Alexey Ponomarenko
email: alexey.ponomarenko.25.12.1993@yandex.ru

Keywords: Sophie Germain prime, safe prime, strong prime, parity problem

A lower bound is given for the number of primes in a special linear form less than N , under the assumption of the weakened Elliott-Halberstam conjecture.

Introduction

Using the weight function of the form (q,p t -prime numbers) it can be proved that there are infinitely many prime numbers in linear form

p r = 2p u + a.
The main role is played by the upper bound for the sum for numbers of the form

2p 1 p 2 ∈ A = {p-a; p ≤ N, p ∈ P, z ≤ p 1 < y ≤ p 2 , |a| ≤ (log N ) B , B ≥ 3}.

Main results

Theorem 2.1. Assuming (1.2) there are infinitely many primes of the form

p r = 2p u + a,
where a is an arbitrary fixed odd integer.

The proof of Theorem 2.1 is given at the end of the paper. We now give several intermediate theorems and lemmas.

Theorem 2.2. (See Theorem 9.7 (Jurkat-Richert) [START_REF] Fouvry | On the switching principle in sieve theory[END_REF])

Let J = {a(n)} ∞ n=1 be an arithmetic function such that a(n) ≥ 0 f or all n and |J| = ∞ n=1 a(n) < ∞.
Let P be a set of prime numbers (2 ∈ P) and, for z ≥ 2, let

P (z) = p∈P p<z p. Let S(J, P, z) = ∞ n=1 (n,P (z))=1 a(n).
For every n ≥ 1, let g n (d) be a multiplicative function such that 0 ≤ g n (p) < 1 f or all p ∈ P.

Define r(d) by

|J d | = ∞ n=1 d|n a(n) = ∞ n=1 a(n)g n (d) + r(d).
Let Q be a finite subset of P, and let Q be the product of the primes in Q. Suppose that, for some satisfying 0 < < 1 200 , the inequality

p∈P\Q u≤p<z (1 -g n (p)) -1 < (1 + ) log z log u
holds for all n and 1 < u < z. Then for any D ≥ z there is the upper bound (2.1) S(J, P, z) < (F (s) + e 14-s )X + R, and for any D ≥ z 2 there is the lower bound

(2.2) S(J, P, z) > (f (s) -e 14-s )X -R, where s = log D log z , F (s
) and f (s) are the continuous functions defined as

F (s) = 1+ ∞ n=1 n≡1(mod2) f n (s) f or s ≥ 1; f (s) = 1- ∞ n=2 n≡0(mod2) f n (s) f or s ≥ 2, X = ∞ n=1 a(n) p|P (z) (1 -g n (p)),
and the remainder term is

R = d|P (z) d<DQ |r(d)|. If there is a multiplicative function g(d) such that g n (d) = g(d) for all n, then X = V (z)|J|, where V (z) = p|P (z) (1 -g(p)).
λ(d) = rs=d δ r η s . Let 0 < < 1 8 , 2 ≤ z ≤ D. Then from Theorem 4 [3] it follows (2.3) S(J, P, z) ≤ X(F ( log D log z ) + E) + l<L d|P (z) λ + l (d)r(J, d).
J,P,z,X are defined as in Theorem 2.2. In this formula, L depends only on and λ + l -is well factorable coefficient of order 1 and of level D, and the constant E satisfies

E = O( + -8 e K (log D) -1 3 ),
where K is some constant > 1. Using the definition given in Theorem 2.2

F (s) = 2e γ s f or 0 < s ≤ 2 with γ -Euler-Mascheroni constant.
α h β m - 1 ϕ(d) (hm,d)=1 α h β m   = O B N (log N ) B
is true for B ≥ 3, uniformly for |a| ≤ (log N ) B , for any positive , for any ν ( ≤ ν ≤ 1 -), with D = N θ(ν)-, where the function θ(ν) has the following value:

           2 3 -ν 3 f or 1 4 < ν ≤ 2 7 , 1 2 + ν 4 f or 2 7 ≤ ν ≤ 2 5 , 1 -ν f or 2 5 ≤ ν ≤ 1 2 . Lemma 2.5. To estimate the sum 1 2 2n∈A 2n=2p 1 p 2 z≤p 1 <y≤p 2 1
we pass from one set A to another F (switching principle), we obtain

(2.5) 1 2 S(F, P, f ) ≤ 0.1773748 e γ 2 N log N V (z) + O N (log N ) B ,
where

F = {2p 1 p 2 +a : z ≤ p 1 < y ≤ p 2 , 2p 1 p 2 < N, (2p 1 p 2 , N ) = 1, |a| ≤ (log N ) B , B ≥ 3}.
Proof. The remainder term of the sieving function S(F, P, f ) by Lemma 2.3 will be equal to

d λ(d) |F d | - |F | φ(d) = O N (log N ) B
with (2, d) = 1 and D = N θ(ν)-. The minimum value for θ(ν) is defined in Lemma 2.4, i.e

θ log p 1 log N ≥ 4 7 f or z ≤ p 1 ≤ y. Since V (f ) V (z) = log z log f 1 + O 1 log N = 0.50002 + O 1 log N
(using the definition for V (z) in Theorem 2.2) we have

1 2 S(F, P, f ) ≤ 0.50002 7 8 e γ 2 1/3 0.25001 dt t(1 -t)) N log N V (z) + O N (log N ) B .

Proof of Theorem 2.1

Let z(2p + a, N ) be the number of primes of the form p r = 2p u + a ≤ N , where a is an arbitrary fixed odd integer and N > e |a| We give a lower bound for z(2p + a, N ) using the weight function (1.1).

z(2p + a, N ) ≥ 2n∈A n∈{1,p≥z} 1 ≥ 2n∈A (2n,P (z))=1 n∈{1,p≥z} 1 ≥ 2n∈A (2n,P (z))=1 v(2n).
Now open the last sum and applying the switching principle for the set A we obtain

z (2p+a, N ) = S(A, P, z)- 1 2 z≤q<y S(A q , P, z)- 1 2 S(B, P, f )- 1 2 S(F, P, f )-S(E, P, f )+O(N 3 4 ),
where z(2p + a, N ) ≥ z (2p + a, N ),

B = {2p 1 p 2 p 3 +a : z ≤ p 1 < y ≤ p 2 ≤ p 3 , 2p 1 p 2 p 3 < N, (2p 1 p 2 p 3 , N ) = 1, |a| ≤ (log N ) B , B ≥ 3} and E = {2p 1 p 2 +a : y ≤ p 1 < f ≤ p 2 , 2p 1 p 2 < N, (2p 1 p 2 , N ) = 1, |a| ≤ (log N ) B , B ≥ 3}.
The first two sums in z (2p + a, N ) are estimated using the Theorem 2.2 and the weakened Elliott-Halberstam conjecture (1.2) with

f (s) = 2e γ log(s -1) s f or s = log D log z ∈ [3, 4]; F (s q ) = 2e γ s q f or s q = log D q log z ∈ (0, 3].
Acting as in Theorem 10.4 [START_REF] Nathanson | Additive Number Theory: The Classical Bases[END_REF] and Theorem 10.5 [START_REF] Nathanson | Additive Number Theory: The Classical Bases[END_REF], only with the value |a| ≤ (log N ) B , B ≥ 3, z = N 0.25001 and D = N 1-C , C ≈ 0.002 we obtain

S(A, P, z) ≥ (f (s)-e 14-s )V (z) ∞ n=1 a(2n)+O N (log N ) B ≥ 1.0981287 e γ 2 N log N V (z), respectively 1 2 z≤q<y S(A q , P, z) ≤ 0.50002 * e γ N z≤q<y 1 φ(q) log D q + O N (log N ) B ≤ ≤ 0.50002 * e γ 1/3 0.25001 dt t(0.998 -t)) N log N V (z).
The third sum in z (2p + a, N ) is estimated as for Theorem 10.6 [START_REF] Nathanson | Additive Number Theory: The Classical Bases[END_REF], only with the value |a| ≤ (log N ) B , B ≥ 3 and z = N 0.25001 and since

V (f ) V (z) = log z log f 1 + O 1 log N = 0.50002 + O 1 log N
(using the definition for V (z) in Theorem 2.2) we obtain An estimate for the fourth sum is given in Lemma 2.5.

( 1 . 1 ) 2 z≤q<y q k 2n k - 1 2 2p 1 p 2 p 3 =2n z≤p 1 <y≤p 2 ≤p 3 1 - 1 2 2p 1 p 2 =2n z≤p 1 <y≤p 2 1 - 2p 1 p 2 =2n y≤p 1 ≤f ≤p 2 1 ,

 11223211 v(2n) = 1 -1 (such a weight function v(2n) leaves only prime numbers when sifting (i.e. v(2n) = 1 with n = p ≥ z and v(2n) ≤ 0 for other values of n) x) B , where |a| ≤ (log N ) B ; D = N 1-C ; C ≈ 0.002; B ≥ 3,

Lemma 2 . 3 .

 23 (See Theorem 4, Theorem 1 [3] and Lemma 2 [1]) An arithmetic function λ(d) is said to be well-factorable of level D ≥ 1 if for any R, S ≥ 1 with RS = D there are functions δ r , η s with |δ r |, |η s | ≤ 1 supported on r < R, s < S, such that

Lemma 2 .

 2 4. (See Lemma 6 [1]) We denote by |α h |, |β m | ≤ 1 two sequences with h ∈ [H, 2H) and m ∈ [M, 2M ), also define ν = log H log N , N = 2HM and the following equality

1 B

 1 ; B = 3. We also denote A = {p-a; p ≤ N, p ∈ P, |a| ≤ (log N ) B , B ≥ 3} and P (z) = p∈P p<z p.

1 2 S<y≤p 2 ≤p 3 2p 1 p 2 p 3 ≤N

 123 (B, P, f ) ≤ 0.50002 * e γ V (z) z≤p 1

  It remains to estimate the last sum in z (2p + a, N ). Acting as in Lemma 2.5, we choose the minimum value of the functionθ log p 1 log N ≥ 1 2 f or y ≤ p 1 ≤ f, so we have S(E, P, f ) ≤ 0.50002 * e γ N ) B ≤

Acknowledgements

None.

≤ 0.693175 e γ 2 N log N V (z).

Putting together estimates for the sums in z (2p + a, N ), we obtain z(2p+a, N ) ≥ (1.0981287-0.2032878-0.0240915-0.1773748-0.693175)

for sufficiently large N , this proves Theorem 2.1.


Samy Skander Bahoura

Samy Skander Bahoura

ABOUT BREZIS-MERLE PROBLEM WITH LIPSCHITZ CONDITION

Keywords: Mathematics Subject Classification: 35J60 35B45 35B50 blow-up, boundary, Dirichlet condition, a priori estimate, Lipschitz condition, smooth or C 2, α domain

published or not. The documents may come     About Brezis-Merle Problem with Lipschitz condition.

INTRODUCTION AND MAIN RESULTS

We set ∆ = -(∂ 11 + ∂ 22 ) on open set Ω of R 2 with a smooth (or C 2,α , α > 0) boundary.

