William Dimbour
email: william.dimbour@espe-guyane.fr

Mawaki Solym

Manou-Abi
email: mawaki.manou-abi@univ-mayotte.fr

S-ASYMPTOTICALLY ω-PERIODIC SOLUTION FOR A NONLINEAR DIFFERENTIAL EQUATION WITH PIECEWISE CONSTANT ARGUMENT VIA S-ASYMPTOTICALLY ω-PERIODIC FUNCTIONS IN THE STEPANOV SENSE

Keywords: S-Asymptotically ω-periodic functions, differential equations with piecewise constant argument, evolutionnary process

In this paper, we show the existence of function which is not S-asymptotically ω-periodic, but which is S-asymptotically ω-periodic in the Stepanov sense. We give sufficient conditions for the existence and uniqueness of S-asymptotically ω-periodic solutions for a nonautonomous differential equation with piecewise constant argument in a Banach space when ω is an integer. This is done using the Banach fixed point Theorem. An example involving the heat operator is discussed as an illustration of the theory.

Introduction

In this paper, we study the existence and uniqueness of S-asymptotically ω-periodic solution of the following differential equation with piecewise constant argument

x ′ (t) = A(t)x(t) + f (t, x([t])), x(0) = c 0 , (1)
where X is a banach space, c 0 ∈ X, [•] is the largest integer function, f is a continuous function on R + × X and A(t) generates an exponentially stable evolutionnary process in X. The study of differential equations with piecewise constant argument (EPCA) is an important subject because these equations have the structure of continuous dynamical systems in intervals of unit length. Therefore they combine the properties of both differential and difference equations. There have been many papers studying EPCA, see for instance [START_REF] Wiener | A Second-Order delay differential equation with multiple Periodic solutions[END_REF], [START_REF] Wiener | Boundary Value Problems for the diffusion equation with piecewise continuous time delay[END_REF], [START_REF] Wiener | A survey of partial differential equations with piecewise continuous arguments[END_REF], [START_REF] Wiener | Excitability of a second-order delay differential equation[END_REF], [START_REF] Wiener | Generalized solutions of functional differential equations[END_REF] and the references therein.

Recently, the concept of S-asymptotically ω-periodic function has been introduced in the litterature by Henríquez, Pierri and Táboas in [START_REF] Henríquez | On S asymptotically ω-periodic function on Banach spaces and applications[END_REF], [START_REF] Henríquez | Existence of S-asymptotically ω-periodic solutions for abstract neutral equations[END_REF]. In [START_REF] Blot | Guérékata S asymptotically ω-periodic functions and applications to evolution equations[END_REF], the authors studied properties of S-asymptotically ω-periodic function taking values in Banach spaces including a theorem of composition. They applied the results obtained in order to study the existence and uniqueness of S-asymptotically ω-periodic mild solution to a nonautonomous semilinear differential equation. In [START_REF] Wu | Asymptotic periodicity for a class of fractional integrodifferential equations[END_REF], the authors established some sufficient conditions about the existence and uniquenes of S-asymptotically ω-periodic solutions to a fractionnal integro-differential equation by applying fixed point theorem combined with sectorial operator, where the nonlinear pertubation term f is a Lipschitz and non-Lipschitz case. In [START_REF] Caicedo | Asymptotically behavior of solutions of some semilinear functional differential and integro-differential equations with infinite delay in Banach spaces[END_REF], the authors prove the existence and uniqueness of mild solution to some functional differential equations with infinite delay in Banach spaces which approach almost automorphic function ( [START_REF] Diagana | Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces[END_REF], [START_REF] N'guérékata | Almost Automorphic and Almost Periodic Functions in Abstract Spaces[END_REF]) at infinity and discuss also the existence of S-asymptotically ω-periodic mild solu-tions. In [START_REF] Xia | Asymptotically periodic of semilinear fractional integro-differential equations[END_REF], the author discussed about the existence of S-asymptotically ω-periodic mild solution of semilinear fractionnal integro-differential equations in Banach space, where the nonlinear pertubation is S-asymptotically ω-periodic or S-asymptotically ωperiodic in the Stepanov sense ( [START_REF] Henríquez | Asymptotically periodic solutions of abstract differential equations[END_REF], [START_REF] Xia | Asymptotically periodic of semilinear fractional integro-differential equations[END_REF], [START_REF] Xie | Criteria of asymptotic ω-periodicity and their applications in a class of fractional differential equations[END_REF]). The reader may also consult [START_REF] Cuevas | S asymptotically ω-periodic solutions of semilinear fractional integro-differential equations[END_REF], [START_REF] Cuevas | S asymptotically ω-periodic solutions of semilinear Volterra equations[END_REF], [START_REF] Cuevas | Existence of S asymptotically ω-periodic solutions for two-times fractional order differential equations. Southeast[END_REF], [START_REF] Dimbour | S asymptotically ω-periodic solution for a nonlinear differential equation with piecewise constant argument in a Banach space[END_REF], [START_REF] Pierri | On S-Asymptotically ω-periodic functions and applications[END_REF] in order to obtain more knowledge about S-asymptotically ω-periodic functions. Motivated by [START_REF] Blot | Guérékata S asymptotically ω-periodic functions and applications to evolution equations[END_REF] and [START_REF] Dimbour | S asymptotically ω-periodic solution for a nonlinear differential equation with piecewise constant argument in a Banach space[END_REF], we will show the existence and uniqueness of Sasymptotically ω-periodic solution for [START_REF] Blot | Guérékata S asymptotically ω-periodic functions and applications to evolution equations[END_REF] where the nonlinear pertubation term f is a S-asymptotically ω-periodic function in the Stepanov sense. The work has four sections. In the next section, we recall some properties about S-asymptotically ω-periodic functions. We study also qualitative properties of Sasymptotically ω-periodic functions in the Stepanov sense. In particular, we will show the existence of functions which are not S-asymptotically ω-periodic but which are S-asymptotically ω-periodic in the Stepanov sense. In section 3, we study the existence and uniquenes of S-asymptotically ω-periodic mild solutions for (1) considering S-asymptotically ω-periodic functions in the Stepanov sense. In section 4, we deal with the existence and uniqueness of S-asymptotically ω-periodic solution for a partial differential equation.

Preliminaries

Definition 2.1.

([8]) A function f ∈ BC(R + , X)
is called S-asymptotically ω periodic if there exists ω such that lim t→∞ (f (t + ω) -f (t)) = 0. In this case we say that ω is an asymptotic period of f and that f is S-asymptotically ω periodic. The set of all such functions will be denoted by SAP ω (R + , X).

Definition 2.2. ([8]

) A continuous function f : R + × X → X is said to be uniformly S-asymptotically ω periodic on bounded sets if for every bounded set ) Assume f : R + × X → X be a function which is uniformly S-asymptotically ω periodic on bounded sets and satisfies the Lipschitz condtion, that is, there exists a constant L > 0 such that

K * ⊂ X, the set {f (t, x) : t ≥ 0, x ∈ K * } is bounded and lim t→∞ (f (t, x) -f (t + ω, x)) = 0 uniformly in x ∈ K * .
||f (t, x) -f (t, y)|| ≤ L||x -y||, ∀t ≥ 0, ∀x, y ∈ X.
If u ∈ SAP ω (R + , X), then the function t → f (t, u(t)) belongs to SAP ω (R + , X).

Let p ∈ [0, ∞[. The space BS p (R + , X) of all Stepanov bounded functions, with the exponent p, consists of all measurable functions f :

R + → X such that f b ∈ L ∞ (R, L p ([0, 1]; X)), where f b is the Bochner transform of f defined by f b (t, s) := f (t + s), t ∈ R + , s ∈ [0, 1]. BS p (R + , X) is a Banach space with the norm ||f || S p = ||f b || L ∞ (R + ,L p ) = sup t∈R + t+1 t ||f (τ )|| p dτ 1 p . It is obvious that L p (R + , X) ⊂ BS p (R + , X) ⊂ L p loc (R + , X) and BS p (R + , X) ⊂ BS q (R + , X) for p ≥ q ≥ 1. We denote by BS p 0 (R + , X) the subspace of BS p (R + , X) consisting of functions f such that t+1 t ||f (s)|| p ds → 0 when t → ∞.
Now we give the definition of S-asymptotically ωperiodic functions in the Stepanov sense.

Definition 2.4. [10] A function f ∈ BS p (R + , X) is called S-asymptotically ω-periodic in the Stepanov sense (or S p -S-asymptotically ω-periodic)if lim t→∞ t+1 t ||f (s + ω) -f (s)|| p = 0. Denote by S p SAP ω (R + , X) the set of such functions. Remark 2.1. It is easy to see that SAP ω (R + , X) ⊂ S p SAP ω (R + , X). Lemma 2.4. Let u ∈ SAP ω (R + , X) where ω ∈ N * , then the step function t → u([t]) satisfies lim t→∞ u( t + ω ) -u( t ) = 0.
Remark 2.2. The proof of the above Lemma is contained in the lines of the proof of the Lemma 2 in [START_REF] Dimbour | S asymptotically ω-periodic solution for a nonlinear differential equation with piecewise constant argument in a Banach space[END_REF].

Corollary 2.5. Let u ∈ SAP ω (R + , X) where ω ∈ N * , then the function t → u([t]) is S-asymptotically ω-periodic in the Stepanov sense but is not S- asymptotically ω-periodic.
Proof. By the above Lemma we have :

∀ǫ 1/p > 0, ∃T > 0; t ≥ T ⇒ ||u([t+ω])-u([t])|| ≤ ǫ 1/p . The function t → u[t] is a step function therefore it is measurable on R + . Then for t ≥ [T ] + 1, we have t+1 t ||u([s + ω]) -u([s])|| p ≤ t+1 t ǫds ≤ ǫ.

Therefore the function t → u([t]

) is S-asymptotically ω-periodic in the Stepanov sense. Now since the function t → u([t]) is not continuous on R + , it can't be S-asymptotically ω-periodic. Definition 2.5. [10] A function f : R + × X → X is said to be uniformly S-asymptotically ω-periodic on bounded sets in the Stepanov sense if for every bounded set B ⊂ X,there exist positive functions

g b ∈ BS p (R + , R) and h b ∈ BS p 0 (R + , R) such that f (t, x) ≤ g b (t) for all t ≥ 0, x ∈ B and ||f (t + ω, x) -f (t, x)|| ≤ h b (t) for all t ≥ 0, x ∈ B.
Denote by S p SAP ω (R + × X, X) the set of such functions.

Definition 2.6. [START_REF] Henríquez | Asymptotically periodic solutions of abstract differential equations[END_REF] A function f : R + × X → X is said to be asymptotically uniformly continuous on bounded sets in the Stepanov sense if for every ǫ > 0 and every bounded set B ⊂ X, there exists t ǫ ≥ 0 and

δ ǫ > 0 such that t+1 t ||f (s, x) -f (s, y)|| p ds ≤ ǫ p , for all t ≥ t ǫ and all x, y ∈ B with ||x -y|| ≤ δ ǫ . Lemma 2.6. [10] Assume that f ∈ S p SAP ω (R + × X, X) is an asymptotically uniformly continuous on bounded sets in the Stepanov sense function. Let u ∈ SAP ω (R + , X), then v(.) = f (., u(.)) ∈ S p SAP ω (R + × X, X).
Lemma 2.7. Let ω ∈ N * . Assume f : R + × X → X be a function which is uniformly S-asymptotically ω periodic on bounded sets and satisfies the Lipschitz condition, that is, there exists a constant L > 0 such that

||f (t, x) -f (t, y)|| ≤ L||x -y||, ∀t ≥ 0, ∀x, y ∈ X. If u ∈ SAP ω (R + , X), then (1) the bounded piecewise continuous function t → f (t, u( t )) satisfies lim t→∞ (f (t + ω, u( t + ω )) -f (t, u( t )) = 0.
(2) the function t → f (t, u( t )) belongs to S p SAP ω (R + , X).

