Arnaud Debussche
email: arnaud.debussche@bretagne.ens-cachan.fr

WEAK APPROXIMATION OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS: THE NON LINEAR CASE

Keywords: 1991 Mathematics Subject Classification. 35A40, 60H15, 60H35 Weak order, stochastic heat equation, Euler scheme

We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that as it is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case a semilinear stochastic heat equation driven by a space-time white noise.

Introduction

When one considers a numerical scheme for a stochastic equation, two types of errors can be considered. The strong error measures the pathwise approximation of the true solution by a numerical one. This problem has been extensively studied in finite dimension for stochastic differential equations (see for instance [START_REF] Kloeden | Numerical solution of stochastic differential equations[END_REF], [START_REF] Milstein | Numerical integration of stochastic differential equations[END_REF], [START_REF] Milstein | Stochastic numerics for mathematical physics[END_REF], [START_REF] Talay | Probabilistic numerical methods for partial differential equations: elements of analysis, Probabilistic models for nonlinear partial differential equations[END_REF]) and also more recently in infinite dimension for various types of stochastic partial differential equations (SPDEs) (see among others [START_REF] Allen | Finite element and difference approximation of some linear stochastic partial differential equations[END_REF], [START_REF] De Bouard | Debussche A semi-discrete scheme for the stochastic nonlinear Schrödinger equation[END_REF], [START_REF] Davie | Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations[END_REF], [START_REF] Greksch | Time-discretised Galerkin approximations of parabolic stochastic PDEs[END_REF], [START_REF] Gyöngy | Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I[END_REF], [START_REF] Gyöngy | Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II[END_REF], [START_REF] Gyöngy | On discretization schemes for stochastic evolution equations[END_REF], [START_REF] Gyöngy | Rate of Convergence of Implicit Approximations for stochastic evolution equations, Stochastic Differential Equations: Theory and Applications[END_REF], [START_REF] Gyöngy | Rate of Convergence of Space Time Approximations for stochastic evolution equations[END_REF], [START_REF] Gyöngy | Implicit scheme for stochastic parabolic partial differential equations driven by spacetime white noise[END_REF], [START_REF] Hausenblas | Numerical analysis of semilinear stochastic evolution equations in Banach spaces[END_REF], [START_REF] Hausenblas | Approximation for Semilinear Stochastic Evolution Equations[END_REF], [START_REF] Lord | A Numerical Scheme for Stochastic PDEs with Gevrey Regularity[END_REF], [START_REF] Millet | On implicit and explicit discretization schemes for parabolic SPDEs in any dimension[END_REF], [START_REF] Printems | On the discretization in time of parabolic stochastic partial differential equations[END_REF], [START_REF] Shardlow | Numerical methods for stochastic parabolic PDEs[END_REF], [START_REF] Yan | Galerkin finite element methods for stochastic parabolic partial differential equations[END_REF], [START_REF] Yan | Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise[END_REF], [START_REF] Walsh | Finite element methods for parabolic stochastic PDE's[END_REF]). Another way to measure the error is the so-called weak order of convergence of a numerical scheme which is concerned with the approximation of the law of the solution at a fixed time. In many applications, this error is more relevant. Pioneering work by Milstein ([24], [START_REF] Milstein | Weak approximation of solutions of systems of stochastic differential equations[END_REF]) and Talay ([33]) have been followed by many articles (see references in the books cited above). Very few works exist in the literature for the weak approximation of solution of SPDEs. A delayed stochastic differential equation has been studied in [START_REF] Buckwar | Weak approximation of stochastic differential delay equations[END_REF]. Weak order for a SPDE has been studied only recently in [START_REF] De Bouard | Debussche Weak and strong order of convergence of a semi discrete scheme for the stochastic Nonlinear Schrodinger equation[END_REF], [START_REF] Debussche | Printems Weak order for the discretization of the stochastic heat equation[END_REF], [START_REF] Hausenblas | Weak Approximation of Stochastic Partial Differential Equations[END_REF]. In order to explain the novelty of the present article, let us focus on a specific example.

We consider a stochastic nonlinear heat equation in a bounded interval I = (a, b) ⊂ R with Dirichlet boundary conditions and driven by a space-time white noise:

(1.1)

             ∂X ∂t = X ξξ + f (X) + σ(X) η, ξ ∈ I, t > 0,
X(a, t) = X(b, t) = 0, t > 0, X(ξ, 0) = x(ξ), ξ ∈ I.

Where f and σ are smooth Lipschitz functions from R to R.

We introduce the classical abstract framework extensively used in the book [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]. We set H = L 2 (I), A = ∂ ξξ , D(A) = H 2 (I) ∩ H 1 0 (I), W is a cylindrical Wiener process so that the space-time white noise is mathematically represented as the time derivative of W . We set f (x)(ξ) = f (x(ξ)), x ∈ H and define σ : H→L(H) by σ(x)h(ξ) = σ(x(ξ))h(ξ), x, h ∈ H. We then rewrite (1.1) as (1.2) dX = (AX + f (X))dt + σ(X)dW, X(0) = x.

It is well known that this equation has a unique solution. We investigate the error committed when approximating this solution by the solution of the Euler scheme

(1.3) X k+1 -X k = ∆t (AX k+1 + f (X k )) + σ(X k ) (W ((k + 1)∆t) -W (k∆t)) , X 0 = x,
where ∆t = T /N , N ∈ N, T > 0.

The study of the weak error aims to prove bounds of the type:

|E(ϕ(X(n∆t))) -E(ϕ(X n )| ≤ c∆t δ ,
with a constant c which may depend on ϕ, x, N and on the various parameter in the equation. Also ϕ is assumed to be a smooth function on H. If such a bound is true, we say that the scheme has weak order δ. In comparison, the strong error is given by E(|(X(n∆t)) -X n |) or E(sup n=0,...,N |(X(n∆t)) -X n |). Clearly, if the scheme has strong order δ then it has weak order δ ≥ δ. Indeed, the test functions ϕ are Lipschitz. In general, it is expected that the weak order is larger than the the strong order.

In the case of the Euler scheme applied to a stochastic differential equation, it is well known that the strong order is 1/2 whereas the weak order is 1 (see [START_REF] Talay | Probabilistic numerical methods for partial differential equations: elements of analysis, Probabilistic models for nonlinear partial differential equations[END_REF]). The classical proof of this uses the Kolmogorov equation associated to the stochastic equation. The main difficulty to generalize this proof to the infinite dimensional equation (1.2) is that this Kolmogorov equation is then a partial differential equation with an infinite number of variables and involving unbounded operators (see (3.6) below). The delayed stochastic differential equation studied in [START_REF] Buckwar | Weak approximation of stochastic differential delay equations[END_REF] is an infinite dimensional problem but since the equation does not contain differential operators the Kolmogorov equation is simpler to study. In [START_REF] Hausenblas | Weak Approximation of Stochastic Partial Differential Equations[END_REF], a SPDE similar to (1.2) is considered but very particular test functions ϕ are used. They are allowed to depend only on finite dimensional projections of the unknown and the bound of the weak error involves a constant which strongly depends on the dimension. In [START_REF] De Bouard | Debussche Weak and strong order of convergence of a semi discrete scheme for the stochastic Nonlinear Schrodinger equation[END_REF], [START_REF] Debussche | Printems Weak order for the discretization of the stochastic heat equation[END_REF], the Kolmogorov equation is not used directly. A change of variable is used in order to simplify it. In [START_REF] De Bouard | Debussche Weak and strong order of convergence of a semi discrete scheme for the stochastic Nonlinear Schrodinger equation[END_REF], the stochastic nonlinear Schrödinger equation is considered and the fact that the linear Schrödinger equation generates an invertible group is used in an essential way. This is obviously wrong for the heat equation considered here. The same change of unknown works in the case of a linear equation with additive noise as shown in [START_REF] Debussche | Printems Weak order for the discretization of the stochastic heat equation[END_REF] but there it is used that the solution can be written down explicitly. We have not been able to generalize this idea to the non linear equation considered here.

We use in fact the original method developed by Talay in the finite dimensional case. The weak error is decomposed thanks to the Kolmogorov equations on each time step. Each term represents the error between the solution of the Kolmogorov equation on one time step and the approximation given by the numerical solution. Due to the presence of unbounded operators, this apparently requires a lot of smoothness on the numerical solution. The main idea here is to observe that the non smooth part of the solutions of (1.2) and (1.3) are contained in a stochastic integral. We get rid of this stochastic integral thanks to Malliavin calculus and an integration by part. We are thus able to prove that as expected the weak order is twice the strong order without artificial assumption except from a technical one on σ. We restrict our presentation to the abstract equation above, a nonlinear heat equation driven by a space-time white noise. However, our method is general and can be used for more general equations as will be shown in future articles. Also, we only consider a semi-discretization in time. A full discretization will treated in forthcoming works.

