Gabriele Mancini
email: gabriele.mancini@math.unipd.it

Pierre-Damien Thizy
email: pierre-damien.thizy@univ-lyon1.fr

NON-EXISTENCE OF EXTREMALS FOR THE ADIMURTHI-DRUET INEQUALITY

Keywords: Mathematics Subject Classification. 35B33, 35B44, 35J15, 35J61

] that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi-Druet inequality does not admit any extremal, when the perturbation parameter α approaches λ 1 . Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler-Lagrange equation, which take into account the fact that the problem becomes singular as α → λ 1 .

Introduction

Let Ω be a smooth bounded domain of R 2 . We let H 1 0 be the usual Sobolev and Hilbert space of functions in Ω, endowed with the scalar product u, v H 1 0 = Ω ∇u.∇v dy , and with the associated norm denoted by • H 1 0 . For all α ≥ 0, we let C α (Ω) be given by C α (Ω) = sup {u∈H 1 0 s.t. ∇u 2 =1} Ω exp 4πu(y) 2 1 + α u 2 2 dy .

(1.1)

Then, the Adimurthi-Druet [START_REF] Adimurthi | Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality[END_REF] inequality claims that

C α (Ω) < +∞ ⇔ α < λ 1 ,
where λ 1 > 0 is the first eigenvalue of ∆ = -∂ xx -∂ yy in Ω with zero Dirichlet boundary condition on ∂Ω.

While the existence of an extremal function for α = 0, i.e. for the standard Moser-Trudinger inequality, was obtained by Carleson-Chang [START_REF] Carleson | On the existence of an extremal function for an inequality of J. Moser[END_REF], Struwe [START_REF] Struwe | Critical points of embeddings of H 1,n 0 into Orlicz spaces[END_REF] and Flucher [START_REF] Flucher | Extremal functions for the Trudinger-Moser inequality in 2 dimensions[END_REF], Yang and Lu [START_REF] Lu | Sharp constant and extremal function for the improved Moser-Trudinger inequality involving L p norm in two dimension[END_REF] were able to prove that there exists an extremal function for (1.1) for all α ≥ 0 sufficiently close to 0. More recently, still concerning the original Adimurthi-Druet inequality (1.1), it was explained in Yang [START_REF] Yang | Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two[END_REF] that the existence of extremals for more general α's closer to λ 1 is left open. We prove here that, surprisingly, there is no extremal function for (1.1) for all α < λ 1 sufficiently close to λ 1 . Then, our main result is stated as follows.

Theorem 1.1 (Non existence of extremals). Let Ω be a smooth, bounded and connected domain of R 2 . Let λ 1 > 0 be the first eigenvalue of ∆ with zero Dirichlet boundary condition. Then there exists α 0 ∈ (0, λ 1 ) such that, for all α ∈ [α 0 , λ 1 ), there is no extremal function for (1.1).

The proof of Theorem 1.1 relies on the recent progresses concerning the blow-up analysis of Moser-Trudinger equations (see [START_REF] Druet | Multi-bumps analysis for Trudinger-Moser nonlinearities I-Quantification and location of concentration points[END_REF][START_REF] Malchiodi | Critical points of the Moser-Trudinger functional on a disk[END_REF]). The difficulty in this problem is a cancellation of the first terms in the Dirichlet energy expansions of Section 2, which enforces to carry out in Section 3 a very precise blow-up analysis. For instance, the estimates obtained in [START_REF] Adimurthi | Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality[END_REF] and [START_REF] Lu | Sharp constant and extremal function for the improved Moser-Trudinger inequality involving L p norm in two dimension[END_REF] are far from being sufficient to conclude here. Note that a similar cancellation was already observed by Martinazzi-Mancini [START_REF] Mancini | The Moser-Trudinger inequality and its extremals on a disk via energy estimates[END_REF] in the radial case, namely when Ω is the unit disk D 2 of R 2 . Even in this more particular case, the authors had to carry out a very careful blow-up analysis of the next lower order terms in order to conclude. To be able to deal with the general (non necessarily radial) situation, we use here the techniques developed in Druet-Thizy [START_REF] Druet | Multi-bumps analysis for Trudinger-Moser nonlinearities I-Quantification and location of concentration points[END_REF]. But, a new additional serious difficulty here is that the problem becomes singular when α gets close to λ 1 . By singular, we mean here that the kernel of the operator obtained by linearizing the limiting equation at 0 does not only contain the zero function. Here (see (2.4), (2.6) and (2.9)), this operator is ∆ -λ 1 with zero Dirichlet boundary condition and we have to compute carefully (see Step 3.4) what happens in its kernel.

As already observed by Del Pino-Musso-Ruf [START_REF] Manuel Del Pino | New solutions for Trudinger-Moser critical equations in R 2[END_REF] in the non-singular case, the critical exponential non-linearity exp(u 2 ) in dimension 2 is more difficult to handle than the Sobolev critical non-linearity u n+2 n-2 in higher dimensions n > 2, and getting sharp energy expansions of positive blow-up solutions reveals to be delicate in this case. Besides, even for Sobolev critical problems in higher dimensions, understanding the behavior of positive blow-up solutions turned out to be very challenging in the singular case. This difficulty was overcome while solving Lin-Ni's conjecture (see Druet-Robert-Wei [START_REF] Druet | The Lin-Ni's problem for mean convex domains[END_REF], Rey-Wei [START_REF] Rey | Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity[END_REF] and Wei-Xu-Yang [START_REF] Wei | On Lin-Ni's conjecture in dimensions four and six[END_REF] and the references therein), where the limiting linearized operator is ∆ with zero Neumann boundary condition, whose kernel is the set of the constant functions.

As far as we know, Theorem 1.1 is the first result proving the non-existence of extremals for an explicit Moser-Trudinger type inequality with critical exponent on bounded domains. Indeed, similar results had so far been proven only for implicit perturbations of the Moser-Trudinger inequality [START_REF] Pruss | Nonexistence of maxima for perturbations of some inequalities with critical growth[END_REF], or for sub-critical inequalities on R 2 [START_REF] Ishiwata | Existence and nonexistence of maximizers for variational problems associated with Trudinger-Moser type inequalities in R N[END_REF], where blow-up of maximizing sequences cannot occur.

The paper is organized as follows. Theorem 1.1 is proved in Section 2. This proof relies on the key energy estimates of Proposition 3.1, whose proof is given in Section 3.

Proof of Theorem 1.1

Assume by contradiction that there exists a sequence (α i ) i such that α i → λ - 1 and such that there exists an extremal function u αi ≥ 0 for C αi (Ω). For simplicity, we drop the indexes i's. Then the u α 's satisfy

           ∆u α = A α u α + 2β α λ α u α exp(β α u 2 α ) in Ω , u α = 0 in ∂Ω , ∇u α 2 2 = 1 , β α = 4π 1 + α u α 2 2 , A α = α 1+2α uα 2 2 < λ 1 , (2.1)
for some positive λ α 's, and in particular, the u α 's are smooth. Indeed, the Moser-Trundinger inequality gives that

u ∈ H 1 0 =⇒ exp(u 2 ) ∈ L p ,
u α ⇀ 0 in H 1 0 , u α → 0 in L p for all p < +∞ , u α L ∞ → +∞ (2.3)
and thus that

β α → 4π and A α → λ - 1 , (2.4
) as α → λ - 1 . Now we rephrase everything in terms of v α := β α u α .

(2.5)

We have that

       ∆v α = v α A α + Λ α exp(v 2 α ) in Ω , v α = 0 in ∂Ω , β α = 4π 1 + α βα v α 2 2 , A α = α 1+2 α βα vα 2 2 (< α < λ 1 ) , (2.6)
where

Λ α = 2β α λ α > 0. Moreover, ∇u α 2 2 = 1 implies ∇v α 2 2 = β α .
(2.7)

We also get that v α 2 → 0 as α → λ - 1 and the second line of (2.6) implies

β α = 2π   1 + 1 + α Ω v 2 α dy π   = 4π 1 + α Ω v 2 α dy 4π - α 2 Ω v 2 α dy 2 16π 2 + O Ω v 2 α dy 3 .
(2.8)

Now, we have that

Ω exp(v 2 α )dy = Ω exp β α u 2 α dy = C α (Ω) → +∞ ,
and, independently, that

Λ α Ω v 2 α exp(v 2 α )dy = Ω |∇v α | 2 dy -A α Ω v 2 α dy = 4π + o(1) ,
so that there must be the case that

Λ α → 0 (2.9)
as α → λ - 1 , since e t ≤ 1 + te t for t ≥ 0. We are now in position to use Proposition 3.1 below: we have that

∇v α 2 2 = 4π 1 + A α 4π Ω v 2 α dy + o Ω v 2 α dy 2 (2.10) as α → λ - 1 .
Then, expanding the third line of (2.6), we get

A α = α - α 2 Ω v 2 α dy 2π + O Ω v 2 α dy 2 .
(2.11)

Now, (2.10) and (2.11) give

β α = 4π 1 + α 4π Ω v 2 α dy - α 2 8π 2 Ω v 2 α dy 2 + o Ω v 2 α dy 2 .
(2.12) But (2.8) and (2.12) have to match, then we get

- λ 2 1 16π 2 Ω v 2 α dy 2 = o Ω v 2 α dy 2 , (2.13)
which is the contradiction we look for.

3. Blow-up analysis on (2.6) Proposition 3.1. Let (v α ) α be a sequence of smooth solutions of

∆v α = v α A α + Λ α exp(v 2 α ) , v α > 0 in Ω , v α = 0 in ∂Ω , (3.1)
for A α ∈ [0, λ 1 ) and Λ α > 0, for all α slightly smaller than λ 1 . We let β α > 0 be given by (2.7) and we assume that (2.4) and (2.9) hold true. We also assume that the v α 's blow-up, namely that

γ α := max Ω v α = v α (x α ) → +∞ , (3.2)
as α → λ - 1 , for x α ∈ Ω. Then, we have that

γ 2 α Ω v 2 α dy → +∞ , (3.3)
that

Λ α = o 1 γ 2 α (3.4)
and that (2.10) hold true as α → λ - 1 . Note that (3.3) and (3.4) (proved in Step 3.4) are specific to our singular case A α → λ - 1 : they would not hold true if the limit of the A α 's were in [0, λ 1 ). Now we turn to the proof of this result. In order to prove Proposition 3.1, we study the asymptotic behavior of the v α 's as α → λ - 1 . We make the assumptions of Proposition 3.1. First, by these assumptions on (Λ α ) α and (A α ) α , the family (f α ) α of functions, given by

f α (t) = t(A α + Λ α exp(t 2 )) ,
is of uniform critical growth in the sense of [START_REF] Druet | Multibumps analysis in dimension 2: quantification of blow-up levels[END_REF]Definition 1]. Also, as in [START_REF] Druet | Multibumps analysis in dimension 2: quantification of blow-up levels[END_REF] (see also the original argument in [START_REF] Adimurthi | Global compactness properties of semilinear elliptic equations with critical exponential growth[END_REF]), if µ α is given by

µ -2 α := Λ α 4 γ 2 α exp(γ 2 α ) → +∞ , (3.5)
then there exists a sequence of positive numbers

(R α ) α such that R α → +∞, R α µ α ≪ d(x α , ∂Ω), and
γ α (γ α -v α (x α + µ α •)) -T 0 C 2 (B0(Rα)) → 0 (3.6) as α → λ - 1 , where T 0 := log 1 + | • | 2 .
We recall that T 0 solves the Liouville equation

∆T 0 = 4 exp(-2T 0 ) (3.7) in R 2 . Note that (3.1), (3.2) and v α H 1 0 = O(1) imply that Λ α exp(γ 2 α ) → +∞ , (3.8)
as α → λ - 1 . Moreover, the PDE in (3.1) is autonomous and the f α 's are increasing in [0, +∞). Therefore, as pointed out in [START_REF] Adimurthi | Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality[END_REF], the arguments in de Figueiredo-Lions-Nussbaum [START_REF] De Figueiredo | A priori estimates and existence of positive solutions of semilinear elliptic equations[END_REF] and Han [START_REF] Han | Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent[END_REF] give that the x α 's do not go to the boundary of Ω.

