F Barthe

P Wolff

Volume properties of high-dimensional Orlicz balls

We prove asymptotic estimates for the volume of families of Orlicz balls in high dimensions. As an application, we describe a large family of Orlicz balls which verify a famous conjecture of Kannan, Lovász and Simonovits about spectral gaps. We also study the asymptotic independence of coordinates on uniform random vectors on Orlicz balls, as well as integrability properties of their linear functionals.

Let ε > 0. As above, since Ψ ≥ 0, Ψe -λΨ e -λΨ ≤ ε + Ψ>ε Ψe -λΨ e -λΨ .

Next, using x ≤ e x and for λ > 2,

and e -λΨ ≥ Ψ≤ε/2 e -λΨ ≥ e -λε/2 Vol({x; Ψ(x) ≤ ε/2}).

Since Ψ(0) = 0 and Ψ is continuous, the latter quantity is positive. Combining the above three estimates, we get

Letting λ → ∞ yields lim sup λ→∞ R(λ) ≤ ε, for all ε > 0.

Lebesgue spaces play a central role in functional analysis, and enjoy remarkable structural properties. A natural extension of this family is given by the class of Orlicz spaces, which also enjoy a wealth of remarkable properties, see e.g. [START_REF] Rao | Theory of Orlicz spaces[END_REF]. Similarly, for p ≥ 1, the unit balls of R n equipped with the ℓ p -norm, often denoted by B n p = {x ∈ R n ; i |x i | p ≤ 1}, are well studied convex bodies, and usually the first family of test cases for new conjectures. Their simple analytic description allows for many explicit calculations, for instance of their volume. A simple probabilistic representation of uniform random vectors on B n p , given in terms of i.i.d. random variables of law exp(-|t| p ) dt/K p is available, see [START_REF] Barthe | A probabilistic approach to the geometry of the l n p -ball[END_REF]. It allows to investigate various fine properties of the volume distribution on B n p . The study of general Orlicz balls is more difficult, due to the lack of explicit formulas, in particular for the volume of the set itself.

In this note, we show that probabilistic methods allow to derive precise asymptotic estimates of the volume of Orlicz balls when the dimension tends to infinity, and rough estimates which are valid in every dimension. This allows us to complement a result of Kolesnikov and Milman [START_REF] Kolesnikov | The KLS isoperimetric conjecture for generalized Orlicz balls[END_REF] on the spectral gap of uniform measures on Orlicz balls, by giving an explicit description of the range of parameters where their result applies, see Section 5. In Section 6, we show, among other results, the asymptotic independence of a fixed set of coordinates of uniform random vectors on some families of Orlicz balls of increasing dimensions. This is a natural extension of a classical observation (going back to Maxwell) about uniform vectors on Euclidean spheres and balls. The last section deals with properties of linear functionals of random vectors on Orlicz balls.

After this research work was completed, we learned by J. Prochno of his independent work [START_REF] Kabluchko | The maximum entropy principle and volumetric properties of Orlicz balls[END_REF] with Z. Kabluchko about similar volume asymptotics for Orlicz balls. Their paper uses sophisticated methods from the theory of large deviations, which have the potential to give more precise results for a given sequence of balls in increasing dimensions. Our approach is more elementary and focuses on uniform convergence over some wide range of parameters, as required by our applications to the spectral gap conjecture.

Notation and statement

Throughout this paper, a Young function is a non-negative convex function on R which vanishes only at 0. Note that we do not assume symmetry at this stage. For a given Young function Ψ : R → R + , denote

B n Ψ = x ∈ R n : n i=1 Ψ(x i ) ≤ 1 the corresponding n-dimensional Orlicz ball. Our aim is to estimate the asymptotic volume of B n Ψ/En = {x ∈ R n : n i=1 Ψ(x i ) ≤ E n } for relevant sequences E n of linear order in the dimension.
Let λ > 0. Consider the following probability measure on R,

µ λ (dt) = e -λΨ(t) dt Z λ ,
with Z λ being a normalization constant. Let X be a random variable with the distribution

µ λ . Set m = m λ = EΨ(X), σ 2 = σ 2 λ = Var Ψ(X) . Our aim is to prove Theorem 1.1. Consider a Young function Ψ and λ > 0. Let n ≥ 1 be an integer and α ∈ R. Set E := m λ n + ασ λ √ n, then Vol B n Ψ/E = (Z λ e λm λ ) n 1 λσ λ √ 2πn e -α 2 /2 e λσ λ √ nα (1 + O(n -1/2 )) = Z n λ e λE λσ λ √ 2πn e -α 2 /2 (1 + O(n -1/2 )),
where the term O(n -1/2 ) depends on λ, Ψ and, non-decreasingly in |α|.

Corollary 1.2. Consider a Young function Ψ and λ > 0. Let (a n ) n≥1 be a bounded sequence, and

E n := m λ n + a n √ n, Then when the dimension n tends to ∞, Vol B n Ψ/En ∼ Z n λ e λEn λσ λ √ 2πn e -a 2 n /(2σ 2 λ ) .
Let us mention that the above results can be applied to B n Ψ/En when E n = mn + a n √ n where m > 0 is fixed and (a n ) n is a bounded sequence. Indeed the next lemma ensures the existence of a λ > 0 such that m = m λ . Lemma 1.3. Let Ψ as above. Then the map defined (0, +∞) to (0, +∞) by

λ → R(λ) := Ψ(t)e -λΨ(t) dt e -λΨ(t) dt is onto.
Proof. By hypothesis, exp(-λΨ) < ∞ for all λ > 0. This fact allows us to apply the dominated convergence theorem, and to show that the ratio R(λ) is a continuous function of λ > 0. Let us show that lim λ→0 + R(λ) = ∞ and lim λ→∞ R(λ) = 0. The claim will then follow by continuity. Consider an arbitrary K > 0. Since Ψ ≥ 0,

Ψe -λΨ e -λΨ ≥ K Ψ≥K e -λΨ e -λΨ = K 1 -Ψ<K e -λΨ e -λΨ ≥ K 1 - Vol({x; Ψ(x) < K}) e -λΨ .
By monotone convergence, lim λ→0 + e -λΨ = ∞. Hence, lim inf λ→0 + R(λ) ≥ K. Since this holds for every K > 0, we conclude that lim λ→0 + R(λ) = ∞.

Probabilistic formulation

We start with a formula relating the volume with an expectation expressed in terms of independent random variables. Let λ > 0. Let (X i ) i∈N * be i.i.d. r.v.'s with the distribution µ λ (dt) = e -λΨ(t) dt/Z λ . Recall that m λ = EΨ(X i ) and σ 2 λ = Var Ψ(X i ) . We denote by S n the normalized central limit sums:

S n = 1 σ λ √ n n i=1 (Ψ(X i ) -m λ ).
With this notation, we get the following representation for any λ > 0

Vol B n Ψ/E = 1 { n i=1 Ψ(x i )≤E} dx = 1 { n i=1 Ψ(x i )≤E} Z n λ e λ n i=1 Ψ(x i ) n i=1 µ λ (dx i ) = Z n λ E e λ n i=1 Ψ(X i ) 1 { n i=1 Ψ(X i )≤E} = (Z λ e λm λ ) n E e λσ λ √ nSn 1 {Sn≤ E-m λ n σ λ √ n } . (2.1)
By the Central Limit Theorem, S n converges in distribution to a standard Gaussian random variable when n tends to infinity. Such Gaussian approximation results allow to estimate the asymptotic behaviour of the above expectations. Nevertheless, a direct application of the CLT or the Berry-Esseen bounds does not seem to be sufficient for our purposes. A more refined analysis is required, built on classical results and techniques on the distribution of sums of independent random variables which go back to Cramér [START_REF] Cramér | Sur un nouveau théorème-limite de la théorie des probabilités[END_REF] (see also [START_REF] Bahadur | On deviations of the sample mean[END_REF]). Theorem 1.1 is a direct consequence of the following one, applied to Y i = (Ψ(X i )m λ )/σ λ . For a random variable V , let P V and ϕ V denote the distribution and the characteristic function.

