Vincent Jf Koziarz
email: koziarz@iecn.u-nancy.fr

Extensions with estimates of cohomology classes

Introduction

Let Y be a complex submanifold of a Kähler manifold X and let L ′ be a Hermitian line bundle on X. First consider the following Problem. Let f be a smooth D ′′ -closed section of Λ 0,q T ⋆ X ⊗ L ′ over Y satisfying a suitable L 2 condition. Can we find a smooth D ′′ -closed extension F of f to X together with a good L 2 estimate for F on X?

The first result of this kind was obtained by T. Ohasawa and K. Takegoshi [OT] in the case when Y is a hyperplane of a bounded pseudoconvex domain X in C n , L ′ is the trivial bundle and q = 0. It was further generalized by L. Manivel [Ma] (with a simplified proof by J.-P. Demailly [START_REF] Demailly | On the Ohsawa-Takegoshi-Manivel L 2 extension theorem, Complex analysis and geometry[END_REF]) in the following setting: X is a weakly pseudoconvex manifold, Y is the zero set of a holomorphic section of a rank r Hermitian bundle over X, L ′ = K X ⊗ L where L is a Hermitian line bundle whose curvature satisfies appropriate positivity properties, K X is the canonical bundle of X, and q = 0. When q ≥ 1, the method leads to a new technical difficulty occurring in the regularity argument for (0, q) forms. In [START_REF] Demailly | On the Ohsawa-Takegoshi-Manivel L 2 extension theorem, Complex analysis and geometry[END_REF], Demailly suggests an approach to overcome this difficulty but, to our knowledge, the complete arguments did not appear anywhere. In this paper, we rather consider the Modified problem. Let q ≥ 1 and f be a smooth D ′′ -closed section of Λ 0,q T ⋆ X ⊗ L ′ over Y satisfying a suitable L 2 condition. Can we find a smooth D ′′ -closed extension F of f to X as a cohomology class (i.e.[F |Y ] = [f ] ∈ H q (Y, L ′ )) together with a good L 2 estimate for F on X?

Observe that if Y is a Stein submanifold (this happens e.g. when X is a Stein manifold), the modified problem is not relevant when q ≥ 1 since the Dolbeault group H q (Y, L ′ ) vanishes. In contrast, we will focus here on the case when Y is a smooth compact hypersurface of a weakly pseudoconvex Kähler manifold X. This situation naturally happens, for example, when X is a compact Kähler manifold, or when X is a holomorphic family of projective algebraic manifolds fibered over the unit disc.

Theorem 1.1. Let (X, ω) be a weakly pseudonconvex n-dimensional Kähler manifold, and let Y ⊂ X be the zero set of a holomorphic section s ∈ H 0 (X, E) of a Hermitian line bundle (E, h E ); the subvariety Y is assumed to be compact and nonsingular. Let L be a line bundle Date: June 24, 2010.

endowed with a smooth Hermitian metric h

L such that √ -1Θ(L) + √ -1d ′ d ′′ log |s| 2 ≥ 0, (1.1) √ -1Θ(L) + √ -1d ′ d ′′ log |s| 2 ≥ α -1 √ -1Θ(E) for some α ≥ 1, (1.2)
|s| 2 ≤ e -α (1.3) on X. Let 0 < κ ≤ 1 and let Ω ⊂ X be a relatively compact open subset containing Y . Then, for any q ≥ 0 and every smooth D ′′ -closed (0, q)-form f with values in K X ⊗ L over Y , there exists a smooth extension F of f to Ω as a cohomology class (i.e.

[F |Y ] = [f ] ∈ H q (Y, K X ⊗L)) such that Ω |F | 2 |s| 2(1-κ) dV ω ≤ C κ Y |f | 2 |ds| 2 dV Y,ω
where C is a numerical constant depending only on Ω, E, L and q.

The norm of the forms with values in bundles will always be computed with respect to the one induced by ω, h E and h L . Also, Θ(E) (resp. Θ(L)) will always denote the curvature of the Hermitian line bundle (E, h E ) (resp. (L, h L )). When the metrics will be twisted by some positive functions, the weights will appear explicitely in the formulae.

Our proof follows many of the ideas outlined in [START_REF] Demailly | On the Ohsawa-Takegoshi-Manivel L 2 extension theorem, Complex analysis and geometry[END_REF]. First, using the weight bumping technique (and the adapted Bochner-Kodaira-Nakano inequality) initiated by Ohsawa and Takegoshi, for all ε > 0, we build extensions of f of class C 1 whose L 2 norm is controlled, and which are "approximately" D ′′ -closed (in the sense that the L 2 norm of their D ′′ -derivative is bounded by a constant times ε). Philosophically, passing to the limit as ε → 0 should provide the desired extension but the limiting elliptic differential system is singular along Y and this forbids the direct use of elliptic regularity arguments. Then, at this point, our strategy differs from Demailly's. Instead, we construct "approximate" q-cocycles ζ ε in Čech cohomology corresponding to the previous extensions via an effective Leray's isomorphism, in a similar fashion as Y.-T. Siu in [Si]. During the process, we solve local D ′′ -equations by standard techniques of L. Hrmander. Then, we can take the limit as ε → 0 and use the ellipticity of the Laplacian in bidigree (0, 0) to ensure the smoothness of the extending cocycle ζ. Finally, reversing the process, we get a smooth extension F of f as a cohomology class. Notice that the constant C in Theorem 1.1 is mainly related to a finite covering of Ω ⊃ Y by Stein open subsets (which is used to apply Leray's isomorphism) and the norm of the derivatives of a partition of unity subordinate to this finite covering. This explains in part why we need Y to be compact.

A consequence of Theorem 1.1 is a qualitative surjectivity theorem for restriction morphisms in Dolbeault cohomology: Corollary 1.2. Let X, Y , E and L be as in Theorem 1.1 i.e. satisfying (1.1), (1.2) and (1.3). Then the restriction morphism

H q (X, K X ⊗ L) -→ H q (Y, (K X ⊗ L) |Y )
is surjective for any q ≥ 0.

Applying Theorem 1.1 to E = C and to any semi-positive line bundle L (for instance L = C), we also easily get the following corollary which contains a special case of the invariance of the Hodge numbers for a family of compact Kähler manifolds (a result due to K. Kodaira and D. Spencer):

Corollary 1.3. Let π : X → ∆ be a proper holomorphic submersion over the unit disc and L a semi-positive line bundle on X. Assume that X is a Kähler manifold of dimension n+1. Then, for any q ≥ 0, h n,q (X t , L) := dim H n,q (X t , L) is independent of t ∈ ∆ (where X t = π -1 (t)).

Acknowledgments. I would like to thank Mihai Pȃun for many valuable discussions. I would also like to thank Jean-Pierre Demailly for explaining me details on his article [START_REF] Demailly | On the Ohsawa-Takegoshi-Manivel L 2 extension theorem, Complex analysis and geometry[END_REF], as well as Benot Claudon and Dror Varolin for useful comments on an earlier version of this paper.

Preliminary material

From now on, we assume that X, Y , L and E satisfy the hypotheses of Theorem 1.1. Let c ∈ R such that Ω ⊂ X c := {x ∈ X , ψ(x) < c}, where ψ is the plurisubharmonic exhaustion of X. Let U = {U j } j∈J be a finite covering of the closure of Ω by coordinate charts φ j : B -→ U j where B is the unit ball in C n , and such that U j ⊂ X c for all j. Denoting by ν the standard Hermitian norm on C n , we assume that the functions

ϕ j := ν • φ -1 j satisfy (2.1) √ -1Θ(L) -β √ -1Θ(E) + √ -1d ′ d ′′ ϕ j ≥ ω
on U j for any β ∈ [0, 1] (this is always possible if the U j 's are chosen small enough). For any multi-index (j 0 , . . . , j ℓ ), we shall denote by ϕ j 0 ,...,j ℓ the function ℓ i=0 ϕ j i which is defined on the intersection U j 0 ,...,j ℓ :=

U j 0 ∩ • • • ∩ U j ℓ .
If F is a smooth Hermitian vector bundle over X and U ⊂ X is an open subset then, for all integer k, we denote by E k (U, F ) the space of sections of F over U which are of class C k and by E k c (U, F ) those with compact support. We also denote by W k (U, F ) the Sobolev space of sections whose derivatives (in the sense of distribution theory) up to order k are in L 2 .

Let us recall three useful results taken from [START_REF] Demailly | Estimations L 2 pour l'opérateur ∂ d'un fibré vectoriel holomorphe semi-positif audessus d'une variété kählérienne complète[END_REF] (Remark 1.6, Lemma 3.3 and Lemma 6.9):

Proposition 2.1. (a) X c \Y is complete Kähler. (b)
Let ω and ω ′ be two Hermitian forms on T X such that ω ≤ ω ′ . Let E be a Hermitian vector bundle on X. Then, for any q ≥ 0 and any u

∈ Λ n,q T ⋆ X ⊗ E, |u| 2 ω ′ dV ω ′ ≤ |u| 2 ω dV ω . (c) Let Ω be an open subset of C n and Y a complex analytic subset of Ω. Assume that v is a (p, q -1)-form with L 2 loc coefficients and w a (p, q)-form with L 1 loc coefficients such that d ′′ v = w on Ω\Y (in the sense of distribution theory). Then d ′′ v = w on Ω.
The following lemma is a consequence of a classical result (see [START_REF] Demailly | L 2 vanishing theorems for positive line bundles and adjunction theory[END_REF], Corollary 5.3):

Lemma 2.2. Let m and p be positive integers.

(a) Let v ∈ W m (U j 0 ,...,j ℓ , Λ n,p T ⋆ X ⊗ L ⊗ E -1 ) such that D ′′ v = 0 and U j 0 ,...,j ℓ
|v| 2 e -ϕ j 0 ,...,j ℓ dV ω < +∞.

Then there exists a (n, p -1)

form u ∈ W m+1 (U j 0 ,...,j ℓ , Λ n,p-1 T ⋆ X ⊗ L ⊗ E -1 ) such that D ′′ u = v and U j 0 ,...,j ℓ
|u| 2 e -ϕ j 0 ,...,j ℓ dV ω ≤ 1 p U j 0 ,...,j ℓ |v| 2 e -ϕ j 0 ,...,j ℓ dV ω .

(b) Let 0 < κ ≤ 1 and ε > 0. Let v ∈ W m (U j 0 ,...,j ℓ , Λ n,p T ⋆ X ⊗ L) such that D ′′ v = 0 and U j 0 ,...,j ℓ |v| 2 (|s| 2 + ε 2 ) 1-κ e -ϕ j 0 ,...,j ℓ dV ω < +∞.
Then there exists a (n, p -1)

form u ∈ W m+1 (U j 0 ,...,j ℓ , Λ n,p-1 T ⋆ X ⊗ L) such that D ′′ u = v and U j 0 ,...,j ℓ |u| 2 (|s| 2 + ε 2 ) 1-κ e -ϕ j 0 ,...,j ℓ dV ω ≤ 1 p U j 0 ,...,j ℓ |v| 2 (|s| 2 + ε 2 ) 1-κ e -ϕ j 0 ,...,j ℓ dV ω .
Proof. We only check the hypotheses of Corollary 5.3 in [START_REF] Demailly | L 2 vanishing theorems for positive line bundles and adjunction theory[END_REF]. The open subset U j 0 ,...,j ℓ ⊂ X is Stein and on U j 0 ,...,j ℓ , the line bundle L ⊗ E -1 , resp. L, endowed with its metric twisted by e -ϕ j 0 ,...,j ℓ , resp. (|s| 2 + ε 2 ) -(1-κ) e -ϕ j 0 ,...,j ℓ , has curvature

√ -1Θ(L) - √ -1Θ(E) + √ -1d ′ d ′′ ϕ j 0 ,...,j ℓ , resp. √ -1Θ(L) + (1 -κ) √ -1d ′ d ′′ log(|s| 2 + ε 2 ) + √ -1d ′ d ′′ ϕ j 0 ,...,j ℓ ,
which, by inequality (3.2) and assumption (2.1), is bounded from below by