(3) the function t → f (t, u( t )) does not belongs to SAP ω (R + , X).

Proof.

(1) Since R(u) = {u( t )|t ≥ 0} is a bounded set, then for every ǫ 2 > 0, there exists a constant L ǫ > 0 such that ||f (t + ω, x) -f (t, x)|| ≤ ǫ 2 for every t > L ǫ and x ∈ R(u). By Lemma 2.4, for every ǫ 2L > 0, there exist T ǫ > 0 such that for all t > T ǫ

||u( t + ω ) -u( t )|| ≤ ǫ 2L .
We have

||f (t + ω, u( t + ω )) -f (t, u( t )|| ≤ ||f (t + ω, u( t + ω )) -f (t, u( t + ω ))|| + ||f (t, u( t + ω )) -f (t, u( t )|| ≤ ||f (t + ω, u( t + ω )) -f (t, u( t + ω ))|| + L||u( t + ω ) -u( t )||.
We put T = max(T ǫ , L ǫ ). Then for all t > T we deduce that

||f (t + ω, u( t + ω )) -f (t, u( t )|| ≤ ǫ 2 + L ǫ 2L ≤ ǫ.
(2) According to (1) we have

lim t→∞ (f (t + ω, u([t + ω]) -f (t, u([t]))) = 0, meaning that ∀ǫ 1/p > 0, ∃T > 0, t ≥ T ⇒ ||f (t + ω, u([t + ω])) -f (t, u([t]))|| ≤ ǫ 1/p . The function t → f (t, u[t]) is continuous on every in- tervals ]n, +1[ but lim t→n -f (t, u( t )) = f n, u(n -1) and lim t→n + f (t, u([t]) = f (n, u(n)). Therefore the func- tion t → f (t, u[t]
) is a piecewise continuous function and it is measurable on R + . Then for t ≥ [T ] + 1, we have

t+1 t ||f (s, u([s + ω])) -f (s, u([s]))|| p ≤ t+1 t ǫ ds ≤ ǫ.
(3) Since the function t → f (t, u([t])) is not continuous on R + , it can't be S-asymptotically ωperiodic.

Lemma 2.8. Let ω ∈ N * . Assume that f : R + × X → X is uniformly S-asymptotically ω-periodic on bounded sets in the Stepanov sense and asymptotically uniformly continuous on bounded sets in the Stepanov sense. Let u : R + → X be a function in SAP ω (R + , X), and let

v(t) = f (t, u([t])). Then v ∈ S p SAP ω (R + , X). Proof. Set B =: R(u) = {u[t], t ≥ 0} ⊂ X.
Since f is uniformly S-asymptotically ω-periodic on bounded sets in the Stepanov sense, there exist functions g B ∈ BS p (R + , R) and h B ∈ BS p 0 (R + , R) satisfying the properties involved in Definition 2.6 and 2.8 in relation with the set B =: R(u). The function v belongs to BS p (R + , X) because

t+1 t ||v(τ )|| p dτ = t+1 t ||f (τ, u([τ ]))|| p dτ ≤ t+1 t ||g B (τ )|| p dτ ≤ sup t≥0 t+1 t ||g B (τ )|| p dτ . Therefore ||v b || L ∞ (R + ,L p ) ≤ ||g B || S p .
We have for all t ≥ 0 :

t+1 t ||f (s + ω, u([s + ω])) -f (s, u([s + ω]))|| p ds ≤ t+1 t ||h B (s)|| p ds.
Note that h B ∈ BS p 0 (R + , R); this implies that for ǫ > 0 there exists t ′ ǫ > 0 such that for all t ≥ t ′ ǫ we have

t+1 t ||h B (s)|| p ds ≤ ǫ p /2. Thus t+1 t ||f (s + ω, u([s + ω])) -f (s, u([s + ω]))|| p ds ≤ ǫ p /2 for all t ≥ t ′ ǫ ( * )
. Furthermore since f is asymptotically uniformly continuous on bounded sets in the Stepanov sense, thus for all ǫ > 0, theres exists t ǫ ≥ 0 and δ ǫ > 0 such that

t+1 t ||f (s, u([s + ω])) -f (s, u([s]))|| p ds ≤ ǫ p /2 for all t ≥ t ǫ ( * * ) because ||u([s + ω]) -u([s])|| ≤ δ ǫ .
The estimates ( * ) and ( * * ) lead to

t+1 t ||v(s + ω) -v(s)|| p ds = t+1 t ||f (s + ω, u([s + ω])) -f (s, u([s]))|| p ds ≤ t+1 t ||f (s + ω, u([s + ω])) -f (s, u([s + ω]))|| p ds + t+1 t ||f (s, u([s + ω])) -f (s, u([s]))|| p ds ≤ ǫ p /2 + ǫ p /2 = ǫ p .
Therefore for all ǫ > 0 there exists T ǫ = M ax(t ǫ , t ′ ǫ ) > 0 such that for all t ≥ T ǫ we have

t+1 t ||v(s + ω) -v(s)|| p ds 1/p ≤ ǫ.
We conclude that v ∈ S p SAP ω (R + , X).

Main Results

Definition 3.1. A solution of (1) on R + is a function x(t) that satisfies the conditions:

(1) x(t) is continuous on R + .

(2) The derivative x ′ (t) exists at each point t ∈ R + , with possible exception at the points [t], t ∈ R + where one-sided derivatives exists.

(3) The equation ( 1) is satisfied on each interval

[n, n + 1[ with n ∈ N.
Now we make the following hypothesis:

(H1) : The function f is uniformly S-asymptotically ω-periodic on bounded sets in the Stepanov sense and satisfies the Lipschitz condition

||f (t, u) -f (t, v)|| ≤ L||u -v||, u, v ∈ X, t ∈ R + .
We assume that A(t) generates an evolutionary process (U (t, s)) t≥s in X, that is, a two-parameter family of bounded linear operators that satisfies the following conditions:

1. U (t, t) = I for all t ≥ 0 where I is the identity operator.

2. U (t, s)U (s, r) = U (t, r) for all t ≥ s ≥ r.

3. The map (t, s) → U (t, s)x is continuous for every fixed x ∈ X.

Then the function g defined by g(s) = U (t, s)x(s), where x is a solution of (1), is differentiable for s < t.

dg(s) ds

= -A(s)U (t, s)x(s) + U (t, s)x ′ (s) = -A(s)U (t, s)x(s) + U (t, s)A(s)x(s) + U (t, s)f (s, x([s])) = U (t, s)f (s, x([s])). dg(s) ds = U (t, s)f (s, x([s])). (2)
The function x([s]) is a step function. By (H1),

f (s, x([s])) is piecewise continuous. Therefore f (s, x([s])) is integrable on [0, t] where t ∈ R + . Inte- grating (2) on [0, t] we obtain that x(t) -U (t, 0)c 0 = t 0 U (t, s)f (s, x([s]))ds.
Therefore, we define Definition 3.2. We assume (H1) is satisfied and that A(t) generates an evolutionary process (U (t, s)) t≥s in X. The continuous function x given by

x(t) = U (t, 0)c 0 + t 0 U (t, s)f (s, x([s]))ds
is called the mild solution of equation ( 1). Now we make the following hypothesis.

(H2): A(t) generates a ω-periodic (ω > 0) exponentially stable evolutionnary process (U (t, s)) t≥s in X, that is, a two-parameter family of bounded linear operators that satisfies the following conditions:

1. For all t ≥ 0, U (t, t) = I where I is the identity operator.

2. For all t ≥ s ≥ r, U (t, s)U (s, r) = U (t, r).

3. The map (t, s) → U (t, s)x is continuous for every fixed x ∈ X.

4. For all t ≥ s,

U (t + ω, s + ω) = U (t, s)
(ω-periodicity).

5. There exist K > 0 and a > 0 such that

||U (t, s)|| ≤ Ke -a(t-s)
for t ≥ s.

Theorem 3.1. We assume that (H2) is satisfied and that f ∈ S p SAP ω (R + , X). Then

(∧f )(t) = t 0 U (t, s)f (s)ds ∈ ω (R , X), t ∈ R + . Proof. Let u(t) = t 0 U (t, s)f (s)ds.
For n ≤ t ≤ n + 1, n ∈ N, we observe that

||u(t)|| ≤ t 0 ||U (t, s)f (s)|| ds ≤ n 0 ||U (t, s)f (s)|| ds + t n ||U (t, s)f (s)|| ds ≤ n 0 M e -a(t-s) ||f (s)|| ds + t n M e -a(t-s) ||f (s)|| ds ≤ n 0 M e -a(n-s) ||f (s)|| ds + t n M ||f (s)|| ds ≤ n-1 k=0 k+1 k M e -a(n-s) ||f (s)|| ds + t n M ||f (s)|| ds ≤ n-1 k=0 k+1 k M e -a(n-k-1) ||f (s)|| ds + n+1 n M ||f (s)|| ds ≤ n-1 k=0 M e -a(n-k-1) k+1 k ||f (s)|| ds + M n+1 n ||f (s)|| ds ≤ n-1 k=0 M e -a(n-k-1) k+1 k ||f (s)|| p ds 1 p + M n+1 n ||f (s)|| p ds 1 p ≤ M ∞ j=0 e -aj + 1 ||f || S p ≤ M 2 -e -a 1 -e -a ||f || S p .
Therefore u is bounded. Now, show that lim

t→∞ u(t + ω) -u(t) = 0.
We have

u(t + ω) -u(t) = t+ω 0 U (t + ω, s)f (s)ds - t 0 U (t, s)f (s)ds = ω 0 U (t + ω, s)f (s)ds + t+ω ω U (t + ω, s)f (s)ds - t 0 U (t, s)f (s)ds = I 1 (t) + I 2 (t),
where

I 1 (t) = ω 0 U (t + ω, s)f (s)ds,
and

I 2 (t) = t+ω ω U (t + ω, s)f (s)ds - t 0 U (t, s)f (s)ds.
We note that

I 1 (t) = U (t+ω, ω) ω 0 U (ω, s)f (s)ds = U (t+ω, ω)u(ω),
and by using the fact that (U (t, s)) t≥s is exponentially stable, we obtain

||I 1 (t)|| ≤ Ke -at ||u(ω)||, which shows that lim t→∞ I 1 (t) = 0. Let ǫ > 0. Since f ∈ S p SAP ω (R + , X), there exists m ∈ N such that for t ≥ m t+1 t ||f (s + ω) -f (s)|| p ds 1 p < ǫ.
For m ≤ n ≤ t ≤ n + 1, we have

I 2 (t) = t 0 U (t, s) f (s + ω) -f (s) ds ≤ I 2,1 (t) + I 2,2 (t) + I 2,3 (t),
where

         I 2,1 (t) = m 0 U (t, s) f (s + ω) -f (s) ds I 2,2 (t) = n-1 k=m k+1 k U (t, s) f (s + ω) -f (s) ds, I 2,3 (t) = t n U (t, s) f (s + ω) -f (s) ds.
We observe that

||I 2,1 (t)|| ≤ m 0 ||U (t, s)|| ||f (s + ω) -f (s)|| ds ≤ M e -a(t-m) m 0 ||f (s + ω) -f (s)||ds. Therefore, there exists ν m ∈ N, ν m ≥ m such that for t ≥ ν m ||I 2,1 (t)|| ≤ ǫ.
Using Holder's inequality, we observe also that

||I 2,2 (t)|| ≤ n-1 k=m k+1 k ||U (t, s)|| ||f (s + ω) -f (s)|| ds ≤ n-1 k=m M k+1 k e -a(t-s) ||f (s + ω) -f (s)|| ds ≤ n-1 k=m M k+1 k e -a(n-k-1) ||f (s + ω) -f (s)|| ds ≤ M n-1 k=m e -a(n-k-1) k+1 k ||f (s + ω) -f (s)|| ds ≤ M n-1 k=m e -a(n-k-1) k+1 k ||f (s + ω) -f (s)|| p ds 1 p
≤ M e -a(n-m-1) + e -a(n-m-2) + ...