Then, up to a subsequence,

x α → x (3.9) as α → λ - 1 , for some x ∈ Ω. Let B α be the radially symmetric solution around x α of ∆B α = B α A α + Λ α exp B 2 α , B α (x α ) = γ α . (3.10)
Let vα be given by

vα (z) = 1 2π|x α -z| ∂Bx α (|xα-z|) v α (y) dσ(y) (3.11)
for all z ∈ Ω\{x α } and vα (x α ) = v α (x α ). Also we let t α be given by

t α (y) = log 1 + |y -x α | 2 µ 2 α = T 0 y -x α µ α . (3.12)
By abuse of notations, we will write sometimes B α (r), t α (r) or vα (r) instead of B α (z), t α (z) or vα (z) respectively, for |z -x α | = r. For any δ ∈ (0, 1), we let r α,δ > 0 be given by

t α (r α,δ ) = δγ 2 α . (3.13) Observe that (3.13) implies r 2 α,δ = µ 2 α exp δγ 2 α + o(1) ≫ µ 2 α , (3.14)
as α → λ - 1 . At last, by [6, Proposition 2], there exists D 0 > 0 such that | • -x α ||∇v α |v α ≤ D 0 in Ω (3.15) for all 0 < λ 1 -α ≪ 1.
The first rather elementary step is as follows.

Step 3.1.

As α → λ - 1 , we have that |log Λ α | = o(γ 2 α ). (3.16)
Moreover, for all δ ∈ (0, 1) and all sequences (z α ) α of points

z α ∈ B xα (r α,δ ), we have that v α (z α ) ≥ γ α (1 -δ + o(1)), (3.17)
and in particular r α,δ < d(x α , ∂Ω).

Proof of Step 3.1. Let R 0 > 0 be such that Ω ⊂ B x(R 0 ). Let also Ω α be given by Ω α = B xα (R 0 )\B xα (µ α ).
We extend v α by 0 outside Ω. Let V α be the unique harmonic function in Ω α such that V α = v α in ∂Ω α . Then, by construction of V α , we know that

Ωα |∇V α | 2 dy ≤ Ωα |∇v α | 2 dy , (3.18)
for all α. Let now Ǎα > 0 be such that Ψ α := Ǎα log R0 |•-xα| and γ α -tα γα coincide on ∂B xα (µ α ). Then, we easily get from (3.5) that

Ǎα = γ 2 α -log 2 γ α log R0 µα = γ α (1 + o(1)) log 1 µα . (3.19)
By (3.6) and elliptic estimates, we get that

|∇V α -∇Ψ α | ≤ o 1 γ α | • -x α | in Ω α (3.20)
for all 0 < λ 1 -α ≪ 1. Then, we get from (3.19) and (3.20) that

Ωα |∇V α | 2 dy = π Ǎ2 α log 1 µ 2 α (1 + o(1)) = 4πγ 2 α (1 + o(1)) log 1
log 1 µ 2 α ≥ (1 + o(1))γ 2 α ,
which concludes the proof of (3.16) using also (2.9) and (3.5). Now we prove (3.17

Ãα = 2 + o(1) γ α . (3.22) But since 0 = ∆ Ψα ≤ ∆v α in Ωα := B xα ( δ)\B xα (µ α ) and since Ψα ≤ v α + o(γ -1 α ) in ∂ Ωα , we get from (3.

6) and the maximum principle that

Ψα ≤ v α + o(γ -1 α ) in Ωα . ( 3
Ψα (z α ) ≥ 1 + o(1) γ α log 1 r 2 α,δ = γ α (1 -δ + o(1)).
This concludes the proof of (3.17), in view of (3.6) and (3.23). Now, we fix δ ∈ (0, 1) and we expand B α up to a distance r α,δ of x α , as α → λ - 1 . As a consequence of Step 3.1, we expand B α up to a distance r α,δ of x α , as α → λ - 1 . Let S 0 be the radial solution around 0 ∈ R 2 of

∆S 0 -8 exp(-2T 0 )S 0 = 4 exp(-2T 0 ) T 2 0 -T 0 , (3.24)
such that S 0 (0) = 0. By [START_REF] Mancini | The Moser-Trudinger inequality and its extremals on a disk via energy estimates[END_REF], the explicit formula for S 0 is

S 0 (r) = -T 0 (r) + 2r 2 1 + r 2 - 1 2 T 0 (r) 2 + 1 -r 2 1 + r 2 1+r 2 1 log t 1 -t dt ,
and in particular,

S 0 (r) = A 0 4π log 1 r 2 + B 0 + O log(r) 2 r -2
where

A 0 = 4π, B 0 = π 2 6 + 2 , (3.25)
as r → +∞. Note that A 0 = R 2 (∆S 0 )dy. For 0 < λ 1 -α ≪ 1, we let S α be given by

S α (z) = S 0 z -x α µ α . (3.26)
Step 3.2. For all sequence (z α ) α such that z α ∈ B xα (r α,δ ), we have that

B α (z α ) = γ α - t α (z α ) γ α + S α (z α ) γ 3 α + O 1 + t α (z α ) γ 5 α , (3.27
)

for all 0 < λ 1 -α ≪ 1.
As a by-product of Step 3.2, B α is radially decreasing in B xα (r α,δ ).

Proof of Step 3.2. Let w 1,α be given by

B α = γ α - t α γ α + w 1,α γ 3 α , (3.28)
and let ρ 1,α > 0 be defined as

ρ 1,α = sup {r ∈ (0, r α,δ ] s.t. |S α -w 1,α | ≤ 1 + t α in [0, r]} . (3.29)
First, we give precise asymptotic expansions of ∆w 1,α in B xα (ρ 1,α ), as α → λ - 1 . We start by proving that the term A α B α is well controlled in B xα (ρ 1,α ), using Step 3.1. Indeed, (3.5), (3.13) and (3.16) give

exp(t α (-2 + (t α /γ 2 α ))) µ 2 α = exp log Λ α + o(γ 2 α ) exp γ α - t α γ α 2 ≥ exp (1 -δ) 2 γ 2 α + o(γ 2 α ) (3.30) in B xα (r α,δ ). Since B α > 0 in B xα (ρ 1,α ), we get from (3.10) that B α ≤ γ α in this ball. Then (3.30) implies A α B α ≤ λ 1 γ α = o exp(t α (-2 + (t α /γ 2 α ))) γ 5 α µ 2 α in B xα (ρ 1,α ) . ( 3
+ t α ) in B xα (ρ 1,α ).
In particular, from (3.28) we get

B α = γ α - t α γ α + O 1 + t α γ 3 α , (3.32)
and

B 2 α = γ 2 α -2t α + t 2 α + 2w 1,α γ 2 α + O 1 + t 2 α γ 4 α (3.33) in B xα (ρ 1,α ). Since t α = O(γ 2 α ) in B xα (r α,δ ), applying the useful inequality exp(x) - k-1 j=0 x j j! ≤ |x| k k! exp(|x|) ,
for all x ∈ R and all integer k ≥ 1, we obtain that

exp t 2 α + 2w 1,α γ 2 α + O 1 + t 2 α γ 4 α = 1 + t 2 α + 2w 1,α γ 2 α + O (1 + t 4 α ) exp(t 2 α /γ 2 α ) γ 4 α (3.34) in B xα (ρ 1,α
). Then, using (3.5), (3.32), (3.33), (3.34), we get that

Λ α B α exp(B 2 α ) = 4 exp(-2t α ) µ 2 α γ α 1 + 2w 1,α + t 2 α -t α γ 2 α + O (1 + t 4 α ) exp(t 2 α /γ 2 α ) γ 4 α (3.35) in B xα (ρ 1,α ). Now,
∆w 1,α = 4 exp(-2t α ) µ 2 α 2w 1,α + t 2 α -t α + O (1 + t 4 α ) exp(t 2 α /γ 2 α ) γ 2 α (3.36) in B xα (ρ 1,α ).
Next, we estimate the growth of the function w

(∆(w 1,α -S α )) dy = -2πr(w 1,α -S α ) ′ (r) , (3.38)
and, from (3.24) and (3.36), that

∆ (w 1,α -S α ) = 8 exp(-2t α ) µ 2 α (w 1,α -S α ) + O (1 + t 4 α ) exp(t 2 α /γ 2 α ) γ 2 α , (3.39)
for all 0 ≤ r ≤ ρ 1,α . By (3.37), we get that

Bx α (r) 8 exp(-2t α + t 2 α /γ 2 α )(1 + t 4 α ) µ 2 α dy ≤ 8π κ -1 1 -(1 + (r/µ α ) 2 ) 1-κ , (3.40) and, since |(w 1,α -S α )(r)| ≤ (w 1,α -S α ) ′ L ∞ ([0,ρ1,α]) r, that Bx α (r) 8 exp(-2t α ) µ 2 α |w 1,α -S α | dy ≤ µ α h(r/µ α ) (w 1,α -S α ) ′ L ∞ ([0,ρ1,α]) , (3.41)
where

h(s) = 8π arctan s - s 1 + s 2 , s ≥ 0 .
Then, by (3.38), (3.39), (3.40), and (3.41), there exists a constant C ′ > 1 such that

r|(w 1,α -S α ) ′ (r)| C ′ ≤ (r/µ α ) 2 γ 2 α (1 + (r/µ α ) 2 ) + µ α (w 1,α -S α ) ′ L ∞ ([0,ρ1,α]) (r/µ α ) 3 1 + (r/µ α ) 3 (3.42)
for all 0 ≤ r ≤ ρ 1,α and all 0 < λ 1 -α ≪ 1. Now we prove that

µ α (w 1,α -S α ) ′ L ∞ ([0,ρ1,α]) = O 1 γ 2 α . (3.43)
Otherwise, we assume by contradiction that

γ 2 α µ α (w 1,α -S α ) ′ L ∞ ([0,ρ1,α]) = γ 2 α µ α |(w 1,α -S α ) ′ |(s α ) → +∞ (3.44) as α → λ - 1 , for s α ∈ (0, ρ 1,α ].
Up to a subsequence, we may assume that