Note that the method developed here does allow to recover the result of [START_REF] Debussche | Printems Weak order for the discretization of the stochastic heat equation[END_REF]. Indeed, in the Euler scheme (1.3), the linear term is fully implicit and we cannot consider a scheme where it is partially implicit such as the theta-scheme considered in [START_REF] Debussche | Printems Weak order for the discretization of the stochastic heat equation[END_REF]. Note also that the proof below are much more complicated than in [START_REF] Debussche | Printems Weak order for the discretization of the stochastic heat equation[END_REF] and [START_REF] De Bouard | Debussche Weak and strong order of convergence of a semi discrete scheme for the stochastic Nonlinear Schrodinger equation[END_REF].

Malliavin calculus has already been used for the numerical analysis of stochastic equations. In [START_REF] Bally | The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function[END_REF], it is used to prove an expansion of the error of the Euler scheme for a stochastic differential equation under minimal assumptions on the test functions ϕ. This is a completely different idea and the Malliavin calculus is used completely differently. It is not clear that such ideas could be used for a SPDE. In a different spirit, Malliavin calculus is used in [START_REF] Szepessy | Adaptive weak approximation of stochastic differential equations[END_REF] to analyse adaptive schemes for the weak approximation of stochastic differential equations.

Our method is much closer to the method developped in [START_REF] Kohatsu-Higa | Weak approximations. A Malliavin calculus approach[END_REF]. There, the Malliavin calculus is also used to get rid of a stochastic integral which appears when writting down the weak error. However, it is done in a global way and the error is not decomposed as in the present article. A fundamental feature of Kohatsu-Higa's method is that the Kolmogorov equation is not used so that more general stochastic equation can be can considered. The solution does not need to be markovian. However, no SPDE have been considered with this method.

Preliminaries and main result

We consider the following stochastic partial differential equation written in an abstract form in a Hilbert space H with norm | • | and inner product (•, •):

(2.1) dX = (AX + f (X))dt + σ(X)dW, X(0) = x,
where the unknown X is a random process on a probability space (Ω, F, P) depending on t > 0 and on the initial data x ∈ H. The operator A is a negative self-adjoint operator on H with domain D(A) and has a compact inverse. We assume that

(2.2) Tr((-A) -α ) < ∞, for all α > 1/2.
We define classically the domain D((-A) β ), β ∈ R, of fractional powers of A and set

|x| β = |(-A) β x|, x ∈ D((-A) β ).
The nonlinear function f takes values in H and is assumed to be C 3 with bounded derivatives up to order 3. We denote by L f a constant such that for x, y ∈ H

(2.3) |f (x)| ≤ L f (|x| + 1), |f (x) -f (y)| ≤ L f |x -y|, |f ′ (x) -f ′ (y)| L(H) ≤ L f |x -y|.
The noise is written in terms of a cylindrical Wiener process W on H (see [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]) associated to a filtration (F t ) t≥0 . The nonlinear mapping acting on the noise maps H onto L(H), it is also assumed to be C 3 with bounded derivatives up to order 3. We denote by L σ a constant satisfying

(2.4) |σ(x)| L(H) ≤ L σ (|x| + 1), |σ(x) -σ(y)| L(H) ≤ L σ |x -y|.
We need a stronger assumption on this mapping, we require

(2.5) |σ ′′ (x) • (h, h)| L(H)) ≤ L σ |h| 2 -1/4 , x ∈ H, h ∈ H.
Note that this implies a strong restriction on σ. (See Remark 2.3 below for some comments on this assumptions).

Recall that the cylindrical Wiener process can be written as

W = ℓ∈N β ℓ e ℓ
where here and in the following (e ℓ ) ℓ∈N is any orthonormal basis of H and (β ℓ ) ℓ∈N is an associated sequence of independent brownian motions. This series does not converge in H but in any larger Hilbert space U such that the embedding H ⊂ U is Hilbert-Schmidt. Similarly, given a linear operator Φ from H to a possibly different Hilbert space K, the Wiener process ΦW = ℓ∈N β ℓ Φe ℓ is well defined in K provided Φ ∈ L 2 (H, K), the space of Hilbert-Schmidt operators from H to K. (See the definition just below).

Recall also that the stochastic integral T 0 Ψ(s)dW (s) is defined as an element of K provided that Ψ is an adapted process with values in L 2 (H, K) such that

T 0 |ψ(s)| 2
L 2 (H,K) ds < ∞ a.s. (see [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]).

If L ∈ L(H) is a nuclear operator, Tr(L) denotes the trace of the operator L, i.e.

Tr(L) = i≥1 (Le i , e i ) < +∞.
It is well known that the previous definition does not depend on the choice of the Hilbertian basis. Moreover, the following properties hold for L nuclear and M bounded Hilbert-Schmidt operators play also an important role. An operator L ∈ L(H) is Hilbert-Schmidt if L * L is a nuclear operator on H. We denote by L 2 (H) the space of such operators. It is a Hilbert space for the norm

L L 2 (H) = (Tr(L * L)) 1/2 = (Tr(LL * )) 1/2 . It is classical that if L ∈ L 2 (H), M ∈ L(H), N ∈ L(H) then N LM ∈ L 2 (H) and (2.8) N LM L 2 (H) ≤ N L(H) L L 2 (H) M L(H) .
See [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF], appendix C, or [START_REF]Introduction to the theory of linear nonselfadjoint operators in Hilbert space[END_REF] for more details on nuclear and Hilbert-Schmidt operators. Note that (2.2) implies that (-A) -β is Hilbert-Schmidt for any β > 1/4.

Our assumptions imply that for any x ∈ H, there exists a unique solution X(t) to equation to (2.1) (see for instance [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF], chapter 7). In the sequel, we often recall the dependence of the solution on the initial data by using the notation X(t, x).

We approximate equation (2.1) by an implicit Euler schemes. Let ∆t = T N > 0 be a time step, we define the sequence (X k ) k=0,...,N by (2.9)

X k+1 = S ∆t X k + ∆tS ∆t f (X k ) + √ ∆tS ∆t σ(X k )χ k+1 , X 0 = x.
We have set χ k+1 = (W ((k + 1)∆t) -W (k∆t))/ √ ∆t. The operators S ∆t is defined by

S ∆t = (I -∆tA) -1 .
This is the classical fully implicit Euler scheme. It will be convenient to use the integral form of (2.1)

(2.10)

X(t) = S(t)x + t 0 S(t -s)f (X(s))ds + t 0 S(t -s)σ(X(s))dW (s), t ≥ 0,
where S(t) = e tA is the semigroup generated by A. Similarly, (2.9) can be rewritten as (2.11)

X k = S k ∆t x + ∆t k-1 ℓ=0 S k-ℓ ∆t f (X ℓ ) + √ ∆t k-1 ℓ=0 S k-ℓ ∆t σ(X ℓ )χ ℓ+1 .
It will be convenient in the following to use the notation:

t k = k∆t, k = 0, . . . , N.
The following inequalities are classical and easily proved using the spectral decomposition of A (2.12)

(-A) β S k ∆t L(H) ≤ ct -β k , k ≥ 1, β ∈ [0, 1]. (2.13) (-A) β S(t) L(H) ≤ ct -β , t > 0, β ≥ 0. (2.14) (-A) β S ∆t L(H) ≤ c∆t -β , β ∈ [0, 1]. (2.15) (-A) -β (I -S ∆t ) L(H) ≤ c∆t β , β ∈ [0, 1].
Note that in (2.11) and (2.9), the noise term makes sense in H. Indeed, by (2.14), (2.2) and (2.8), we know that S ∆t is a Hilbert-Schmidt operator on H.

We are interested in the approximation of the law of the solution of (2.1). More precisely, we wish to prove an estimate on the error committed when approximating E(ϕ(X(T, x))) by E(ϕ(X N (x))). The function ϕ is a smooth function on H.

In all the article, we use the notation Dϕ(x) for the differential of a C 1 function on H at the point x. If ϕ : H → K, where K is another Hilbert space, Dϕ(x) ∈ L(H, K) the space of continuous linear operator from H to K. When K = R, we identify the differential with the gradient thanks to Riesz identification theorem. We use the same notation and have the identity for x, h ∈ H: Dϕ(x).h = (Dϕ(x), h).

