V Eymet
email: eymet@sphinx.ups-tlse.fr

R Fournier

S Blanco

J L Dufresne

A Boundary-Based Net Exchange Monte-Carlo Method for absorbing and scattering thick media

Keywords: Monte Carlo, net-exchange formulation, scattering, numerical optimization, convergence

A boundary-based net-exchange Monte Carlo method was introduced in [1] that allows to bypass the difficulties encountered by standard Monte Carlo algorithms in the limit of optically thick absorption (and/or for quasi-isothermal configurations). With the present paper, this method is extended to scattering media. Developments are fully 3D, but illustrations are presented for plane parallel configuration. Compared to standard Monte Carlo algorithms, convergence qualities have been improved over a wide range of absorption and scattering optical thicknesses. The proposed algorithm still encounters a convergence difficulty in the case of optically thick, highly scattering media.

Introduction

The Monte-Carlo Method (MCM) has been widely used in the field of transport phenomena simulation, and more specifically in the field of radiative transfer computing [START_REF] Hammersley | Monte-Carlo methods[END_REF][START_REF] Howell | Application of monte carlo to heat transfer problems[END_REF][START_REF] Howell | The monte-carlo method in radiative heat transfer[END_REF]. In this particular case, the method mainly consists in simulating numerically the physical statistical model of photons transport, from their emission to their absorption in a potentially scattering medium. A well known advantage of this method is that the corresponding computing code is easy to set up and to modify. Another main advantage is that it is a reference method : as the MCM is a statistical method, a standard deviation may be computed in addition to each result, that may be interpreted as a numerical uncertainty. Also, it has recently been shown that the MCM allows the computation of parametric sensitivities with no extra significant computing [START_REF] De Lataillade | Monte-carlo method and sensitivity estimations[END_REF]. This can be helpful for design needs, or when radiative transfer is coupled with other physical processes. Finally, the MCM is known to be well adapted to the treatment of configurations with a high level of complexity (complex geometries, complex spectral properties, ...). However, in spite of these advantages over other methods and in spite of the regular increase of available computational powers, the computational effort requirement of MCM often remains a significant drawback.

Different works in the last fifteen years tried to preserve the main advantages of the method, in particular its strict analogy with physical processes, and the ability to solve complex problems, while trying to improve convergence qualities. There are mainly two ways MCM convergence can be enhanced : formulation changes and adaptation of sampling laws [START_REF] Hammersley | Monte-Carlo methods[END_REF]. As far as formulation is concerned, most attention has been devoted to reverse Monte Carlo algorithms [START_REF] Walters | Rigorous development for radiation[END_REF], that make use of reciprocal transport formulations (application of the reciprocity principle to the integral form of the radiative transfer equation), and to net-exchange Monte Carlo algorithms [START_REF] Cherkaoui | Monte-carlo simulation of radiation in gases with a narrow-band model and a net-exchange formulation[END_REF][START_REF] Cherkaoui | Radiative net exchange formulation within 1d gaz enclosures with reflective surfaces[END_REF][START_REF] Dufresne | Méthode de monte-carlo par échanges pour le calcul des bilans radiatifs au sein d'une cavité 2d remplie de gaz[END_REF][START_REF] Tessé | Radiative transfer in real gases using reciprocal and forward monte carlo methods and a correlated-k approach[END_REF], that make use of net exchange transport formulations (combination of forward and reciprocal formulations, photons being followed both ways along each optical path). Netexchange Monte Carlo algorithms allowed in particular to bypass the problem of standard Monte Carlo algorithms for quasi-isothermal configurations. As far as sampling laws optimization is concerned, numerous works have successfully used the biasing of sampled directions toward the parts of the system that most contribute to the addressed radiative quantity [START_REF] Martin | Monte carlo analysis of the backscattering of radiation from a sphere to a plane[END_REF], or the biasing of sampled frequencies as function of temperature field and spectral properties [START_REF] Cherkaoui | Monte-carlo simulation of radiation in gases with a narrow-band model and a net-exchange formulation[END_REF][START_REF] Dufresne | Inverse gaussian kdistributions[END_REF].

Recently, the combination of formulation efforts and sampling laws adaptations permitted to solve the well known convergence problem of traditional Monte-Carlo algorithms in the case of strong optical thickness configurations [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF]. If a gas volume is optically very thick, most emitted photons are absorbed very close to their emission position, and thus do not take part to the exchange of the gas volume with the rest of the system. Consequently, very large numbers of statistical realizations are required to reach satisfactory convergences. This problem could be solved in the case of purely absorbing systems thanks to a net-exchange formulation in which emission positions are sampled, starting form the volume boundary, along an inward oriented sampled direction (a formulation that will be named here a "boundary-based net-exchange formulation"). All sampling laws (frequency, boundary position, direction and emission position) where also finely optimized in order to insure that the al-gorithm automatically adapts to system optical thickness in the whole range from the optically thin to the optically thick limits.

The present paper is one of a series that seek to improve the MCM through such methodological developments. It proposes techniques to take into account scattering in the above mentioned boundary-based net-exchange algorithm. The formulation used in [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF] has been generalized and clarified in order to take into account the scattering phenomena. Developments are fully 3D, but convergence illustrations are presented for plane parallel configurations that are specifically meaningful in the atmospheric science community.

Sec. 2 of this article puts the emphasis on the multiple integral theoretical developments on which our Monte-Carlo algorithm is based. Sec. 3 presents gas volume emission results in a simple test case, thus revealing the algorithm convergence qualities together with its limits of applicability. Finally, Sec. 4 completes this convergence illustration in terms of radiative flux divergence profiles.

Theoretical developments

The three next paragraphs deal with improvements that were brought to the standard bundle transport MCM during the last few years, through a number of different methodological developments.

Exchange Formulation

Let us consider that, for the purpose of a 3D radiative transfer computation, the considered system is divided into volume and surface elements. Until further mention, this geometric division is only motivated by the required level of analysis and it therefore implies no physical assumption : the volume and surface elements have any geometrical shapes and are inhomogeneous.

The energy rate E i,j emitted by an arbitrary gas volume i and absorbed by an arbitrary gas volume j may be expressed as : where B(P ) is the monochromatic blackbody intensity at point P . Γ (P,u 0 ) represents the space of (infinite) optical paths γ originated from point P , in the direction u 0 , distributed according to p(γ; P, u 0 ). Every such path will finally reach volume V j and will even cross it an infinite number of times. l γ is the curvilinear coordinate along the optical path γ and l γ,n are the values of this curvilinear coordinate at the positions of the n th intersection between optical path γ and gas volume V j : l - γ,n stands for the n th entry point coordinate, and l + γ,n stands for the n th exit point coordinate. σ n is the curvilinear abscissa of point P n in the n th intersection interval between γ and V j . T γ,n is the transmitivity between point P and the position l - γ,n : it is a product of exponential attenuations and of surface reflectivities for surface reflexions.

E ij = V i dV i (P ) 4π dω(u 0 ) Γ (P,u 0 ) p(γ; P, u 0 )dγ k a (P ) ∞ n=1 T γ,n l + γ,n l - γ,n dσ n (P n ) k a (P n )B(P )exp - σn l - γ,n dσ k a (σ ) (1)
The integral over Γ (P,u 0 ) , according to the distribution p(γ; P, u 0 ), will not be detailed in this paper, because the purpose of this work is to put the emphasis on the Net Exchange Formulation itself, and not to deal with the physical model used for optical paths representation. These paths are purely random walk optical paths, and all the complexities associated with the formulation of scattering angles and free path length are deported into the expression of p(γ; P, u 0 ), that represents formally the existence probability density of a given optical path γ.

This formulation may be used to derive a standard path integrated Monte-Carlo algorithm, that may be described as follows :

• First, the emission point P is randomly chosen in the gas volume V i , and the emission direction u 0 is randomly chosen in the unit sphere (4π st). • The optical path γ is generated with a standard random walk technique.

• Each time this optical path reaches the gas volume V j , a point P is randomly chosen along the part of γ that intersects V j .

• Finally, the optical path γ ends when it is long enough for the energy bundle to be considered as totally attenuated (as function of the required level of accuracy).

Using a Monte-Carlo algorithm based on a traditional formulation, the Net Radiative Budget is expressed as the difference between approximate emitted and absorbed energy rates that are computed separately, and that can be close the one to the other for nearly isothermal configurations, inducing numerical convergence difficulties.

Net Exchange Formulation

The Net Exchange Formulation is based on the Net Exchange Rates Ψ ij between all pairs of elements (either surface or volume elements) i and j, which is defined as the difference between the energy rate emitted from element i and absorbed by element j, and the energy emitted from element j and absorbed by element i.

The advantages of formulating radiative transfers in terms of Net Exchange Rates have been shown by Green [START_REF] Green | Division of radiative streams into internal transfer and cooling to space[END_REF]. The Net Exchange Formulation has been introduced in the MCM by [START_REF] Cherkaoui | Monte-carlo simulation of radiation in gases with a narrow-band model and a net-exchange formulation[END_REF][START_REF] Cherkaoui | Radiative net exchange formulation within 1d gaz enclosures with reflective surfaces[END_REF]. This radiative transfer formulation solved the convergence problem encountered by the MCM in nearly isothermal configurations. A general formulation of a Net Exchange Rate Ψ ij between two gas volumes i and j may be directly deduced from the energy rate equation Eq. 1, replacing B(P ) by B(P ) -B(P ) [START_REF] Dufresne | Méthode de monte-carlo par échanges pour le calcul des bilans radiatifs au sein d'une cavité 2d remplie de gaz[END_REF] :

Ψ ij = V i dV i (P ) 4π dω(u 0 ) Γ (P,u 0 ) p(γ; P, u 0 )dγ k a (P ) ∞ n=1 T γ,n + γ,n l - γ,n dσ n (P n ) k a (P n ) B(P ) -B(P n ) exp - σn l - γ,n dσ k a (σ ) (2)
where B(P n ) is the monochromatic blackbody intensity at point P n in volume V j (see Fig. 1). Similarity of equations Eq. 1 and Eq. 2 makes fairly easy the implementation of a Net Exchange Formulation in a standard Monte Carlo algorithm : the monochromatic blackbody intensity B(P ) has to be replaced by B(P ) -B(P ) .