ρ 1,α µ α → δ 0 (3.45)
as α → λ - 1 , for some δ 0 ∈ (0, +∞]. Note that (3.42) with (3.44) gives s α = O(µ α ), µ α = O(s α ) and then δ 0 > 0. Let wα be given by

wα (s) = (w 1,α -S α )(µ α s) µ α (w 1,α -S α ) ′ L ∞ ([0,ρ1,α])
, so that, by (3.42) and (3.44), there exists a constant C ′′ > 0 such that

| w′ α (s)| ≤ C ′′ 1 + s in [0, ρ 1,α /µ α ] , (3.46)
for all 0 < λ 1 -α ≪ 1. Then, by (3.39), (3.46) and elliptic theory, we get that there exists w such that

wα → w in C 1 loc (B 0 (δ 0 )) as α → λ - 1 , (3.47
Bx α (ρ1,α) exp(-2t α ) µ 2 α |w 1,α -S α |dy = o(µ α (w 1,α -S α ) ′ L ∞ ([0,ρ1,α]) ) . ( 3
r|(w 1,α -S α ) ′ (r)| = o µ α (w 1,α -S α ) ′ L ∞ ([0,ρ1,α]) (3.51)
for all 0 ≤ r ≤ ρ 1,α and as α → λ - 1 . But (3.51) is clearly not possible at s α . This concludes the proof of (3.43). Now, plugging (3.43) in (3.42), using that w 1,α (0) = S α (0) = 0 and the fundamental theorem of calculus, we get that

w 1,α -S α L ∞ ([0,ρ1,α]) = O 1 + t α γ 2 α as α → λ - 1 ,
which, in view of (3.29), gives ρ 1,α = r α,δ and concludes the proof of Step 3.2. Now, we compare the behavior of v α and B α in B xα (r α,δ ). Let κ be any fixed number in (0, 1). Let r α be given by

r α = sup r ∈ (0, r α,δ ] s.t. |v α -B α | ≤ κ γ α in B xα (r) . (3.52)
We get from (3.15) and (3.17) that

| • -x α ||∇v α | ≤ D 0 (1 -δ + o(1))γ α in B xα (r α,δ ) . (3.53)
Then letting w α be given by v α = B α + w α , (3.54) we get from (3.52) and (3.53) that

|w α | ≤ κ + D 0 π (1 -δ + o(1)) 1 γ α in B xα (r α ) . (3.55)
Then, we obtain from (3.1), (3.10) and (3.55) that there exists a constant Step 3.3. Let δ ∈ (0, 1) be given. Then we have that r α = r α,δ and, in other words,

D 1 > 0 such that |∆w α | ≤ λ 1 + D 1 1 + 2B 2 α exp(B 2 α ) |w α | , ≤ D 1 1 + 2B 2 α exp(B 2 α )(1 + o( 1
vα -B α L ∞ (Bx α (r α,δ )) = o 1 γ α , as α → λ - 1 .
Moreover, we have that

∇(v α -B α ) L ∞ (Bx α (r α,δ )) = O 1 γ α r α,δ
and then, there exists a constant C > 0 such that

|v α -B α | ≤ C | • -x α | γ α r α,δ in B xα (r α,δ ) . (3.57)
As a direct consequence of Steps 3.2 and 3.3 we get the asymptotic expansion

f α (v α ) = O(γ α ) + Λ α B α + O | • -x α | r α,δ γ α exp B 2 α + O | • -x α | r α,δ = O(γ α ) + Λ α B α exp(B 2 α ) 1 + O | • -x α | r α,δ .
Then, expanding as in (3.31) and (3.35), we find that

f α (v α ) = 4 exp(-2t α ) µ 2 α γ α 1 + 2S α + t 2 α -t α γ 2 α + O (1 + t 4 α ) γ 4 α exp t 2 α γ 2 α + 1 + t 2 α γ 2 α | • -x α | r α,δ (3.58)
in B xα (r α,δ ). Since δ < 1, we can argue as in (3.37) to estimate the exponential in the error term. Specifically, we can find κ > 1 such that

f α (v α ) = 4 exp(-2t α ) µ 2 α γ α 1 + 2S α + t 2 α -t α γ 2 α + O 1 + t 2 α γ 2 α | • -x α | r α,δ + O exp (-κt α ) µ 2 α γ 4 α . (3.59)
Similarly, we obtain

v α f α (v α ) = 4 exp(-2t α ) µ 2 α 1 + 2S α + t 2 α -2t α γ 2 α + O 1 + t 2 α γ 2 α | • -x α | r α,δ + O exp (-κt α ) µ 2 α γ 4 α . (3.60)
in B xα (r α,δ ). Now we focus on the behavior of the v α 's in Ω\B xα (r α,δ ). Assume that 0 < δ ′ < δ < 1. We let ṽα be given by

ṽα = v α in Ω\B xα (r α,δ ) , min (v α , (1 -δ ′ )γ α ) in B xα (r α,δ ) . (3.61) Note that v α < (1 -δ ′ )γ α in ∂B xα (r α,δ ) (3.62) by (3
.27) and (3.57). Then we have that ṽα ∈ H 1 0 and that v α = ṽα + ṽ1,α , where ṽ1,α := 1 Bx α (r α,δ ) (v α -(1 -δ ′ )γ α ) + and t + = max(t, 0). Now, by (3.62) and continuity, we have that ṽ1,α is zero in a neighborhood of ∂B xα (r α,δ ). Then, for any given R > 0, we can compute

Bx α (r α,δ ) |∇ṽ 1,α | 2 dy = Bx α (r α,δ ) ∇ṽ 1,α (y).∇v α dy = Bx α (r α,δ ) (∆v α )ṽ 1,α dy ≥ Bx α (Rµα) f α (v α )ṽ 1,α dy ≥ δ ′ (1 + o(1)) B0(R) 4 (1 + |z| 2 ) 2 dz
for 0 < λ 1 -α ≪ 1, since r α,δ /µ α → +∞ and using (3.5) and (3.6). Since R > 0 is arbitrary, we obtain ṽ1,α

H 1 0 ≥ 4πδ ′ (1 + o(1)).
Since (2.4) implies ṽα + ṽ1,α

2 H 1 0 = 4π + o(1)
, and since ṽ1,α and ṽα are H 1 0orthogonal, we get that

ṽα 2 H 1 0 ≤ 4π(1 -δ ′ + o(1)) . (3.63)
Moreover, since δ and δ ′ may be arbitrarily close to 1 in the above argument, we can check that, up to a subsequence, v α ⇀ 0 weakly in H 1 0 and then that

v α → 0 strongly in L p , (3.64)
for any p ≥ 1, as α → λ - 1 . Furthermore, by (3.63) and Moser's inequality, there exists p ′ > 1 such that (exp(ṽ 2 α )) α is bounded in L p ′ (Ω). Using (3.27) and (3.57), we can also check that ṽα = v α in Ω\B xα (r α,δ /2). Then, we get that

(exp(v 2 α )) α is bounded in L p ′ (Ω\B xα (r α,δ /2)) . (3.65)
From now on, we fix p ≥ 2 and r > 1 such that

1 p ′ + 1 p + 1 r = 1 . (3.66)
In the sequel, v is the unique function characterized by

     ∆v = λ 1 v , v > 0 in Ω , v = 0 in ∂Ω , v p = 1 . (3.67)
Step 3.4. For all sequence (z α ) α of points such that z α ∈ Ω\B xα (r α,δ ), we have that . First, integrating (3.59) and using the dominated convergence theorem, we get that

v α (z α ) = v α p v(z α ) + o ( v α p ) + 1 γ α log 1 |x α -z α | 2 + O 1 γ α , ( 3
Bx α ( r α,δ 2 ) f α (v α )dy = Bx α ( r α,δ 2 ) 4 exp (-2t α ) µ 2 α γ α dy + O 1 γ 3 α + O µ α r α,δ γ α . ( 3
f α (v α )dy = 4π γ α + O 1 γ 3 α , (3.73)
for all 0 < λ 1 -α ≪ 1. Independently, by (3.69), we get that there exists C > 0 such that

|G zα (y) -G zα (x α )| ≤ C|y -x α | r α,δ (3
G zα (y)f α (v α (y))dy = 4π γ α + O 1 γ 3 α G zα (x α ) + Bx α ( r α,δ 2 )
f α (v α (y))|y -x α |dy .

f α (v α (y))|y -x α |dy = O Bx α ( r α,δ 2 ) exp (-κt α ) |y -x α | γ α µ 2 α r α,δ dy = o 1 γ α . ( 3
) c G zα (y)f α (v α (y))dy ≤ C G zα L r v α L p λ 1 + Λ α exp(v 2 α ) L p ′ (Bx α ( r α,δ 2 ) c ) = O ( v α L p ) (3
v α (z α ) ≤ (1 + o(1)) log C |xα-zα| 2 γ α + C v α L p , ( 3
(λ 1 -A α ) Ω vv α dy = Λ α Ω vv α exp(v 2 α )dy ≥ 4πv(x)(1 + o(1)) γ α (3.81)
as α → λ - 1 , where x is as in (3.9), using (3.6). Since A α → λ - 1 by (2.4), we get (3.3) and (3.80) from (3.81) and the Cauchy-Schwarz inequality. Now we prove that

v α v α p → v in C 1 loc ( Ω\{x}) , (3.82)
as α → λ - 1 . By (2.9), (3.1), (3.64), (3.79), (3.80) and elliptic theory, we get that (v α / v α p ) α converges in C 1 loc ( Ω\{x}) to some ṽ solving ∆ṽ = λ 1 ṽ ,

in Ω\{x}. But, by (3.79) and (3.80) again, we get that 0 ≤ ṽ ≤ C in Ω\{x} for C as in (3.79), that ṽ solves (3.83) in Ω and that ṽ p = 1. Then ṽ = v and (3.82) is proved. Since r α,δ → 0 and by (3.82), we can find a sequence (δ α ) α of positive real numbers converging to 0 such that δ α ≥ r α,δ /2 and such that

v α v α p -v C 0 (Ω\Bx α (δα)) = o(1) (3
G zα (y)f α (v α (y))dy = 1 γ α log 1 |x α -z α | 2 + O 1 γ α . ( 3
α (δα) c G zα (y)f α (v α (y))dy= v α p λ 1 Ω G zα (y)v(y)dy + o( v α p ) = v α p v(z α ) + o( v α p ) , (3.90) as α → λ - 1 . Now we denote Ω α = B xα (δ α )\B xα r α,δ 2
. On the one hand, using (3.69), (3.79), (3.80) and δ α → 0 as α → λ - 1 , we get that In order to conclude the proof of Proposition 3.1, it remains to prove (2.10).

Ωα G zα (y)A α v α dy = O δ 2 α log 1 δ α v α p + O 1 γ α = o( v α p ). ( 3
Proof of Proposition 3.1 (ended). By (3.1), in order to get (2.10), it is sufficient to prove that

Λ α Ω v 2 α exp(v 2

1 0≤

 1 18), (3.21), and since v α 2 H 4π + o(1), we get that

  ). Observe that (3.5),(3.14) and (3.16) imply r α,δ → 0, as α → λ - 1 . Let δ be given in (0, d(x, ∂Ω)), for x as in (3.9). Let now Ãα > 0 be such that Ψα := Ãα log δ |•-xα| and γ α -tα γα coincide on ∂B xα (µ α ). Using (3.5) and (3.16), we easily get that

  .23) But by (3.5), (3.13), (3.14) and (3.22), for z α ∈ B xα (r α,δ ), we have that

  .31) Next we observe that (3.25) and (3.29) imply w 1,α = O(1

  by (3.7), (3.10), (3.31), and (3.35), we get that

  8 exp(-2T 0 ) w in B 0 (δ 0 ) , w(0) = 0 , w radially symmetric around 0 ∈ R 2 ; (3.48) but (3.48) implies w ≡ 0 in B 0 (δ 0 ) . (3.49) By (3.46), (3.47), (3.49) and the dominated convergence theorem we get

  ))|w α | (3.56) in B xα (r α ), using also (3.13), (3.16), (3.25) and (3.27) to get exp(B 2 α ) ≫ λ 1 . Summarizing, the v α 's satisfy (3.1) and (3.53), and the B α 's satisfy (3.27) in B xα (r α,δ ), while (3.25) holds true. Moreover, the w α 's satisfy (3.56) in B xα (r α ). Then, arguing exactly as in [8, Section 3] dealing with the case A α = 0, we get the following result.