Similarly, if ϕ ∈ C 2 (H, R), D 2 ϕ(x)
is a bilinear operator from H ×H to R and can be identified with a linear operator on H through the identity:

D 2 ϕ(x).(h, k) = (D 2 ϕ(x)h, k), x, h, k ∈ H.
Sometimes, we also use the notations ϕ ′ , ϕ ′′ instead of Dϕ or D 2 ϕ. Given two Banach spaces K 1 and K 2 , we denote by

• k the norm on C k b (K 1 , K 2 )
, the space of k times continuously differentiable mapping from K 1 to K 2 with derivatives bounded up to order k.

We use Malliavin calculus in the course of the proof. We now recall the basic definitions. (See [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]). Given a smooth real valued function F on H n and ψ 1 , . . . , ψ n ∈ L 2 (0, T, H), the Malliavin derivative of the smooth random variable F (

T 0 (ψ 1 (s), dW (s)), . . . , T 0 (ψ n (s), dW (s))) at time s in the direction h ∈ H is given by D h s F T 0 (ψ 1 (s), dW (s)), . . . , T 0 (ψ n (s), dW (s)) = n i=1 ∂ i F T 0 (ψ 1 (s), dW (s)), . . . , T 0 (ψ n (s), dW (s)) (ψ i (s), h) .
We also define the process DF by (DF (s), h) = D h s F . It can be shown that D defines a closable operator with values in L 2 (Ω, L 2 (0, T, H)) and we denote by D 1,2 the closure of the set of smooth random variables as above for the topology defined by the norm

F D 1,2 = E(|F | 2 ) + E( T 0 |D s F | 2 ds 1/2 .

We define similarly the Malliavin derivative of random variables taking values in

H. If G = i∈N F i e i ∈ L 2 (Ω, H) where F i ∈ D 1,2 for all i ∈ N and i∈N T 0 |D s F i | 2 ds < ∞, we set D h s G = i∈N D h s F i e i , D s G = i∈N D s F i e i .
We define D 1,2 (H) as the set of such random variables.

When h = e m , we write

D em = D m .
The chain rule is valid and given

u ∈ C 1 b (R), F ∈ D 1,2 then u(F ) ∈ D 1,2 and D(u(F )) = u ′ (F )DF . Also if G = i∈N F i e i ∈ D 1,2 (H) and u ∈ C 1 b (H, R) then u(G) ∈ D 1,2 and D(u(G)) = Du(G).DG = (Du, DG), or equivalently D h s (u(G)) = i∈N ∂ i uD h s F i = (Du, D h s G).
Note that as already mentionned, we identify the differential of a function in C 1 (H, R) with its gradient.

For F ∈ D 1,2 and ψ ∈ L 2 (Ω × [0, T ]; H) such that ψ(t) ∈ D 1,2 for all t ∈ [0, T ] and T 0 T 0 |D s ψ(t)| 2 dsdt < ∞,
we have the integration by part formula:

E F T 0 (ψ(s), dW (s)) = E T 0 (D s F, ψ(s))ds ,
where the stochastic integral is a Skohorod integral which is in fact defined by duality. In this article, we only need to consider the Skohorod integral of adapted processes in which case it corresponds with the Itô integral. Moreover, the integration by part formula above holds for

F ∈ D 1,2 and ψ ∈ L 2 (Ω × [0, T ]; H) when ψ is an adapted process. Recall that if F is F t measurable then D s F = 0 for s ≥ t.
We will often use the following form of the integration by part formula whose proof is left to the reader.

Lemma 2.1. Let F ∈ D 1,2 (H), u ∈ C 2 b (H) and ψ ∈ L 2 (Ω × [0, T ], L 2 (H)) be an adapted process then E Du(F ) • T 0 ψ(s)dW (s) = E m∈N T 0 D 2 u(F ) • (D m s F, ψ(s)e m )ds = E T 0 Tr ψ * (s)D 2 u(F )D s F ds .
Also we remark that this Lemma remains valid if u is not assumed to be bounded but only u ∈ C 2 (H) provided the expectations and the integral above are well defined. This is easily seen by approximation of u by bounded functions.

We now state our main result.

Theorem 2.2. Assume that f and σ are C 3 b functions from H to H and L(H) and that σ satisfies (2.4), then for any x ∈ H, T > 0, ε > 0, the Euler Scheme (2.9) satisfies the following weak error estimate

|E(ϕ(X(T, x))) -E(ϕ(X N ))| ≤ C(T, |ϕ| C 3 b , |x|, ε)∆t 1/2-ε , ϕ ∈ C 3 b (H).
Remark 2.3. Assumption (2.4) is quite restrictive. It is void for an additive noise or a noise of the form BX dW where B is a linear operator from H to L(H). Otherwise, it implies that the noise is a perturbation of such noise. An example of a noise satisfying this is

σ(x) = Bx + σ((-A) -1/4 x)
where B ∈ L(H) and σ : H→L(H) is a C 3 function with derivatives bounded up to order 3.

This assumption is crucial in our proof. It is used in essential way in Lemma 4.5 which is used at many points of the proof. Apart from this point, our result is optimal. If the noise is assumed to satisfied some non degeneracy assumptions, the smothness assumption on the test function ϕ can be weakened. This will be investigated in a future work.

In all the article, C or c denote constants which may depend on A, f, σ, Q or T but not on ∆t. Their value may change from one line to another. The initial data x is fixed and the constant may also depend on |x|. Note also that we assume that ∆t ≤ 1, we could also assume ∆t ≤ ∆t 0 for some ∆t 0 > 0. In this case, the different constants would depend on ∆t 0 . Finally, ε is a small positive number.

Proof of the main result

The proof uses different tools from stochastic calculus such as Itô formula, Kolmogorov equations, Malliavin calculus. Sometimes, it may be very lengthy and technical to justify rigorously their use in infinite dimension. We avoid these tedious justifications by using Gakerkin approximations. We replace equation (2.1) by the finite dimensional stochastic equation

dX m = (AX m + f m (X m ))dt + σ m (X m )dW, X m (0) = P m
where P m is the eigenprojector on the m first eignevectors of A, f m (x) = P m f (x), σ m (x) = P m σ(x)P m . It is not difficult to prove that X m converges to X in various senses.

Similarly, we replace the discrete unknown X k by a finite dimensional sequence defined in an obvious way.

We prove the result for these finite dimensional objects with constants that do not depend on the dimension m. It is then easy to deduce the result for our infinite dimensional equation.

In order to lighten the notation, we omit to explicit the dependence on m below and write

X, f , σ instead of X m , f m , σ m .
Step 1: We first define a continuous interpolation of the discrete unknown. We rewrite (2.9) as follows:

X k+1 = X k + t k+1 t k A ∆t X k + S ∆t f (X k )ds + t k+1 t k S ∆t σ(X k )dW (s)
where A ∆t = S ∆t A. Note that A ∆t is in fact a Yosida regularization of A and is a bounded operator:

(3.1) |A ∆t | L(H) ≤ c∆t -1 .
It is then natural to define X on [0, T ] by

(3.2) X(t) = X k + t t k A ∆t X k + S ∆t f (X k )ds + t t k S ∆t σ(X k )dW (s), t ∈ [t k , t k+1 ).
Clearly, X is a continuous and adapted process. Given a smooth function G on [0, T ] × H, Itô formula implies for t ∈ [t k , t k+1 ) (see [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]):

(3.3) G(t, X(t)) = G(t k , X(t k )) + t t k dG dt (s, X(s)) + L k,∆t G(s, X(s))ds + t t k (DG(s, X(s)), σ(X k )dW (s)).
Where for ψ ∈ C 2 (H, R)

L k,∆t ψ(x) = 1 2 Tr (S ∆t σ(X k )) (S ∆t σ(X k )) * D 2 ψ(x) + (A ∆t X k + S ∆t f (X k ), Dψ(x)).
Step 2: Decomposition of the error.

Let us define

(3.4) u(t, x) = E(ϕ(X(t, x))), t ∈ [0, T ].
Then the weak error at time T is equal to

(3.5) u(T, x) -E(ϕ(X N )) = E(u(T, x) -u(0, X N ) = N -1 k=0 E (u(T -t k , X k ) -u(T -t k+1 , X k+1 )) .
It is well known that u is a solution to the forward Kolmogorov equation:

(3.6) du dt (t, x) = Lu(t, x) = 1 2 Tr{σ(x)σ * (x)D 2 u(t, x)} + (Ax + f (x), Du(t, x)).
Therefore, Itô formula (3.3) implies

E(u(T -t k+1 , X k+1 )) = E(u(T -t k , X k )) + E t k+1 t k L k,∆t u(T -t, X(t)) -Lu(T -t, X(t))dt.
The first term in (3.5) will be treated separately and we decompose the error as follows

(3.7) u(T, x) -E(ϕ(X N )) = u(T, x) -E(u(T -∆t, X 1 )) + N -1 k=1 a k + b k + c k .
Where

a k = E t k+1 t k A X(t) -A ∆t X k , Du(T -t, X(t)) dt, b k = E t k+1 t k f ( X(t)) -S ∆t f (X k ), Du(T -t, X(t)) dt, c k = 1 2 E t k+1 t k Tr σ( X(t))σ * ( X(t)) -(S ∆t σ(X k )) (S ∆t σ(X k )) * D 2 u(T -t, X(t)) dt.
In the next steps, we estimate separately the different terms in (3.7).