√ -1Θ(L) - √ -1Θ(E) + √ -1d ′ d ′′ ϕ j 0 ≥ ω, resp. √ -1Θ(L) -(1 -κ) √ -1Θ(E)s, s |s| 2 + ε 2 + √ -1d ′ d ′′ ϕ j 0 ≥ ω.
The fact that u can be chosen in the Sobolev space W m+1 comes from the ellipticity of the Laplacian. Let us explain why in the case (a), the case (b) being completely similar. In fact, the D ′′ -equation is solved using complete metrics ω ε on U j 0 ,...,j ℓ , such that ω ε ≥ ω and ω ε → ω as ε → 0. For any ε > 0, the corresponding minimal solution u ε (i.e. the one satisfying D ′′ u ε = v and u ε ∈ (Ker D ′′ ) ⊥ω ε ) is such that U j 0 ,...,j ℓ |u ε | 2 ωε e -ϕ j 0 ,...,j ℓ dV ωε ≤ 1 p U j 0 ,...,j ℓ |v| 2 ωε e -ϕ j 0 ,...,j ℓ dV ωε ≤ 1 p U j 0 ,...,j ℓ |v| 2 e -ϕ j 0 ,...,j ℓ dV ω where the latter inequality comes from Proposition 2.1 (b). Then, there exists a sequence (ε µ ) converging to 0 such that u εµ converges weakly to some u in L 2 loc as µ → +∞: u satisfies

D ′′ u = v, U j 0 ,...,j ℓ
|u| 2 e -ϕ j 0 ,...,j ℓ dV ω ≤ 1 p U j 0 ,...,j ℓ |v| 2 e -ϕ j 0 ,...,j ℓ dV ω but also u ∈ (Ker D ′′ ) ⊥ω = Im (D ′′ ) ⋆ω (and therefore (D

′′ ) ⋆ω u = 0). Indeed, L 2 (ω) ⊂ L 2 (ω ε ) because | . | 2 ωε dV ωε ≤ | . | 2
dV ω and since ω ε → ω, we get u ∈ (Ker D ′′ ) ⊥ω by the dominated convergence theorem. Finally, u satisfies D ′′ u = v and (D ′′ ) ⋆ u = 0 which is an elliptic differential system and standard arguments give

u ∈ W m+1 if v ∈ W m .
Notice that we can skip the extraction of a weak limit if we only need a solution with the same estimate on a slightly smaller relatively compact open subset of U j 0 ,...,j ℓ : all we have to do is take a complete metric on U j 0 ,...,j ℓ which coincides with ω on the smaller subset.

Finally, we also select a smooth partition of unity {σ j } j∈J subordinate to U (i.e. for each j, σ j ∈ E ∞ c (U j , [0, 1]) and j∈J σ j (x) = 1 for any x ∈ Ω).

Proof of the theorem

Recall that, by assumption, Y ⊂ X is a smooth hypersurface so that E ≃ O X (Y ).

3.1. Construction of smooth extensions. In this section, we prove the following Lemma 3.1. For any k ≥ 0, there exists a smooth section

f ∞ ∈ E ∞ (X, Λ n,q T ⋆ X ⊗ L) such that (a) f ∞ coincides with f in restriction to Y , (b) | f ∞ | = |f | at every point of Y , (c) D ′′ f ∞ = 0 at every point of Y , (d) s -1 D ′′ f ∞ ∈ E k (X, Λ n,q+1 T ⋆ X ⊗ L ⊗ O X (-Y )).
Proof . Let us cover Y by coordinate patches W j ⊂ X biholomorphic to polydiscs and with the following property: if we denote the corresponding coordinates by (z j , w j ) ∈ ∆ × ∆ n-1 , where w j = (w 1 j , . . . , w n-1 j ), then W j ∩ Y = {z j = 0}. On each W j , we fix some holomorphic

σ j ∈ Γ(W j , K X ⊗ L) which trivializes K X ⊗ L.
As explained in [START_REF] Demailly | On the Ohsawa-Takegoshi-Manivel L 2 extension theorem, Complex analysis and geometry[END_REF], the restriction map (Λ 0,q T ⋆ X ) |Y -→ Λ 0,q T ⋆ Y can be viewed as an orthogonal projection onto a C ∞ subbundle of (Λ 0,q T ⋆ X ) |Y . One might extend this subbundle from W j ∩ Y to W j and then extend f on W j by some smooth form f j ∈ E ∞ (W j , Λ n,q T ⋆ X ⊗ L). Using a smooth partition of unity θ j ∈ E ∞ c (W j , R), j θ j = 1 on a neighbourhood of Y , we get a global smooth extension f = j θ j f j of f which fulfills conditions (a) and (b). Since

(D ′′ f ) |Y = (D ′′ f |Y ) = D ′′ f = 0,
we can write D ′′ f = dz j ∧ g j on W j ∩ Y for some smooth (0, q)-forms g j which we extend arbitrarily to W j . Then

f ∞ := f - j θ j zj g j
coincides with f on Y and satisfies (c).

We proceed by induction to get (d). Assume that on each W j ,

D ′′ f ∞ = z j f j (z j , w j ) + zk j dz j ∧ |I|=q a I (w j )σ j d wI j + |I ′ |=q+1 b I ′ (w j )σ j d wI ′ j + zk+1 j h j (z j , w j ) for some f j , h j ∈ E ∞ (W j , Λ n,q+1 T ⋆ X ⊗ L), a I , b I ′ ∈ E ∞ (∆ n-1 , C), k ≥ 1
, and where the multiindices I, I ′ are increasing. We say that f ∞ enjoys property (P k ). Remark that such an equality implies that s

-1 D ′′ f ∞ ∈ E k-2 (X, Λ n,q+1 T ⋆ X ⊗ L ⊗ O X (-Y )) if k ≥ 2. Moreover, the extension f ∞ we just constructed satisfies property (P 1 ) because D ′′ f ∞ = 0 along Y .
Of course, D ′′ (D ′′ f ∞ ) = 0, but also the direct computation gives

D ′′ (D ′′ f ∞ ) = z j D ′′ f j (z j , w j ) + kz k-1 j dz j ∧ |I ′ |=q+1 b I ′ (w j )σ j d wI ′ j + zk j h ′ j (z j , w j ) for some h ′ j ∈ E ∞ (W j , Λ n,q+2 T ⋆ X ⊗ L), hence the b I ′ 's must vanish identically. So if we take f ′ ∞ = f ∞ - j θ j zk+1 j k + 1 |I|=q a I (w j )σ j d wI j ,
we have

D ′′ f ′ ∞ = D ′′ f ∞ - j zk+1 j k + 1 d ′′ θ j + θ j zk j dz j ∧ |I|=q a I (w j )σ j d wI j - j θ j zk+1 j k + 1 D ′′ |I|=q a I (w j )σ j d wI j = j θ j D ′′ f ∞ -zk j dz j ∧ |I|=q a I (w j )σ j d wI j + j zk+1 j h ′′ j (z j , w j ) = j z j θ j f j (z j , w j ) + j zk+1 j θ j h j (z j , w j ) + h ′′ j (z j , w j ) for some h ′′ j ∈ E ∞ c (W j , Λ n,q+1 T ⋆ X ⊗ L). Then, f ′ ∞ enjoys property (P k+1 ).
3.2. Construction of approximate extensions with control. Let θ : R -→ [0, 1] be a smooth function with support in (-∞, 1), such that θ ≡ 1 on (-∞, 1/2] and |θ ′ | ≤ 4, and consider the truncated extension of f Y )) for some k ≥ 1 which will be determined later. We wish to solve on X the equation

f ε := θ(ε -2 |s| 2 ) f ∞ where f ∞ is the extension provided by Lemma 3.1, such that s -1 D ′′ f ∞ ∈ E k (X, Λ n,q+1 T ⋆ X ⊗ L ⊗ O X (-
D ′′ u ε = D ′′ f ε
with estimate, and the additional constraint that u ε vanishes along Y . We also expect some regularity on u ε in order to justify that f ε -u ε is a (D ′′ -closed) extension of f . In general, we are not able to get this by the method we use here, and we can only produce approximate solutions.

The fundamental tool is the following existence result (see [START_REF] Demailly | On the Ohsawa-Takegoshi-Manivel L 2 extension theorem, Complex analysis and geometry[END_REF], [Pa]):

Theorem 3.2. Let X be a complete Kähler manifold of dimension n equipped with a (not necessarily complete) Kähler metric ω, and let L be a line bundle endowed with a smooth Hermitian metric. Assume that there exist two smooth bounded functions η, λ > 0 on X satisfying

(3.1) η √ -1Θ(L) - √ -1d ′ d ′′ η - √ -1 d ′ η ∧ d ′′ η λ ≥ √ -1τ d ′ µ ∧ d ′′ µ
for some positive function τ and some function µ. Let us consider the (densely defined) modified D ′′ operators T u := D ′′ ( η + λu) and Su := √ η(D ′′ u)

acting on forms with values in L. Let g = d ′′ µ ∧ g 0 + g 2 be a L 2 form of (n, q + 1) type (q ≥ 0) with values in L such that (a)

D ′′ g = 0, (b) g 0 ∈ L 2 (X, Λ n,q T ⋆ X ⊗ L), (c) C(g 0 , τ ) := X 1/τ |g 0 | 2 dV ω < +∞, (d) |g 2 | 2 ≤ γ C(g 0 , τ
) almost everywhere for some positive constant γ.

Then, for any u ∈ Dom T ⋆ ∩ Dom S, we have

X g, u dV ω 2 ≤ C(g 0 , τ ) T ⋆ u 2 + Su 2 + γ u 2 .
In particular, there exist v ∈ L 2 (X, Λ n,q T ⋆ X ⊗ L) and w ∈ L 2 (X, Λ n,q+1 T ⋆ X ⊗ L) such that T v + γ 1/2 w = g together with the estimate

X |v| 2 dV ω + X |w| 2 dV ω ≤ C(g 0 , τ ).
As before, let c ∈ R be such that Ω ⊂ X c . For simplicity, we will assume in the sequel that X = X c . We are going to apply Theorem 3.2 to D ′′ f ε on X\Y . By Proposition 2.1 (a), X\Y can be equipped with a complete Kähler metric. As for the bundle L, we endow it with its original metric multiplied with the weight |s| -2 in order to force the vanishing of the approximate solution along Y .

For any ε > 0, we set σ ε :=log(ε 2 + |s| 2 ). Remark that, because of the condition |s| ≤ e -α , this function is positive for ε small enough. Let χ : R + -→ R + be any strictly concave function whose derivative satisfies 1 ≤ χ ′ ≤ 2 and such that χ(log(ε 2 + e -2α )) ≥ 2α for any ε > 0 small enough (in [START_REF] Demailly | On the Ohsawa-Takegoshi-Manivel L 2 extension theorem, Complex analysis and geometry[END_REF] and [Pa], χ(t) = t + log(1 + t) but we will choose other functions picked in [MV]). Let us define the two positive functions (again ε is assumed to be small enough)

η ε := χ(σ ε ) and λ ε := - χ ′ (σ ε ) 2 χ ′′ (σ ε ) .
Although this is done carefully in [START_REF] Demailly | On the Ohsawa-Takegoshi-Manivel L 2 extension theorem, Complex analysis and geometry[END_REF] and [Pa], we check quickly that η ε and λ ε fulfill condition (3.1) in Theorem 3.2. It is easy to see that

(3.2) - √ -1d ′ d ′′ σ ε ≥ √ -1 ε 2 |s| 2 d ′ σ ε ∧ d ′′ σ ε - √ -1Θ(E)s, s ε 2 + |s| 2
and it is straightforward that

d ′ η ε = χ ′ (σ ε )d ′ σ ε , d ′′ η ε = χ ′ (σ ε )d ′′ σ ε , d ′ d ′′ η ε = χ ′ (σ ε )d ′ d ′′ σ ε + χ ′′ (σ ε )d ′ σ ε ∧ d ′′ σ ε . Thus, since χ ′ is positive, - √ -1d ′ d ′′ η ε ≥ 1 χ ′ (σ ε ) ε 2 |s| 2 + 1 λ ε √ -1d ′ η ε ∧ d ′′ η ε - χ ′ (σ ε ) ε 2 + |s| 2 √ -1Θ(E)s, s .
If ε is small enough, then for any x ∈ X, η ε (x) ≥ χ(log(ε 2 + e -2α )) ≥ 2α. Taking into account the curvature assumptions (1.1) and (1.2) in Theorem 1.1 as well as the fact that χ ′ ≤ 2, we obtain

η ε ( √ -1Θ(L) + √ -1d ′ d ′′ log |s| 2 ) ≥ η ε α √ -1Θ(E) ≥ χ ′ (σ ε ) ε 2 + |s| 2 √ -1Θ(E)s, s .
Finally, summing up the two latter inequalities, we get

η ε ( √ -1Θ(L) + √ -1d ′ d ′′ log |s| 2 ) - √ -1d ′ d ′′ η ε - √ -1 λ ε d ′ η ε ∧ d ′′ η ε ≥ 1 χ ′ (σ ε ) ε 2 |s| 2 √ -1d ′ η ε ∧ d ′′ η ε which proves that (3.1) is fulfilled with τ = 1 χ ′ (σ ε )
ε 2 |s| 2 and µ = η ε . Now, we can write

D ′′ f ε = d ′′ η ε ∧ g ε + θ |s| 2 ε 2 D ′′ f ∞ where g ε := 1 + |s| 2 ε 2 θ ′ |s| 2 ε 2 f ∞ χ ′ (σ ε )
.