Using a Net Exchange Formulation, the Net Radiative Budget of a gas volume i may be expressed as a sum of Net Exchange Rates :

Ψ i = j Ψ ij (3)
Here, using a Monte-Carlo algorithm based on a Net Exchange Formulation means that all Net Exchange Rates are computed separately (as pondered sums of blackbody intensity differences B(P ) -B(P ) , which induces no numerical difficulty) and are then added to produce the Net Radiative Budget.

In the limit case of nearly isothermal configurations, no convergence difficulty will be encountered : as it does no longer compute the difference between two very close approximate values, a Monte-Carlo algorithm based on a Net Exchange Formulation will lead to much better accuracies than traditional Monte-Carlo algorithms. [START_REF] Cherkaoui | Monte-carlo simulation of radiation in gases with a narrow-band model and a net-exchange formulation[END_REF][START_REF] Cherkaoui | Radiative net exchange formulation within 1d gaz enclosures with reflective surfaces[END_REF].

Boundary-Based Net Exchange Formulation

A typical difficulty that is encountered by any standard MCM (both bundle transport and path integrated MC algorithms) 1 is the problem of optically thick systems. Let us consider the computation of the emission E from a given gas volume, using a standard path integrated Monte-Carlo algorithm. In the case of an optically thick gas volume, the computation of E will suffer from a convergence problem : most bundles emitted into the gas volume will be totally attenuated when they cross the volume boundary. Only those emitted very close to the boundary will have a chance to leave the gas volume with a significant computational weight. Thus, the computation of E will require a great number of statistical realizations N in order to get a good accuracy over E. This convergence difficulty is due to the fact that emission positions are chosen uniformly among the gas volume. A possible way to solve this problem would be to sample more often emission positions close to the volume boundary, so that most bundles would leave the gas volume with a significant energy, thus contributing more significantly to E. Modifying the way emission positions are sampled means to modify sampling laws used in the algorithm, without modifying the result of the multiple integral ; in order to do this, we choose to use a net exchange formulation different from the initial formulation presented in Eq. 2. This reformulation -that brings forward the distance between emission point and first exit point -is the purpose of the present subsection.

Eq. 2 starts with an integration over all locations P within volume V i , then one integrates over all optical paths γ starting at P and γ happens to cross the boundary of V i (here noted S i ) at a location Q (see Fig. 2) : this boundary does not appear as an explicit integration domain. On the contrary, the following formulation (that will be referred as "Boundary-Based Net Exchange Formulation") starts with an integration over all exit locations Q on S i , then one integrates over the exit hemisphere at Q and then over all the optical paths initiating within V i and crossing its boundary at the retained exit location and exit direction : the boundary of V i appears as an explicit integration domain, but not the volume V i itself.

Ψ ij = S i dS i (Q) 2π dω(u 0 ) u 0 .n L i (Q, u 0 ) Γ (Q,u 0 ) p(γ; Q, u 0 )dγ ∞ n=1 T γ,n l + γ,n l - γ,n dσ n (P n )k a (P )k a (P n ) B(P ) -B(P n ) exp - σn l - γ,n dσ k a (σ ) (4)
where L i (Q, u 0 ) is the fraction of the intensity at Q in direction u 0 that corresponds to photons emitted within V i and crossing S i for the first time. Using the optical path reciprocity principle, it is possible to formulate L i (Q, u 0 ) as :

L i (Q, u 0 ) = Γ (Q,-u 0 ) p(γ; Q, -u 0 )dγ l+ γ,1 0 dσ(P ) exp - σ 0 d σ k a ( σ ) (5)
where Γ (Q,-u 0 ) is the space of optical paths originated from point Q, in the direction -u 0 and l+ γ,1 stands for the point at which γ first exits V i (see Fig. 2).

The Monte-Carlo algorithm that was derived from this new formulation of Net Exchange Rates Ψ ij may be described as follows :

• First, a point Q is randomly chosen on the boundary S i surrounding gas volume V i , and the initial direction u 0 is randomly chosen in the exit unit hemisphere (2π st). • Starting at Q in the direction -u 0 , the optical path γ is generated with a standard random walk technique until it first exits V i and P is then randomly chosen within V i along this truncated path. • Starting at Q in the direction u 0 , the optical path γ is generated with a standard random walk technique. • Each time γ reaches volume V j , a point P is randomly chosen along the part of γ that intersects V j . • Finally, the optical path γ ends when it is long enough for the net-exchange bundle to be considered as totally attenuated (as function of the required level of accuracy).

At this point of the developments, only the boundary-based reformulation of net exchange rates has been achieved. In the next subsection, it will be shown how the sampling laws that arise from this formulation (Monte Carlo computation of the corresponding multiple integrals) may be optimized in order to solve convergence difficulties in the optically thick limits.

Optimization of sampling laws

The above described algorithm principle requires successive random generations 2 of an exit position Q, an exit direction u 0 , a first exchange position P (via the curvilinear abscissa σ), and second exchange positions P (via the curvilinear abscissa σ). It can be easily shown that any non zero probability density function may be used for each such sampling insuring the same integral solution at the limit of an infinite number of bundles. One way of illustrating this point is to rewrite Net Exchanges Rates Ψ ij , starting from 2 The random walk sampling laws corresponding to the generations of γ and γ are left apart in the present article because no optimization is proposed concerning this part of the algorithm. Such an optimization process is non trivial and none of the attempts made at date have enough generality to be implemented on a standard basis. Among the most successful attempts, a specific mention can be made to the work of Berger and al. reported in [START_REF] Hammersley | Monte-Carlo methods[END_REF] for simulation of optically thick radiation shields.

Eq. 4 and transforming all successive integrals into statistical averages :

Ψ ij = < ∞ n=1 I n 1 β n > = S i pdf S (Q)dS i (Q) 2π pdf Ω (u 0 )dω(u 0 ) Γ (Q,-u 0 ) p(γ; Q, -u 0 )dγ l+ γ,1 0 pdf Σ(σ)dσ(P ) Γ (Q,u 0 ) p(γ; Q, u 0 )dγ ∞ n=1 l + γ,n l - γ,n pdf Σn (σ n )dσ n (P ) ∞ n=1 I n 1 β n (6)
where I n is the net-exchange density :

I n = (u 0 .n) k a (P )exp - σ 0 k a ( σ )d σ T γ,n k a (P n ) B(P )-B(P n ) exp - σn l - γ,n dσ k a (σ ) (7
) and β the correction term :

β n = pdf S (Q)pdf Ω (ω)pdf Σ(σ)pdf Σn (σ n ) (8)
Eq. 6, Eq. 7 and Eq. 8 insure a continuous link between the retained photon transport model (with a given formulation choice, here Eq. 4) and the Monte Carlo algorithm : successive integral averages are translated into successive random sampling events and for each set of sampled variables the retained quantity is the sum of all I n 1 βn (whose average value will be an approximation of Ψ ij ). Once the transport model and the integral formulation have been chosen the only remaining question is the choice of the sampling probability density functions : this last choice does not modify the algorithmic structure, neither does it change the solution after convergence, but it strongly affects algorithmic convergence via the variance of ∞ n=1 I n 1 βn . The more physical knowledge is introduced in these probability density functions, the smaller the variance of ∞ n=1 I n 1 βn and the faster the convergence [START_REF] Hammersley | Monte-Carlo methods[END_REF]. The probability density functions proposed hereafter are designed to insure satisfactory convergence speeds for a wide range of absorption and scattering optical thicknesses. The main objective was generality, hopping that such a set of probability density functions can serve as a start basis for more detailed adjustments when addressing specific configurations families.

• Sampling of exit points Q

The boundary sampling law pdf S (Q) has been chosen as uniform : pdf S (Q) = 1/S i . In the general case, having no information concerning the parts of S i through which V i exchanges most radiative energy with its environment, no better pdf adjustment could be proposed. Obviously, for specific configurations where V i exchanges radiation with hot spots at identified locations, this information can be directly used to modify pdf S (Q) so that the areas of stronger net-exchanges are more frequently sampled.

• Sampling of exit directions u 0

In the work of De Lataillade and al.( [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF]), the angular sampling law pdf Ω (u 0 ) was optimized for the case of a purely absorbing medium. The lambertian distribution was used for strong optical thicknesses, whereas the isotropic distribution was used in case of optically thin gas volumes. The limit between weak and strong optical thicknesses was set to τ a = 1 where τ a is the absorption optical thickness of the considered volume :

pdf Ω (u 0 ) = 1 2π if τ a < 1 pdf Ω (u 0 ) = u 0 .n π if τ a ≥ 1 (9)
In the present work, the limit criteria is modified in order to account for the effect of scattering.

pdf

Ω (u 0 ) = 1 2π if τ eq = τ a + (1 -g)τ s < 1 pdf Ω (u 0 ) = u 0 .n π if τ eq = τ a + (1 -g)τ s ≥ 1 (10)
where τ s is the scattering optical thickness of the considered volume and g is the phase function asymmetry parameter. In the case of a purely absorbing medium, τ eq = τ a and we are back to the proposition of [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF] : when τ a > 1, the absorption mean free path λ a = 1 ka is smaller than the system size which insures that the specific intensity of emitted photons (photons emitted within the gas volume that reach the boundary) is close to isotropy. Multiple scattering also induces an isotropic distribution of specific intensity at the volume boundary, but here the relevant scale is not the scattering mean free path λ s = 1 ks but the scattering transport mean free path λs 1-g which accounts for the shape of the scattering phase function (forward scattering induces higher values of the transport mean free path) [START_REF] Case | Linear Transport Theory[END_REF]. When the medium both absorbs and scatters, the relevant scale is the total transport mean free path λ eq defined as 1 λeq = 1 λa + 1-g λs which leads to the proposition of Eq. 10.

