Yamina Ouedjedi
email: oudjediyamina@gmail.com

Arnaud Rougirel
email: rougirel@math.univ-poitiers.fr

Khaled Benmeriem
email: benmeriem@univ-mascara.dz

GALERKIN METHOD FOR TIME FRACTIONAL SEMILINEAR EQUATIONS

Keywords: Riemann-Liouville derivatives, Galerkin method, Fractional semilinear equations. May 2, 2019

This paper gathers the tools for solving Riemann-Liouville time fractional non-linear reaction-diffusion equations by the Galerkin method. As a model problem, existence and uniqueness is proved for semilinear heat equations with polynomial growth at infinity.

Introduction

The Galerkin approximation method in an efficient and robust tool for solving partial differential equations (see for instance [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF], [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF], [START_REF] Roubicek | Nonlinear partial differential equations with applications[END_REF]). In that paper, only low order Riemann-Liouville time fractional derivatives are considered; that is to say, the order of the fractional derivatives is less than 1. In order to implement this method for solving time fractional non-linear problems, three tools have to be developed: (i) a suitable functional framework; (ii) time fractional inequalities; (iii) an Aubin-Lions theory. Let us introduce these elements.

Usually, fractional Gagliardo-Sobolev spaces are used. However, they are not very suitable for time fractional problems since the connection between these spaces and time fractional derivatives is not straightforward. The consequence is that a trivial initial condition is needed. Moreover, these spaces are quite complicated to handle.

Recently, suitable and simpler fractional spaces appear in the literature. See for instance [START_REF] Djilali | Galerkin method for time fractional diffusion equations[END_REF], [START_REF] Li | Some compactness criteria for weak solutions of time fractional PDEs[END_REF], [dF19]. These spaces are natural generalizations of the spaces involved in the integer case. See Section 3 for details.

By time fractional inequalities, we means for example 1 2 D α 0,t Ω u(t, x) 2 dx ≤ Ω D α 0,t u(t, x)u(t, x) dx.

(1.1)

According to the integer setting, we cannot expect such an relationship to hold without imposing a zero initial condition. Under some smoothness conditions, (1.1) is proved in [Zac12, Theorem 2.4], and in [LL18b, Proposition 2.18] by a simple and smart convexity argument. In order to apply (1.1) to nonlinear problems, we have relaxed the smoothness assumptions by a density argument: see Corollary 3.3 and Proposition 3.4 where a integral version of (1.1) is featured. We refer also to [dF19], where time fractional inequalities hold for hilbert valued functions. However, Aubin-Lions theory allows to get point-wise convergence by compactness arguments, and to pass to the limit in non-linear terms. By adapting the arguments of [START_REF] Li | Some compactness criteria for weak solutions of time fractional PDEs[END_REF], we obtain the compactness result stated in Corollary 3.5.

In the two forcoming sections, we recall or develop the tools for solving fractional equations. The Galerkin method is implemented in section 4 for solving time fractional semilinear heat equations.

Preliminaries

As far as integrable functions are concerned, convolution is a basic tool in fractional calculus. However, in order to obtain a density result, namely Theorem 3.1, we will need to make the convolution of non absolutely integrable functions. That can be achieved following [START_REF] Gelfand | Generalized functions[END_REF], for causal functions.

Let (X, . ) be a real Banach space, and T be a positive number. Let us recall that

f ∈ L 1 loc (R; X) is said to be causal if f = 0 a.e. on (-∞, 0). Definition 2.1. Let f ∈ L 1 loc (R; X), g ∈ L 1 loc (R) be causal functions.
Then the convolution of f and g is the causal function of L 1 loc (R; X) defined, for a.e. t ∈ R, by

g * f (t) = R g(t -y)f (y) dy.
Classically, fractional derivatives involve another kind of convolution, since the functions are defined on [0, T ].

Definition 2.2. Let f ∈ L 1 (0, T ; X) and g ∈ L 1 (0, T ). Then the convolution of g and f is the element of L 1 (0, T ; X) defined, for a.e. t ∈ [0, T ], by

g * T f (t) := t 0 g(t -y)f (y) dy.
Of course these two definitions are consistent. Indeed, the following result hods true.

Proposition 2.1. Let f ∈ L 1 loc (R; X) and g ∈ L 1 loc (R) be causal functions. Then g | [0,T ] * T f | [0,T ] = (g * f ) | [0,T ] in L 1 (0, T ; X). Above, f | [0,T ] denotes the restriction of f to [0, T ].
The elementary proof of that proposition is omitted. Owing to the above result, we will write g * f instead of g * T f , if no confusion can occur.

The following standard inequality will be useful. Let I = [0, T ] or R. If f ∈ L p (I; X) with 1 ≤ p ≤ ∞, and g ∈ L 1 (I) then g * f belongs to L p (I; X) and

g * f L p (I;X) ≤ g L 1 (I) f L p (I;X) .
(2.1)

Let us turn our attention to fractional derivatives. The following kernels are of fundamental importance in the theory of fractional derivatives. For α ∈ (0, ∞), we denote by g α the causal function of L 1 loc (R) defined, for a.e. t > 0, by

g α (t) = 1 Γ(α) t α-1 .
These Kernels satisfy the following semi-group property : for α > 0, β > 0,

g α * g β = g α+β in L 1 loc (R). (2.2)
Now, we are able to introduce the fractional Riemann-Liouville derivative of vectorvalued functions. In the sequel, α ∈ (0, 1) will denote the fractional order of differentiation.

Definition 2.3. Let 1 ≤ q < ∞ and u ∈ L q (0, T ; X). We say that u admits a fractional derivative of order α in L q (0, T ; X) if

g 1-α * u ∈ W 1,q (0, T ; X).
In this case, the fractional derivative of order α of u is the function of L q (0, T ; X) defined by

D α 0,t u := d dt {g 1-α * u}.
Above, W 1,q (0, T ; X) denotes the space of functions belonging to L q (0, T ; X) whose first order derivative (in the sense of distributions) belongs to L q (0, T ; X). Proposition 2.2. Let 1 ≤ q < ∞, α ∈ (0, 1) and u ∈ L q (0, T ; X). If u admits a fractional derivative in L q (0, T ; X) then

u = (g 1-α * u)(0)g α + g α * D α 0,t u in L q (0, T ; X). (2.3) Moreover, if α ≤ 1 -1 q then (g 1-α * u)(0) = 0.
Any function u satisfying the assumptions of Proposition 2.2 has, in some sense, a weak singularity at t = 0. Indeed, let us assume that u = vg β where v ∈ X \ {0} and β > 0.

Then by (2.2), u belongs to L q (0, T ; X) and has a α-derivative in L q (0, T ;

X) iff β > 1 -1 q or β = α.
Proof of Proposition 2.2. Equality (2.3) is well known (see for instance [DR18, Proposition 3.4]). In order to prove the second assertion, we observe that g α does not belong to L q (0, T ) if α ≤ 1 -1 q . Thus we must have (g 1-α * u)(0) = 0.

Let us now focus on weak fractional derivative of vector-valued functions. The starting point is the following standard integration by-part formula.

Proposition 2.3. [DR18, Proposition 3.1] Let α ∈ (0, 1), f ∈ L 1 (0, T ; X) and ψ ∈ C 1 ([0, T ]). Assume that f admits a derivative of order α in L 1 (0, T ; X).

Then T 0 D α 0,t f (t)ψ(t) dt = - T 0 f (t)D α t,T ψ(t) dt + g 1-α * f ψ T 0 in X, (2.4)
where

D α t,T ψ(t) := T t g 1-α (y -t)ψ (y) dy, ∀t ∈ [0, T ],
and ψ := d dt ψ denotes the derivative of ψ. Moreover, if in addition, ψ(0

) = ψ(T ) = 0 then T 0 f (t)D α t,T ψ(t) dt ≤ g 2-α (T ) f L q (0,T ;X) ψ L ∞ (0,T ) .
(2.5)

That property allows us to define fractional derivative in the sense of distributions. Indeed, (2.5) shows that the linear map

D(0, T ) → X, ϕ → - T 0 f (t)D α t,T ϕ(t) dt
is a vector-valued distribution, whose order is (at most) 1. The set of distributions with values in X is denoted by D (0, T ; X). That allows us to set this definition.

Definition 2.4. Let α ∈ (0, 1), q ∈ [1, ∞) and f ∈ L q (0, T ; X). Then the weak derivative of order α of f is the vector-valued distribution, denoted by D α 0,t f , and defined, for all ϕ ∈ D(0, T ), by

D α 0,t f, ϕ = - T 0 f (t)D α t,T ϕ(t) dt in X.
If we want to highlight the duality taking place in the above bracket, we will write D α 0,t f, ϕ D (0,T ;X),D(0,T ) instead of D α 0,t f, ϕ . Clearly, weak fractional derivatives is a natural extension of (first order) weak derivatives, also called derivatives in the sense of distribution.

Of course, the concept of weak derivative extends that of fractional derivative given in Definition 2.3. See [DR18, Proposition 3.2] for details. Finally, we recall a proposition useful for passing to the limit in fractional derivatives.

Proposition 2.4. [DR18, Proposition 3.3] Let α ∈ (0, 1), V be a real Banach space, q ∈ [1, ∞), and f ∈ L q (0, T ; V ). We assume that f admits a derivative of order α in L q (0, T ; V ). Then, for each v in V , f, v V ,V admits a derivative of order α in L q (0, T ) and

D α 0,t f (•), v V ,V = D α 0,t f, v V ,V , in L q (0, T ). (2.6)
Here, V denotes the dual space of V and •, • V ,V the corresponding duality bracket.

Fractional spaces

In this section, we introduce the functional framework for solving fractional semilinear equations. Let X, Y be real Banach spaces such that X is continuously embedded into Y . Also, let T > 0, α ∈ (0, 1) and p, q ∈ [1, ∞). Then we introduce the following space

W α p,q (0, T ; X, Y ) := u ∈ L p (0, T ; X) : D α 0,t u ∈ L q (0, T ; Y ) (3.1) and 0 W α p,q (0, T ; X, Y ) := {u ∈ W α p,q (0, T ; X, Y ) : (g 1-α * u)(0) = 0 in Y . (3.2)
W 1 p,q (0, T ; X, Y ) is the standard Sobolev space used for solving non-linear PDE's by the Galerkin method (see for instance [START_REF] Roubicek | Nonlinear partial differential equations with applications[END_REF]). Therefore, W α p,q (0, T ; X, Y ) is the "simplest" spaces we can think of when solving (4.1).

In (3.1), D α 0,t u is understood in the sense of distribution, i.e. in the sense of Definition 2.4. Alternatively, W α p,q (0, T ; X, Y ) may be defined trough Definition 2.3, as the set of functions in L p (0, T ; X) which admits a fractional derivative in L q (0, T ; Y ).

Equipped with the norm

u W α := u 2 L p (0,T ;X) + D α 0,t u 2 L q (0,T ;Y ) 1/2 , (3.3)
it is clear that W α p,q (0, T ; X, Y ) and 0 W α p,q (0, T ; X, Y ), are Banach spaces. We start by a density result. As far as the above fractional spaces are considered, such results are quite uncommon in the literature (see however [START_REF] De | On the fractional version of Leibniz rule[END_REF]Theorem 39]). This Theorem allows to extend the coercivity result of [START_REF] Li | Some compactness criteria for weak solutions of time fractional PDEs[END_REF] from an hilbertian setting into Banach setting.

Theorem 3.1. Let X, Y be real Banach spaces such that X is continuously embedded into Y and, for p, q ∈ [1, ∞), u ∈ 0 W α p,q (0, T ; X, Y ).