  .68) for all 0 < λ 1 -α ≪ 1, where p is as in (3.66) and v is as in (3.67). Moreover, (3.3) and (3.4) hold true. Proof of Step 3.4. Let (z α ) α be a sequence of points such that z α ∈ Ω\B xα (r α,δ ) for all α. Let G be the Green's function of ∆ in Ω with zero Dirichlet boundary conditions. Then (see for instance [8, Appendix B]), there exists a constant C > 0 such that0 < G x (y) ≤ 1 2π log C |x -y| |∇G x (y)| ≤ C |x -y| (3.69)for all x = y in Ω. By the Green's representation formula and (3.1), we get thatv α (z α ) = Ω G zα (y)f α (v α (y))dy . (3.70) Now, we split the integral in (3.70) according to Ω = B xα r

  .78) for all α. Putting together (3.69), (3.70), (3.77) and (3.78), we have obtained that there exists C, C > 0 such that

  .91) On the other hand, using (2.9), (3.70) (3.79), and the dominated convergence theorem we have thatΩα G zα (y)Λ α v α exp(v 2 α α -y| 2 + o(1) 2 dy = o ( v α p ) .

  α -•| 2 ≤ 1 -δ + o(1) < 1 (3.93) in B xα (r α,δ /2)c , for all 0 < λ 1 -α ≪ 1. Combining (3.89), (3.90), (3.91) and (3.92), we get (3.68). At last we prove(3.4). By (3.5), (3.27), (3.57), we have thatv α (z α ) = B α (z α ) + O α, where (z α ) α is given such that zα ∈ ∂B xα (r α,δ ). But picking z α = zα in (3.68), we get from (3.80), (3.86) and (3.94) that log 1 γ 2 α Λ α =γ α v α p v(x)(1 + o(1)) → +∞ , (3.95) as α → λ - 1 , which concludes the proof of (3.4) and that of Step 3.4.

4 2 α dz + O 1 γ 4 α 2 ∆S 0 dz = R 2 4

 42422 (3.60) and the dominated convergence theorem, we get thatΛ α Bx α (r α,δ ) Bx α ( r α,δ µα ) exp(-2T 0 ) 1 + (2S 0 + T 2 0 -2T 0 ) γ + µ α r α,δ.(3.97)Note that the term of order γ -2 α in (3.97) vanishes since, by (3.24) and (3.25), we getA 0 = R exp(-2T 0 )T 0 dz .Then (3.14) and (3.97) imply Λ α Bx α (r α,δ ) in (3.92), using (3.64), (3.68), (3.93) and the dominated convergence theorem. At last, we easily get from (3.68) and (3.80) that Ω conclude from (3.98)-(3.100) with (3.4) and (3.80) that (3.96) holds true, which concludes the proof of Proposition 3.1.

  for all 1 ≤ p < +∞, and then standard elliptic theory applies. Since C λ1 (Ω) = +∞,

	we get that
	C α (Ω) → +∞	(2.2)
	as α → λ -1 , by the monotone convergence theorem. Then, by Lions [12, Theorem
	I.6], we have that, up to a subsequence,

  1,α -S α . In the sequel, restricting to B xα (r α,δ ) gives that 2 -tα

	error term (1 + t 4	γ 2 α	≥ 2 -δ > 1 and then, a sufficiently good decay of the

α ) exp(t α (-2 + (t α /γ 2 α ))). Namely, we can find κ > 1 and C > 0 such that

(1 + t 4 α ) exp(t α (-2 + (t α /γ 2 α ))) ≤ C exp(-κt α ) (3.37) in B xα (r α,δ

). Now, we observe that Bx α (r)

  .89) Now, since ∆v = λ 1 v and A α → λ - 1 , we get by (2.9), (3.64), (3.69) and (3.84) that

	Bx

Acknowledgements

This work was initiated at the department of Mathematics and Computer Science of the university of Basel, where the first author was employed until September 2017. The authors warmly thank Luca Martinazzi for having invited the second author in Basel and for many fruitful discussions on these topics. The first author was

1 supported by Swiss National Science Foundation, projects nr. PP00P2-144669 and PP00P2-170588/1.


Mirjam Czjzek

Sylvie Létoffé

Cécile Wandersman

Muriel Delepierre

Anne Lecroisey

Nadia Izadi-Pruneyre

The crystal structure of the secreted dimeric form of the hemophore HasA reveals a domain swapping with an exchanged heme ligand

Keywords: GdnHCl, guanidine HCl, OD, optical density, PDB, Protein Data Bank, rmsd, root mean square deviation

come

The crystal structure of the secreted dimeric form of the hemophore HasA reveals a domain swapping with an exchanged heme ligand.

INTRODUCTION

Iron is essential for many metabolic pathways. However despite its abundance, it is practically insoluble in the presence of oxygen at a physiological pH and is thus scarcely available. Heme-containing compounds, which are widespread among vertebrate hosts, are advantageous iron reservoirs for bacterial pathogens. Therefore several Gram-negative bacteria use a heme uptake system involving an extracellular heme-binding protein called hemophore. The function of hemophores is to acquire free or hemoprotein-bound heme and to deliver it to a specific receptor at the cell surface whereby the heme is internalized and then used as iron source.

The hemophore secreted by the opportunistic pathogen Serratia marcescens, HasA, was the first to be identified [START_REF] Létoffé | Iron acquisition from heme and hemoglobin by a Serratia marcescens extracellular protein[END_REF] and is now very well characterized. It is secreted under iron deficiency condition by an ABC-type pathway via a C-terminal secretion signal. This 19kDa protein binds one b heme molecule with an affinity of 5.3 10 10 M -1 [START_REF] Izadi | Purification and Characterization of an Extracellular Heme-Binding Protein, HasA, Involved in Heme Iron Acquisition[END_REF][START_REF] Deniau | Thermodynamics of heme binding to the HasA(SM) hemophore: effect of mutations at three key residues for heme uptake[END_REF]. HasA is able to bind free heme or to capture it from other hemoproteins, like hemoglobin. The heme transfer from HasA to its receptor HasR which has a much lower affinity for heme (Ka 5 10 6 M -1 ) occurs by protein-protein interaction [START_REF] Izadi-Pruneyre | The heme transfer from the soluble HasA hemophore to its membrane-bound receptor HasR is driven by protein-protein interaction from a high to a lower affinity binding site[END_REF]. The next steps of heme internalization, including the heme transfer from the receptor to the periplasm and the recycling and/or ejection of the empty hemophore from the receptor are dependent on the energy given by an inner membrane complex composed of three proteins TonB-ExbB-ExbD [START_REF] Létoffé | Free and hemophore-bound heme acquisitions through the outer membrane receptor HasR have different requirements for the TonB-ExbB-ExbD complex[END_REF].

The structure of HasA shows an original +fold, with an unusual arrangement of the hemebinding site. The helices are all packed on one side of the molecule and the  strands on the other side. The heme is held by two loops at the interface of the  and  parts of the molecule and is highly exposed to solvent. The iron atom is ferric and coordinated by the N of a histidine residue on one side (His32) and to the O of a tyrosine residue (Tyr75) on the other [START_REF] Arnoux | The crystal structure of HasA, a hemophore secreted by Serratia marcescens[END_REF].

We report here the identification, the characterization and the X-ray structure of a dimeric form of the hemophore present in the culture medium: DHasA. Our results reveal that DHasA is a domain-swapped dimer in which the same structural element is exchanged between the two subunits. It is noteworthy that the exchanged part contains one of the heme ligands, His32. Domain swapping was first described at the atomic level by Eisenberg and coworkers in 1994 in the crystal structure of diphtheria toxin [START_REF] Bennett | Refined structure of monomeric diphteria toxin at 2.3 A resolution[END_REF]. It is a mechanism by which one protein molecule exchanges a structural "domain" with an identical partner ("domain" is not used in a strict sense). The "domain" can be as large as an entire tertiary globular domain, or as small as a single element of secondary structure. Currently more than 60 proteins have been identified as domain-swapped protein [START_REF] Yang | Domain swapping is a consequence of minimal frustration[END_REF]. It has been advanced that the domain swapping plays a role in the regulation of function, in an evolutionary strategy to create protein complexes, or in misfolding and aggregation, but its biological relevance is still unclear [START_REF] Liu | 3D domain swapping: as domains continue to swap[END_REF]. In this work, we show that DHasA could function as an efficient heme reservoir in the hemophore system.

MATERIAL AND METHODS

Expression and purification of the HasA proteins

ApoHasA proteins, HasA and DHasA, were obtained from the culture supernatant of Escherichia coli strain POP3 (pSYC34PAM) grown at 30°C in M9 glycerol medium and purified as described previously [START_REF] Izadi-Pruneyre | Backbone NMR assignment and secondary structure of the 19 kDa hemophore HasA[END_REF]. Serratia marcescens SM365 was also grown at 30°C in M9 glycerol supplemented with 0.2 mM 2,2"-dipyridyl to reduce available iron.

The holoforms were prepared by adding a freshly prepared hemin solution to the apo-proteins at a ratio of about 120%. The excess of heme was eliminated by gel filtration. All of the samples were in 50 mM sodium phosphate buffer, pH 7 otherwise mentioned.

The HasA concentration was calculated using the previously determined  277nm value of 19500 M -1 cm -1 [START_REF] Létoffé | Haemophore-mediated bacterial haem transport: evidence for a common or overlapping site for haem-free and haem-loaded haemophore on its specific outer membrane receptor[END_REF]. The same  277nm value was used to determine the concentration of DHasA sample in monomeric protein. All absorption measurements were done in a Perkin-Elmer Lambda 2 spectrophotometer using 1 or 0.2 cm path length cuvettes.

Hemin solution

Hemin solutions were prepared just before use to prevent heme aggregation. Bovine hemin (Sigma) was dissolved in 150 l of 0.1 N NaOH, diluted to volume desired by the adequate buffer, and centrifuged. The concentrations of the hemin solutions were measured in 0.1 N NaOH using the  385 (0.1N NaOH) value of 58440 M -1 cm -1 (12)

Analytical Ultracentrifugation

Ultracentrifugation studies were performed in an XLA (Beckmann) analytical ultracentrifuge using 12mm path length cells. Purified DHasA samples were 2 10 -5 M (monomer concentration) in 20 mM sodium phosphate, pH 7. The sedimentation/diffusion equilibrium experiments were performed as previously described [START_REF] Izadi | Purification and Characterization of an Extracellular Heme-Binding Protein, HasA, Involved in Heme Iron Acquisition[END_REF].

Dimer/monomer formation

The presence of dimeric and monomeric proteins in various culture supernatant was tested by native-PAGE, followed by immunodetection with anti-HasA antibodies and revealed by ECL+ (Amersham) as described previously by Létoffé et al [START_REF] Létoffé | Ligand delivery by haem carrier proteins: the binding of Serratia marcescens haemophore to its outer membrane receptor is mediated by two distinct peptide regions[END_REF]. The immunodetection was used to increase the detection threshold of the samples concentration. Purified monomeric and dimeric HasA of known concentrations of 10 -5 , 10 -6 and 10 -7 M were used as standard in order to estimate the concentration of secreted protein.

Heme binding

Heme binding experiments were performed by UV-visible spectroscopy by following the Soret band at 407 nm. The concentrations of DHasA and of the hemin solution were respectively 1.8 10 -6 M (monomer concentration) and 2.25 10 -5 M.

Isothermal titration calorimetry (ITC)

Titrations were performed at 25°C using a MicroCal MCS titration calorimeter (MicroCal Inc., Northampton, MA). Proteins and hemin were in 20 mM sodium phosphate, pH 7.