• Sampling of first exchange position P

As in [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF], the first exchange position P along γ is sampled by use of a randomly generated abscissa σ between 0 and l+ γ,1 (see Fig. 2). The main interest of the proposed boundary based formulation is that the sampling law can be chosen as function of the absorption optical thickness in order to favor emission positions close to the boundary in the optically thick limit. This is done using an exponential probability density function for γ, which corresponds to an ideal adaptation for isothermal gas volumes :

pdf Σ(σ) = k a exp(-k a σ) 1 -exp(-k a l+ γ,1 ) (11)
Random generation of γ is simply performed on the basis of a uniform random generation of r in the unit interval according to :

σ = - 1 k a ln   1 -r 1 -exp(-k a l+ γ,1 )   (12)
For small values of absorption coefficient k a (optically thin limit), the above expression reduces to σ ≈ r l+ γ,1 , which is equivalent to choosing uniformly σ within [0, l+ γ,1 ]. The physical significance of this, is that each point of lγ contributes the same way to the radiative transfer, because the energy emitted at each point is totally transmitted out of the gas volume. On the contrary, for strong values of k a (optically thick limit), Eq. 12 reduces to σ ≈ -1 ka ln(1 -r). l+ γ,1 is no longer taken into account and most exchange positions P are sampled in the immediate vicinity of the boundary : most statistical events have a significant contribution to the net-exchange and the statistical variance is reduced.

• Sampling of second exchange positions P n Similarly, second exchange positions are generated along γ by use of randomly generated abscissa σ n according to :

pdf Σn (σ n ) = k a exp(-k a (σ n -l - γ,n )) 1 -exp(-k a (l + γ,n -l - γ,n )) (13)
Unlike in [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF], when the medium is both absorbing and scattering, the impact of these sampling laws on the behavior of the associated Monte-Carlo algorithm is configuration dependent : sampling law adaptation is not satisfactory in the whole parameter range. The leading parameter is the single scattering albedo : ω = ks ka+ks • For ω << 1, scattering is negligible compared to absorption. In this case, the medium may be considered as purely absorbing, and it has been shown in [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF] that the proposed sampling laws are suitable for such configurations.

In particular, they solve the convergence difficulty encountered by Monte-Carlo algorithms in optically thick absorption configurations. • For usual values of ω (ω ∈]0, 1[ except for values very close to unity), scattering increases optical path lengths, and the use of the presented sampling laws results in a correct sampling of both exchange positions P and P . • For ω ≈ 1, absorption is negligible compared to scattering. In this particular case, the proposed sampling laws fail to sample efficiently the optical path space. The difficulty may be described as follows : when scattering is the dominant process, the medium may be considered as optically thin on the point of view of absorption. In this case, all points into a given gas volume contribute equally to the exchange between this gas volume and the rest of the system. Even if the use of the proposed law for pdf Σ(σ) will result in a uniform sampling of first exchange positions P along all generated optical paths, most of these paths will be very short, because of the medium strong scattering properties (intense backscattering from point Q). First exchange positions P will therefore be mainly sampled in the vicinity of the volume boundary which is not in accordance with the physics of radiative net-exchanges in little absorbing and highly scattering configurations. The proposed algorithm will therefore encounter convergence difficulties. We will see however that this difficulty is partly compensated by a reduction of the average number of scattering events to be numerically generated, the overall cost of the algorithm remaining satisfactory up to high albedo levels.

Convergence illustration : non-isothermal slab emission

As in [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF], the proposed algorithm is first tested using the academic problem of monochromatic slab emission. A single horizontal slab is considered, constituted of semi-transparent medium, with uniform absorbing and scattering optical properties, between two black boundaries at 0K. The slab physical thickness is H and the z-axis is downward-positive. The temperature profile across the slab is such that the blackbody intensity at the considered frequency follows a linear profile B(z) from 0 at the top to B 0 at the bottom of the slab. The addressed quantity is the downward slab emission, which is also the net-exchange rate between the slab and the bottom boundary. In each figure, N is displayed for three different Monte-Carlo algorithms :

• 1 -A standard Monte-Carlo algorithm, in which bundles are generated uniformly within the layer, with isotropic directions, and are attenuated along their multiple scattering optical paths until they leave the layer (algorithm based on an exchange formulation with a uniform law for volume sampling and an isotropic law for angular sampling, see Eq. 1). • 2 -The boundary-based net-exchange algorithm proposed in Sec. 2.

• 3 -The same algorithm except that the angular sampling law of [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF] is used (see Eq. 9), instead of that in which we attempted to account for scattering (see Eq. 10).

It can be seen in Fig. 3(a) that for small values of the single-scattering albedo (ω = 0.01), N is stabilizing for algorithms 2 and 3 (boundary based algorithms) as the slab total optical thickness τ H increases, while for algorithm 1 (standard MC algorithm), N keeps increasing for large values of τ H . In the It is no longer the case for extremely strong single-scattering albedoes (Fig. 6(a), ω = 0.9999) ; this convergence difficulty for high albedoes was explained in the previous section : for a high value of ω, the medium is optically thin for absorption, and first exchange points P should be sampled uniformly within the slab. This is what the standard algorithm does, whereas most optical paths sampled by algorithms 2 and 3 (starting from the slab boundaries) are very short (because of the medium's strong scattering coefficient) thus first exchange positions P are mainly sampled close to the boundaries. Altogether, in the limit of extremely high albedoes, algorithms 2 and 3 require a greater number of statistical realizations because of a non-adapted P sampling law. However, the numerical cost of the algorithm is not directly the number of required statistical realizations N , but the product N < N s > where < N s > is the average number of scattering events. Concerning < N s >, Fig. 3(b) -6(b) illustrate that :

• For low values of τ H and ω, the mean number of scattering events < N s > required for each statistical realization is of the same order of magnitude for all three algorithms. • In the special case of both high τ H and high ω, < s > can be about 10 times greater for the standard algorithm than for algorithms 2 and 3.

This may be explained, making the assumption that < N s >≈ <L> λs =< L > k s with < L > the average path length and λ s = 1 ks the scattering mean free path. For algorithms 2 and 3, it has been shown (see. [START_REF] Blanco | An invariance property of diffusive random walks[END_REF]) that < L > is independent of scattering properties : < L >= 2H. For algorithm 1, it can be easily shown that < L > is proportional to Hτ s . 3 At high values of ω, this finally gives < N s >∼ τ 2 H for algorithm 1 and < N s >≈ 2τ H for algorithms 2 and 3.

These two competing effects combine at high albedo and results are shown in Fig. 7 and Fig. 8. These figures display the product N < N s > for τ H = 10 and τ H = 100, as a function of the single scattering albedo ω. It appears that the two effects previously emphasized for high albedo (N lower for algorithm 1 than for algorithms 2 and 3, and < N s > greater for algorithm 1 than for algorithms 2 and 3) result in the fact that algorithm 2 remains faster than algorithm 1 up to relatively high values of ω, and becomes slower above a critical value of ω. The value ω c at which both algorithms converge at the same speed depend on τ H , ω c increasing as τ H increases (ω c ≈ 0.91 for τ H = 10 and ω c ≈ 0.998 for τ H = 100).

4 Convergence illustration : radiative flux divergence within a nonisothermal slab

In the preceding example a linear blackbody intensity profile was used for convergence tests concerning slab emission. This kind of blackbody intensity profile is not relevant for radiative flux divergence computations in the limit of strong optical thicknesses : with the underlying idea of Rosseland (diffusion) approximation, the radiative budget is indeed only function of the black-body profile second order derivative. Fig. 9-11 therefore present convergence tests with the same slab configuration as above, but with a parabolic blackbody intensity profile(B 0 at slab boundaries and B 0 + ∆B at slab center) :

B(z) = B 0 + ∆B 1 -4 z H -1 2 2
. Computations are performed using a slab discretization into 20 layers same thickness, with N = 10000 statistical realizations per layer. Presented results are the average value of the radiative flux divergence within each layer. Results concerning layer 3 and layer 10 are presented in logarithmic scale in order to highlight the behaviors in the optically thin and optically thick limits where Monte Carlo algorithms commonly encounter convergence difficulties.

In the optically thin limit, the radiative flux divergence is proportional to k a , and therefore to τ H (when both layer width H and single scattering albedo ω are fixed). In the optically thick limit, short-distance energy redistribution processes are dominant and the radiative flux divergence follows the diffusion approximation. In the case of a parabolic blackbody intensity profile, it is constant across the slab and (for fixed values of H and ω) inversely proportional to τ H (see Appendix A). Analytical results corresponding to the diffusion approximation are superimposed to the Monte Carlo results in Fig. 9c-11c and Fig. 9d-11d. Also presented are the analytical results corresponding to the pure absorption approximation (neglecting scattering) : these analytical solutions are available, in the specific case of a parabolic blackbody intensity profile, thanks to the 4th and 5th exponential integral functions (see Appendix A).

The results of Fig. 9 lead to the same conclusions as those of figure 78in [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF] : for low albedoes, the convergence qualities of the present algorithm are similar to those of the previous algorithm designed for purely absorbing media 4 . This is compatible with the fact that, for ω = 0.01, the pure absorption approximation appears as accurate for all optical thicknesses from 10 -2 to 10 2 . Using 10000 statistical realizations per layer, the statistical uncertainty (more precisely the standard deviation) remains lower than a few percents for layer 10 ; it reaches 10% for layer 3 at τ H = 10 and is independant of optical thickness above τ H = 10. As explained in [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF], the fact that the uncertainty becomes independant of optical thickness at high optical thicknesses (whereas it diverges for standard Monte Carlo algorithms) comes from the fact that the boundary-based sampling of emission positions is idealy adapted to optical thickness and that the only remaining task is to perform the integration over the blackbody intensity profile, which is independant of optical thickness. The fact that higher uncertainties are observed for layer 3 than for layer 10 is due to symetry reasons : the radiative balance of layer 10 is the sum of the net-exchanges through its bottom and top interfaces, that are of same sign, whereas the radiative balance of layer 3 is the difference between a heating and a cooling term, all net-exchanges being computed with similar uncertainties.