(3.4) Then there exists a sequence (u n ) n≥1 in C ∞ ([0, T ]; X) such that u n (0) = 0 for each n, and

u n → u, in 0 W α p,q (0, T ; X, Y ). (3.5)
To prove this theorem we use the following lemma, whose proof can be found in [Oue].

Lemma 3.2. Let T > 0 and u ∈ L 1 loc (R; X) be a causal function such that u | [0,T ] belongs 0 W α p,q (0, T ; X, Y ). Then, for each h > 0, u(• -h) lies in 0 W α p,q (0, T ; X, Y ) and

u(• -h) --→ h→0 u, in W α p,q (0, T ; X, Y ).
Proof of Theorem 3.1. By lemma 3.2 we may assume that there exists h > 0 such that u = 0 a.e. on [0, h]. For each integer n ≥ 1, let us choose ρ n : R → R to be a mollifier function such that supp ρ n ⊆ [0, h], ∀n ≥ 1, (3.6) where supp ρ n denotes the support of ρ n . Let

u := u on [0, T ] 0 elsewhere , (3.7)
and define

u n := ρ n * u ∈ L p (R, X).
(3.8) We observe that u n ∈ C ∞ (R; X) and u n (0) = 0 by (3.6). Firstly, it is well known that the restriction of u n converges towards u in L p (0, T ; X). Secondly, let us show that

D α 0,t u n| [0,T ] → D α 0,t u in L q (0, T ; Y ). (3.9)
For, let

F := D α 0,t u on [0, T ] 0
elsewhere .

For each n ≥ 1, the function

v n := ρ n * g α * F (3.10) lies in L 1 loc (R; Y ) and satisfies, according to [LL18a, Lemma 2.2] or [GS77, Chap I.5], g 1-α * v n = g 1-α * ρ n * g α * F = g 1-α * g α * ρ n * F = g 1 * ρ n * F.
Recalling that g 1 is the causal function equal to 1 a.e. on [0, ∞), we get by differentiability

d dt {g 1-α * v n } = ρ n * F in L q (R; Y ). (3.11) Whence, since F = D α 0,t u on [0, T ], D α 0,t v n| [0,T ] = d dt {g 1-α * v n } | [0,T ] ---→ n→∞ D α 0,t u in L q (0, T ; Y ). (3.12) Besides, since g α * F is supported in [0, ∞[, one has for a.e. t ∈ [0, T ], v n (t) = t 0 ρ n (t -y)(g α * F )(y) dy,
and, by definition of F ,

(g α * F ) | [0,T ] = g α * D α 0,t u = u in L q (0, T ; Y ),
thanks to Proposition 2.2 and the fact that (g 1-α * u)(0) = 0, since u = 0 a.e. on [0, h]. Thus, in view of (3.8),

v n| [0,T ] = u n| [0,T ] ;
so that (3.9) follows from (3.12). That completes the proof of the theorem.

Let us now turn to time fractional inequalities. For, let X be a real Banach space densely and continuously embedded into a real Hilbert space H. Then X is a Banach subspace of its dual space X and v, • X ,X = (v, •) H , ∀v ∈ X, where the bracket denotes the duality between X and X, and (•, •) H the inner product of H. By [LL18b, Proposition 2.18], any u in W 1,1 (0, T ; H) with u(0) = 0, satisfies

1 2 g 1-α * u(•) 2 H (t) ≤ t 0 D α 0,t u(s), u(s) H ds, ∀t ∈ [0, T ]. (3.13)
Then combining Theorem 3.1 and (3.13), we get easily the following result.

Corollary 3.3. For X, H as above, let p ≥ 2 whose conjugate exponent is denoted by p . Assume that u ∈ 0 W α p,p (0, T ; X, X ).

(3.14)

Then, for each t ∈ [0, T ], 1 2 g 1-α * u(•) 2 H (t) ≤ t 0 D α 0,t u(s), u(s) X ,X ds. (3.15)
That corollary will be useful to get uniqueness results. Regarding existence, the following proposition will be used.

Proposition 3.4. Let X, H as above, α ∈ (0, 1) and p ≥ 2 be such that α > 1/p . Assume

u ∈ W α p,p (0, T ; X, X ), (3.16
)

and (g 1-α * u)(0) ∈ X. Then T 0 D α 0,t u(t), u(t) -(g 1-α * u)(0)g α (t) X ,X dt ≥ 0.
Proof. Set for simplicity v := (g 1-α * u)(0). Since α > 1/p , the function u -vg α belongs to L p (0, T ; X). Moreover, for each t ∈ [0, T ], there holds

g 1-α * (u -vg α )(t) = g 1-α * u(t) -v ---→ t→0 + 0, in X .
Hence, u(t) -vg α lies in 0 W α p,p (0, T ; X, X ) and D α 0,t (u -vg α ) = D α 0,t u. Then the assertion follows from Corollary 3.3.

The following compactness result is proved in [START_REF] Li | Some compactness criteria for weak solutions of time fractional PDEs[END_REF] for Caputo's derivatives. We just adapt their proof to our framework.

Corollary 3.5. Let X ⊂ X 0 ⊂ Y be Banach spaces such that X is compactly embedded into X 0 . Let α ∈ (0, 1) and p > 1. Then W α p,1 (0, T ; X, Y ) is compactly embedded into L r (0, T ; X 0 ), for all r ∈ [1, p).

Proof. Let B(0, R) denote the closed ball of W α p,1 (0, T ; X, Y ) with radius R > 0 and center 0. Since B(0, R) is bounded in L p (0, T ; X), it is enough to prove, according to classical Simon's result [Sim87, Theorem 6], that for each τ ∈ (0, T ) and h

∈ [0, T -τ ], sup u∈B(0,R) u(• + h) -u(•) L 1 (0,τ ;Y ) --→ h→0 0 (3.17)
For, by Proposition 2.2, we have

u = g α (•) g 1-α * u(0) + g α * D α u in L 1 (0, T ; Y ).
Thus, for all h ∈ [0, T -τ ] such that h ≤ 1, one has, for a.e. t ∈ [0, τ ],

u(t + h) -u(t) = g α (t + h) -g α (t) g 1-α * u(0) + t 0 g α (t + h -y) -g α (t -y) D α u(y) dy + t+h t g α (t + h -y)D α u(y) dy. (3.18)
Let us estimate the first term in the right hand side of the above equation. Since

W 1,1 (0, T ; X, Y ) is embedded into C([0, T ], Y ), we have g 1-α * u(0) 2 C([0,T ],Y ) ≤ C g 1-α * u 2 L 1 (0,T,X) + C D α u 2 L 1 (0,T,Y )
. With (2.1) and u ∈ B(0, R), we get

g 1-α * u(0) C([0,T ],Y ) ≤ C g 1-α 2 L 1 (0,T ) + 1) 1/2 R. Besides, τ 0 g α (t + h) -g α (t) dt = g α+1 (τ ) -g α+1 (τ + h) + g α+1 (h) ≤ g α+1 (h),
since g α+1 is increasing. There result that the first term is bounded in L 1 (0, τ, Y ) by C(R)g α+1 (h), for some constant C(R) independent of u and h.

Regarding the second term, its L 1 (0, τ, Y )-norm is bounded by

τ 0 t 0 g α (t + h -y) -g α (t -y) D α u(y) Y dy dt ≤ τ 0 D α u(y) Y dy τ y g α (t -y) -g α (t + h -y) dt, by Fubini's Theorem. Moreover, τ y g α (t -y) -g α (t + h -y) dt = g α+1 (τ -y) -g α+1 (τ + h -y) + g α+1 (h) ≤ g α+1 (h),
since g α+1 is increasing. Thus second term is bounded in L 1 (0, τ, Y ) by Rg α+1 (h).

We proceed in the same way for the third term of (3.18). Its L 1 (0, τ, Y )-norm is bounded by

τ +h 0 D α u(y) Y dy y y-h g α (t + h -y) dt ≤ Rg α+1 (h).
Finally, (3.17) holds which completes the proof of the theorem.

Time fractional semilinear heat equations

Let n be a positive integer, Ω be a bounded open subset of R n , T > 0, and 0 < α < 1. The problem under consideration is

         Find u : [0, T ] × Ω → R such that D α 0,t u -∆u + f (u) = 0 on [0, T ] × Ω u = 0 on [0, T ] × ∂Ω (g 1-α * u)(0, •) = v on ∂Ω.
(4.1)

Here v : Ω → R is the initial condition and f : R → R is a non-linear function with polynomial growth at infinity. We will assume that f has the convenient sign at ±∞ in order to avoid blow-up phenomena and get global in time solutions. These assumptions on f are standard in pattern formation equations (see [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF]). We will solve (4.1) by the Galerkin method. In the standard case where α = 1, that method consists in solving first a finite dimensional approximated problem, and then pass to the limit. In the fractional case, it turns out that some extra condition is needed for the solvability of the approximated problem. Roughly speaking, that condition looks like a growth condition on the derivative of f (see [Oue]). However, it is not needed to pass to the limit. Also such assumption is not needed in the case α = 1.

In order to avoid that extra assumption, we truncate the non-linear term f . Then we solve the truncated fractional PDE by projecting onto a finite dimensional space. Finally, we pass to the limit in the truncated problem. Hence, we will first solve (4.1) for sub-linear f . The general case will be investigated in the second subsection.

4.1. The hilbertian case. When f : R → R has a sub-linear growth, we may work with Hilbert spaces. Thus we have only to control the fractional derivative. More precisely, we will assume that there exists a positive constant C such that

f (u) -f (v) ≤ C u -v (4.2) f (u)u ≥ -C, ∀u, v ∈ R. (4.3)
Recalling the notation (3.1) for fractional spaces, the problem under consideration is then

     Find u ∈ W α 2,2 0, T ; H 1 0 (Ω), H -1 (Ω) such that D α 0,t u -∆u + f (u) = 0 in L 2 0, T ; H -1 (Ω) (g 1-α * u)(0) = v in L 2 (Ω). (4.4) Theorem 4.1. Assume v ∈ H 1 0 (Ω) and f satisfies (4.2), (4.3). (i) If α ∈ ( 1 2 , 1) then (4.2) has a unique solution. (ii) If α ∈ (0, 1 2 ] then (a) if v = 0 then (4.2) has no solution; (b) if v = 0 then (4.
2) has a unique solution.

Proof. By Proposition 2.2, we derive that (4.4) has no solution if α ≤ 1/2 and v = 0. On the other hand, if v = 0 then the solvability of (4.4) can be achieved as in the case α ∈ ( 1 2 , 1). Thus we will only consider in the sequel the case where α > 1/2. Existence of a solution. We will implement the Galerkin approximation method. For, let us introduce some notation. Denote by (•, •) 0 the inner product of L 2 (Ω) and

A : H 1 0 (Ω) → H -1 (Ω), u → -∆u.
For k = 1, 2, . . . , let (w k , λ k ) ∈ H 1 0 (Ω) × (0, ∞) be a k th mode of A such that (w k ) k≥1 forms an hilbertian basis of L 2 (Ω). For n = 1, 2, . . . , we denote by F n the vector space generated by w 1 , . . . , w n . Finally, we decompose the initial condition v, by writing

v = k≥1 b k w k in H 1 0 (Ω),
and we set

v n := n k=1 b k w k . (4.5) Whence v n ∈ F n and v n → v in H 1 0 (Ω).
(i) An approximated problem . For each integer n ≥ 1, our approximated problem takes the form

     Find u n ∈ L 2 (0, T ; F n ) such that D α 0,t u n ∈ L 2 (0, T, F n ) D α 0,t u n , w) 0 + ∇u n , ∇w) 0 + f (u n ), w) 0 = 0 in L 2 (0, T ), ∀w ∈ F n (g 1-α * u n )(0) = v n . (4.6)
Thanks (4.2), we may show by standard methods in fractional calculus (see [Oue] for details) that the ordinary fractional differential equation (4.6) is uniquely solvable.