We consider the following equation:

(P ) ∆u = V e u in Ω ⊂ R 2 , u = 0 in ∂Ω.
Here, we assume that: 1 (Ω) and u ∈ W 1,1 0 (Ω).

0 ≤ V ≤ b < +∞, e u ∈ L
We can see in [START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF] a nice formulation of this problem (P ) in the sense of the distributions. This Problem arises from geometrical and physical problems see for example [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF][START_REF] Bandle | Isoperimetric Inequalities and Applications[END_REF][START_REF] Nagasaki | Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities[END_REF][START_REF] Tarantello | Multiple condensate solutions for the Chern-Simons-Higgs theory[END_REF]. The above equation was studied by many authors, with or without the boundary condition, also for Riemannian surfaces, see [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF][START_REF] Bandle | Isoperimetric Inequalities and Applications[END_REF][START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF][START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF][START_REF] Brezis | Nonlinear elliptic equations with measures revisited[END_REF][START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF][START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF][START_REF] Chang | Scalar curvature equation on 2and 3-spheres[END_REF][START_REF] Chen | A priori estimates for solutions to nonlinear elliptic equations[END_REF][START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF][START_REF] De Figueiredo | A priori Estimates and Existence of Positive Solutions of Semilinear Elliptic Equations[END_REF][START_REF] Ding | The differential equation ∆u = 8π -8πhe u on a compact Riemann surface[END_REF][START_REF] Gilbarg | Elliptic Partial Differential Equations of Second order[END_REF][START_REF] Yy | Harnack Type Inequality: the method of moving planes[END_REF][START_REF] Yy | Blow-up analysis for solutions of -∆u = V e u in dimension two[END_REF][START_REF] Ma | Convergence for a Liouville equation[END_REF][START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF][START_REF] Nagasaki | Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities[END_REF][START_REF] Tarantello | Multiple condensate solutions for the Chern-Simons-Higgs theory[END_REF], where one can find some existence and compactness results. In [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] we have the following important Theorem, Theorem A(Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]).For (u i ) i and (V i ) i two sequences of functions relative to (P ) with,

0 < a ≤ V i ≤ b < +∞ then, for all compact subset K of Ω it holds, sup K u i ≤ c,
with c depending on a, b, K and Ω.

One can find in [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] an interior estimate if we assume a = 0, but we need an assumption on the integral of e u i , namely, we have: Theorem B(Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]).For (u i ) i and (V i ) i two sequences of functions relative to the problem (P ) with, 0 ≤ V i ≤ b < +∞ and Ω e u i dy ≤ C, then, for all compact subset K of Ω it holds;

sup K u i ≤ c,
with c depending on b, C, K and Ω.

We look to the uniform boundedness on all Ω of sequences of solutions of the Problem (P ). Remark that, when a = 0 the boundedness of Ω e u i is a necessary condition in the problem (P ) as showed in [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] by the following counterexample.

Theorem C(Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]).There are two sequences (u i ) i and (V i ) i of the problem (P ) with,

0 ≤ V i ≤ b < +∞ and Ω e u i dy ≤ C, such that, sup Ω u i → +∞.
To obtain the two first previous results (Theorems A and B) Brezis and Merle used an inequality (Theorem 1 of [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]) obtained by an approximation argument with the Fatou's lemma and they applied the maximum principle in W 1,1 0 (Ω) which arises from Kato's inequality. Also this weak form of the maximum principle is used to prove the local uniform boundedness result by comparing a certain function and the Newtonian potential. We refer to [START_REF] Brezis | Nonlinear elliptic equations with measures revisited[END_REF] for a topic about the weak form of the maximum principle.

Remarks: 1) Theorem 1 of [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF], can be obtained by the usual maximum principle and Agmon regularity theorem which require C 2 regularity on the domain.

2) The duality Theorem which we use require C 2 regularity on the domain, see Gilbarg-Trudinger books.

Note that for the problem (P ), by using the Pohozaev identity, we can prove that Ω e u i is uniformly bounded when 0 < a ≤ V i ≤ b < +∞ and ||∇V i || L ∞ ≤ A and Ω starshaped, when a = 0 and ∇ log V i is uniformly bounded, we can bound uniformly Ω V i e u i . In [START_REF] Ma | Convergence for a Liouville equation[END_REF] Ma-Wei have proved that those results stay true for all open sets not necessarily starshaped.

In [START_REF] Chen | A priori estimates for solutions to nonlinear elliptic equations[END_REF] Chen-Li have proved that if a = 0 and ∇ log V i is uniformly bounded, then the functions are uniformly bounded near the boundary.

In [START_REF] Chen | A priori estimates for solutions to nonlinear elliptic equations[END_REF] Chen-Li have proved that if a = 0 and Ω e u i is uniformly bounded and ∇ log V i is uniformly bounded, then we have the compactness result directly. Ma-Wei in [START_REF] Ma | Convergence for a Liouville equation[END_REF], extend this result in the case where a > 0.

If we assume V more regular, we can have another type of estimates called sup + inf type inequalities. It was proved by Shafrir see [START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF] that, if (u i ) i , (V i ) i are two sequences of functions solutions of the previous equation without assumption on the boundary and, 0 < a ≤ V i ≤ b < +∞, then we have the following interior estimate:

C a b sup K u i + inf Ω u i ≤ c = c(a, b, K, Ω).
One can see in [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF] an explicit value of C a b = a b . In his proof Shafrir has used a blow-up function, the Stokes formula and an isoperimetric inequality see [START_REF] Bandle | Isoperimetric Inequalities and Applications[END_REF]. For Chen-Lin, they have used the blow-up analysis combined with some geometric type inequality for the integral curvature.

Now, if we suppose (V i ) i uniformly Lipschitzian with A its Lipschitz constant then C(a/b) = 1 and c = c(a, b, A, K, Ω) see Brezis-Li-Shafrir [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF]. This result was extended for Hölderian sequences (V i ) i by Chen-Lin, see [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF]. Also, one can see in [START_REF] Yy | Harnack Type Inequality: the method of moving planes[END_REF] an extension of the Brezis-Li-Shafrir result to compact Riemannian surfaces without boundary. One can see in [START_REF] Yy | Blow-up analysis for solutions of -∆u = V e u in dimension two[END_REF] explicit form, (8πm, m ∈ N * exactly), for the numbers in front of the Dirac masses when the solutions blow-up. Here, the notion of isolated blow-up point is used.

In [START_REF] Chang | Scalar curvature equation on 2and 3-spheres[END_REF] we have some a priori estimates on the 2 and 3-spheres S 2 , S 3 .

Here we give the behavior of the blow-up points on the boundary and a proof of Brezis-Merle Problem with Lipschitz condition.

The Brezis-Merle Problem (see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]) is:

Problem. Suppose that V i → V in C 0 ( Ω) with 0 ≤ V i . Also, we consider a sequence of solutions (u i ) of (P ) relative to (V i ) such that, Ω e u i dx ≤ C,
is it possible to have:

||u i || L ∞ ≤ C?
Here, we give a caracterization of the behavior of the blow-up points on the boundary and also, in particular we extend Chen-Li theorems, indeed, the result of Chen-Li holds for analytic domains and our result holds for smooth of C 2,α domains. For the behavior of the blow-up points on the boundary, the following condition is enough,

0 ≤ V i ≤ b.
The condition V i → V in C 0 ( Ω) is not necessary, but for the proof of the compactness for the Brezis-Merle problem we assume that:

||∇V i || L ∞ ≤ A.
Our main results are: Theorem 1.1. Assume that max Ω u i → +∞, where (u i ) are solutions of the problem (P ) with:

0 ≤ V i ≤ b and Ω e u i dx ≤ C, ∀ i,
then, after passing to a subsequence, there is a function u, there is a number N ∈ N and there are N points x 1 , . . . , x N ∈ ∂Ω, such that,

∂ ν u i → ∂ ν u + N j=1 α j δ x j , α j ≥ 4π, in the sense of measures on ∂Ω. u i → u in C 1 loc ( Ω -{x 1 , . . . , x N }). Theorem 1.2.
Assume that (u i ) are solutions of (P ) relative to (V i ) with the following conditions:

0 ≤ V i ≤ b, ||∇V i || L ∞ ≤ A and Ω e u i ≤ C, we have, ||u i || L ∞ ≤ c(b, A, C, Ω).
In the previous theorem we have a proof of the global a priori estimate which concern the problem (P ). The proof of Chen-Li and Ma-Wei [START_REF] Chen | A priori estimates for solutions to nonlinear elliptic equations[END_REF][START_REF] Ma | Convergence for a Liouville equation[END_REF], use the moving-plane method for the case ∇ log V i uniformly bounded near the boundary (and C 2,α domain, 1 ≥ α > 0) and for analytic domain for the case ∇V i uniformly bounded.

To prove Theorem 1.2, we argue by contradiction and use Theorem 1.1.

PROOF OF THE THEOREMS

Proof of theorem 1.1:

We have:

u i ∈ W 1,1 0 (Ω).
Since e u i ∈ L 1 (Ω) by the corollary 1 of Brezis-Merle's paper (see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]) we have e u i ∈ L k (Ω) for all k > 2 and the elliptic estimates of Agmon and the Sobolev embedding (see [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]) imply that:

u i ∈ W 2,k (Ω) ∩ C 1,ǫ ( Ω).
We denote by ∂ ν u i the inner normal derivative. By the maximum principle we have, ∂ ν u i ≥ 0.

By the Stokes formula we have,

∂Ω ∂ ν u i dσ ≤ C,
We use the weak convergence in the space of Radon measures to have the existence of a nonnegative Radon measure µ such that,

∂Ω ∂ ν u i ϕdσ → µ(ϕ), ∀ ϕ ∈ C 0 (∂Ω).
We take an x 0 ∈ ∂Ω such that, µ(x 0 ) < 4π. For ǫ > 0 small enough set I ǫ = B(x 0 , ǫ) ∩ ∂Ω. We choose a function η ǫ such that,

           η ǫ ≡ 1, on I ǫ , 0 < ǫ < δ/2, η ǫ ≡ 0, outside I 2ǫ , 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (I 2ǫ ) ≤ C 0 (Ω, x 0 ) ǫ .
We take a ηǫ such that,

∆η ǫ = 0 in Ω ⊂ R 2 , ηǫ = η ǫ in ∂Ω.
Remark: We use the following steps in the construction of η ǫ :

We take a cutoff function η 0 in B(0, 2) or B(x 0 , 2):

1-We set η ǫ (x) = η 0 (|x -x 0 |/ǫ) in the case of the unit disk it is sufficient.