Fig. 10 and Fig. 11 lead to very similar observations which means that in terms of required numbers of statistical realizations, the conclusions of Sec. 3 are still valid for radiative flux divergence calculations : no specific difficulty is encountered with the proposed algorithm up to extreme values of both absorption and scattering optical thicknesses (except for extreme cases where both optical thickness τ H and single scattering albedo ω are very high, typically τ H = 100 and ω = 0.9999). The average numbers of scattering events are not displayed in these figures as no additional observation can be made compared to those made in the preceding section : it increases less rapidly with the present algorithm than with a standard Monte Carlo algorithm, which partially compensates the convergence limit at high τ H and high ω.

Conclusion

The above presented algorithm is an extension to scattering media of the algorithm introduced in [START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF] as a way to bypass the difficulties encountered by standard Monte Carlo algorithms in the optically thick limit. It is based on a boundary-based net-exchange formulation together with a detailed optimization of optico-geometric sampling laws. It is little sensitive to optical thickness up to both extreme values of absorption optical thickness and scattering optical thickness, two major difficulties of standard Monte Carlo algorithms. As it is based on a net-exchange formulation, it also encounters no difficulty when applied to quasi-isothermal configurations. As will be presented in a forthcoming publication, this algorithm is in particular suitable for detailed analysis of infrared radiation in the terrestrial atmosphere, in which are simultaneously encountered wide ranges of absorption optical thicknesses (because of the line spectra of atmospheric gases) and wide ranges of scattering optical thicknesses (from optically thin dust clouds to optically thick water clouds) [START_REF] Eymet | Longwave radiative exchange analysis of cloudy atmospheres with a net exchange formulation[END_REF][START_REF] Eymet | A monte carlo method to develop radiative transfer parametrizations for terrestrial gcm[END_REF].

Structurally speaking, the proposed algorithm is very much similar to most standard Monte Carlo algorithms, except for the sampling of emission posi- tions that is modified according to the boundary-based approach. All optimized sampling laws are also mathematically very simple and corresponding random generation procedures introduce no specific difficulty. Altogether, the proposed algorithm should therefore be easy to implement on the basis of any existing Monte Carlo code. We also hope that the presented formal derivations should allow that the reader derives its own sampling laws for best optimization in front of specific configurations. Finally, a difficulty remains in the limit of very high scattering optical thicknesses combined with very low absorption optical thicknesses. We believe that this difficulty (that was already well identified and intensively explored for nuclear shielding applications [START_REF] Hammersley | Monte-Carlo methods[END_REF][START_REF] Berger | Reflection and transmission of gamma radiation by barriers : semianalytic monte carlo calculation[END_REF] ) can only be faced working on the diffusive random walk itself, using formulation efforts and sampling laws adaptations. This point was not addressed in the present paper and it will undoubtedly require further detailed analysis of the statistics of multiple scattering optical paths in finite size systems. A Appendix A: radiative flux divergence expressions at the scattering optically thin and optically thick limits.

A.1 Diffusion approximation in a planes parallel configuration.

In the case of optically thick configurations, the diffusion approximation (which is equivalent to the Rosseland approximation) may be used. The radiative flux q r (z) can be written as : , the radiative flux becomes :

q r (z) = 32π∆B (k a + k s )H D z H - 1 2 (A.2)
And its divergence is :

div(q r )(z) = 32π∆B (k a + k s )H 2 D (A.3)
Finally, the average radiative flux divergence between altitudes z i-1 and z i may be written as :

< div(q r ) >= z i z i-1 div(q r (z))

z i -z i-1 = 32π∆B (k a + k s )H 2 D = 1 τ H 32π∆B H D (A.4)
Note that even in the optically thick limit, the diffusion approximation is not valid for the computation of the average flux divergence in the bottom and top layers (layers 1 and 20 in the text). The diffusion approximation is only valid far from the boundaries.

A.2 Absorption approximation in a plane parallel configuration with black boundaries and a parabolic black intensity profile.

The average radiative flux divergence in layer i (between altitudes z i-1 and z i ) may be expressed as :

< div(q r ) >= 2π with I + (z, µ) and I -(z, -µ) respectively the upward and downward specific intensities at altitude z, in the zenithal direction θ with µ = cos(θ). Under the pure absorption approximation, these intensities may be written as : into the above expressions leads to :

< div(q r ) >= 2π z i -z i-1   4∆B Hk a   E 4 k a z i -E 4 k a z i-1 -E 4 k a (H -z i ) + E 4 k a (H -z i-1 )   + 8∆B (Hk a ) 2   E 5 k a z i -E 5 k a z i-1 -E 5 k a (H -z i ) + E 5 k a (H -z i-1 )   + 16∆B(z i -z i-1 ) 3k a H 2   (A.8)

Fig. 1 .

 1 Fig. 1. Discretization of an absorbing and scattering semi-transparent medium into volume elements

Fig. 2 .

 2 Fig. 2. Boundary-based reformulation of net exchange rates

Fig. 3a -

 3a Fig.3a-6a display the number of statistical realizations N needed in order to get a 1 percent standard deviation over the slab emission value, as a function of slab total optical thickness τ H , for 4 different values of the single scattering albedo ω = ks ka+ks . Correspondingly, Fig.3b-6b display the mean number of scattering events < N s > along each sampled optical path.

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. (a) : Number of statistical realizations N required to compute slab emission with a relative standard deviation of 1 percent as function of slab total optical thickness τ H . (b) : Average number of scattering events < N s > as function of slab total optical thickness τ H . Calculations held with ω = 0.01. Presented results correspond to three different algorithms : standard Monte Carlo algorithm (algo #1), boundary-based net-exchange algorithm (algo #2), boundary-based net-exchange algorithm without the optimization of angular sampling as function of scattering (algo #3).

Fig. 6 .

 6 Fig.6. Same as Fig.3, except that ω = 0.9999

Fig. 9a -

 9a Fig. 9a-11a display the radiative flux divergence profile for different values of the slab total optical thickness τ H . In these successive three figures, the single scattering albedo is respectively equal to 0.01, 0.50 and 0.90. For the same values of single scattering albedo, Fig. 9c -11c and Fig. 9d -11d display radiative flux divergence averages in layers 3 and 10 respectively, as function of slab total optical thickness τ H . Standard deviations are presented in Fig. 9b -11b, Fig. 9e -11e and Fig. 9f -11f.

10 Fig. 9 .

 109 Fig. 9. Average value of the radiative flux divergence within each of the 20 layers using N = 10000 statistical realizations per layer. The slab width is H = 1m, scattering is isotropic and the single scattering albedo is ω = 0.01. (a): radiative flux divergence profile for three values of the slab total optical thickness τ H ; (b): standard deviations corresponding to (a); (c): radiative flux divergence average in layer 3 as a function of τ H ; (d): radiative flux divergence average in layer 10 as a function of τ H ; (e): standard deviations corresponding to (c); (f): standard deviations corresponding to (d).

Fig. 10 .

 10 Fig.10. Same as Fig.9with ω = 0.5

Fig. 11 .

 11 Fig.11. Same as Fig.9with ω = 0.9

1 3( 1 -

 11 q r (z) = -hνc k a + k s D ∂G ∂z (A.1)with G(z) = 1 hνc 4π I(z, u)dω(u) the local photon density, where I(z, u) is the specific intensity at altitude z in direction u and D = ωg) . In optically thick systems, we can make the assumption that G(z), the local photon density, is equal to the equilibrium intensity at the local temperature : G(z) = 4π hνc B(z), with B(z) the local blackbody intensity. With the assumption of a parabolic blackbody intensity profile B(z) = B 0 + ∆B 1 -4 z H -

Fig. A. 1 .I 7 )

 17 Fig. A.1. Plane-parallel slab with n homogeneous layers and parabolic black intensity profile.

The term "bundle transport algorithm" is used for algorithms in which the photon bundle's energy is totally absorbed at a stochastically determined position. The term "path integrated algorithm" is used for algorithms in which the photon bundle's energy is exponentially attenuated along the photon bundle's optical path

This property may be derived directly from Markov theory with absorbing states[START_REF] Feller | An introduction to probability theory and its applications[END_REF] in a one-dimensional case, with constant free path length (problem well known as the "Gambler's ruin problem"). Extension to exponentially distributed free path length is tedious but is accessible without any specific mathematical difficulty. To our knowledge, extension to three dimensions is not available, but it may easily be observed experimentally that the proportionality property remains valid, at least for qualitative reasonings such as those made in the present text.

Note that a scaling error was made in[START_REF] De Lataillade | A net exchange monte carlo approach to radiation in optically thick systems[END_REF] : results of figure8were presented omitting to divide by a factor 25 corresponding to the narrow band width dη = 25cm -1 with which computations were held

with E n the n th exponential integral :


Carl Safi
email: carl.safi@ensiacet.fr

Michaël Charton

Olivier Pignolet

Pierre-Yves Pontalier

Carlos Vaca-Garcia

: O Pignolet

P.-Y Pontalier

P.-Y Pontalier

Evaluation of the protein quality of Porphyridium cruentum

Keywords: Amino acid profile, Food consumption, Porphyridium cruentum, Rhodophyta, Protein extract

The amino acid profile of the red microalga Porphyridium cruentum and its protein extract have been determined in order to assess the nutritional quality of this biomass for human consumption. Total protein determined by elemental analysis represented 56 % of its dry weight. Hydro-soluble proteins extracted at pH 12 and 40 °C were analysed by the Lowry method giving 47 %, which represented 84 % of total protein per dry weight. The amino acid sequence of the biomass and the protein extract was composed of a set of essential (39 % for the former and 37 % for the latter) and non-essential amino acids (61 % for the former and 63 % for the latter) that compares favourably with the standard protein/amino acid requirements proposed by Food and Agricultural Organisation and World Health Organisation.

Introduction

Microalgae have been thoroughly studied for their impressive richness in multiple important components that can be valorised for numerous products such as biofuel, animal feed, food supplements, dyes and pharmaceuticals. For instance, in Japan and Chile, Chlorella vulgaris is added to food, such as noodles and pasta [START_REF] Fradique | Incorporation of Chlorella vulgaris and Spirulina maxima biomass in pasta products. Part 1: preparation and evaluation[END_REF], to improve the nutritional quality of the meal. Spirulina platensis is also used as food supplement in France, Mexico, USA and other countries (FAO 2008).

