from the single Ppccd8 mutant were transformed with a construct carrying the same flanking sequences as PpMAX2-KO2, subcloned into the pJET1.2 vector, with a Hygromycin resistance cassette from pMBLH8a [START_REF] Knight | Moss Gene Technology[END_REF]. Transformants were selected on 30 mg l -1 Hygromycin B. Stable transformants of the PpMAX2 gene were confirmed by PCR using specific primers (Fig. S1 and Table S1).

Protoplast assays

Protoplasts were isolated as described in [START_REF] Trouiller | MSH2 is essential for the preservation of genome integrity and prevents homeologous recombination in the moss Physcomitrella patens[END_REF], counted, and kept overnight in the dark at 24°C, in liquid 8.5 % mannitol PP-NH 4 . The next day, drops of 750 protoplasts gently mixed with 0.7% top agar (v/v) were transferred on 8.5 % mannitol PP-NH 4 plates, 6 with various (0 to 3 µM) concentrations of (±)-GR24 for 5 days, prior to transfer onto plates without mannitol (but with (±)-GR24).

Molecular cloning and subcellular protein localization

Generating the PPpMAX2:GUS lines

The ZmUbi-1 promoter was eliminated from the pMP1300 vector

[http://labs.biology.ucsd.edu/estelle/Moss_files/pMP1300-K108N+Ubi-GW-GUS.gb] by PCR amplification using primers Ubi-pr and Ubi-exp (Table S1) and the plasmid backbone was self-ligated and renamed pMP1301. A 1961 bp promoter region for PpMAX2 was amplified from P. patens gDNA using primers PPpMAX2_F and PPpMAX2_R (Table S1).

The product was purified and cloned into the vector pCR ® 8/GW/TOPO ® (Life Technologies ® , USA-CA). An LR-clonase reaction between the pMP1301 and pCR8::PPpMAX2 plasmids yielded PPpMAX2:GUS, which was used to transform WT P. patens. A stable G418 resistant line was used for subsequent histochemical analysis to determine GUS localisation.

Generating the ZmUbi:gfp:PpMAX2 lines

Single-stranded P. patens cDNA was used as template to amplify the PpMAX2 coding sequence using the PpMAX2_F and PpMAX2_R primer set (Table S1). The 2493 bp product was cloned into the pCR ® 8/GW/TOPO ® . pCR8::PpMAX2 was recombined with the pMP1335 vector [http://labs.biology.ucsd.edu/estelle/Moss_files/ pK108N+Ubi-mGFP6-GW.gb] to get pMP1335::PpMAX2. pMP1335::PpMAX2 was linearised by SfiI digestion and transformed into WT P. patens. Stable G418 resistant lines were screened for insertion by PCR using the GFP_F and PpMAX2_R primers (Table S1). For one of these positive GFP:PpMAX2 lines the localisation of the recombinant GFP:PpMAX2 was determined by visualising protonemal tissue on a confocal microscope (Carl Zeiss Confocal LSM 780 Elyra with SR-SIM superresolution plasform). For analysis, protonemal tissue was fixed in 4%

(v/v) formaldehyde for 10 min and then stained with a 0.0125% (w/v) Hoescht33342 solution.

Images were analysed by the ZEN 2012 (blue edition) software package (Carl Zeiss, Germany).

Arabidopsis complementation and phenotyping experiments

Constructs in which the PpMAX2 coding sequence was constitutively expressed alone or in a GFP fusion were introduced into the max2-3 (N592836) T-DNA insertion mutant. The pUbi10 promoter, corresponding to the first 634 base pairs immediately upstream of the ubiquitin-10 gene from Arabidopsis (At4g05320) was used to drive PpMAX2 expression [START_REF] Grefen | A ubiquitin-10 promoter-based vector set for fluorescent protein tagging facilitates temporal stability and native protein distribution in transient and stable expression studies[END_REF]. Expression of the PpMAX2 mRNA and/or fluorescence of the GFP were checked in the corresponding transformed lines (Fig. S2). Results for two independent lines carrying each PpMAX2 construct are shown. Hypocotyl length under low fluence experiments were carried out as previously described [START_REF] De Saint Germain | An histidine covalent receptor and butenolide complex mediates strigolactone perception[END_REF].

RNA extraction and gene expression analyses

Gene expression analyses were done by reverse-transcription quantitative PCR (RT-qPCR) as previously described [START_REF] Hoffmann | Strigolactones inhibit caulonema elongation and cell division in the moss Physcomitrella patens[END_REF]Lopez-Obando et al. 2016a), with primers listed in Table S1.

Statistical analyses

For statistical analyses, ANOVA and Kruskal-Wallis tests were used (R Commander version 1.7-3).