2-Or, in the general case: we use a chart (f, Ω = f (B r (0))), for r > 0 small enough and f (0) = x 0 and we take µ ǫ (x) = η 0 (f (|x|/ǫ)) to have connected sets I ǫ and we take η ǫ (y

) = µ ǫ (f -1 (y)). Because f, f -1 are Lipschitz, |f (x) -x 0 | ≤ k 2 |x| ≤ 1 for |x| ≤ 1/k 2 and |f (x) -x 0 | ≥ k 1 |x| ≥ 2 for |x| ≥ 2/k 1 > 1/k 2 , the support of η is in I (2/k 1 )ǫ .            η ǫ ≡ 1, on f (I (1/k 2 )ǫ ), 0 < ǫ < δ/2, η ǫ ≡ 0, outside f (I (2/k 1 )ǫ ), 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (I (2/k 1 )ǫ ) ≤ C 0 (Ω, x 0 ) ǫ .
3-Also, we can take:

µ ǫ (x) = η 0 (|x|/ǫ) and η ǫ (y) = µ ǫ (f -1 (y)), we extend it by 0 outside f (B 1 (0)). We have f (B 1 (0)) = D 1 (x 0 ), f (B ǫ (0)) = D ǫ (x 0 ) and f (B + ǫ ) = D + ǫ (x 0 ) with f and f -1 smooth diffeo- morphism.            η ǫ ≡ 1, on the connected set J ǫ = f (I ǫ ), 0 < ǫ < δ/2, η ǫ ≡ 0, outside J ′ ǫ = f (I 2ǫ ), 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (J ′ ǫ ) ≤ C 0 (Ω, x 0 ) ǫ .
And,

H 1 (J ′ ǫ ) ≤ C 1 H 1 (I 2ǫ ) = C 1 4ǫ, because f is Lipschitz.
Here H 1 is the Hausdorff measure.

We solve the Dirichlet Problem:

∆η ǫ = ∆η ǫ in Ω ⊂ R 2 , ηǫ = 0 in ∂Ω.
and finaly we set ηǫ = -η ǫ + η ǫ . Also, by the maximum principle and the elliptic estimates we have :

||∇η ǫ || L ∞ ≤ C(||η ǫ || L ∞ + ||∇η ǫ || L ∞ + ||∆η ǫ || L ∞ ) ≤ C 1 ǫ 2 ,
with C 1 depends on Ω.

We use the following estimate, see [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF][START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF][START_REF] Tarantello | Multiple condensate solutions for the Chern-Simons-Higgs theory[END_REF],

||∇u i || L q ≤ C q , ∀ i and 1 < q < 2.
We deduce from the last estimate that, (u i ) converge weakly in W 1,q 0 (Ω), almost everywhere to a function u ≥ 0 and Ω e u < +∞ (by Fatou's lemma). Also, V i weakly converge to a nonnegative function V in L ∞ . The function u is in W 1,q 0 (Ω) solution of :

∆u = V e u ∈ L 1 (Ω) in Ω ⊂ R 2 , u = 0 in ∂Ω.
According to the corollary 1 of Brezis-Merle result, see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF], we have e ku ∈ L 1 (Ω), k > 1. By the elliptic estimates, we have u ∈ C 1 ( Ω).

For two vectors v, w of R 2 we denote by v • w the inner product of v and w.

We can write,

∆((u i -u)η ǫ ) = (V i e u i -V e u )η ǫ -2∇(u i -u) • ∇η ǫ . (1)
We use the interior esimate of Brezis-Merle, see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF],

Step 1: Estimate of the integral of the first term of the right hand side of (1).

We use the Green formula between ηǫ and u, we obtain,

Ω V e u ηǫ dx = ∂Ω ∂ ν uη ǫ ≤ C ′ ǫ||∂ ν u|| L ∞ = Cǫ (2)
We have,

∆u i = V i e u i in Ω ⊂ R 2 , u i = 0 in ∂Ω.
We use the Green formula between u i and ηǫ to have:

Ω V i e u i ηǫ dx = ∂Ω ∂ ν u i η ǫ dσ → µ(η ǫ ) ≤ µ(J ′ ǫ ) ≤ 4π -ǫ 0 , ǫ 0 > 0 (3)
From (2) and (3) we have for all ǫ > 0 there is i 0 = i 0 (ǫ) such that, for i ≥ i 0 ,

Ω |(V i e u i -V e u )η ǫ |dx ≤ 4π -ǫ 0 + Cǫ (4)
Step 2: Estimate of integral of the second term of the right hand side of (1).

Let Σ ǫ = {x ∈ Ω, d(x, ∂Ω) = ǫ 3 } and Ω ǫ 3 = {x ∈ Ω, d(x, ∂Ω) ≥ ǫ 3 }, ǫ > 0. Then, for ǫ small enough, Σ ǫ is a manifold. The measure of Ω -Ω ǫ 3 is k ǫ 3 ≤ meas(Ω -Ω ǫ 3 ) = µ L (Ω -Ω ǫ 3 ) ≤ k 1 ǫ 3 .
Here µ L is the Lebesgue measure.

Remark: for the unit ball B(0, 1), our new manifold is B(0, 1 -ǫ 3 ).

( Proof of this fact; let's consider d(x, ∂Ω) = d(x, z 0 ), z 0 ∈ ∂Ω, this imply that (d(x, z 0 )) 2 ≤ (d(x, z)) 2 for all z ∈ ∂Ω which it is equivalent to (z -z 0 ) • (2x -z -z 0 )
≤ 0 for all z ∈ ∂Ω, let's consider a chart around z 0 and γ(t) a curve in ∂Ω, we have;

(γ(t) -γ(t 0 ) • (2x -γ(t) -γ(t 0 )) ≤ 0 and it is clear that, γ ′ (t 0 ) • (x -γ(t 0 )) = 0, which imply that x = z 0 -sν 0 where ν 0 is the outward normal of ∂Ω at z 0 ))
With this fact, we can say that S = {x, d(x, ∂Ω) ≤ ǫ} = {x = z 0 -sν z 0 , z 0 ∈ ∂Ω, -ǫ ≤ s ≤ ǫ}. It is sufficient to work on ∂Ω. Let's consider a charts (z, D = B(z, 4ǫ z ), γ z ) with z ∈ ∂Ω such that ∪ z B(z, ǫ z ) is cover of ∂Ω . One can extract a finite cover (B(z k , ǫ k )), k = 1, ..., m, by the area formula the measure of S ∩ B(z k , ǫ k ) is less than a kǫ (a ǫ-rectangle). For the reverse inequality, it is sufficient to consider one chart around one point on the boundary).

We write,

Ω |∇(u i -u) • ∇η ǫ |dx = Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx + Ω-Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx.
(5)

Step 2.1:

Estimate of Ω-Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx.
First, we know from the elliptic estimates that

||∇η ǫ || L ∞ ≤ C 1 /ǫ 2 , C 1 depends on Ω
We know that (|∇u i |) i is bounded in L q , 1 < q < 2, we can extract from this sequence a subsequence which converge weakly to h ∈ L q . But, we know that we have locally the uniform convergence to |∇u| (by Brezis-Merle's theorem), then, h = |∇u| a.e. Let q ′ be the conjugate of q.

We have, ∀f ∈ L q ′ (Ω)

Ω |∇u i |f dx → Ω |∇u|f dx
If we take f = 1 Ω-Ω ǫ 3 , we have:

for ǫ > 0 ∃ i 1 = i 1 (ǫ) ∈ N, i ≥ i 1 , Ω-Ω ǫ 3 |∇u i | ≤ Ω-Ω ǫ 3 |∇u| + ǫ 3 . Then, for i ≥ i 1 (ǫ), Ω-Ω ǫ 3 |∇u i | ≤ meas(Ω -Ω ǫ 3 )||∇u|| L ∞ + ǫ 3 = ǫ 3 (k 1 ||∇u|| L ∞ + 1).
Thus, we obtain,

Ω-Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx ≤ ǫC 1 (2k 1 ||∇u|| L ∞ + 1) (6)
The constant C 1 does not depend on ǫ but on Ω.

Step 2.2: Estimate of

Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx.
We know that, Ω ǫ ⊂⊂ Ω, and ( because of Brezis-Merle's interior estimates)

u i → u in C 1 (Ω ǫ 3 ). We have, ||∇(u i -u)|| L ∞ (Ω ǫ 3 ) ≤ ǫ 3 , for i ≥ i 3 = i 3 (ǫ).
We write,

Ω ǫ3 |∇(u i -u) • ∇η ǫ |dx ≤ ||∇(u i -u)|| L ∞ (Ω ǫ 3 ) ||∇η ǫ || L ∞ ≤ C 1 ǫ for i ≥ i 3 , For ǫ > 0, we have for i ∈ N, i ≥ max{i 1 , i 2 , i 3 }, Ω |∇(u i -u) • ∇η ǫ |dx ≤ ǫC 1 (2k 1 ||∇u|| L ∞ + 2) (7)
From (4) and (7), we have, for ǫ > 0, there is i

3 = i 3 (ǫ) ∈ N, i 3 = max{i 0 , i 1 , i 2 } such that, Ω |∆[(u i -u)η ǫ ]|dx ≤ 4π -ǫ 0 + ǫ2C 1 (2k 1 ||∇u|| L ∞ + 2 + C) (8)
We choose ǫ > 0 small enough to have a good estimate of (1).

Indeed, we have:

∆[(u i -u)η ǫ ] = g i,ǫ in Ω ⊂ R 2 , (u i -u)η ǫ = 0 in ∂Ω. with ||g i,ǫ || L 1 (Ω) ≤ 4π -ǫ 0 /2.
We can use Theorem 1 of [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] to conclude that there is q ≥ q > 1 such that:

Vǫ(x 0 ) e q|u i -u| dx ≤ Ω e q|u i -u|ηǫ dx ≤ C(ǫ, Ω).
where, V ǫ (x 0 ) is a neighberhooh of x 0 in Ω. Here we have used that in a neighborhood of x 0 by the elliptic estimates, 1 -Cǫ ≤ ηǫ ≤ 1. (We can take, f (B ǫ 3 (0)) and we have

B k 2 ǫ 3 (x 0 ) ⊂ f (B ǫ 3 (0)) ⊂ B k 1 ǫ 3 (x 0 ) for a chart (f, B 1 (0)) around x 0 ).
Thus, for each x 0 ∈ ∂Ω -{x 1 , . . . , xm } there is ǫ x 0 > 0, q x 0 > 1 such that:

B(x 0 ,ǫx 0 ) e qx 0 u i dx ≤ C, ∀ i. (9)
By the elliptic estimates (see [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second order[END_REF]) (u i ) i is uniformly bounded in W 2,q 1 (V ǫ (x 0 )) and also, in C 1 (V ǫ (x 0 )). Finaly, we have, for some ǫ > 0 small enough,

||u i || C 1,θ [B(x 0 ,ǫ)] ≤ c 3 ∀ i.
We have proved that, there is a finite number of points x1 , . . . , xm such that the squence (u i ) i is locally uniformly bounded (in C 1,θ , θ > 0) in Ω -{x 1 , . . . , xm }.

Proof of theorem 1.2:

We know that:

u i ∈ W 2,k (Ω) ∩ C 1,ǫ ( Ω).
We can do integration by parts. The first Pohozaev identity applied around each blow-up point see for example [START_REF] Ma | Convergence for a Liouville equation[END_REF] gives :

∂Ωx k [(∂ ν u i )∇u i - 1 2 ||∇u i | 2 ν]dx = Ωx k ∇V i e u i - ∂Ωx k V i e u i ν, (10)
Here Ω x k is a neighborhood of x k on which we can use the integration by part obtained by a chart around x k .