Step 3: Estimate of u(T, x) -E(u(T -∆t, X 1 )). By the Markov property

u(T, x) = E(ϕ(X(T, x))) = E(u(T -∆t, X(∆t))).
Therefore, by Lemma 4.4, for any ε > 0,

|u(T, x) -E(u(T -∆t, X 1 ))| ≤ c(T -∆t) -1/2+ε ϕ 1 E |X(∆t) -X 1 | -1/2+ε . Moreover X(∆t) -X 1 = (S(∆t) -S ∆t )x + ∆t 0 S(t -s)f (X(s, x))ds -∆tS ∆t f (x) + ∆t 0 S(t -s)σ(X(s, x))dW (s) - √ ∆tS ∆t σ(x)χ 1 .
It is easy to prove that

|(-A) -1/2+ε (S(∆t) -S ∆t )| L(H) ≤ c∆t 1/2-ε .
Since (S(t)) t≥0 is a contraction semigroup and |(-A) -1/2+ε

S(t -s)f (X(s, x))ds -1/2+ε ≤ ∆tL f E( sup s∈[0,∆t] |X(s, x)| + 1) ≤ c∆t(|x| + 1).
Similarly

|∆tS ∆t f (x)| -1/2+ε ≤ c∆t(|x| + 1
). We then have

E | ∆t 0 S(t -s)σ(X(s, x))dW (s)| 2 -1/2+ε = E ∆t 0 |(-A) -1/2+ε S(t -s)σ(X(s, x))| 2 L 2 (H) ds ≤ E ∆t 0 |(-A) -1/2+ε | 2 L 2 (H) |S(t -s)| 2 L(H) |σ(X(s, x))| 2 L(H) ds
and by (2.2), (2.4), Lemma 4.2

E | ∆t 0 S(t -s)σ(X(s, x))dW (s)| 2 -1/2+ε ≤ c∆t(|x| + 1). Similarly E | √ ∆tS ∆t σ(x)χ 1 | 2 ≤ c∆t(|x| + 1).
Gathering these estimate and using Cauchy-Schwartz inequality , we obtain

(3.8) |u(T, x) -E(u(T -∆t, X 1 ))| ≤ c(T -∆t) -1/2+ε ∆t -1/2+ε ≤ c∆t 1/2-ε
where, as mentionned above, the constant is allowed to depend on T , x, ϕ, f , σ . . .

Step 4: Estimate of a k , k ≥ 1.

We split a k as follows:

a k = a 1 k + a 2 k with a 1 k = E t k+1 t k (A -A ∆t )X k , Du(T -t, X(t)) dt, a 2 k = E t k+1 t k A( X(t) -X k ), Du(T -t, X(t)) dt.
Note that A ∆t -A = θ∆tS ∆t A 2 . By Lemma 4.4 below, we know that Du(T -t, X(t)) is in D((-A) γ ) for γ < 1/2 and it is easy to see that X k belongs to D((-A) δ ) for δ < 1/4. It is impossible to compensate the presence of A 2 by such arguments. The idea is to recall (2.11) and to observe that the irregularity of X k is contained in the stochastic integral. We thus further decompose a 1 k in three terms according to (2.11). The first two terms are easy to treat. The third one involves the stochastic integral and is estimated thanks to Malliavin calculus. We set

a 1,1 k = -θ∆tE t k+1 t k S ∆t A 2 S k ∆t x, Du(T -t, X(t)) dt, a 1,2 k = -θ∆tE t k+1 t k S ∆t A 2 ∆t k-1 ℓ=0 S k-ℓ ∆t f (X ℓ ), Du(T -t, X(t)) dt, a 1,3 k = -θ∆tE t k+1 t k S ∆t A 2 √ ∆t k-1 ℓ=0 S k-ℓ ∆t σ(X ℓ )χ ℓ+1 , Du(T -t, X(t)) dt, so that a 1 k = a 1,1 k + a 1,2 k + a 1,3
k . By (2.14), (2.12) and Lemma 4.4, we have for k = 1, . . . , N -2 and ε > 0 (3.9)

|a 1,1 k | ≤ c∆tE t k+1 t k |S ∆t (-A) 1/2+2ε | L(H) |(-A) 1-ε S k ∆t | L(H) |(-A) 1/2-ε Du(T -t, X(t))| |x|dt ≤ c∆t 1/2-2ε t -1+ε k t k+1 t k (T -t) -(1/2-ε) dt.
The estimate of a 1,2 k is similar. We have by (2.3), (2.12)

∆t(-A) 1-ε k-1 ℓ=0 S k-ℓ ∆t f (X ℓ ) ≤ L f ∆t k-1 ℓ=0 (-A) 1-ε S k-ℓ ∆t L(H) (|X ℓ | + 1) ≤ c∆t k-1 ℓ=0 t -1+ε k-ℓ (|X ℓ | + 1) . Since ∆t k-1 ℓ=0 t -1+ε k-ℓ ≤ ε -1 T ε ,
we deduce thanks to Lemma 4.4 and Lemma 4.1

(3.10) |a 1,2 k | ≤ c∆t t k+1 t k S ∆t (-A) 1/2+2ε L(H) (T -t) -1/2+ε dt ≤ c∆t 1/2-2ε t k+1 t k (T -t) -(1/2-ε) dt.
To treat a 1,3 k , we first rewrite it in terms of a stochastic integral and then use Lemma 2.1

a 1,3 k = θ∆tE t k+1 t k t k 0 S ∆t A 2 S k-ℓs ∆t σ(X ℓs )dW (s), Du(T -t, X(t)) dt = θ∆tE t k+1 t k t k 0 Tr σ * (X ℓs )S ∆t A 2 S k-ℓs ∆t D 2 u(T -t, X(t))D s X(t) ds dt
where ℓ s = [s/∆t] is the integer part of s/∆t. By the chain rule and (3.2), we have for

s ∈ [0, t k ], h ∈ H, t ∈ [t k , t k+1 ), D h s X(t) = D h s X k + t t k A ∆t D h s X k + S ∆t f ′ (X k ) • D h s X k ds + t t k S ∆t σ ′ (X k ) • D h s X k dW (s).
Fro β < 1/4, we have, by (2.14), (2.2), (2.4), (2.8)

E t t k S ∆t σ ′ (X k ) • D h s X k dW (s) 2 β = E t t k (-A) β S ∆t σ ′ (X k ) • D h s X k 2 L 2 (H) ds ≤ E t t k (-A) -1/4-ε 2 L 2 (H) (-A) β+1/4+ε S ∆t 2 L(H) σ ′ (X k ) • D h s X k 2 L(H) ds ≤ c∆t 1/2-2β-2ε E |D h s X k | 2 .
We then use (3.1), (2.3) to bound the other terms above and obtain thanks to Poincaré inequality

(3.11) E |D h s X(t)| 2 β ≤ cE |D h s X k | 2 β , s ∈ [0, t k ], t ∈ [t k , t k+1 ).
By Lemma 4.3, we obtain for β < 1/4

E (-A) β D s X(t) 2 L(H) ≤ ct -2β k-ℓs .
We are now ready to conclude the estimate of a 1,3 k . We choose ε > 0 and write thanks to (2.4), (2.14), (2.12), Lemma 4.5 and (2.2)

|a 1,3 k | ≤ θ∆tE t k+1 t k t k 0 |σ * (X ℓs )| L(H) S ∆t A 1/2+2ε L(H) (-A) 1-3ε/2 S k-ℓs ∆t L(H) × (-A) 1/2-ε/2 D 2 u(T -t, X(t))(-A) 1/2-ε/2 L(H) Tr (-A) -1/2-ε/2 (-A) ε D s X(t) L(H) ds dt ≤ c∆tE t k+1 t k t k 0 ∆t -1/2-2ε t -1+3ε/2 k-ℓs (T -t) -1+ε t -ε k-ℓs ds dt. Since t k 0 t -1+ε/2 k-ℓs ds ≤ 2 ε T ε/2 , we deduce (3.12) |a 1,3 k | ≤ c∆t 1/2-2ε t k+1 t k (T -t) -1+ε dt.
Gathering (3.9), (3.10) and (3.12), we obtain for k = 1, . . . , N -1