+ 1 ǫ ≤ M 1 -e -a ǫ.
We observe also that

||I 2,3 (t)|| ≤ t n ||U (t, s)|| ||f (s + ω) -f (s)|| ds ≤ t n M e -a(t-s) ||f (s + ω) -f (s)|| ds ≤ M t n ||f (s + ω) -f (s)|| ds ≤ M n+1 n ||f (s + ω) -f (s)|| ds ≤ M n+1 n ||f (s + ω) -f (s)|| p ds 1 p ≤ M ǫ. Finally, for t ≥ ν m ||I 2 (t)|| ≤ ||I 2,1 (t)|| + ||I 2,2 (t)|| + ||I 2,3 (t)|| ≤ 1 + M 1 -e -a + M ǫ, thus lim t→∞ I 2 (t) = 0.
We conclude that u ∈ SAP ω (R + , X). Now we make the following hypothesis.

Theorem 3.2. Let ω ∈ N * . We assume that the hypothesis (H1) and (H2) are satisfied. Then (1) has a unique S-asymptotically ω-periodic mild solution provided that

Θ := LM a < 1.
Proof. We define the nonlinear operator Γ by the expression

(Γφ)(t) = U (t, 0)c 0 + t 0 U (t, s)f (s, φ([s]))ds = U (t, 0)c 0 + (∧ 1 φ)(t), where (∧ 1 φ)(t) = t 0 U (t, s)f (s, φ([s])).
According to the hypothesis (H2), we have

||U (t + ω, 0) -U (t, 0)|| ≤ ||U (t + ω, 0)|| + ||U (t, 0)|| ≤ Ke -a(t+ω) + Ke -at . Therefore lim t→∞ ||U (t + ω, 0) -U (t, 0)|| = 0.
According to the Lemma 2.7 (resp. lemma 2.8) the function t → f (t, φ( t )) belongs to S p SAP ω (R + , X).

According to the Theorem 3.1 the operator ∧ 1 maps SAP ω (R + , X) into itself. Therefore the operator Γ maps SAP ω (R + , X) into itself.

We have

||(Γφ)(t) -Γψ)(t)|| = t 0 U (t, s) f (s, φ([s])) -f (s, ψ([s])) ds ≤ t 0 ||U (t, s)|| ||f (s, φ([s])) -f (s, ψ([s]))||ds ≤ L t 0 ||U (t, s)|| ||φ([s]) -ψ([s])||ds ≤ LM t 0 e -a(t-s) ||φ([s]) -ψ([s])||ds ≤ LM t 0 e -a(t-s) ||φ -ψ|| ∞ ds ≤ LM 1 -e -at a ||φ -ψ|| ∞ ≤ LM a ||φ -ψ|| ∞ .
Hence we have :

||Γφ -Γψ|| ∞ ≤ LM a ||φ -ψ|| ∞ .
This proves that Γ is a contraction and we conclude that Γ has a unique fixed point in SAP ω (R + , X). The proof is complete.

Application

Consider the following heat equation with Dirichlet conditions: + (-3 + sin(πt))u(t, x) + f (t, u([t], x)), u(t, 0) = u(t, π) = 0, t ∈ R + , u(0, x) = c 0 ,

where c 0 ∈ L 2 [0, π] and the function f is uniformly Sasymptotically ω-periodic on bounded sets and satisfies the lipschitz condition, that is, there exists a constant L > 0 such that ||f (t, x) -f (t, y)|| ≤ L||x -y||, ∀t ≥ 0, ∀x, y ∈ X.

Let X = L 2 [0, π] be endowed with it's natural topology. Define

D(A) = {u ∈ L 2 [0, π] such that u ′′ ∈ L 2 [0, π]
and u(0) = u(π) = 0} Au = u ′′ f or all u ∈ D(A).

Let φ n (t) = 2 π sin(nt) for all n ∈ N. φ n are eigenfunctions of the operator (A, D(A)) with eigenvalues λ n = -n 2 . A is the infinitesimal generator of a semigroup T (t) of the form

T (t)φ = ∞ n=1 e -n 2 t φ, φ n φ n , ∀φ ∈ L 2 [0, π]
and ||T (t)|| ≤ e -t , f or t ≥ 0 (see [START_REF] Rong-Hua | Stepanov-like pseudo-almost automorphic mild solutions for some abstract differential equations[END_REF], [START_REF] Xia | Weighted pseudo asymptotically periodic mild solutions of evolutions equations[END_REF]). Now define A(t) by: D(A(t)) = D(A) A(t) = A + q(t, x), where q(t, x) = -3 + sin(πt). Note that A(t) generates an evolutionnary process U (t, s) of the form U (t, s) = T (t -s)e t s q(,v,x)dx .

Since q(t, x) = -3 + sin(πt) ≤ -2, we have U (t, s) ≤ T (t -s)e -2(t-s) and ||U (t, s)|| ≤ ||T (t -s)||e -(t-s) ≤ e -3(t-s) .

Since q(t + 2, x) = q(t, x), we conclude that U (t, s) is a 2-periodic evolutionnary process exponentially stable.

The equation ( 3) is of the form

x ′ (t) = A(t)x(t) + f (t, x([t])), x(0) = c 0 .

By Theorem 3.2, we claim that Theorem 4.1. If L < 3 then the equation ( 3) admits an unique mild solution u(t) ∈ SAP ω (R + , X).


Olivier Pignolet

Sébastien Jubeau

Carlos Vaca-Garcia

Philippe Michaud
email: philippe.michaud@univ-bpclermont.fr

Highly valuable microalgae: biochemical and topological aspects

Keywords: Biorefinery, Cyanobacteria, Eukaryotic rnicroalgae, Biofuel, Lipid, Polysaccharide, Pigment, Protein

The past decade has seen a surge in the interest in microalgae culture for biodiesel production and other applications as renewable biofuels as an alternative to petroleum transport fuels. The development of new tech nologies for the culture of these photosynthetic microor ganisms and improved knowledge of their biochernical composition has spurred innovation in the field of high value biomolecules. These developments are only eco nornically viable if all the rnicroalgae fractions are valo rized in a biorefinery strategy. Achieving this objective requires an understanding of microalgae content and the cellular localization of the main biomolecular farnilies in order to develop efficient harvest and sequential recovery technologies. This review summarizes the state of the art in rnicroalgae compositions and topologies using some examples of the main industrially farmed rnicroalgae.

Introduction

Microalgae are unicellular and multicellular photosynthetic microorganisms from freshwater and marine systems that offer extremely interesting industrial potential. Definitions of rnicroalgae include prokaryotic rnicroalgae, i.e., cyano bacteria (Chloroxybacteria), and eukaryotic rnicroalgae, i.e., diatoms (Bacillariophyta), green algae (Chlorophyta) and red algae (Rhodophyta). Eukaryotic rnicroalgae are recognized as primitive plants and can be either autotrophic or heterotrophic. Autotrophic microalgae require only inorganic compounds such as CO 2 , N, S, P, and light as an energy source for their growth and development. They convert captured solar energy into biomass (photosynthesis) with efficiencies that generally exceed those of terrestrial plants (3 % reported for marine rnicroalgae against 0.2-2 % for terrestrial plants) [108]. Sorne photosynthetic rnicroal gae are rnixotrophic, meaning they are able to perform photosynthesis and to catabolize exogenous organic nutri ents. For autotrophic microalgae, photosynthesis is essential to convert solar radiation and CO 2 into reducing power, adenosine triphosphate (ATP), 0 2 , and 3-phosphoglycerate, which are then used to support growth. With these simple growth requirements, microalgae can sustainably generate lipids, proteins, and carbohydrates at a large scale, offering promising environmentally friendly alternatives to the current consumer products. The most prorninent examples are biofuels, human diet and animal feed supplements, and applications in the pharmaceutics, nutraceutics, and cos metics industries. Sorne macroalgal compounds also show interesting functional properties (free radical scavengers) and could provide new treatments against diseases such as cancer or even AIDS [START_REF] Raposo | Bioactivity and applications of sulphated polysaccharides from marine microalgae[END_REF]. However, these high-value compounds (primary products) are generally extracted individually through expensive processes that waste most of the microalgal material (co-products). The microalgae industry's future hinges on enabling first-generation biore fineries to valorize the main primary products from a bio mass and not only one of them. In a schematic way, first generation biorefineries work to a three-step concept high lighted in a schematic flowsheet shown in Fig. 1. These steps are disintegration of cells, fractionation of main component families, and purification of valuable biomole cules. Nevertheless, microalgal biorefineries are hard to set up due to the broad chemical and morphological diversity of microalgae and the high-value fractions that need to be obtained for specific applications. Indeed, each unit opera tion has to preserve the specific properties of the obtained fractions and the stability of the target compounds in the remaining materials. Given that the main molecular content groups are often associated with specific intracellular compartments, unit operations could be performed on a deconstructed biomass in a conceptual biorefinery (Fig. 2). These main families are lipids, polysaccharides, proteins, and pigments. Even though pigments are biochemically varied, they can be considered a single biochemical family for the purposes of a valorization strategy. All these 8 8 Photobioreactor Harvesting Cell disruption Fractionation Purification Fig. 1 A microalgae biorefinery compounds are qualitatively and quantitatively dependent on microalgal species used. However, some species have risen to prominence in the industry and are dedicated to specific applications due to their select biochemical com position. The aim of this review is to describe the bio chemical composition and topology of selected microalgae and to highlight the compounds to be valorized in a first generation biorefinery strategy.