(ii) Estimates. Since α > 1/2 and v n ∈ H 1 0 (Ω), w := u n -g α v n is a suitable test-function for (4.6), hence D α 0,t u n , u n -g α v n ) 0 + Ω |∇u n | 2 dx + Ω f (u n )u n dx = g α Ω ∇u n ∇v n dx + Ω f (u n )g α v n dx,
in L 2 (0, T ). Since u n belongs to W α 2,2 (0, T ; L 2 (Ω), L 2 (Ω)), Proposition 3.4 yields that the time integral of the term involving the fractional derivative is non negative. Then, using (4.2), (4.3) and the boundedness of (v n ) in H 1 0 (Ω), we derive by standard estimates

u n L 2 (0,T ;H 1 0 (Ω)) ≤ C (4.7) D α 0,t u n (t) L 2 (0,T ;H -1 (Ω)) ≤ C (4.8) f (u n ) L 2 (0,T ;L 2 (Ω)) ≤ C. (4.9)
(iii) Passage to the limit. According to (4.7), there exists u ∈ L 2 (0, T ;

H 1 0 (Ω)) such that up to a subsequence u n u in L 2 (0, T ; H 1 0 (Ω))-weak. Moreover, by Corollary 3.5, W α 2,1 (0, T ; H 1 0 (Ω), H -1 (Ω)
) is compactly embedded into to L 1 (0, T ; L 2 (Ω)). Thus, up to a subsequence,

u n → u a.e on [0, T ] × Ω.
Then, by continuity of f ,

f (u n ) → f (u) a.e on [0, T ] × Ω.
Thus, using also (4.9), Lion's Lemma [Lio69, lemma I-1.3] yields

f (u n ) f (u) in L 2 (0, T ; L 2 (Ω)).
(iv) Solvability of the equation of (4.4). Let us show that

D α 0,t u -∆u + f (u) = 0 in L 2 (0, T ; H -1 (Ω)).
For, let k ≥ 1 be fixed and n ≥ k. For each ϕ ∈ D(0, T ), we derive from (4.6), Proposition 2.4 and Proposition 2.3 that

T 0 -u n (t)D α t,T ϕ(t) + Au n -f (u n ) ϕ(t) dt, w k H -1 (Ω)
,H 1 0 (Ω) = 0. Passing to the limit in n and using Definition 2.4, we get

D α u + Au + f (u) = 0 in D (0, T ; H -1 (Ω)).
Since Au and f (u) belong to L 2 (0, T ; H -1 (Ω)), we derive that u lies in

W α 2,2 0, T ; H 1 0 (Ω), H -1 (Ω) and D α u + Au + f (u) = 0 in L 2 0, T ; H -1 (Ω) . (4.10) (v) Initial condition. Let 1 ≤ k ≤ n and ψ ∈ H 1 (0, T ) with ψ(T ) = 0.
Then, starting from (4.10) and using Proposition 2.4 and Proposition 2.3, we derive

- T 0 u(t), w k 0 D α t,T ψ(t) dt -(g 1-α * u)(0), w k 0 ψ(0) + T 0 Au, w k H -1 (Ω),H 1 0 (Ω) ψ(t) dt + T 0 f (u(t)), w k 0 ψ(t) dt = 0. (4.11)
Besides, going back to the equation of (4.6) and proceeding in the same way, we get

- T 0 (u n (t), w k ) 0 D α t,T ψ(t) dt -(g 1-α * u n )(0), w k 0 ψ(0) + T 0 Au n , w k H -1 (Ω),H 1 0 (Ω) ψ(t) dt + T 0 f (u n (t)
), w k 0 ψ(t) dt = 0.

We pass to the limit to get

- T 0 (u(t), w k ) 0 D α t,T ψ(t) dt -(v, w k ) 0 ψ(0) + T 0 Au, w k H -1 (Ω),H 1 0 (Ω) ψ(t) dt + T 0 f (u(t)), w k 0 ψ(t) dt = 0. (4.12)
Comparing (4.11) with (4.12), we deduce that (g 1-α * u)(0) = v.

Uniqueness for Problem (4.4). Let u 1 , u 2 be two solutions to (4.4). Then u := u 1 -u 2 satisfies

D α 0,t u -∆u + f (u 1 ) -f (u 2 ) = 0 in L 2 (0, T ; H -1 (Ω)) (4.13) (g 1-α * u)(0) = 0. (4.14)
Let τ ∈ (0, T ]. Testing (4.13) with u, we derive with the Lipschitz assumption (4.2)

τ 0 D α 0,t u(t), u(t) H -1 (Ω),H 1 0 (Ω) dt ≤ C τ 0 u(t) 2 L 2 (Ω) dt. (4.15)
By (4.14), u lies in 0 W α 2,2 (0, T ; H 1 0 (Ω), H -1 (Ω)). Then Corollary 3.3 yields

g 1-α * u(.) 2 L 2 (Ω) (τ ) ≤ 2C τ 0 u(t) 2 L 2 (Ω) dt. (4.16)
If there exists τ ∈ (0, T ] such that u = 0 a.e. on [0, τ ] then we set t 0 := sup{τ ∈ (0, T ] : u = 0 a.e. on [0, τ ]}.

Otherwise, we put t 0 := 0. Now, in order to get uniqueness, it is enough to show that t 0 = T . Arguing by contradiction, let us assume that t 0 ∈ [0, T ). Then for each τ ∈ (t 0 , T ], we have

τ t 0 u(t) 2 L 2 (Ω) dt = τ 0 u(t) 2 L 2 (Ω) dt = 0.
Then going back to (4.16) and using the decay of g 1-α , we derive

g 1-α (τ -t 0 ) τ t 0 u(t) 2 L 2 (Ω) dt ≤ 2L τ t 0 u(t) 2 L 2 (Ω) dt.
The condition t t 0 u(y) 2 L 2 (Ω) dy = 0 leads to the boundedness of τ → g 1-α (τ -t 0 ) on (t 0 , T ]. That impossibility shows that t 0 = T . The proof of the theorem is now completed. 4.2. The polynomial growth case. We will assume that the reaction term f has a polynomial growth at infinity. Thus we cannot work no more with fractional Hilbert spaces. However, our functional framework remains, in some sense hilbertian, since the initial condition is constrained to stay in a subspace of L 2 (Ω).

Let n be a positive integer, Ω be a bounded open subset of R n , and 0 < α < 1. Let f : R → R satisfy, for some positive constants C, c and p f (u)u ≥ c|u| p+1 -C (4.17)

|f (u)| ≤ C|u| p + C, ∀u ∈ R (4.18)
f is non decreasing on some neighborhood of -∞ and ∞ (4.19)

f ∈ W 1,1 loc (R). (4.20)
The latter condition means that f is a locally Lipschitz function on R. Also, (4.19) means that there exists some M 0 ∈ (0, ∞) such that f is non decreasing on (-∞, -M 0 ] and on [M 0 , ∞).

Let us denote by (p + 1) the conjugate exponent of p + 1 i.e. (p + 1) ∈ (1, ∞) and

(p + 1) := p + 1 p ⇐⇒ 1 (p + 1) + 1 p + 1 = 1.
The problem under consideration is

           Find u ∈ L 2 0, T ; H 1 0 (Ω) ∩ L p+1 0, T ; L p+1 (Ω) such that D α 0,t u ∈ L 2 0, T ; H -1 (Ω) + L (p+1) 0, T ; L (p+1) (Ω) D α 0,t u -∆u + f (u) = 0 in L 2 0, T ; H -1 (Ω) + L (p+1) 0, T ; L (p+1) (Ω) (g 1-α * u)(0) = v. (4.21)
Let us notice that (4.21) is a natural extension of the standard case where α = 1. See for instance [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF], [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF] or [START_REF] Chipot | Elements of nonlinear analysis[END_REF].

Moreover, for each u ∈ L p+1 0, T ; L p+1 (Ω) ,

where C is the constant appearing in (4.17). Then, according to Theorem 4.1, the following truncated problem

     Find u M ∈ W α 2,2 0, T ; H 1 0 (Ω), H -1 (Ω) such that D α 0,t u M -∆u M + f M (u M ) = 0 in L 2 (0, T ; H -1 (Ω)) (g 1-α * u M )(0) = v in L 2 (Ω).
(4.26) has a unique solution since α > p p+1 ≥ 1/2. Observe that f M converges toward f uniformly on compact sets of R. Thus it is expected that, in the limit M → ∞, u M gives a solution to (4.21).

(ii) Estimates. Arguing as in the proof of Theorem 4.1, we get, using in particular Proposition 3.4

T 0 Ω |∇u M | 2 dx + f M (u M )u M dx dt ≤ C + C T 0 Ω f M (u M )g α v dx dt.
Young inequality and Lemma 4.3 below lead to

f M (u M )g α v ≤ ε f M (u M ) (p+1) + C ε g α v p+1 ≤ εC 0 f M (u M )u M + εC 0 + C ε g α v p+1 , ∀M ≥ M 0 ,
where C 0 and M 0 are the constants appearing in Lemma 4.3. Thus Let us now show that the sequence (D α u M ) M ≥0 remains bounded in L 1 (0, T, V p ). Indeed, testing (4.26) with w ∈ V p , and using Hölder inequality, we arrive to

T 0 Ω |∇u M | 2 + f M (u M )u M dx dt ≤ C. ( 4
D α u M , w H -1 (Ω) ≤ u M H 1 0 (Ω) w H 1 0 (Ω) + f M (u M ) L (p+1) (Ω) w L p+1 (Ω)
. Moreover, by density, H -1 (Ω) is a subspace of V p . Thus, with (4.27) and (4.28) Thus, with (4.17), we get that u ∈ L p+1 (0, T ; L p+1 (Ω)), and (see (4.22)) that f (u) lies in L (p+1) (0, T ; L (p+1) (Ω)). Then Lion's Lemma yields f M (u M ) f (u) in L (p+1) (0, T ; L (p+1) (Ω)).

T 0 D α u M V p dt ≤ C, ∀M ≥ M 0 . ( 4
For any w ∈ V p and ϕ ∈ D(0, T ), testing (4.26) with wϕ and using Proposition 2.4, we get in a standard way D α 0,t u -∆u + f (u) = 0 in L 2 0, T ; H -1 (Ω) + L (p+1) 0, T ; L (p+1) (Ω) . (4.31) (v) Initial condition. Since (p + 1) ≤ 2, (4.31) holds in L 1 (0, T ; V p ). Then testing (4.31) with wψ for any w ∈ V p and ψ ∈ H 1 (0, T ) such that ψ(T ) = 0, we derive, by applying Proposition 2.4 in L 1 (0, T ; V p ), Comparing (4.32) with (4.33), we deduce that (g 1-α * u)(0) = v.

Uniqueness for Problem (4.21). We proceed as in the proof of Theorem 4.1. The nonlinear term is controlled in a usual way, by using the following property. There exists M > 0 such that f (u) -f (v) (u -v) ≥ 0, ∀|u| ≥ M, ∀v ∈ R.

See [Oue] for details. The proof of the theorem is now completed.