Results

Physcomitrella patens contains a single MAX2 homologue

The single P. patens MAX2 homologue (Pp3c17_1180v3) was named PpMAX2 [START_REF] Delaux | Origin of strigolactones in the green lineage[END_REF][START_REF] Li | OaMAX2 of Orobanche aegyptiaca and Arabidopsis AtMAX2 share conserved functions in both development and drought responses[END_REF]. Phylogenetic analysis of full-length predicted MAX2 proteins indicated that, in contrast to previously published phylogenies that used a higher number of EST and full-length sequences, MAX2 from P. patens, Marchantia polymorpha and Selaginella moellendorffii formed a separate clade to seed plant proteins (Fig. S3a) [START_REF] Delaux | Origin of strigolactones in the green lineage[END_REF][START_REF] Bythell-Douglas | Evolution of strigolactone receptors by gradual neofunctionalization of KAI2 paralogues[END_REF]. Thus, the precise relationships between MAX2 homologues in vascular plants and those in non-vascular plants remain ambiguous. Nevertheless, the lack of any other close homologue in moss and the fact that MAX2 is present as a single copy gene in a large majority of plant genomes suggest that PpMAX2 is likely orthologous to AtMAX2.

PpMAX2 has no intron (Fig. S3b), and the predicted PpMAX2 protein is larger than vascular plant MAX2 proteins, containing C terminal insertions (Fig. S3c). Alignment of several predicted MAX2 protein sequences from vascular plants and bryophytes showed that PpMAX2 has a conserved F-box domain and similar LRR repeats composition to AtMAX2, with the exception that LRR13 is longer and consequently could not be modeled to existing Fbox structures in this region (Fig. S3c-d).

PpMAX2 is expressed in most cells, and PpMAX2 localizes to the nucleus

To characterize the expression profile of PpMAX2, a 1961 bp promoter fragment was cloned upstream of the GUS coding sequence and introduced into the neutral Pp108 locus of wildtype (WT) moss plants by targeted insertion [START_REF] Schaefer | Efficient gene targeting in the moss Physcomitrella patens[END_REF]. Expression of the GUS reporter was observed in protonemal filaments, but not at the very tips of caulonema (Fig. 1a). Expression was also observed in gametophore axes and leaves (Fig. 1a-d), with stronger staining in older leaves than in young leaves at the top of the gametophore (Fig. 1c,d). This pattern was corroborated by expression data from the P. patens eFP-Browser and Genevestigator public databases [START_REF] Hiss | Large-scale gene expression profiling data for the model moss Physcomitrella patens aid understanding of developmental progression, culture and stress conditions[END_REF][START_REF] Ortiz-Ramirez | A Transcriptome Atlas of Physcomitrella patens Provides Insights into the Evolution and Development of Land Plants[END_REF], that also indicated strong expression in sporophytes (Fig. S4). To determine the sub-cellular localization of PpMAX2, a GFP sequence was inserted in-frame and upstream of the PpMAX2 protein coding sequence, and introduced into WT plants. In accordance with knowledge of F-box protein function from flowering plants [START_REF] Stirnberg | MAX2 participates in an SCF complex which acts locally at the node to suppress shoot branching[END_REF]), PpMAX2 localized to nuclei in protonemal cells (Fig. 1e-h).

Ppmax2 mutants are small plants with few but large gametophores, and show converse phenotypes to Ppccd8 mutants

To determine the role of PpMAX2, Ppmax2 mutants were engineered by targeted replacement using two replacement strategies, and two independent knockout lines were obtained (Fig. S1a,b). Whilst regeneration efficiencies were very low relative to WT plants (not shown), both Ppmax2-1 and Ppmax2-2 mutants showed the same phenotype (Fig. 2a) with very few protonema and rapid differentiation of large gametophores relative to WT plants (Fig. 2a).

Ppmax2 mutants were small with limited growth after several weeks of culture (Fig. 2b,c).

When grown on soil plugs, plant diameter and the number of gametophores per plant were considerably reduced (Fig. 2c-d) and no sporophytes were found. We also tested the effect of the Ppmax2 mutation on gametophore branch patterning [START_REF] Coudert | Three ancient hormonal cues co-ordinate shoot branching in a moss[END_REF]. Although the size of the apical inhibition zone (the apical portion of gametophores devoid of branches) was slightly smaller and the overall branch number was slightly higher in Ppmax2-1 mutants than in WT plants, the spacing between branches was similar in both genotypes (Fig. 2e, Fig S5

Ppmax2 mutants can elicit a strigolactone response

As SL molecules are very difficult to quantify, we used an indirect approach to determine whether Ppmax2 overproduces SL and quantified expression of PpCCD7, a SL-responsive gene whose transcript levels decrease following (±)-GR24 application [START_REF] Proust | Strigolactones regulate protonema branching and act as a quorum sensing-like signal in the moss Physcomitrella patens[END_REF].