A quick computation shows that

(3.3) lim ε→0 1 ε 2 X\Y θ ′ |s| 2 ε 2 2 | f ∞ | 2 dV ω = c 0 Y |f | 2 |ds| 2 dV Y,ω
for some "universal" constant c 0 . Therefore, since θ(ε -2 |s| 2 ) is supported in {|s| < ε}, and since D ′′ f ∞ = 0 on Y , for any γ > 0,

θ |s| 2 ε 2 D ′′ f ∞ ≤ γ X\Y 1 τ |g ε | 2 |s| 2 dV ω = γ ε 2 X\Y 1 + |s| 2 ε 2 2 θ ′ |s| 2 ε 2 2 | f ∞ | 2 dV ω if ε > 0 is small enough.
Hence, we can apply Theorem 3.2: we find u ε,γ = √ η ε + λ ε v ε,γ and w ε,γ which satisfy the equation

D ′′ u ε,γ + γ 1/2 w ε,γ = D ′′ f ε on X\Y and such that X\Y |u ε,γ | 2 |s| 2 (η ε + λ ε ) dV ω + X\Y |w ε,γ | 2 |s| 2 dV ω ≤ 1 ε 2 X\Y 1 + |s| 2 ε 2 2 θ ′ |s| 2 ε 2 2 | f ∞ | 2 χ ′ (σ ε ) dV ω ≤ 4 ε 2 X\Y θ ′ |s| 2 ε 2 2 | f ∞ | 2 dV ω . (3.4) 3.3. Regularization of the approximate solution. Recall that Y is a divisor such that E ≃ O X (Y ).
Here we use a trick of Demailly: we consider s -1 u ε,γ (resp. s -1 w ε,γ ) as a L 2 (0, q)-form (resp. (0, q + 1)-form) with values in the twisted line bundle K X ⊗ L ⊗ O X (-Y ) equipped with a smooth Hermitian metric. By Proposition 2.1 (c), we can write

D ′′ (s -1 u ε,γ ) + γ 1/2 s -1 w ε,γ = s -1 D ′′ f ε not only on X\Y but also on X because s -1 u ε,γ is locally L 2 , s -1 w ε,γ is L 2 hence locally L 1 , and s -1 D ′′ f ε is of class C k hence locally L 1 (recall that f ∞ , as chosen in Lemma 3.1, is such that s -1 D ′′ f ∞ is of class C k , k ≥ 1)
. However, we do not know much about the regularity of u ε,γ and w ε,γ .

But

E ∞ c (X, Λ n,q T ⋆ X ⊗ L ⊗ O X (-Y ))
is dense in Dom D ′′ for the graph norm, where we consider D ′′ as an operator acting on (n, q) forms on X with values in L ⊗ O X (-Y ). More precisely, the density holds when X is endowed with a complete metric. If X = X c as we assumed above, we can work instead on X c ′ for some c ′ > c, and there exists on X c ′ some complete Kähler metric which coincides with ω on X c .

Then, we can find some

t ε,γ ∈ E ∞ (X, Λ n,q T ⋆ X ⊗ L ⊗ O X (-Y )), which is L 2 , such that (3.5) X |t ε,γ | 2 η ε + λ ε dV ω - X |s -1 u ε,γ | 2 η ε + λ ε dV ω ≤ ε
(recall that η ε is bounded by 2α from below), and D ′′ (t ε,γ -s -1 u ε,γ ) has L 2 norm bounded by γ 1/2 from above. As a consequence,

D ′′ t ε,γ = s -1 D ′′ f ε + r ε,γ on X, with r ε,γ ∈ E k (X, Λ n,q+1 T ⋆ X ⊗ L ⊗ E -1 ) since D ′′ t ε,γ and s -1 D ′′ f ε are of class C k . Moreover, r ε,γ satisfies (3.6) X |r ε,γ | 2 ≤ C 2 1 γ
for some positive constant C 1 depending on f , but not on ε and γ (see (3.4) and (3.3)).

Finally, s -1 D ′′ f ε is D ′′ -closed on X\Y , hence on X by Proposition 2.1 (c), and therefore D ′′ r ε,γ = 0.

3.4. The choice of η ε and λ ε . Let us come now to the choice of η ε and λ ε (see [MV] for more details). For any 0 < κ ≤ 1, we define for t ≥ 0 the functions

g κ (t) = κ -1 e κt , h κ (t) = t 0 1 2e κy -1 dy and χ κ (t) = 1 + t + h κ (t). One checks immediatly that 1 ≤ χ ′ κ ≤ 2 and χ ′′ κ < 0. Moreover, χ κ (-log(ε 2 + e -2α )) ≥ 1 -log(ε 2 + e -2α ) ≥ 1 + 2α -log(1 + ε 2 e 2α ) ≥ 2α
when ε is small enough. Clearly, χ κ (t) ≤ 1 + 2t and it follows that

χ κ (t) g κ (t) ≤ κ(1 + 2t) e κt ≤ 2
as is seen from a simple computation. Moreover,

- χ ′ κ (t) 2 χ ′′ κ (t) ≤ 2g κ (t).
As a consequence, if we fix κ ≤ 1 and take χ = χ κ , we have

(3.7) η ε + λ ε ≤ 4 κ(|s| 2 + ε 2 ) κ .
3.5. Construction of q-cochains via Leray's isomorphism. Recall that we have fixed a finite open covering U = {U j } j∈J of Ω. We endow the group

C ℓ 2 (U , E 1 (Λ n,p T ⋆ X ⊗ L ⊗ E -1 )) of (alternate) ℓ-cochains with values in E 1 (Λ n,p T ⋆ X ⊗ L ⊗ E -1
) which are L 2 with the norm

ς ℓ 2 = max j 0 <•••<j ℓ U j 0 ,...,j ℓ
|ς ℓ j 0 ,...,j ℓ | 2 e -ϕ j 0 ,...,j ℓ dV ω and for all 0 < κ ≤ 1 and ε > 0, we endow

C ℓ 2 (U , E 1 (Λ n,p T ⋆ X ⊗ L)) with the norm ς ℓ 2 κ,ε = max j 0 <•••<j ℓ U j 0 ,...,j ℓ
|ς ℓ j 0 ,...,j ℓ | 2 (|s| 2 + ε 2 ) 1-κ e -ϕ j 0 ,...,j ℓ dV ω .

Remark that in the case when ς 0 is the 0-cocycle associated to a section

ς of E 1 (Λ n,p T ⋆ X ⊗ L ⊗ E -1 )) (resp. E 1 (Λ n,p T ⋆ X ⊗ L))), ς 0 2 ≤ X |ς| 2 dV ω resp. ς 0 2 κ,ε ≤ X |ς| 2 (|s| 2 + ε 2 ) 1-κ dV ω
since the ϕ j 's are nonnegative. Now, we construct a (q + 1)-cocycle in Z q+1 (U , O(K X ⊗ L ⊗ E -1 )) corresponding to r ε,γ via Leray's isomorphism between the Dolbeault and the Čech cohomology groups. In fact, we are mostly interested in the intermediate cochains which appear during the process and the control we have on their norm. The extension f ∞ of f is supposed to be sufficiently regular (i.e. k is large enough in Proposition 3.1) in order that r ε,γ , and every cochain obtained by solving local D ′′ -equations below, is at least of class C 1 (see section 3.3, Lemma 2.2 and use Sobolev lemma:

W m (U ) ⊂ E 1 (U ) for any open subset U ⊂ X if m > 1 + n
2 ). For notational simplicity, we denote r ℓ ε,γ,(j 0 ,...,j ℓ ) ∈ Γ(U j 0 ,...,j ℓ , E 1 (Λ n,q-ℓ T ⋆ X ⊗ L ⊗ E -1 )) by r ℓ j 0 ,...,j ℓ . First, we solve the equation D ′′ r 0 j = r ε,γ on the U j 's, then we solve the equations D ′′ r ℓ+1 j 0 ,...,j ℓ+1 = (δr ℓ ) j 0 ,...,j ℓ+1 on

U j 0 ∩ • • • ∩ U j ℓ+1 (0 ≤ ℓ ≤ q -1) using each time Lemma 2.2 (a). Finally, δr q ε,γ ∈ Z q+1 (U , O(K X ⊗L⊗E -1 )) is a representative in Čech cohomology of [r ε,γ ] ∈ H q+1 (X, K X ⊗ L ⊗ E -1 ).
Lemma 3.3. For any ℓ, we have r ℓ ε,γ 2 ≤ (ℓ + 1) . . . 2.1 (q + 1).q . . . (q -ℓ + 1) r ε,γ 2 ≤ (ℓ + 1) . . . 2.1 (q + 1).q . . . (q -ℓ + 1) X |r ε,γ | 2 dV ω .

Proof. For any ℓ ≥ 1,

r ℓ ε,γ 2 ≤ 1 q -ℓ + 1 δr ℓ-1 ε,γ 2 ≤ ℓ + 1 q -ℓ + 1 r ℓ-1 ε,γ 2 and r 0 ε,γ 2 ≤ 1 q + 1 r ε,γ 2 ≤ 1 q + 1 X |r ε,γ | 2 dV ω
by the estimate in Lemma 2.2 (a).

In a similar manner, we produce a q-cochain ζ ε,γ ∈ C q (U , O(K X ⊗ L)) corresponding to the "approximately" D ′′ -closed extension f ε -st ε,γ ∈ Γ(X, E 1 (Λ n,q T ⋆ X ⊗ L)) of f . More precisely, on the U j 's, we solve the equation D ′′ h 0 j = f ε -st ε,γ + sr 0 ε,γ,j , then we solve D ′′ h ℓ+1 j 0 ,...,j ℓ+1 = (δh ℓ ) j 0 ,...,j ℓ+1 + (-1) ℓ+1 sr ℓ+1 j 0 ,...,j ℓ+1 on U j 0 ∩ • • • ∩ U j ℓ +1 (0 ≤ ℓ ≤ q -2) using Lemma 2.2 (b). This is indeed possible since the right-hand side is D ′′ -closed: if D ′′ h ℓ j 0 ,...,j ℓ = (δh ℓ-1 ) j 0 ,...,j ℓ + (-1) ℓ sr ℓ j 0 ,...,j ℓ then D ′′ (δh ℓ ) j 0 ,...,j ℓ+1 = (δ(D ′′ h ℓ )) j 0 ,...,j ℓ+1 = (δ 2 h ℓ-1 ) j 0 ,...,j ℓ+1 + (-1) ℓ s(δr ℓ ) j 0 ,...,j ℓ+1 = 0 + (-1) ℓ sD ′′ r ℓ+1 j 0 ,...,j ℓ+1 .

Finally, let ζ ε,γ := δh q-1 ε,γ + (-1) q sr q ε,γ ∈ C q (U , O(K X ⊗ L)) (in particular D ′′ ζ ε,γ = 0 and ζ ε,γ is actually smooth by ellipticity of D ′′ in bidegree (0, 0)).

In the next proposition, we denote by V the finite Stein covering

{V j } j∈J = {U j ∩ Y } j∈J of Y . Proposition 3.4. Let 0 < κ ≤ 1. The cochain ζ ε,γ enjoys the following properties: (a) ζ ε,γ 2 κ,ε ≤ 16(q + 1)c 0 κ Y |f | 2 |ds| 2 dV Y,ω + β(ε, γ)
where β is a positive function such that β(ε, γ) → 0 as ε, γ → 0 (recall that for any γ > 0,

ζ ε,γ only exists if ε > 0 is small enough). (b) For any ε, γ, ζ ε,γ |Y ∈ Z q (V, O((K X ⊗ L) |Y )) is a representative in Čech cohomology of the cohomology class [f ] ∈ H q (Y, (K X ⊗ L) |Y ).
Proof. Let χ = χ κ be as in section 3.4. We use the corresponding η ε and λ ε .