Porphyridium cruentum is a unicellular microalga with cell diameter ranging between 6 and 10 μm [START_REF] Durmaz | The effect of low temperature on fatty acid composition and tocopherols of the red microalga, Porphyridium cruentum[END_REF]. It grows in saline water and lacks a cell wall, but instead, it excretes a sulphurized polysaccharide that encapsulates the cell [START_REF] Arad | The potential of production of sulfated polysaccharides from Porphyridium[END_REF][START_REF] Arad | Effect of nitrogen on polysaccharide production in a Porphyridium sp[END_REF][START_REF] Adda | Effect of nitrate on growth and production of cell-wall polysaccharide by the unicellular red alga Porphyridium[END_REF][START_REF] Geresh | The extracellular polysaccharides of the red microalgae: chemistry and rheology[END_REF][START_REF] Geresh | Sulfation of extracellular polysaccharides of red microalgae: preparation, characterization and properties[END_REF], therefore accumulating a high amount of polysaccharides 40-57 % [START_REF] Fuentes | Biomass nutrient profiles of the microalga Porphyridium cruentum[END_REF]. The source of red colour of this microalga is phycoerythrin, which plays an important role together with other phycobiliproteins present as an accessory photosynthetic pigment [START_REF] Ley | Isolation and function of allophycocyanin B of Porphyridium cruentum[END_REF]. In addition, P. cruentum is a rich source of lipids and polyunsaturated fatty acids, such as eicosapentanoic acid, docosahexaenoic acid and arachidonic acid (AA) [START_REF] Shiran | Biosynthesis of eicosapentaenoic acid in the microalga Porphyridium cruentum: the use of externally supplied fatty acids[END_REF][START_REF] Cohen | Fatty acid unsaturation in the red alga Porphyridium cruentum. Is the methyleneinterrupted nature of polyunsaturated fatty acids an intrinsic property of the desaturases?[END_REF][START_REF] Khozin-Goldberg | Triacylglycerols of the red microalga Porphyridium cruentum can contribute to the biosynthesis of eukaryotic galactolipids[END_REF] that have health benefits by reducing the risk of cardiovascular disease, high cholesterol levels [START_REF] Ginzberg | Chickens fed with biomass of the red microalga Porphyridium sp. have reduced blood cholesterol level and modified fatty acid composition in egg yolk[END_REF] and inflammation and also reduce the risk of cancer [START_REF] Patil | Fatty acid composition of 12 microalgae for possible use in aquaculture feed[END_REF]). Moreover, this microalga accumulates high quantity of proteins (39 % according to [START_REF] Fuentes | Biomass nutrient profiles of the microalga Porphyridium cruentum[END_REF].

Some studies have focused on amino acid profile of phycocyanin [START_REF] Ducret | The complete amino acid sequence of R-phycocyanin-I α and β subunits from the red alga Porphyridium cruentum[END_REF] or on extracellular protein-polysaccharide linkage analysis [START_REF] Heaney-Kieras | Structural studies on the extracellular polysaccharide of the red alga, Porphyridium cruentum[END_REF]. Other studies worked on the nutritional importance of this microalga without taking into consideration its protein quality and only relying on fatty acids, carbohydrates and phycobilisomes [START_REF] Fuentes | Biomass nutrient profiles of the microalga Porphyridium cruentum[END_REF][START_REF] Berge | In vitro anti-inflammatory and anti-proliferative activity of sulfolipids from the red alga Porphyridium cruentum[END_REF]. González [START_REF] López | Protein measurements of microalgal and cyanobacterial biomass[END_REF] measured the nitrogen to protein conversion factor of five different microalgae, including P. cruentum, by taking into account protein nitrogen by Lowry method and total nitrogen by elemental analysis and Kjeldahl method. Nevertheless, no study has been carried out on the amino acid profile of the crude P. cruentum in order to evaluate the quality of proteins present in the biomass.

Therefore, this study focuses on evaluating the protein quality of the microalga by investigating the amino acid profile of P. cruentum to compare it with standard protein source proposed by World Health Organisation (WHO) and Food and Agricultural Organisation (FAO) (WHO 2007;[START_REF] Becker | Microalgae biotechnology and microbiology[END_REF]). The profile of the protein extract of this microalga was also performed to evaluate the potential of both substrates as an unconventional protein source for food consumption and food supplement.

Materials and methods

The microalga Porphyridium cruentum (strain UTEX 161) was grown in batch mode in an indoor tubular air-lift photobioreactor (PBR, 10 L) at 25 °C [START_REF] Loubiere | Investigations in an external-loop airlift photobioreactor with annular light chambers and swirling flow[END_REF], inoculated from a prior culture in a flat panel air-lift PBR (1 L). Hemerick culture medium was used [START_REF] Hemerick | Culture methods and growth measurements[END_REF]. Culture circulation was by sterile air injection at the bottom of the PBR. The pH and temperature were recorded by a pH/temperature probe, and the pH was regulated at pH 7.5 with CO 2 bubbling. P. cruentum was harvested during the exponential phase and concentrated by centrifugation (11 % dry matter) and was furnished as frozen paste from Alpha Biotech (Asserac, France).

The frozen paste of crude microalga was freeze dried in a Fisher Bioblock Scientific Alpha 2-4 LD Plus device (France). The pressure was reduced to 0.010 bar, and the temperature was further decreased to -80 °C, and freezedrying was under vacuum for 48 h.

Protein extraction

For protein determination, stock solutions were prepared with approximately 500 mL of ultrapure water and some drops of NaOH 2 N to adjust the pH to the desired value from 7 to 14. A sample of 1 g of freeze-dried biomass was added to 50 mL of stock solution. The mixture was heated at 40 °C with stirring for 1 h. The sample was then centrifuged at 10,000 × g for 10 min. Samples were taken for analyses: colorimetric method of Lowry, elemental analysis and amino acid analysis.

Lowry method [START_REF] Lowry | Protein measurement with the Folin phenol reagent[END_REF] The procedure involves reaction of proteins with cupric sulphate and tartare in an alkaline solution, leading to the formation of tetradentate copper protein complexes. The addition of the Folin-Ciocalteu reagent leads to the oxidation of the peptide bonds by forming molybdenum blue with the copper ions. Therefore, a calibration curve was prepared using a concentration range of bovine serum albumin from 0 to 1,500 μg mL -1 . In order to measure the protein content, 0.2 mL of each standard or samples containing the crude protein extract were withdrawn, and 1 mL of modified Lowry reagent was added to each sample. Each sample was then vortexed and incubated for exactly 10 min. After incubation, 100 μL of Folin-Ciocalteu Reagent (1 N) were added and again vortexed and incubated for exactly 30 min. The blue colour solution is then measured at 750 nm.

Elemental analysis

Total nitrogen of the freeze-dried biomass was evaluated using a PerkinElmer 2400 series II elemental analyser. Samples of 2 mg were placed in thin capsules and then heated at 925 °C using pure oxygen as the combustion gas and pure helium as the carrier gas, then evaluating the nitrogen percentage and converting it into protein percentage using the standard 6.25 as the conversion factor.

Amino acid analysis

Amino acid composition of the biomass was determined according to the method of [START_REF] Moore | Photometric ninhydrin method for use in the chromatography of amino acids[END_REF]. The samples were hydrolysed with 6 N HCl at 103 °C for 24 h. Then, the hydrolysed material was adjusted to pH 2.2 with 6 N NaOH and then stabilised with a pH 2.2 citrate buffer solution. The final solution was then filtered through a 0.45-μm PTFE membrane to remove any residual solid remaining in the solution. The analysis was performed using an amino acid analyser Biochrom Ltd 30 (Cambridge, UK) equipped with a system "column + pre-column (PEEK)" (size 200×4.6 mm) with ion exchange resins containing sodium. The separation of amino acids is carried out by elution with loading buffers at different pH, with a flow rate of 20 μL. After reaction with ninhydrin, amino acids are detected at a wavelength of 570 nm, with the exception of proline, for which detection was at 440 nm. Ammonia was automatically added to compensate the value of some less resistant amino acids that disappeared after the strong acid hydrolysis.

Results

All results in this section are presented as percent dry weight of the biomass. Aqueous extraction of P. cruentum was carried out at different pH. The highest hydro-soluble protein content obtained was 47 % at pH 12 and 47 % for pH 14, and the lowest was 8 % at pH 7 followed by 33 % at pH 10 (Fig. 1). On the other hand, total protein obtained by elemental analysis was 56 % from which hydro-soluble proteins represented 84 % with 16 % as non-soluble proteins.

The amino acid sequence of P. cruentum and its protein extract obtained at pH 12 (Table 1) included a set of essential and non-essential amino acids (Table 2) representing a total of 38.7 % essential, 61.0 % non-essential and 0.3 % ornithine, which is an unclassified amino acid. As for the protein extract, 36.8 % were essential, 62.9 % non-essential and 0.2 % ornithine. The major amino acids of the crude biomass were aspartic acid 11.2 % and serine 8.1 % as well as glutamic acid 8.2 %. The major amino acids of the protein extract were aspartic acid 10.7 %, glutamic acid 9.0 % and alanine 10.5 %. It is noteworthy that the alanine content in the protein extract was 36 % higher compared to crude microalga.

Discussion

Total nitrogen content obtained by elemental analysis was required to calculate total protein content and, then through it, the exact percentage of hydro-soluble and non-hydrosoluble proteins content of the biomass. Surprisingly, total protein content (56 %) was higher than the results (34 %) reported by [START_REF] Fuentes | Biomass nutrient profiles of the microalga Porphyridium cruentum[END_REF] and (39 %) [START_REF] Becker | Microalgae biotechnology and microbiology[END_REF], where the standard conversion factor 6.25 was used. This is probably due to the growing conditions of the microalga or to the standard conversion factor used, which can overestimates the exact protein content. For instance, multiple studies [START_REF] Lourenço | Distribution of intracellular nitrogen in marine microalgae: basis for the calculation of specific nitrogen-to-protein conversion factors[END_REF][START_REF] Lourenço | Distribution of intracellular nitrogen in marine microalgae: calculation of new nitrogen-to-protein conversion factors[END_REF][START_REF] Mariotti | Converting nitrogen into protein: beyond 6.25 and Jones' factors[END_REF][START_REF] López | Protein measurements of microalgal and cyanobacterial biomass[END_REF]) recommended a conversion factor lower than the standard 6.25.