Theorem 2.1. Let (Y i ) i≥1 be a sequence of i.i.d. real random variables such that E|Y i | 3 < ∞, EY i = 0 and Var(Y i ) = 1. Suppose ε, δ > 0 are such that so-called Cramér's condition is satisfied for Y i :

|ϕ Y i (t)| ≤ 1 -ε for |t| > δ.
(2.2)

For n ≥ 1, let S n = (Y 1 + • • • + Y n )/ √ n. Then for ℓ > 0 and α ∈ R, E e ℓ √ n Sn 1 Sn≤α = 1 ℓ √ 2πn e ℓ √ nα-α 2 /2 1 + O(n -1/2 ) .
Remark 2.2. The term O(n -1/2 ) involves an implicit dependence in ℓ, α and the law of Y 1 .

For n ≥ 16ℓ 2 + (2|α| + 1) 2 ℓ -2 , our argument provides a term O(n -1/2 ) which depends only on (ℓ, δ, 1/ε, ν 3 := E|Y i | 3 , |α|). Moreover the dependence is continuous in the parameters, and non-decreasing in all the parameters but ℓ. This allows for uniform bounds when the parameters are in compact subsets of their domain.

Remark 2.3. Note that non-trivial ε and δ exist by the Riemann-Lebesgue lemma as soon has the law of Y i is absolutely continuous.

Probabilistic preliminaries

We start with some useful lemmas. The first one is a key estimate for quantitative central limit theorems, quoted from Petrov's book [START_REF] Petrov | Sums of independent random variables[END_REF].

Lemma 3.1 ([18], Lemma V.2.1, p. 109). Let X 1 , . . . , X n be independent random variables,

EX j = 0, E|X j | 3 < ∞ (j = 1, . . . , n). Denote B n = n j=1 EX 2 j , L n = B -3/2 n n j=1 E|X j | 3 and S n = B -1/2 n n j=1 X j . Then |ϕ Sn (t) -e -t 2 /2 | ≤ 16L n |t| 3 e -t 2 /3
for |t| ≤ 1 4Ln . Lemma 3.2 ([18], Lemma I.2.1, p. 10). For any characteristic function ϕ,

1 -|ϕ(2t)| 2 ≤ 4(1 -|ϕ(t)| 2 )
holds for all t ∈ R. Lemma 3.3. Let (S n ) be as in Theorem 2.1. Let T be independent of (S n ) and assume that its characteristic function ϕ T is Lebesgue integrable. Then for all n ≥ 1, the density of S n + T n is bounded by a number

C = C(1/ε, δ, ν 3 , ϕ T 1 ), which is non-decreasing in each of its parameters. Proof of Lemma 3.3. Since ϕ Sn+T /n = ϕ Sn ϕ T (•/n) is Lebesgue integrable, the inversion formula ensures that the density of S n + 1 n T at x equals g Sn+ 1 n T (x) = 1 2π ∞ -∞ e -itx ϕ Sn (t)ϕ T t/n dt = 1 2π ∞ -∞ e -itx e -t 2 /2 ϕ T t/n dt + 1 2π ∞ -∞ e -itx (ϕ Sn (t) -e -t 2 /2 )ϕ T t/n dt ≤ 1 √ 2π + 1 2π ∞ -∞ |ϕ Sn (t) -e -t 2 /2 | |ϕ T t/n | dt.
To bound the last integral, we apply Lemma 3.1 with B n = n and L n = ν 3 n -1/2 . We get

g Sn+T /n (x) ≤ 1 √ 2π + 1 2π |t|≤ √ n 4ν 3 16ν 3 √ n |t| 3 e -t 2 /3 dt + 1 2π |t|> √ n 4ν 3 (|ϕ Sn (t)| + e -t 2 /2 )|ϕ T t/n | dt ≤ 1 √ 2π + 72ν 3 π √ n + 1 2π |t|> √ n 4ν 3 |ϕ Sn (t)| |ϕ T t/n | dt I + 1 2π |t|> √ n 4ν 3 e -t 2 /2 dt.
For integral (I) from the last line we use (2.2) which implies

|ϕ Sn (t)| = ϕ Y 1 t/ √ n n ≤ (1 -ε) n for |t| ≥ δ √ n.
However, δ might be larger than 1 4ν 3 , i.e. 4ν 3 δ ≥ 1. If this is so, we use Lemma 3.2 on characteristic functions: since (2.2) implies

1 -|ϕ Y i (t)| 2 ≥ ε for |t| ≥ δ, Lemma 3.2 implies that for any non-negative integer k, 1 -|ϕ Y i (t)| 2 ≥ 4 -k ε for |t| ≥ 2 -k δ. Taking k = ⌈log 2 (4ν 3 δ)⌉) implies 2 -k δ ≤ 1 4ν 3 and 4 -k ≥ 1 (8ν 3 δ) 2 and hence |ϕ Y i (t)| 2 ≤ 1 - ε (8ν 3 δ) 2 for |t| ≥ 1 4ν 3 .
In any case, we obtain that

|ϕ Sn (t)| ≤ 1 - ε max(1, (8ν 3 δ) 2 ) n/2 for |t| ≥ √ n 4ν 3 . (3.1)
Using the above we estimate the integral (I) as follows. Using the rough estimate

(1 -x) m = e m log(1-x) ≤ e -mx = 1 e mx ≤ 1 mx ,
valid for any m > 0 and x ∈ (0, 1), we get

I ≤ 1 - ε max(1, (8ν 3 δ) 2 ) n/2 n ∞ 0 |ϕ T (u)| du ≤ n 2 max(1, (8ν 3 δ) 2 ) nε ϕ T 1 ≤ 2 ϕ T 1 1 + (8ν 3 δ) 2 ε .
Finally we obtain that the density of

S n + 1 n T is bounded by C 1 + C 2 ν 3 + C 3 ϕ T 1 1+(ν 3 δ) 2 ε for some constants C 1 , C 2 , C 3 > 0.
Denote by φ the density of the standard normal distribution on R and let Φ be its cumulative distribution function. Our last two preliminary statements are easy consequences of the equality e γt φ(t) = e γ 2 /2 φ(tγ) satisfied by the Gaussian density Lemma 3.4. Let Z be a standard normal random variable. For any Borel set A ⊂ R,

Ee γZ 1 Z∈A = e γ 2 /2 P(Z ∈ A -γ).
(3.2) Lemma 3.5. For any s > 0 and α ∈ R, and λ such λs

-α s > 1, it holds ∞ 0 λe -λx 1 √ 2πs e -(x-α) 2 2s 2 dx = 1 √ 2πs e -α 2 2s 2 1 + O 1 + |α| s λs -α s .
In particular if s and α stay bounded in the sense that s ∈ [1/S, S], |α| ≤ A holds for some A, S > 0, then for λ > 2AS -2 + S -1 , the last factor simplifies to