(a) We have

ζ ε,γ κ,ε ≤ δh q-1 ε,γ κ,ε + sr q ε,γ κ,ε ≤ q + 1 h q-1 ε,γ κ,ε + e -ακ 2 r ε,γ by Lemma 3.3, since |s| 2 (|s| 2 + ε 2 ) -(1-κ) ≤ |s| κ ≤ e -ακ 2
according to assumption (1.3). In the same way, for any ℓ ≥ 1,

h ℓ ε,γ κ,ε ≤ 1 √ q -ℓ ( δh ℓ-1 ε,γ κ,ε + sr ℓ ε,γ κ,ε ) ≤ √ ℓ + 1 √ q -ℓ h ℓ-1 ε,γ κ,ε + (ℓ + 1) . . . 2.1 (q + 1).q . . . (q -ℓ) e -ακ 2 r ε,γ
where we also used the estimate in Lemma 2.2 (b). Finally,

h 0 ε,γ κ,ε ≤ 1 √ q f ε -st ε,γ κ,ε + sr 0 ε,γ κ,ε ) ≤ 1 √ q f ε -st ε,γ κ,ε + 1 (q + 1).q e -ακ 2 r ε,γ .
Collecting all these inequalities, we get

ζ ε,γ κ,ε ≤ q + 1 f ε -st ε,γ κ,ε + qe -ακ 2 r ε,γ ≤ q + 1 f ε -st ε,γ κ,ε + q C 1 γ 1/2 . Now, it is easy to see that X | f ε -st ε,γ | 2 (|s| 2 + ε 2 ) 1-κ dV ω ≤ X |st ε,γ | 2 (|s| 2 + ε 2 ) 1-κ dV ω + C 2 ε 2κ
for some constant C 2 , as f ε is uniformly bounded with support in {|s| < ε}. Finally, by (3.7),

X |st ε,γ | 2 (|s| 2 + ε 2 ) 1-κ dV ω ≤ X (|s| 2 + ε 2 ) κ |t ε,γ | 2 dV ω ≤ 4 κ X |t ε,γ | 2 η ε + λ ε dV ω
and the desired inequality follows from (3.4), (3.5) and (3.3). (b) It is clear since on U j 0 ,...,j ℓ +1 ∩ Y , the restriction of (δh ℓ ) j 0 ,...,j ℓ is always D ′′ -closed, hence we construct an "exact" representative in restriction to Y .

3.6. Passing to the limit and reversing the process to get the extension. Now, we just have to make γ and ε go to zero and extract a weak limit of ζ ε,γ . This weak limit ζ is an element of Z q (U , O(K X ⊗ L)) since D ′′ ζ ε,γ = 0 for any ε, γ, and δζ ε,γ = (-1) q δ(sr q ε,γ ) which, by (3.6) and Proposition 3.3, has L 2 norm bounded by some constant times γ 1/2 , thus δζ = 0. Moreover, ζ |Y ∈ Z q (V, O((K X ⊗ L) |Y )) is a representative in Čech cohomology of the cohomology class [f ] ∈ H q (Y, (K X ⊗L) |Y ) since this is the case of any ζ ε,γ |Y by Proposition 3.4 (b). For all j 0 , . . . , j q , as the ϕ j 's are bounded from above by 1, we obtain U j 0 ,...,jq

|ζ| 2 |s| 2(1-κ) dV ω ≤ 16e q+1 (q + 1)c 0 κ Y |f | 2 |ds| 2 dV Y,ω
if we take the limit in the inequality of Proposition 3.4 (a). Finally, we construct the desired extension F in the following way. For 0 ≤ ℓ ≤ q -1, we produce ξ ℓ ∈ C ℓ (U , E ∞ (Λ n,q-ℓ-1 T ⋆ X ⊗ L)) such that (i) (δξ q-1 ) j 0 ,...,jq = ζ j 0 ,...,jq on U j 0 ∩ • • • ∩ U jq , (ii) (δξ ℓ ) j 0 ,...,j ℓ+1 = D ′′ ξ ℓ+1 j 0 ,...,j ℓ+1 on U j 0 ∩ • • • ∩ U j ℓ+1 (0 ≤ ℓ ≤ q -2). These δ-equations are solved by using the partition of unity {σ j } j∈J subordinate to U in the following way: ξ q-1 j 0 ,...,j q-1 = i σ i ζ i,j 0 ,...,j q-1 , ξ ℓ j 0 ,...,j ℓ = i σ i D ′′ ξ ℓ+1 i,j 0 ,...,j ℓ (0 ≤ ℓ ≤ q -2). Finally, we set F = D ′′ ξ 0 j = q! j 1 <•••<jq j i =j ζ jq,...,j 1 ,j d ′′ σ j 1 ∧ • • • ∧ d ′′ σ jq on U j . Then, F defines a D ′′ -closed section in Γ(Ω, E ∞ (Λ 0,q T ⋆ X ⊗ K X ⊗ L)) such that [F |Y ] = [f ] ∈ H q (Y, (K X ⊗ L) |Y ) and the estimate

Ω |F | 2 |s| 2(1-κ) dV ω ≤ 16e q+1 (q + 1)c 0 C σ |J|! κ(|J| -q -1)! Y |f | 2 |ds| 2 dV Y,ω
holds for some constant C σ depending only on the partition of unity {σ j } j∈J and q (one can take C σ = max j∈J |d ′′ σ j | 2q ).


D Platis

N E Labrou
email: lambrou@aua.gr

Application of a PEG/salt aqueous two-phase partition system for the recovery of monoclonal antibodies from unclarified transgenic tobacco extract

Keywords: Aqueous two phase system, molecular farming, downstream processing, therapeutic monoclonal antibody, transgenic plants Aqueous two phase system, downstream processing, molecular farming, therapeutic monoclonal antibody, transgenic plants CCD, central composite design, ELISA, enzyme-linked immunosorbent assay, mAb polyethylenoglycol, Pi, phosphate, SDS-PAGE, sodium sulphate dodecyl polyacrylamide gel electrophoresis, TBS, Tris buffered saline, TCA, trichloroacetic

Aqueous two phase partition systems (ATPS) have been widely used for the separation of large variety of biomolecules. In the present report, the application of a polyethyleneglycol/phosphate (PEG/Pi) ATPS for the separation of anti-HIV monoclonal antibodies 2G12 (mAb 2G12) and 4E10 (mAb 4E10) from unclarified transgenic tobacco crude extract was investigated. Optimal conditions that favour opposite phase partitioning of plant debris/mAb as well as high recovery and purification were found to be 13.1% w/w [PEG 1500], 12.5% w/w [Pi] at pH 5 with a phase ratio of 1.3 and 8.25% w/w unclarified tobacco extract load. Under these conditions, mAb 2G12 and mAb 4E10 were partitioned at the bottom phosphate phase with 85 and 84% yield and purification 2.4-and 2.1-fold, respectively. The proposed ATPS was successfully integrated in an affinity-based purification protocol, using Protein A, yielding antibodies of high purity and yield. From this study, ATPS was shown to be suitable for initial protein recovery and partial purification of mAbs from unclarified transgenic tobacco crude extract.

Introduction

The application of aqueous two-phase systems (ATPS) in protein recovery has received considerable attention. ATPS offer low-cost, easily scalable, environmentally benign as well as multifunctional operations [START_REF] Ferreira | Purification of human immunoglobulin G by thermoseparating aqueous two-phase systems[END_REF][START_REF] Johansson | Protein partitioning in poly(ethylene glycol)/sodium polyacrylate aqueous twophase systems[END_REF][START_REF] Pericin | Optimization of conditions for acid protease partitioning and purification in aqueous two-phase systems using response surface methodology[END_REF][START_REF] Salgado | Prediction of the partitioning behaviour of proteins in aqueous two-phase systems using only their amino acid composition[END_REF]. PEG/salt systems, in particular, are widely applied for bioseparation of proteins because they are cost-effective low viscocity systems, suitable for continuous large scale operations, present shorter sepration times and able to handle high biomass load [START_REF] Albertsson | Partition of cell particles and macromolecules in polymer twophase systems[END_REF][START_REF] Flanagan | Application of aqueous twophase partition in PEG-phosphate systems. Design and implementation of a prototype process for recovery of bulk protein fractions from waste brewer's yeast[END_REF][START_REF] Hustedt | Partitioning in Aqueous Two-Phase Systems: Theory, Methods, Uses and Applications to Biotechnology[END_REF][START_REF] Kula | Technical aspects of extractive enzyme purification[END_REF]. The bulk of both phases consist mainly of water (80-90%). PEG has a stabilising effect on the protein tertiary structure, offering thus a gentle biocompatible nontoxic environment for labile biomolecules at which biological activity and protein solubility is maintained [START_REF] Dreyer | Ionic liquids for aqueous two-phase extraction and stabilization of enzymes[END_REF]. A variety of operational conditions, such as choice and concentration of polymer, salt, pH can be manipulated in order to achieve differential partitioning of the protein of interest. Differential partitioning mainly depends on surface properties (such as the size, charge and hydrophobicity) of the protein as well as on the physicochemical traits of the two phases [START_REF] Andrews | Correlation for the partition behavior of proteins in aqueous two-phase systems: effect of surface hydrophobicity and charge[END_REF][START_REF] Huddleston | The molecular basis of partitioning in aqueous two-phase systems[END_REF][START_REF] Rosa | Affinity partitioning of human antibodies in aqueous two-phase systems[END_REF]. Despite the fact that recovery of monoclonal antibodies from complex biological tissues has been reported in several studies, the wide-scale application of ATPS has been hindered by the poorly-understood partitioning mechanism [START_REF] Azevedo | Chromatography-free recovery of biopharmaceuticals through aqueous two-phase processing[END_REF][START_REF] Huddleston | The molecular basis of partitioning in aqueous two-phase systems[END_REF].

Future progress in utilizing transgenic plants for biopharmaceuticals production will depend on the efficiency of the purification methods, since purification represents most of the cost of biopharmaceutical production in transgenic plants [START_REF] Hassan | Considerations for extraction of monoclonal antibodies targeted to different subcellular compartments in transgenic tobacco plants[END_REF][START_REF] Lienard | Pharming and transgenic plants[END_REF].

Therefore, new approaches are necessary to minimize or eliminate purification costs.

Several chromatographic strategies have been investigating so far. These include cation exchange chromatography [START_REF] Zhang | Process engineering strategy for recombinant protein recovery from canola by cation exchange chromatography[END_REF], anion exchange chromatography [START_REF] Wang | Cored anion-exchange chromatography media for antibody flow-through purification[END_REF], [START_REF] Gupta | Immobilized metal affinity chromatography without chelating ligands: purification of soybean trypsin inhibitor on zinc alginate beads[END_REF][START_REF] Zhang | Suitability of immobilized metal affinity chromatography for protein purification from canola[END_REF], polyelectrolyte precipitation [START_REF] Menkhaus | Host selection as a downstream strategy: polyelectrolyte precipitation of beta-glucuronidase from plant extracts[END_REF],

Protein A affinity chromatography [START_REF] Ramessar | Cost-effective production of a vaginal protein microbicide to prevent HIV transmission[END_REF], and biomimetic affinity chromatography [START_REF] Platis | Lock-and-key motif as a concept for designing affinity adsorbents for protein purification[END_REF][START_REF] Platis | Affinity chromatography for the purification of therapeutic proteins from transgenic maize using immobilized histamine[END_REF]. ATPS is a pre-purification/concentration method that offers potential for product capture integrated with solids removal. ATPS has been evaluated for purifying egg white lysozyme [START_REF] Balasubramaniam | Tobacco protein separation by aqueous two-phase extraction[END_REF] and mAb 2F5 and 2G12 using clarified tobacco extracts [START_REF] Platis | Development of an aqueous two-phase partitioning system for fractionating therapeutic proteins from tobacco extract[END_REF][START_REF] Platis | New downstream processing strategy for the purification of monoclonal antibodies from transgenic tobacco plants[END_REF]. In this latter study, the effect of PEG molecular weight, pH and tie line length, as well as the overall potential use of PEG/phosphate aqueous twophase systems for the recovery and purification of mAbs 2F5 and 2G12 were examined in Platis et al., 2006 [25]. Similarities in the partitioning behaviour of monoclonal antibodies, especially of the same subclass (IgG1) have been observed in the literature probably due to overall similarities in general structural and physicochemical characteristics [START_REF] Andrews | Partitioning and purification of monoclonal antibodies in aqueous two-phase systems[END_REF][START_REF] Wingren | Comparison of surface properties of human IgA, IgE, IgG and IgM antibodies with identical and different specificities[END_REF]. Therefore, in the present study, ATPS optimization was achieved by analyzing the factors influencing the partitioning of mAb 2G12 and tobacco proteins, based on initial parameter range discovery as published previously [START_REF] Platis | Development of an aqueous two-phase partitioning system for fractionating therapeutic proteins from tobacco extract[END_REF]. This allowed the use of a standard two-parameter Central Composite Design (CCD) for a narrow range of PEG and phosphate concentrations on multiple PEG MWs, in order to predict conditions of optimal differential plant debris separation, purification and yield [START_REF] Li | Optimization of Conditions for Bacteriocin Extraction in PEG/Salt Aqueous Two-Phase Systems Using Statistical Experimental Designs[END_REF][START_REF] Rosa | Application of central composite design to the optimisation of aqueous two-phase extraction of human antibodies[END_REF]. All pertaining experiments were conducted at pH 5, which was found to be optimal for extraction and partitioning [START_REF] Platis | Development of an aqueous two-phase partitioning system for fractionating therapeutic proteins from tobacco extract[END_REF].