(3.13) |a 1 k | ≤ c∆t 1/2-2ε t -1+ε k + 1 t k+1 t k (T -t) -1+ε dt + 1 .
We now estimate a 2 k . Let us set

a 2,1 k = E t k+1 t k (t -t k ) AA ∆t X k , Du(T -t, X(t)) dt, a 2,2 k = E t k+1 t k (t -t k ) AS ∆t f (X k ), Du(T -t, X(t)) dt, a 2,3 k = E t k+1 t k t t k AS ∆t σ(X k )dW (s), Du(T -t, X(t)) dt, so that thanks to (3.2), we have a 2 k = a 2,1 k + a 2,2 k + a 2,3 k . The first term a 2,1 k is similar to a 1 k
above and is majorized in the same way

(3.14) |a 2,1 k | ≤ c∆t 1/2-2ε t -1+ε k + 1 t k+1 t k (T -t) -1+ε dt + 1 .
for k = 1, . . . , N -1. The second one is not difficult to treat, we have using similar arguments as above

(3.15) |a 2,2 k | ≤ c∆t|(-A) 1/2+ε S ∆t | L(H) E(|f (X k )|) t k+1 t k (T -t) -(1/2-ε) dt ≤ c∆t 1/2-ε t k+1 t k (T -t) -(1/2-ε) dt for k = 1, . . . , N -1.
The estimate of a 2,3 k requires the use of Lemma 2.1. It implies

a 2,3 k = E t k+1 t k t t k Tr σ * (X k )S ∆t AD 2 u(T -t, X(t))D s X(t) ds dt.
Since, X k is F t k measurable, we have from (3.2)

(3.16) D s X(t) = S ∆t σ(X k ) s ∈ (t k , t k+1 ], t k ≤ s ≤ t < t k+1 .
It follows, thanks to (2.4), (2.14), (2.2) and Lemma 4.5,

a 2,3 k = E t k+1 t k (t -t k )Tr σ * (X k )S ∆t AD 2 u(T -t, X(t))S ∆t σ(X k ) dt ≤ c∆tE t k+1 t k |σ(X k )| L(H) |S ∆t (-A) 1/2+ε/2 | L(H) |(-A) 1/2-ε/2 D 2 u(T -t, X(t))(-A) 1/2-ε/2 | L(H) ×Tr((-A) -1/2-ε/2 )|(-A) ε S ∆t | L(H) |σ(X k )| L(H) dt ≤ c∆t 1/2-3ε/2 t k+1 t k (T -t) -1+ε dt for k = 1, . . . , N -1 .
Finally, we obtain

(3.17) |a 2 k | ≤ c∆t 1/2-2ε (t -1+ε k + 1) t k+1 t k (T -t) -1+ε dt + 1
for k = 1, . . . , N -1. Together with (3.13) this yields the estimate of a k

|a k | ≤ c∆t 1/2-2ε (t -1+ε k + 1) t k+1 t k (T -t) -1+ε dt + 1 .

It follows easily (3.18)

N -1

k=1 |a k | ≤ c∆t 1/2-2ε .
Step 5: Estimate of b k . This term seems easier to treat since we do not have the unbounded operator A. However, since it involves the nonlinear term, we need to use Ito formula (3.3) to control f ( X(t))-f (X k ), this introduces many terms. For some of them we again use Malliavin integration by parts.

First, we get rid of S ∆t . We have thanks to (2.15), (2.3), Lemma 4.4 and Lemma 4.1:

b 1 k = E t k+1 t k (I -S ∆t ) f (X k ), Du(T -t, X(t)) dt ≤ cE t k+1 t k (1 + |X k |)|(-A) -1/2+ε (I -S ∆t )| L(H) |(-A) 1/2-ε Du(T -t, X(t))|dt ≤ c∆t 1/2-ε E t k+1 t k (T -t) -1/2+ε dt for k = 0, . . . , N -1. We now estimate b 2 k = b k -b 1 k = E t k+1 t k f ( X(t) -f (X k ), Du(T -t, X(t)) dt = E t k+1 t k i∈N (f i ( X(t) -f i (X k ))∂ i u(T -t, X(t))dt,
where f i = (f, e i ) and ∂ i = (D•, e i ). We choose (e i ) i∈N as the orthonormal basis of eigenvectors of A. By (3.3), we have for i ∈ N

f i ( X(t) = f i (X k ) + t t k 1 2 Tr (S ∆t σ(X k ))(S ∆t σ(X k )) * D 2 f i ( X(s)) ds + t t k A ∆t X k + S ∆t f (X k ), Df i ( X(s)) ds + t t k Df i ( X(s)), σ(X k ) dW (s).
With obvious notations, this defines the decomposition

b 2 k = b 2,1 k + b 2,2 k + b 2,3 k + b 2,4 k .
To treat the first term, we rewrite it as follows 1 :

b 2,1 k = 1 2 E t k+1 t k t t k i∈N Tr (S ∆t σ(X k ))(S ∆t σ(X k )) * D 2 f i ( X(s)) ∂ i u(T -t, X(t))ds dt = 1 2 E t k+1 t k t t k Tr {(S ∆t σ(X k ))(S ∆t σ(X k )) * A(s, t)} ds dt where A(s, t) ∈ L(H) is defined by (A(s, t)h, k) = i∈N D 2 f i ( X(s)).(h, k)∂ i u(T -t, X(t)) = D 2 f ( X(s)).(h, k), Du(T -t, X(t)) , h, k ∈ H. Obviously |A(s, t)| L(H) ≤ D 2 f ( X(s)) L 2 (H×H,H) Du(T -t, X(t)) ,
where L 2 (H × H, H) denotes the space of bilinear operators from H × H to H. By (2.3) and Lemma 4.4, we deduce:

|A(s, t)| L(H) ≤ c.
Then, we write thanks to (2.4), (2.14), (2.2),

|Tr {(S ∆t σ(X k ))(S ∆t σ(X k )) * A(s, t)}| ≤ Tr (-A) -1/2-ε (-A) 1/2+ε S ∆t L(H) |σ(X k )| 2 L(H) |A(s, t)| L(H) ≤ c∆t -1/2-ε (1 + |X k |) 2 .
We deduce by Lemma 4.1

(3.19) b 2,1 k ≤ c∆t 3/2-ε .
1 Recall that we in fact work with Galerkin approximations so that all sums below are finite sums.

The second term b 2,2 k involves the same difficulty as a 1 k above. We rewrite it using (2.11). This gives

b 2,2 k = E t k+1 t k t t k i∈N A ∆t S k ∆t x + A ∆t ∆t k-1 ℓ=0 S k-ℓ ∆t f (X ℓ ), Df i ( X(s)) ∂ i u(T -t, X(t))dsdt +E t k+1 t k t t k i∈N A ∆t t k 0 S k-ℓτ ∆t σ(X ℓτ )dW (τ ), Df i ( X(s)) ∂ i u(T -t,
A ∆t S k ∆t x + A ∆t ∆t k-1 ℓ=0 S k-ℓ ∆t f (X ℓ ), Df i ( X(s)) ∂ i u(T -t, X(t))dsdt = E t k+1 t k t t k Df ( X(s)) • A ∆t S k ∆t x + A ∆t ∆t k-1 ℓ=0 S k-ℓ ∆t f (X ℓ ) , Du(T -t, X(t)) dsdt ≤ cE t k+1 t k t t k |(-A) ε S ∆t | L(H) (-A) 1-ε S k ∆t x + k-1 ℓ=0 (-A) 1-ε S k-ℓ ∆t L(H) |f (X ℓ )| ds dt ≤ c∆t 2-ε (t -1+ε k + 1).
The second term of b 2,2 k requires an integration by parts, we obtain

E t k+1 t k t t k i∈N A ∆t t k 0 S k-ℓτ ∆t σ(X ℓτ )dW (τ ), Df i ( X(s)) ∂ i u(T -t, X(t))dsdt = E t k+1 t k t t k i,j,m∈N A ∆t t k 0 S k-ℓτ ∆t σ(X ℓτ )e m , e j dβ m (τ )∂ j f i ( X(s))∂ i u(T -t, X(t))dsdt = E t k+1 t k t t k t k 0 i,j,m,n∈N A ∆t S k-ℓτ ∆t σ(X ℓτ )e m , e j ∂ j,n f i ( X(s)) D m τ X(s), e n ∂ i u(T -t, X(t)) +∂ j f i ( X(s))∂ i,n u(T -t, X(t)) D m τ X(t), e n dτ dsdt = E t k+1 t k t t k t k 0 i,m∈N D 2 f i ( X(s)) A ∆t S k-ℓτ ∆t σ(X ℓτ )e m , D m τ X(s) ∂ i u(T -t, X(t)) + B i (s, t)A ∆t S k-ℓτ ∆t σ(X ℓτ )e m , D m τ X(t) dτ dsdt = E t k+1 t k t t k t k 0 i∈N Tr D τ X(s) * D 2 f i ( X(s))A ∆t S k-ℓτ ∆t σ(X ℓτ ) ∂ i u(T -t, X(t)) +Tr D τ X(t) * B i (s, t)A ∆t S k-ℓτ ∆t σ(X ℓτ ) dτ dsdt
where, for i ∈ N, B i (s, t) is defined by

(B i (s, t)g, h) = (Df i ( X(s)), g) n∈N ∂ i,n u(T -t, X(t))(h, e n ), g, h ∈ H.
The first term above is estimate as b 2,1 k . For the second term, we write

i∈N (B i (s, t)g, h) = D 2 u(T -t, X(t)) • (Df ( X(s)) • g, h) = (D 2 u(T -t, X(t))h, Df ( X(s)) • g), g, h ∈ H. Therefore i∈N B i (s, t) L(H) ≤ Df ( X(s)) L(H) D 2 u(T -t, X(t)) L(H)
.