We use the boundary condition on Ω and the boundedness of u i and ∂ j u i outside the x k , to have:

∂Ω (∂ ν u i ) 2 dx ≤ c 0 (b, A, C, Ω). ( 11
)
Thus we can use the weak convergence in L 2 (∂Ω) to have a subsequence ∂ ν u i , such that:

∂Ω ∂ ν u i ϕdx → ∂Ω ∂ ν uϕdx, ∀ ϕ ∈ L 2 (∂Ω), (12)
Thus, α j = 0, j = 1, . . . , N and (u i ) is uniformly bounded.

ACKNOWLEDGEMENT.

The work was supported by YY. Li's Grant and Pr Jiguang Bao. The author would like to thank Pr. YY.Li for his support.


Audrey De Jong

Jonathan Casey

Sheila Nainan

HAL is a


P Shi

Z Oger

Yang

Yuxiao Wang

Dr Likui Zhang
email: lkzhang@yzu.edu.cn

Xinyuan Zhu

Yuting Li

Haoqiang Shi

Philippe Oger

Zhihui Yang

Biochemical characterization of a thermostable endonuclease V from the hyperthermophilic euryarchaeon Thermococcus barophilus Ch5

Keywords: Endonuclease V, Thermococcus barophilus, DNA repair, DNA

nicking; Thermostability 1

Introduction

Deamination of DNA is a common lesion caused by spontaneous hydrolysis, endogenous or environmental factors as well as deaminase enzymes . Three of the four DNA bases, cytosine, adenine, and guanine, are deaminated to the corresponding base analogs uracil, hypoxanthine, and xanthine, respectively. While the rate of adenine deamination in single-stranded DNA (ssDNA) is much lower than that of cytosine deamination under physiological conditions in vitro , adenine deamination in double-stranded DNA (dsDNA) induced by nitrous acid occurs at a similar rate as observed in cytosine deamination . Similar to the background levels of 1-10 per 10 6 nucleotides in tissues or cells , deoxyinosine in E. coli and Saccharomyces cerevisiae cells is estimated to be 1.2 and 2.0 per 10 6 nucleotides, respectively. Since deoxyinosine has a strong ability to form mismatch, an A-T base pair would be subsequently converted to a G-C base pair during DNA replication before the deoxyinosine is repaired , potentially leading to mutations in the genome. Increased AT to GC mutations are detrimental to the cells, which may cause genome instability or cancer occurence. However, cells have evolved the corresponding excision repair pathway to counteract potential mutations generated by deoxyinosine replication. Endonuclease V (endo V) is considered to play an important role in initiating deoxyinosine repair in a base excision repair pathway. Endo V, encoded by the nfi gene, was first identified in E. coli as an endonuclease that nicks damaged DNA and was subsequently proved to nick the second phosphodiester bond on the 3′ side of the deaminated base lesion .

Biochemical studies on E. coliE. coli endo V demonstrated that the enzyme is capable for recognizing all three deaminated products, deoxyuridine, deoxyinosine and deoxyxanthosine, and nicking the corresponding damaged DNA. Moreover, E. coliE. coli endo V displays broad substrate specificity towards apurinic/apyrimidinic (AP) site, insertion/deletion mismatches (IDM), flaps, loops and pseudo-Y DNA structures in vitro ; however, genetic studies revealed that the enzyme is able to protect cells from mutagenic detriments and nitrosative deamination during normal growth in vivo . Recently, Zhang et al. solved the crystal structure of E. coli endo V, demonstrating the substrate recognition and catalytic mechanism of the enzyme . In combination with biochemical, genetic and structural studies, E. coli endo V is proposed to recognize deoxyinosine in DNA and initiating repair pathway by hydrolyzing the second phosphodiester bond on the 3′ side of the deoxyinosine. Endo V homologues are ubiquitously distributed in all three domains of life, bacteria, eukarya and archaea . In addition to E. coli endo V, the endo V homologue from the hyperthermophilic bacterium Thermotoga maritima has been well studied .

The crystal structure of T. maritima endo V (Tma endo V) in complex with DNA was reported . Biochemical and structural characterization suggest that Tma endo V is a specific endonuclease for cleaving deoxyinosine-containing DNA substrate. By contrast, the mouse and human endo Vs lacked significant activity towards deoxyuridine, and displayed strong cleaving activity for inosine-containing RNA .

Currently, limited studies on archaeal endo V enzymes have been reported. Similar to the mammalian endo V, the endo V from Archaeoglobus fulgidus only recognizes deoxyinosine-containing DNA substrates . However, Ferroplasma acidarmanus endo V is fused downstream of an O6-alkylguanine-DNA alkyltransferase, and exhibits cleavage activities for DNA substrates containing uracil, hypoxanthine and xanthine bases in vitro . Recently, the endo V from Pyrococcus furiosus displays deoxyinosine endonuclease activity , but its substrate specificity remains unknown. Overall, archaeal endo V enzymes harbor diverse mechanisms for recognizing various DNA lesions; however, knowledge of these enzymes is relatively limited. indicated that this strain possesses an endonuclease V (Tba endo V) . In this study, the Tba endo V gene was cloned and expressed in E. coli. The recombinant Tba endo V was purified and biochemically characterized. We also report that Tba endo V has a strong affinity to bind to a deoxyinosine-containing ssDNA, and specifically cleaves deoxyinosine-containing DNA. This is the first report on biochemical characterization of a thermostable endonuclease V from Thermococcus species.

Materials and methods

Cloning, expression and purification of Tba endo V

The genomic sequence of the T. barophilus Ch5 encodes an endonuclease V (GenBank accession number: ALM74570). The genomic DNA of the T. barophilus Ch5 was prepared as described by [START_REF] Oger | Complete Genome Sequence of the Hyperthermophilic and Piezophilic Archaeon Thermococcus barophilus Ch5, Capable of Growth at the Expense of Hydrogenogenesis from Carbon Monoxide and Formate[END_REF]. By using the genomic DNA of T. barophilus Ch5 as a template, Phusion DNA polymerase (Thermo Scientific, Waltham, MA, USA) was used to amplify the gene encoding Tba endo V in the presence of two primers (see Table 1). The amplified DNA product was cloned into vector pET-30a(+) (Novagen, Merck, Darmstadt, Germany) and the recombinant plasmid containing a sequence encoded a 6 × His-tag at the C-terminus of Tba endo V was sequenced and then transformed into E. coli BL21 (DE3) pLysS cells (Transgene, Beijing, China) for protein expression.

The expression strain E. coli harboring the recombinant plasmid was cultured into LB medium containing 100 µg/ml kanamycin and 34 µg/ml chloramphenicol at 37°C until the OD 600 reached 0.5. And then, isopropyl thiogalactoside (IPTG) was added to the culture with a final concentration of 0.1 mM, and the culture was shaken for another 10 h at room temperature until the OD 600 reached 1.4. The cells were collected by centrifugation (5,000 × g for 20 min at room temperature) and resuspended with Ni column buffer A (20 mM Tris-HCl pH 8.0, 1 mM dithiothreitol (DTT), 500 mM NaCl, 50 mM imidazole and 10% glycerol). The cells were disrupted by sonication in an ice bath. After centrifugation (16,000 × g for 30 min at 4°C), the supernatant was harvested and heated at 80°C for 20 min to remove unthermostable E. coli proteins. The resulting supernatant was applied to a HisTrap FF column (GE Healthcare, Uppsala, Sweden) and purified with AKTA fast protein liquid chromatography (GE Healthcare) by elution with a linear gradient of 50-500 mM imidazole after centrifugation (16,000 × g for 30 min at 4°C). Fractions containing Tba endo V protein were collected and visualized by Coomassie-staining, 10% sodium dodecyl sulfate-polyacrylamide gel electrophoresis (SDS-PAGE) analysis. The purified Tba endo V protein was dialyzed in a storage buffer containing 50 mM Tris-HCl pH8.0, 0.1 mM ethylenediaminetetraacetic acid (EDTA), 1 mM DTT and 50% glycerol, and was stored at -80°C. The protein concentration was determined using a Bradford Protein Assay Kit (Bio-Rad, Hercules, CA, USA).

DNA cleavage assays

The p45 deoxyoligonucleotides containing I and U at the 25 th base, p22, t22 and t45 deoxyoligonucleotides were synthesized at Biotech company, China. The sequences of these deoxyoligonucleotides are shown in Table 1. The deoxyoligonucleotides (p45U, p45I and p22) were labeled with Cy3 at the 5′-terminus (shown by asterisks). The Cy3 labeled deoxyoligonucleotides were annealed to the complementary deoxyoligonucleotides to prepare ds DNA(*p22/t22, *45U/45T and *45I/45T) in an annealing buffer (20 mM Tris-Cl pH 8.0 and 1M NaCl). The endonuclease activity of Tba endo V was performed in the reactions (10 µl) containing 20 mM Tris-HCl pH 8.0, 1 mM DTT, 0.1 mg/ml BSA, 8% glycerol and the purified Tba endo V. The reactions were kept at 85°C for 20 min and stopped by the addition of 10 µl of stop solution containing 98% formamide and 20 mM EDTA.

Samples were heated at 100°C for 5 min and chilled rapidly on ice for 5 min, and then loaded onto a denaturing 15% polyacrylamide gel containing 8M urea. After electrophoresis, the gels were scanned and Cy3-labelled DNA was visualized with a Molecular Image analyser (PharosFx System, BioRad). ImageQuant software was used for the quantitative analysis. The amounts of substrates and enzymes are described in the figure legend.

Assays of optimal temperature and pH for DNA cleavage

Assays to determine the optimal temperature for DNA cleavage by Tba endo V were performed in the reactions (10 µl) containing 200 nM of Cy3-labeled ssDNA (*p45I) and dsDNA (*p45I/t45), 20 mM Tris-HCl pH 8.0, 1 mM DTT, 1 mM MgCl 2 , 0.1mg/ml BSA, 8% glycerol and 300 nM of Tba endo V. The reactions were kept at 30, 40, 50, 60, 70, 80, 90, and 100°C for 20 min, respectively. The reactions were stopped and treated as described above.

The optimal pH for the endonuclease activity Tba endo V was examined using a pH range of 5.0-11.0 in reactions (10 µl) containing 200 nM of Cy3-labeled ssDNA (*p45I), 20 mM buffer with varied pHs, 1 mM DTT, 1 mM MgCl 2 , 0.1 mg/ml BSA, 8% glycerol and 300 nM Tba endo V. Five different buffers (all at 20 mM concentrations) were used to prepare different pHs: acetate-sodium acetate (pH 5.0), sodium phosphate-NaOH (pH 6.0, pH 7.0, and pH 7.5), Tris-HCl (pH 8.0 andpH 8.5), Gly-NaOH (pH 9.0), and, NaHCO 3 -NaOH (pH 10.0, andpH 11.0). The reactions were kept at 85°C for 20 min. The reactions were stopped and treated as described above.