Protein isolation and purification from plants is a difficult task, owing to the complexity of the plant system and the presence of phenolic compounds, pigments, and mucilages [START_REF] Hassan | Considerations for extraction of monoclonal antibodies targeted to different subcellular compartments in transgenic tobacco plants[END_REF][START_REF] Jervis | Purification technologies for plant proteins[END_REF]. The presence of such compounds is potential interfering factor in downstream processing and has to be addressed early in the isolation process. For example, the presence of phenols and anionic polysaccharides can cause problems during chromatographic purification since they bind strongly to the chromatographic matrices and may give them ion-exchange characteristics [START_REF] Loomis | Plant phenolic compounds and the isolation of plant enzymes[END_REF][START_REF] Loomis | Overcoming problems of phenolics and quinones in the isolation of plant enzymes and organelles[END_REF][START_REF] Smith | Improved methods for the extraction of polyphenol oxidase from d'anjou pears[END_REF] or potentially reduce their half-life [START_REF] Platis | Development of an aqueous two-phase partitioning system for fractionating therapeutic proteins from tobacco extract[END_REF]. The presence of these compounds may strongly bind, inactivate and modify recombinant proteins expressed in transgenic plants [START_REF] Hagerman | The specificity of proanthocyanidin-protein interactions[END_REF][START_REF] Spencer | Polyphenol complexation--some thoughts and observations[END_REF].

This can have a detrimental effect on the structural integrity, purity and function of the isolated protein. The economic impact of these metabolites on the downstream process, especially on an industrial scale, has not been adequately addressed.

Plant biotechnology has demonstrated that transgenic plants are suitable hosts for expressing recombinant biomolecules. Several reagent-grade recombinant proteins from transgenic corn (trypsin and avidin) [START_REF] Hood | Molecular farming of industrial proteins from transgenic maize[END_REF][START_REF] Woodard | Maize (Zea mays)-derived bovine trypsin: characterization of the first large-scale, commercial protein product from transgenic plants[END_REF] and rice (lysozyme and lactoferin) [START_REF] Suzuki | Expression, characterization, and biologic activity of recombinant human lactoferrin in rice[END_REF][START_REF] Yang | Expression and localization of human lysozyme in the endosperm of transgenic rice[END_REF] have been commercialized, while clinical trials for plant-derived therapeutic proteins (e.g. interferon alpha-2b) are underway [START_REF] Arlen | Field production and functional evaluation of chloroplastderived interferon-alpha2b[END_REF].

In the present study, the application of ATPS for the recovery of mAbs from unclarified crude tobacco extract was evaluated. This is the first report of using ATPS as a pre-processing plant debris removal tool in the downstream processing procedure for the purification of monoclonal antibodies from transgenic tobacco. The results showed that the system offers significant advantages by minimizing intervening processing steps, such as centrifugation or filtration. The high cost of filtration or centrifugation for pre-chromatography processing, necessary to remove plant debris, makes it prohibitive at a large scale [START_REF] Hart | Large scale, in situ isolation of periplasmic IGF-I from E. coli[END_REF]. At the same time the system acts as a guard, increasing the performance and capacity of subsequent chromatographic steps.

Methods

Plant growth

Transgenic Nicotiana tabacum seeds were pre-germinated on plates, on filter paper, Whatman 3MM (watered with distilled water). The plates were kept for 72 hrs at 30°C. After germination, the plants were transferred into plastic pots (1-2 plants in each pot), in soil. The plants were grown in a controlled environment (12 h day/12 h night cycle, at 25°C day/21°C night-regime and 65% humidity. The plants were watered with deionized water every 2 days. Aqueous extraction of mAbs 2G12 and 4E10 from transgenic tobacco leaves was performed by mixing 5 g of tobacco leaves with 15 mL of 50 mM sodium phosphate buffer pH 5.0. Subsequently, the sample was disintegrated in a blender (total 10 min, breaking every 15 s) and the crude mixture placed on a rotary mixer for 60 min at 4°C. The amount of protein and monoclonal antibody extracted was determined by the method of Bradford [START_REF] Bradford | A rapid and sensitive method for the quantitation of microgram quantities of protein utilizing the principle of protein-dye binding[END_REF] and ELISA [START_REF] Platis | New downstream processing strategy for the purification of monoclonal antibodies from transgenic tobacco plants[END_REF], respectively. The expression level of mAbs 2G12 and 4E10 in transgenic tobacco were approximately 0.5-1% w/w in the total soluble protein (TSP) extracted fraction.

Monoclonal antibody

Aqueous Two Phase System (ATPS)

Phase diagrams for PEG-phosphate systems have been previously reported [START_REF] Snyder | Phase Compositions, Viscosities and Densities of Aqueous Two-Phase Systems Composed of Polyethylene Glycol and Various Salts at 25°[END_REF][START_REF] Zaslavsky | Aqueous Two-Phase Partitioning[END_REF]. Preparation of the phase systems was done from stock solutions of PEG 50% w/w and phosphate 40% w/w. In the case of the phosphate stock solution the appropriate working solution was achieved by mixing appropriate amounts of K 2 HPO 4 and NaH 2 PO 4 until the desired pH was achieved. ATPS systems (20 g) were prepared by mixing the sample (final concentration: 8.25% w/w crude transgenic tobacco extract), water and phosphate buffer, followed by gradual addition of the appropriate amount of 50% (w/w) PEG. Gradual addition of PEG was necessary in order to avoid precipitation of the sample due to increased PEG concentration.

Following thorough mix of the systems they were allowed to equilibrate at room temperature for 1-24 hrs without centrifugation. Samples were removed at specific time points from the top and bottom phases in order to calculate total protein and mAb 2G12 partitioning. Total plant protein was added in the samples to a final concentration of 0.

Full Central Composite Design

A two-level full Central Composite Design (CCD) was used in order to study the partitioning behaviour of tobacco proteins and mAb 2G12 and determine the effect of PEG and phosphate (Pi) concentrations at different PEG MW conditions. The variables were studied at two levels and for each variable studied, high (coded value +1) and low (coded value: -1) set points were selected based on results obtained from previous experiments [START_REF] Platis | Development of an aqueous two-phase partitioning system for fractionating therapeutic proteins from tobacco extract[END_REF] and considering the experimental conditions required for phase and debris separation. The ranges of factors are PEG molecular weight (1500-6000), PEG concentration (11-15% w/w) and phosphate concentration (11-15% w/w) at pH 5 and crude transgenic tobacco extract load 8.25% w/w. To set up a design, the inputs of each factor and its level as well as all responses are required. The responses were percentage of mAb 2G12 bottom phase recovery and purification factor. All experiments were analyzed using MINITAB (edition 14) software for statistical analysis and response optimization.

Affinity chromatography on Protein A column.

Affinity chromatography on Protein A column was conducted according to Platis et al., 2006 [25]. Briefly, a column packed with protein A adsorbent (0.5 mL) was equilibrated with potassium phosphate buffer (20 mM, 150 mM NaCl, pH 7.0, 10 mL) and the antibody sample (bottom phase ATPS, approximately 0.5 ml, diluted to 10 mL with equilibration buffer). The column was washed with potassium phosphate buffer (20 mM, 0.15 mM NaCl, pH 7.0, 10 mL). Bound antibody was eluted with glycine-HCl buffer (0.1 M, pH 2.5, 2 mL).

Determination of total phenolics and alkaloids.

Determination of total phenolics and alkaloids in samples from tobacco extract was carried out as described in Platis & Labrou (2006) [START_REF] Platis | Development of an aqueous two-phase partitioning system for fractionating therapeutic proteins from tobacco extract[END_REF].

Results and Discussion

The previously designed ATPS systems [START_REF] Platis | Development of an aqueous two-phase partitioning system for fractionating therapeutic proteins from tobacco extract[END_REF][START_REF] Platis | New downstream processing strategy for the purification of monoclonal antibodies from transgenic tobacco plants[END_REF] did not account for the influence of interfering factors such as, cell fragments and debris that are present in the crude tobacco leaf extract, since the added tobacco extract had been clarified by centrifugation, following by filtration. In order to take full advantage of the partitioning potential of ATPS, the ATPS conditions were adopted so that mAb separation would be possible using unclarified crude extract, therefore avoiding initial centrifugation and filtration steps. All experiments were analyzed using MINITAB (edition 14) software for statistical analysis and response optimization. Selection of optimal ATPS system conditions was based on differential plant debris separation, purification and yield. Based on previous bibliography, a series of pilot experiments was designed to determine whether such a favourable plant debris partitioning was possible. Initial experiments showed that ATPS systems based on PEG 3000 and PEG 6000 (data not shown) were unable to separate plant debris concentrate from mAb 2G12 (tested at a variety of conditions at both short and long TLLs) and were subsequently dismissed from further investigation. It was found that only PEG 1500 was able to achieve opposite phase partitioning of plant debris/mAb 2G12, which is in accordance with previous published results [START_REF] Cisneros | Recovery in aqueous two-phase systems of lutein produced by the green microalga Chlorella protothecoides[END_REF][START_REF] Srinivas | Aqueous two-phase extraction in combination with ultrafiltration for downstream processing of Ipomoea peroxidase[END_REF]. For this reason, as well as the previous preference of PEG 1500-containing ATPS systems for mAb 2G12 partitioning [START_REF] Platis | New downstream processing strategy for the purification of monoclonal antibodies from transgenic tobacco plants[END_REF] which exhibited higher yield and purification ability than PEG 3000-and PEG 6000containing ATPS systems, the attention was focused towards the analysis of ATPS conditions in PEG 1500-containing systems that favour opposite phase partitioning of plant debris/mAb 2G12. We have previously shown that ATPS conditions close to the binodal curve (short TLLs) are optimal for mAb 2G12 partitioning [START_REF] Platis | Development of an aqueous two-phase partitioning system for fractionating therapeutic proteins from tobacco extract[END_REF]. Pilot screening experiments showed that ATPS conditions with low phosphate concentrations (lower than 11% w/w) despite exhibiting high yield and purifying ability also exhibited significant plant debris and polyphenol leakage to the bottom phase and were therefore not selected. Increasing the phosphate concentration (>12% w/w) substantially improved plant debris separation (Figure 1). Therefore, initially we have methodically scanned the putative, predicted area of interest identifying not only the crossover point of the binodal curve but also roughly examining the effect of PEG Based on this initial information a standard two-parameter full CCD was chosen, concentrating of ATPS conditions close to the tie line, which proved to favour plant debris separation (Table 1). The contour plots as well as response optimization indicated the desirable combination of variables to obtain the most efficient conditions for maximum simultaneous mAb 2G12 recovery in the bottom phase and purification (Figure 2). Although V t /V b ratio was also considered as a factor for the design of the CCD, however, previous attempts to design ATPS systems with very high V t /V b ratios resulted in significant problems with mAb 2G12 precipitation (i.e. low TLL conditions with [PEG]>20% w/w), therefore resulting in a lower yield [START_REF] Platis | New downstream processing strategy for the purification of monoclonal antibodies from transgenic tobacco plants[END_REF]. For this reason the CCD was designed so that it would produce the highest possible V t /V b ratios without compromising yield. In order to maximize the processing capacity of the ATPS the maximum possible transgenic tobacco extract load of 8.25% w/w was adopted for all ATPS experiments. Higher protein load in ATPS reduced the purifying ability of the ATPS system.