1 + O A,S 1 λ .
Proof. Using a standard Gaussian random variable Z, we rewrite the left-hand side as

T := ∞ 0 λe -λx 1 √ 2πs e -(x-α) 2 2s 2 dx = λEe -λ(sZ+α) 1 Z>-α s = λe -λα e λ 2 s 2 /2 P Z > λs - α s = λe -α 2 2s 2 e (λs-α s ) 2 /2 1 -Φ λs - α s
where the second equality follows from (3.2). Next we use the classical bound, for t > 0,

1 t ≥ √ 2π e t 2 /2 1 -Φ(t) ≥ 1 √ t 2 + 2 , which implies that for t > 1, √ 2π te t 2 /2 1 -Φ(t) = 1 + O(1/t 2 ). When λs -α s > 1 we obtain that √ 2πs e α 2 2s 2 T = λs λs -α s 1 + O 1 (λs -α s ) 2 = 1 + α s λs -α s • 1 + O 1 (λs -α s ) 2 = 1 + O 1 + |α| s λs -α s .
The case when α and s are bounded readily follows.

4 Proof of Theorem 2.1

Our aim is to show that for any α ∈ R,

I = J × (1 + O(n -1/2
)) where

I = Ee ℓ √ nSn 1 {Sn≤α} and J = 1 ℓ √ 2πn e ℓ √ nα e -α 2 /2 .
Let Z be a standard Gaussian random variable, independent of the Y i 's. The first step is to introduce the modified quantity

I 2 = Ee ℓ √ n(Sn+n -1 Z) 1 {Sn+n -1 Z≤α} ,
and to check that it is enough for our purpose to establish

I 2 = J × (1 + O(n -1/2
)). In order to do so we estimate the difference between I and I 2 . By the triangle inequality:

|I 2 -I| ≤ Ee ℓ √ nSn |e ℓn -1/2 Z -1|1 {Sn≤α} + Ee ℓ √ n(Sn+n -1 Z) 1 {Sn+n -1 Z≤α} -1 {Sn≤α} = I 3 + I 4 + I 5 ,
where

I 3 = Ee ℓ √ nSn |e ℓn -1/2 Z -1|1 {Sn≤α} I 4 = Ee ℓ √ n(Sn+n -1 Z) 1 {α<Sn≤α-n -1 Z} I 5 = Ee ℓ √ n(Sn+n -1 Z) 1 {α-n -1 Z<Sn≤α} .
By independence

I 3 = I • E|e ℓn -1/2 Z -1|. Next, we use that for t ∈ [0, 1], E|e tZ -1| ≤ E e 2tZ -2e tZ + 1 = e 2t 2 -2e t 2 /2 + 1 ≤ 3t.
Thus, under the hypothesis n ≥ 16ℓ 2 we obtain that

I 3 ≤ 3ℓ √ n I ≤ 3I/4.
For the term I 4 , we introduce T = U + U ′ where U and U ′ are independent random variables uniformly distributed in (-1, 1) and note that ϕ T (u) = (sin(u)/u) 2 is Lebesgue integrable. Since |T | ≤ 2 a.s.,

I 4 ≤ e ℓ √ nα ∞ 0 P(α < S n ≤ α + n -1 x)φ(x) dx ≤ e ℓ √ nα ∞ 0 P(α -2/n < S n + T /n ≤ α + (x + 2)/n)φ(x) dx.
By Lemma 3.3, S n + T /n has a density which is bounded by a constant, say C > 0. Then

I 4 ≤ e ℓ √ nα ∞ 0 C x + 4 n φ(x) dx = C n e ℓ √ nα (π -1/2 + 2) = C √ n • ℓ √ 2π e α 2 /2 J • (π -1/2 + 2) = J • O(n -1/2 ).
The term I 5 is estimated in a similar way:

I 5 ≤ e ℓ √ nα ∞ 0 e ℓxn -1/2 P(α -x/n < S n ≤ α)φ(x) dx ≤ e ℓ √ nα ∞ 0 e ℓxn -1/2 P(α -(x + 2)/n < S n + T /n ≤ α + 2/n)φ(x) dx ≤ e ℓ √ nα ∞ 0 e ℓx C x + 4 n φ(x) dx = J • O(n -1/2 ).
This concludes the first step of the proof, which guarantees that for n ≥ 16ℓ 2

|I 2 -I| ≤ 3ℓ √ n I + O 1 √ n J . (4.1)
Our next task is to prove that I 2 = J × (1 + O(n -1/2 )). We use the Fourier transform approach. It relies on the Parseval formula, which ensures that whenever random variables V and W have square integrable densities g V and g W , their characteristic functions are also square integrable and the following relation holds:

∞ -∞ g V (x)g W (x) dx = 1 2π ∞ -∞ ϕ V (t)ϕ W (t) dt. (4.2)
Given n, set W = α -(S n + 1 n Z). Then

I 2 = Ee ℓ √ n(Sn+n -1 Z) 1 {Sn+n -1 Z≤α} = e ℓ √ nα Ee -ℓ √ nW 1 W ≥0 = e ℓ √ nα ℓ √ n ∞ 0 ℓ √ ne -ℓ √ nx dP W (x).
Let V a random variable having exponential distribution with parameter ℓ √ n. We have proved that

I 2 := ℓ √ ne -ℓ √ nα I 2 = g V (x) dP W (x).
Observe that our goal is to establish that

I 2 = 1 √ 2π e -α 2 /2 (1 + O(n -1/2
)). Since P W is given by the convolution of a probability measure and of the bounded density of Z/n, it is absolutely continuous with bounded (and thus square-integrable) density. Hence, we may apply the Parseval formula (4.2) to V and W . Since ϕ W (t) = e iαt ϕ Sn (t)e -t 2 /(2n 2 ) , we obtain

I 2 = 1 2π ∞ -∞ 1 1 -it ℓ √ n e -iαt ϕ Sn (t)e -t 2 /(2n 2 ) dt = M + E 2π ,
where

M = ∞ -∞ e -iαt 1 -it ℓ √ n e -t 2 /2 e -t 2 /(2n 2 ) dt E = ∞ -∞ e -iαt 1 -it ℓ √ n (ϕ Sn (t) -e -t 2 /
2 )e -t 2 /(2n 2 ) dt.

Applying Parseval's formula as before, but replacing S n with and independent standard Gaussian variable

G yields M/(2π) = g V dP W where W = α -(G + Z/n) has N (α, 1 + n -2 ) distribution. Therefore M 2π = ∞ -∞ ℓ √ ne -ℓ √ n e -(x-α) 2 2(1+n -2 ) 2π(1 + n -2 ) dx.
Lemma 3.5 with λ := ℓ √ n and

s 2 := 1 + n -2 yields, provided ℓ √ n ≥ 2|α| + 1, M 2π = 1 2π(1 + n -2 ) e - α 2
2(1+n -2 ) (1 + O(n -1/2 )) = 1 √ 2π e -α 2 2 (1 + O(n -1/2 )).
It remains to bound the error term:

|E| = ∞ -∞ e -iαt 1 -it ℓ √ n (ϕ Sn (t) -e -t 2 /2 )e -t 2 /(2n 2 ) dt ≤ ∞ -∞ ϕ Sn (t) -e -t 2 /2 e -t 2 /(2n 2 ) dt ≤ |t|≤ √ n/(4ν 3 ) 16ν 3 n -1/2 |t| 3 e -t 2 /3 dt + |t|> √ n/(4ν 3 ) ϕ Sn (t) e -t 2 /(2n 2 ) dt + |t|> √ n/(4ν 3 ) e -t 2 /2 dt ≤ Cν 3 n -1/2 + I + II,
where the second inequality follows from Lemma 3.1. The estimate of the term II is immediate:

II ≤ 2e -n/(32ν 2
3 ) .