Furthermore, cell disruption was not necessary since P. cruentum does not have a cell wall, and instead, it is encapsulated with a fragile polysaccharide layer surrounding the cell. Thus, the extraction of the internal proteins was not mechanically hindered. Only the chemical environment controls the extraction process. Hence, extraction of proteins was optimal at pH 12 since it gave similar results at pH 14, 1.4 times higher than pH 10 and 6.5 times higher than pH 7. Therefore, pH 12 was sufficient to extract the maximum amount of proteins that have been quantified by the Lowry method. This colorimetric method is considered as one of the most accurate methods to quantify proteins. Nevertheless, it takes only into consideration hydro-soluble proteins [START_REF] Crossman | Determination of protein for studies of marine herbivory: a comparison of methods[END_REF][START_REF] Diniz | Gross chemical profile and calculation of nitrogen-to-protein conversion factors for five tropical seaweeds[END_REF]), which represents the major part of proteins present in the microalga. In our case, soluble protein content (84 %) was higher than results reported by González [START_REF] López | Protein measurements of microalgal and cyanobacterial biomass[END_REF], where it was estimated to represent 75 % of total protein. The variability observed in total and hydro-soluble protein content, compared to literature values, depends on environmental and culture conditions, which are key factors for the growth of the microalga and its nutrient content. Moreover, the high value for hydro-soluble proteins is explained by the possible presence of small peptides and free amino acids in the reaction with Lowry's reagents [START_REF] Barbarino | An evaluation of methodologies for extraction and quantification of protein of marine macro-and microalgae[END_REF]. Protein nutritional quality for food consumption is evaluated by the amino acid profile and by the proportion and availability of amino acids [START_REF] Becker | Microalgae biotechnology and microbiology[END_REF]). It should be noted that plants are capable of synthesising essential and non-essential amino acids, while humans and other animals are only capable of synthesising non-essential amino acids, and the rest are obtained from food [START_REF] Mahan | Krause's Food, nutrition and diet therapy[END_REF]. P. cruentum shows an interesting combination of ten essential and nine non-essential amino acids in its proteins. A difference has been detected between the fractions of amino acids of the crude biomass and its protein extract, indicating that the extractable hydro-soluble proteins have a different primary structure. Fifteen out of 19 amino acids of the protein extract showed statistical significant difference (α00.05) compared to the amino acid profile of the crude biomass (Table 1). There was no high difference between the total percentage of essential and non-essential amino acids of crude P. cruentum (Table 2) and its protein extract, proving again that the cell wall of this microalga was not a barrier that prevents the extraction of intracellular proteins. Furthermore, the fragility of its pseudo-cell wall favours the digestibility of the proteins compared to C. vulgaris and Scenedesmus sp., which are considered as important protein sources but have a rigid cell wall mainly composed of cellulose. Hence, P. cruentum does not require a previous unit operation to liberate the internal proteins.

The protein quality assessed in this study compares favourably to standard protein food recommended by WHO and FAO [START_REF] Becker | Microalgae biotechnology and microbiology[END_REF], and it can be considered as an additional key element for this unicellular microalga, and together with its remarkable richness in polyunsaturated fatty acids [START_REF] Berge | In vitro anti-inflammatory and anti-proliferative activity of sulfolipids from the red alga Porphyridium cruentum[END_REF] accompanied by its anticancer and immunomodulating activities of the phycobiliproteins (Bermejo [START_REF] Roman | Recovery of pure B-phycoerythrin from the microalga Porphyridium cruentum[END_REF] and its high content in polysaccharides [START_REF] Fuentes | Biomass nutrient profiles of the microalga Porphyridium cruentum[END_REF], it could make P. cruentum a special unconventional food source that covers all the necessary nutritional primary requirements. Of course, other aspects, such as toxicity, must be evaluated to give a final assessment on appropriateness for human consumption.

Fig. 1

 1 Fig. 1 Hydro-soluble protein percent per 100 g dry biomass at different pH and 40 °C. Results are based on three replicates for three experiments at each point ± SD (n06)

Table 1

 1

	Amino acid percent per 100 g of P. cruentum and its	Amino acids	P. cruentum	P. cruentum	Significant	FAO/WHO a	Eggs a	Soya a
	protein extract (aqueous alkaline			protein extract	difference (α00.05)
	extraction) compared to standard
	food protein source	Aspartic acid	11.21±0.45	10.71±0.02	No	NA	11	1.30
		Threonine	6.25±0.25	4.45±0.01	Yes	4	5	4.00
		Serine	8.11±0.29	7.49±0.02	Yes	NA	6.9	5.80
		Glutamic acid	8.17±0.29	9.05±0.01	Yes	NA	12.6	19.00
		Glycine	6.86±0.28	7.68±0.01	Yes	NA	4.2	4.50
		Alanine	6.67±3.67	10.46±0.02	No	NA	NA	5.0
		Cysteine	0.33±0.01	0.27±0.01	Yes	3.5	2.3	1.90
		Valine	2.50±0.10	3.15±0.01	Yes	5	7.20	5.30
		Methionine	2.78±0.11	2.37±0.01	Yes	NA	3.20	1.30
		Isoleucine	5.25±0.24	5.34±0.02	No	4	6.60	5.30
		Leucine	5.83±0.21	7.30±0.01	Yes	7	7.00	7.70
		Tyrosine	4.43±0.18	3.69±0.01	Yes	6	4.20	3.20
		Phenylalanine	5.00±0.20	4.12±0.01	Yes	NA	5.80	5.00
		Histidine	1.11±0.04	0.79±0.03	Yes	NA	2.40	2.60
		Lysine	5.50±0.21	5.60±0.01	No	5.5	5.30	6.40
		Arginine	7.78±0.29	6.63±0.02	Yes	NA	6.20	7.40
	Results are based on three repli-	Tryptophan	1.39±0.05	0.72±0.02	Yes	1	1.70	1.40
	cates for three experiments at each point ± SD (n03) NA not available a Becker 1994	Ornithine Proline Ammoniac	0.27±0.01 2.53±0.17 8.02±0.30	0.22±0.01 1.87±0.02 8.05±0.04	Yes Yes No	NA NA NA	NA 4.2 NA	NA 5.3 NA

Table 2

 2 Content of essential and non-essential amino acid of the biomass and the protein extract without taking ammoniac into considerationAmino acidsP. cruentum (%) P. cruentum protein extract (%)

	Essential AA	38.7	36.8
	Non-essential AA	61.0	63.0
	Unclassified	0.3	0.2

Acknowledgments The authors would like to thank Alpha Biotech for providing the biomass. This work was supported by the French National Research Agency (ANR) in the context of Algoraffinerie project.


Carl Safi
email: carl.safi@ensiacet.fr

Michaël Charton

Olivier Pignolet

Françoise Silvestre

Carlos Vaca-Garcia

Pierre-Yves Pontalier

P.-Y Pontalier

P.-Y Pontalier

Influence of microalgae cell wall characteristics on protein extractability and determination of nitrogen-to-protein conversion factors

Keywords: Amino acid profile, Cell wall, Conversion factor, Nitrogen, Protein extract

Additional evidence about the influence of the cell wall physical and chemical characteristics on protein extractability was determined by calculating the conversion factors of five different microalgae known to have different cell wall composition, and their protein extracts. The con version factors obtained for crude rigid cell walled Chlore/la vulgaris, Nannochloropsis oculata and Haematococcus plu vialis were 6.35, 6.28 and 6.25, respectively, but for their protein extracts the values were lower with 5.96, 5.86 and 5.63. On the other band, conversion factor obtained for fragile cell walled microalgae Porphyridium cruentum and Athrospira platensis was 6.3 5 for the former and 6.27 for the latter, with no significant difference for their protein extract with 6.34 for the former and 6.21 for the latter. In addition, the highest hydro-soluble protein percentage recovered from total protein was for P. cruentum 80.3 % and A. platensis 69.5 % but lower for C. vulgaris with 43.3 %, N oculata with 33.3 % and H pluvialis with 27.5 %. The study spotted the light on the influence of the cell wall on evaluating the conversion factor and protein extractability. In addition, it showed the necessity of finding the conversion factor every

Introduction

Microalgae have been consumed long time ago by the Aztecs and other Mesoamericans who used this biomass as an important food source [START_REF] Vonshak | Spirulina platensis (Arthrospira). Physiology, cell biology and biotechnology[END_REF]. Nowadays, in Japan for instance, Chlore/la vulgaris is added to food such as noodles and pasta [START_REF] Fradique | Incorporation of Chlorella vulgaris and Spirulina maxima biomass in pasta products. Part 1: preparation and evaluation[END_REF] to improve the nutritional quality of the meal.

Microalgae are gaining interest due to their capacity to accumulate important amounts of multiple components (proteins, lipids, carbohydrates and pigments) compared to any other sources, and therefore protein content is consid ered as one of the cardinal components determining their nutritional value. For instance, Arthrospira (Spirulina) max ima can accumulate proteins up to 71 % dry weight [START_REF] Becker | Microalgae biotechnology and microbiology[END_REF]. Thus, analysing and quantifying the protein content are key factors that should be thoroughly investigated. A capital point is to calculate precisely the nitrogen-to-protein conversion factor (NTP). While the standard value of 6.25 is used, Kjeldahl or elemental analysis may lead to an overesti mation or underestimation of the protein quantity. Moreover, these two methods take into account the totality of the nitrogen present in the biomass from which 59-98 % [START_REF] Lourenço | Distribu tion of intracellular nitrogen in marine microalgae: basis for the calculation of specific nitrogen-to-protein conversion factors[END_REF][START_REF] Fujihara | Nitrogen-to-protein conversion factors for common vegetables in Japan[END_REF][START_REF] Lourenço | Distribution of intracellular nitrogen in marine microalgae: calculation of new nitrogen-to-protein conversion factors[END_REF][START_REF] Lopez | Protein measurements of microalgal and cyanobacterial biomass[END_REF]) of total nitrogen belongs to protein and the rest cornes from pigments, nucleic acids and other inorganic components. It is true that the colorimetric method of Lowry [START_REF] Lowry | Protein mea surement with the Folin phenol reagent[END_REF]) is an accurate method for protein quantification [START_REF] Peterson | Review of the Folin phenol protein quantitation method of Lowry, Rosebrough, Farr and Randall[END_REF][START_REF] Lopez | Protein measurements of microalgal and cyanobacterial biomass[END_REF]) and it does not require a conversion factor. Nevertheless, this method determines only the hydro soluble proteins [START_REF] Crossman | Determination of protein for studies of marine herbivory: a comparison of methods[END_REF][START_REF] Diniz | Gross chemical profile and calculation of nitrogen-to-protein con version factors for five tropical seaweeds[END_REF] and not the total protein content.