Solutions were thoroughly degassed by stirring under vacuum before use. Due to heme adsorption, the calorimeter cell and the microsyringe used for injections were extensively washed after each experiment. For all titrations, the heat of dilution of the ligand was measured by injecting the ligand into the buffer alone or by injecting an excess of ligand into the cell after saturation. The value obtained was subtracted from the heat of reaction to obtain the effective heat of binding. All of the injections were carried out with 3 min intervals.

The theoretical titration curves were fitted to the experimental data using the ORIGIN software (Micorcal). The molar binding stoechiometry and molar binding enthalpy (H) were determined directly from the fitted data. The affinity constant of DHasA for heme, which is higher than 10 8 M, was too large to be measured by ITC direct titration. Therefore, it was determined by the displacement method (14) using the H32A-H83Q double mutant of the heme binding pocket with Ka=2.6 x10 7 M -1 and H= -29.2 kJ.mol -1 as previously described by Deniau et al [START_REF] Deniau | Thermodynamics of heme binding to the HasA(SM) hemophore: effect of mutations at three key residues for heme uptake[END_REF]. The Gibbs (binding) free energy (G) and molar entropy of binding (S) were calculated using the equation: G=RT ln Ka = H  TS, where R is the gaz constant and T is the temperature in Kelvin.

E.coli growth assays

Growth stimulation of C600 hemA::km att ::hasR by exogenously supplied purified protein was tested as follow. Cells expressing HasR were grown in LB medium supplemented with 20 g/ml of delta aminolevulinic acid (LBD) and appropriate antibiotics to an OD 600 of 1 and 100 l of the resulting culture was mixed with 3 ml of top agar and plated on LBD 0.02 % arabinose. Aliquots of 50 l of serial dilutions of various apo/holo monomer/DHasA were added to wells punched in the solidified agar. Growth around the wells was noted after overnight incubation at 37°C.

NMR spectroscopy

NMR samples contained about 1.5 mM HasA or DHasA (monomer concentration) in 50 mM Sodium phosphate buffer, pH 5.6. NMR spectra were recorded on Varian Inova 500 or 600 MHz spectrometers equipped with the VNMR 6.1 software and with Sun Sparc Ultra 2 and Ultra 5 respectively. 1 H- 15 N HSQC experiments [START_REF] Kay | Pure absorption gradient enhanced heteronuclear single quantum correlation with improved sensitivity[END_REF] were recorded at 30°C using 1024 x 128 data points and 64 scans. The one-dimensional 1 H NMR spectra were acquired at 50°C with a spectral width of 200 ppm, 8k data points and a recycle delay of 40 ms. Water suppression was achieved by presaturation.

Protein crystallization

The protein concentration was about 4.0 mg/ml in 50 mM phosphate buffer at pH 7 and 0.02% NaN 3 . Initial crystallization trials were performed with MDL (Molecular Dimensions Limited) and Decode Genetics ("Wizard" 1&2, Emerald Biostructures), that is, a total of 192 trials in two 96 multi-well crystallization plates from Greiner. These trials were set up using a Cartesian crystallization robot, mixing 200 l of protein with 100 l of reservoir solution.

Crystals were grown by the hanging-drop vapor-diffusion method using 24-welled Linbro plates. Hanging drops on siliconized cover slips were prepared by mixing 2 l of the protein solution with 1 l of the reservoir solution that contained 100 mM Tris buffer at pH 8.5, 2 M NaH 2 PO 4 and 2 M K 2 HPO 4 . They were then placed over 0.5 ml of the reservoir solution. All crystallization trials were carried out at 20° Celsius.

Data collection and processing

X-ray diffraction data of a DHasA crystal were collected at 100 K at the European Synchrotron Research Facilities (ESRF, Grenoble, France) beamline ID29 using an ADSC Quantum 4R CCD detector. The crystal was flash-cooled in a liquid-nitrogen stream with 10 %(v/v) glycerol as cryoprotectant. The wavelength of the synchrotron X-rays was 0.979 Å.

The crystal was rotated through 100° with a 1.0° oscillation range per frame. The raw data were processed using the program MOSFLM ( 16) and then merged and scaled using the program SCALA [START_REF]Collaborative Computational Project, Number 4, CCP4[END_REF].

Structure determination and refinement

The molecular replacement with the program AMoRe [START_REF] Navaza | AmoRe: an automated package for molecular replacement[END_REF] in the resolution range of 9 to 3.1 Å with the model of monomeric HasA (PDB ID: 1B2V) gave two solutions corresponding to the two monomeric subunits that form the dimer, with an overall correlation coefficient of 0.59 and an R-factor of 37.4 %. The refinement was performed using REFMAC5, part of the CCP4-package [START_REF]Collaborative Computational Project, Number 4, CCP4[END_REF]. Water molecules were added using CCP4/warp [START_REF] Perrakis | wARP: improvement and extension of crystallographic phases by weighted averaging of multiple-refined dummy atomic models[END_REF]. The stereochemistry of the final structures was evaluated using PROCHECK [START_REF] Laskowski | PROCHECK: a program to check the stereochemical quality of protein structures[END_REF]. All refinement statistics are summarized in Table 1.

RESULTS

Identification of a dimeric form of HasA in solution -DHasA-

Gel filtration chromatography of the supernatant of POP3 (pSYC34PAM) over-expressing

HasA in minimum medium at 30°C, gives two peaks (Figure 1). Their ratio varies depending on the culture from 10 to 50%. Analysis of the fractions corresponding to these peaks by SDS-PAGE shows that they both contain one protein that migrates like HasA, HasA being the unique protein secreted in E. coli overexpressing HasA. The slower migrating peak corresponds to the monomeric form of HasA, previously characterized and studied [START_REF] Izadi | Purification and Characterization of an Extracellular Heme-Binding Protein, HasA, Involved in Heme Iron Acquisition[END_REF]. The faster migrating peak has an apparent molecular weight corresponding to twice that of HasA, it will be called DHasA hereafter.

The molecular weight of DHasA was determined by analytical ultracentrifugation with the sedimentation/diffusion equilibrium experiment. The experimental data fit in a satisfactory manner using a one-solute ideal solution model, with a molecule of molecular weight 36900 + 412 (data not shown). This value is in agreement with that of 38302 for a dimeric form of HasA, the variation remaining within the limits of methods.

DHasA is formed in vivo

DHasA is secreted by Serratia marcescens in comparable proportions to that observed in E.coli. This observation proves that it is not an artefact of over-expression in E.coli.

DHasA is observed by native-PAGE in the culture supernatant. The formation of DHasA is thus not dependent on purification conditions, especially not on the Ammonium Sulfate precipitation leading to high protein concentration. As observed in Figure 2, DHasA and HasA secreted by S. marcescens exist both in cleaved and uncleaved forms. The conditions of cleavage have already been studied for HasA and the study revealed that cleavage is performed by S. marcescens major protease, PrtSM, and concerns the twelve C-terminal residues of HasA [START_REF] Izadi-Pruneyre | NMR studies of the C-terminal secretion signal of the haembinding protein, HasA[END_REF].

The presence of DHasA is not the result of an accumulation of the monomeric form during the culture. Indeed, it is detected during the exponential phase of culture when cell density reaches an OD 600 of 1 and when the total HasA protein concentration is estimated to be 10 -6 M and not only during the stationary phase.

DHasA, like HasA, is secreted in minimum medium in the apoform as shown by the absence of the Soret band at 407 nm, characteristic of the holoform, in the UV-visible spectrum. The holoform can be obtained by adding a solution of hemin to the culture medium. The amount of dimeric form obtained is comparable to that obtained when the medium is not supplemented with hemin. The presence (or absence) of heme in the culture medium has thus no influence on DHasA formation.

Structural comparison of DHasA and another dimeric form obtained in vitro

Three different crystal forms of HasA were obtained under different crystallization conditions [START_REF] Arnoux | Functional aspects of the heme bound hemophore HasA by structural analysis of various crystal forms[END_REF]. The overall structure of HasA in the three crystals is the same. Nevertheless, even though all were obtained from a purified monomeric sample, HasA is in a dimeric form in one of the crystal (PDB ID: 1DK0). Therefore the question arises: are the two dimeric proteins obtained in vitro and in vivo identical? In order to answer this question we have analyzed the structure of DHasA by NMR to determine if it is equivalent to that of 1DK0.

The 15 N-1 H HSQC spectrum is a fingerprint of the backbone conformation of a protein. Each 15 N-1 H cross-peak reflects amino acid local environment in protein. The comparison of 15 N-1 H HSQC spectrum of ApoDHasA with that of ApoHasA for which the complete assignment is known [START_REF] Izadi-Pruneyre | Backbone NMR assignment and secondary structure of the 19 kDa hemophore HasA[END_REF] allowed us to obtain structural informations. This spectral comparison was done at two concentrations, namely 0.4 and 1.5 mM. At both concentrations, the number of 15 N-1 H amide connectivities present in the ApoDHasA spectrum is comparable to that usually found in the ApoHasA spectrum at the limit of resolution (169 vs 172 respectively). The high similarity between both spectra suggests a high structural homology between the two protein forms.

Most of the residues could be easily assigned by homology. Residues 48-58, 71, 72, 73, 75 and 77 could not be assigned in DHasA spectra. This observation indicates that while the global structure of the protein is conserved, the environment and/or the structure of theses residues are changed indicating that they are involved in dimerisation. Residues 42, 43, 45, 67, 69, 82, 83, 86, 99 which display slight differences in their chemical shifts as compared to the monomeric form (up to 1 ppm in 15 N and 0.2 ppm in 1 H dimension) may be located at the immediate sequential and/or spatial vicinity (backbone and side-chain) of residues at interface. (Figure 3)

The crystal structure of the dimeric form of HasA, 1DK0, could not explain all the differences between the ApoHasA and ApoDHasA spectra. Conversely, residues 88-107 located at the interface of the monomeric subunits of HasA in 1DK0 do not show significant variation of chemical shifts. These observations are indicative of substantial differences in the quaternary structure of DHasA with respect to that of 1DK0. Thus, the dimeric form observed in the crystal structure of 1DK0 might be an intermediate of aggregation, stabilized in the crystal. Indeed, at high ionic strength (i.e 150mM NaCl) the presence of some aggregated HasA has been detected.

No dissociation of DHasA into monomers was observed by NMR at 30°C at a protein concentration of 1.5 mM during one week and inversely no dimerisation of the monomeric sample was observed at the same or higher concentrations (up to 1.8 mM). The lack of dissociation and the very high stability of DHasA, generally observed in domain-swapped dimers, suggest that DHasA might be this type of dimer. In order to check this hypothesis, we have thus determined the crystal structure of DHasA and studied properties and activity of this new HasA dimeric form by different methods.

DHasA crystallizes as a domain swapped dimer

DHasA has successfully been crystallized. The optimized reservoir conditions are 100 mM Tris buffer at pH 8.5, 2 M K 2 HPO 4 and 2 M NaH 2 PO 4 . The brownish crystals belonged to space group P2 1 , with unit parameters highly resembling those of 1DK0 (Table 1), and diffracted to a maximum resolution of 2.3 Å. Two molecules are present in the asymmetric unit, giving a crystal volume per protein mass (V M ) of 2.17 Å 3 Da -1 and a solvent content of 43% by volume [START_REF] Matthews | Solvent content of protein crystals[END_REF]. The coordinates describing two protein molecules with two heme groups per asymmetric unit were refined to a final R-factor and R free of 16.9% and 22.3%.