In order to estimate I, we use (3.1) and a variant of its previous application using the bound (1x) m ≤ 1/e mx ≤ 2/(mx) 2 for x ∈ (0, 1):

I ≤ 1 - ε max(1, (8ν 3 δ) 2 ) n/2 n √ 2π = O ν 3 , 1 ε ,δ (n -1/2 ). Hence E = O(n -1/2 ) = e -α 2 /2 O(e α 2 /2 n -1/2 ) = e -α 2 /2 O |α| (n -1/2
). This ends the proof of the second step, asserting I 2 = J × (1 + O(n -1/2 )). Combining the latter with (4.1) yields the claim of the theorem.

Application to spectral gaps

Our volume asymptotics for Orlicz balls allow to complement a result of Kolesnikov and Milman [START_REF] Kolesnikov | The KLS isoperimetric conjecture for generalized Orlicz balls[END_REF] about a famous conjecture by Kannan, Lovász and Simonovits, which predicts the approximate value of the Poincaré constants of convex bodies (a.k.a. inverse spectral gap of the Neumann Laplacian). More precisely if µ is a probability measure on some Euclidean space, one denotes by C P (µ) (resp. C Lin P (µ)) the smallest constant C such that for all locally Lipschitz (resp. linear) functions f , it holds

Var µ (f ) ≤ C |∇f | 2 dµ.
Obviously C Lin P (µ) ≤ C P (µ), and the KLS conjecture predicts the existence of a universal constant c such that for any dimension n and any convex body K ⊂ R n , C P (λ K ) ≤ c C Lin P (λ K ), where λ K stands for the uniform probability measure on K. The conjecture turned out to be central in the understanding in high-dimension volume distributions of convex sets. We refer to e.g. [START_REF] Alonso | Approaching the Kannan-Lovász-Simonovits and variance conjectures[END_REF][START_REF] Brazitikos | Geometry of isotropic convex bodies[END_REF][START_REF] Kolesnikov | The KLS isoperimetric conjecture for generalized Orlicz balls[END_REF][START_REF] Yin | The Kannan-Lovász-Simonovits conjecture[END_REF][START_REF] Chen | An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture[END_REF] for more background and references, and to [START_REF] Klartag | Bourgain's slicing problem and KLS isoperimetry up to polylog[END_REF] for a recent breakthrough. Kolesnikov and Milman have verified the conjecture for some Orlicz balls. We state next a simplified version of their full result on generalized Orlicz balls. Part of the simplification is unessential, as it amounts to reduce by dilation and translations to a convenient setting. A more significant simplification, compared to their work, is that we consider balls where all coordinates play the same role.

Theorem 5.1 ([13]

). Let V : R → R + be a convex function with V (0) = 0 and such that dµ(x) = e -V (x) dx is a probability measure. We also assume that the function x → xV ′ (x), defined almost everywhere, belongs to the space L 2 (µ). For each dimension n ≥ 1, let

Level n (V ) := E ≥ 0; e -E Vol n B n V /E ≥ 1 e
n n e -n n! .

Then there exists a constant c, which depends only on V (through xV ′ (X) L 2 (µ) ) such that for all E ∈ Level n (V ),

C P (λ B n V /E ) ≤ c C Lin P (λ B n V /E ).
Moreover, Level n (V ) is an interval of length at most e n!e n n n = e √ 2πn(1 + o(1)) as n → ∞, and

1 + n R V (x)e -V (x) dx ∈ Level n (V ).
We can prove more about the set Level n (V ) and in particular we show that its length is of order √ n:

Proposition 5.2. Let V : R → R + be a Young function such that dµ(x) = e -V (x) dx is a probability measure. Let m 1 = V e -V be the average of V with respect to µ, and σ 2 1 its variance. For every ε ∈ (0, 1) there exists an integer n 0 = n 0 (V, ε) depending on V such that for all n ≥ n 0 ,

m 1 n -σ 1 (1 -ε) √ 2n ; m 1 n + σ 1 (1 -ε) √ 2n ⊂ Level n (V ).
Proof. We apply Theorem 1.1, with Ψ = V and λ = 1. With the notation of the theorem µ = µ 1 and Z 1 = e -V = 1. We choose E of the following form:

E = m 1 n + ασ 1 √ n with |α| ≤ (1 -ε) √ 2. The theorem ensures that Vol B n V /E = e E σ 1 √ 2πn e -α 2 /2 1 + O 1 √ n , where the O(n -1/2 ) is uniform in α ∈ [-(1 -ε) √ 2, (1 -ε) √ 2]
. A sharp inequality due to Nguyen and Wang ensures that σ 2 1 = Var e -V (V ) ≤ 1 (see [START_REF] Nguyen | Inégalités fonctionnelles et convexité[END_REF][START_REF] Wang | Heat capacity bound, energy fluctuations and convexity[END_REF], [START_REF] Van Hoang Nguyen | Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa's theorem[END_REF] and for a short proof [START_REF] Fradelizi | Optimal concentration of information content for log-concave densities[END_REF]). Therefore

e -E Vol B n V /E ≥ 1 √ 2πn e -(1-ε) 2 1 + O 1 √ n , whereas 1 e n n e -n n! = e -1 √ 2πn (1 + o(1)).
Hence for n large enough and for all α in the above interval e -E Vol B n V /E ≥ 1 e n n e -n n! .

Corollary 5.3. Let V : R → R + be a Young function such that dµ(x) = e -V (x) dx is a probability measure. Let m 1 and σ 2 1 denote the average and the variance of V with respect to µ. We also assume that the function x → xV ′ (x) belongs to the space L 2 (µ). Let ε ∈ (0, 1). Then there exists c = c(V, ε) such that for all n ≥ 1 and all E ∈

m 1 n -σ 1 (1 -ε) √ 2n ; m 1 n + σ 1 (1 -ε) √ 2n , C P (λ B n V /E ) ≤ c C Lin P (λ B n V /E ).
Proof. Combining the later proposition and theorem yields the result for n ≥ n 0 (V, ε).

In order to deal with smaller dimensions, we simply apply known dimension dependent bounds: e.g. Kannan, Lovász and Simonovits [START_REF] Kannan | Isoperimetric problems for convex bodies and a localization lemma[END_REF] proved that C P (λ K ) ≤ κnC Lin P (λ K ) for all convex bodies K in R n , with κ a universal constant.

Asymptotic independence of coordinates

A classical observation, going back to Maxwell, but also attributed to Borel and to Poincaré, states that for a fixed k, the law of the first k coordinates of a uniform random vector on the Euclidean sphere of R n , centered at the origin and of radius √ n, tends to the law of k independent standard Gaussian random variables as n tends to infinity. Quantitative versions of this asymptotic independence property where given by Diaconis and Freedman [START_REF] Diaconis | A dozen de Finetti-style results in search of a theory[END_REF], as well as a similar result for the unit sphere of the ℓ 1 -norm, involving exponential variables in the limit. Extensions to random vectors distributed according to the cone measure on the surface of the unit ball B n p were given by Rachev and Rüschendorf [START_REF] Rachev | Approximate independence of distributions on spheres and their stability properties[END_REF], while Mogul'skiȋ [START_REF] Mogul | skiȋ. De Finetti-type results for l p[END_REF] dealt with the case of the normalized surface measure. Explicit calculations, or the probabilistic representation put forward in [START_REF] Barthe | A probabilistic approach to the geometry of the l n p -ball[END_REF], easily yield asymptotic independence results for the first k coordinates of a uniform vector on the set B n p itself, when k is fixed and n tend to infinity.