We used Ppccd8 mutant plants for these experiments as the SL response is easier to observe in mutants than in WT plants [START_REF] Hoffmann | Strigolactones inhibit caulonema elongation and cell division in the moss Physcomitrella patens[END_REF], and plants were transferred onto media containing no exudate, 1 µM (±)-GR24, WT, Ppccd8 or Ppmax2-1 exudate. PpCCD7 transcript levels were assayed 6 h after transfer (Fig. 3). Transfer of plants onto medium containing Ppccd8 exudate led to PpCCD7 transcript levels similar to those observed following transfer onto fresh medium. However, transfer onto medium containing Ppmax2-1 exudate led to a significant decrease of PpCCD7 transcript level, as was observed following transfer onto media containing (±)-GR24 or WT exudate (Fig. 3). Thus Ppmax2-1 exudate affects PpCCD7 transcript levels in a similar way to WT exudate, and Ppmax2-1 is likely to produce SL at similar levels to WT plants.

Ppmax2 mutants show growth responses to (±)-GR24 application

The response of Ppmax2 mutants to exogenously applied (±)-GR24 was tested and compared to the response of Ppccd8 mutants to identify any roles for PpMAX2 in SL signaling. These experiments were carried out using dark-grown caulonema where differences in growth are most pronounced [START_REF] Hoffmann | Strigolactones inhibit caulonema elongation and cell division in the moss Physcomitrella patens[END_REF], and plants were grown vertically so that caulonema extending with a negative gravitropism on the medium could be directly measured. Under these conditions both Ppmax2-1 and Ppccd8 mutant caulonema showed significant and dosedependent growth suppression (Fig. 4a). The relative decrease in caulonema length was greater in the Ppmax2-1 mutant than in Ppccd8 in all tested conditions (Fig. 4a). We also assayed SL responsiveness using a protoplast regeneration assay, and found that fewer plants regenerated in WT and Ppccd8 and Ppmax2 mutant plants following (±)-GR24 application, with the response being dose-dependent (Fig. 4b). Thus, Ppmax2 mutants can respond to SL application, and the response is pronounced in caulonema when mutants are grown in the dark, or in protoplasts regenerating in the light.

Ppmax2 mutants show transcriptional responses to (±)-GR24 application

Ppmax2 responses to SL were further analyzed using SL-responsive genes as molecular markers. The PpCCD7 transcript level was very low in Ppmax2-1 mutants relative to levels in Ppccd8 mutants and WT plants, and in contrast to a significant response observed in WT and Ppccd8, no significant decrease was noted in Ppmax2-1 mutants 6 h after transfer on medium with (±)-GR24, (Fig. 4c). We also measured transcript abundance of the PpKUF1LA gene (Pp3c2_34130v3.1), a moss homologue of Arabidopsis KAR-UP F-BOX1 (KUF1). KUF1

transcript levels are sensitive to (±)-GR24 application in Arabidopsis SL biosynthesis mutants [START_REF] Nelson | F-box protein MAX2 has dual roles in karrikin and strigolactone signaling in Arabidopsis thaliana[END_REF][START_REF] Waters | Specialisation within the DWARF14 protein family confers distinct responses to karrikins and strigolactones in Arabidopsis[END_REF][START_REF] Stanga | Functional redundancy in the control of seedling growth by the karrikin signaling pathway[END_REF]. PpKUF1LA (Pp3c2_34130v3.1) transcript levels increased 6 h after transfer on medium containing (±)-GR24 in both lightgrown WT and Ppccd8 mutants, but no response was detected in light-grown Ppmax2-1 mutants (Fig. S6a). As the bioassay suggested a Ppmax2-1 response to SL in the dark (Fig. 4a), we tested gene expression in dark grown plants. In contrast to WT and Ppccd8 mutant plants, no decrease of the PpCCD7 transcript level was observed in Ppmax2 mutants following transfer on (±)-GR24 (Fig. S6b). However, in dark-grown Ppmax2-1 plants, transcript levels of PpKUF1LA significantly increased following transfer on (±)-GR24 as in WT and Ppccd8 mutant plants (Fig. 4d). Thus Ppmax2 mutants remain responsive to exogenously-applied SL.