Since the goal was to use ATPS for plant debris removal, avoiding centrifugation/filtration steps, the systems were allowed to separate under gravity at room temperature. The time that is required for complete phase separation of the system (20-50 g) was approximately between 1-3 h, depending on phase composition.

Between 1-3 h mAbs remained stable as determined by ELISA assay [START_REF] Platis | Development of an aqueous two-phase partitioning system for fractionating therapeutic proteins from tobacco extract[END_REF]. To simulate this equilibration period the systems were also left for 24 h at room temperature in order to test the stability of mAbs in solution as well as the stability of the ATPS. Results showed that key parameters, such mAb bottom phase yield and purification factor, remained relatively unaltered during this time period (Table 1), although it is obvious that prolonged exposures of the antibody with the unclarified tobacco extract should be avoided in order to minimize detrimental effects such as protease degradation or polyphenol-antibody interactions. Optimal recovery and purification conditions were predicted, based on the CCD analysis presented and taking into account the additional screening of the area close to the binodal, to be 13.1% w/w [PEG 1500], 12.5% w/w [Pi] at pH 5 with a V t /V b phase ratio of 1.3 and 8.25% w/w transgenic tobacco extract load. These conditions were evaluated using an 80 g aqueous two phase system and the results are presented in Table 2. Under these conditions, mAb 2G12 and mAb 4E10 were partitioned at the bottom phosphate phase with 85 and 84% yield and purification factor of 2.4 and 2.1, respectively. The final chosen ATPS conditions for debris separation provide similar mAbs yields as in the case of mAb purification from clarified plant extract [START_REF] Platis | New downstream processing strategy for the purification of monoclonal antibodies from transgenic tobacco plants[END_REF]. In addition, plant debris does not appear to have any effect on mAb stability in the crude mixture for up to 3 hours tested. It should be noted, that mAb partitioning at the bottom phase with simultaneous debris partitioning seriously minimizes the number of useful conditions for efficient partitioning. However, by carefully screening the area close to the binodal we have managed to isolate a set of conditions that provide satisfactory results.

To further demonstrate the usefulness and necessity of the ATPS, we have attempted to incorporate this step into the common purification strategy that is routinely used by antibody manufacturers e.g. affinity chromatography on Protein A, aiming at purifying the mAbs to homogeneity. Affinity chromatography with Protein A is the most selective method for mAb purification and is conventionally used for the initial antibody. The affinity chromatography step was carried out according to the standard procedures commonly employed by antibody manufacturers [START_REF] Platis | Development of an aqueous two-phase partitioning system for fractionating therapeutic proteins from tobacco extract[END_REF]. The results showed that the incorporation of the ATPS step before affinity chromatography does not affect the effectiveness of the whole procedure. This is because the antibody fraction obtained from the ATPS requires no dialysis or ultrafiltration (for buffer exchange) prior to loading onto the affinity column. The initial and final concentration of mAbs 2G12 and 4E10 was approximately 5-7 µg/ml (in the plant extract) and 3-5 µg/ml (after Protein A purification) respectively. The initial and final concentration of total protein were 800-1000 µg/ml and <0.5 µg/ml. Analysis of purified antibody by SDS-PAGE showed the absence of any major contaminating proteins or degraded antibody variants (Figure 3).

In conclusion, ATPS offers unique advantages for large scale processing e.g. low cost, easy scale up, continuous processing and high biocompatibility. In addition, the system allows fast approach to equilibrium and therefore high separation rates, a crucial factor for minimizing the time that the heterologous protein is exposed to deleterious plant extract. Thus, the present method provides a simple and effective way of preparing therapeutic antibodies in two steps (ATPS, affinity chromatography) suitable for analytical or clinical purposes. Central Composite design for contour surface study of the effect of PEG and phosphate concentration on the bottom phase yield and purification factor of mAb 2G12 at pH 5. The observed V t /V b ratio has also been documented. Precise measurements of the K part for mAb 2G12 and tobacco proteins was not possible when crude unclarified tobacco extract was used due to the increase amount of debris. ND, not determined due to the inability to form biphasic systems.

F

Transgenic

  Nicotiana tabacum seeds expressing mAb 4E10 were kindly donated by Prof. J. Ma (St. George's, University of London, UK). Transgenic Nicotiana tabacum seeds expressing mAb 2G12 were kindly donated by Prof. R. Fischer (Institute for Molecular Biotechnology, Aachen, Germany). CHO-derived and purified human monoclonal anti-HIV 2G12 and 4E10 antibodies were kindly donated by Dr. Dietmar Katinger at Polymun Scientific. PEG (molecular masses 1500, 3000 and 6000), potassium phosphate monobasic, potassium phosphate dibasic, polyclonal antibody [goat anti-human-IgG (γ-chain) (GAH-IgG)], polyclonal antibody [goat anti-human-IgG (κ-chain) (GAH-IgG & AP)] conjugated to alkaline phosphatase, was obtained from Sigma-Aldrich (St. Louis, MO, USA). All other chemicals were of analytical grade.

  2G12 and 4E10 extraction from transgenic tobacco leaves

  3-0.35 g/L. Control systems without protein (containing extraction buffer) were used for reference and comparison. Estimation of the volume of the top bottom phases was done visually in graduated centrifuge tubes and the volume ratio was subsequently calculated (V r = volume of top phase/volume of bottom phase). Total protein and antibody concentration was determined by Bradford assay and ELISA respectively. The following parameters were evaluated : (a) Yield of mAb 2G12 (the amount of mAb 2G12 in the bottom phase/initial mAb 2G12 added into the whole system) and (b) Purification factor Purification factor, defined as the ratio: ([mAb 2G12]/[total tobacco protein]) bottom phase /([mAb 2G12]/[total tobacco protein]) crude extract .

3. 2 . 5 .

 25 Sodium dodecyl sulphate polyacrylamide gel electrophoresis (SDS-PAGE).

10 A 12 .

 1012 5% (w/v) reducing SDS-PAGE was carried out as described inPlatis et al., 2008 [26].

  on plant debris separation and leakage to the bottom phase.

  facilitate rapid volume reduction as well as separation of the

  The authors thank the European Union for the financial assistance provided. This work was performed within the framework of the Pharma-Planta research project (No 503565).

Figure 1 .

 1 Figure 1. Photograph showing differential partitioning of tobacco plant debris to the

Figure 2 .

 2 Figure 2. Contour plots for mAb 2G12 showing the effect of PEG 1500 and phosphate

Figure 3 .

 3 Figure 3. SDS-PAGE analysis of mAb 2G12 and 4E10 fractionation from transgenic

Table 2 .

 2 Purification of mAb 2G12 (A) and mAb 4E10 (B) from transgenic tobacco extract using ATPS. Conditions: 13.1% w/w [PEG 1500], 12.5% w/w [Pi] at pH 5 with a V t /V b phase ratio of 1.3 and 8.25 % w/w transgenic tobacco extract load.

	21
	A
	Step	Yield	Purification	K part	tobacco	K part	mAb 2G12	V t /V b
		(%)	Factor
			(fold)
	Crude extract	100	1
	ATPS
	(top phase)	0.92	0.046			0.45	0.015
	ATPS								1.3
	(bottom phase) ATPS bottom phase after dialysis	r 95 o 85	1.37 2.4
	Step Crude extract ATPS	P Yield (%) e Purification Factor (fold) e 100 1 r	K part	tobacco	K part	mAb 4E10	V t /V b
	(top phase) (bottom phase) ATPS	0.92 92	0.15 1.83 R			0.38	0.033	1.3
	ATPS bottom phase after dialysis	84	2.1	e
				v i e
							w

Wiley-VCH Biotechnology Journal


Damien Durville

Simulation of the mechanical behaviour of woven fabrics at the scale of fibers

Keywords: Woven fabrics, Mechanical properties, Textile materials, Contact-friction interactions, Finite element simulation

published or not. The documents may come

Simulation of the mechanical behaviour of woven fabrics at the scale of fibers

Damien Durville

Introduction

As textile fibers are more used to manufacture technical fabrics or to reinforce composite materials, the need of mechanical characterization of textile materials is growing. If the global mechanical behaviour of fabrics can be investigated and identified through experiments, the simulation becomes a new means to explore the complex behaviour of such structures.

A woven fabric is constituted by components of two levels. At a first level, a fabric is made of interlacing tows, whereas at a second level tows are constituted by bundles D. Durville Laboratoire de Mécanique des Sols Structures et Matériaux (LMSSM) CNRS UMR8579 -Ecole Centrale Paris Grande Voie des Vignes -92290 Chatenay-Malabry (France) Tel.: +33-141131290 E-mail: damien.durville@ecp.fr of fibers. The modeling of woven fabrics can be tackled at three different scales. At a macroscopic scale, homogeneized models can be derived to represent the fabric as a plate or a shell. By developing models for yarns or tows, considered as continuous media, it is possible to build intermediate approaches to the behaviour of fabrics at a mesoscopic scale, considering the fabric as an assembly of interlacing tows. However, going down one more level, the mechanical behaviour of individual fibers has to be taken into account to describe the behaviour of woven structures at a microscopic scale.

Many approches are available in the literature concerning the modeling of fabrics at macroscopic and at mesoscopic scale [START_REF] Ben Boubaker | Mesoscopic fabric models using a discrete massspring approach: Yarn-yarn interactions analysis[END_REF][START_REF] Bridgens | Membrane material behaviour: concepts, practice and develoments[END_REF][START_REF] Ballhause | A microstructure model for fabric-reinforced membranes based on discrete element modelling[END_REF][START_REF] Boisse | A mesoscopic approach for the simulation of woven fibre composite forming[END_REF][START_REF] Badel | Computational determination of in-plane shear mechanical behaviour of textile composite reinforcements[END_REF]. One of the limits of these approaches is that they have to assume and to identify intermediate models to describe the behaviour of the threads, or the interactions between them.

To avoid to have to make assumptions about the behaviour of intermediate components, the approach presented here focuses on the microscopic scale, in order to understand and to predict mechanisms taking place at the scale of the fibers. This approach considers small samples of woven fabrics as collections of individual fibers undergoing large deformations, and needs to model not only the behaviour of all individual fibers, but also contact-friction interactions developed between them. The only characteristics required for the modeling are those of fibers, that are readily available. This approach, by describing the mechanical state of all individual fibers involved in the sample, is also useful for the prediction of damage, in particular due to fiber breakage.

The determination of the initial configuration of the woven fabric, which is unknown a priori, is a critical issue for for simulation at microscopic scale, since the initial geometry of each fiber has to be described. To obtain this initial configuration, it is necessary to simulate the weaving process. This preliminary task provides as results many useful informations about the arrangement of fibers in the fabric.

When dealing with woven fabrics at the scale of fibers, the difficulties lie in the detection and modeling of numerous contact-friction interactions taking place within the collection of fibers. Within the framework of large deformations, this leads to nonlinear problems that require optimized algorithms to be solved efficiently, especially when high numbers of degrees of freedom and contacts are considered. The recent improvements in computing capacities make the finite simulation at the scale of fibers a possible way to explore the complex behaviour of textile fabrics.

Similar approaches at microscopic scale can be found in literature. At the scale of fibers, for the computation of the initial configuration of braided structures or 3D interlock woven fabric, digital elements have been used by Miao [START_REF] Miao | Mechanics of textile composites: Microgeometry[END_REF]. Since these digital elements have neither bending nor torsional stiffness, fibers must be tightened to find a solution. Finckh [START_REF] Finckh | Numerical simulation of mechanical properties of fabrics -weaving / numerische simulation der mechanischen eigenschaften textiler flächengebilde -gewebeherstellung[END_REF] proposed to simulate the weaving process and to apply dynamic loading cases using an explicit solver.