In this section we study marginals of a random vector ξ (n) uniformly distributed on B n Ψ/En , where E n and n tend to ∞.

Let us start with the simple case when E n = mn for some m > 0, which can be written as m = m λ for some λ > 0. Let k ≥ 1 be a fixed integer, then the density at (x 1 , . . . , x k ) ∈ R k of the first k coordinates (ξ

(n) 1 , . . . , ξ (n) k ) is equal to Vol n-k B n Φ/En ∩ {y ∈ R n ; y i = x i , ∀i ≤ k} Vol B n Φ/En = Vol B n-k Ψ/(En-k i=1 Ψ(x i )) Vol B n Ψ/En
We apply Corollary 1.2 twice: once for the denominator, and once for the numerator after writing

m λ n - i≤k Ψ(x i ) = m λ (n -k) + m λ k -i≤k Ψ(x i ) √ n -k √ n -k.
We obtain that the above ratio is equivalent to

Z n-k λ e λ(En-i≤k Ψ(x i )) λσ λ 2π(n -k) • λσ λ √ 2πn Z n λ e λEn ∼ e -λ k i=1 Ψ(x i ) Z k λ •
Thus we have proved the convergence in distribution of (ξ

(n) 1 , . . . , ξ (n)
k ) to µ ⊗k λ as n tends to infinity. In other words the first k coordinates of ξ (n) are asymptotically i.i.d. of law µ λ . This is true for more general balls and for a number of coordinates going also to infinity:

Theorem 6.1. Let E n = m λ n + α n σ λ √ n, where (α n ) n≥1 is bounded. Let the random vector ξ (n) be uniformly distributed on B n Ψ/En . For any k n = o( √ n), lim n→∞ d T V (ξ (n) 1 , . . . , ξ (n) kn ), µ ⊗kn λ = 0.
Proof. Below, we simply write ξ i for ξ

(n)

i . Recall that (X i ) are i.i.d. r.v.'s with the distribution µ λ . Set

t n := n 1/4 k 1/2 n so that t n = o( √ n) and k n = o(t n ).
The total variation distance between the law of (ξ

(n) 1 , . . . , ξ (n)
kn ) and

µ ⊗kn λ is R kn 1 Vol(B n Ψ/En ) R n-kn 1 {(x,y)∈B n Ψ/En } dy - 1 Z kn λ e -λ(Ψ(x 1 )+•••+Ψ(x kn )) dx ≤ B kn Ψ/tn Vol B n-kn Ψ/(En-kn i=1 Ψ(x i ))
Vol(B n Ψ/En ) -1

Z kn λ e -λ(Ψ(x 1 )+

Ψ(X i ) > t n ≤ E kn i=1 Ψ(X i ) t n = k n m λ t n = o(1)
Similarly, and since by definition n i=1 Ψ(ξ i ) ≤ E n and the ξ i 's are exchangeable

P kn i=1 Ψ(ξ i ) > t n ≤ E kn i=1 Ψ(ξ i ) t n ≤ k n E n nt n = k n m λ t n = o(1).
In order to estimate (6.1), we use Theorem 1.1. Since

k n = o( √ n) and t n = o( √ n), we know that E n -t = m λ (n -k n ) + β n σ λ √ n -k n ,
where Taking the quotient gives lim n P(S n ≥ t) = e -λt .

β n := α n n n -k n + m λ k n -t σ λ √ n -k n is a bounded sequence such that β n -α n = o(
λ • E n - n i=1 Φ ξ (n) i -→ Exp(1). Proof. Let S n := E n -n i=1 Φ ξ (n) i ≥ 0. For t ≥ 0, P(S n ≥ t) = P n i=1 Φ ξ (n) i ≤ E n -t = Vol B n Ψ/(En-t) Vol B n Ψ/En

Integrability of linear functionals

Linear functionals of uniform random vectors on convex bodies are well studied quantities. Their density function, known as the parallel section function, measures the volume of hyperplane sections in a given direction. We refer e.g. to the book [START_REF] Brazitikos | Geometry of isotropic convex bodies[END_REF], and in particular to its sections 2.4 and 8.2 about the ψ 1 and ψ 2 properties, which describe uniform integrability features (exponential integrability for ψ 1 , Gaussian type integrability for ψ 2 ). They can be expressed by upper bounds on the Laplace transform.

In this section, we deal with even Young functions Ψ, so that the corresponding sets B n Ψ are origin-symmetric, and actually unconditional. The forthcoming study is valid for any dimension, without taking limits, so we consider the dimension n fixed, and write ξ = (ξ 1 , . . . , ξ n ) for a uniform random vector on B n Ψ . We show that the arguments of [3] for ℓ n p unit balls extend to Orlicz balls. Ee a i ξ 1 .

Proof. Let ε 1 , . . . , ε n be i.i.d. random variables with P(ε i = 1) = P(ε i = -1) = 1 2 , and independent of ξ. Then by symmetry of Ψ, (ε 1 ξ 1 , . . . , ε n ξ n ) has the same distribution as ξ. Hence,

Ee a,ξ = E n i=1 e a i ε i ξ i = E E n i=1 e a i ε i ξ i ξ = E n i=1 cosh(a i ξ i ).
Next by the subindependence property of coordinates, due Pilipczuk and Wojtaszczyk [START_REF] Pilipczuk | The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball[END_REF], and using the symmetry again as well as exchangeability:

Ee a,ξ ≤ n i=1 E cosh(a i ξ i ) = n i=1 Ee a i ξ i = n i=1 Ee a i ξ 1 .
The above lemma shows that the Laplace transform of any linear functional a, ξ can be upper estimated using the Laplace transform of the first coordinate ξ 1 . Therefore it is natural to study the law of ξ 1 . For t ∈ R consider the section of B n Ψ :

S(t) := {y ∈ R n-1 ; (t, y) ∈ B n Ψ }. and f (t) := Vol n-1 (S(t)). Then P ξ 1 (dt) = f (t)dt/Vol n (B n Ψ )
. By the Brunn principle, f is a log-concave function. It is also even by symmetry of the ball, therefore it is non-increasing on R + . We observe that a slightly stronger property holds: Averaging these two inequalities and using the convexity of Ψ, we get for any θ ∈ (0, 1):

(1θ)Ψ(t) + θΨ(u)

+ n-1 i=1 Ψ ((1 -θ)a i + θb i ) ≤ 1. (7.1)
This can be rewritten as (1θ)a + θb ∈ S Ψ -1 ((1θ)Ψ(t) + θΨ(u)) .

Hence we have shown that (1θ)S(t) + θS(u) ⊂ S Ψ -1 ((1θ)Ψ(t) + θΨ(u)) ,

and by the Brunn-Minkowski inequality, in multiplicative form f (t) 1-θ f (u) θ ≤ f Ψ -1 ((1θ)Ψ(t) + θΨ(u)) .