Assays for Tba endo V Thermostability

For its thermostability assays, Tba endo V were heated at 100°C for 20, 40, 60, 90, 120, and 180 min, respectively. The heated Tba endo V were employed to examine DNA cleavage activity in the reactions (10 µl) containing 200 nM of Cy3labeled ssDNA (*p45I), 20 mM Tris-HCl pH 8.0, 1 mM DTT, 1 mM MgCl 2 , 0.1 mg/ ml BSA, 8% glycerol and 300 nM of the heated Tba endoV. The reactions were kept at 85°C for 20 min. The reactions were stopped and treated as described above.

Effects of divalent metal ions and NaCl on DNA cleavage

The effect of divalent metal ions on Tba endo V activity was investigated by the addition of 1 mM of Ca 2+ , Mg 2+ , Zn 2+ , Mn 2+ , Ni 2+ and Co 2+ (analytical purity) in the reactions (10 µl) containing 200 nM of Cy3-labeled ssDNA (*p45I), 20 mM Tris-HCl pH 8.0, 1 mM DTT, 1 mM MgCl 2 , 0.1mg/ml BSA, 8% glycerol and 300 nM Tba endo V. The reactions were kept at 85°C for 20 min. The reactions were stopped and treated as described above.

In addition, we determined the effects of Mg 2+ and Mn 2+ concentrations ranging from 10 nM to 3000 µM on Tba endo V activity in the reactions (10 µl) containing 200 nM of Cy3-labeled ssDNA (*p45I), 20 mM Tris-HCl pH 8.0, 1 mM DTT, 1 mM MgCl 2 , 0.1mg/ml BSA, 8% glycerol and 300 nM of Tba EndoV. The reactions were kept at 85°C for 20 min. The reactions were stopped and treated as described above.

Effect of NaCl on DNA cleavage

To determine the effect of NaCl on Tba endo V activity, DNA cleavage assays were performed using various NaCl concentrations ranging from 100 to 100 mM in the reactions (10 µl) containing 200 nM of Cy3-labeled ssDNA (*p45I), 20 mM Tris-HCl pH 8.0, 1 mM DTT, 1 mM MgCl 2 , 0.1mg/ml BSA, 8% glycerol and 300 nM Tba endo V. The reactions were kept at 85°C for 20 min. The reactions were stopped and treated as described above.

DNA binding by electrophoresis mobility shift assays

The DNA binding reaction mixtures (10 µl) contained 50 nM Cy3-labeled DNA substrate (*p22, *p22/t22, *p45I, and *p45/t45), 20 mM Tris-HCl pH 8.0, 1 mM DTT, 0.1 mg/ml BSA, 8% glycerol and various Tba endo V concentrations (50-3000 nM). The reactions were carried out at 25°C for 10 min. Samples were electrophoresed on a 4% native polyacrylamide gel in 0.1 × TBE (Tris-borate-EDTA) buffer. After electrophoresis, the gels were scanned and Cy3-labeled DNA was visualized with a Molecular Image analyser (BioRad). ImageQuant software was used for the quantitative analysis.

Results

Overexpression and purification of Tba endo V

The complete genome of the hyperthermophilic archaeon T. barophilus Ch5 has been sequenced, and was shown to encode an endonuclease V [GenBank accession number: ALM74570]. We used Clustal X V2.0 to align endonuclease V sequences, which were retrieved from the protein sequence database at the National Center for Biotechnology Information (http://www.ncbi.nlm.nih.gov/protein/). The alignment results suggested that Tba endo V shares seven highly conserved motifs with other 1A). Homology analysis on endo V protein sequences demonstrated that Tba endo V is a member of the enonuclease V family that can cleave deoxyinosine-containing DNA.

To characterize the enzyme biochemically, the gene of Tba endo V was cloned into the pET-30a(+) expression vector, and expressed in E. coli BL21(DE3)pLys cells. The recombinant Tba endo V protein was successfully expressed as a His-tag fusion protein (Fig. 1B). After sonication, heat treatment at 85°C for 25 min and purification by affinity chromatography with a Ni column, a band of about 25 kDa that corresponded to Tba endo V protein was observed (Fig. 1B).

Cleavage of deoxyinosine-containing DNA by Tba endo V

We employed normal DNA (*p22), deoxyinosine-containing DNA (*p45I) and U-containing DNA (*p45U) as reaction substrates to perform the enzymatic analysis of Tba endo V at 85°C, which is optimal growth temperature of T. barophilus Ch5 .

When using 50-300 nM of Tba endo V in the DNA cleavage reactions in the presence of 1 mM Mg 2+ and 10 nM DNA (*p45I) at 85°C and at pH 8.0 for 20 min, a short DNA product was observed (Fig. 2A), however, no cleavage product was formed using the normal DNA (*p22) as a template, suggesting that the enzyme could cleave deoxyinosine-containing DNA. In addition, we found no cleavage product in the reaction with U-containing DNA (*p45U) (Fig. 2A), suggesting that Tba endo V cannot cleave U-containing DNA.

Heat treatment at 80°C for 25 min during the purification of Tba endo V would inactivate the enzymatic activity of E. coli endo V; however, there was a slight possibility of E. coli endo V contamination, which would interfere with our observations. To rule out this possibility, we expressed the empty pET-30a(+) vector in the same expression strain as that one used for Tba endo V, disrupted the cells , and heated the supernatant. No cleavage product was observed (data not shown), when we used the heated supernatant produced from the empty vector ruling out the possibility of an E. coli endo V contamination during purification of Tba endo V. Furthermore, we observed that the Tba endo V is inactive to U-containing DNA, which also indirectly supports the lack of E. coli endo V contamination during purification of Tba endo V. Taken together, our results suggest that Tba endo V can cleave deoxyinosinecontaining DNA at 85°C.

In addition, we determined the effect of Tba endo V concentration on DNA cleavage activity by the addition of varied enzyme concentration ranging from 5 nM to 500 nM. As increasing the enzyme concentration, the DNA cleavage product gradually increased. When 50 nM enzyme concentrations was employed, more than 95% of DNA substrate was cleaved (Fig. 2B), suggesting that the enzyme has strong activity for cleaving deoxyinosine-containing DNA.

Optimal reaction temperature and thermostability of Tba endo V Since T. barophilus Ch5 is a hyperthermophilic archaeon and Tba endo V can cleave deoxyinosine-containing DNA at 85°C; therefore, we first determined the optimal temperature for cleaving deoxyinosine-containing DNA using Cy3-labeled p45I and p45I/t45 as DNA substrates, respectively.

When reaction temperatures increased from 30 to 100°C, Tba endo V cleaved deoxyinosine-containing DNA with varied cleavage efficiency using the Cy3-labeled ssDNA as a substrate. The cleavage efficiency of Tba endo V reached more than 95%, when reaction temperatures were 50-90 o C (Fig. 3A). However, Tba endo V had reduced cleavage efficiency when the reaction temperature was less than 50 o C or more than 90 o C. These observations suggest that 50-90 o C is the optimal temperature range for Tba endo V to cleave deoxyinosine-containing ssDNA.

In addition, the Tba endo V also displayed cleavage activity on deoxyinosinecontaining ds DNA as a template over the 30 to 100°C temperature range, showing more than 95% of cleavage efficiencybetween 70 and 90 o C. The Tba endo V exhibited less than 90% cleavage efficiency when reaction temperature was higher than 90 o C or less than 70 o C (Fig. 3B), suggesting that 70-90 o C is the optimal reaction temperature range for the enzyme to cleave deoxyinosine-containing dsDNA.

In the lower temperature range, e.g. 30-50 o C, Tba endo V showed a higher cleavage efficiency for deoxyinosine-containing ssDNA than for deoxyinosinecontaining dsDNA demonstrating that Tba endo V has a preference for single stranded vs. double stranded deoxyinosine-containing DNA, which cleavage may be impaired by the hydrogen bond and base stacking in the dsDNA.

To further examine the thermostability of the enzyme, Tba endo V was heated for 20-180 min at 100°C prior to activity assays. When heated for 20-90 min, Tba endo V retained more than 95% of cleavage activity (Fig. 3C). This activity decreased to 62% after 120mn of heating (Figure 3C). After 180 min, only 10% of cleavage activity reamined (Fig. 3C). Taken together, these observations suggest that Tba endo V is thermostable.

Optimal reaction pH for cleavage activity of Tba endo V

The impact of pH on the endonuclease activity of Tba endo V was assayed for pHs ranging from 5.0 to 11.0 in the standard DNA cleavage reactions. At the lowest pH (pH=5.0) and the highest pH (pH>10.0) tested, we measured a very limited cleavage efficiency (4% and ~20% respectively, Fig. 4), suggesting that low pH and high pHs are unfavorable for the enzyme to cleave DNA. Between pH=6.0 and pH=9.0, Tba endo V exhibited more than 86% cleavage efficiency with an optimal pH range between 7.5 and 9.0.

Effect of divalent metal ions on Tba endo V activity

Previous studies suggested that a divalent metal ion is required for endo Vs to cleave deoxyinosine-containing DNA. Here, we determined the effects of various divalent metal ions (Mg 2+ , Mn 2+ , Cu 2+ , Ca 2+ , Zn 2+ , Ni 2+ and Co 2+ ) on the DNA cleavage activity of Tba endo V. In absence of divalent ion (assay in the presence of 10 mM EDTA), Tba endo V displayed no cleavage activity (Fig. 5A), suggesting that a divalent metal ion is indeed required for the enzyme to cleave deoxyinosinecontaining DNA. In the absence of EDTA to chelate free ions in the assays (no divalent ion lane), Tba endo V displayed 12% of cleavage efficiency (Fig. 5A), which may be caused by copurification of a divalent metal ion bound to the protein during the purification of the enzyme. Both observations suggest that Tba endo V is dependent on a divalent metal ion for its endonuclease activity.

As expected, the DNA cleavage efficiency of Tba endo V varied with the different divalent metal ions tested. In presence of 2 mM Ca 2+ , Zn 2+ , Co 2+ , or Cu 2+ no significant activity was observed. In presence of Ca 2+ , Zn 2+ and Co 2+ , the Tba endo V was almost inactive, indicating that these ions could displace the ion copurified with the protein and inhibit its activity. In presence of Cu 2+ , the activity observed was equivalent to that of the control (14% vs. 12%), indicating that this ion could probably not substitute to the native ion in the Tba endo V structure and inhibit the cleavage activity. Tba endo V was the most efficient in presence of 2 mM Mg 2+ and Mn 2+ .

while the enzyme retained partial cleavage activity (65%) in the presence of 2 mM Ni ++ . Thus, the order of divalent metal ions for DNA cleavage by Tba endo V would be Mg 2+ ≈ Mn 2+ > Ni 2+ > Cu 2+ > Ca 2+ ≈ Zn 2+ ≈ Co 2+ . Tba endo V exhibits cleavage activity over a large concentration of Mg 2+ and Mn 2+ concentrations ranging from 10 µM to 3 mM (Fig. 5B and5C). Tba endo V displayed complete nicking activity in the presence of Mn 2+ and Mg 2+ concentrations equal to or higher than 10µM and 100µM respectively; At 500nM Mn 2+ , and 10µM Mg 2+ the activity measured was only slightly higher than controls. At high ion concentrations, Mn 2+ begins to be less efficient in comparison to Mg 2+ , which allows full protein activity to atl east 3mM (not shown).