The present work [START_REF] Durville | Numerical simulation of entangled materials mechanical properties[END_REF][START_REF] Durville | Recent Advances in Textile Membranes and Inflatable Structures, chap. Finite element simulation of the mechanical behaviour of textile composites at the mesoscopic scale of individual fibers[END_REF] is based on the use of an implicit solver, in accordance with the quasi-static framework in which the problem is set. Compared to an explicit solvers, an implicit offers a good control of the quality of the solution, but needs efficient numerical algorithms to solve large loading increments and save computation time.

To introduce the global approach, a first section is dedicated to the beam model employed to represent fibers. The contact modeling, including geometrical and mechanical aspects, is then described. The application of boundary conditions to the ends of tows and fibers, using rigid bodies and average binding conditions are presented in the next section. The simulation of the weaving process is explained and results on two samples with different weave architectures are shown. The initial configuration of these Fig. 1 The three kinematical vectors used to define the placement in the kinematical beam model samples is first computed by simulating the weaving process, and biaxial and shear tests are then simulated to identify their mechanical properties.

3D finite strain beam model

Introduction

Beam models are usually based on the assumption that beam cross-sections remain rigid. The motion of cross-sections is described by two vector fields (six degrees of freedom) : one translation for the center of the section, and one rotation. These models present two main disadvantages : the handling of large rotations requires complex formulations, and they do not take any account of possible deformations of crosssections. To overcome these limits, an enriched kinematical model, based on three vector fields and nine degrees of freedom was developed.

Enriched kinematical beam model

The enriched kinematical beam model is based on a first order Taylor expansion of the placement of any material particle of the beam with respect to the line of centroids. Denoting ξ(ξ 1 , ξ 2 , ξ 3 ) a material particle defined in the reference configuration (see Fig. 1), a first order expansion of its placement x(ξ) can be expressed as follows :

x(ξ) = x(0, 0, ξ 3 ) + ξ 1 ∂x ∂ξ 1 (0, 0, ξ 3 ) + ξ 2 ∂x ∂ξ 2 (0, 0, ξ 3 ) + o(ξ 1 , ξ 2 ). (1)
Denoting respectively x 0 , g 1 and g 2 the three vectors fields introduced in the above expansion, which depend only on the curvilinear abscissa ξ 3 of the particle, a kinematical model can be based on the following expression of the placement of any particle :

x(ξ) = x 0 (ξ 3 ) + ξ 1 g 1 (ξ 3 ) + ξ 2 g 2 (ξ 3 ). ( 2
)
In this model, three vectors are used to describe any cross-section of the beam : the vector x 0 which defines the position of its center, and two other vectors, g 1 and g 2 , called section vectors, which determine the orientation of the section. According to this expression for the placement, the displacement u(ξ) of any particle is derived as follows :

u(ξ) = u 0 (ξ 3 ) + ξ 1 h 1 (ξ 3 ) + ξ 2 h 2 (ξ 3 ). (3)
With this kinematical model based on three vector fields, beam cross-sections are assumed to remain plane. However, since the norm of section vectors and the angle they form can vary, planar deformations of cross-sections can be considered.

Strains and constitutive law for the beam model

With this model, because of the consideration of planar deformations of cross-sections, no term of the Green-Lagrange strain tensor is constrained to be zero, and fully threedimensional effects, such as Poisson's effect, are taken into account. The fact that the strain tensor is fully three-dimensional allows to use usual three-dimensional constitutive laws, taking into account strains in all directions.

3 Contact and friction modeling

Introduction

The high number of contacts involved in the considered collections of fibers makes the detection and modeling of contact-friction interactions a crucial issue of the problem. The taking into account of contact-friction interactions requires first to detect contacts which are occuring within the collection of beams and which are continuously evolving due to the large relative displacements allowed between fibers. Once contacts have been detected, interactions forces developed in normal and tangential directions need to be introduced by means of appropriate contact and friction models, allowing the use of efficient algorithms.

Geometrical detection of contact

Because of the continuous evolution of contact during the loading, the contact detection has to be repeated during the solving of the problem, and the process to detect contact needs therefore to be efficient in order not to consume too much computational time.

Contact configurations between fibers can be very different depending on their relative positions. Very localized and reduced contact areas can be observed at crossings between fibers of two different tows, whereas continuous contact lines may appear between two neighboring fibers of the same tow.

Methods to detect contact between 3D beams can be found in literature [START_REF] Wriggers | On contact between three-dimensional beams undergoing large deflections[END_REF][START_REF] Zavarise | Contact with friction between beams in 3-d space[END_REF]. They often lie on the search of nearest distances between beams, which may consume much computation time when a high number of fibers is considered. In order to consider the various possible configurations between fibers, even with a high number of fibers, we propose a method based on the determination, in regions where contact is likely to occur, of discrete contact elements, constituted by pairs of material particles predicted to enter into contact (see Fig. 2). A contact element, denoted E ij c (x G ), is defined by the pair of material particles, ξ (i) and ξ (j) , belonging to two different beams i and j, that are predicted to enter into contact at x G :

E ij c (x G ) = ξ (i) , ξ (j) such that x(ξ (i) ) = x(ξ (j) ) = x G .
The process of determination of contact elements follows three successive stages to determine these contact elements. First, regions where contact is likely to occur, and which are called proximity zones, are looked for. A proximity zone is defined as two parts of beam centroidal lines which are stated to be close enough (Fig. 3). These proximity zones are obtained for each pair of beams in the collection, using a coarse discretization on one of the beam, and searching for the lowest distance to the other beam.

Once zones of proximity have been determined, an intermediate geometry is defined for each zone, as the average between the two parts of lines close to each other (Fig. 3). The purpose of this intermediate geometry is to approximate the geometry of the contact area. Contact elements are the defined at discrete points regularly distributed along this intermediate geometry, by looking for the two material particles on the surface of beams which are likely to enter into contact at these discrete positions. To do this, a plane orthogonal to the intermediate geometry is used to determine two cross-sections whose centers are located at the interesection between these planes and the centroidal lines (Fig. 4). Particles candidate to contact are then accurately located on the edge of the cross-sections using geometrical criteria (Fig. 5).

Formulation of contact conditions

In order to formulate linearized contact conditions, a normal direction, denoted N(E ij c ), is associated to each contact element E ij c . This direction sets the direction according to which distance between interacting particles will be measured (Fig. 7). This contact direction is generally taken as the one between the two centers of cross-sections involved in contact.

In order to prevent interpenetration between beams, kinematical conditions are set for all contact elements. These conditions prescribe that the distance between particles of a contact element, measured according to the normal contact direction, must remain positive. This distance, defined as the gap, is expressed for each contact element as follows :

gap

E ij c (x G ) = x(ξ (i) ) -x(ξ (j) ), N(E ij c ) ≥ 0.

Constitutive laws for contact and friction

A constitutive law for normal contact is considered in order to link normal reactions to the gaps measured at contact elements. This constitutive law has the form of a penalty law, and is regularized by a quadratic part for very small penetrations in order to stabilize the contact algorithm (see Fig. 8). The penalty coefficient is adjusted for each proximity zone in order to limit the maximum penetration for each zone. As far as the friction is concerned, we use a Coulomb's law including a small reversible elastic displacement before sliding occurs.

Global algorithm to solve the nonlinear problem

To solve the problem for each loading increment, imbricated loops are introduced to iterate on the different nonlinearities. For each increment, with a first level loop, iterations are made on the process of contact determination to generate contact elements. Then, these contact elements being fixed, iterations are made at a second level on the normal contact directions. The third level loop is dedicated to iterations of a Newton-Raphson algorithm on all other nonlinearities (contact status, friction direction, finite strains). Because of these three embedded loops, the total number of iterations may be high if a good rate of convergence is not obtained for the Newton-Raphson algorithm. Models characteristics need to be carefully adjusted for this purpose.

4 Driving of boundary conditions

Introduction of rigid bodies

As the considered samples of woven fabrics are made of components of different levels (tows and fibers), the application of boundary conditions to the edges of the sample is a delicate issue. The difficulty is that we want to apply globally loadings to each component (edge or end of tow), while allowing the subcomponents (tows for the edges, and fibers for tows) the possibility to move and rearrange, in order not to introduce artificial stiffnesses at the boundaries. In fact, if ends of tows and fibers were fixed respectively to the edges an the tows, this would rigidify the boundaries of the sample and would modify the global solution of the problem.

The goal of our boundary conditions is to distribute loads applied globally to the edges, to the subcomponents attached to these edges (respectively tows attached to edges, and fibers attached to tows) while allowing these subcomponents the possibility to move and rearrange.

To meet this objective, rigid bodies are introduced and attached to each edge and to each end of tow. These rigid bodies are used to apply proper boundary conditions to each edge and each end of tow. Boundary conditions to be applied to subcomponents are then defined with respect to these rigid bodies. Standard conditions can be used to prescribe a displacement or a force to each individual subcomponent. However, in order to allow subcomponents to move and rearrange, new conditions are set in average for all subcomponents attached to the same component. This way, it is possible to prescribe that all ends of tows follow in average the motion of a edge, or that the averaged motion of all ends of fibers corresponds to that of the tow. This combination of rigid bodies for components at different levels, and of averaged binding conditions provides a versatile way of driving boundary conditions, enabling to apply globally forces or displacements to the components of different levels, while allowing them the possibility to rearrange.

Formulation of averaged binding conditions to the rigid bodies

Averaged conditions are aimed at binding globally the average displacements of a set of subcomponents to those of the moving rigid body they are attached to. As the motion of a rigid body is defined by six degrees of freedom (three translations and three rotations), six averaged binding conditions have to be set in order to control the average displacement and the average rotation of subcomponents. Let the rigid body be defined by the position of its center, R n and its three moving directions (D n i ) at step n, and the set of N subcomponents be defined by the positions of corresponding nodes (x n 1 , ..., x n N ). The condition on the average displacement means that the displacement of the barycenter of nodes of subcomponents to bind to the rigid body must be the same as the displacement of the rigid body, with respect to the moving frame attached to this rigid body, and is expressed as follows :

1 N N λ=1 x n λ -R n , D n i = 1 N N λ=1 x n-1 λ -R n-1 , D n-1 i , ∀i ∈ (1, 2, 3). ( 4
)
In order to prevent the rotation of the set of subcomponents around the direction D i of the rigid body, a condition is prescribed for the average of orthoradial displacements defined with respect to this direction. A radial direction d n-1 rad,λ and an orthoradial direction d n-1 ortho,λ are defined in the previous configuration for each subcomponent in the following way :

d n-1 rad,λ = x n-1 λ -R n-1 x n-1 λ -R n-1 d n-1 ortho,λ = D n-1 i × d n-1 rad,λ D n-1 i × d n-1 rad,λ
The orthoradial direction d n ortho,λ in the current configuration is then obtained by a coordinate system transformation :

d n ortho,λ = d n-1 ortho,λ , D n-1 i D n i . (5)
The global condition on the average of orthoradial displacement around direction D i is then expressed as follows :

1

N N λ=1 x n λ -x n-1 λ , d n ortho,λ = 0. ( 6
)
Equations 4 and 6 are of the same kind. They link together degrees of freedom of subcomponents and of the master rigid body by the means of a quadratic condition involving products of displacement field component. Thanks to these conditions, displacements of bond subcomponents are globally controled whereas each component remains free to move individually with respect to the whole set. The trajectories of individual fibers in a piece of fabric can not be known a priori, but need to be computed by simulating in some way the weaving process. This task is performed starting from an arbitrary configuration where all tows are laying on the same plane, and are assumed to be straight and to have elliptical cross-sections. In this configuration (Fig. 9) tows are interpenetrating each other at crossings. According to the chosen weave pattern, it is possible to determine which tow must be below or above the other at each crossing. The weaving process is simulated by making fibers from different tows fulfil progressively the superimposing order defined by the weave pattern. For this purpose, during steps dedicated to the simulation of the weaving, the normal contact direction for contacts between fibers of different tows is oriented according to the order prescribed by the weave pattern at a given crossing. By this way, fibers from different tows are step-by-step separated until there is no more interpenetration between tows. Usual normal contact directions are then applied for all contacts to carry out a relaxation step in order to obtain the equilibrium configuration of the initial woven structure.

Computation of the initial configuration for a plain weave and a twill weave

To illustrate the simulation of the weaving process, two examples are presented here.