Lipids

Biological lipids are chemically diverse groups that nev ertheless share the common defining feature of being insoluble in water. Their biological fonctions, like their chemistry, are equally diverse. Oil-rich algae (called ole aginous species) can be grown either autotrophically and heterotrophically. In a biorefinery concept, this class of compounds, and more especially reserve lipids, are at the origin of biofuel production. Microalgae for biofuel pro duction can be grown in several places all year round with significant advantages such as good oil productivity exceeding the yields of the best oilseed crops, non-use of arable land, low water consumption compared to crop irrigation, and low environmental impact [START_REF] Mata | Microalgae for biodiesel production and other applications: a review[END_REF]. Microalgae growing under normal conditions actually produce signif icant amounts of lipids, but this lipid accumulation decreases when cultivation conditions are optimized to increase biomass yield [START_REF] Mandal | Microalga Scenedesmus obliquus as a potential source for biodiesel production[END_REF]109,124]. Microalgae contain primarily polar lipids such as phospholipids and glycolipids together with neutral storage lipids such as monoglycerides (MAGs), diglycerides (DAGs), and triglycerides (TAGs), free fatty acids, hydrocarbons, and pigments [START_REF] Hu | Microalgal triacylglycerols as feedstocks for biofuel production: perspectives and advances[END_REF]. Polar lipids are confined to cell organelle membranes such as the thylakoid membranes of the chloroplast [START_REF] Hu | Microalgal triacylglycerols as feedstocks for biofuel production: perspectives and advances[END_REF]. TAGs are neutral lipids mainly stored in vacuoles within the cell [START_REF] Brown | Effects of harvest stage and light on the biochemical composi tion of the diatom Thalassiosira pseudonana[END_REF][START_REF] Hu | Microalgal triacylglycerols as feedstocks for biofuel production: perspectives and advances[END_REF]. Most microalgae accumulate very few TAGs during the exponential growth phase but can produce and store substantial amounts of T AGs during the stationary phase or under adverse environmental conditions [START_REF] Berge | Reassess ment of lipid-composition of the diatom, skeletonema costatum[END_REF][START_REF] Hu | Microalgal triacylglycerols as feedstocks for biofuel production: perspectives and advances[END_REF]114]. Fatty acid (FA) composition may differ between lipid classes. As a rule, saturated fatty acids (SAFA) and monounsaturated fatty acids (MUF A) comprise the storage lipid fraction while polyunsaturated fatty acids (PUFA) largely comprise the structural lipid fraction [START_REF] Hu | Microalgal triacylglycerols as feedstocks for biofuel production: perspectives and advances[END_REF]. How ever, culture processes and species selection have a major influence on lipid composition, accumulation, and distri bution. For many microalgae, growth stressors such as excessive solar energy captured by photosynthesis, nitro gen depletion, salinity, or high temperature lead to pri marily lipid storage, with high TAGs levels [START_REF] Chen | Culti vation, photobioreactor design and harvesting of microalgae for biodiesel production: a critical review[END_REF][START_REF] Cohen | The production potential of eicosapentaenoic and arachidonic acids by the red alga Porphyridium cruentum[END_REF][START_REF] Khozin-Goldberg | Unraveling algal lipid metabolism: recent advances in gene identification[END_REF]108,114] or to a shift in FA composition [START_REF] Durmaz | The effect of low temperature on fatty acid composition and tocopherols of the red microalga Porphyridium cruentum[END_REF]. Sorne microalgae are well-known lipid producers extensively

C

studied for biofuel and more especially for biodiesel pro duction. Biodiesel is defined as monoalkyl esters of vege table oils or animal fats. lt is produced by the transesterification of T AGs with methanol ( or other alco hols) in the presence of an appropriate catalyst, with glycerol as by-product. Biodiesel standards around the world, including the USA (ASTM D 6751) and Europe (EN 14214), also address storage and handling issues linoleic and linolenic acid esters, for example, are prone to oxidation and so need to be chemically reduced by a hydrogenation process [START_REF] Knothe | Analyzing biodiesel: standards and other methods[END_REF]. Hydrogenation converts dou ble bonds to single bonds by H addition. Other microalgae producing lipids are exploited for food and feed applica tions, notably due to their unsaturated FA or pigment contents. A majority of the high-value lipid compounds extracted from these algae are long-chain PUFA and carotenoid pigments. Although all microalgae are candidate lipid-producers, the "winners" for industry applications are the eukaryotic microalgae Chlorella vulgaris (up to 58 % dw) [START_REF] Amaro | Advances and perspectives in using microalgae to produce biodiesel[END_REF], Nan nachloropsis oculata (up to 69 % dw) [START_REF] Bondioli | Oil production by the marine microalgae Nannochloropsis sp. F&M-M24 and Tetraselmis suecica F&M-M33[END_REF], Botryococcus braunii (up to 75 % dw) [START_REF] Chisti | Biodiesel from microalgae[END_REF], and Scenedesmus obliquus (up to 50 % dw) [START_REF] Mata | Microalgae for biodiesel production and other applications: a review[END_REF]. Nonetheless, another eukaryotic microalgae such as Porphyridium cruentum may have futures for lipid-based industrial applications as they have original and/or highly specific lipid profiles, rich in long chain PUFA. Indeed, Porphyridium cruentum is rich in high-value FA such as arachidonic acid (20:4co6) and ei cosapentaenoic acid (20:5co3) [START_REF] Cohen | The role of triacylglycerol as a reservoir of polyunsaturated fatty acids for the rapid production of chloroplastic lipids in certain microalgae[END_REF] for a global lipid con tent of between 9 and 19 % [START_REF] Mata | Microalgae for biodiesel production and other applications: a review[END_REF].

Chlorella vulgaris

Lipid accumulation as chloroplastic and cytoplasmic droplets by the green microalga C. vulgaris in freshwater (Fig. 2) requires a pre-extraction cell lysis step. Growth on nitrogen-limiting media resulted in higher lipid accumu lation (20--53 % ) but lower growth rate, whereas mixo trophic cultivation yielded higher lipid content (40-53 %) and higher lipid productivity (67-144 mg/1/day) in all culture media [124]. The FA composition of microalgal lipids comprises over 60-68 % SAFA [124]. More spe cifically, the FA most accumulated are described as C18, C18:l, C18:2, and C18:3 (3.4, 16.3, and 79.4 % of total FA [START_REF] Lee | Comparison of several methods for effective lipid extraction from rnicroalgae[END_REF]. Choi et al. [START_REF] Choi | Effects of methanol on cell growth and lipid production from mixotrophic cultivation of Chlore/la sp[END_REF] compared the lipid production of mixotrophic and autotrophic cultures of Chlorella sp. Cultures were carried out in photobioreactors using various light periods and various CO 2 supplementations [START_REF] Choi | Effects of methanol on cell growth and lipid production from mixotrophic cultivation of Chlore/la sp[END_REF]. Methanol (1 % v/v) was used as sole organic carbon source in the mixotrophic conditions. The highest lipid accumu lation (18.6 % dw) with high proportions of C16:0, C18:0, and C18:1 was obtained with 5 % CO 2 and 450 µE/m 2 /s of 12: 12 (h) irradiation cycles in mixotrophic mode. It is interesting to note that in this study, mixotrophy gave better results than autotrophy. Previous data [START_REF] Choi | Effects of methanol on cell growth and lipid production from mixotrophic cultivation of Chlore/la sp[END_REF][START_REF] Lee | Comparison of several methods for effective lipid extraction from rnicroalgae[END_REF]124] suggest the lipid composition of C. vulgaris is suitable for biodiesel production. C. vulgaris was also grown in a column aeration photobioreactor filled with artificial wastewater in semicontinuous mode [START_REF] Feng | Lipid production of Chlore/la vulgaris cultured in artificial wastewater medium[END_REF]. The highest lipid production and content were 147 mg/1/day and 42 % dw, respectively, in the optimal growth conditions, and the system was characterized as highly efficiency at removing nutrients from wastewater [START_REF] Feng | Lipid production of Chlore/la vulgaris cultured in artificial wastewater medium[END_REF].

Nannochloropsis oculata

Nannochloropsis oculata is well known in marine bio technology for its potential interest in biodiesel production. N. oculata NCTU-3 was grown in a semicontinuous system aerated with 2 % CO 2 and operated by one-day replace ment with the objective of optimizing culture for long-term biomass and lipid productions [START_REF] Chiu | Lipid accumulation and CO 2 utilization of Nannochloropsis oculata in response to CO 2 aeration[END_REF]. A two-stage cultivation strategy bas also been proposed to enhance the lipid pro duction of Nannochloropsis oculata [109]. Biomass growth and lipid production were experimented in two separate stages. Final lipid yield was 2.82-fold higher in these conditions than in traditional single-stage batch cultivation systems [109]. Microalgae lipid content was strongly influenced by the physical-chemical environment and composition of the culture media. For example, a temper ature increase from 20 to 25 °C doubled the lipid content of N. oculata (from 7.90 to 14.92 % dw), while a 75 % decrease in nitrogen concentration in the medium increased N. oculata lipid content from 7.90 to 15.31 % dw without impact on growth [START_REF] Converti | Effect of temperature and nitrogen concentration on the growth and lipid content of Nannochloropsis oculata and Chlore/la vulgaris for biodiesel production[END_REF]. The authors identified high amounts of palmitic acid (16:0), which constituted about 60 % (mol/ mol) of the overall lipid fraction, along with C18:l, C18:2, and C18:3. This result confirmed the finding that storage lipids in Nannochloropsis sp. consist largely of the SAFA palmitic acid (C16:0) and the MUFA palmitoleic acid (C16:1) [110]. However, Tonon et al. [114] found that 68 % of TAGs in Nannochloropsis oculata consisted of the PUFA eicosapentaenoic acid (20:5 co3, EPA) at the end of stationary phase compared to just 8 % in the exponential phase. Similarly, under nitrogen deprivation, the average biomass and lipid productivities of Nannochloropsis sp. reached 9.9 and 6.5 g/m 2 /day, respectively. Lipid content was 68.5 % in starved biomass compared to 39.1 % in non starved biomass, and the two cultivation conditions led to significant differences in lipid classes and FA composition [START_REF] Bondioli | Oil production by the marine microalgae Nannochloropsis sp. F&M-M24 and Tetraselmis suecica F&M-M33[END_REF]. Effectively, under nitrogen deprivation, polar lipid content decreased whereas neutral lipid content increased. The major lipids represented under nitrogen starvation were TAGs, polar lipids, and hydrocarbons (79, 9, and 2.5 % of total lipids, respectively) [START_REF] Bondioli | Oil production by the marine microalgae Nannochloropsis sp. F&M-M24 and Tetraselmis suecica F&M-M33[END_REF]. The FA composi tion of the different lipid classes (neutral, polar, and phospholipids) includes mainly C16:0, C16:l, and C18:1. Note that polar lipids and phospholipids have a significant ratio of C20:5m3 (eicosapentaenoic acid) in their structure. Lipid content, FA profiles, MAGs, DAGs, and TAGs were analyzed for 2 years in Nannochloropsis oculata biomasses grown outdoors in closed vertical fiat-panel photobioreac tors [START_REF] Olofsson | Seasonal variation of lipids and fatty acids of the microalgae Nannochloropsis oculata grown in outdoor large-scale photobioreactors[END_REF]. Results showed that the highest lipid content was recorded during autumn, signaling an optimal non-linear response to light and temperature. The authors hypothe sized that enhanced thylakoid stacking under reduced light conditions resulted in more structural lipids concomitantly with the increase in glycerides due to released photo-oxi dative stress. The relative increase in MUFA during auturnn suggested a synthesis either of structural FA as MUFA or a relative increase of C16:1 incorporated into TAGs and DAGs.