The overall fold of each monomeric subunit of DHasA is very similar to that of HasA but formed by parts coming from the two different polypeptide chains present in the asymmetric unit. Each monomeric subunit is composed of seven beta strands packed on one side and four -helices on the other side. But for each subunit residues 2-49 does not belong to the same polypeptide chain than other residues (Figure 4a-c). This crossing over of polypeptide chains to form a multimer assemblage is frequently observed in the so-called domain-swapped dimers. Given the fact that both the monomer and the domain swapped dimer of HasA exist in solution and in stable form, DHasA can thus be considered to be a bona fide example of 3D swapping as defined by Liu & Eisenberg [START_REF] Bennett | Refined structure of monomeric diphteria toxin at 2.3 A resolution[END_REF]. The backbone superimposition of one subunit of DHasA and the monomeric form of HasA shows a rmsd of 0.93. Significant structural changes occur at the dimer interface of the swapped domain, as compared to the monomeric form. As expected, the highest rmsd value is observed in the hinge loop (43-48) linking the swapped domain to the rest of its subunit. Besides to this, two residues, Asn34 and Gly35 also show high rmsd values, compared to the overall mean value, this being valid for the two subunits composing the dimer as shown in Figure 5a. The overall average B-factor, however shows that these regions do not display particular high B-factors as compared to monomeric HasA (Figure 5b). On the contrary, dimeric form of HasA appears to be less flexible, especially in the region of the heme binding pocket.

The heme binding sites are similar to that of the monomeric form, and are composed of two loops positioned at the interface of the  and side that fix the heme in a relatively solventexposed environment. The swapping makes both heme sites less accessible than that of the monomeric form. The resolution of the data unambiguously revealed that the heme can take two orientations, related by a 180° rotation about the , -meso axis, as has already been observed in the previously determined HasA structures [START_REF] Arnoux | Functional aspects of the heme bound hemophore HasA by structural analysis of various crystal forms[END_REF]. The heme iron of each monomeric part is coordinated by the O of a tyrosine (Tyr75) and to the Nof a histidine (His32) with the difference that the latter belongs to the swapped region, and the heme molecule of each subunit is actually bound by residues coming from the two polypeptide chains.

Heme environment of DHasA characterization by NMR

The 1 H NMR spectra of HoloDHasA shows the presence of two sets of resonances with relative intensity of 70:30 corresponding to the two possible orientations of the heme observed in the crystal structure (Figure 6). This ratio is well visible in a couple of isolated peaks at 72.5 and 75.7 ppm. The same ratio was equally found for the monomeric form as reported by Caillet-Saguy et al [START_REF] Caillet-Saguy | Direct-detected 13C NMR to investigate the iron(III) hemophore HasA[END_REF]. The methyl resonances of the two heme molecules are observed between 25 and 50 ppm. These chemical shifts are intermediate between those typical for a high spin S=5/2 iron (III) heme and those for a low spin S=1/2 iron (III) heme. This presumes the existence of a high spin / low spin equilibrium as that observed for the monomeric form [START_REF] Caillet-Saguy | Direct-detected 13C NMR to investigate the iron(III) hemophore HasA[END_REF]. The chemical shifts are however slightly different to that of the monomeric form at the same temperature and pH, indicating that the heme iron properties and environment in solution and/or the high spin/low spin species ratio are not exactly the same for the two protein forms.

Activity of DHasA

The function of HasA is to take up heme and to shuttle it to a specific receptor. To determine the implication of DHasA in the heme acquisition pathway mediated by the hemophore, if any, we have studied the heme binding properties and its delivery to the receptor HasR.

Heme binding: To study the heme binding, we carried out heme titration experiments by UVvisible spectroscopy and by ITC microcalorimetry as described in our previous works [START_REF] Izadi | Purification and Characterization of an Extracellular Heme-Binding Protein, HasA, Involved in Heme Iron Acquisition[END_REF][START_REF] Deniau | Thermodynamics of heme binding to the HasA(SM) hemophore: effect of mutations at three key residues for heme uptake[END_REF][START_REF] Létoffé | Haemophore-mediated bacterial haem transport: evidence for a common or overlapping site for haem-free and haem-loaded haemophore on its specific outer membrane receptor[END_REF]. The titration curve obtained by UV-visible spectroscopy shows that DHasA binds one molecule of heme/monomer (data not shown). The ITC titration data can be fitted without any constraint to a single set of sites model with a stoichiometry of 1 mol /mol of monomer and an affinity constant higher than 10 8 M -1 for each site. However, at the minimal concentration that could be used in the calorimetric cell to obtain a reliable heat emission signal, 10 -5 M, the curve was too steep at the equivalence point to determine precisely the affinity constants (Figure 7). Hence, we determined their values by displacement titration using a low affinity mutant the H32A-H83Q: 9 10 8 M -1 for each site.

From the results of the ITC titration experiments, we could deduce an enthalpy value of -110.5 + 2 kJ.mol -1 and an entropy value of -59.4 kJ.mol -1 for the heme binding. The interaction of heme with DHasA is thus strongly enthalpy driven as observed in our previous work [START_REF] Deniau | Thermodynamics of heme binding to the HasA(SM) hemophore: effect of mutations at three key residues for heme uptake[END_REF] for the monomeric form of HasA (-105.4 + 3.8 kJ.mol -1 ). The decrease in binding entropy in DHasA (TS=-59.4 kJ.mol -1 ) compared to HasA (TS=-44.3 kJ.mol -1 ) is consistent with a decrease of the solvent exposed hydrophobic surface area around the heme from 14 Å 2 to 10 Å 2 .

The difference of heme iron properties in solution deduced from NMR analysis and the slight difference of some residues position in the loop containing His32, in particular Asn34 and Gly35, linked to the heme propionate groups by hydrogen bonds, can explain the lower affinity of DHasA for the heme molecule. Indeed, besides the residues of the hinge loop, these are the residues that display the highest rmsd values in the superposition of each monomer of DHasA with the crystal structure of monomeric HoloHasA (1B2V). The difference in the positions of Asn34 and Gly35 is only seen in the presence of heme (Figure 5a).

Heme delivery of DHasA to HasR: Added exogenously, the HoloDHasA does not allow heme acquisition of C600 hemA::km at ::hasR. It has recently been shown that the heme of HasA is transferred to the receptor HasR by protein-protein interaction (4). The inability of HoloDHasA to transfer directly its heme to HasR can be due either to an absence of interaction between these two proteins or to an inability of HoloDHasA to deliver its heme to HasR. The first hypothesis was tested using the solid-phase binding assay. Aliquots of 10 7 CFU of POP3 (pam238-hasR) HasR producing bacteria and of the control strain POP3 (pam238) were spotted onto nitrocellulose membranes and incubated with serial dilutions of apo or HoloDHasA protein. The interaction was probed with a polyclonal antiHasA serum [START_REF] Létoffé | Interactions of HasA, a bacterial haemophore, with haemoglobin and with its outer membrane receptor HasR[END_REF]. As shown in Figure 8, no binding affinity above background was observed with samples incubated with 10 -6 , 10 -7 or 10 -8 M apo or HoloDHasA indicating that the DHasA is unable to bind to the receptor HasR.

Nevertheless in our growth test, HoloDHasA mixed with ApoHasA allows the heme acquisition by bacteria and become thus a source of heme for bacteria. This result indicates that the heme of DHasA is transferred to HasA who delivers it, in turn, to the specific receptor (Table 2).

DISCUSSION

In this work we have identified a secreted dimeric form of HasA, DHasA. Its structure reveals that DHasA is a domain-swapped dimer. Three-dimensional domain swapping is an event by which a monomer exchanges part of its structure with an identical monomer to form an oligomer, where each subunit has a similar structure to the monomer. While each subunit structure of DHasA compared to that of HasA monomeric form is entirely conserved significant structural changes occur at the dimer interface of the domain-swapping. As expected, the highest rmsd value is observed in the hinge loop (43-48), composed of GGFYGG. Although the relation between the hinge loop sequence and swapping phenomenon is not actually demonstrated, the presence of prolines and residues generating, some flexibility have been put forward in facilitating the domain swapping process. Here, the presence of 4 glycine residues in the hinge loop ensures the flexibility needed to perform the swapping. Furthermore, the presence of two hydrophobic residues (FY) constrained at the dimer interface due to both steric hindrance and their hydrophobic character may be a factor that drive the hinge loop to swap or not. The hinge loop sequence is strictly conserved in Erwinia caratovora hemophore (http://www.sanger.ac.uk/Projects/E_caratovora) which is not yet characterized. It is conserved in the form of GXFXXG in all of the other hemophores found in Pseudomonas aeruginosa, P. fluorescence, Yersinia pestis, and Y. enterocolitica.

The swapping brings residues Phe45 and Tyr46 close to the 75-85 loop of the same polypeptide chain, creating supplementary hydrophobic interactions among residues of the same chain. As a consequence, Phe78 is in a more hydrophobic environment than that observed in the monomeric form where it is exposed to the solvent.

The swapping in DHasA conserves the overall structure of the heme binding site as well as the ligands of the heme group, although it is noteworthy that one of them, His 32, is provided by the swapped domain. Nevertheless, some differences are observed in the heme binding pocket of DHasA as compared to that of the monomeric form: i) both heme sites of DHasA are less accessible to solvent because of the swapping and the heme binding pocket appears to be less flexible by the protein-protein interface (lower B-factors in Figure 5b) ii) residues Asn34 and Gly35, which belong to the swapped domain and are hydrogen bonded to the heme propionate groups, show higher rmsd than that of the rest of protein (except for the hinge loop, Figure 5a). These differences can explain the variations of heme chemical shifts observed in solution by NMR and the lower affinity of DHasA for heme compared to that of monomeric form (9. 10 8 M -1 vs 5,3 10 10 M -1 ).

We have shown here that DHasA exists in the culture supernatant and its presence does not result from over-expression or purification conditions which might expose the protein to destabilizing condition at high concentration. This observation and the very high stability of HasA -Tm of 60°C and half unfolding concentration of GdnHCl of 2.7 M- [START_REF] Wolff | Antifolding activity of the SecB chaperone is essential for secretion of HasA, a quickly folding ABC pathway substrate[END_REF], allow to assume that DHasA folds directly from an unfolded state and by an independent pathway, and not via the initial formation of the monomer. This model has also been proposed by Rousseau and co-workers for suc1, a cell cycle regulatory protein, with two native states, a monomer and a domain swapped dimer (27,[START_REF] Rousseau | The unfolding story of threedimensional domain swapping[END_REF]. Assuming that a single polypeptide sequence gives access to the two forms of HasA under the same conditions, both in E.coli and in S.marcescens and this apparently independently of any particular bacterial signal, two questions arise: what are the folding pathways for the monomer and the domain swapped dimer and how are they connected to each other? The fact that HasA of Serratia marcescens is secreted via its C-terminal secretion signal means that the polypeptide needs to be fully synthesized before being secreted. Moreover, the cytoplasmic chaperone SecB has been shown to be essential for HasA secretion in vivo [START_REF] Delepelaire | The SecB chaperone is involved in the secretion of the Serratia marcescens HasA protein through an ABC transporter[END_REF]. In its absence, folded HasA accumulates in the cytoplasm and is unable to be secreted [START_REF] Debarbieux | Folded HasA inhibits its own secretion through its ABC exporter[END_REF]. Together, these data show that the folding of HasA should be achieved during or after secretion. Furthermore, in vitro study of refolding of denatured HasA has demonstrated that Sec B slows down HasA folding. In this study, refolding of denatured HasA has been analyzed by stopped-flow measurements of the fluorescence of its unique tryptophan [START_REF] Wolff | Antifolding activity of the SecB chaperone is essential for secretion of HasA, a quickly folding ABC pathway substrate[END_REF]. DHasA had not been characterized before and consequently the mono/oligomeric state of refolded HasA has not been checked in that study.