Note that if in (7.1) we had used convexity in the form Ψ((1-θ)t+θu) ≤ (1-θ)Ψ(t)+θΨ(u), then we would have derived the Brunn principle from the Brunn-Minkowski inequality.

The next result shows that Ψ is more convex than the square function, the corresponding Orlicz balls enjoy the ψ 2 property. This applies in particular to B n p for p ≥ 2, a case which was treated in [START_REF] Barthe | Extremal slabs in the cube and the Laplace transform[END_REF] .

Lemma 7.2 ensures that there exists a concave function c such that for all u ≥ 0, log f (u) = c(Ψ(u)). Note that c is also non-increasing on R + since the section function f is. Hence

u ≥ 0 → log f ( √ u) = c(Ψ( √ u))
is concave. Theorem 12 of [START_REF] Barthe | Extremal slabs in the cube and the Laplace transform[END_REF] ensures that t ≥ 0 → R e u √ t f (u) du is log-concave. In other words,

t ≥ 0 → log L ξ 1 ( √ t)
is concave. From Lemma 7.1, using symmetry and the above concavity property

Ee a,ξ ≤ n i=1 L ξ 1 (a i ) = n i=1 L ξ 1 a 2 i ≤ L ξ 1 1 n i a 2 i n = L ξ 1 |a| √ n n .
To conclude we need the bound L ξ 1 (t) ≤ e t 2 E(ξ 2 1 )/2 (it follows from the fact that t ≥ 0 → log L ξ 1 ( √ t) is concave, hence upper bounded by its tangent application at 0, which is easily seen to be tE(ξ 2 1 )/2). We obtain Ee a,ξ ≤ e

Lemma 7 . 1 .

 71 Let a ∈ R n , and ξ be uniform on B n Ψ , then Ee a,ξ ≤ n i=1

Lemma 7 . 2 .

 72 Let Ψ be an even Young function and f (t) = Vol n {y ∈ R n-1 ; (t, y) ∈ B n Ψ } . Then the function log f • Ψ -1 is concave and non-increasing on R + . Here Ψ -1 is the reciprocal function of the restriction of Ψ to R + . Proof. Let t, u ≥ 0. Let a ∈ S(t) and b ∈ S(u). Then by definition Ψ(t) + n-1 i=1 Ψ(a i ) ≤ 1 and Ψ(u) + n-1 i=1 Ψ(b i ) ≤ 1.

1 2 |a| 2 E(ξ 2 1 )

 121 , and we conclude using the symmetries of ξ sinceE a, ξ 2 = i a 2 i E(ξ 2 i ) + i =j a i a j E(ξ i ξ j ) = |a| 2 E(ξ 2 1).

  ••+Ψ(x kn )) dx + P (ξ 1 , . . . , ξ kn ) ∈ B kn Ψ/tn + P (X 1 , . . . , X kn ) ∈ B kn

								Ψ/tn
	=	0	tn	Vol B n-kn Ψ/(En-t) Vol(B n Ψ/En )	-	e -λt Z kn λ	d dt	Vol(B kn Ψ/t ) dt	(6.1)
				kn			kn
		+ P	Ψ(ξ i ) > t n + P	Ψ(X i ) > t n .
				i=1			i=1
	By Markov's inequality,
				kn
				P
				i=1

  1), both properties holding uniformly in t ∈ [0, t n ]. Therefore, Theorem 1.1 applied to B n-kn Ψ/(En-t) gives The next result gives the asymptotic distribution of a sort of distance to the boundary for high-dimensional Orlicz balls. Theorem 6.2. Let E n = m λ n+α n σ λ √ n, where (α n ) n≥1 is bounded. Let the random vector ξ (n) be uniformly distributed on B n Ψ/En . Then the following convergence in distribution occurs as n goes to infinity:

	uniformly in t ∈ [0, t n ]. On the other hand, Theorem 1.1 applied to B n Ψ/En yields
				Vol B n Ψ/En =	Z n λ e λEn λσ λ √ 2πn	e -α 2 n /2 (1 + o(1)).
	Combining the above two asymptotic expansions, we obtain
						Vol B n-kn Ψ/(En-t) Vol(B n Ψ/En )	=	e -λt λ Z kn	(1 + o(1))
	uniformly in t ∈ [0, t n ]. Therefore the term (6.1) equals
	o(1)	0	tn	e -λt Z kn λ	d dt	Vol(B kn Ψ/t ) dt = o(1) • P	kn i=1	Ψ(X i ) ≤ t n = o(1).
			Vol B n-kn Ψ/(En-t) =	Z n-kn λ λσ λ 2π(n -k n ) e λ(En-t)	e -β 2 n /2 (1 + o(1))

.

  Theorem 7.3. Let Ψ be an even Young function, such that t > 0 → Ψ( √ t) is convex. Let ξ be a uniform random vector on B n Ψ . Then for all a ∈ R n , Proof. Let L X (t) = Ee tX denote the Laplace transform of a real valued random variable. Then with the notation of Lemma 7.2, L ξ 1 (t) = e tu f (u) du Vol(B n Ψ )

	Ee a,ξ ≤ Ee	|a| √ n ξ 1	n	≤ e	1 2 E a,ξ 2 .

Acknowledgements: We are grateful to Emanuel Milman and Reda Chhaibi for useful discussions on related topics. We also thank Joscha Prochno for communicating his recent work to us.


Clément Dégut

Veronika Schwarz

Luc Ponchon

Pierre Barraud

Ronald Micura

Carine Tisné
email: carine.tisne@cnrs.fr

Design of cross-linked RNA/protein complexes for structural studies

Keywords: methyltransferase, TrmI, tRNA, cross-link, RNA-protein complex, covalent trapping

Crystallographic studies of RNA/protein complexes are primordial for the understanding of recognition determinants and catalytic mechanisms in the case of enzymes. However, due to the flexibility and propensity to conformational heterogeneity of RNAs, as well as the mostly electrostatic interactions of RNA/protein complexes, they are difficult to crystallize. We present here a method to trap the two interacting partners in a covalent complex, based on a modified reactive RNA allowing the use of the full range of common crystallogenesis tools.

We demonstrate the practicability of our approach with the production of a covalent complex of the Thermus thermophilus m 1 A58 tRNA modification enzyme, and a modified stem loop mimicking the natural substrate of the enzyme.

Introduction

Structural studies of tRNA modification enzymes are motivated by two main questions: what is the catalytic mechanism and what are the determinants of tRNA recognition? X-ray crystallography is the method of choice to address these points. However, crystal structures of modification enzymes in complex with tRNAs are difficult to obtain. Considering the mostly electrostatic interaction surface of protein/tRNA complexes, many high-salt crystallization conditions are inappropriate, since they drastically decrease protein/RNA affinity. Moreover, due to their flexibility and propensity to conformational heterogeneity, RNAs are difficult to crystallize. For these reasons, structural information is currently sparse for RNA modification enzymes. To overcome the difficulties of protein/RNA crystallization, covalent trapping strategies based on the catalytic mechanism of particular RNA modification enzymes have been developed [START_REF] Guelorget | Mechanism-based strategies for trapping and crystallizing complexes of RNA-modifying enzymes[END_REF]. However, natural enzyme-RNA covalent intermediates are rare and do not appear in many reactions, including the methylation of the N1-atom of adenosine (m 1 A) at position 58 of tRNAs [START_REF] Oerum | m1A Post-Transcriptional Modification in tRNAs[END_REF]. Inspired by the structure-based covalent trapping strategies that have been implemented for studying protein/DNA complexes [START_REF] Verdine | Covalent trapping of protein-DNA complexes[END_REF], we sought to determine whether similar strategies could be applied to RNA modification enzymes for which no natural covalent intermediate can be isolated. As a proof of principle, we chose the TrmI enzyme from Thermus thermophilus (TtTrmI), which constitutes a perfect model system. No TtTrmI/tRNA structure is indeed available, but fairly good models of the complex can be obtained from the crystal structure of its human homologue in complex with tRNA [START_REF] Finer-Moore | Crystal Structure of the Human tRNA m(1)A58 Methyltransferase-tRNA(3)(Lys) Complex: Refolding of Substrate tRNA Allows Access to the Methylation Target[END_REF]. Such modeling can indeed be used to drive the rationale design of the cross-linking sites. In detail, we investigated the possibility to covalently link TtTrmI to an RNA hairpin mimicking the Tarm of its substrate tRNA Asp , and determine whether covalently trapped enzyme-RNA complexes are amenable to crystallography.