Thus, Tba endo V has a preference for Mn 2+ , while at high concentrations, Mg 2+ is prefarable.

Inhibition of Tba endo V activity by NaCl

To reveal the effect of NaCl on endonuclease activity of Tba endo V, we added various NaCl concentrations (100-1000 mM) in the DNA cleavage reactions. In absence of NaCl, Tba endo V cleaved almost completely DNA substrate (Fig. 6). As

NaCl concentration increased, the amount of DNA cleavage product gradually decreased. Specifically, the addition of 500 mM NaCl to the DNA cleavage reaction enabled Tba endo V to cleave DNA with approximately 50% cleavage percent. At concentrations over 800 mM, only residual endonuclease activity comparable to controls remaiend. These results show that the endonuclease activity of Tba endo V is inhibited by high NaCl concentrations.

Kinetic assays of DNA cleavage by Tba endo V

Here, we performed kinetic assays of DNA cleavage by Tba endo V under the optimal reaction condition where 300 nM of the enzyme, 1 mM Mg 2+ and pH 8.0 were used. As the reaction time extended, increasing DNA cleavage product was formed by Tba endo V. When reaction time was 8 min, the cleavage percent of Tba endo V reached 92%, suggesting that the enzyme has strong endonuclease activity (Fig. 7).

DNA-binding of Tba endo V

In this work, we employed EMSA to determine the binding of Tba endo V to normal ssDNA and dsDNA, and deoxyinosine-containing ssDNA and dsDNA (Fig. 8). As increasing DNA substrate concentration, free DNA substrates were gradually bound by Tba endo V, suggesting that the enzyme is able to bind all four DNA substrates. When using deoxyinosine-containing ssDNA (p45I) as a substrate, the whole free DNA substrate was bound at 400 nM of Tba endo V (Fig. 8A), while at least 800 nM of enzyme was needed for complete binding to normal ssDNA (p22) (Fig. 8C). Similarly, when using deoxyinosine-containing dsDNA as DNA substrate, 800 nM of Tba endo V was required to bind all the free DNA substrate (Fig. 8B) while 2000 nM of the enzyme was required for the complete binding of the normal dsDNA substrate (Fig. 8D). These observations show that Tba endo V preferably binds ssDNA, regardless of the presence of deoxyinosine, and that it binds preferentially to deoxyinosine containing substrates regarless of their being single or double stranded. In fine, Tba endo V displays a higher affinity for single-stranded, deoxyinosine-containing DNA.

Discussion

In this contribution, we present the first biochemical characterization of a thermostable endonuclease V from te hyperthermophilic archaeon T. barophilus Ch5.

Here, we demonstrated that Tba endo V can cleave deoxyinosine-containing DNA at temperatures ranging from 30 to 100°C. The optimal temperature of the enzyme was 60-90°C, which is similar to those of Pfu endo V and A. fulgidus endo V . Tba endo V still had cleavage activity even when heated at 100°C for 120 min, suggesting that the enzyme is a thermostable endonuclease. Our results provides a support for the hypothesis that hyperthermophilic nature is a unique feature of enzymes from hyperthermophiles, as well as a necessity for these organisms often experiencing temperature above 100°C in their natural environments.

The Tba endo V cleaved only deoxyinosine, which is similar to the A. fulgidus endo V but contrasts with that of E. coli, further confirming that the cleavage spectra of endo Vs is organism specific. Similar to the E. coli endo V, the Tba endo V can bind unsubstituted ssDNA and dsDNA without sequence specificity. However, the Tba endo V exhibits stronger affinity for deoxyinosine-containing DNA than normal DNA as observed for human endo V, as well as a preference for single stranded targets.

The reported endo Vs are dependent on divalent metal ions, typically Mg 2+ , Mn 2+ , and Ni 2+ , as metal cofactors for their endonuclease activity, with a marked preference for one divalent metal ion. In the case of the Tba endo V, we demonstrated its activity in the presence of Mn 2+ , Mg 2+ , and Ni 2+ , but not in presence of Cu 2+ , Zn 2+ , Ca 2+ , and Co 2+ . Mn 2+ and Mg 2+ are the divalent ions allowing the complete cleaveage of deoxyinosine-containing DNA, which resembles the Pfu and human endo Vs.

However, Mn 2+ is the preferred divalent metal ion for Tba endo V, while Mg 2+ and to a much lesser extent, Mn 2+ , Ni 2+ , Co 2+ can support the endonuclease activity of the human endo V. The optimal deoxyinosine-specific endonuclease activity of Tba endo

V was observed at 10µM Mn 2+ and 0.1 mM Mg 2+ while 5 mM Mg 2+ is required for A.

fulgidus endo V to cleave deoxyinosine-containing DNA at maximum efficiency.

Analyses of single-molecule studies on endo V revealed that these enzymatic activities are precursors of DNA repair and are fueled by two metal ions such as Ca 2+

and Mg 2+ , with the former being associated with protein binding and the latter with DNA cleavage, supporting the hypothesis that endo V has two metal binding sites to regulate its enzymatic activities . .

The Tba endo V shares several similar features with Pfu endo V, the other characterized endo V protein from the closely related hyperthermophilic archaeon

Pyrococcus furiosus, which belongs to the same family as Thermococcus barophilus.

Both endo V have optimal cleavage activity ca. neutral pH, and are inhibited by high NaCl concentrations. However, Tba endo V displays higher specific enzyme activity than Pfu endo V at pH 6.5. Furthermore, the Tba endo V is inhibited by much higher NaCl concentrations (400mM, Fig. 8) than the one from Pyrococcus furiosus

(200mM).
Since the rate of deamination of DNA increases with temperature, significant amount of deoxyinosine can be generated at 85°C, which is the optimal growth temperature of T. barophilus Ch5. Thus, the activity of Tba endo V might be essential for mutation prevention in this organism in response to the known mutatgenic ptential of deoxyinosine. Recently, a novel endonuclease, designated as endonuclease Q (Endo Q) was biochemically characterized in two other Thermococcales, Pyrococcus furious and Thermococcus kodakarensis. Similar to EndoV, EndoQ has a preference for deoxyinosine containing DNA substrate, and is abile to cleave DNA in 5' to the damage base, while Endo V is cleaving in 3'. Both enzymes have been shown ot be essential for deoxyinosine repair. We have found an homologue of endo Q in the genome of T. barophilus Ch5, and its biochemical characterization is under investigation in our laboratory. The analysis of the Tba endo V and endo Q will provide important information on the deoxyinosine repair mechanisms in

Thermococcales.

Biochemical and structural data support that endonuclease V is an unusual and interesting repair enzyme, which allows its use in biotechnology. For example, the combined use of E. coli T/G DNA glycosylase and E. coli endo V has been employed to detect mutations in the BRCA1 gene . In addition, Tma endo V is used to scan Various Tba endo V concentrations ranging from 5 nM to 500 nM were added in the DNA cleavage reactions using the Cy3-labeled p45I as substrate. Reaction mixtures were electrophoresed through a 15 % denaturing PAGE.

Fig. 3 Optimal temperature and thermostability of Tba endo V. A. DNA cleavage reactions of Tba endo V were performed at various temperatures (30, 40, 50, 60, 70, 80, 90 and 100°C, respectively) in the presence of 1 mM Mg 2+ using the Cy3-labeled p45I as the substrate for 20 min. B. DNA cleavage reactions of Tba endo V were performed at various temperatures (30, 40, 50, 60, 70, 80, 90 and 100°C, respectively) in the presence of 1 mM Mg 2+ using the Cy3-labeled p45I /t45 as the substrate for 20 min. C. To evaluate its thermostability, Tba endo V was heated at 100°C for 20, 40, 60, 90, and 120 min, respectively. DNA cleavage reactions of the heated Tba endo V were performed in the presence of 1 mM Mg 2+ using the Cy3-labeled p45I as substrate at 85°C for 20 min. Reaction mixtures were electrophoresed through an 15 % denaturing PAGE. It is important to adress the issue of the background level here since it will bring comments from the reviewers.

Figure 8

  Hyperthermophilic archaea are facing severe challenges for deamination of DNA due to their inhabiting in high temperature environment . Deamination of DNA occurs at a higher rate at high temperature than at room temperature . To maintain their genome stability, hyperthermophilic archaea have evolved several repair pathways including enzymes to eliminate potentially mutational effects created by deamination of DNA. Thermococcus is an important branch of euryarchaea which thrive in hyperthermophilic environments. Increasing Thermococcus species have been isolated and studied. Thermococcus barophilus Ch5, which was isolated from a deep-sea hydrothermal field of the Mid-Atlantic Ridge (Logachev field chimney, 3,020 m depth) , is a new member of the piezophilic hydrothermal vent archaeal species. T. barophilus Ch5 has a pressure optimum of 40 Mpa and a temperature optimum of 85 o C . Analysis of the completed whole genome sequences of T. barophilus Ch5

endo

  Vs from archaea (Archaeoglobus fulgidus [GenBank accession number: KUK05292], Pyrococcus furiosus [GenBank accession number: WP_011012124], Sulfolobus solfataricus [GenBank accession number: AKA78077]), eukaryotes (Homo sapiens [GenBank accession number: Q8N8Q3], and Schizosaccharomyces pombe [GenBank accession number: Q10348]) and bacteria (Escherichia coli [GenBank accession number: EDV65061] and Thermotoga maritima [GenBank accession number: AKE27746]) (Fig.

Figure legends Fig. 1 Fig. 2

 legends12 Figure legends

Fig. 4

 4 Fig. 4 Effect of pH on the DNA cleavage reactions of Tba endo V. DNA cleavage

Fig. 5

 5 Fig. 5 Effect of divalent metal ions on DNA cleavage reactions of Tba endo V. A.

Fig. 6

 6 Fig. 6 Effect of NaCl on DNA cleavage reactions of Tba endo V. DNA cleavage

Fig. 7

 7 Fig. 7 Time course of DNA cleavage by Tba endo V. DNA cleavage reactions of

Acknowledgements

This work was supported by the National Natural Science Foundation of China Grant (No. 41306131), the Provincial Natural Science Foundation Grant of Jiangsu Province, China (No. BK20130440), academic leader of middle and young people of Yangzhou University Grant to L.Z., and the practice innovation training program for college students in Jiangsu to Y.L (No. 201711117059Y).

Conflict of Interests

The authors declare no conflict of interests regarding the publication of this paper.