Nevertheless, as the tryptophan, located in the swapped part of DHasA, has exactly the same environment in both monomeric HasA and in DHasA, the formation or the presence of DHasA, if any, could not be detected by the fluorescence signal. The folding/unfolding study of both forms of HasA at different concentrations, while checking the presence and the relative amount of each form for example by analysis of the fluorescence anisotropy decay, will provide further information on the folding pathway and contribute to a better understanding of the domain swapping phenomenon.

Currently more than 60 proteins have been identified as domain-swapped oligomers with no apparent structural or functional connection between them. Despite the increasing number of examples, clear evidence for a functional role of domain swapping is lacking [START_REF] Rousseau | The unfolding story of threedimensional domain swapping[END_REF]. Several examples exist for which only the domain swapped version of the protein has been described or only the swapped multimeric form is functional. In some cases, however, both swapped and unswapped forms of the protein have been characterized and domain swapping has been proposed to have a regulatory function. For example, Glyoxalase I from Pseudomonas aeruginosa exists both in an active domain-swapped dimeric state and as a metastable less active monomer [START_REF] Saint-Jean | Active monomeric and dimeric forms of Pseudomonas putida Glyoxalase I : evidence for 3D domain swapping[END_REF]. In vitro conversion from dimeric to metastable monomeric Glyoxalase I can be triggered upon addition of glutathation. After removal of glutathione, glyoxalase I slowly reverts to a more stable and active domain-swapped form. However, no evidence exists for such a mechanism in vivo. Another interesting finding is that formation of domainswapped oligomers can introduce allosteric regulation of the protein activity already present in the monomer, as shown in bovine seminal Rnase [START_REF] Vitagliano | A potential allosteric subsite generated by domain swapping in bovine seminal ribonuclease[END_REF]. It is intersting to mention the example of Cyanovirin-N for which an identical activity and binding affinity towards gp120 was observed for both the pure monomeric or dimeric protein. Both forms are stable, however the elevated value of the dimerization constant (2.5mM at 50°C) suggests that the dimeric form does not exist under physiological conditions [START_REF] Barrientos | The highly specific carbohydrate-binding protein cyanovirin-N: structure, anti-HIV/Ebola activity and possibilities for therapy[END_REF]. In our case, both the monomeric and the domain swapped dimeric forms exist in vivo and are stable and active. The monomeric form of HasA binds one molecule of free or hemoprotein-bound heme and delivers it to a specific outer membrane receptor, HasR, by protein-protein interaction (4). The swapped dimeric form, DHasA, binds two molecules of heme with an affinity constant which remains high, even if it is lower than that of the monomeric form. While it is not capable to deliver its hemes directly to the extracellular receptor HasR, since it does not interact with this receptor, it can, however, transfer them to the monomeric form of HasA that, in turn, will interact with HasR to deliver heme. Létoffé et al have shown that two regions of the hemophore are involved in the interaction with HasR, namely the beta strand composed of residues 51 to 60 and the region consisting of residues 95 to 107 [START_REF] Létoffé | Ligand delivery by haem carrier proteins: the binding of Serratia marcescens haemophore to its outer membrane receptor is mediated by two distinct peptide regions[END_REF]. These two regions in DHasA become partially inaccessible by dimerization. Furthermore, steric hindrance due to the bigger size of DHasA can be an obstacle for recognition and interaction with HasR. Therefore, we propose that DHasA could serve as a heme reservoir, stocking it and keeping it more accessible for bacteria by the Has system. The efficiency and advantage of DHasA for the Has system would result in its ability to bind two heme molecules with an affinity lower than that of the monomeric form but still high compared to other heme binding proteins in serum, like albumin (the two highest affinities determined for heme-albumine are 7.8 10 7 M -1 and 2.3 10 8 M -1 [START_REF] Gattoni | Stability of the heme-globin linkage in alphabeta dimers and isolated chains of human hemoglobin. A study of the heme transfer reaction from the immobilized proteins to albumin[END_REF]. Furthermore, as the Kd for heme binding of heme receptors from other bacteria is estimated to be in the order of M, the capacity of DHasA to serve as a heme reservoir for the Has system makes the heme acquisition via hemophore efficient and advantageous for Serratia marcescens in the presence of other bacteria sharing the same space. (monomer concentration). Injections of 10 l of a hemin solution at 5.9 10 -5 M were performed with an interval of 3 min. The ITC titration data were fitted without any constraint to a single set (or class) of sites model with a stoichiometry of 1.024 + 0.005 (mol/mol of monomer), an affinity constant higher than > 10 8 M -1 and a H observed of -108.8 (+ 1.3) kJ.mol -1 (or -26.1 Kcal.mol -1 + 0.3) for each site.

Figure 1 :

 1 Figure 1:Gel filtration on a HiPrep 16/60 Sephacryl S-100 High resolution equilibrated with

Figure 2 :

 2 Figure 2:DHasA is directly observed in the culture supernatant of S.marcescens in iron

Figure 3 :

 3 Figure 3: Superimposition of 600 MHz 1 H-15 N HSQC spectra of HasA (in black) and DHasA

Figure 4 :

 4 Figure 4: (A) Ribbon representation of the backbone of domain swapped dimeric DHasA

Figure 5 :

 5 Figure 5: (A): R.m.s. deviation calculated by superimposing the backbone atoms of the

Figure 6 : 1 H

 61 Figure 6: 1 H NMR spectra of HoloHasA (A) and HoloDHasA (B) recorded at 500 MHz and at

Figure 7 :

 7 Figure 7: Isothermal Titration Calorimetry (ITC) of the interaction of DHasA with hemin in

Figure 8 :

 8 Figure 8: Solid phase binding assays. Each sample was tested in triplicate at concentration

Acknowledgement

We are indebted to the European synchrotron research facilities (ESRF, Grenoble, France) for beam time allocation for this project and to the staff on beamline ID29 for technical assistance during data collection. We thank Robert Gilli for help with ITC experiments.


Increasing aesthetic experience, installing films

That was what Roland Barthes regreted the most. In the flow of the viewing of a film, it seems that no place has been left for experiencing thoughtfullness or what I shall call a pensitivity (pensitivité). The words are well known, which Raymond Bellour quoted in an essay dedicated to the "pensive spectator" 1 . "Do I add to the images in movies ?-I don't think so ; I don't have time : in front of the screen, I'm not free to shut my eyes ; Otherwise, opening them again I would not discover the same image ; I am compelled to a continuous voracity ; a multitude of other qualities, but no pensitivity : that's why the photogram interests me." 2 That default that will lead Barthes to prefer the photogram would be to find in the temporal form of the film, which favor a psychological activity mainly "protentional" and substantially caused by the hurry of the scrolling, as voracious as Cronos himself. Yet, in order to undo the grip of this temporal form of the film, there are other ways than to rely on the photogram or on the freeze frame-which strongly distort the temporal experience of the film-and that is to rely on excerpts. In the eyes of Barthes, the photogram was interesting because it materializes something similar to the hermeneutic pause-and the possibility to "add to the images", namely to supplement the images with a vagabond semiosis. The excerpt, which carry out duration, allows to materialize the aesthetic "moment"-or would I name it better by saying the momentum since the latin word evokes more substantially the idea of an impulse or of a potency.

Filmic studies : from inception to intensification

The "filmic studies" 3 which I produce are meant to arouse something of the aesthetic momentum implied by the excerpts, to reinforce the call to states not only of pensivity but also and, for the most, of sensitivity in the spectator's experience which is usually and routinely "compelled to a continuous voracity". Indeed, this kind of temporal voracity will regularly stamp out the induction of an aesthetic relation as much as a critical relation for the reason that the "attentional resources" of the spectator are commonly focused on these significant elements in regard to understanding a narrative. But we need more that the only process of extracting excerpts and restoring the momentum it carries if we wish to undo the habits of perceptual anticipations. We need the apparatus of formal processes and, whenever possible, of scenographic setups.

My pieces are conceived and designed to be installed and staged following a kind of "grammar" inspired by the multi--channel video installations we can find in galleries an contemporary art exhibitions. The purpose is to benefit from the possibilities offered by ever the use of multiple displays and their "staging". It is also to suggest an "aesthetic attitude". Besides the fact that exhibiting these "critical" installations in academic or cultural institutions such as libraries will ease the inception of an "aesthetic attitude", a few formal processes, which I oftenly make use of, will ease not only the inception of that "aesthetic attitude" but its intensification. Taking into account the limited time I have today, I will choose to focus on two of these formal processes : First, the looping of the excerpts. And then, the use of multiple displays or split screen. Extracting an excerpt from a feature film will allways deviate the initial experience of shots and sequences that are "natively" constitutive parts of a more larger temporal form, but nevertheless the looping of the excerpt allows to preserve something similar to the momentum of the excerpt. It also happens to outsource it and "open" its temporality (to use Jean--Marie Schaeffer's word). Yet, looping the excerpt, opening it to an unbounded temporality, it is to a certain extent "compelling" the spectator in reshaping his attention in the sense of an intensification not only of a semiosic pensitivity, but also and for the most, of an aesthetic-or shall I say aesthesic-sensitivity. If we follow up the precise and precious psychological analysis made recently by Jean--Marie Schaeffer about the aesthetic experience in his namesake book, we have to notice how it involves the exchange between a kind of "standard" attention for this other kind of attention, called "aesthetic". Standard attention is based on an efficient perceptual processing of the stimuli, on the use of subconscious "short reactional loops" and, above all, "transitive" ones that are goal--oriented toward a quick schematization and categorization or toward an urgent understanding of, let's say, a narrative situation for example. On the contrary, aesthetic attention is characterized by its circular dimension : it aims at its renewal, and nothing else, regulated by the two polarities of curiosity and "fluence"-In Schaeffers vocabulary, the fluence identify with a pleasant ease in the perceptual or conceptual processing, that is experimented at the peril of boredom : "The circle formed by this complex attention exercised for its own sake and by the hedonic calculation evaluating this exercise and modulating the attentionnal processing in the aftermath define the aesthetic experience. [...] the aesthetic experience is this circle and nothing else than this circle." 4 Looping excerpts is then an appropriate way to enhance this circling. The looping of excerpts is the first among the formal processes that I make use of in my installations, but there are other factors that will enhance the aesthetic experience of these excerpts. One is the use of multiple displays or, on the occasion of a mono--channel viewing, the use of the split screen device. Schaeffer himself underlines that a few films using this device of the split screen are aimed at arousing a kind of attention which he names "parallel", that is to say a kind of attention which targets several focal points at the same time. He first comments on Jacques Tati's strategy who, in his famous feature film Playtime, offers many long shots in which the composition will drive a kind of tabular contemplation and then discusses the split screen device itself : "A more classical form to invite to a parallel attention is the technical device of the split screen which is in use since 1930 [...] It is a less demanding strategy in terms of attentional resources that the one used for Playtime, and this is because of at least three reasons. First of all, the number of inlayed screens, and then of focal points combinated is limited to two, three or four. Thereafter, each of the inlayed screens is generally highly focused ; finally, and mainly, the split screen is geared as to pre-structure the dispatching of attention, since every focal point has is own frame, which will simplify the attention effort."5 Conducting the spectator to experience a parallel attention, it's at the same time damming up the logics of an attention called, on the opposite, "serial" (schematizing and "ranked"), to reintroduce this kind of back and forth dynamic between a focused attention and a distributed attention, between this kind of commonly targeted attention on which, for example, the continuity editing system of classical montage is builded and a tabular attention which randomly scatters the focal points. Moreover, this reinforced dynamic between focalized attention and distributed attention, between serial and parallel attention, leads to a state in which "top--down" perceptual processing will be much more prominent and will determine an attentional investigation of "low level" stimuli whereas these very same stimuli are usually processed through "pre--attention" ways, which means subconsciously. In other words, during an aesthetic experience, the limit of perceptual responsiveness is "lowered" (meaning increased), sometimes considerably, and the ability to discriminate the smaller elements that objects or textures put before our senses is extensively enhanced. It is what Jean--Marie Schaeffer is talking about when he comes to designate the saturated-or overloaded-dimension of the aesthetic relation which opposes to the goal--orientated standard attention, much more "economic". He takes on this example of a painting by Cy Twombly titled Leda and the Swan (1962) 6 of which, once you have elapsed the "first encounter" of an effort to categorize which will promptly be accomplished, you spectator will need whole of your attentional resources, and maybe also some prior knowledge, to maintain without displeasure a succesfull continuation of an aesthetic experience. By discovering this scribbled surface where only a few graphic signs are understandable at first sight-a window up to the right, the word Leda down to the right next to other strikethrough words-, you will be driven to open widly your eyes in front of this graphical senseless entanglement, which may be displeasant unless your curiosity shall be awakened by the oddity of these eratic traces or/and by a prior knowledge of the works of the artist, leading you to voluntarily "lower" your sensibility treshold. "These expectations will lead you in a second stage to come down towards lower levels of perceptual processing in which you will pay attention to the layouts as layouts (orientation, continuity and discontinuity...). Your attention will anchor on visual features of low level : you will seek for figurative indices, microstructures of designs, 'twomblyan' graphic signatures so to speak. Your attention will engage in an endogenous manner in a descending dynamic that will lead you to consciously process numerous elements which, when you first looked at the painting were wether preattentionally processed, wether ruled at as mere 'noise'." 7 The example of Wendy's TV 1 & 2 (Loig Le Bihan, 2017) I now would like to draw on an example with my last installation, a diptych, which I have newly produced, with the precious help of my colleague Claire Chatelet, in the aftermath of the completion and the publishing of my book dedicated to Shining (Stanley Kubrick, 1980) 8 . This filmic study, called Wendy's TV 1 & 2, aims at producing an intensification of that kind of aesthetic relationship one could freely adopt when viewing the "source" feature film of Kubrick.