Botryococcus braunii

B. braunii is a colonial hydrocarbon-rich green microalga [START_REF] Banerjee | Botryo coccus braunii: a renewable source of hydrocarbons and other chemicals[END_REF] widespread in freshwater and brackish lakes at tem perate, tropical, and Arctic latitudes. B. braunii has tre mendous potential as a renewable biomass feedstock and high potential for the production of renewable liquid hydrocarbons. The lipids of B. braunii can be repeatedly extracted from wet biomass without the usual harvesting and dewatering steps [18]. By exposing B. braunii to bio compatible organic solvent, a substantial fraction of hydrocarbons was obtained without impairing cell viability [START_REF] Frenz | Hydrocarbon recovery and biocompatibility of solvents for extraction from cultures of Botryococcus braunii[END_REF], as the majority of biosynthesized lipids in B. braunii are located in the outer cell wall of the algal material [18]. Note that a high percentage of hydrocarbons in some petroleum fuels were probably derived exclusively from this specie [122]. Depending on the type of hydrocarbons synthesized, B. braunii is classified into A, B, and L strains but some strains of this species are not classified [START_REF] Kawachi | Relationship between hydrocarbons and molecular phy logeny of Botryococcus braunii[END_REF]. For example, strain B. braunii AP103 grown on open raceway pond produced a biomass concentration of 1.8 ± 0.13 g/1. lts biochemical analysis revealed 19 % lipids, 33 % car bohydrates, 18 % proteins, and 11 % hydrocarbons [START_REF] Ashokkumar | Mass culture of Botryo coccus braunii Kutz. under open raceway pond for biofuel production[END_REF]. The hydrocarbon profile showed the presence of hepta decane (34 %) and hexadecane (12.5 %). The major FA detected in the lipids extracted were oleic (25. 7 % ), lino lenic (34.26 %), and palmitic (9.42 %) acid [START_REF] Ashokkumar | Mass culture of Botryo coccus braunii Kutz. under open raceway pond for biofuel production[END_REF]. B. braunii race A is a ubiquitous colonial green microalga that is an important synthesizer of lipids in fresh and brackish water ecosystems [START_REF] Metzger | Structure and chemistry of a new chemical race of Botryococcus braunii (Chlorophyceae) that produces lycopadiene, a tetrat erpenoid hydrocarbon[END_REF]. Race A produces C23-C33 odd-num bered n-alkadienes, mono-, tri-, tetra-, and pentanes from FA. Linear olefins can account for up to 61 % of the dry matter of green active-state colonies. Note that the growth of B. braunii (race A) and its production of hydrocarbons and FA were influenced by environmental salinity [START_REF] Rao | Effect of salinity on growth of green alga Botryococcus braunii and its constituents[END_REF]. The L race produces a single tetraterpene hydrocarbon called lycopadiene (C 40 H 78 ) and it accounts for up to 2-8 % of d ry biomass. The B race produces polyunsatu rated and branched C30--C37 terpenoid hydrocarbons called polymethylated botryococcenes that are promising renewable energy sources as they accumulate at ve ry high levels of 26-86 % dry weight in the algae [START_REF] Metzger | Structure and chemistry of a new chemical race of Botryococcus braunii (Chlorophyceae) that produces lycopadiene, a tetrat erpenoid hydrocarbon[END_REF]. Photo synthesis, growth, and metabolic productivity of B. braunii BOT-22 cells (race B) were saturated at approximately 1,000, 100, and 200 µmol/m 2 /s, respectively, under monochromatic red light. Lipid and hydrocarbon produc tion was not directly supported by an increase in photo synthetic activity [102].

Scenedesmus obliquus

The green microalgae Scenedesmus obliquus is an oleagi nous alga well known for its ability to accumulate lipids, notably in fresh wastewater [START_REF] Mandal | Waste utilization and biodiesel production by the green rnicroalga Scenedesmus obliquus[END_REF]. The lipids of S. obliquus grown under controlled conditions have been characterized as 7 .24 % neutral lipid, 2.45 % glycolipid and 1.48 % phospholipid on a d ry weight basis. The major neutral lipids were di-and triacylglycerol, free and esterified sterols, and hydrocarbons. The glycolipids were monoga lactosyl diglyceride, digalactosyl diglyceride, free and esterified sterol glycoside. The phospholipids included phosphatidyl ethanolamine, phosphatidyl glycerol and phosphatidyl choline. The 14 FA identified in the lipid fractions were led by C18:2, C16:0, C18:3(alpha), C18:1, C16:3, C16:l, and C16:4. Unsaturated and essential FA were quantified at 80 and 38 % of total algal lipids, respectively [START_REF] Choi | Lipid content and fatty acid composition of green algae Scenedesmus obliquus grown in a constant cell density apparatus[END_REF]. The C16/C18 FA groups of S. obliquus lipids obtained from cultivation under nitrogen-starved conditions could reach up to 92.4 % of total FA [START_REF] Ho | Effect of light intensity and nitrogen starvation on CO 2 fixation and lipid/carbohydrate pro duction of an indigenous microalga Scenedesmus obliquus CNW-N[END_REF]. Oleic acid, an important indicator of biodiesel quality, accounted for 35.10 % of total FA. The main FA of S. obliquus cul tivated in nutrient-rich media with 10 % CO 2 were C16:0 (15-16 %), C16:1 (2-3 %), C18:0 (17-18 %), C18:1 (15-16 %), C18:2 (13-14 %), C18:3 (3-4 %), and several minor component FA with a carbon number lower than C16 and higher than C18 [START_REF] Ho | Effect of light intensity and nitrogen starvation on CO 2 fixation and lipid/carbohydrate pro duction of an indigenous microalga Scenedesmus obliquus CNW-N[END_REF][START_REF] Mandal | Microalga Scenedesmus obliquus as a potential source for biodiesel production[END_REF]. These data were con firmed by [119]. Note also that under P-deficiency and thiosulphate supplementation, the lipid content of S. obli quus increased up to 30 % [START_REF] Mandal | Microalga Scenedesmus obliquus as a potential source for biodiesel production[END_REF].

Polysaccharides

Polysaccharides are defined as polymers of monosaccha rides linked by glycosidic bonds. As for other organisms, microalgae polysaccharides demonstrate strong structural variability and biological fonctions [START_REF] Raposo | Bioactivity and applications of sulphated polysaccharides from marine microalgae[END_REF]. Based on their role in the microalgae physiology, they can be split in three classes: structural polysaccharides generally associated to cell walls, energy polysaccharides like starch, and poly saccharides implicated in cellular communication and recognition sites. Structural investigations into polysac charides are complicated by the very broad diversity of their constitutive monosaccharides (pentoses and hexoses), non-sugar substituents (e.g., sulfate, acetate, or pyruvate) and glycosidic linkages. If we exclude starch, exopoly saccharides (also known as exopolymeric substances or exopolymers when associated to proteins) have gained significant attention in the published literature. This pop ularity is explained by their extractability, as these bio polymers are produced by microalgae in the extracellular medium without covalent linkages with cell walls. In some cases, it is difficult to distinguish these exopolysaccharides from the cell wall in which they are embedded (e.g., exo polysaccharides of Porphyridium sp.) [START_REF] Arad | Red microalgal cell-wall polysaccharides: biotechnological aspects[END_REF]. Their physio logical role is unclear, but some authors have hypothesized that they prevent desiccation [START_REF] Ramus | The capture and transduction of light energy[END_REF] or offer a mechanical protection enabling the cells to develop in a range of environments [START_REF] Arad | Red microalgal cell-wall polysaccharides: biotechnological aspects[END_REF]. Microalgal production of polysaccharide and especially exopolysaccharide (EPS) could be also one way to control photosynthetic activity. Algae colony for mation is linked to polysaccharide production [113], which is catalyzed by various factors including C/N ratio. Reports suggest glyoxylate, a stimulator of carbon metabolism, is able to inhibit photorespiration and increase photosynthesis in higher plants [START_REF] Oliver | Metabolic regulation of glycolate synthesis, photorespiration, and net photosynthesis in tobacco by L-glutamate[END_REF] and some cyanobacteria [START_REF] Bergman | Glyoxylate decreases the oxygen sensitivity ofnitrogenase activity and photosynthesis in the cyanobacterium Anabaena cylindrical[END_REF]. Adding glyoxylate to algae like Anabaena cylindrica [START_REF] Bergman | Glyoxylate induced changes in the carbon and nitrogen metabolism of the cyanobacterium Anabaena cylindrical[END_REF] and Cyanospira capsulate [START_REF] Philippis | Response of an exo polysaccharide-producing heterocystous cyanobacterium to changes in metabolic carbon flux[END_REF] triggers an excess of carbon flux resulting in intracellular polysaccharide accumulation and soluble extracellular polysaccharide release. A culture study by Liu et al. [START_REF] Liu | PSII-effi ciency, polysaccharide production, and phenotypic plasticity of Scenedesmus obliquus in response to changes in metabolic carbon flux[END_REF] found that Scenedesmus obliquus supplemented with glyoxylate increased exopolysaccharide production.

In recent years, there has been of surge of interest in polysaccharide-producing bacteria, cyanobacteria, and microalgae for industrial applications. Effectively, EPSs are regarded as an abundant source of structurally diverse polysaccharides and some may possess unique properties for special applications. Excluding the EPSs from Por phyridium sp. [START_REF] Gloaguen | The extracellular polysaccharide of Porphyridium sp.: an NMR study of lithium-resistant oligo saccharidic fragments[END_REF] or Arthrospira platensis [116], which have been partially characterized, and starch (known as floridean starch) located in cytoplasm and chloroplasts, structural characterization data on microalga-produced polysaccharides remains scarce, even on cell-wall poly saccharides. Vilchez et al. [120] tried to document a global composition of microalgae and described mono-and oli gosaccharides as mainly represented by trehalose, glucose, sucrose, and mannose but also polyols. Polysaccharides were more or less ignored, and the authors only gave a global composition for monosaccharides (glucose, galact ose, xylose, methylxylose, and glucuronic acid).

Chlorella vulgaris

The most important substance in C. vulgaris for human health is a cell-wall �-(1,3)-glucan that has potential as an antitumor agent [START_REF] Nomoto | Anti-tumor effect by oral administration of Chlorella extract, PCM-4 by oral admission[END_REF]. This class of polysaccharide is well known as an active immunostimulator, free radical scav enger, and reducer of blood lipids [START_REF] Laroche | New developments and pro spective applications for � (1,3) glucans[END_REF]. To our knowledge, the exopolysaccharides or other cell wall polymers from this microalgae have not been described. The only excep tion is an acidic polysaccharide containing high contents of glucuronic acid and rhamnose identified as a cell wall component [START_REF] Ogawa | A new trisaccharide, alpha o-glucopyranuronosyl-(1 ---> 3)-alpha-L-rhamnopyranosyl-(1 -+ 2)-alpha-L-rhamnopyranose from Chlorella vulgaris[END_REF].

Arthrospira platensis

Arthrospira platensis, (Spirulina) is a Gram-negative cya nobacteria species. A. platensis grows in tropical and subtropical waters with pH between 8 and 11 and high carbonate and bicarbonate concentrations. The EPS of A. platensis is generally considered as a polyanionic sulfated heteropolymer containing proteins, carbohydrates, and sulfated groups. Contents were estimated at 55, 13, and 0.5 %, respectively in A. platensis Compére 1968/3786 strain grown in photoautotrophic-mode for 25 days. Car bohydrates are mainly composed of seven neutral sugars [galactose (14.9 %), xylose (14.3 %), glucose (13.2 %), fructose (13.2 %), rhamnose (3.7 %), arabinose (1 %), mannose (0.3 %)], and two uronic acids [galacturonic and glucuronic acids; 13.5 % together] [116]. Trabelsi et al.