PpMAX2 expression is light responsive, and Ppmax2 has impaired light responses

To further investigate roles for PpMAX2 in light-regulated development, WT tissues were grown in the light for 7 days and then placed in the dark for 5 days prior to transfer into red light for increasing lengths of time. PpMAX2 transcript levels were higher in the dark than in the light (Fig. 5a). One hour of red light treatment led to a significant decrease in PpMAX2

transcript levels, and a 3-hour treatment resulted in a minimal expression level that was comparable to PpMAX2 expression levels in white light (Fig. 5a), thus PpMAX2 expression is light regulated. In white light, gametophores with the same number of leaves as WT, Ppccd8

or Ppmax2-2 mutant plants were taller in Ppmax2-2 mutants (Fig. 5b), showing an etiolation phenotype associated with light regulated development in other plants. To investigate a potential role for PpMAX2 in photomorphogenesis, Ppmax2-1 mutants were grown under continuous red light for 25 days. A strong etiolation phenotype was observed in Ppmax2-1 mutant gametophores but not in WT or Ppccd8 (Fig. 5c). We analyzed the transcript levels of genetic markers for light response in WT versus Ppmax2-1 mutant tissues. Ppmax2-1 mutant and WT plants were first grown in white light for 2 weeks and then transferred into the dark for 4 days prior to exposure to red light for increasing time periods (0.5h to 24h). After red light treatment, transcript levels of both ELONGATED HYPOCOTYL 5a (PpHY5a) and NADPH-PROTOCHLOROPHYLLIDE OXIDOREDUCTASE 1 (PpPOR1) were measured by RT-qPCR (Fig. 5d,e). The transcript levels of PpHY5a showed a transient and rapid increase with red light exposure in WT whilst remaining almost unchanged in Ppmax2-1. PpPOR1

transcript levels also increased with red light exposure in WT but remained lower in Ppmax2-1. The Ppmax2 mutants thus have an impaired response to red light.

Ppmax2 is epistatic to Ppccd8

To examine the genetic interaction between PpMAX2 and PpCCD8, Ppmax2 mutants were engineered in the Ppccd8 mutant background (Fig. S1c). Ppccd8-Ppmax2 double mutants had a phenotype similar to that of Ppmax2 mutants, with no additive effects on plant extension or gametophore development, indicating that the Ppmax2 mutation can override the effect of the Ppccd8 mutation (Fig. 6a,b). Whilst up-regulated PpCCD7 transcript levels are a genetic marker of Ppccd8 mutants, PpCCD7 expression was down-regulated in both Ppmax2-1 and Ppccd8-Ppmax2 double mutants (Fig. 6c), further suggesting that the Ppmax2 mutation is epistatic to the Ppccd8 (Fig. 6a,b).

PpMAX2 cannot complement Atmax2 mutant phenotypes

The data above suggest that roles for MAX2 in SL signaling are not conserved between P.

patens and Arabidopsis. To test this hypothesis, we heterologously expressed PpMAX2 in the Atmax2-3 mutant background, and used AtMAX2 as a control (Fig. 7,FigS2). Whilst AtMAX2 expression was able to restore WT plant phenotypes, PpMAX2 expression failed to complement the reduced height, higher branching and elongated hypocotyl under low fluence mutant phenotypes of Atmax2-3 (Fig. 7). Some partial complementation of the branching phenotype was observed in the lines where the PpMAX2 gene was fused to the GFP, with intermediate branching between WT and Atmax2-3 (Fig. 7c). However, as these lines are smaller in size than the Atmax2-3 mutant (Fig. 7a), one cannot conclude that these were complemented lines. Thus PpMAX2 and AtMAX2 are not functionally equivalent.

Discussion

Phylogenetic studies have suggested that SL biosynthesis and signaling pathways are conserved amongst land plants [START_REF] Proust | Strigolactones regulate protonema branching and act as a quorum sensing-like signal in the moss Physcomitrella patens[END_REF][START_REF] Delaux | Origin of strigolactones in the green lineage[END_REF][START_REF] Waters | Specialisation within the DWARF14 protein family confers distinct responses to karrikins and strigolactones in Arabidopsis[END_REF][START_REF] Bowman | Insights into Land Plant Evolution Garnered from the Marchantia polymorpha Genome[END_REF]. SLs or SL-like compounds are found in bryophytes and in the moss, P.

patens, both the PpCCD7 and PpCCD8 proteins have been shown to have in vitro enzymatic activities that are conserved with seed plants, indicating probable conservation of at least the early steps in SL biosynthesis [START_REF] Decker | Strigolactone biosynthesis is evolutionarily conserved, regulated by phosphate starvation and contributes to resistance against phytopathogenic fungi in a moss, Physcomitrella patens[END_REF] Arabidopsis, the max2 mutation is considerably more pleiotropic in comparison to the ccd8 (max4) mutation. SL-independent seed germination and photomorphogenesis phenotypes are observed in Atmax2 mutants [START_REF] Nelson | F-box protein MAX2 has dual roles in karrikin and strigolactone signaling in Arabidopsis thaliana[END_REF][START_REF] Shen | MAX2 Affects Multiple Hormones to Promote Photomorphogenesis[END_REF]. As both SL and the unidentified KAI2-Ligand (KL) signal through AtMAX2, the mutant combines the effect of alteration of several pathways. It is possible that in moss the Ppmax2 mutation is also highly pleiotropic and that the strong effect of the Ppmax2 mutation masks or overrides the Ppccd8 phenotype. This hypothesis is supported by the Ppccd8-Ppmax2 double mutant phenotype that resembles the Ppmax2 phenotype.