Projection :

Wendy's TV1 (The Shining/Summer of '42, 1'12'', looped) [Fig. 1, 2, 3, 4] It seems to me that the most prominent aesthetical effect of this first part of the diptychthe ignitor of a curiosity that will counterbalance the fluence eventually produced by the piece9 -has to be found in the sound modulations of the echo effect between the dialogue lines enunciated by the characters of the film quoted in The Shining the way they are restituted in its mix (audible on the left speaker) and via the soundtrack of source film Summer of 42 (1971) by Robert Mulligan (audible on the right speaker). The slightly imperfection of the synchronization between the two excerpts produces a reverb effect and a spatialization that gives a kind of monumentality to a scene which, in the film, is altogether rather minor (and has been ruled out of the european version of the film). But it also leads the spectator to a "refining" of his listening which will eventually conduct to a better appreciation of the sound mix, of the overlaying and masking effects between the dialogue lines of the two films.

Projection : Wendy's TV2 (Shining/Carson City, 1'38'', looped) [Fig. 5, 6, 7, 8] In a first viewing of this second piece of the diptych, chances are great that the "voracity" of the film will overrule the backgrounds of the images in the fringe of the visible. But chances are great also that the pragmatic context and the experience of the first piece of the diptych will engage us to try to identify the link between the two excerpts, and to renew our attention. We will succeed in establishing this link more or less quickly by catching up the image broadcasted on a TV screen in the background of the Torrance's apartment (which his an excerpt from the western Carson City, 1952, by André de Toth10 ). All of these will play a part in this first stage of the reception of the work, a stage in which the attention of the viewer might be of a "divergent" style, as Schaeffer would put it, that is to say an attentional activity dimly selective, unranked, segmented which will delay the stage of integration of the datum and of categorization. If the "hedonic valency" (as Schaeffer calls it) that is produced by the aesthetic experience of this work is sufficient-and that's my strong wish-, then the experience will extend and wether move on to a state of pensitivity (in which maybe the figurative echoes between an elevator and a tunnel will resonate...) or wether simply go on in an increased state of sensitivity... And maybe, then, will we finally recognize, behind Jack's character, the TV in the hallway, the same prop that we saw in the other piece of the diptych... And maybe will we incidentally, after a period of time, notice this woman with a bun who change her position in the armchair from one shot to the other...

  Fig.1

Jean--Marie Schaeffer, L'expérience esthétique, Paris : Gallimard, 2015, pp. 249--250. Here and after, my translations.

Ibidem, p.

6 http://www.christies.com/media--library/images/features/articles/2017/03/21/twombly/twombly.jpg?w=780 7 J.--M. Schaeffer, op. cit., pp. 85--86. 8 Cf. Loig Le Bihan, Shining au miroir. Surinterprétations, Aix--en Provence : Rouge profond, 2017.

On the counterbalance of fluence with curiosity in aesthetic relationship, cf. "De la fluence à la curiosité et à l'intérêt"in J.--M.Schaeffer, op. cit.,

For a critical explanation of the use of this excerpt from Carson City, cf. the section of my book titled "Carson City et le double tunnel" (Loig Le Bihan,Shining au miroir, op. cit.,


David D Morrison

J Ames

D Rrley

John F Zancanaro

Multiple Shooting Method for Two-Point Boundary Value Problems*

Introduction

The common techniques for solving two-point boundary value problems can be classified as either "shooting" or "finite difference" methods. Central to a shooting method is the ability to integrate the differential equations as an initial value problem with guesses for the unknown initial values. This ability is not required with a finite difference method, for the unknowns are considered to be the values of the true solution at a number of interior mesh points. Each method has its advantages and disadvantages. One serious shortcoming of shooting becomes apparent when, as happens altogether too often, the differential equations are so unstable that they "blow up" before the initial value problem can be completely integrated. This can occur even in the face of extremely accurate guesses for the initial values. Hence, shooting seems to offer no hope for some problems. A finite difference method does have a chance for it tends to keep a firm hold on the entire solution at once. The purpose of this note is to point out a compromising procedure which endows shooting-type methods with this particular advantage of finite difference methods. For such problems, then, all hope need not be abandoned for shooting methods. This is desirable because shooting methods are generally faster than finite difference methods.

The organization is as follows: I. The two-point boundary value problem is stated in quite general form. II. A particular shooting method is described which is designed to solve the problem in this form.

---*Work supported by a research contract at Space Technology Laboratories, Inc.

Ill. The two-point boundary value problem is then restated in such a way that: (a) the restatement still falls within the general form, and (b) the shooting method now has a better chance of success when the equations are unstable.

The Method

I. The two-point boundary problem may be stated as follows:

if= g(y,t), j[y(to), y(T)] = 0 (1)
where y, g and fare vectors of order n; f and g may be nonlinear.

II. A shooting method which has been successful in many cases is the following: Let y(t) be the true solution of (1), and let y 0 = y(to). Let fjo be an approximation to yo , and define y( t) to be the solution of the initial value problem:

fj = g(y, t), y(to) = Yo • (2)
Thus y satisfies the differential equation, but not (in general) the boundary conditions; that is, j(Y(to), y( T)] = E.

(

) 3
Assume the existence of the partial derivatives necessary to construct the matrices P, JYI and N, where

ag;(t) af; af; Pu = ayj(t) ' 1ni,j = ayi(to)' ni,j = ayi(T) ( 4) (i, j = 1, 2, • • • , n). Let 'f'J(t) = y(t) -y(t). Finding y(t) is then equivalent to finding 'f'J(t) so that fj + iJ = g(y + ' f' J, t), j[y(to) + 'f'J(to), y(T) + 'f'J(T)] = 0.
(5) Expanding these expressions in a first order Taylor series and using (2) gives the approximate formula (6)

The partial derivatives are to be evaluated for the approximate solution y.

This suggests the following iterative procedure: where Uo is the identity matrix of order n, except that the ith diagonal element of Uo may be suppreEsed if the ith component of y(t0) is a known eondition. If k components of y(t0 ) are known, then only n(n -k) variational equations need to be integrated instead of n 2 • 2. Compute M, Nand f.

3. Solve the linear algebraic system [MU0 + NU(T) ]a =f. Then 11 = Ua satisfies the required equations (6) as may be verified by direct substitution. 4. Form 'lo = U 0 a which provides a correction to y(t0) = iio . Then iio + 'lo replaces iio ; stop or go back to 1. This process may be iterated until appropriate convergence tests are satisfied. (This method, incidentally, is similar to that in [START_REF] Goodman | The numerical integration of two-point boundary value problems[END_REF] but has the advantages that the problem solved is more general and no backwards integration is needed.)

A modification often helpful at step 3 is, instead, to minimize II[MUo -1-NU(T)]a + ~11 2 under the constraints that lajl :::=;; Mi where the J.l!Ij are prescribed constants.

This enables one to remain close to the current iterant y(O) in case the unconstrained solution is too large to give a useful correction.

Ill. The multiple-shooting method can be described as a compromise between shooting and solving implicit finite difference equations. Its only devices are a general shooting method, such as the one introduced above, and a restatement of the problem. Without loss of generality let to = 0. The interval from 0 to T is split into k equal subintervals. In each subinterval, the dependent variables are denoted by new symbols. There are now nk dependent variables, but the interval of integration is 0 to T jk. By changing the sign of iJ in an interval, the direction of integration in that interval is reversed. Thus, if the integration is unstable in only one direction in the original problem and the sign of iJ is changed in half the subintervals, then half the integrations will be in the stable direction. Moreover, the subintervals can be made short enough so that the integration will end before any instability takes over. As in finite difference methods a firm hold is kept on the solution at the splitting points. Also, all boundary conditions given in the original problem can b3 satisfied at once. The new boundary conditions are that individual pieces of the same original variable must match up at the interior splitting points. Thus, if Yi denotes a variable in the original interval from (p -1) T /k to pT /k and y i denotes the same variable in the interval

Example

The above multiple shooting method was successful in a case in which simple shooting failed with a more accurate guess. More specifically, the integration from 0 to T was sufficiently unstable to cause overflow when the initial guess was correct to three significant figures. Yet splitting the interval into four subintervals and using guesses correct to only two significant figures led to convergence. The specific problem was: A value for y;(O) is then guessed for i = 2, 3, 4, 8 (thus determining the guesses for i = 5 and 6) and the general shooting procedure can be applied.

1 .

 1 Simultaneously integrate from to to T U = PU, U(to) = Uo (7) and l~ = g(Y, t), y(to) = iio, (8)

  from pT /k to (p + 1) T jk and if the direction of integration in the later interval is reversed, then in general y;(O) and Yi(O) are guessed and the condition introduced is that y;(T/k) == Yi(T/k). If Ym denotes the same variable in the interval from (p + l)T/k to (p + 2)T/k, another condition is that Yi(O) = Ym(O), etc.

Y1( 0 )

 0 = 1, YI(4) = e4, with solution Yi = et . Y~i+ 2 J Y2i+1 and reversing the direction of integration in the second and fourth intervals yields the boundary conditions:YI(O) = 1 YI ( 1) = Y3 ( 1) y2(1) = Y4(1) Ya(O) = ys(O) Y4(0) = Ys(O) ys(l) = Y1( 1)ys(1) = Ys(1)y1(0) = e 4