[115] evaluated EPS production in relation to culture mode: high EPS levels were obtained in culture medium using mixotrophy with a short cultivation period whereas photoautotropohy and heterotrophy cultures produced low EPS levels at longer cultivation periods (290; 219 and 30 mg/1, respectively; 1.5 g/1 glucose and/or 100 µmol photon m-2 s-1 ). Optimizing light and glucose concentra tions led to a peak production level of 369 mg EPS/1 in mixotrophic mode.

EPS from A. platensis exhibits interesting functional properties. It has value to industry due a non-Newtonian pseudoplastic behavior, but the main applications remain pharmaceutical in nature due to its wide range of biological properties. The sulfated EPS from A. platensis acts as anticoagulant, anti-thrombogenic and anti-atherogenic [START_REF] Raposo | Bioactivity and applications of sulphated polysaccharides from marine microalgae[END_REF]. In addition, calcium and sodium spirulans isolated from hot water-extracted EPS interfere with the adsorption and penetration of viruses like herpes or HIV [START_REF] Hayashi | A natural sulfated polysaccharide, calcium spirulan, isolated from Spirulina plat ensis: in vitro and ex vivo evaluation of anti-Herpes Simplex virus and anti-human innnunodeficiency virus activities[END_REF]. Chal louf et al. [START_REF] Challouf | Evaluation of cytotoxicity and biological activity in extracellular polysaccharides released by cyanobacterium Arthrospira plat ensis[END_REF] ran cytotoxicity and antioxidant assays on solvent-isolated EPS and concluded that its biological activity is closely tied to the molecular weight and con formation of EPS in terms of solvent-polymer interactions and the presence of sulfated moieties.

Race-A and race-B strains produce EPS at up to 250 g/m 3 whereas L-race strains produce EPS at up to 1 kg/m 3 [START_REF] Banerjee | Botryo coccus braunii: a renewable source of hydrocarbons and other chemicals[END_REF]. However, EPS production level varies with strain and culture conditions such as culture media, illumination, agitation, and above all salinity [START_REF] Dayananda | Isolation and characterization of hydrocarbon producing green alga Botryococcus braunii from Indian freshwater bodies[END_REF][START_REF] Dayananda | Autotrophic cultivation of Botryococcus braunii for the production of hydrocarbons and exopolysac charides in various media[END_REF][START_REF] Rao | Effect of salinity on growth of green alga Botryococcus braunii and its constituents[END_REF]. B. braunii appears to produce carbohydrates as an osmoprotectant. The authors failed to describe the structures of these exo polymers and only quantified them after ethanol precipi tation of culture medium supematant and/or phenol sulfuric assay [START_REF] Dayananda | Isolation and characterization of hydrocarbon producing green alga Botryococcus braunii from Indian freshwater bodies[END_REF][START_REF] Dayananda | Autotrophic cultivation of Botryococcus braunii for the production of hydrocarbons and exopolysac charides in various media[END_REF][START_REF] Rao | Effect of salinity on growth of green alga Botryococcus braunii and its constituents[END_REF].

Dunaliella salina

Dunaliella salina is a type of halophile pink microalgae especially found in sea salt fields. EPS production by this strain increases with salt concentration in the culture medium, and peak production (944 mg/1) was obtained at 5 M NaCl [START_REF] Mishra | Characterization of extra cellular polymeric substances produced by micro-algae Dunal iella salina[END_REF]. Isolated EPS showing emulsifying activity slightly reduces with salinity. 1 H NMR, FT-IR, and HPLC have given structural dues on the presence of uronic acids, halides, sulfides, alkyl amine, and/or cyclic amine with polysaccharides. Galactose, glucose, xylose, and fructose were detected as constitutive monosaccharides [START_REF] Mishra | Characterization of extra cellular polymeric substances produced by micro-algae Dunal iella salina[END_REF]. Fine grained monosaccharide composition analysis in five fractions of a polysaccharidic crude extract fractionated by DEAE chromatography using reverse-phase HPLC showed that PDl and PD4a fractions were acidic heteropolysac charides mainly containing glucose and galactose, respec tively, and that PD4a contained sulfated groups. PD2 and PD3 were identified as a glucan. PD4b was a complex of polysaccharide covalently linked with nucleic acids. Compositional analysis of PD4a confirmed the deduction made by Fabregas et al. [START_REF] Fabregas | In vitro inhibition of the replication of haemorrhagic septicaemia virus (VHSV) and African swine fever virus (ASFV) by extracts from marine microalgae[END_REF] that water extracts with antivirus activity from D. salina contained sulfated poly saccharides with a molecular weight of a hundred-thousand Da [START_REF] Dai | Sugar composition deterrnination of polysaccharides from Du nalielle salina by modified RP-HPLC method of precolumn derivatization with 1-phenyl-3-methyl-5-pyrazolone[END_REF]. These EPS proved thermostable up to 270 °C and their dynamic viscosity was significantly high at pH 3.0 but decreased with shear rate, confirming pseudoplastic rheo logical properties [START_REF] Mishra | Characterization of extra cellular polymeric substances produced by micro-algae Dunal iella salina[END_REF].

Porphyridium sp.

Porphyridium sp. is a red marine microalga (Rodophyta) with spherical cells whose cell walls lack cellulose microfibrillar components. These microalgae are encapsu lated within a sulfated polysaccharide gel (red microalgae cell wall). During growth, the outer gel dissolves in the growth medium while 50--70 % of the polysaccharide stays attached to the cell [106], forming an outer layer that shields the microalgae against extreme environment con ditions (e.g., desiccation or salinity). The polysaccharides Fig. 3 Structures of oligosaccharides isolated from the exopolymeric fraction of Porphyridium sp. [START_REF] Gloaguen | The extracellular polysaccharide of Porphyridium sp.: an NMR study of lithium-resistant oligo saccharidic fragments[END_REF] have a molecular weight in the range 2-7 x 10 6 Da and are negatively charged due to the presence of glucuronic acid and sulfate groups. They are composed of up to ten monosaccharides, the most abundant being xylose, glucose, and galactose. Note that proteins represent 5.5 % of poly mer dw. The structure of EPS from Porphyridium sp. was partially characterized by Gloagen et al. [START_REF] Gloaguen | The extracellular polysaccharide of Porphyridium sp.: an NMR study of lithium-resistant oligo saccharidic fragments[END_REF] with oligo saccharides obtained after polysaccharide degradation by lithium (Fig. 3). This polysaccharide has various concrete applications in the cosmetics, drugs, crop growth promot ers, and health food markets. A feature of these EPSs is their fluid-dynamic behavior that gives highly viscous solutions at relatively low polymer concentrations in a wide range of pH and temperature values, yielding com parable rheological properties to industrial polysaccharides [START_REF] Arad | Red microalgal cell-wall polysaccharides: biotechnological aspects[END_REF][START_REF] Patel | Separation and frac tionation of exopolysaccharides from Porphyridium cruentum[END_REF]. Sun et al. [111] tested the antioxidant activity of degraded (microwave) polysaccharides from P. cruentum and found that smaller-sized polysaccharides have a better antioxidant effect. It was also shown that these polysac charides exhibit antiviral (anti-herpes) and antitumor activities [106, 112].

Pigments

Pigments are complex molecules or macromolecules that have the ability to change the color of reflected or trans mitted light as a result of wavelength-selective absorption. Chlorophylls are green pigments with polycyclic planar structures esterified by a long phytol side-chain where Mg 2+ occupies the central position. The chlorophylls differ by the substituents on their ring. The chlorophylls identi fied in microalgae are a, b, and c and are located in the thylakoids of chloroplasts [START_REF] Lehninger | Lehninger princi ples of biochemistry[END_REF]. Depending on microalgae species, some chlorophylls are not always present in pho tosynthetic structures. Cyanobacteria and red algae employ phycobylins such as phycoerythrobilins and phycocyano bilins [START_REF] Beale | Biosynthesis of phycobilins[END_REF]. These open-chain tetrapyrols have the polyene system of chlorophylls but not their cyclic structure or central Mg 2+ . They are covalently linked to specific pro teins, forming phycobiliproteins. Phycobiliproteins are associated in order complexes called phycobilisomes. Phycobilisomes are the primary light-harvesting structures of these microorganisms and are attached in regular arrays to the outer surface of the thylakoids [START_REF] Basaca-Loya | Extraction and purification of B-phycoerythrin from the red microalga Rhodosorus marinus[END_REF]. Phycobilisomes absorb light over a wide range of wavelengths and transfer the energy to the reaction centers in the photosynthetic membranes for conversion into chemical energy [START_REF] Bermejo | Recovery of pure B-phycoerythrin from the microalga Porphyridium cruentum[END_REF]. In addition to chlorophylls, thylakoid membranes contain accessory photosynthetic pigments called carotenoids. Carotenoids are a class of hydrocarbons (carotenes) and their oxygenated derivatives (xanthophylls) consist of eight isoprenoid units (Fig. 4). Carotenoids may be purple, yel low, or red. The accumulation of secondary carotenoids is generally thought to be a survival strategy employed by the algae under photo-oxidative stress or other adverse envi ronmental conditions [START_REF] Hu | Enhanced protection against oxidative stress in an astaxanthin-overproduction Haematococcus mutant (Chlorophyceae)[END_REF]. The most important are red orange P-carotene, an isoprenoid, and the yellow lutein. This section focuses on xanthophylls and phycobilipro teins, which are high-value compounds for industrial applications. Among the microorganisms detailed for these molecular families in the literature, the following eukary otic microalgae emerge as the most significant: Chlorella vulgaris, Dunaliella salina, Haematococcus pluvialis, Porphyridium cruentum, Arthrospira platensis, and Haslea ostrearia.

Chlorella vulgaris

Like other species of microalgae, C. vulgaris is able to activate metabolic pathways and accumulate carotenoids in response to high irradiance stresses, nitrogen availability, and other changes in environmental conditions. Gouveia et al. [START_REF] Gouveia | Evolution of pigment composition in Chlorella vulgaris[END_REF] studied carotenoid biosynthesis by C. vulgaris grown in a reactor for 22 days under a light intensity of 38/ µE/s/m 2 , saline stress (30 g/1 NaCl), and nitrogen starva tion. They observed that P-carotene and lutein were first biosynthesized along with chlorophyll-a during the expo nential phase. Carotenogenesis was activated just before the stationary phase, and P-carotene and lutein concentra tions decreased while canthaxanthin, astaxanthin, and as taxanthin esters were biosynthesized in stressed cells. It was suggested that the accumulation of canthaxanthin resulted from the oxidative transformation of P-carotene and that hydroxylated/oxidized astaxanthin was biosyn thesized from zeaxanthin/lutein precursors towards hy droxylative pathways. Lutein and P-carotene content in [START_REF] Cha | Antiproliferative effects of carotenoids extracted from Chlore/la ellipsoidea and Chlore/la vulgaris on human colon cancer cells[END_REF]. Carote noids from C. vulgaris have also been tested for their coloring and antioxidant properties in egg yolk, fish, food emulsions, and soybean oil [START_REF] Gouveia | Functional food oil coloured by pigments extracted from microalgae with supercritical CO 2[END_REF][START_REF] Gouveia | Evolution of pigment composition in Chlorella vulgaris[END_REF].