If p > 1 then the theorem tells nothing when 1 p+1 < α ≤ p p+1 . More regularity on D α 0,t u allows to fill that gap. Indeed, let α ≤ p p+1 and u be a solution to (4.21) such that D α 0,t u ∈ L p+1 0, T ; H -1 (Ω) + L (p+1) (Ω) . Then, according to the proof of Theorem 4.2, v = 0; so that (4.21) has a unique solution for v = 0 and α ≤ p p+1 . The following result is used in the proof of Theorem 4.2. We refer to [Oue] for its proof.

  .27) Also, by (4.27) and Lemma 4.3 again, T 0 Ω |f M (u M )| (p+1) dx dt ≤ C. (4.28)

  .29) (iii) Passage to the limit. According to (4.27) and (4.28), Corollary 3.5 yields the existence of some u ∈ L 1 (0, T ; L 2 (Ω)) such that up to a subsequence,u M ----→ M →∞ u a.e. on [0, T ] × Ω. (4.30)Since f M converges toward f uniformly on compact sets of R, there holds f M (u M ) → f (u) a.e. on [0, T ] × Ω. Thus, using also (4.25) and (4.27), Fatou's Lemma leads to T 0 Ω f (u)u dx dt ≤ C.

0

  ), w V p ,Vp D α t,T ψ(t) dt -(g 1-α * u)(0), w V p ,Vp ψ(0) + T Au, w H -1 (Ω),H 1 0 (Ω) ψ(t) dt + T 0 Ω f u(t) w ψ(t) dx dt = 0. (4.32)On the other hand arguing as in the proof of Theorem 4.1, we obtain-T 0 (u(t), w) 0 D α t,T ψ(t) dt -(v, w) 0 ψ(0) T 0 Au, w H -1 (Ω),H 1 0 (Ω) ψ(t) dt + T 0 Ωf u(t) w ψ(t) dx dt = 0. (4.33)

Lemma 4. 3 .

 3 Let f : R → R satisfy (4.17), (4.18) with p > 0 and f M defined by (4.24). Then there exist M 0 > 0 and C 0 > 0 such that|f (u)| p+1 p ≤ C 0 f M (u)u + C 0 , ∀u ∈ R, ∀M ≥ M 0 .

(4.18) and the Hölder inequality yield that f (u) ∈ L (p+1) 0, T ; L (p+1) (Ω) .

(4.22)

Hence the three terms involved in the equation of Problem (4.21) belong to

then u has a fractional derivative in L (p+1) (0, T ;

Therefore the initial condition in Problem (4.21) is meaning full. Now, we may state our main result.

Theorem 4.2. Let us assume the following.

For sake of simplicity, we set

and denote its dual space by V p .

Proof of Theorem 4.2. Let α ≤ 1 p+1 and u be a solution to (4.21). Since p ≥ 1, we have (p + 1) ≤ 2. Thus u and D α u belong to L (p+1) (0, T ; V p ). Hence Proposition 2.2 yields

Since g α ∈ L (p+1) (0, T ), the initial condition v must be trivial. In that case, existence and uniqueness may be achieved as in the case α > p p+1 . So, in the sequel of that proof, we will assume that α > p p+1 . Existence of a solution. (i) A truncated problem . For all positive integer M , we define


Nicolas S Merle

Romain Paule

Juliette Leon

Marie Daugan

Tania Robe-Rybkine

Victoria Poillerat

Carine Torset

Veronique Fremeaux-Bacchi

Ph.D Jordan D Dimitrov
email: jordan.dimitrov@crc.jussieu.fr

Ph.D Lubka T Roumenina
email: lubka.roumenina@crc.jussieu.fr

Juliette Léon

Tania Robe- Rybkine

Véronique Frémeaux-Bacchi

P-selectin drives complement attack on endothelium during intravascular hemolysis in TLR-4/heme-dependent manner

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

Introduction

Intravascular hemolysis is a hallmark of a large spectrum of pathologies, including socially significant diseases as sickle cell disease (SCD) or malaria, characterized by the presence of a large amount of red blood cells (RBC) degradation products in the circulation [START_REF] Hoshino | Cutting edge: Toll---like receptor 4 (TLR4)---deficient mice are hyporesponsive to lipopolysaccharide: evidence for TLR4 as the Lps gene product[END_REF][START_REF] Wessels | Studies of group B streptococcal infection in mice deficient in complement component C3 or C4 demonstrate an essential role for complement in both innate and acquired immunity[END_REF]. Cell-free hemoglobin, heme and RBC microvesicles act by different mechanisms to promote vascular and tissue injury (3). Heme is a danger-associated molecular pattern molecule, which recruits and activates neutrophils [START_REF] Merle | Characterization of Renal Injury and Inflammation in an Experimental Model of Intravascular Hemolysis[END_REF][START_REF] Tolosano | Defective Recovery and Severe Renal Damage After Acute Hemolysis in Hemopexin---Deficient Mice[END_REF][START_REF] Roumenina | A prevalent C3 mutation in aHUS patients causes a direct C3 convertase gain of function[END_REF] and macrophages (7,8), induces vasoconstriction (9) and activates endothelial cells (EC) (10,11). Moreover, heme modulates the activity of different plasma systems(12), including the innate immune complement cascade (11,(13)(14)(15)(16)(17).

In hemolytic diseases, signs of complement system (18,19) activation are detected in circulation, on RBC and on endothelium, but the mechanisms and the pathological relevance are still not well understood (13,20,21). Heme triggers Toll-like receptor 4 (TLR4) signaling (7,10) on EC, but the contribution of this innate immune receptor in transforming resting endothelium to a complement-activating surface is unclear. The liver is highly vascularized and impacted by hemolysis, being the second site of RBC fragments clearance and hemoglobin and heme detoxification. Furthermore, hepatic involvement is observed in 10%-40% cases of SCD crisis (22) and in models of hemolytic diseases (17,[START_REF]DEF$-+,$0&/1+,")$6#7I![END_REF], but the contribution of complement system remains unexplored.

We describe a non-conventional mechanism of complement activation in hemolytic conditions. In it, the fluid phase component C3(H 2 O) (or a C3(H 2 O)-like) binds to the EC membranes via P-selectin (P-sel), expressed after heme-triggered TLR4 signaling.

Deficiency/blockade of complement, TLR4, P-sel and heme attenuated the liver stress response in mice with intravascular hemolysis. These revealed the critical impact of the triad &" " " TLR4/P-sel/complement in the liver injury under hemolytic conditions, and its relevance for hemolytic diseases.

Results

C3 activation fragments are deposited on vascular endothelium in the liver of mouse models of intravascular hemolysis

Well-characterized models for heme injection and phenylhydrazine (PHZ)-induced intravascular hemolysis (20,24) were used to determine to what extend hemolysis is responsible for complement deposits in the liver. We used an antibody (Ab), detecting the C3 activation fragments (C3b, iC3b, C3(H 2 O)-like) but not native C3. Treatment with PHZ or heme significantly increased C3 activation fragments deposition on endothelium in the liver, compared to PBS in WT mice (Fig. 1A andB). The double staining of C3 activation fragments/vWF was localized on macrovessels, mainly central hepatic veins, as well as on liver sinusoids (Fig. S1A: Fig. 1 from the SI Appendix), confirmed by co-localization with vascular CD31 (Fig. S1B). Bile ducts were not impacted by complement activation, as shown by the absence of co-localization with an epithelial marker cytokeratine (Fig. S1C). The C3 activation fragments staining in the liver sinusoids co-localized, though, with macrophages/Kupffer cells CD68 (Fig. S1D), granulocytes/neutrophils Ly6G (Fig. S1E) and with platelets' marker CD41a (Fig. S1F). The presence of neutrophils and platelets here supports the concept that complement plays a critical role in the induction of plateletneutrophil aggregates on endothelium(25).

As expected, no C3 activation fragments deposits were detected in C3 -/-mice treated with PHZ (Fig. S2). Importantly, in TLR4 -/-mice, the complement deposits on vessels were strongly reduced, compared to WT mice, despite treatments (Fig. 1A andB). These vascular injection of heme at different regiments (weak effect with 40 !mol/kg i.v. and strong one with two i.p. injections, Fig. 2C and D, Fig. S3A). Of note, i.v. injection of 100!mol/kg of heme was lethal. Therefore, the i.p. injection was selected for the rest of the study, since it was well tolerated by the mice(24, 31) and triggered complement deposits (Fig. 1A andB). The hememediated NGAL expression was attenuated by pretreatment with Hx (Fig. S3B). The NGAL staining in hemolytic mice was with punctiform appearance in the liver sinusoids (likely the EC and Kupffer cells) as well as on macrovessels walls in WT mice (ramification of hepatic arteries, portal veins and central hepatic veins), while in TLR4 -/-, only a fraction of the sinusoidal staining was detected with no signal in the macrovessels.

Having shown the contribution of the C3 activation fragments in the liver injury, we studied the role of C5a and C5b-9. No staining for C5b-9 was detected in the liver sections of PHZ-(" " " injected mice. Nevertheless, C5a showed increased plasmatic levels 24h after PHZ injection (Fig. 2F). To find out if C5a contributes to the pathological process, we injected C5 blocking Ab BB5.1 (known to prevent mouse C5 cleavage(32)), which prevented the C5a generation (Fig. 2E). The treatment resulted in a partial decrease of the ALT levels (Fig. 2F). The NGAL liver staining by immunofluorescence (IF) showed significant decrease (Fig. 2G, Fig. S3C), despite that there was no significant difference in NGAL mRNA levels (Fig. S3D). As expected, the C3 activation fragment deposits were not modified by BB5.1 (Fig. S3E) Taken together, these results demonstrate the critical importance of the TLR4 signaling and complement activation at the level of C3 and to less extent -of C5 for the liver injury during intravascular hemolysis.

Heme triggers TLR4 activation on EC in vitro

Having shown TLR4-dependent EC activation and complement deposits in presence of heme and hemolysis, we studied this process in vitro. TLR4 staining was positive on resting macrovascular EC (primary human umbilical vein endothelial cells, HUVEC) by flow cytometry (Fig. S4A), as described for multiple EC, including of liver origin(33). After 30 min incubation with heme, TLR4-inhibitor TAK-242 or TLR4-ligand LPS, TLR4 expression remained stable (Fig. S4B-D). CD14 staining showed a double peak on resting state, distributed at 50% of both dim (CD14 low ) and bright cells (CD14 high ) (Fig. S4E andF). Bright population of CD14 increased up to 80% after heme treatment, only partially prevented by TAK-242 (Fig. S4F). LPS had no effect (Fig. S4G andH). The absence of TLR4 internalization (occurring in macrophages(34)) despite the rapid CD14 upregulation from intracellular stores (as azurophilic granules of neutrophils (35,36)) could be explained by the lack of TRAM in EC(37).

)" " " TLR4 activation was studied by phosphorylation of its signaling pathway intermediates and was in agreement with previous data(10). Phosphorylation of p-38 increased within 6 h with 25 !M of heme in a time-dependent manner, measured by IF (Fig. S4I). Moreover, translocation of P-p65 in the nucleus increased up to 6 h, measured by western blot and IF (Fig. S4J andK). Similarly, LPS induced an increase of the p38 and p65 phosphorylations.

These results indicate that heme triggers TLR4 signaling on EC.

Inhibition of TLR4 attenuates heme-induced complement deposits on EC.