We deduce by Lemma 4.

A ∆t t k 0 S k-ℓτ ∆t σ(X ℓτ )dW (τ ), Df i ( X(s)) ∂ i u(T -t, X(t))dsdt ≤ c∆t 3/2-ε . Therefore b 2,2 k ≤ c∆t 3/2-ε . It is also easy to see that b 2,3 k = E t k+1 t k t t k i∈N S ∆t f (X k ), Df i ( X(s)) ∂ i u(T -t, X(t))dsdt = E t k+1 t k t t k Du(T -t, X(t)) • Df ( X(s)) • S ∆t f (X k ) dsdt ≤ c∆t 2 .
It remains to estimate b 2,4 k . We again integrate by parts the stochastic integral and obtain by Lemma 2.1:

b 2,4 k = E t k+1 t k t t k i∈N Df i ( X(s)), σ(X k )dW (s) ∂ i u(T -t, X(t))dt = E t k+1 t k t t k Tr D s X(t) * D 2 u(T -t, X(t))Df ( X(s))σ(X k ) ds dt = E t k+1 t k t t k Tr σ * (X k )S ∆t D 2 u(T -t, X(t))Df ( X(s))σ(X k ) ds dt ≤ c∆t 3/2-ε , thanks to (3.16), (2.

2) and (2.14).

We conclude this step by gathering the previous estimates. This enables us to write

N -1 k=1 |b k | ≤ c∆t 1/2-ε
Step 6: Estimate of c k . Using the symmetry of Du, we introduce the decomposition of c k :

c k = 1 2 E t k+1 t k Tr σ( X(t))σ * ( X(t)) -(S ∆t σ(X k )) (S ∆t σ(X k )) * D 2 u(T -t, X(t)) dt = 1 2 E t k+1 t k Tr (I -S ∆t )σ( X(t)) (I -S ∆t )σ( X(t)) * D 2 u(T -t, X(t)) dt +E t k+1 t k Tr S ∆t σ( X(t)) (I -S ∆t )σ( X(t)) * D 2 u(T -t, X(t)) dt + 1 2 E t k+1 t k Tr S ∆t σ( X(t)) -σ(X k ) S ∆t σ( X(t)) * D 2 u(T -t, X(t)) dt + 1 2 E t k+1 t k Tr S ∆t σ(X k ) S ∆t σ( X(t)) -σ(X k ) * D 2 u(T -t, X(t)) dt = c 1 k + c 2 k + c 3 k + c 4 k .
The first two terms are easy to treat, we use similar arguments as in the previous steps and write thanks to (2.7), Lemma 4.5, Lemma 4.1, (2.15)

c 1 k ≤ cE t k+1 t k Tr (-A) -1/2+ε (I -S ∆t )σ( X(t)σ * ( X(t))(I -S ∆t )(-A) -1/2+ε (T -t) -1+2ε dt ≤ cE t k+1 t k Tr (-A) -1/2+ε (I -S ∆t )(I -S ∆t )(-A) -1/2+ε (T -t) -1+2ε dt ≤ c∆t 1/2-3ε t k+1 t k (T -t) -1+ε dt
The second term is similar, we have

c 2 k ≤ E t k+1 t k (-A) -1/2+ε (I -S ∆t ) L(H) σ( X(t) 2 L(H) (-A) 2ε S ∆t L(H) Tr (-A) -1/2-ε (-A) 1/2-ε D 2 u(T -t, X(t))(-A) 1/2-ε L(H) dt ≤ c∆t 1/2-3ε t k+1 t k (T -t) -1+ε dt
The estimate of the next term is much more complicated. It is based on similar arguments as before but the computations are much longer. We use (3.3) and obtain for h, k ∈ H:

σ( X(t)) -σ(X k ) h, k = 1 2 t t k Tr (S ∆t σ(X k )) (S ∆t σ(X k )) * D 2 (σ(•)h, k) ( X(s)) dt + 1 2 t t k A ∆t X k + S ∆t f (X k ), D (σ(•)h, k) ( X(s)) dt = (Ah, k) + (Bh, k) + (Ch, k).
Thus we may write

c 3 k = 1 2 E t k+1 t k Tr S ∆t A S ∆t σ( X(t)) * D 2 u(T -t, X(t)) dt + 1 2 E t k+1 t k Tr S ∆t B S ∆t σ( X(t)) * D 2 u(T -t, X(t)) dt + 1 2 E t k+1 t k Tr S ∆t C S ∆t σ( X(t)) * D 2 u(T -t, X(t)) dt = c 3,1 k + c 3,2 k + c 3,3 k . Note that (Ah, k) = 1 2 t t k ℓ∈N
σ ′′ ( X(s)).(S ∆t σ(X k )e ℓ , S ∆t σ(X k )e ℓ ) h, k ds.

By (2.5), for u, v ∈ H, σ ′′ ( X(s)).(u, v) h, k ≤ L σ |u| -1/4 |v| -1/4 |h| |k| ≤ c|u| |v| |h| |k|.
We deduce, thanks to (2.2), (2.14),

(Ah, k) ≤ c∆t 1/2-ε (1 + |X k |) 2 |h| |k|, and
|A| L(H) ≤ c∆t 1/2-ε (1 + |X k |) 2 .
Then, by Lemma 4.1, Lemma 4.5, (2.14) and again (2.2)

c 3,1 k ≤ c∆t 1/2-3ε t k+1 t k (T -t) -1/2+ε dt.
The term c 3,2 k involves the same difficulty as a k and b 2,2 k . We use (2.11) to replace X k by a sum of three terms:

(Bh, k) = 1 2 t t k A ∆t S k ∆t x + ∆tA ∆t k-1 ℓ=0 S k-ℓ ∆t f (X ℓ ) + t k 0 A ∆t S k-ℓτ ∆t σ(X ℓτ )dW (τ ), D (σ(•)h, k) ( X(s)) ds = 1 2 t t k σ ′ ( X(s)) • A ∆t S k ∆t x + ∆tA ∆t k-1 ℓ=0 S k-ℓ ∆t f (X ℓ ) + t k 0 A ∆t S k-ℓτ ∆t σ(X ℓτ )dW (τ ) h, k ds = (B 1 h, k) + (B 2 h, k) + (B 3 h, k).
We then write thanks to (2.4), (2.12) and (2.14)

(B 1 h, k) = 1 2 t t k σ ′ ( X(s)) • A ∆t S k ∆t x h, k ds ≤ c t t k σ ′ ( X(s)) • A ∆t S k ∆t x L(H) |h| |k|ds ≤ c∆t|A ∆t S k ∆t x| |h| |k| ≤ c∆t 1-ε t 1-ε k |h| |k|. Similarly (B 2 h, k) ≤ c∆t 1-ε |h| |k|.
It follows, thanks to Lemma 4.5, (2.14) and (2.2)

1 2 E t k+1 t k Tr S ∆t (B 1 + B 2 ) S ∆t σ( X(t)) * D 2 u(T -t, X(t)) dt ≤ c∆t 1-3ε (t 1-ε k + 1) t k+1 t k (T -t) -1/2+ε dt.
The estimate of the part of c 3,2 k involving B 3 is very technical. As before, we get rid of the stochastic integral thanks to an integration by parts. This results in a supplementary trace term. In order to work with the double trace, we write everything in terms of the components of the operators and vectors. Given an operator G on H, we set G i,j = (Ge i , e j ). We thus write