Materials and methods

Samples

The genes encoding the different TtTrmI mutants were purchased from GeneCust in pET28a vectors. The TtTrmI mutants were expressed and purified as described for the wild-type [START_REF] Barraud | Crystal structure of Thermus thermophilus tRNA m1A58 methyltransferase and biophysical characterization of its interaction with tRNA[END_REF].

RNAs with site-specific disulfide crosslinker units were prepared following the convertible nucleoside strategy for RNA solid-phase synthesis as previously described [START_REF] Allerson | A chemical method for sitespecific modification of RNA: the convertible nucleoside approach[END_REF] Associates, Inc., Dexter, MI 48130 USA (order number BA 0263 and BA 0279). The synthesis of the oligoribonucleotides followed standard procedures. Their transformation and deprotection was based on the procedures described previously [START_REF] Allerson | A chemical method for sitespecific modification of RNA: the convertible nucleoside approach[END_REF] and further optimized for the RNA containing the inosine building block. In short, the beads were treated 1) with a 1.0 M solution of cystamine in dimethylsulfoxide overnight, 2) washed with acetonitrile, 3) treated with a 1.0 M solution of 1,8-diaazabicyclo[6.4.0]undec-7-ene (DBU) in acetonitrile for 2 hours, and 4) treated with a 1:1 mixture of methylamine in water (40%) and aqueous ammonia (30%) for 30 min at 65°C. The cleaved RNA was filtered into a round bottom flask and the solvents evaporated. The residue was treated with 1.0 M TBAF in THF containing 10 vol% H2O and stored for 12 hours at 37 °C. The reaction was quenched by adding 1.0 M triethylammonium acetate buffer and the mixture was applied to a Sephadex G25 size exclusion column (HiPrep TM 26/10 Desalting, 2.6x10 cm, GE Healthcare). The crude RNA was purified by anion exchange chromatography using Dionex DNAPac PA-100 columms as previously described in reference [START_REF] Riml | Synthesis, Thermodynamic Properties, and Crystal Structure of RNA Oligonucleotides Containing 5-Hydroxymethylcytosine[END_REF].

TtTrmI/RNA covalent complexes tests

The modified RNAs were dissolved at 500 μM in buffer A containing 50 mM Tris-HCl pH

TtTrmI Q200C/RNA G57* covalent complex

The covalent complex between the TrmI Q200C mutant and the RNA G57* was formed in 400 µL with the protein and the RNA at 100 μM in buffer A for 1h at 35°C. Purification was conducted at 4°C on an AktaPurifier system with a 8 mL MonoQ column (GE Healthcare) with a 100-500 mM KCl gradient over 500 mL at 4 mL/min. Crystallization assays were set up using a Mosquito robot (TTPLabtech) as 200 nL sitting drops on 96-well plates.

CrystalScreen1&2 (HamptonResearch) and Nucleix (Qiagen) crystallization suite were tested.

Results and discussion

Rational design of modified RNAs and TtTrmI mutants

The most effective technique for maintaining RNA in the active site of the enzyme is to form a covalent complex between this RNA and the enzyme [START_REF] Castello | System-wide identification of RNA-binding proteins by interactome capture[END_REF][START_REF] Cusack | RNA-protein complexes[END_REF][START_REF] Graifer | Photoactivatable RNA derivatives as tools for studying the structural and functional organization of complex cellular ribonucleoprotein machineries[END_REF]. We adapted to RNA/protein complex a method previously described to crystallize the HIV-1 reverse transcriptase covalently linked to its primer DNA [START_REF] Huang | Structure of a covalently trapped catalytic complex of HIV-1 reverse transcriptase: implications for drug resistance[END_REF][START_REF] Huang | Trapping of a catalytic HIV reverse transcriptase*template:primer complex through a disulfide bond[END_REF][START_REF] Bec | Thermodynamics of HIV-1 reverse transcriptase in action elucidates the mechanism of action of non-nucleoside inhibitors[END_REF]. The formation of the covalent complex relies on disulfide-bond exchange between a cystamine group on the RNA and a spatially close free cysteine in the protein (Fig 1A). The sulfur of the cystamine group being already oxidized, any non-specific complex formation such as a RNA-S-S-RNA complex is unlikely.

As a proof of concept, we investigated the formation of covalent complex between TtTrmI, a m 1 A58 tRNA methyltranferase and a substrate hairpin (Fig. 1B, Table S1) mimicking the Tarm of T. thermophilus tRNA Asp . The nucleotide derivatives are available for guanosine and cytosine. Around the A58 target nucleoside, C56, G57 and G59 could be modified (Fig. 1B).

Three RNA were produced, each bearing one modification either on C56, G57 or G59 (see Table S1).

Only one cysteine per enzyme should be present to produce a homogeneous covalent complex. No cysteine residues are present in the wild-type TtTrmI. A model of the enzyme/substrate interaction (Fig. 1C) was generated by alignment of the catalytic cores of TtTrmI [START_REF] Barraud | Crystal structure of Thermus thermophilus tRNA m1A58 methyltransferase and biophysical characterization of its interaction with tRNA[END_REF] and human Trm61 in complex with tRNA [START_REF] Finer-Moore | Crystal Structure of the Human tRNA m(1)A58 Methyltransferase-tRNA(3)(Lys) Complex: Refolding of Substrate tRNA Allows Access to the Methylation Target[END_REF] (Fig. S1). This model reveals that the substrate can be accommodated similarly in the homotetrameric bacterial protein, therefore helping to design appropriate cross-linking sites. Ten cysteine mutants of TtTrmI, namely K71C, R72C, S73C, A74C, R127C, H130C, M172C, E173C, Q200C and P231C, were produced based on their proximity to the modified nucleosides and their position in the TtTrmI structure in order to avoid structure alteration as much as possible.

M172C and R127C could not be expressed in E. coli, and R72C and E173C do not adopt the tetrameric form of TtTrmI essential for its enzymatic activity [START_REF] Barraud | Crystal structure of Thermus thermophilus tRNA m1A58 methyltransferase and biophysical characterization of its interaction with tRNA[END_REF][START_REF] Guelorget | Structural comparison of tRNA m(1)A58 methyltransferases revealed different molecular strategies to maintain their oligomeric architecture under extreme conditions[END_REF][START_REF] Dégut | The m1A(58) modification in eubacterial tRNA: An overview of tRNA recognition and mechanism of catalysis by TrmI[END_REF] (Table S2 and Fig. S2). As a result, these four mutants were discarded from further analysis. Unwanted intermolecular disulfide bridge between TtTrmIs have been checked on non-reducing SDS-PAGE, and no significant amount of intermolecular adducts has been observed except for P231C (Fig. S3).