Our in vivo experiments suggested a direct link between the heme-mediated TLR4 signaling and the EC complement deposits. Therefore, we used an in vitro model, where exposure of EC to heme and normal human serum (NHS) resulted in an increase of C3 activation fragments and C5b-9 deposition within 30 min (Fig. S5A andC). This was associated with increased Bb release (1,8-fold at 100!M heme, compared to medium), confirming the implication of the alternative pathway (AP) (11,38). Inhibition of TLR4 by TAK-242 prevented about 50% of both C3 activation fragments and C5b-9 deposition (Fig. S5B andD).

However, LPS treatment did not induce complement deposits (SI Fig. S5E-H). Moreover, the effect of heme was not related to LPS contamination, since the deposits on heme-exposed EC were unaffected despite pre-incubation with LPS-inhibitor Polymixin B (Fig. S5I andJ).

To find out why LPS did not promote complement deposits in the studied conditions, we incubated different doses of heme or LPS in NHS. At 2 !M, LPS did not trigger complement activation, as measured by release of Bb or sC5b-9 (Fig. S5K andL). In contrast, exposure to 100 !M resulted in release of Bb and sC5b-9 from both LPS and heme (Fig. S5K andL).

Inhibition of TLR4 does not impact the expression/binding of complement regulators and receptors.

*" " "

A defect of complement regulation subsequent to TLR4 activation by heme could explain the partial protective effect of TAK-242 against complement deposition on EC. Thus, we studied the impact of heme-treatment on the factor H (FH) binding on EC and the MCP expression, the two major complement regulators, as well as the expression of C3a and C5a receptors (C3aR and C5aR, respectively) within 30 min.

Binding of FH increased on EC surface in presence of heme, and was not prevented by TAK-242 (Fig. S6A andB). Heme caused a dose-dependent decrease of MCP (up to 50%), in agreement with Frimat et al.(11), which was not prevented by TAK-242 (Fig. S6C andD).

Treatment with LPS did not modulate FH binding or MCP expression (Fig. S7A-D).

To find out whether the downregulation of MCP contributes to the complement deposits on heme-exposed EC, we performed gene silencing, which reduced the expression of MCP to up to 80% (Fig. S6G andH). Upon exposure to heme, the EC with silenced MCP presented more C3 activation fragments deposition, compared with a control siRNA (Fig. S6G andH).

Expression of both C3aR and C5aR revealed a double peak on resting state, distributing 50% of dim and bright cells. Heme-treatment rapidly increased both C3aR and C5aR expressions, enhancing the bright cells population (Fig. S6I-L). However, neither C3aR-nor C5aRincreased expression was prevented by TAK-242 (Fig. S6I-L). In comparison, LPS-treatment did not affect C3aR and C5aR (Fig. S7E-H). These data point towards a second mechanism, mediating complement activation on human EC, which is independent on TLR4 and related to the expression of complement regulators.

A rapid increase of P-sel expression is the causal link between TLR4 and complement activation in presence of heme on EC in vitro.

!+" "

P-sel expression induced by heme could participate to complement deposition on EC(39). First, we observed that both LPS and heme induced Weibel Palade bodies (WPB) mobilization within 30 min, translated by the release of vWF and the rapid expression of P-sel (Fig. S8A-F). We did not detect properdin binding on EC (Fig. S8E andF).

Staining of C3b and P-sel by IF on HUVEC revealed a co-localization in the presence of heme (Fig. S9A). Investigation of the interaction between C3b and P-sel by SPR gave a Kd about 100nM (Fig. 3A). Moreover, C3(H 2 O) was also able to bind recombinant P-sel (Fig. 3B). Presence of heme did not affect the binding of both C3b and C3(H 2 O) on P-sel (Fig. S9B andC). C3(H 2 O) was detected non-covalently bound to heme-exposed EC, as acidic wash abolished C3a staining (Fig. 3C). The acidic wash reduced the C3 fragments binding to the same level as did the pre-incubation with TAK242, showing that the non-covalent binding was TLR-4 dependent (Fig. 3D).

In order to understand whether these interactions have functional consequences on EC, we tested the capacity of a blocking Ab against P-sel to prevent complement deposition. Blocking of P-sel prevented 50% of C3 fragments deposition, compared to cells treated with an irrelevant Ab. This inhibition was equivalent to TLR4 blocking by TAK-242, and no additive effects of TAK-242 and Ab against P-sel were observed (Fig. 3E andF).

P-sel blockade attenuates complement deposits and liver stress response in vivo

Finally, to test the hypothesis that P-sel is responsible for the complement deposits in vivo, we pre-treated the WT mice with blocking anti-P-sel Ab, followed by heme or PHZ. This treatment prevented liver stress response in the PHZ-treated mice, as measured by the decrease of NGAL protein staining (Fig. S10A-C) and mRNA expression (Fig. 4A). This indicates that the injected dose shows some efficiency, despite that it did not rescue the ALT !!" " levels (Fig. S11). Importantly, this P-sel blockade prevented complement deposits on liver endothelium in PHZ-and heme-treated WT mice (Fig. 4B andC), both on large vessels (Fig. 4D left) and on liver sinusoids (Fig. 4D right). Together, the in vitro and in vivo data demonstrate that the P-sel expression is a causal link between TLR4 and complement activation during intravascular hemolysis.

Discussion

Here we demonstrate that intravascular hemolysis triggers complement-dependent liver injury. We found a direct link between heme-triggered TLR4 signaling on endothelium and complement system activation (Fig. S12). These complement deposits are mediated by P-sel expression, causing recruitment of C3b and C3(H 2 O) (or a heme-promoted C3(H 2 O)-like form) on the cell surface. TLR4 signaling-mediated complement activation triggers liver stress response in hemolytic conditions, relevant for SCD.

Despite the clear evidence of complement activation in hemolytic diseases, its pathological relevance remains unclear. After hemolysis induction, C3 activation fragments deposits occurred on liver endothelium and in the sinusoidal vessels. Moreover, the increase of the ALT levels and the overexpression of the inflammation and cell damage marker NGAL were largely prevented in C3 -/-mice. The terminal pathway was also activated, as measured by upregulation of plasmatic C5a, despite the lack of detectable C5b-9 deposits in the liver. At least in part the liver injury was C5-dependent, since ALT and NGAL staining partially decreased after blockade of C5. These results place complement, and especially the C3 activation fragments, as a key mediator of liver tissue damage in hemolytic conditions, such as SCD.

!#" "

The process behind the acquisition of complement-activating phenotype by the endothelium is not well understood. Although predominant in the kidney, this phenomenon is not restricted to glomerular microvasculature(20) and is detected on liver endothelium of heme-injected mice(31) and here in mice with PHZ-induced hemolysis. Here we establish that the complement deposits on endothelium are mediated by TLR4 and triggered by the TLR4 ligand heme. Furthermore, TLR4 deficiency partially prevented the liver stress response in our hemolysis model. Our results support the findings of Bozza and colleagues (7) for the involvement of TLR4 in heme sensing under hemolytic conditions, here in a system exempt from certain heme-related artefacts occurring in vitro (40). Further, we investigated the molecular and cellular mechanism explaining complement activation on EC under hemolytic conditions.

Belcher et al. demonstrated that heme triggers EC activation and WPB mobilization via TLR4(10). Both P-sel(39, 41) and vWF(42, 43) modulate complement activation. P-sel promotes anchoring of C3b to EC membrane(39, 41) and, indeed, here we detected C3b/P-sel interaction. Together with the covalent and non-covalent binding of C3b to the cell surface, we found a non-covalent attachment of C3(H 2 O) (or a heme-promoted C3(H 2 O)-like form) to heme-exposed EC. C3(H 2 O) is the fluid phase activation product of C3, critical for the so called "tick over" of the AP(44). Here we discovered a non-conventional mechanism of complement deposits triggering the AP, where P-sel is expressed on heme-treated EC surface via TLR4-mediated process and serves as a platform for C3(H 2 O) attachment. This explains why C3(H 2 O), was found on heme-exposed EC membrane. Another platform molecule for recruitment of C3 fragments is properdin(45, 46), but we did not detect it on heme-exposed EC. Here, blocking P-sel/C3 activation fragments interaction drastically reduced complement activation and the endothelium injury. Together, these results demonstrate a central role of Psel to tag endothelium as a target for complement activation in vivo, and provide the missing !$" " link between TLR4 and complement activation. This phenomenon is of a particular relevance, since P-sel is a critical player in the vaso-occlusion process of SCD patients(47-50). Indeed, P-sel inhibitor crizanlizumab was associated with a lower frequency of sickle cell-related pain crises in patients(51). Complement activation is also observed in SCD(52-54) and its blockade prevents stasis in a mouse model( 55). Therefore, we postulate that a P-sel blockade will reduce complement activation on endothelium of SCD patients, and therefore the complement-mediated endothelial lesions.

At the trace amount used, sufficient for signaling pathway activation and P-sel expression, as shown here and described for microvascular EC(56) and in mice(41), LPS failed to induce complement activation. Only very high doses of LPS were able to activate complement, in line with previous works performed in vitro (57, 58) and in vivo( 59). In contrast, heme exposes more abilities to promote complement activation on EC. This disparity with LPS lies in the capacity of heme to bind C3 and to generate C3(H 2 O)(11). The concomitant expression of P-sel and C3(H 2 O) generation by heme may explain its capacity to activate complement on endothelium in our model.

Our in vitro results indicate that TLR4 stimulation is not the only mechanism, contributing to complement-activating phenotype of heme-exposed human EC, while it is dominant in mice.

The loss of MCP contributes to the C3 activation fragments deposits in TLR4-independent way, as seen on the MCP-silenced, heme-exposed EC. The decreased MCP levels will hamper the inactivation of C3b to iC3b, promoting this generation of novel convertases. FH binds to the heme-exposed cells to strengthen this regulatory capacity, but is not sufficient to compensate the regulatory function in case of severe MCP loss. Of note, mice lack expression of MCP on EC, pointing at the species differences in the mechanism of complement regulation on endothelium(60) and might explain the dominant effect observed in mice. In the mouse model, the complement activation by RBC microvesicles and the hypoxia-mediated !%" " cell and tissue injury during hemolysis could account for the TLR-4 independent, complement-dependent part of the liver injury.

C3aR and C5aR were rapidly upregulated on EC in a TLR4-independent manner, most likely through exocytosis from still unidentified granules, different from WPB(61) and likely related to lysosomes, as in T-cells(20). Heme triggers C3a and C5a release in serum and on EC surface( 11) and we detected it in the plasma of PHZ-injected mice. This could induce C3aR and C5aR signaling and the subsequent amplification of complement deposits(41, 61). C3a and C5a are potent pro-inflammatory mediators, playing a key role in vascular and tissue injury(19). Hemolysis, TLR4 signaling and the complement anaphylatoxins would synergize to trigger vascular stress response and tissue damage in hemolytic diseases. Moreover, the liver injury is a complex process. Additional hemolysis-derived factors or the ischemia/reperfusion injury are also well known for activating complement system and for involving TLR4. Heme could synergize also with other endogenous TLR4 ligands, which could be released during hemolysis such as HMGB1(62, 63), and could contribute to the liver injury and stress response. RBC microvesicles, activating complement and carrying adhesive C3b deposits, as well as hemoglobin-mediated effects through NO scavenging contribute to these hemolysis-mediated pathologic effects (64,65).

In conclusion, our study demonstrated that complement triggers tissue injury during intravascular hemolysis, by a novel mechanism of acquisition of a heme-dependent, complement-sensitive activated phenotype by endothelium (Fig. S12). The sequence of reactions involves: 1) heme-mediated signaling through TLR4, 2) P-sel expression and 3) interaction of C3b and C3 (H 2 O). This process triggers the amplification loop of the AP, generating local inflammation. This mechanism could operate at any type of endothelium, since EC generally express TLR4. On the basis of these results, targeted therapies on heme, TLR4 pathway or P-sel could prevent organ dysfunction in hemolytic diseases, including by !&" " limiting complement activation. Moreover, complement appears as a novel therapeutic target, at C3 and to some extend at C5 level, to prevent organ injury in hemolytic diseases.