E t k+1 t k Tr S ∆t B 3 S ∆t σ( X(t)) * D 2 u(T -t, X(t)) dt = E t k+1 t k i,j,m∈N B i,j 3 σ m,j ( X(t))) S ∆t D 2 u(T -t, X(t))S ∆t m,i dt = i,j,m,n,r∈N E t k+1 t k t t k t k 0 ∂ r σ i,j ( X(s)) A ∆t S k-ℓτ ∆t σ(X ℓτ )e n , e r dβ n (τ )σ m,j ( X(t))) S ∆t D 2 u(T -t, X(t))S ∆t m,i ds dt.
It is important to recall here that in fact we work with finite dimensional approximations of the solutions so that all the above sums are finite. We now use the Malliavin integration by parts and obtain

E t k+1 t k Tr S ∆t B 3 S ∆t σ( X(t)) * D 2 u(T -t, X(t)) dt = i,j,m,n,r∈N E t k+1 t k t t k t k 0 p∈N ∂ r,p σ i,j ( X(s)) D n τ X(s), e p A ∆t S k-ℓτ ∆t σ(X ℓτ )e n , e r σ m,j ( X(t))) S ∆t D 2 u(T -t, X(t))S ∆t m,i + p∈N
∂ r σ i,j ( X(s)) A ∆t S k-ℓτ ∆t σ(X ℓτ )e n , e r ∂ p σ m,j ( X(t))) D n τ X(t), e p S ∆t D 2 u(T -t, X(t))S ∆t m,i +∂ r σ i,j ( X(s)) A ∆t S k-ℓτ ∆t σ(X ℓτ )e n , e r σ m,j ( X(t)))

S ∆t D 3 u(T -t, X(t)) • D n τ X(s) S ∆t m,i
dτ ds dt

= I + II + III.
We then write

I = i,j,m,n∈N E t k+1 t k t t k t k 0 D 2 σ i,j ( X(s)) • D n τ X(s), A ∆t S k-ℓτ ∆t σ(X ℓτ )e n σ m,j ( X(t)) S ∆t D 2 u(T -t, X(t))S ∆t m,i dτ ds dt = j∈N E t k+1 t k t t k t k 0 D 2 u(T -t, X(t))
• φ 1 (s, τ, k)e j , S ∆t σ( X(t))e j dτ ds dt

= E t k+1 t k t t k t k 0 Tr σ * ( X(t))S ∆t D 2 u(T -t, X(t))φ 1 (s, τ, k) dτ ds dt
where we have set

φ 1 (s, τ, k)h 1 = n∈N S ∆t D 2 σ( X(s)) • D n τ X(s), A ∆t S k-ℓτ ∆t σ(X ℓτ )e n h 1 , h 1 ∈ H. Let us define Σ s,h 1 ,h 2 by (Σ s,h 1 ,h 2 u, v) = S ∆t D 2 σ( X(s)) • (u, v) h 1 , h 2 , u, v ∈ H.
Then by (2.5)

|Σ s,h 1 ,h 2 | L(H) ≤ c |h 1 | |h 2 |.
We deduce by (2.4), (2.2), (2.14), (2.12), (3.11) and Lemma 4.3

(φ 1 (s, τ, k)h 1 , h 2 ) = Tr σ * (X ℓτ )S k-ℓτ ∆t A ∆t Σ s,h 1 ,h 2 D τ X(s) ≤ |σ * (X ℓτ )| L(H) Tr(-A) -1/2-ε |(-A) 1/2+ε S k-ℓτ ∆t A ∆t | L(H) |Σ s,h 1 ,h 2 | L(H) |D τ X(s)| L(H) ≤ c∆t -1/2-2ε t -1+ε k |h 1 | |h 2 |(1 + |X k |)
and by Lemma 4.1, Lemma 4.5 and (2.2)

I ≤ c∆t 1/2-2ε t -1+ε k t k+1 t k (T -t) -1/2-ε dt.
Similarly, we may write

II = n∈N E t k+1 t k t t k t k 0 Tr Dσ( X(s)) • A ∆t S k-ℓτ ∆t σ(X ℓτ )e n Dσ( X(t)) • D n τ X(t) * S ∆t D 2 u(T -t, X(t))S ∆t dτ ds dt ≤ c∆t -2ε E t k+1 t k t t k t k 0 |φ 2 (τ, s, t, k)| L(H) (T -t) -1/2+ε dτ ds dt with φ 2 (τ, s, t, k) = n∈N Dσ( X(s)) • A ∆t S k-ℓτ ∆t σ(X ℓτ )e n Dσ( X(t)) • D n τ X(t) * .
We use similar arguments to estimate its norm. For u, v ∈ H, we write

(φ 2 (τ, s, t, k)u, v) = n∈N Dσ( X(t)) • D n τ X(t) * u, Dσ( X(s)) • A ∆t S k-ℓτ ∆t σ(X ℓτ )e n * v = Tr σ * (X ℓτ )S k-ℓτ ∆t A ∆t a * v b u with a v h = S ∆t Dσ( X(s)) • h * v, b u h = S ∆t Dσ( X(t)) • D h τ X(t) * u. Since |a v | L(H) ≤ c |v|, |b u | L(H) ≤ c |u|, we deduce |φ 2 (τ, s, t, k)| L(H) ≤ cTr{S k-ℓτ ∆t A ∆t } ≤ c∆t -1/2-2ε t -1+ε k-ℓτ . and II ≤ c∆t 1/2-4ε t k+1 t k (T -t) -1/2-ε dt. Finally III = 1 2 t k+1 t k t t k t k 0 n∈N
Tr{γ n S ∆t σ( X(t))}dτ ds dt

where for u, v ∈ H n∈N (γ n u, v) = n∈N D 3 u(T -t, X(t)) D n τ X(s), u, S ∆t Dσ( X(s)) • A ∆t S k-ℓτ ∆t σ(X ℓτ )e n v = Tr κ(u, v)(-A) -1/2-ε (-A) 2ε D n τ X(s) , and for h 1 , h 2 ∈ H (κ(u, v)h 1 , h 2 ) = D 3 u(T -t, X(t))• (-A) 1/2-ε h 1 , u, S ∆t Dσ( X(s)) • A ∆t S k-ℓτ ∆t σ(X ℓτ )h 2 v . By Lemma 4.6 |κ(u, v)| L(H) ≤ c(T -t) -1/2+ε
(γ n u, v) ≤ c(T -t) -1/2+ε t -1+ε k-ℓτ ∆t -3ε |u| |v|. It follows |γ n | L(H) ≤ c(T -t) -1/2+ε t -1+ε
k-ℓτ ∆t -3ε and by (2.2), (2.14)

III ≤ c∆t 1/2-4ε t k+1 t k (T -t) -1/2+ε dt.
We can now conclude

|c 3,2 k | ≤ c∆t 1/2-4ε (t -1+ε k + 1)( t k+1 t k (T -t) -1/2+ε dt + 1). Finally, it is easy to check |C| L(H) ≤ c∆t(1 + |X k |) and |c 3,3 k | ≤ c∆t 1-2ε t k+1 t k (T -t) -1/2+ε dt.
We deduce

|c 3 k | ≤ c∆t 1/2-4ε (t -1+ε k + 1) t k+1 t k ((T -t) -1/2-ε + 1)dt,
and, since c 4 k is majorized in exactly the same way,

|c k | ≤ c∆t 1/2-4ε (t -1+ε k + 1) t k+1 t k ((T -t) -1+ε + 1)dt. It follows N -1 k=1 |c k | ≤ c∆t 1/2-4ε .
Step 7: Conclusion.

It is now easy to gather all previous estimates in (3.7) and deduce

|u(T, x) -E (ϕ(X N )) | ≤ c∆t 1/2-4ε .
Recall that all the above computations have been done on the Galerkin approximations of X and X k ). The constant c above does not depend on m so that we can easily let m→∞ in this estimate and obtain the result.

Auxiliary Lemmas

In this section, we state and prove technical Lemmas used in the preceeding section. Again, the various estimates used here could be difficult to justify rigorously on the infinite dimensional equation and we in fact work with Galerkin approximations. Taking the limit m→∞ at the end of the proofs gives the results rigorously.

The first two Lemmas are very classical and we state them without proof.