In addition, the extraction of proteins can be dimin ished by the cell wall barrier, which can prevent the solubilisation of all the intracellular proteins affecting thus the value of the nitrogen-to-protein conversion factor. Therefore, the impact of the cell wall character istics on protein extractability should be taken into account and analysed in order to prevent an incorrect estimation of the protein content.

Multiple studies have focused on finding a method to rec ommend the right conversion factor; for instance, Gonzalez [START_REF] Lopez | Protein measurements of microalgal and cyanobacterial biomass[END_REF] focused on obtaining the conversion factor of five microalgae a: fter breaking the cell wall, and then finding a correlation between protein content and total nitrogen content ( elemental analysis or Kjeldahl). As a result, among five micro algae a new mean conversion factor was estimated to 4.44 (elemental analysis) and 5.95 (Kjeldahl). Another study [START_REF] Lourenço | Amino acid composition, protein content and calcu lation of nitrogen-to-protein conversion factors for 19 tropical seaweeds[END_REF] determined the conversion factor for 19 tropical seaweeds ha:rvested directly from the beach; and in a second study [START_REF] Lourenço | Distribu tion of intracellular nitrogen in marine microalgae: basis for the calculation of specific nitrogen-to-protein conversion factors[END_REF], 12 marine microalgae were analysed under different growth phases and a mean value of 4.58 was found.

The following study assesses the impact of the cell wall on the protein extractability and the evaluation of the NTP for five microalgae intensively grown worldwide and having wide taxonomie diversity.

Materials and methods

The microalgae used a:re: the cyanobacterium Arthrospira platensis (strain PCC 8005), two Chlorophyceae C. vulgaris (strain SAG 211-19), Haematococcus pluvialis (unknown strain), one Rhodophyta Porphyridium cruentum (strain UTEX 161) and the Eustigmatophyceae Nannochloropsis oculata (unknown stra:in).

Each alga was cultivated in a different culture medium: Hemerick medium for P. cruentum, Sueoka medium for C. vulgaris, Basal medium for H pluvialis, Conway medium for N oculata, and Zarrouk medium for A. platensis. All were grown in batch mode in an indoor tubula:r air-lift photobioreactor (PBR, 10 L) at 25 °C [START_REF] Loubiere | Investigations in an extemal-loop airlift photobioreactor with annular light chambers and swirling flow[END_REF] after inoculation from a prior culture in a fiat panel air-lift PBR (1 L). Culture mixing was by sterile air injection at the bottom of the PBR. The pH and temperature were recorded by a pH/temperature probe and pH was regulated at 7 .5 with CO 2 • The algae were harvested during the exponential growth phase and concentrated by centrifuga tion, and then supplied as a frozen paste from Alpha Biotech (Asserac, France). The biomass contained 20 % dry weight. The frozen paste of crude microalgae was freeze-dried in a Fisher Bioblock Scientific Alpha 2-4 LD Plus device (Illkirch, France).

Protein extraction

Stock solutions were prepared with approximately 500 mL of ultrapure water and some drops of 2 N NaOH to adjust the solution to pH 12. A sample of 1 g of freeze-dried biomass was added to 50 mL of stock solution. The mixture was heated to 40 °C with stirring for 1 h followed by centrifugation at 5,000xg for 10 min. Samples were taken for analysis by the colorimetric method of [START_REF] Lowry | Protein mea surement with the Folin phenol reagent[END_REF], elemental analysis and amino acid analysis.

Lowry method

A calibration curve was prepared using a concentration range of bovine serum albumin from O to 1,500 µg mL-1 . In order to measure the protein content, 0.2 mL of each standard or samples containing the crude protein extract were withdrawn and then 1 mL of modified Lowry reagent was added to each sample. Each sample was then vortexed and incubated for exactly 10 min. After incubation, 100 µL of Folin-Ciocalteu reagent (1 N) were added and again vortexed and incubated for exactly 30 min. The absorbance was then measured at 750 nm.

Elemental analysis

Total nitrogen of the freeze-dried biomass was evaluated using a PerkinElmer 2400 series II elemental analyser. Samples of2 mg were placed in tin capsules and then heated at 925 °C using pure oxygen as the combustion gas and pure helium as the carrier gas, then evaluating the nitrogen per centage and converting it into protein percentage by using the conversion factors calculated for each microalga in this study.

Amino acid analysis and NTP calculation

The determination of the amino acid composition of the biomass was performed according to a widely used standard method [START_REF] Moore | Photometric ninhydrin method for use in the chromatography of amino acids[END_REF]. The samples were hydro lysed with 6 N hydrochloric acid at 103 °C for 24 h. Then, the hydrolysed material was adjusted to pH 2.2 with 6 N NaOH and stabilised with a pH 2.2 citrate buffer solution. The final solution was then filtered with 0.45 µm PTFE membrane to remove any residual solids remaining in the solution. The analysis was performed by using an amino acid analyser Biochrom Ltd 32+ (Cambridge, UK.) equipped with a high pressure PEEK "column+precolumn" (size, 200x4.6 mm) packed with Ultropac cation exchange resin containing sodium. The separation of amino acids is carried out by elution with loading buffers (flow rate 25 mL h-1 ) at different pH. After reaction with ninhydrin (flow rate 3 5 mL h -l ), amino acids are detected with a UV detector at a wavelength of 570 nm, with the exception of proline, for which detection occurs at 440 nm. Calculation of NTP was carried out according to the method (k A ) of [START_REF] Mossé | Nitrogen-to-protein conversion factor for ten cereals and six legumes or oilseeds. A reappraisal of its definition and determination. Variation according to species and to seed protein content[END_REF] and [START_REF] Sriperm | Evaluation of the fixed nitrogen-to-protein (N:P) conversion factor (6.25) versus ingredi ent specific N:P conversion factors in feedstuffs[END_REF]. It should be mentioned that ammonia was added to compensate the value of some less resistant amino acids that disappeared after the strong acid hydrolysis. In addition, the strong hydrolysis of the peptide bonds generates one molecule of water from each amino acid and therefore during the quantification of total amino acid one molecule of water was subtracted from each amino acid in order to get the total amino acid residue which represents the exact quantity of all amino acids [START_REF] Lourenço | Distribu tion of intracellular nitrogen in marine microalgae: basis for the calculation of specific nitrogen-to-protein conversion factors[END_REF][START_REF] Fujihara | Nitrogen-to-protein conversion factors for common vegetables in Japan[END_REF].

Confocal laser scanning microscopy

Cells were observed with an SP2-AOBS confocal laser scanning microscope (Leica). The fluorochrome calcofluor white that binds to the ,8---+ 1--4 linkages in the cell wall polysaccharides was added to the samples. Excited at 488 nm, the cells are identified coloured in light blue.

Statistical analysis

Three experiments were conducted separately on all micro algae and their protein extract. Statistical analyses were carried out on Microsoft Excel. Measurements of three replicates for each sample were reproducible for ±5 % of the respective mean values.

Results

An amino acid analyser was used to obtain the amino acid profile of the crude microalgae (Table 1) and their protein extract (Table 2). The protein primary composition was recon stituted in order to find the conversion factor (Table 3) that takes into account only the protein nitrogen. The highest conversion factor evaluated for the crude biomass was 6.35 for C. vulgaris and P crnentum and the lowest was 6.25 for H pluvialis. If we compare the NTP value of the crude micro algae and the protein extract, we observe that there is no significant difference for P cruentum and A. platensis. However, a significant difference was measured for the other three species, which correspond to green microalgae (Table 3). The total nitrogen content of crude microalgae was determined by elemental analysis. The hydro-soluble proteins were quantified by the Lowry method. By using the NTP values from Table 3, the total proteins and the corresponding fraction of hydro-soluble proteins were accu rately determined starting from the results of elemental anal ysis (Table 4).

A small difference was observed between the essential and non-essential amino acids total percentage for P cruen tum, A. platensis and C. vulgaris and their protein extract, and an important difference was observed for N oculata and H. pluvialis with a noticeable increase in non-essential amino acids percentage in the protein extract and a decrease in essential amino acids percentage (Table 5).

Finally, the laser scanning confocal microscopie images presented in Fig. 1 showed that in the case of P cruentum and A. platensis a total disruption of the cell wall occurred after the alkaline treatment, whereas C. vulgaris, N oculata and H. pluvialis maintained their globular form indicating that at least a part of their cell wall was intact.