Several bioassays were used to test the SL response of the Ppmax2 mutant, and the Ppmax2 mutant is sensitive to (±)-GR24 applications under protoplast regeneration and early growth in light conditions, as well as during caulonemal growth in the dark. Furthermore, a transcriptional response of SL-responsive genes in the Ppmax2 mutant is observed in dark conditions. We observed that the scale of the Ppmax2 response to (±)-GR24 was variable compared to that of WT or Ppccd8 mutants (Fig. 4). This may be related to the use of racemic (±)-GR24 that could induce SL-independent effects [START_REF] Scaffidi | Strigolactone Hormones and Their Stereoisomers Signal through Two Related Receptor Proteins to Induce Different Physiological Responses in Arabidopsis[END_REF], not yet characterized in moss. The fact that PpMAX2 expression does not restore the Arabidopsis max2 phenotypes also argues against a role in SL response, although the moss PpMAX2 Fbox protein may not be able to recognize Arabidopsis protein interaction partners in transformed lines due to differences in C-terminus protein structure (Fig. S3,c).

Our conclusion that PpMAX2 is not crucial for SL signaling in moss leads us to hypothesize that other factors (e.g. F-box proteins) may be required. Interestingly, MAX2-independent SL responses have previously been hypothesized for roots of seed plants [START_REF] Ruyter-Spira | Physiological effects of the synthetic strigolactone analog GR24 on root system architecture in Arabidopsis: another belowground role for strigolactones?[END_REF][START_REF] Shinohara | Strigolactone can promote or inhibit shoot branching by triggering rapid depletion of the auxin efflux protein PIN1 from the plasma membrane[END_REF][START_REF] Walton | The Response of the Root Proteome to the Synthetic Strigolactone GR24 in Arabidopsis[END_REF]) and high doses of (±)-GR24 (5-10 µM) can induce a response in Arabidopsis max2 mutants [START_REF] Ruyter-Spira | Physiological effects of the synthetic strigolactone analog GR24 on root system architecture in Arabidopsis: another belowground role for strigolactones?[END_REF]. Furthermore, MAX2-independent promotion of stromule formation can be induced by (±)-GR24 [START_REF] Vismans | Low-Phosphate Induction of Plastidal Stromules Is Dependent on Strigolactones But Not on the Canonical Strigolactone Signaling Component MAX2[END_REF]). An unknown factor involved in SL signaling could thus be conserved between moss and vascular plants and able to signal with more subtle effects than the MAX2 pathway. SL signaling in moss could also be F-box protein independent, implicating different downstream mechanisms to those so far described in vascular plants in signaling. Investigation of the roles of PpSMXL genes, and putative degradation of PpSMXL proteins should clarify this point in the future.

Do Ppmax2 and Ppccd8 mutants really have opposite phenotypes?

The response to SL of the Ppmax2 mutant was difficult to pinpoint because Ppmax2 mutants have a converse phenotype to Ppccd8 mutants. Whilst Ppmax2 mutant plants are small and have few protonemal filaments, Ppccd8 plants produce many protonemata and spread across the substrate. We previously showed that whilst WT plants cease protonemal spread in response to near neighbors, Ppccd8 mutants are insensitive to neighbors in Petri cultures [START_REF] Proust | Strigolactones regulate protonema branching and act as a quorum sensing-like signal in the moss Physcomitrella patens[END_REF]. This phenomenon leads to small plant size as in the Ppmax2 mutants and WT plants grown on high (non-physiological) doses of (±)-GR24 are also small with comparable size to Ppmax2 plants (Fig. S7). Another line of evidence supporting the interpretation that Ppmax2 and Ppccd8 mutant phenotypes are converse is the transcript level of several SL-responsive genes, conversely affected in Ppccd8 and Ppmax2 mutants. For instance, PpCCD7 transcript levels are very low in Ppmax2 but much higher in Ppccd8 mutants (Fig. 4c).

If the phenotypes of Ppccd8 and Ppmax2 mutants are converse, Ppmax2 plants may overproduce and/or over-accumulate SLs. This hypothesis was tested indirectly by monitoring the Ppccd8 mutant response to Ppmax2 exudate versus Ppccd8 or WT exudates or (±)-GR24 treatment (Fig. 3), and the results suggest that Ppmax2 does not over-produce SLs, but verification by SL quantification is required, and these assays are challenging in moss.