Materials and methods

For fully detailed procedures, please refer to SI Appendix Materials and Methods.

Mouse treatment.

All experiments were conducted in accordance with the recommendations for the care and use of laboratory animals and with the approval APAFIS#7135-" 2016100520465430v5 of the French Ministry of Agriculture. Eight week old C57Bl/6, C3 -/- and TLR4 -/-mice were injected i.p. with 100!L of PBS, freshly prepared heme ( 40!mol/kg(10)), or with PHZ (900 !mol/kg). In a set of experiments, mice were pre-treated with i.p. injection of 40 !mol/kg of human Hx 1 h before heme or PHZ injection. Blocking Ab against P-selectin, anti-C5 or their isotype controls were administered in a set of experiments.

mRNA level analyses. Snap-frozen liver sections were recovered in RLT buffer (Qiagen)+ 1% !-mercaptoethanol and used for mRNA extraction and gene expression analyzed by RTqPCR as described(24). NGAL expression was normalized by actin.

IF.

Five !m thick frozen sections of mouse livers were stained for C3 activation fragments or NGAL. Co-localization with vWF staining was used to quantify the complement deposits on vascular endothelium in the liver as % of double positive area, expressed as FC.

Gene silencing. At 80% confluency, HUVEC were transfected in Opti-MEM medium supplemented with lipotransfectamine and 4 nM of MCP siRNA for 20 min at RT.

!'" "

ALT activity quantification. ALT activity in mouse plasma was quantified using a colorimetric assay, expressed in nmol/min/!L following the protocol provided by the manufacturer.

Surface Plasmon Resonance.

Binding studies were performed using the ProteOn XPR36.

Recombinant human P-selectin was covalently coupled to a sensor chip. Interaction were measured with increased concentration of C3b, and C3(H 2 O). WT, TLR4 -/-and C3 -/-mice were injected with PBS, heme or PHZ. Livers were recovered after 24 hours. A) ALT activity was measured in WT and TLR4 -/-mice, presented as fold change (FC), compared to the PBS group (n"4). B) ALT activity was measured in WT and C3 -/-mice (n"3). C) Gene expression of NGAL in livers from WT and TLR4 -/-mice, FC (n"4). D) Gene expression of NGAL in livers from WT and C3 -/-mice, FC (n"3). E-G) Efficacy of C5 blockade to prevent liver injury. F) C5a levels in the plasma of mice, injected with PHZ or pre-treated with irrelevant IgG (Irr IgG) or anti-C5 blocking Ab BB5.1 ("-C5).

The levels of C5a are compared at day -3 (D-3) and day 1 (D-1) post -PHZ injection. The prevention of C5a release indicated that the Ab exerted its blocking effect (n"4). chosen as described [START_REF] Merle | Characterization of Renal Injury and Inflammation in an Experimental Model of Intravascular Hemolysis[END_REF][START_REF] Tolosano | Defective Recovery and Severe Renal Damage After Acute Hemolysis in Hemopexin---Deficient Mice[END_REF]. At the endpoint, organs were recovered and frozen in nitric liquid and stock to -80°C. We used male and female C57Bl6/J and TLR4-/-or C3-/-mice; sexmatched animals were used for each experiment.

WT mice were pre-treated with blocking antibody against P-selectin or irrelevant IgG by i.v.

with monoclonal rat anti-mouse CD41a--0411-82 MWReg30). Stained slides were scanned by slide scanner Axio Scan (Zeiss). Quantification of the staining was performed using Visiopharm software (Visiopharm A/S, Hørsholm, Denmark). The colocalization of vWF and the C3 activation fragments was used to quantify the complement deposits on vascular endothelium in the liver as % of double positive area. The results were presented as fold change (FC), compared to the results of the PBS-injected mice. This organ has high amount of easily identifiable macro-vessels to allow robust analyses of macro and microvasculature. The NGAL is quantified as % of positive area and presented as FC compared to the results of the PBS-injected mice A L T activity quantification. ALT activity in mouse plasma was quantified using a colorimetric assay (Sigma, MAK052-1KT), expressed in nmol/min/µL following the protocol provided by the manufacturer.

Detection of C5a in mouse plasma. C5a levels in mouse plasma was quantified using a ELISA assay, using Purified Rat Anti-Mouse C5a (BD Pharmingen, Clone I52-1486) as capture and Biotin Rat Anti-Mouse C5a (BD Pharmingen, Clone I52-278) as detection antibody. The levels of C5a were compared for every mice at basal level (day 3) and 24h after injection of PHZ.

The prevention of the PHZ-mediated increase of C5a in the plasma after injection of the anti-C5 blocking antibody Bb5.1 was used as marker to validate the antibody efficacy.

Complement activation and complement receptors on E C F low cytometry. HUVEC were cultured in 24-wells plates as previously described [START_REF] Roumenina | A prevalent C3 mutation in aHUS patients causes a direct C3 convertase gain of function[END_REF]7), exposed to heme at the indicated doses for 30 min at 37 °C in serum-free M199 cell culture

Figure 1 :

 1 Figure 1: Pattern of staining for C3 activation fragments in the liver of mice with intravascular hemolysis and their dependence on TLR4 and heme. A-B) C3 fragments deposition on vascular endothelium. A) Double staining for vWF (green) and C3b/iC3b (red) of WT and TLR4 -/-mice liver frozen sections treated with PBS, heme or PHZ. Co-localization is in orange. Focus on macrovessels, in particular central hepatic veins. White arrows point to C3 activation fragments deposition along endothelium. B) Staining quantification, presented as fold change (FC) of the double positive C3 activation fragments (C3 act fr)/vWF area of each liver section, normalized to the staining of the PBS-injected group of the WT mice. Comparison of WT and TLR4 -/-(n"3) mice. *: p<0.05; **: p<0.005, Two-way ANOVA with Tukey's test for multiple comparisons. C) Frozen liver sections of WT mice, treated with PHZ +/-Hx, were stained for vWF (green) and C3 activation fragments (red).

Figure 2 :

 2 Figure 2: Hemolysis triggers liver injury in a TLR4-and complement-dependent manner. WT, TLR4 -/-and C3 -/-mice were injected with PBS, heme or PHZ. Livers were

  F) ALT activity levels (FC) or G) quantification of the % of NGAL positive area in the liver (FC) in the mice, injected with PBS or PHZ, pretreated with Irr IgG or anti-C5. *: p<0.05; **: p<0.005, ***: p<0.001, ****: p<0.0001; Two-way ANOVA with Tukey's test for multiple comparisons. Values are box plots with median and Min/Max points in (A), (B), (E), (F), (G), and mean +/-SEM in (C) and (D). White arrowheads point to NGAL staining in parenchyma, white arrows point to NGAL staining along endothelium.

Figure 3 :

 3 Figure 3: P-selectin is induced secondary to heme-mediated TLR4 activation and serves as an anchoring platform for C3 activation fragments. A,B) Interaction between C3b/Pselectin and C3(H 2 O)/P-selectin were studied by surface plasmon resonance by injecting

Figure 4 :

 4 Figure 4: P-selectin blockade prevents complement deposits and liver stress response in mice with intravascular hemolysis. Anti-P-selectin administration prevents complement activation on endothelium and increases of liver injury marker NGAL in hemolytic conditions. WT mice were injected with PBS, heme or PHZ after first administration of an irrelevant (Irr Ig) or blocking Ab against P-selectin ("P-sel). A) Gene expression of NGAL in livers. B,C) Staining quantification (n"4) for vascular C3 activation fragments deposits (C3 act fr). B) after injection of heme or C) after induction of hemolysis by PHZ. D) Examples of double staining for vWF (green) and C3 activation fragments (red) of liver frozen sections by IF after PHZ injection. Focus on macrovessels, in particular central hepatic veins (left) and on sinusoidal capillaries (right). White arrows point to colocalization between C3 activation fragments and vWF (orange). *: p<0,05, **: p<0.005, ***: p<0.001. Two way ANOVA with Tukey's test for multiple comparisons. Values are represented as mean +/-SEM in (A) and box plots with median and Min/Max points in (B,C). FC: fold change, compared to PBSinjected mice.

  Figure 1

  Figure 2

  Figure 3

Figure 4 prepared

 4 Figure 4

Acknowledgements

This work was supported by grants from: Agence Nationale de la Recherche (ANR JCJC -INFLACOMP 2015-2018 ANR-15-CE15-0001) to LTR; European Research Council (ERC Starting Grant, ERC-StG-2015-678905 CoBABATI) to JDD; by a grant from CSL Behring France to LTR, and by INSERM. The cytometric and microsocpy analysis were performed at the «" Centre d'Histologie, d'Imagerie et de Cytométrie», (CHIC), Centre de Recherche des Cordeliers UMRS1138, (Paris, France), UPMC Flow Cytometry network (RECYF). We are grateful to the CHIC and CEF teams and to Amélie DiGiovanni for the excellent technical assistance.

Supplementary M aterials and methods

Supplementary F igures P-selectin drives complement attack on endothelium during intravascular hemolysis in TLR-4/heme-dependent manner Nicolas S. Merle 1,2,3 , Romain Paule 1,3,4 , Juliette Léon 1,2,3 , Marie Daugan 1,2,3 , Tania Robe-Rybkine 1,2,3 , Victoria Poillerat 1,2,3 , Carine Torset 1,2,3 , Véronique Frémeaux-Bacchi 1,2,3,5 , Jordan D. Dimitrov 1,2,3,* , Lubka T. Roumenina Immunofluorescence on mouse tissue.

Five µm thick frozen sections of mouse livers were cut with Cryostat Leica AS-LMD and fixed in acetone on ice for 10 min. Complement deposition was studied using rat anti-mouse C3 fragments (C3bi/C3b/C3c) (Hycult, HM1078), visualized with goat anti-rat AF555 (Thermoscientific, A-21434). NGAL expression was studied using goat anti-mouse NGAL (R&D system, AF1857), revealed by donkey anti-goat AF555 or AF647 (Thermoscientific, A21432 and A21447). Endothelial cells were visualized with two EC markers, vWF using polyclonal sheep anti-human vWF (Abcam, ab11713) revealed by donkey anti-sheep AF647

(Thermoscientific, A21448), and CD31 using polyclonal rabbit anti-mouse CD31 (Abcam, ab124432) revealed by goat anti-rabbit AF647 (Thermoscientific, A21245). Granulocytes and macrophages were respectively labeled with rat monoclonal anti-mouse Ly6G-PE (BD and rat monoclonal anti-mouse CD68--0681- MgCl2 to a final concentration of 25 µg/ml or with a polyclonal sheep antibody as an irrelevant antibody (R&D systems, 5-001). HUVEC were stained for C3 deposition and C5b-9 formation as described below.

Gene silencing. At 80% confluency, HUVEC have been cultured in F12K medium, supplemented with 2 mM L-glutamine, 0.1 mg/mL heparin, 0.05 mg/mL ECGS (Merck, 02-102) and 10% FCS. HUVEC were transfected in Opti-MEM medium (Gibco) supplemented Surface Plasmon Resonance. Binding studies were performed using the ProteOn XPR36 Protein Interaction Array system (Bio-Rad Laboratories), based on SPR technology.