Discussion

The present study contributes original individual NTP con version factors for five current species of microalgae. Indeed, direct comparison of our results with literature values can only be done for N oculata [START_REF] Lourenço | Distribu tion of intracellular nitrogen in marine microalgae: basis for the calculation of specific nitrogen-to-protein conversion factors[END_REF]; the report from Gonzalez Lopez et al. ( 2010) combines all the cyanobacteria and P cruentum in a global result. To our knowledge, NTP data for C. vulgaris and H. pluvialis have never been reported. Moreover, according to [START_REF] Mossé | Nitrogen-to-protein conversion factor for ten cereals and six legumes or oilseeds. A reappraisal of its definition and determination. Variation according to species and to seed protein content[END_REF], there are three kinds of NTP conversion factors called k, k A and k p [START_REF] Yeoh | Protein contents, amino acid composi tions and nitrogen-to-protein conversion factors for cassava roots[END_REF][START_REF] Sriperm | Evaluation of the fixed nitrogen-to-protein (N:P) conversion factor (6.25) versus ingredi ent specific N:P conversion factors in feedstuffs[END_REF]. In this study, k A was calculated for all species, and this kind of conversion factor is larger than kp and k. For this reason, there is a significant difference between the conversion factors calculated for N oculata in this study (6.28) and the one calculated for the same microalga (4.87) in the [START_REF] Lourenço | Distribu tion of intracellular nitrogen in marine microalgae: basis for the calculation of specific nitrogen-to-protein conversion factors[END_REF] study. The NTP conversion factor of the crude microalgae and their protein extract depended on the type of microalgae. If we calculate the relative difference between these two values for a single species, we observe that there is almost no difference in the case of red algae and cyanobacteria, whereas it reached almost 10 % for green microalgae (Table 3). These differences can be correlated to the rigidity of the cell wall. Indeed, it is generally accepted that green microalgae possess a more rigid cell wall than red algae or cyanobacteria. More in detail, P cruentum does not have a true cell wall but instead encapsu lated by a layer of sulfurized polysaccharides [START_REF] Arad | The potential of production of sulfated polysaccharides from Porphyridium[END_REF][START_REF] Arad | Effect of nitrogen on polysac charide production in a Porphyridium sp[END_REF][START_REF] Adda | Effect of nitrate on growth and production of cell-wall polysaccharide by the unicellular red alga Porphyridium[END_REF][START_REF] Geresh | The extracellular polysaccharides of the red microalgae: chemistry and rheology[END_REF][START_REF] Geresh | Sulfation of extracellular polysaccharides of red microalgae: preparation, characterization and properties[END_REF][START_REF] Sobczuk | Effects of agitation on the microalgae Phaeodactylum tricornutum and Porphyridium cruentum[END_REF]. A. platensis has a relatively fragile cell wall mainly composed of murein without cellulose [START_REF] Lee | Phycology[END_REF][START_REF] Lu | Preven tive effects of Spirulina platensis on skeletal muscle damage under exercise-induced oxidative stress[END_REF]. As far as the green microalgae, the chlorophycean C. vulgaris and the eustigma tophycean N. oculata, both have a cell wall mainly composed of cellulose and hemicelluloses [START_REF] Payne | Evaluation of diets for culture of the calanoid copepod Gladioferens imparipes[END_REF]. The highest difference perceived was for H pluvialis, which has a thick trilaminar cell wall composed of cellulose and sporopollenin in the aplanospore stage [START_REF] Mendes-Pinto | Evaluation of different cell disruption processes on encysted cells of Haematococcus pluvialis: effects on astaxanthin recovery and implications for bioavailability[END_REF][START_REF] Montsant | Presence of a nonhydrolyzable biopolymer in the cell wall of vegetative cells and astaxanthin-rich cysts of Haematococcus pluvialis (Chlorophyceae)[END_REF][START_REF] Aflalo | On the relative efficiency of two-vs. one-stage production of astaxanthin by the green algaHaematococcus pluvialis[END_REF]. The composition of its cell wall makes this microalga less permeable and extremely mechanically resistant [START_REF] Hagen | Ultrastructure and chemical changes in the cell wall of Haematococcus pluvialis (Volvocales, Chlorophyta) during aplanospore formation[END_REF].

Crude microalgae

Taking into account the standard deviation of the three samples considered for each microalga, we could affirm that, at 95 % of confidence level, all the values of total protein content shown in Table 4 are statistically equivalent. This fact is noteworthy, because when we consider the extracted hydro-soluble proteins (Table 4), we observe that their amounts decrease following the same correlation than the differences in NTP value: high extraction for the micro algae with no real cell wall and very low extraction for the most rigid of all. Therefore, the hydro-soluble protein capa ble to be extracted under alkaline conditions from inside the cell is correlated to the cell wall characteristics and to the freeze-drying process that conserves well the samples but it makes the protein extraction more difficult for some algal species. In addition, the high values of hydro-soluble proteins extracted especially for P cruentum and A. platensis could be explained by the possible presence of peptides and free amino acids because no precipitation was carried out [START_REF] Nguyen | A rapid micro-scale method for the extraction and analysis of protein in marine samples[END_REF][START_REF] Barbarino | An evaluation ofmethodologies for extraction and quantification of protein of marine macro-and microalgae[END_REF].

Further evidence was found in the variation in composition relative to essential and non-essential amino acids (Table 5) in the protein extract of the green microalgae. There was a significant drop in the percentage of essential amino acids ranging from 13.9 % for C. vulgaris to 49.1 % for N oculata and an increase in non-essential amino acids from 6.7 % for the former to 25.2 % for the latter. On the contrary, only small changes in the com position of the proteins were detected for fragile cell wall microalgae, with a percentage difference ranging from 4.5 % for A. platensis to 4.8 % for the essential amino acids of P cruentum, whereas non-essential amino acids difference ranged from 3.5 % for the former to 4.8 % for the latter. These correlations with the relative hardness of the cell wall can integrate a chemical dimension as well. Indeed, the sporopollenin contained in the most rigid cell wall (H pluvia lis) is known to be extremely resistant to chemical agents [START_REF] Hagen | Ultrastructure and chemical changes in the cell wall of Haematococcus pluvialis (Volvocales, Chlorophyta) during aplanospore formation[END_REF]. But in the case of cellulose-rich cell walls, such as C. vulgaris and N oculata, the sodium hydroxide is able to penetrate the microcrystalline structure of cellulose to form alcoholates in a process similar to mercerisation. Sodium hydroxide can also dissolve the hemicelluloses attached to cellulose. The partial permeation of this kind of cell wall can therefore occur by alkaline action. Finally, A. platensis has a cell wall rich in amino sugars cross-linked with oligopeptide chains. The former are labile in alkaline conditions by deami dation of the N-acetylglucosamine and the latter are soluble in alkaline conditions. The cell wall becomes therefore very permeable allowing the alkaline extraction of proteins. In summary, the chemical action acts in synergy with the me chanical characteristics of the cell wall.

Protein extract

Extraction of proteins together with the evaluation of the conversion factor brought additional evidence that the cell wall of any microalga plays an important role in protein quantification. This means that not taking it into consider ation may lead to wrong quantification of the protein con tent. In addition, for microalgae, there is not a universal conversion factor that can be recommended for all species as demonstrated by comparison of our study with many studies which have been carried out on dozens of different microalgae. Therefore, every time accurate quantification of protein is needed, it will be required to evaluate the conver sion factor. In addition, this study showed a correlation between the cell wall rigidity and/or the chemical structure and the differences in NTP conversion value. Microalgae with fragile cell wall did not show significant differences with their protein extract, which was the complete opposite for the microalgae having a rigid cell wall that showed noticeable difference on evaluating the conversion factor of their protein extract, and therefore, breaking or permea bilize the rigid cell wall of C. vulgaris, N oculata and H pluvialis is strongly required to prevent underestimation of the protein content after extraction. Logically, there is no need for cell wall breaking for P cruentum since it does not have a cell wall and conceming A. platensis soft cell wall disruption technique is needed to make sure that its fragile cell wall will not hinder the extraction of any intracellular components.

Fig. 1

 1 Fig. 1 Laser scanning confocal microscopie observation before (left) and after (right) alkaline treatment for each microalga. a and b P. cruentum, c and d A. platensis, e and f C. vulgaris, g and h N oculata, i and j H pluvialis

Table 2

 2 Results of total amino acids of the protein extractProtein extract amino acid composition extracted at pH 12 and 40 °C of five microalgae expressed in Amino acids

	P. cruentum

Table 3

 3 Nitrogen-to-protein conversion factors for the crude micro algae and their protein isolate based on three replicates for three experiments±SD (n = 3)

	NTP
	Microalgae	Crude	Protein	% Relative
		microalgae	extract	difference
	P. cruentum	6.35±0.03	6.34±0.04	0.16
	A. platensis	6.27±0.02	6.21±0.07	0.96
	C. vulgaris	6.35±0.07	5.96±0.23	6.14
	N. oculata	6.28±0.06	5.86±0.32	6.69
	H. pluvialis	6.25±0.07	5.63±0.18	9.92

Table 4

 4 Different protein contents in crude microalgae Lo wry, hydro-soluble protein % (dw) at pH 12 and 40 °C calculated by Lowry method

		Nea" (%)	Total protein = NeaxNTP b (%)	Hydro-soluble	Proportion of hydro
				protein = PLowr/ (%)	soluble protein in total
					protein =	N :0 ";.Tfrp x 100(%)
	P. cruentum	9.04±0.69	57.33±3.84	46.06±0.97	80.34±1.69
	A. platensis	8.53±0.20	53.51±1.10	37.19±2.67	69.50±5.00
	C. vulgaris	7.81±0.18	49.59±1.04	21.50±0.34	43.35±0.62
	N. oculata	7.41±0.39	46.55±2.14	15.52±0.42	33.34±0.90
	H. pluvialis	8.27±0.07	51.73±0.43	14.22±0.69	27.48±1.34
	"Nea, total nitrogen % (dw) obtained by elemental analysis
	b NTP, nitrogen-to-protein conversion factor from Table 3

c P

Table 5

 5

	Percentage (dw) of
	essential and non-essential
	amino acids for each microalga and its protein extract	Essential N on-essential Not identified Essential Non-essential Not identified
	P. cruentum	46.81	44.89	8.29	44.58	47.15	8.27
	A. platensis	41.76	49.67	8.57	39.87	51.50	8.62
	C. vulgaris	41.02	51.03	7.94	35.32	54.70	9.97
	N. oculata	41.00	50.46	8.53	20.88	67.47	11.64
	H. pluvialis	43.91	48.50	7.58	29.01	60.17	10.81

Acknowledgments

The authors would like to thank Alpha Biotech for providing the biomass and also we are thankful to Dr Philippe Michaud, Dr Celine Laroche, Mr Jean-François Fabre and Mrs Cécile Pouzet for their support and expertise. This work was supported by the French National Research Agency (ANR) in the context of Algoraffinerie project.