Ethical Statement


Marta Salvador Castell

Nick Brooks

Judith Peters

Philippe Oger

Induction of non-lamellar phases in archaeal lipids at high temperature and high hydrostatic pressure by apolar polyisoprenoids

It is now well established that cell membranes are much more than a barrier that separate the cytoplasm from the outside world. Regarding membrane's lipids and their self-assembling, the system is highly complex, for example, the cell membrane needs to adopt different curvatures to be functional. This is possible thanks to the presence of non-lamellar-forming lipids, which tend to curve the membrane. Here, we present the effect of squalane, an apolar isoprenoid molecule, on an archaea-like lipid membrane. The presence of this molecule provokes negative membrane curvature and forces lipids to self-assemble under inverted cubic and inverted hexagonal phases. Such non-lamellar phases are highly stable under a broad range of external extreme conditions, e.g. temperatures and high hydrostatic pressures, confirming that such apolar lipids could be included in the architecture of membranes arising from cells living under extreme environments.

Introduction

Since Singer and Nicolson 1 first defined the membrane as a mosaic of proteins and lipids in a lamellar fluid liquid-crystalline (L ) phase, our view of the membrane ultrastructure has become more sophisticated, recognizing the existence of membrane domains of specific lipid composition in natural membranes. Each composition corresponds to a particular phase, which play important roles in cell membrane function and cell physiology as they have very specific physicochemical properties, such as different lateral motions, rotations, fluidity, and phase structuration, and promote the formation of lamellar or non-lamellar phase 2,3 .

Lamellar, cubic and inverted hexagonal phases possess different bending fluctuations and intrinsic curvatures4, which are mostly defined by the shape of the lipids present in the membrane. This has been rationalized using the critical geometric packing parameter (P) concept, which defines P as v(a*l) -1 where v is the molecular volume, l is the molecular length and a is the molecular area at the lipid-water interface 5,6 . For P<1, molecules are inverted cones and tend to produce positive curvatures, i.e. the hydrocarbon chains are closest to each other.

For P>1, lipids present conical shapes and adopt negative curvatures and self-assemble under the so-called inverted phases, such as the cubic (Q) and hexagonal (HII) phases, simply to release their curvature frustration (Figure 1). Changes between lamellar and non-lamellar former lipid compositions tune the flexibility and curvature of the membrane 7 which is used by cell to tune its membrane functionality by adjusting its membrane lipid compositions to adapt itself and maintain essential functions under different environmental conditions. Archaeal phospholipids present unique characteristics: a sn-glycerol-1-phosphate (G-1-P) backbone, ether linkages, and isoprenoid hydrocarbon chains 8 . The methyl groups along the acyl chains are responsible for the fact that both phospholipid's rotamers are nearly energetically equivalent9, thus allowing archaeal lipids to occur in the L phase for a broad range of temperature (-120°C to 120°C) 9,10 . Isoprenoid chains pack densely, increase membrane viscosity and decrease its water permeability [11][START_REF] Shinoda | AIP Conf. Proc[END_REF][START_REF] Tristram-Nagle | [END_REF] . The presence of ether bounds, isoprenoid hydrocarbon chains and transmembrane lipids is thought to be the reason of the incredibly high resistance of archaeal cell membranes to extreme conditions 14 . For example, some Archaea have their optimal growth conditions at low pH, high salinity, high temperature (HT) and/or high hydrostatic pressure (HHP), including a highly stable and selective permeable cell membrane.

Intriguingly, liposomes made from archaeal lipids present highly irregular shapes 15 and indeed, extracellular membrane vesicles are highly produced by Archaea 16 . All these findings indicate that non-lamellar-forming lipids may play a particular important role in the Archaea domain.

The geometric packing parameter predicts that most archaeal phospholipids will have zero or positive curvatures which will not favour the formation of the non-lamellar phases indispensable for biological activity. This is for example the case for the membrane lipids of Thermococcus barophilus which have phosphatidyl inositol polar headgroup and derivatives of this polar headgroup. Recently, Cario and colleagues 17 proposed a novel archaeal membrane architecture for the hyperthermophile and piezophile Archaeon T. barophilus in order to explain its ability to thrive at temperatures close to 100°C 18 . According to this hypothesis, apolar isoprenoid molecules would populate the hydrophobic space between the two bilayer leaflets, thus modifying the membrane properties, decreasing permeability to proton and water and membrane rigidity 19 . In addition, the presence of the interleaving molecules within the plane of the membrane is proposed to be implied in modifying the phases transition in the membrane. Indeed, the partitioning of non-polar molecules, such as alkanes, aromatic compounds and isoprenoid chains into the hydrocarbon region, increases the tendency of the lipid layer to adopt a negative interfacial curvature [20][21][22] . Therefore, lipids reduce their hydrocarbon packing stress self-assembling within inverted non-lamellar phases. For example, the addition of 5% alkane to a mixture of PC/PE-lipids reduces the L -HII transition by 55°C 23 . To date, the effect of apolar lipids on archaeal membranes has not been studied.

Using Small Angle X-ray Scattering (SAXS), we have determined the lipid phases from a mixture of archaeal lipids in presence of different molar percentages (mol%) of the polyisoprenoid squalane at different temperatures and pressures. Here we show that in absence of the apolar intercalant, the archaeal lipids cannot form non-lamellar phases, and that the presence of the apolar molecule imposes highly stable non-lamellar type II Q and H phases.

Experimental section I. Chemicals

The high purity (>99%) synthetic archaeal lipids 1,2-di-O-phytanyl-sn-glycero-3-phosphocholine (DoPhPC) and 1,2-di-O-phytanyl-sn-glycero-3-phosphoethanolamine (DoPhPE) were purchased from Avanti Polar Lipids in the lyophilized form and utilized without further purification.

2,6,10,15,19,23-Hexamethyltetracosane (squalane) was bought from Alfa Aesar.

II. Small angle X-ray diffraction

SAXS experiments were performed on beamline I22 24 at the Diamond Light Source (United Kingdom). DoPhPC and DoPhPE were used in a proportion 9:1 molar. Phospholipids with contents of 20% (w/v water) and the adequate quantity of squalane (1%, 2.5%, 5% and 10% molar) were placed inside a pressure chamber 25 . The pressure and temperature dependent experiment was carried out at six different temperatures (10°C, 25°C, 40°C, 55°C, 70°C and 85°C) in pressure jumps of 50 bar from 0 bar to 1000 bar.

The X-ray energy was 18 keV, the momentum transfer is defined as q= 4 , where 2

is the scattering angle. The small-angle regime was 0.005 < q < 0.35 Å-1. The type of phase can be distinguished by the characteristic SAXS peak ratios26: for a lamellar phase they are equidistant, for a hexagonal phase they are: 1, 3, 2, 7…, for a cubic phase Pn3m they are: 2, 3, 2, 6, 8, 3… and for a cubic phase Im3m they are: 2,2, 6, 8, 3... The lattice parameter a is defined as alam= 2 /qn and ahex= (2 /qn)*2/3 for a lamellar and a hexagonal phase, respectively. To determine the lattice parameter for the cubic phase, the experimentally = q/2 , is plotted vs ( + + ) , where h,k,l are the Miller indices by which the peaks are indexed. The plot should intercept through the origin and be linear with a slope of 1/acubic.

Results and discussion

I. Archaeal phospholipids self-organize under lamellar phases

Lipid phases in excess of water can be identified by the relative Bragg peak positions in the SAXS patterns [START_REF] Tyler | Methods in Membrane Lipids[END_REF] . Collected data points and phase diagrams are shown in Figure 2. In absence of the apolar molecule, only lamellar phases can be identified in the SAXS spectra (Salvador Castell, unpublished results). At 10°C and ambient pressure, the lattice parameter of the lamellar phase is 62.7 ± 0.6 Å. As expected, the lamellar phase appears sensitive to HT (Figure S1). The lattice parameter increases at least to the highest temperature tested of 85°C. However, starting from 70°C the sample loses its correlation between bilayers and becomes so disordered that only the first diffraction order can be fitted with accuracy. The lattice parameter increases from 62.7 ± 0.6 Å at 10°C to 73.2 ± 1 Å. The lamellar phase appears little sensitive to hydrostatic pressure.

Indeed, the lattice parameter varies only within the error bars of the measure. In contrast to other lipid systems in which hydrostatic pressure exert a structuring impact on the bilayers [START_REF] Mccarthy | Advances in Biomembranes and Lipid Self-Assembly[END_REF] ,

we see no such effect on DoPhPC:DoPhPE-based membranes. In fact, at the highest combined pressures and temperatures (85°C and >750 bar), no diffraction peaks are visible anymore.

II. Addition of 1 mol% squalane induces the formation of inverted non-lamellar phases

Upon addition of squalane, even at the smallest quantity of 1 mol%, we observed the presence of non-lamellar phases of type II, in co-existence or not with lamellar phases, which suggests that the presence of squalane increases the negative membrane curvature (Salvador Castell, unpublished results). At all temperatures, we observe a cubic phase, in coexistence with a lamellar phase up to 40°C and a hexagonal phase above. In contrast to the lipid system without squalane, the system with 1mol% squalane is both pressure and temperature sensitive. The alam increases as a function of HHP from 67.6 ± 0.3 Å at ambient pressure to 69.8 ± 0.3 Å at 1000 bar at 10°C for example, but also as a function of temperature where it reaches 69.6 ± 0.5 Å at 40 C and 1 bar (Figure S2). Temperature is known to favour the negative intrinsic curvature of the lipid self-assembly. At temperatures of 55 C and above, a mixture of a Pn3m cubic phase and an inverted hexagonal phase coexist. At 85 C, three different Pn3m phases with very slightly different lattice values coexist, which is indicative that they have most likely different hydration levels (Figure 3). Note the substantial lattice parameters of the cubic phases (aPn3m = 254 ± 2 Å and aIm3m = 332 ± 3 Å) which reflect the high potential of squalane to reduce chain packing frustration. Values for the lattice parameter for bacterial lipid cubic phases are substantially smaller, such as 105 Å for monoolein (MO) 28,29 or 122 Å for 1,2-dioleoyl-sn-glycero-3phosphoethanolamine (DOPE) 30 . However, under certain conditions this lattice parameter can reach values higher than those measured here, an example is the case of a diphytanoyl PCcholesterol mixture, in which the QII phase has a acubic of 510 Å 31 . At 85 C, we observe the replacement of the Pn3m by the Im3m phase (Figure 3). Thus remarkably, HHP leads to the replacement of a Pn3m cubic phase with a lower acubic and higher negative curvature to a Im3m's cubic phase with a higher acubic and lower negative membrane curvature. In accordance with geometric considerations for those coexisting phases 32,33 , we found that the ratio between their lattice parameters is 1.28. The hexagonal phase has a lattice parameter ranging from 87.9 ± 0.5 Å at 55 C to 80.1 ± 0.5 Å at 85 C under ambient pressure. This ahex is highly sensitive to pressure, ranging from 82.2 ± 0.5 Å at 1 bar to 90.1 ± 0.5 Å at 1000 bar at 70 C for example. This is in accordance with the fact that non-lamellar self-assemblies easily swell at HHP 34 . In addition, such ahex is much higher than the ahex usually obtained for familiar lipids which lie at 52.9 Å at 97 C for MO 35 or 65.8 Å at 70 C for DOPE 36 , for example. This may be directly related to the ability of squalane to release the chain frustration at HII phases with high ahex.