Recombinant human P-selectin (R&D Systems) was covalently coupled via amine coupling in 50 mM sodium acetate at pH 4,8 to a sensor chip. The immobilization of concentration of Pselectin resulted in 1,600 resonance units, corresponding to a surface density of 1,3 ng/mm 2 .

C3(H2O) was generated by freezing/thawing native C3. For this reason, exact quantity of C3(H2O) could not be determined, thus preventing calculation of binding affinity. P-selectin coated channel was corrected for non-specific binding to the control channel (no protein; between 5 and 10%). P-Selectin binding at equilibrium was determined at increased concentrations of C3b (A112, CompTech), and freeze/thaw C3 (A113, CompTech) (0.0625, 0.125, 0.25, 0.5 and 1 µM). The set of sensorgrams were best fit to the 1:1 Langmuir binding model using the BIA evaluation software provided by the manufacturer.

Complement activation in serum. Heme or LPS were diluted to final concentration of 0, 2 and 100 µM in 33% NHS and incubated for 30 mins at 37°C. Bb and sC5b-9 levels in the serum were evaluated by ELISA kit following the instructions of the manufacturer (MicroVue Bb Plus EIA and MicroVue SC5b-9 Plus EIA, Quidel).

Statistics

Results were analyzed using a statistical software package (GraphPad Prism 5) as indicated in the figures legends.


Yanlei Xu
email: yanleixu@163.com

Qi Zhang

Chenxiao Li
email: lichenxiao2000@163.com

Xindong Wang

Xiaotian Meng
email: mengxiaotian11@163.com

Non-Invasive Edge Detection of Leaves based on Order Morphology

Keywords: order morphology, plant leaf, edge detection

Non-invasive edge detection of leaves is the key step of leaf feature extraction. Traditional algorithms for edge detection of leaves are usually invasive, since the detection is carried out after the leaves are picked. We apply order morphology in this study to non-invasive edge detection of leaves.

First we analyze the algorithm for order morphology edge detection of leaves. In particular, the impact of structural elements and percentile on the detection is discussed. Based on the theory of order morphology transform of leaf images, the operator for detecting the leaf edge is constructed. Finally simulation experiment is carried out on leaf images under the conditions of natural illumination and artificial noise, respectively. Results show that the proposed algorithm is accurate and fast in leaf edge extraction and not sensitive to noise.

Introduction

In agricultural field, non-contact detection and measurement of plants as a way of monitoring plant growth status is an emerging topic. Leaves are the organs where photosynthesis takes place, and the morphological features of leaves represent an important aspect of plant growth status [START_REF] Allen J Coombes | The Book of Leaves [M[END_REF]. Shape is one major morphological feature of leaves [START_REF] Gheng | Mean-shift-based color segmentation of images containing green vegetation[END_REF][START_REF] Lopez-Molina | Generating fuzzy edge images from gradient magnitudes[END_REF][START_REF] Wen | Application of sobel algorithm in edge detectionof images[END_REF][START_REF] Dong | Color image recognition method based on the prewitt operator[END_REF][START_REF] Jinyong | Segmentation of medical images with canny operator and GVF snake model[END_REF][START_REF] Tizhoosh | Fast fuzzy edge detection[END_REF], and edge extraction of leaves is one step towards leaf shape extraction.

Traditional edge detection algorithms are invasive and involve the use of picked leaves. In non-invasive detection and measurement, leaf images may present the leaves as overlapped with complex shape features. It is difficult to determine the curvature at different positions along the edge. Moreover, illumination also has a great impact on the clarity of the extracted edge. It is now urgent to establish an algorithm for non-invasive edge detection of leaves in open field.

Edge detection algorithms generally depend on the discontinuity of pixel features in different regions. Edges exist between adjacent regions that have different pixel features, and the grayscale values of the pixels along the edge are discontinuous. Considering this fact, edge detection can be carried out using differential operators. Good segmentation can be achieved with Roberts operator, Sobel operator, Prewitt operator and Canny operator where the edges between different regions are clear and the grayscale values of the pixels vary more obviously. But these operators are sensitive to noise [START_REF] Xianmin | A fast edge detection roberts algorithm of coal gangue binary image[END_REF][START_REF] Wen | Application of sobel algorithm in edge detectionof images[END_REF]. Leaf images collected under natural light usually contain considerable noise and the edge information extracted is obscure. Conditional differential operators are not applicable in that case.

More studies are concerned with edge detection of picked leaves than with non-invasive edge detection of leaves. Z. Husin et al. [START_REF] Husin | Embedded portable device for herb leaves recognition using image processing techniques and neural network algorithm[END_REF]proposed threshold-based segmentation for edge detection of leaves, but the algorithm was applied to the entire leaf rather than to the edge. Therefore, this method is not conducive to shape extraction, and only specific types of crops can be detected. More importantly, the leaves were detected only after being picked. Li et al. [START_REF] Han | Cotton leaf image edge detection using Mean-shift algorithm and lifting wavelet transform[END_REF]proposed the algorithm for edge detection of leaves in cotton images based on Mean-shift and lifted wavelet transform. This algorithm reduced the non-edge noise and more effectively extracted the edge of overlapping leaves. But this algorithm was more fit for images collected under laboratory conditions than under natural light. Lin et al. [START_REF] Kaiyan | Plant leaf edge detection based on fuzzy logic[J][END_REF] described the method of edge detection of leaves based on fuzzy logics, which reduced the pseudo-edge of leaves. But this method was confined to leaf images collected in greenhouse environment. Xu et al. [START_REF] Yanlei | Plant leaf vein edge detection based on fuzzy order morphology [J][END_REF] put forward the method of edge detection of leaf veins based on fuzzy order morphology. Hu et al. [START_REF] Hu Chunhua | Edge detection of hardwood seedlings leaves based on intuitionistic fuzzy set [J][END_REF]established the method of edge detection of seedlings of broad-leaved trees based on intuitionistic fuzzy set. The above algorithms are based on fuzzy theory and they are capable of extracting clear edges, though the computing load is very large and the complexity is high.

Order morphology theory is the combination of mathematical morphology and order statistics. Order morphology transform of digital images is the ordering of finite data of grayscale values of pixels within the range of structural elements [START_REF] Arce | Detail preserving ranked-order based filters for image processing[END_REF]. Order morphology provides good filtering effect and noise inhibiting effect, especially in the open field. We first analyze the order morphology theory and the order morphology transform of leaf images. Structural elements and percentile are identified as the influence factors of the detection result. Finally, the leaf edge is extracted by order morphology transform. Simulation experiments are carried out under natural light condition, and the extracted leaf edges are clear and accurate and insensitive to noise.

2

Theory of order morphology transform

Basic concepts

The concept and property of order morphology transform are first defined [START_REF] Yan Xue-Qiang | Applications of Order Morphology to Image Edges Detection[END_REF][START_REF] Hu | A Multi-directions Algorithm for Edge Detection Based on Fuzzy Mathematical Morphology[END_REF]. Definition 1: Suppose the set of discrete points is A, B is structural element, 0≤µ(B)=k<+∞ (metricµ(.) is the count of B points). Thus the order morphology transform of A with respect to B is :

A○ PE A B (p=0,1/(k-1), 2/(k-1),…,1), defined as X= AA ○ PE A B { } 1 1 2 1 1 0, = 1 + 1) = ) ∩ : = , k , k p , p (k B A ( μ x  - - (1)
where p is the percentile for order morphology transform. Discrete order morphology transform can be based either on morphology sum and morphology difference.

AA ○ 0E A B= A B A x = = ∩ }} 1 ) ( : { µ ⊙B (2) AA ○ 1E A B= A k B A x = = ∩ } ) ( : { µ ⊕B (3)
The geometrical meaning of discrete is: AA ○ PE A B is the x set of (k-1)p+1 points which B include A The properties of order morphological transformation AA ○ PE A B are: monotonicity, expansibility and inverse expansibility.

Property 1 AA ○ 0E A A B ⊂ A ○ PE A A B ⊂ A ○ 1E A B AA ○ P1E A A B ⊂ A ○ P2E A B, 0≤p1≤p2≤1 Property 2 A ⇒ C ⊂ A A ○ PE A C ⊂ B A ○ PE A B Property 3 (AA ○ pE A B) c =A c A ○ qE A B, p+q=1
Property 3 show that the morphology transformation AA ○ PE A B and AA ○ qE A B,p+q=1, are the dual transformation.

Definition 2: Suppose 0≤p, q≤1, A(p,q)B= AA ○ PE A BA ○ qE A is called as multiplex order morphology transformation, obviously:

B A B A  = ) 1 , 0 ( (4) B A B A • = ) 0 , 1 ( (5) B A A B A ) 0 , 1 ( ) 1 , 0 ( ⊂ ⊂ (6) c c B q p A B q p A ] ) 1 , 1 ( [ ) , ( - - = (7)
Definition 2: n multiplex order morphology is defined as:

AA ○ PE A n B=(…(AA ○ PE A B) A ○ PE A B)A ○ PE A B)…A ○ PE A B (8)

Order morphology transformation of plan leaf

The order morphology is applied to the plant leaf, which is the order transformation of plan leave.

(1) Z(n) is the n-D digital space (2) f or g is the gray image of plant leaf (3) B is the structure element (4) nB=B⊕B⊕B⊕B…………⊕B

(5) f≤g is f(x) ≤g(x), f-g is f(x)-g(x) , x∈Z(n) Definition 1 Supposed plant leaf image f: 0≤ f(x) ≤m(m is the gray value) , B={x1,x2, …xk} is the structure element, 0≤µ(B)=k<+∞ µ(.) is the count of B. The order of k values in structure element B is: f(x1*) ≤ f(x2*) ≤…≤ f(xk*), define the d rank order variable in structure element B: order(d :f| B)= f(x d*) (d=1, 2, …,k) (9) Definition 2 The d rank order variable of f(x) in B (d=(k-1)p+1 is rank)is defined as the order morphological transformation of f about B, marked fA ○ PE A B, that is f g d j i = ) ( A ○ PE A ) 1 , ), 1 /( 2 ), 1 /( 1 , 0 *)( ( ) / : (  - - = = = k k p x f B f d order B d (10) Definition 3 f ( p, q) B = ( fA ○ PE A B )A ○ qE A B, ( p, q = 0,1 / ( k -1) ,…,1) is named duplicate symmetrical mix order morphological transformation of f about structure element B.

Analysis of edge features in order morphology transform

As shown above, order morphology transform can act as a filter using different percentile p on the grayscale images. Thus the grayscale images can be treated by various mathematical operations [START_REF] Serra | Morphological filtering: An overview[END_REF][START_REF] Cousty | Morphological filtering on graphs[END_REF][START_REF] Wilburn | Theory of ranked-order filters with applications to feature extraction and interpretive transforms[END_REF][START_REF] Bin | Study on Application of Order Morphology Filtering for Detection of Small Target and Point Target[J][END_REF][START_REF] Zhao Chun-Hui | A Class of Adaptive Ranked-Order Morphological Filters[J][END_REF][START_REF] Xu | Gray-scale Image Edge Detection Based on Order Morphology Transformation[END_REF][START_REF] Jian-Nan | Gray-scale Image Edge Detection Based on Multi-structuring Elements Order Morphology Transformation[J][END_REF][START_REF] Jian-Nan | Infrared Image Edge Detection Based on Multiplex Order Morphology Transformation[END_REF][START_REF] Haixia | Edge detection method based on adaptive order morphology filter[END_REF][START_REF] Haixia | Edge detection method based on Tsallis entropy difference of remotesensing image[END_REF][START_REF] Bin | Mov ing Small Target Detect ion Based on Order Morphology Filtering in Infrared Image Sequences[END_REF].