Alternatively, Ppmax2 mutants could phenocopy a constitutive SL response.

As PpMAX2 is an F-box protein, putatively involved in degradation processes by the proteasome system, PpMAX2 could target activators of SL signaling for degradation, and such activators so far remain unidentified. SMXL proteins are known targets for degradation in seed plant SL signaling pathways, and SMXLs are considered as repressors of this pathway [START_REF] Soundappan | SMAX1-LIKE/D53 Family Members Enable Distinct MAX2-Dependent Responses to Strigolactones and Karrikins in Arabidopsis[END_REF][START_REF] Wang | Strigolactone Signaling in Arabidopsis Regulates Shoot Development by Targeting D53-Like SMXL Repressor Proteins for Ubiquitination and Degradation[END_REF]. Interestingly, the converse phenotypes of Ppmax2 and Ppccd8 mutants did not hold for gametophore branching, as Ppmax2

gametophores did not lack branches as in a pea CCD8 overexpressor line (PpRMS1OE) [START_REF] Coudert | Three ancient hormonal cues co-ordinate shoot branching in a moss[END_REF]. PpMAX2 may function in protonema and early gametophore development, but not in later development (Fig. 8).

The low levels of PpCCD7 expression in Ppmax2 in comparison to WT suggest that PpMAX2 and SL are not completely independent. However, this could be an indirect effect of reduced gametophore production in the mutant (Fig. 2d) as the highest PpCCD7 transcript levels were observed at the base of the gametophore [START_REF] Proust | Strigolactones regulate protonema branching and act as a quorum sensing-like signal in the moss Physcomitrella patens[END_REF]. There may also be indirect feedback control on transcript levels. In vascular plants, environmental conditions (N, P, drought) or endogenous factors as auxin control the expression levels of SL biosynthesis genes (Al-Babili and Bouwmeester 2015; [START_REF] Ligerot | The pea branching RMS2 gene encodes the PsAFB4/5 auxin receptor and is involved in an auxin-strigolactone regulation loop[END_REF]). It would be interesting to quantify auxin levels in both Ppmax2 and Ppccd8 mutants to test whether differences in IAA levels translate into differences in PpCCD7 transcript levels. Further experiments are needed to have a clear understanding of the moss SL signaling pathway. In particular, biochemistry to test protein interactions and quantification of the levels of other hormones should be very informative.

The role of MAX2 in light response is similar between moss and seed plants

The shoot elongation phenotype of the Ppmax2 mutant under red light and its misregulation of light responsive genes support the notion that PpMAX2 plays a role in light responses (Fig. 5), as does its flowering plant homologue [START_REF] Shen | The F-box protein MAX2 functions as a positive regulator of photomorphogenesis in Arabidopsis[END_REF]). The paucity of caulonemal filaments in Ppmax2 may also be related to a defective light response as a similar phenotype was observed in the light sensing-defective P. patens ∆hy5ab and pubs-hy2 double mutants [START_REF] Yamawaki | Functional characterization of HY5 homolog genes involved in early light-signaling in Physcomitrella patens[END_REF][START_REF] Chen | Distinct phytochrome actions in nonvascular plants revealed by targeted inactivation of phytobilin biosynthesis[END_REF]. In our experiments, tissue used for RNA extraction included a mix of protonemata and gametophores, and the ratio of different tissue types may be different in mutants versus WT plants given their distinct phenotypes. Whilst we interpret the light responsive gene expression data with caution, our results suggest that the ancestral role of MAX2 may be to promote photomorphogenesis. Despite this likely shared role with AtMAX2, PpMAX2 cannot complement the Atmax2 mutant hypocotyl phenotype under low fluence light (Fig. 7d), potentially because Arabidopsis MAX2 and moss PpMAX2 protein partners may not recognize one another. As with the shade avoidance response of vascular plants it is possible that PpMAX2 helps plants to grow in an ideal amount of light. In this instance, PpMAX2 could allow plants to respond to low light, delaying gametophore growth and investing energy in spreading protonemal tissues to find light patches. This regulation could also require HY5, given the similar phenotypes of the mutants (see above) and the misregulation of HY5a transcript levels in the Ppmax2 mutant.

An ancestral role of MAX2 in moss development

Our data and our model for roles for MAX2 in land plants (Fig. 8) open the question of an evolutionary benefit to seed plants in recruiting this F-box protein to SL signaling. We propose that combining the ability of MAX2 to regulate the levels of downstream proteins (e.g. SMXL proteins) would have added a level of fine (endogenous) regulation to photomorphogenesis or aspects of development already under the control of this F-box protein in early land plants. Further studies in other land plants including gymnosperms, lycophytes and other bryophytes will answer this question.