Arthrospira platensis

C-phycocyanin is the major phycobiliprotein of the cya nobacterium A. platensis (also known as Spirulina platen sis). This bright-blue pigment is part of the phycobilisome and is used for photosynthetic light collection and transfer of chernical energy to the photosynthetic apparatus.

C-phycocyanin has an estimated molecular weight of 100--200 kDa according to filtration techniques [START_REF] Jaouen | Clarification and concentration with membrane technology of a phycocyanin solution extracted from Spirulina platensis[END_REF]. It is formed of two sub-units a, and p with molecular weights of 20.5 and 23.5 kDa, respectively [START_REF] Basaca-Loya | Extraction and purification of B-phycoerythrin from the red microalga Rhodosorus marinus[END_REF]. A. platentis has a chernical composition that shows prornising effects on human and animal health. Indeed, the whole alga is used as nutrient ingredients, and C-phycocyanin is used as natural dyes for food and cosmetics. C-phycocyanin also shows promise as a potential therapeutic agent for oxidative stress-induced diseases and as a fluorescent marker for biomedical research. C-phycocyanin content could reach 20 % [103]. However, A. platensis does contain other phycobiliproteins: Patel et al. [START_REF] Patel | Purification and characterization of C-phycocyanin from cyanobacterial species of marine and freshwater habitat[END_REF] determined C-phyco cyanin, allophycocyanins, and phycoerythrin contents to be 17.5, 3.8, and 1.2 % dw, respectively. A. platensis is also rich in carotenoids, 80 % of which are P-carotenes, the rest being physoxanthin and cryptoxanthin. The P-carotene and cryptoxanthin contents are very low under normal growth conditions (just 0.2 and 0.01 % dw, respectively) but these two carotenoids warrant mention as they are both con vertible to vitarnin A by mammals [START_REF] Bujard | Composition and nutritive value of blue green algae (Spirulina) and their possible use in food formula tions[END_REF]. Abd El-Baky et al. [START_REF] El-Baky | Spirulina species as a source of carotenoids and cx-tocopherol and its an ticarcinoma factors[END_REF] studied the influence of nitrogen and NaCl concentra tions on A. platensis growth in batch culture and found that cyanobacterium responded to nitrogen starvation and high NaCl levels by accumulating carotenoids. High carotenoid contents were obtained when A. platensis was grown at a low nitrogen level (51 ppm N), with values ranging from 19.82 to 24.1 mg/g (dw).

Haematococcus pluvialis and Haematococcus sp.

Haematococcus pluvialis is a freshwater unicellular green eukaryotic microalgae belonging to the class Chlorophy ceae. It can grow under autotrophic or heterotrophic con ditions and is commercially well known for its ability to produce massive amounts of astaxanthin (3,3' -dihydroxy p, P-carotene-4,4' -dione ), a red ketocarotenoid accumulated at up to 2.0 % (dw). Astaxanthin is biosynthesized through the isoprenoid pathway [START_REF] Grung | Algal carotenoids: 1. Secondary carotenoids 2. Haematococcus pluvialis aplanospores as a source of (3S,3'S)-astaxanthin esters[END_REF]. lt is a high-value product for pharmaceutical, nutraceutical and animal nutrition appli cations [START_REF] Lorenz | Connnercial potential for Haematococcus rnicroalgae as a natural source of astaxanthin[END_REF]. Astaxanthin from Haematococcus algae is under consideration for clearance by the US FDA and has already been cleared for sale in several European countries as a dietary supplement ingredient for human consumption.

H. pluvialis changes from a motile, flagellated cell to a non-motile, thick-walled aplanospore during the growth cycle [START_REF] Borowitzka | Culture of the astaxanthin-producing green alga Haematococcus pluvialis l. Effects of nutrients on growth and cell type[END_REF][START_REF] Kobyashi | Astaxanthin pro duction by a green alga, Haematococcus pluvialis accompanied with morphological changes in acetate media[END_REF]. The astaxanthin is contained in the apl anospore. Astaxanthin content in the aplanospores is about 1-2 % of dw, and the thick wall requires physical breakage before the astaxanthin can be either extracted or made available to organisms consuming the alga [START_REF] Mendes-Pinto | Evaluation of different cell disruption processes on encysted cells of Haematococcus pluvialis: effects on astaxan thin recovery and implications for bio-availability[END_REF]107]. Zhekisheva et al. [125] suggested that astaxanthin accu mulation under a monoesterified form under high light or nitrogen starvations was linked with FA accumulation. In both cases, the newly formed FA mostly correspond to oleic acid. The production of oleic acid-rich triacylglyce rols and the esterification of astaxantin enable the oil globules to maintain a high content of astaxantin esters. The highest rate of carotenoid biosynthesis was observed almost simultaneously with the highest rates of cell divi sion and chlorophyll biosynthesis after exposure to lirnited nitrogen and high irradiance [START_REF] Hagen | Fosmidomycin as an inhibitor of the non-mevalonate terpenoid pathway depresses synthesis of secondary carotenoids in flagellates of the green alga Haema tococcus pluvialis[END_REF]. A photo-protective role for astaxanthin has been suggested based on comparisons of astaxanthin-rich cells with astaxanthin-free vegetative cells, along with lower lipid peroxidation [START_REF] Hagen | Functional aspects of secondary carotenoids in Haematococcus lacustris [Girod] Ro stafinski (Volvocales). IV. Protection from photodynarnic dam age[END_REF], a higher Fv/ Fm value [START_REF] Hagen | Functional aspects of secondary carotenoids in Haematococcus lacustris (Volvocales). III. Action as a sunshade[END_REF], or a higher oxygen evolution rate [START_REF] Hagen | Fosmidomycin as an inhibitor of the non-mevalonate terpenoid pathway depresses synthesis of secondary carotenoids in flagellates of the green alga Haema tococcus pluvialis[END_REF]121].

Dunaliella salina and Dunaliella sp.

When the halophilic eukaryotic rnicroalgae D. salina is under appropriate (carotenogenic) cultivation conditions, it produces more than 10 % (on a dw basis) P-carotene versus just 0.3 % for other microalgae and higher plants. This massive accumulation of P-carotene seems to be a pro tection mechanism to counteract the negative effects of solar radiations [START_REF] Garcia-Gonzales | Production of Dunaliella salina biomass rich in 9-cis-�-carotene and lutein in closed tubular photobior eactor[END_REF]123]. High P-carotene content makes the cell display bright red coloration owing to the charac teristic absorption spectrum of P-carotene. Dunaliella species concentrate �-carotene in the interthylakoid spaces of the cell' s single chloroplast in the form of lipid globules under carotenogenic conditions. Few or no globules exist when the cell is cultivated under non-induced conditions [START_REF] Ben-Amotz | Accumulation of � carotene in halotolerant algae: purification and characterization of �-carotene-rich globules from D. bardawil (Chlorophyceae)[END_REF]. One analysis found that this membrane-free lipid globule was exclusively composed of �-carotene (more than half), neutral lipids (mainly with triacylglycerols) and a small amount of protein [START_REF] Ben-Amotz | Accumulation of � carotene in halotolerant algae: purification and characterization of �-carotene-rich globules from D. bardawil (Chlorophyceae)[END_REF].

Porphyridium sp.

The main accessory light-harvesting complexes of Por phyridium cruentum are phycobilisomes. P. cruentum phycobilisomes are described as hemiellipsoidal, contain ing a tricylindrical core subassembly in the semi-spherical centre and several peripheral rods radiating out from the center to form the rounded surface of the semi-sphere. The core contains allophycocyanins, while the peripheral rods are mainly composed of phycocyanins and phycoerythrins. It has been shown that P. purpureum (P. cruentum) con tains four phycobiliproteins: allophycocyanins (5 % ), R-phycocyanin (11 %), b-phycoerythrin (42 %), and B-phycoerythrin ( 42 % ) [START_REF] Bermejo | Chromato graphie purification and characterization of b-phycoerythrin from Porphyridium cruentum: semipreparative HPLC separation and characterization of its subunits[END_REF]. Porphyridium sp. is the potential organism for the production of phycoerythrins and especially B-phycoerythrin. B-phycoerythrin is the major light-harvesting pigment and the most valuable of the three main phycoerythrins classes due to its high fluorescence efficiency (intense unique pink color) and therapeutic value (immunomodulatory and anticarcino genic activity). Rebolloso-Fuentes et al. [START_REF] Fuentes | Biomass nutrient profiles of the microalga Porphyridium cruentum[END_REF] studied the influence of dilution rates, residence time, and solar irra diation conditions on the biomass profiles of P. cruentum grown on outdoor photobioreactor systems. The authors reported peak values of 3.6 % dw PB and 2.8 % dw phy coerythrins, but allophycocyanins and phycocyanin were also detected at up to 0.36 and 0.43 % dw, respectively. It has also been demonstrated in P. purpureum that phyco erythrin production is dependent on salt concentration in the culture medium. Sodium chloride seems to possess the maximum effect, and it was established that PB and phy coerythrins production can reach over 4.8 and 3.3 % dw, respectively, under optimal cultivation conditions at lab scale. Further optimization of specific culture parameters (light and sodium bicarbonate concentration) of a culture of P. purpureum led to a pigment-rich content composed of B-phycoerythrin (12.17 % dw), R-phycocyanin (10.2 % dw), and allophycocyanins (2.0 % dw) [118].

Haslea ostrearia

Haslea ostrearia, previously known as Navicula ostrearia, is a diatom involved in the greening of the French oysters called "Fines de Claires" [START_REF] Lankester | On green oysters[END_REF]. This microalga synthesizes a blue-green water-soluble pigment called marennine that is first stored at the apex of the cells then released into the surrounding environment [START_REF] Ranson | L'absorption de matières organiques dissoutes par la surface extérieure du corps chez les animaux aquatiques[END_REF]. Marennine is present under two different location-specific forms: extracellular maren nine (EMn; molecular weight: 9,893 Da) and intracellular marennine (IMn; molecular weight: 10,751 Da) [START_REF] Pouvreau | Purification of the blue-green pigment "marennine" from the marine tychopelagic diatom Haslea ostrearia (Gaillon/Bory) Simonsen[END_REF].

Although the structure of the molecule remains unknown, Pouvreau et al. [START_REF] Pouvreau | Purification of the blue-green pigment "marennine" from the marine tychopelagic diatom Haslea ostrearia (Gaillon/Bory) Simonsen[END_REF] have shown that it is a polyphenolic compound close to the proanthocyanides. The specific absorption coefficients of purified EMn and IMn were 12.13 1/g/cm at 677 nm and 6.88 1/g/cm at 672 nm, respectively. This pigment is not involved in photosyn thesis processes but instead acts as a photoprotector that is produced in response to different stresses such as high light intensity [START_REF] Neuville | Recherche sur le déterminisme de la production de marennine par la diatomée marine Navicula ostrearia (Gaillon) Bory en culture in vitro[END_REF] or nitrogen and silica depletion [START_REF] Neuville | Recherche sur le déterminisme de la production de marennine par la diatomée marine Navicula ostrearia (Gaillon) Bory en culture in vitro[END_REF][START_REF] Rossignol | Pro duction of exocellular pigment by the marine diatom Haslea ostrearia Simonsen in a photobioreactor equipped with immersed ultrafiltration membranes[END_REF]. Marennine possesses antiproliferative, antiviral, and photoprotective activities [START_REF] Gastineau | Biological activities of purified marennine, the blue pigment responsible for the greening of oysters[END_REF][START_REF] Pouvreau | Antioxidant and free radical scavenging properties of marennine, a blue-green polyphenolic pigment from the diatom Haslea ostrearia (Gail lon/Bory) Simonsen responsible for the natural greening of cultured oysters[END_REF].