Filtering and edge detection by order morphology transform

The leaf images contain much noise, which can be filtered by order morphology transform using different percentile. When 0 < p < 1 /2, the positive pulse noise in the image can be inhibited; when 1 /2 < p < 1, the negative pulse noise is inhibited [29] . Two transforms can be performed separately, that is, using p value of 0-1/2 and p of 1/2-1. Thus both positive and negative pulse noise is eliminated. Suppose 0 < p < 1 /2 and 1 /2 < q < 1. Let f ( p, q) B be open operation and f (q, p) B closed operation, both of which can act as filter. The open operation can filter out the high-luminance noise and the closed operation the smaller noise. Therefore, all types of noise can be removed by setting different p and q values.

The noise in the image can be removed and the edge be detected using the above operators. Fig. 1 is the schematic diagram of edge detection based on order morphology transform. When structural element B is located in the flat region, the grayscale values of pixels within B do not vary much. So the difference is not great before and after the transform. When B is located in the region of abrupt change of grayscale values, the grayscale values of the pixels within B vary considerably. The output image will be very different from the original image. When 0<p<1/2, the output image fA ○ PE A B is subtracted from the original image f. Given the anti-extensibility of order morphology transform, the difference f-fA ○ PE A B (0<p<1/2) is the edge information of the original image. When 1/2<p<1, the original image f is subtracted from the output image fA ○ PE A B, and the difference fA ○ PE A B -f (1/2<p<1) is the edge information of the original image.

Fig.1 Schematic diagram of edge detection

The working principle of edge detection of leaves based on order morphology transform can be described briefly as follows: When the structural element B is located within the range of the leaf, the grayscale values of the pixels within B vary slightly. Therefore, the input and output grayscale values differ little. When B is located over the edge of the leaves, the grayscale values within B vary greatly, so the input and output grayscale values also show large difference. By amplifying the changes of grayscale values of the edges through the transform, the leaf edges can be accurately recognized. Thus the edge detection operator based on order morphology has the following form:

Order morphology has unparallel advantages in edge detection by choosing proper structural elements and percentile. We build the operator based on order morphology for edge detection of leaf images.

G(f)=f -fA ○ PE A B ( p<1/2) G(f)=fA ○ PE A B -f (p>1/2) ( 11
)

Building the edge detection operator

If only one structural element is used, then the output image will contain only one type of geometrical information, which means a loss of leaf edge details. To detect the edge more accurately, several structural elements may used. These structural elements may differ in size and shape. We present the edge detection operator based on several structural elements.

G(f)=fA ○ qE A n1B1A ○ PE A n2B2A ○ qE A n3B3 -fA ○ qE A n4B4A ○ PE A n5B5) (12) G(f)=fA ○ qE A n1B1A ○ PE A n2B2 -fA ○ qE A n3B3A ○ PE A n4B4A ○ PE A n5B5) (13) G(f)=fA ○ qE A n1B1A ○ PE A n2B2A ○ qE A n3B3 -fA ○ qE A n4B4A ○ PE A n5B5A ○ PE A n6B6) (14)
In the operator, ni represents different structural element; Bi indicates the shape of the structural element. The outer edge of the image is extracted in formula [START_REF] Kaiyan | Plant leaf edge detection based on fuzzy logic[J][END_REF], and the edge spanning over the Euclidean border is extracted in formula [START_REF] Hu Chunhua | Edge detection of hardwood seedlings leaves based on intuitionistic fuzzy set [J][END_REF]. Different shape of the structural elements corresponds to order morphology operation in different directions of the leaf. The final leaf edge is obtained by combining together the edges extracted in all directions. For a given structural element, different size corresponds to different scale of extraction. Small-size structural elements have lower noise-removing capacity, but they are more sensitive to edge details. On the contrary, large-size structural elements have stronger noise-removing capacity, but the more details may be lost. So the best way is to use several structural elements with varying size and shape. An illustration is carried out using formula [START_REF] Hu Chunhua | Edge detection of hardwood seedlings leaves based on intuitionistic fuzzy set [J][END_REF]. This formula extracts the Euclidean edge of the leaf. Suppose the structural elements n1B1 and n2B2, n4B4 and n5B5 have identical shape and size, respectively. If n3B3 and n6B6 have the same shape but different size, then the larger the n3 and n6, the wider and the brighter the extracted edge; otherwise, the edge will be narrower and darker. If n3B3 and n6B6 have the same size but different shape, it is necessary to choose appropriate shape of the structural element depending on specific leaf so as to extract the edge more accurately.

Impact of structural element and percentile on leaf edge detection

In mathematical morphology, the choice of structural elements has a large impact on the extracted edge [START_REF] Guo Jingfeng | The Analysis and Study of Structure Element of Mathematical Morphology[END_REF]. This is also true in order morphology transform, but no relevant studies have been reported yet. We discuss the impact of structural elements and percentile on leaf edge detection.

The thickness and direction of the extracted edge can be controlled by controlling the shape and size of the structural elements. The structural element is expressed as the morphology sum of several structural element Bj, which is niBj (n is the size and B is the shape). Fig. 2 shows the edges of different thickness when the size of the structural element is 3×3, 4×4 and 5×5, respectively, with the constant shape of a rectangle. It can be seen from Fig. 2 that the larger the size of the structural element, the thicker the extracted edge is. But thicker edges are usually obscure with the generation of some pseudo-edges, which increases the difficulty of further processing. To overcome this defect, the size of the structural element can be varied depending on the needs.

Similarly, by changing the shape of the structural elements, the edges extracted will show different features. Given the same size of the structural elements, different shape of the structural elements will result in different features of edges. Here we choose the constant size of 3×3 and vary the shape of the structural elements. The results are shown in Fig. 3. As shown in Fig. 3, edges with different features are extracted by setting different shape of the structural elements. For example, a cross-like shape results in satisfactory edges in transverse and vertical direction, but poor edges in other directions. In Fig. 3(b), both the cross-like shape and the diagonal shape result in clear edges. Because the leaf edges are approximately elliptical or circular or oblique lines but rarely rectangular, the use of rectangular structural element results in less clear edges (See Fig. 3(a)). It is important to choose the appropriate shape of structural elements for specific leaves, as discussed below.

In order morphology transform, the values of percentile (p and q) are important to edge detection. We discuss the impact of percentile on edge detection by choosing different percentile values. Let 0 < p < 1 /2 and 1 /2 < q < 1 and the structural element is rectangular with size of 6×6. The leaf vein edges are extracted at different p and q values, as shown in Fig. 4. When p=1/3 and q=35/36, the extracted leaf edges are shown in Fig. 4(a); when p=1/36 and q=35/36, the extracted leaf edges are shown in Fig. 4 (b); when p=1/3 and q=35/36, the leaf edges are shown in Fig. 4(c). It is easy to see that if p and q differ greatly, the extracted edges are more clear and thicker, though some details are lost (comparing (a) and (b), the leaf veins are more clear). But pseudo-edges appear in (a). After several experiments, it is found that a large difference existing between p and q is more conducive to edge detection.

Simulation experiment on order morphology edge detection of leaves

We carry out simulation experiment on order morphology detection of leaves under natural light condition. 14), n1 is 5×5 and B1 is rectangular; n2 is 3×3 and B2 is rectangular; n3 is 7×7 and B3 is cross-like shape. n4B4 and n5B5, n1B1 and n2B2 have identical shape and size, respectively. When n6 is 5×5 and B6 is rectangular, the detection result is shown in Fig. (e). Then n1 is 5×5 and B1 is rectangular; n2 is 3×3 and B2 is rectangular; n3 is 5×5 and B3 is diagonal; n4B4 and n5B5, n1B1 and n2B2 have identical shape and size, respectively. When n6 is 5×5 and B6 is cross-like shape, the detection result is shown in One major advantage of order morphology edge detection is that the noise is filtered simultaneously. So we perform another experiment where 10% salt and pepper noise is added into the leaf images. The edge detection of leaves is carried out using Sobel operator, Canny operator and the proposed operator in Formula (11), respectively. The results are shown in Fig. 6. This proves the ability of the proposed operator in filtering noise and its high applicability to natural light condition. Since the proposed operator allows for the adjustment of size and shape of structural elements and the percentile value, it can adapt to a variety of leaf features perfectly.

Conclusion

We propose a method of non-invasive order morphology edge detection of leaves. Starting with thorough analysis of the order morphology theory and order morphology transform of leaf images, we proceed to the noise filtering and edge detection ability of order morphology. Structural elements and percentile have large impact on the detection results. We then build the order morphology edge detection operator based on several structural elements with varying size and shape. The algorithm is verified by simulation experiment.

The experiment consists of two parts. One is the edge extraction from leaf images collected under natural light condition. The other is edge extraction in the presence of artificial noise. Results show that the leaf edges extracted by the proposed operator are more clear and accurate and insensitive to noise. This operator can adapt to natural light condition and can be used for shape recognition of plant leaves and plant growth monitoring in the future.

Fig. 2

 2 Leaf edge of different thickness

Fig. 3

 3 Fig.3 Edge of different shape

3 Fig. 4

 34 Fig.4 Edge of different percent

Fig. 5 (

 5 a) is the original leaf image, and the leaf edges are extracted using traditional Sobel operator and Canny operator. The results are shown in Fig.5(b) and (c). We apply order morphology edge detection operator in formula[START_REF] Han | Cotton leaf image edge detection using Mean-shift algorithm and lifting wavelet transform[END_REF] to extract leaf edges, as shown in Fig.5(d). Using the proposed operator based on several structural elements of different size and shape, the extracted edges are shown in Fig.5(e) and (f).

Fig. 5

 5 Fig.5 Edge detection of plant leafThe leaf edges extracted by Sobel operator are less clear and discontinuous with considerable loss of details. Canny operator fails to extract leaf edges and the resulting image is obscure and does not show the general contour of the leaves. Though the traditional order morphology operators can detect the leaf edges, the edges have low contrast and do not present a clear contour with loss of some of the edge information. Formula (14) is used as the edge detection operator based on several structural elements of varying size and shape. The edges in Fig. (e) and Fig. (f) are extracted using different structural elements. In Formula (14), n1 is 5×5 and B1 is rectangular; n2 is 3×3 and B2 is rectangular; n3 is 7×7 and B3 is cross-like shape. n4B4 and n5B5, n1B1 and n2B2 have identical shape and size, respectively. When n6 is 5×5 and B6 is rectangular, the detection result is shown in Fig.(e). Then n1 is 5×5 and B1 is rectangular; n2 is 3×3 and B2 is rectangular; n3 is 5×5 and B3 is diagonal; n4B4 and n5B5, n1B1 and n2B2 have identical shape and size, respectively. When n6 is 5×5 and B6 is cross-like shape, the detection result is shown in Fig. (f). All edges are clear and intact using different structural elements. The edges in Fig. (3) have the highest clarity and contrast, because the structural elements used are larger and have more variable sizes than those in Fig. f.One major advantage of order morphology edge detection is that the noise is filtered simultaneously. So we perform another experiment where 10% salt and pepper noise is added into the leaf images. The edge detection of leaves is carried out using Sobel operator, Canny operator and the proposed operator in Formula (11), respectively. The results are shown in Fig.6.

  Fig. (f). All edges are clear and intact using different structural elements. The edges in Fig. (3) have the highest clarity and contrast, because the structural elements used are larger and have more variable sizes than those in Fig. f.

Fig. 6

 6 Fig.6 Edge detection of noised plant leaf