Testing covalent complex formation in microvolume

The TtTrmI mutants K71C, S73C, A74C, H130C, Q200C and P231C were next tested for their capacity to produce covalent complex with one of the three modified RNA hairpin. In order to not waste modified RNAs (Table S1), the test reactions were performed in small scale (5 µL). Two salt concentrations were tested in the aim of increasing the specificity of the interaction when increasing salt concentration. Although less contaminant species are visible on the gel at 500 mM NaCl, it is detrimental to the overall yield of reaction (Fig. 1D).

Protein-RNA adducts were disrupted in presence of β-mercaptoethanol, confirming that covalent complexes are formed via disulfide bonds (Fig. S5). Three major trends can be drawn from these tests (Fig. 1D). The RNA C56* reacts with very good yields with all tested mutants. While this RNA could seem promising, the very low specificity of the interaction led us to discard it. The RNA G59* has a good cross-linking capacity with K71C and S73C. The RNA G57* has good cross-linking capacity with K71C, S73C, A74C, Q200C and P231C.

However, only for the Q200C mutant, intermolecular protein disulfide bonds are rarely formed either in the protein in solution or in the presence of RNA (Fig. 1D, Fig. S3).

Considering the specificity of all cross-links, their compatibility with our modeling (e.g. the proximity of the modified nucleoside with the mutated residue, Fig. 1C), and the overall quality of the different mutants (Table S1, Fig. S2 andS3), the cross-linking of the RNA G57* with TtTrmI Q200C was selected for further crystallization assays.

Purification and crystallogenesis assays for a selected covalent complex

First, we performed a large-scale production of the covalent complex formed between the RNA G57* and TtTrmI Q200C, and purified it on a MonoQ anion exchange column (Fig. 2A). An isolated peak corresponds to the complex, and analysis by UV spectra, and SDS-PAGE allowed us to evaluate the RNA:TtTrmI tetramer stoichiometry (Fig. 2B,C). Taking into account their respective extinction coefficient, one can evaluate that one RNA for 4

TtTrmI monomers, i.e. one RNA per TtTrmI tetramer, is present in the fraction of the covalent complex. Then, commercials crystallization screening kits were used to find crystallization conditions.

Small crystals appeared within few days (Fig. 1D), but their small size did not allow to record datasets.

Conclusions

In this work we show that covalent trapping of RNA-protein complexes can be applied to RNA modification enzymes. As a proof of principle, we could obtain covalent complexes between several cysteine mutants of TtTrmI and functionalized RNAs. One of these crosslinked RNA-protein complexes was selected and purified in sufficient quantity to carry out first tests of crystallogenesis. Although crystal hits were obtained, they were too small to enable dataset measurements. Nevertheless, these promising trials validate the fact that covalently trapped RNAs to RNA modification enzymes by disulfide bond chemistry are amenable to crystallographic studies. This approach allows for testing a wide range of conditions with very small amounts of reactants enabling the rapid identification of the best pairs of partners and their cross-linking conditions. This practicability study was focused on a RNA-protein complex for which an interaction model could be easily constructed. This approach is nonetheless applicable in the absence of such a model. Some previous knowledge of the interaction and/or the location of the active site are in such case essential. The disulfide unit is linked to the RNA amino-group by a CH2CH2 tether, which provides a high flexibility and a relatively long reach. The acceptor cysteine residue can be as far as 4 Å from the aminogroup of Gs or Cs, which leaves some leeway in the selection of cross-linking sites, and estimations regarding potential positions are sufficient to select good candidates. Our tests also show that the reaction can be performed in a wide range of temperature and salt concentration, yet low salt concentration is favorable as for any RNA/protein interaction study. Similarly to what we observed here, purification of such complex should be relatively straightforward on anion exchange column, regarding the great difference in overall charge between a free protein and a protein-RNA complex. Finally, we note that chemical synthesis of the required pre-functionalized RNA has a typical size limitation of about 30 to 40 nucleotides. Nevertheless, a large variety of chemical and enzymatic ligation techniques exist [START_REF] Lang | The preparation of site-specifically modified riboswitch domains as an example for enzymatic ligation of chemically synthesized RNA fragments[END_REF][17][18] that allow the integration of the modification into large RNA, and hence, covalent RNA-protein complexes with larger RNA are addressable as well.

  O-(tert-Butyldimethylsilyl)-O 4 -(4-chlorophenyl)-3'-O-[(diisopropylamino)(2-cyanoethoxy)phosphino]-5'-O-(4,4'-dimethoxytrityl)uridine and 3'-O-[(diisopropylamino)(2-cyanoethoxy)phosphino]-5'-O-(4,4'-dimethoxytrityl)-2-fluoro-6-O-(4nitrophenylethyl)-2'-O-(tert-butyldimethylsilyl)inosine were purchased from Berry &

8. 0

 0 and 250 mM KCl. Proteins were prepared at 300 μM in the same buffer. The reactions were carried out in 5 μL at 100 μM of TrmI mutants and 200 μM of modified RNA. The reactions were incubated for 1 hour at 40 °C. The results were analyzed by 10% SDS-PAGE without reducing agent.

Fig. 1 .

 1 Fig. 1. Design and analysis of cross-linked complexes between TtTrmI and modified RNAs. (A) Principle of the reaction between modified RNAs and cysteine residue. (B) Sequences of modified RNAs, the adenine in red is the TtTrmI target base for methylation, modifications were introduced at position 56, 57 or 59. (C) Model placing tRNA Lys 3 from the Trm6/Trm61-tRNA Lys 3 structure (PDB 5CCB) in the structure of TtTrmI (PDB 2PWY). In tRNA Lys 3, nucleoside 57 is an adenine, whereas it is a guanine in tRNA Asp . (D) Non-reducing SDS-PAGE of the microvolume reactions. TrmI/RNA covalent complex are indicated by arrows. S-S crossed lane: control unmodified RNA, Ø lane: protein alone.

Fig. 2 .

 2 Fig. 2. Purification of the covalent complex between TrmI Q200C and the RNA G57*. (A) Chromatogram of the TrmI Q200C/RNA G57* purification on a MonoQ column, free TrmI is in peak A, the covalent complex in B. (B) UV spectra of peaks obtained in A. (C) non-reducing SDS-PAGE, TrmI was loaded in lane P, the mixture before purification in lane I. The arrows show the TrmI/RNA covalent complex. (D) Crystals of 8 µm obtained in the Nucleix screen (Qiagen, condition 29: 0.05M Na.HEPES pH7.0, 4M LiCl2, 0.01M MgCl2).

Acknowledgements

The authors acknowledge financial support from the CNRS and the University Paris Descartes.

Synthesis of spin-labeled riboswitch RNAs using convertible nucleosides and DNA-catalyzed RNA ligation. Bioorg Med Chem 21, 6171-80.

[18] Graber, D., Moroder, H., Steger, J., Trappl, K., Polacek, N., and Micura, R. (2010) Reliable semi-synthesis of hydrolysis-resistant 3'-peptidyl-tRNA conjugates containing genuine tRNA modifications. Nucleic Acids Res 38, 6796-802.