Proteins

Proteins constitute a large fraction of the actively growing eukaryotic microalgae and cyanobacteria. They are gener ally undervalued compared to minor products such as unsaturated FA or pigments for medical, pharmaceutical, and nutraceutical applications. Today, there are still no significant applications of purified proteins from microal gae, since the presence of non-protein components gener ally leads to undesired changes in color, taste, and protein structure [104]. Nevertheless, efforts are now turning to the impressive genetic potential of microalgae on one band and the specific activities of microalgal proteins on the other. However, some algal biomasses are often used directly as a supplement in human diet, animal feed, or aquaculture, largely due to their high protein levels.

Chlorella vulgaris

C

Arthrospira platensis

A. platensis is considered an alternative protein source and a very promising food supplement or additive, as its bio mass contains up to 70 % dw ofproteins [START_REF] Avila-Leon | Arthrospira platensis biomass with high protein content cultivated in continuous process using urea as nitrogen source[END_REF]. The quality of powders from A. platensis ( drum-dried and sun-dried bio masses) was assessed using parameters representative of nutritional quality. Maximum digestion coefficient, net protein utilization, biological value, and protein efficiency ratio values were 77.6, 83.9, 65.0, and 2.10, respectively. These values are relatively close to those of reference nutrition proteins such as casein or egg proteins, high lighting the quality of A platensis as protein source for human nutrition or as a feed supplement [START_REF] Becker | Micro-algae as a source of protein[END_REF]. Additional data indicate that proteins from A. platensis are up to 90 % digestible and contain about 60 % essential AA [START_REF] Dillon | Spirulina as a source of proteins in human nutrition[END_REF]. Depleted culture media have a significant impact on the protein content of A platensis. In a study on batch cultures systems, the crude protein content of S. platensis harvested at the stationary phase was 67.4, 53.5, and 5.6 % dw for controls, 50 %-nitrogen-depleted media and 100 %-nitro gen-depleted media, respectively [ 117].

Scenedesmus obliquus

S. obliquus is commonly used for biofuel production but is also important as a source of proteins, with maximum content reaching 56 % [START_REF] Becker | Micro-algae as a source of protein[END_REF]. However, many authors have described S. obliquus as a good candidate for wastewater treatment and the conversion of nutrients or waste com pounds to biomass. The protein content of S. obliquus grown in outdoor conditions in artificial wastewater during 3 days under seasonal irradiance levels was ranged from 33.5-38 % dw in winter to 51 % dw in summer [START_REF] Gomez-Villa | Biomass production and nutrient budget in outdoor cultures of Scene desmus obliquus (Chlorophyceae) in artificial wastewater, under the winter and summer conditions of Mazatlan, Sinaloa, Mexico[END_REF]. Çelekli and Balci [START_REF] Çelekli | The influence of different phosphate and nitrate concentrations on growth, protein and chlorophyll a content of Scenedesmus obliquus[END_REF] evaluated the production of proteins from Scenedesmus sp. grown in Johnson's medium as a function of P and N concentration in the culture medium, and found that protein production was strongly related to nutrient concentration. Maximum protein yield was 32 mg/1 for 0.3 mM P and 12 mM N. In another experiment, Ruiz Martin et al. [101] showed that a semicontinuous culture of immobilized S. obliquus did not appear to allow sustainable culture for wastewater treatment compared to a free-cell culture. Protein content first increased but then tailed off toward the end of the treatment (250 h), meaning that the exponential phase was maintained for a few cycles before the culture collapsed. However, protein content hit a peak of 30 % dw compared to only 16 % dw in the free-cell culture.

The immobilized system thus appears suitable for separat ing biomass from wastewater whereas the free-cell system appears to be the best option for producing biomasses with high nutritional potential. This nutritional quality of S. obliquus was confirmed by the maximum biological value, digestion coefficient, net protein utilization, and protein efficiency ratio values (75.0, 88.0, 67.3, and 1.99, respec tively) and the essential AA content that ranged from 30 to 45 % of total AA. It was concluded that low concentrations of metallic pollutants stimulated the production of biomass and improved total protein content and AA biosynthesis whereas high concentrations were inhibitory [START_REF] Becker | Micro-algae as a source of protein[END_REF].

Dunaliella sp.

The marine microalgae D. salina is mainly identified as a carotenoid producer, and is widely used in cosmetics.

There bas been little research on the nutritional, functional, and biological properties of the proteins but intensive research on their specific role in the mechanisms of salt tolerance. Certain specific proteins (e.g., two membrane bound proteins: a 60-kDa carbonic anhydrase and a 150-kDa transferrin-like protein) and enzymes (e.g., dihy droxy acetone reductase reduction of dihydroxyacetone in glycerol) can be induced by different salinities in the osmotic response processes [START_REF] Liska | Enhanced photosynthesis and redox energy production contribute to salinity tolerance in Dunaliella as revealed by homology-based proteomics[END_REF]. More, Dunaliella sp. is amenable for genetic manipulation and is a good candidate for the induction, accumulation and production of several novel bioactive compounds, including valuable enzymes. For instance, dihydroxy acetone reductase bas been obtained from D. salina at the industrial scale and subse quently produced for commercialization [11].

P. cruentum is not necessarily known for its nutritional value despite its rich FA composition (eicosapentaenoic acid, docosahexaenoic acid, etc.) and significant protein content (up to 35 %) [START_REF] Gonzalez L6pez | Pro tein measurements of rnicroalgal and cyanobacterial biomass[END_REF], which includes a significant fraction of phycobiliproteins, EPS-embedded proteins, biological membrane or cell wall proteins, and free pro teins, but more crucially, superoxide dismutase. Superox ide dismutase is a free cytoplasmic enzyme that is synthesized in abundance and considered a high-value compound for biomedical applications such as the treat ment of cancers or HIV involving free radicals and other active forms of oxygen [START_REF] Bannister | Aspects of the structure, fonction, and applications of superoxide dismutase[END_REF]. Moreover, among the proteins of the EPS cell wall of Porphyridium sp., the most prom inent is a 66-kDa glycoprotein (consisting of a 58-kDa polypeptide and 8-kDa sugar moieties) that is tightly bound but not covalently linked to EPSs. This glycoprotein merits attention due to its role in biorecognition and its potential biotechnological applications such as for understanding the N-glycosylation mechanisms linked to the production of therapeutic proteins, where the algae could be used as a "cell factory" [START_REF] Levy-Ontman 0 | Unique N-glycan moieties of the 66-kDa cell wall glycoprotein from the red rnicroalga Porphyridium sp[END_REF].

Tetraselmis suecica

Tetraselmis suecica, a marine green microalgae, is widely used as a source of nutrients for invertebrates in aqua culture due to its homogenous chemical composition and high protein content ranging from 26 to 69 % dw, depending on growth conditions [START_REF] Fabregas | Productivity and biochemical composition of cyclostat cultures of the marine microalga Tetraselmis suecica[END_REF]. Bondioli et al. [START_REF] Bondioli | Oil production by the marine microalgae Nannochloropsis sp. F&M-M24 and Tetraselmis suecica F&M-M33[END_REF] recorded a substantial decrease in protein content when T. suecica was grown in semicontinuous-mode in outdoor "green wall panels" under both N and P starvation. Indeed, the protein content of N-sufficient, N-starved, and N-and-P-starved T. suecica were 52, 8, and 9 % dw, respectively, in 14-day cultures. Nevertheless, Laws et al. [START_REF] Laws | Phosphate-limited growth and uptake kinetics of the marine prasinophyte Tetra selmis suecica (Kylin) Butcher[END_REF], studying the relationship between growth rate and biochemical composition, concluded that the control of T. suecica growth rate in a continuous-mode system and phosphate-limited conditions is a convenient way to optimize the biochemical composition of T. suecica and its nutritional value to invertebrate consumers in aqua culture systems. Protein content under this protocol was estimated to range from a minimum of 7 % at zero growth rate to a maximum of 59 % at a relative growth rate of 1.0.

Conclusions

The past two decades have seen huge strides forward in our understanding of the chemical composition, cellular compartments, and culture conditions of eukaryotic mic roalgae and cyanobacteria, culminating in the development of new-generation photobioreactors enabling large-scale microalgae biomass farming. Moreover, the recent industry drive for microalgae applications in the near future, mainly for biodiesel production but also for more specific appli cations in the fields of food, feed, cosmetics, and thera peutics, has prompted a significant increase in funding for this field of science. However, the fractionation of all components of microalgae biomasses under a biorefinery framework is an emergent concept that starts to be trans lated into real-world industrial practice. The complexity of membrane bound organelles, granules, and droplets con taining specific biomolecule classes combined with inade quate methods for separating them has meant that in many cases, only a single biomolecular fraction extracted from specific species actually gets used. A biorefinery approach generally entails upgrading spent biomass after lipid extraction at the expense of alternative bulk or fine chemical production streams. Sorne breakthroughs are still needed in the design and development of technologies that can reduce costs and increase yields, but even more important is the development of a methodology and tech nical processes for the global utilization of microalgae. The strategy can be considered as the eco-design of an inte grated production scheme. Integrated studies will probably attain the goal of using lipid-producing strains under adapted growth conditions and optimized processes to successfully create an economically viable biorefinery of microalgae.

Fig. 2

 2 Fig. 2 Ultrastructure of microalgae: <Emphasis Type= Chlorophyceae, b Rodophyceae, and c cyanobacteria (based on Re f. [70])

Fig. 4

 4 Fig.[START_REF] Ashokkumar | Mass culture of Botryo coccus braunii Kutz. under open raceway pond for biofuel production[END_REF] Structure of the main microalgal carotenoids[START_REF] Guaratini | Comparison of diode array and electrochemical detection in the C 30 reverse phase HPLC analysis of algae carotenoids[END_REF]

  vulgaris grown under normal conditions is about 8 and 5 mg/100 g, respectively[START_REF] Cha | Antiproliferative effects of carotenoids extracted from Chlore/la ellipsoidea and Chlore/la vulgaris on human colon cancer cells[END_REF] 100]. Never theless, Mendes et al.[START_REF] Mendes | Supercritical CO 2 extraction of carotenoids and other lipids from Chlorella vulgaris[END_REF] extracted pigments of "orange" C . vulgaris obtained from a carotenogenesis-optirnized culture. Supercritical CO 2 extraction was estimated to be

	efficient at removing carotenoids from well-crushed bio
	mass (content: 547 mg/100 g dw) compared to solvent
	(acetone) extraction (content: 133 mg/100 g dw). Asta
	xanthin and canthaxanthin composed 2/3 of the extracted
	carotenoids [80]. However, carotenoid content is strain
	dependent. Ruen-ngam et al. [100] estimated the lutein
	content from spray-dried high lutein content C. vulgaris to
	be 7.9 mg/g dw. Carotenoids from C. vulgaris have
	important pharmaceutical and food-industry applications.
	Indeed, serni-purified carotenoid (mainly lutein) extracts
	showed antiproliferative activity in an MTT assay on
	HCTl 16 cells with interesting IC 50 values