The expression of the PpMAX2 gene during all stages of moss development is in agreement with the putative function of PpMAX2 as a component of an SCF complex regulating the homeostasis of multiple targets. Phenotypes of Ppmax2 mutants and the Ppccd8-Ppmax2

double mutant indicate an early and simultaneous role in repressing gametophore/bud differentiation and stimulating the chloronema to caulonema transition. Thus PpMAX2 could act conversely to SLs which repress plant spread [START_REF] Proust | Strigolactones regulate protonema branching and act as a quorum sensing-like signal in the moss Physcomitrella patens[END_REF]. Interestingly, in moss, auxin has been shown to regulate the chloronema to caulonema transition [START_REF] Ashton | Analysis of gametophytic development in the moss, Physcomitrella patens , using auxin and cytokinin resistant mutants[END_REF][START_REF] Prigge | Physcomitrella patens auxin-resistant mutants affect conserved elements of an auxin-signaling pathway[END_REF][START_REF] Jang | Auxin promotes the transition from chloronema to caulonema in moss protonema by positively regulating PpRSL1and PpRSL2 in Physcomitrella patens[END_REF], while cytokinins induce bud differentiation [START_REF] Von Schwartzenberg | Cytokinins in the bryophyte Physcomitrella patens: analyses of activity, distribution, and cytokinin oxidase/dehydrogenase overexpression reveal the role of extracellular cytokinins[END_REF]. It would thus be interesting to investigate both the auxin and cytokinin status of the Ppmax2 mutant. Involvement of all three hormonal pathways in moss gametophore branching has been recently addressed [START_REF] Coudert | Three ancient hormonal cues co-ordinate shoot branching in a moss[END_REF], and this study suggests that auxin, cytokinin and SL signaling may interact, as in vascular plants.

In seed plants, MAX2 has been linked to signaling by a still unknown KL compound "which interacts at some level with auxin and light signaling to regulate growth and development" [START_REF] Waters | KAI2-and MAX2-mediated responses to karrikins and strigolactones are largely independent of HY5 in Arabidopsis seedlings[END_REF][START_REF] Conn | Evidence that KARRIKIN-INSENSITIVE2 (KAI2) Receptors may Perceive an Unknown Signal that is not Karrikin or Strigolactone[END_REF]. As the receptor KAI2 is ancestral, this pathway may be present in bryophytes. It could then be argued that the Ppmax2 phenotype is the consequence of disturbing this second signaling pathway (Fig. 8). Given this scenario, KL signaling could interfere with or mask SL signaling, because the phenotype of the Ppccd8-Ppmax2 double mutant is closer to that of Ppmax2. It has not yet been possible to test this hypothesis as moss does not seem to respond to karrikins [START_REF] Hoffmann | Strigolactones inhibit caulonema elongation and cell division in the moss Physcomitrella patens[END_REF]) and the nature of KL compound is still elusive. The study of interactions of PpMAX2 with some of the 13 PpKAI2-LIKE and/or the four PpSMXL putative targets found in moss genome [START_REF] Bennett | Strigolactone signalling: standing on the shoulders of DWARFs[END_REF]Lopez-Obando et al. 2016a) will be key to confirming the place of PpMAX2 in these signaling events.

Figure legends:

Fig. 1 :Fig. 2 :

 12 Fig. 1: Pattern of Physcomitrella patens PpMAX2 gene expression and subcellular localization of the protein. (a-d): Pattern of PpMAX2 gene expression by staining of a moss line expressing the GUS coding sequence under the control of PpMAX2 gene promoter (inserted in Pp108 locus) (a) protonema cells, (b-d) gametophore leaves and stems; arrow in (c): rhizoids. scale bars: (a-b): 0.1 mm; (c): 1 mm (d): 0.5 mm. (e-h) Nuclear localization of PpMAX2 in a protonemal tip cell of a WT moss line transformed with a mGFP6::PpMAX2 translational fusion by homologous recombination. (e) Nucleus labeling with Hoescht33342. (f) GFP fluorescence (g) chloroplast autofluorescence (h) Merge of all 3 images (e-g), indicating co-localization of Hoescht33342 and GFP to the nucleus. Scale bar: 20 µm.

Fig. 3 :

 3 Fig. 3: The Physcomitrella patens Ppmax2 mutant exudate tested on PpCCD7 expression is similar to WT Three-week-old Ppccd8 plants were transferred for 6h on medium with 0µM (±)-GR24 (Ctl), or 1µM (±)-GR24, or on medium where the WT, or the different mutants had grown (and exuded SLs) for 3 weeks noted as "exud". Data represent means of transcript levels of 3 biological repeats relative to PpAPT expression level, ± SE. Different letters indicate significantly different results based on a post-hoc Kruskal-Wallis test (P < 0.05).