III. Higher negative curvature in presence of 2.5 mol% of squalane

By further adding squalane, the system presents a coexistence of a lamellar and HII phases at low temperature and no longer a coexistence of lamellar and cubic phases, which is indicative of a further increase of the membrane negative curvature. As for the 1 mol% squalane lipid system, the lamellar phase is present up to 40°C, but in contrast, it seems to be far less sensitive to pressure (Figure S3). At 85°C, the hexagonal phase coexists with two cubic phases, probably Im3m. Remarkably, at 55°C and 70 °C, HHP induces a phase transition to a second hexagonal phase coexisting with the one already presented at low pressures (Figure 4a). We propose that this phase separation could result from the rearrangement of squalane between a squalane rich and a squalane poor phase. As observed before, the hexagonal phase is much more sensitive to temperature and HHP than the lamellar phase. The ahex decreases from 94.6 ± 0.5 Å at 10 C to 77.4 ± 0.5 Å at 85 C at 1 bar and increases from 94.6 ± 0.5 Å to 103.1 ± 0.5 Å at 1 bar to 1000 bar at 10 C (Figure S3b). Last, at 85 C, two pressure-sensitive Im3m phases appear in coexistence with the HII phase (Figure 4b). The acubic varies from 248 ± 2 Å and 264 ± 2 Å at 1 bar, to 264 ± 2 Å and 279 ± 2 Å, at 1000 bar for the two phases (Figure S3c). The presence of these two QII phases may be due to different hydration levels as observed at 1 mol% squalane.

IV. A competition between lamellar and inverted hexagonal phase appears by adding 5 mol% squalane

In presence of 5 mol% squalane at 1 bar, the system presents a mixture of one lamellar and one inverted hexagonal phase at temperatures up to 55 C (Figure 5a). As expected, temperature increases the membrane's negative curvature and at 70°C and above only the HII is present. In contrast, pressure promotes the lamellar phase, which can be readily observed at 70 C, where the lamellar phase appears above 350 bar. Higher HHP values further increase the structuration of the lamellar phase (Figure 5b). At their highest temperature tested, e.i. 85 C, the lamellar phase appears at 1000 bar. In presence of 5 mol% squalane, the lamellar-to-hexagonal transition is a direct conversion as no cubic phase appears.

Interestingly, HHP swells much more the HII phase than the lamellar one. At 10 C, alam remains almost constant at 57.8 ± 0.4 Å -58.0 ± 0.4 Å from 1 bar to 1000 bar but, due to the capability of squalane to reduce chain frustration, the hexagonal phase is able to swell from 111.8 ± 0.5 Å at 1 bar to 121.2 ± 0.5 Å at 1000 bar for the same temperature (Figure 5c and Figure S4).

The different pressure sensitivities of the L and HII phases are easily observed when plotting a, e.i. the difference between the lattice parameter at HHP and at ambient pressure (Figure 6a).

The a of the lamellar phase is ca. 0.2 Å*kbar -1 and not significative. In contrast, the HII phases swell up to 9.5 Å*kbar -1 at 10 C and 1.5 Å*kbar -1 at 85 C. Familiar L phases usually swell up to 2 Å*kbar -1 while HII may swell as much as 6 Å*kbar -1 34 . The swelling of HII decreases with temperature. This unexpected behaviour can be explained by the coexistence of the HII and lamellar phases in the system. Indeed, one can observe the change of slope of a at 70 C and 450 bar (Figure 6b), which coincides with the appearance of the lamellar phase in this lipid -1

, in presence of the lamellar phase it increases to 5.2 Å*kbar -1 . At 85°C, at which no lamellar phase is observed below 1000 bar, the -1

. Such differences may be explained by the partition of two polar headgroups and the apolar molecule squalane between the two phases in coexistence.

Indeed, in this lipid mixture we have two lipids with very different curvature. Phospholipids containing PC headgroups present a zero-curvature monolayer and self-assemble in a lamellar phase. DoPhPE easily bends under negative curvatures and therefore induce the formation of inverted non-lamellar phases, such as the HII. Accordingly, when L and HII phases are present, the system may present a "lipid-separation" and DoPhPE may have the tendency to adopt the HII phase with its preferred curvature. When the lamellar phase is present, as at 70 C and pressures above 450 bar, DoPhPE may principally form the HII phase (Figure 6c). This effect was already observed in bacterial-like lipids, which better accommodate in L or L phases. For example, a decrease in temperature induces domains enriched in saturated phospholipids with small quantities of unsaturated lipids 37 . As DoPhPE has a higher lateral compressibility, HII presents a higher a slope when most of DoPhPE is partitioned in the non-lamellar phase and excluded from the L phase. However, the presence of methyls in the hydrocarbon chains could reduce the lateral diffusion of these phospholipids and squalane may aggregate in the chain frustration points and most especially at the interphase points, to fill the voids and release packing tension. The chain-packing frustration increases as the diameter of the hexagonally packed cylindrical inverse micelles increases. When the lamellar phase disappears, the apolar molecule can better distribute along the HII phase and squalane aggregates may disappear (Figure 6c). Therefore, the lattice parameter of the HII phase decreases and consequently its pressure response.

V.

Only the inverted hexagonal phase is present when 10 mol% squalane is added At 10 mol% squalane in the DoPhPC:DoPhPE (9:1) mixture only the HII phase is observed under all temperatures and pressures studied. At ambient pressure, we could detect a slow phase transition between two HII phases. The axially integrated 1-D patterns at 55 C, 70 C and 85 C (Figure 7a) shows the apparition of a HII phase at higher q and the gradual disappearance of the peaks characteristics of the second hexagonal phase. The spectrum at 85 C stills shows shoulders corresponding to the low temperature HII phase. The lattice parameters of HII range from 104.8 ± 0.5 Å at 10 C and 1 bar to 92.8 ± 0.5 Å at 85 C and 1 bar (Figure 7c and Figure S5).

As thermal motions increase lipid dynamics, the HII phase is much more pressure sensitive at HT (Figure 7b and7c). a expansion increases from 0.1 Å*kbar -1 at 10°C to 3.9 Å*kbar -1 at 85 C consistent with a phase transition between two hexagonal phases. This phase transition also possibly occurs at lower temperature, although very close to 1 bar. Such phase transition is not visible at temperatures above 40°C. These low hexagonal phases differ principally by their ability to swell with pressure, which is congruent with a higher pressure-dependent lateral compressibility of the high temperature hexagonal phase.

Conclusions

X-ray diffraction experiments clearly show that the addition of squalane drastically changes membrane curvature of a lipid DoPhPC:DoPhPE (9:1) mixture, an effect comparable to that of cholesterol at biological relevant quantities on eukaryote-like lipids 38,39 . SAXS data displays the presence of highly stable non-lamellar lipid phases dependent on mol% of squalane, temperature and pressure. The complete understanding of the observed structural transitions will require a deeper study as it will be necessary to consider different kind of interactions, such as lipid-water, lipid-lipid and lipid-squalane. However, the outstanding acubic values and resistance revealed by the system may be useful for the stabilization or modification of cubosomes and to adjust lipid membrane properties of archaeosomes 40,41 .

The fact that small quantities of squalane (1 -10 mol%) are capable to impose negative intrinsic curvatures on the system has biological significance. Our results indicate that archaeal membranes simply require to regulate squalane proportions in the cell membrane to tune its curvature and flexibility. Therefore, due to the presence of small amounts of the apolar molecule, archaeal cells would be capable to manage essential cell processes such as fusion and fission 42 .

Furthermore, recognizing that squalane is present in archaeal cell membranes and considering the high metastability of all phase transitions in a large range of temperatures and pressures, we can predict that non-lamellar phases and particularly, inverted cubic phases, may be locally present in archaeal membranes.

Conflicts of interest

There are no conflicts to declare.

  Figures

Figure 1 .

 1 Figure 1. Schematic representation of different lipid phases as a function of lipid curvature. The lipid headgroups are represented in green, the hydrocarbon chains in brown and water location in blue. The arrows indicate the direction for the increase of the different parameters: chain packing parameter (P), negative intrinsic curvature of the membrane, pressure and temperature.

Figure 2 .

 2 Figure 2. Phase diagrams of the DoPhPC:DoPhPE (9:1) in presence of excess water as a function of the proportion of squalane, temperature (Left panel) and pressure (Right panel) Left, sketches of each phase (same colour scheme as in figure 1). Full circles indicate conditions under which pressure-induced phase transitions are observed and detailed in the right panel, and open circles those under which no pressure-induced phase transition is observed between 0 and 1000 bar.

Figure 3 .

 3 Figure 3. SAXS spectra of the mixture DoPhPC:DoPhPE +1 mol% squalane. At 85 C and ambient pressure (black), 500 bar (green) and 1000 bar (red). Top inset, zoom into the cubic phase region.

Figure 4 .

 4 Figure 4. SAXS spectra of the mixture DoPhPC:DoPhPE + 2.5 mol% squalane. a) At 70 C as a function of applied pressure (from 0 to 1000 bar, 50 bar steps). Two HII coexist above 600 bar. b) At 85 C at ambient pressure, QII Im3m and the HII phases coexist.

Figure 5 .

 5 Figure 5. SAXS spectra of the mixture DoPhPC:DoPhPE + 5 mol% squalane. a) As a function of temperature at ambient pressure (0 bar applied). b) at 70°C as a function of pressure, from 0 to 1000 bar of pressure applied, 50 bar steps. A lamellar phase appears at 350 bar. c) At 10°C as a function of pressure. The L and HII phases coexist at ambient pressure (black), 500 bar (green) and 1000 bar (red).

Figure 6 .

 6 Figure 6. Squalane distribution between lamellar and hexagonal phases. a) pressure. The lattice parameter has been normalized to that at ambient pressure for each temperature. L phase (full symbols); HII (empty symbols). Up to 55 C, both, L and HII coexist. b) Detail on a as a function of pressure at 70 C. c) Schemes of phase coexistence and lattice parameter evolution as a function of pressure and temperature in the DoPhPC:DoPhPE:5 mol% squalane system. Squalane is represented in black, black spheres represent squalane aggregates, lipid hydrocarbon chains are in brown, lipid headgroups are shown in green (PC) and in red (PE).

Figure 7 .

 7 Figure 7. SAXS spectra of the mixture DoPhPC:DoPhPE + 10 mol% squalane. a) SAXS spectra at ambient pressure for 55 C (blue), 70 C (yellow) and 85 C (magenta). b-c) SAXS spectra at 10°C (b) and 85°C (c) of HII phase as a function of pressure. d) Normalized lattice parameter of the HII phase as a function of pressure and temperature. Lattice parameter has been normalized to ambient pressure conditions.

Acknowledgements

This work was supported by the French National Research Agency programmes ANR 2010 BLAN 1725-04 and ANR 17-CE11-0012-01 to PO and JP and the Royal Society to PO and NB. MSC was supported by a PhD grant from the French Ministry of Research.

Electronic Supplementary Information