The two examples use the same initial arrangement of ten tows, but two different weave architectures are considered to compute the initial configuration of the woven fabrics. The main data concerning these examples are gathered in Tab. 1. Each tow is made of 48 fibers, and the whole sample consists of 480 fibers.

The configurations obtained after the simulation of the weaving process are shown in Fig. 10 and 11 for the plain weave and the twill weave respectively. Interesting details can be viewed on Fig. 12 steps during the weaving process. The evolution of fibers trajectories, the changes in cross-sections shapes and the amplitude of shrinkage can be seen on these figures. The evolution of the tows cross-sections shape is interesting to notice : while these crosssections were initially elliptical, they take a lenticular shape in the plain weave, and various elongated shapes in the twill weave. In Fig. 14, which shows different slices of the same tow from the twill weave, the various shapes of cross-sections can be observed. This demonstrates the ability of the model to allow the fibers to rearrange in function of the loads applied to them.

The rearrangement of fibers is even allowed at the end of tows (Fig. 15) where boundary conditions are applied. This proves the effectiveness of the binding of ends of fibers to rigid bodies by means of averaged conditions : loads and displacements are globally prescribed to the barycenter of the fiber ends of the group, but they keep the possibility to move and rearrange.

Simulation of loading cases

Various loading cases can be simulated by applying different loads (displacements or forces) to the edges of the woven samples after simulating the weaving process. Two different ratios between the elongations applied in warp and weft directions were used : a ratio α = 1 for equibiaxial tensile tests, and a ratio α = 0.5. Loadings curves for the two ratios are plotted on Fig. 16 for the plain weave and on Fig. 17 for the twill weave. The loading curves exhibit the usual J-shape aspect at the beginning. This nonlinear effect is certainly due to the filling of free gaps between fibers as the tensile load increases, until the arrangement of fibers within tows reaches a maximum density.

Shear tests

To simulate a shear tests, opposite horizontal displacements have been applied to the top and bottom edges of the samples, while lateral edges were left free to rotate around 18 and 19. The shear loading curves (Fig. 20) show similar behaviours for the two samples at the begining of the loading, while the plain weave becomes stiffer as the shear angle increases. This difference between the two weave architectures may be due to the fact that the locking between tows is more severe for the plain weave than for the twill weave. A general approach to the mechanical behaviour of woven fabrics by means of finite element simulation has been presented in this paper. This approach, formulated within a large deformations framework, is based on the representation of all fibers constituting woven structures by means of 3D beam models, and on the taking into account of contact-friction interactions between fibers.

The detection of the numerous contacts occurring within collections of fibers is an essential point of the approach. Based on the determination of proximity zones between fibers, and on the construction of intermediate geometries to approximate the actual contact zones, the presented method generates automatically contact elements made of pairs of material particles which are candidates to contact. The process of determination of contact elements has to be repeated as the solution evolves. It is sufficiently general to apply to very different configurations between fibers, from parallel to crossings fibers.

Robust models and efficient algorithms have been developed to ensure a good convergence for the solution of the global nonlinear problem. These optimized models and algorithms allow to consider samples of woven fabrics made of few hundreds of fibers and involving up to 100 000 contact elements, with a reasonable CPU time.

The ability of the method is demonstrated first by the determination of the initial configuration of woven fabrics by simulating the weaving process. Useful informations about the trajectories and cross-section shapes of tows, and about the trajectories and curvatures of fibers, are obtained as results from this initial simulation. The simulation of various loading cases in order to identify the mechanical properties of fabrics shows that the model can consider large deformations.

The proposed modeling, based on the identification of very few parameters (mechanical properties of fibers, definition of tows arrangement and of the weave pattern), appears as a suited tool to describe and to understand complex phenomena occurring at the scale of fibers, and to predict the complex global mechanical behavior of textile materials at a macroscopic scale. This kind of simulation may be very useful to study the influence on different design parameters on the mechanical behaviour of textile fabrics.

The approach can also be applied to different mechanical problems involving assemblies of fibers. It should be helpful to study the mechanical behaviour of individual tows, or even to simulate the spinning of fibers to form yarns.

Fig. 2 Fig. 3

 23 Fig. 2 Definition of a contact element

Fig. 4 Fig. 6

 46 Fig. 4 Determination of couples of cross-sections candidates to contact

Fig. 7

 7 Fig. 7 Definition of a normal direction for a contact element

Fig. 8

 8 Fig. 8 Models for normal and tangential behaviour

Fig. 9

 9 Fig. 9 Initial configuration of tows before weaving

Fig. 10

 10 Fig. 10 Configuration after weaving for the plain weave sample

Fig. 11

 11 Fig. 11 Configuration after weaving for the twill weave sample

Fig. 12

 12 Fig. 12 Slices at successive steps during the simulation of weaving for the plain weave

Fig. 13

 13 Fig. 13 Slices at successive steps during the simulation of weaving for the twill weave

Fig. 14

 14 Fig. 14 Successive slices on the same tow of the twill weave

Fig. 16 Fig. 17

 1617 Fig. 16 Loading curves for the biaxial tensile test on the plain weave

Fig. 18

 18 Fig. 18 Configuration at the end of the shear test for the plain weave

Fig. 20

 20 Fig. 20 Loading curves for the shear test

Table 1

 1 and 13 which represent slices of the samples at successive Characteristic figures of the simulations

	No. of warp tows	5
	No. of weft tows	5
	No. of fibers per tow	48
	Total no. of fibers	480
	No. of nodes	38 880
	No. of dofs	≈ 350 000
	No. of contact elements	≈ 80.000


Metod Saniga

Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits

Keywords: 51Exx, 81R99 PACS Numbers: 02.10.Ox, 02.40.Dr, 03.65.Ca Finite Projective Spaces, Spreads of Lines, Pauli Groups of N -Qubits

Given a (2N -1)-dimensional projective space over GF(2), PG(2N -1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG(N -1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG(N -1, 4). Under such mapping, a non-degenerate quadric surface of the former space has for its image a non-singular Hermitian variety in the latter space, this quadric being hyperbolic or elliptic in dependence on N being even or odd, respectively. We employ this property to show that generalized Pauli groups of N -qubits also form two distinct families according to the parity of N and to put the role of symmetric Pauli operators into a new perspective. The N = 4 case is taken to illustrate the issue, due to its link with the so-called black-hole/qubit correspondence.

Multiple qubit states play a key role in various fields of quantum information theory like quantum computing, coding and quantum error-correction (see, e. g., [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]). Recently, and rather surprisingly, they have also been recognized to be of great relevance for getting insights into the nature of entropy formulas of a certain class of stringy black hole solutions (see, e. g., [START_REF] Borsten | Black holes, qubits and octonions[END_REF]). It is, therefore, important to deepen our understanding of these fundamental buildings blocks of quantum world. In the present note we do so through the geometry of their associated generalized Pauli groups.

Let PG(d, q) be a d-dimensional projective space over GF(q), q being a power of a prime.1 A t-spread S of PG(d, q) is a set of t-dimensional subspaces of PG(d, q) which partitions its point-set [START_REF] Eisfeld | Partial) t-spreads and minimal tcovers in finite projective spaces[END_REF]. If the elements of S in a subspace V form a t-spread on V , one says that S induces a t-spread on V . A t-spread S is called geometric (or normal) if it induces a t-spread on each (2t + 1)-dimensional subspaces of PG(d, q) spanned by a pair of its elements. It is a well-known fact that PG(d, q) possesses a t-spread iff (t + 1)|(d + 1); moreover, this condition is also sufficient for PG(d, q) to have a geometric t-spread. B. Segre showed [START_REF] Segre | Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane[END_REF] that a geometric t-spread of PG(N (t + 1) -1, q), N ≥ 2, gives rise to a projective space PG(N -1, q t+1 ) as follows: the points of this space are the elements of S and its lines are the (2t + 1)-dimensional subspaces spanned by any two distinct elements of S, with incidence inherited from PG(N (t+1)-1, q). For a particular case of t = 1 (i. e., a spread of lines), Dye [START_REF] Dye | Maximal subgroups of finite orthogonal groups stabilizing spreads of lines[END_REF] demonstrated that a hyperbolic or an elliptic quadric of PG(2N -1, q) has an induced (geometric) spread of lines if and only if N is, respectively, even or odd, in which case it is mapped onto a non-singular Hermitian variety H(N -1, q 2 ) of PG(N -1, q 2 ). We shall now show that this property has for q = 2 a very interesting physical implication.

It is already a firmly established fact [START_REF] Saniga | Multiple qubits as symplectic polar spaces of order two[END_REF][START_REF] Havlicek | Factor-group-generated polar spaces and (multi-)qudits[END_REF][START_REF] Thas | The geometry of generalized Pauli operators of N-qudit Hilbert space, and an application to MUBs[END_REF][START_REF] Vrana | The Veldkamp space of multiple qubits[END_REF] that the commutation relations between the elements of the generalized Pauli group of N -qubits, N ≥ 2, can be completely reformulated in the geometrical language of symplectic polar space of rank N and order two, W(2N -1, 2); the generalized Pauli operators (discarding the identity) answer to the points of W(2N -1, 2), a maximally commuting subset has its representative in a maximal totally isotropic subspace of W(2N -1, 2) and commuting translates into collinear. One of the most natural representations of W(2N -1, 2) is that in terms of the points and the set of totally isotropic subspaces of PG(2N -1, 2) endowed with a symplectic polarity. Employing this representation, it has been found in [START_REF] Havlicek | Factor-group-generated polar spaces and (multi-)qudits[END_REF] that in the real case the symmetric elements/operators of the N -qubit Pauli group always lie on a hyperbolic quadric in the ambient space PG(2N -1, 2). Combining this fact with Dye's result, we arrive at our main observation: it is only for N even when all symmetric generalized Pauli operators of W(2N -1, 2) can be mapped to the points of an Hermitian variety of the space PG(N -1, 4) associated through a geometric spread of lines with the ambient space PG(2N -1, 2). Hence, in this regard, when it comes to generalized Pauli groups 'even-numbered' multi-qubits are found to stand on a slightly different footing than 'odd-numbered' ones.

We shall finish this communication by briefly mentioning an especially interesting even case, N = 4. Here, a hyperbolic quadric Q + (7, 2) of PG [START_REF] Saniga | Multiple qubits as symplectic polar spaces of order two[END_REF][START_REF] Borsten | Black holes, qubits and octonions[END_REF] formed by the symmetric operators is well known for its puzzling triality swapping points and two systems of generators and has for its spread-induced image an Hermitian surface H [START_REF] Hirschfeld | General Galois Geometries[END_REF][START_REF] Eisfeld | Partial) t-spreads and minimal tcovers in finite projective spaces[END_REF] of PG(3, 4) (see, e. g., [START_REF] Havlicek | On Invariant Notions of Segre Varieties in Binary Projective Spaces[END_REF]). This Hermitian surface is, in turn, nothing but the generalized quadrangle GQ(4, 2) in disguise (see, e. g., [START_REF] Payne | Finite Generalized Quadrangles[END_REF]), the dual of which -GQ(2, 4) -was found to play a prominent role in the so-called black-hole-qubit correspondence, by fully encoding the E 6(6) symmetric entropy formula describing black holes and black strings in D = 5 [START_REF] Lévay | Black hole entropy and finite geometry[END_REF]. Our finding thus, inter alia, not only opens up an unexpected window through which also four-qubit Pauli group, like its lower rank cousins, could find its way into some black hole entropy formula(s), but also puts the role of symmetric operators into a new perspective. It is also important to keep in mind this remarkable three-to-one correspondence, i. e., that it is always a triple of (collinear) operators of the ambient space PG(7, 2) which comprises a single point of PG [START_REF] Hirschfeld | General Galois Geometries[END_REF][START_REF] Eisfeld | Partial) t-spreads and minimal tcovers in finite projective spaces[END_REF].

For the standard mathematical nomenclature and notation employed in what follows, see, e. g.,[START_REF] Hirschfeld | General Galois Geometries[END_REF].

Acknowledgements

This work was partially supported by the VEGA grant agency, projects Nos. 2/0092/09 and 2/0098/10. The idea exposed in this paper originated from discussions with Prof. Hans Havlicek, Dr. Boris Odehnal (Vienna University of Technology) and Dr. Petr Pracna (J. Heyrovský Institute of Physical Chemistry, Prague).