Fig. 6

 6 Fig. 6 The Physcomitrella patens Ppmax2 mutation is epistatic to Ppccd8. (a) Bright field photographs of WT, single Ppccd8, single Ppmax2 mutant and Ppccd8-Ppmax2 double mutant. Scale: left, 20 day-old: 1mm; right, 2 month-old: 5mm. (b) Comparison of Ppccd8-Ppmax2 mutant plant diameter to that of WT and Ppccd8 and Ppmax2 mutants after 4 weeks (mean of 3 plates with 16 plants measured per plate, ± SE). Different letters indicate significantly different results between genotypes based on an ANOVA (P < 0.05) (c) Expression of the SL responsive gene PpCCD7 relative to PpAPT and PpACT3 expression in Ppccd8, Ppmax2-1 and Ppccd8-Ppmax2 grown for 3 weeks in the light. Asterisks denote significant differences between Ppccd8 and the other mutants based on a post-hoc Kruskal-Wallis test (P < 0.001). Data represent means of 3 biological repeats ± SE.

Fig. 7 :Fig. 8 :

 78 Fig.7: Expression of Physcomitrella patens PpMAX2 gene in the Arabidopsis max2 mutant does not restore MAX2 function (a) Mean height and (c) mean number of rosette branches , ± SE, from 4-week-old Arabidopsis plants (n=12) of each genotype 10 days after decapitation. (b) Corresponding pictures of one exemplary plant per genotype are shown. (d) Hypocotyl length of 5-day-old Arabidopsis plantlets (n=15) grown in vitro under low light intensity (20-30 µE). Names of the transformed plants indicate the construct harbored. Controls used for all experiments were Arabidopsis WT Columbia (Col-0, white bar), Atmax2-3 mutant (N592836,

Fig. S1

 S1 Fig. S1 Gene targeting of the PpMAX2 gene

  Fig.1 179x178mm (300 x 300 DPI)

  Fig.2 188x224mm (300 x 300 DPI)

  Fig.3 150x117mm (300 x 300 DPI)

  Fig.4 188x206mm (300 x 300 DPI)

  Fig.5 186x252mm (300 x 300 DPI)

  Fig.6 178x243mm (300 x 300 DPI)

  Fig.è 194x205mm (300 x 300 DPI)

  . Homologues of key genes of the SL signaling pathway are found in the P. patens genome, with one PpMAX2, 13 PpKAI2-LIKE and four PpSMXL genes. Whilst it is likely that some of the KAI2 proteins may function as SL receptors in moss(Lopez-Obando et al. 2016a), as yet no functional studies demonstrate their involvement in SL perception. This study focused on the moss PpMAX2 gene and our results indicate that roles in photomorphogenesis are conserved with Arabidopsis MAX2, but that a role of PpMAX2 in SL signaling is unlikely.

SL signaling pathway in moss is distinct from flowering plants, and does not require the

PpMAX2 F-box protein

The Ppmax2 phenotype and the ability of the mutant to respond to SL are evidence that PpMAX2 is not necessary for SL signaling. Gametophore branching

[START_REF] Coudert | Three ancient hormonal cues co-ordinate shoot branching in a moss[END_REF]

and plant spread phenotypes are different between the Ppccd8 and Ppmax2 mutants. These results contrast with mutant phenotypes in seed plants, where shoot branching and plant height are comparable in ccd8 and max2 mutants

[START_REF] Gomez-Roldan | Strigolactone inhibition of shoot branching[END_REF][START_REF] Umehara | Inhibition of shoot branching by new terpenoid plant hormones[END_REF]

. In

New Phytologist

Acknowledgements:

We thank Sonia Canessane for Arabidopsis transformants selection and first characterization, François Didier-Boyer for the generous supply of (±)-GR24, Michel Burtin and Jean-Paul Saint-Drenan for the technical assistance in red light experiments. PH, JK and RdV thank Shaun Peters for valuable discussions and insights.

Funding: This research was supported by the Agence Nationale de la Recherche (contract ANR-12-BSV6-004-01), by the BBSRC (BB/L00224811) and Gatsby (GAT2962), and by the National Research Foundation (SARChi Research Chair "Genetic tailoring of biopolymers") of South Africa. The IJPB benefits from the support of the Labex Saclay Plant Sciences-SPS (ANR-10-LABX-0040-SPS). This article is based upon work from COST Action FA1206 STREAM, supported by COST (European Cooperation in Science and Technology). YC thanks the CNRS ATIP-Avenir programme for ongoing support.