. . , N , s ∈ [0, t k ], we have t 2β k-ℓs E |D h s X k | 2 β ≤ c|h| 2 , h ∈ H.
Proof: By (2.11) and the chain rule, we obtain the following formula for the Malliavin derivative of X k :

D h s X k = S k-ℓs ∆t σ(X ℓs )h + ∆t k-1 ℓ=ℓs+1 S k-ℓ ∆t f ′ (X ℓ ) • D h s X ℓ + √ ∆t k-1 ℓ=ℓs+1 S k-ℓ ∆t (σ ′ (X ℓ ) • D h s X ℓ )χ ℓ+1
for s ∈ [0, t k ] and h ∈ H. By (2.12), (2.2),(2.3), (2.4), we deduce for ε > 0

E |D h s X k | 2 β ≤ c t -2β k-ℓs |h| 2 +   ∆t k-1 ℓ=ℓs+1 (-A) β S k-ℓ ∆t L(H) f ′ (X ℓ ) L(H) D h s X ℓ   2 +∆t k-1 ℓ=ℓs+1 (-A) β S k-ℓ ∆t (σ ′ (X ℓ ) • D h s X ℓ ) 2 L 2 (H) ≤ c t -2β k-ℓs |h| 2 + L 2 F   ∆t k-1 ℓ=ℓs+1 t -β k-ℓ D h s X ℓ   2 +L σ ∆t k-1 ℓ=ℓs+1 t -1/2-ε-2β k-ℓ D h s X ℓ ) 2 L 2 (H)
.

It is now easy to use a discrete Gronwall Lemma and prove max l=ℓs+1,...,k where u is defined in (3.4).

t 2β ℓ-ℓs E |D h s X ℓ | 2 ≤ c|h| 2
Proof: Differentiating (3.4), we obtain for h ∈ H:

Du(t, x) • h = E Dϕ(X(t, x)) • η h,x (t)
where η h,x (t) is the solution of    dη h,x = Aη h,x + f ′ (X(t, x)) • η h,x dt + σ ′ (X(t, x)) • η h,x dW, η h,x (0) = h.

We rewrite this equation in the integral form Proof: We use the same notations as in the proof of Lemma 4.4. We differentiate a second time (3.4) and obtain for h, k ∈ H:

η h,
(4.2) D 2 u(t, x) • (h, k) = E D 2 ϕ(X(t, x)) • (η h,x (t), η k,x (t)) + Dϕ(X(t, x)) • ζ h,k,x (t)
where ζ h,k,x (t) is the solution of

          
dζ h,k,x = Aζ h,k,x + f ′′ (X(t, x)) • (η h,x (t), η k,x (t)) + f ′ (X(t, x)) • ζ h,k,x (t) dt + σ ′′ (X(t, x)) • (η h,x (t), η k,x (t)) + σ ′ (X(t, x)) • ζ h,k,x (t) dW, ζ h,k,x (0) = 0.

We rewrite this equation in the integral form ζ h,k,x (t) = t 0 S(t -s) f ′′ (X(s, x)) • (η h,x (s), η k,x (s)) + f ′ (X(s, x)) • ζ h,k,x (s) ds + t 0 S(t -s) σ ′′ (X(s, x)) • (η h,x (s), η k,x (s)) + σ ′ (X(s, x)) • ζ h,k,x (s) dW (s), t ≥ 0.

Using similar argument as above and (2.5), we prove The result follows easily using (4.4) and this inequality in (4.2).

(
The following Lemma is proved thanks to similar arguments.

(2. 6 )

 6 Tr(LM ) = Tr(M L), and, if L is also positive, (2.7) Tr(LM ) ≤ Tr(L) M L(H) .

Lemma 4 . 1 .Lemma 4 . 2 .Lemma 4 . 3 .

 414243 For any l ∈ N, there exists a constant c l such that max k=0,...,N E(|X k | l ) ≤ c l (|x| l + 1). For any l ∈ N, there exists a constant cl such that sup t E(|X(t, x)|) ≤ cl (|x| l + 1). For any β ∈ [0, 1/4), there exists a constant c such for k = 1, .

Lemma 4 . 4 .

 44 Let ϕ ∈ C 1 b (H, R). For any β < 1/2, there exists a constant c β such that for t > 0, x ∈ H |Du(t, x)| β ≤ c β t -β ϕ 1 ,

Lemma 4 . 5 .

 45 Let ϕ ∈ C 2 b (H, R). For any β, γ < 1/2, there exists a constant c β,γ such that for t > 0, x ∈ H |(-A) β D 2 u(t, x)(-A) γ | L(H) ≤ c β,γ t -(β+γ) ϕ 2, where u is defined in (3.4).

Lemma 4 . 6 .

 46 Let ϕ ∈ C 3 b (H, R). For any β < 1/2, there exists a constant c β such that for t > 0, x ∈ H, h 1 ∈ D((-A) β ), h 2 ∈ H, h 3 ∈ H D 3 u(t, x) • ((-A) β h 1 , h 2 , h 3 ) ≤ c β t -β ϕ 3 |h 1 | |h 2 | |h 3 |,where u is defined in (3.4).

  x (t) = S(t)h+ We deduce |Du(t, x) • h| ≤ c ϕ 1 t -β |h| -β .Taking the supremum over h yields the result.

	Using (2.3) and then Cauchy-Schwarz inequality, we obtain
	E η h,x (t)	2	≤ c t -2β |h| 2 -β + L 2 f E		0	t	|η h,x (s)|ds	2	+ E	0	t	(t -s) -α η h,x (s)	2	ds
			≤ c t -2β |h| 2 -β + c	0	t	E(|η h,x (s)| 2 ds + E	0	t	(t -s) -α η h,x (s)	2	ds.
	It is classical that this implies
	(4.1)			sup t∈[0,T ]	t 2β E η h,x (t)	2		≤ |h| 2 -β .
		0	t	S(t-s)f ′ (X(s, x))•η h,x (s)ds+	0	t	S(t-s)σ ′ (X(s, x))•η h,x (s)dW (s), t ≥ 0.
	By (2.4), (2.2), (2.12), we have for y, k ∈ H and α > 1/2:
	S(t)σ ′ (y) • k L 2 (H) ≤ L σ (-A) -α/2	L 2 (H)	(-A) α/2 S(t)	L(H)	|k| ≤ ct -α/2 |k|.

  4.3) E |ζ h,k,x (t)| 2 ≤ cE t 0 |η h,x (s)| |η k,x (s)| + |ζ h,k,x (s)|ds -s) -α |η h,x (s)| 2 -1/4 |η k,x (s)| 2 -1/4 + |ζ h,k,x (s)| 2 ds, t ≥ 0.Proceeding as in Lemma 4.4, we have thanks to Burkholder inequality and then to Minkowsky inequalityE η h,x (t) -s) -α E(|η h,x (s)| 4 )Plugging these inequalities and similar ones for η k,x in (4.3) yields

	2 Taking the square root of this inequality and using a generalized Gronwall Lemma, we deduce +cE t ≤ c t -4β |h| 4 -β + cE t 0 |η h,x (s)|ds 4 + E t 0 (t -s) -α η h,x (s) 2 ds 2 ≤ c t -4β |h| 4 -β + c t 0 E(|η h,x (s)| 4 ) 1/4 ds 4 +c t 0 (t 1/2 ds 2 . (4.4) sup t∈[0,T ] 2 Therefore, by (4.4), (4.5) E(|η h,x (t)| 4 -1/4 ) ≤ ct 1-4β |h| 4 -β sup (t 4 t∈[0,T ]
	0

t 4β E η h,x (t) 4 ≤ c|h| 4 -β .

Similarly, we have

E(|η h,x (t)| 4 -1/4 ) ≤ ct 1-4β |h| 4 -β + cE t 0 |η h,x (s)|ds 4 + cE t 0 (t -s) -α |η h,x (s)| 2 ds 2 ≤ ct 1-4β |h| 4 -β + c t 0 E |η h,x (s)| 4 1/4 ds 4 +c t 0 (t -s) -α E |η h,x (s)| 4 1/2 ds E ζ h,x (t) 2 ≤ c|h| 2 -β |h| 2 -γ .


The rise of preprints in chemistry

Chemistry is now starting to embrace preprints, with more and more researchers in chemical and materials sciences posting their manuscripts online prior to peer review. Preprints can speed up the dissemination of scientific results and lead to more informal exchanges between researchers, hopefully accelerating the pace of research as a whole.

François-Xavier Coudert

A preprint is a scientific article made available online to the research community by its authors that

has not yet been formally peer reviewed (Fig. 1). Most often it is the version of an article that has just been (or soon will be) submitted to a traditional journal, but it can also sometimes be a working document that its authors wish to share publicly. Preprints allow researchers, amongst other things, to disseminate their latest advances in research more quickly, to stake a claim of priority on the research, and to receive feedback from other researchers in the wider community.

The online pre-publication model was invented by theoretical physicists, with the arXiv preprint server 1 that opened in 1991, hosted at the time by the Los Alamos National Laboratory. The scope of the server was later expanded to include contributions from other fields including astronomy, mathematics, computer science and quantitative biology to name just a few. In January 2015, arXiv passed the significant milestone of hosting one million articles and continues to grow, with 155,866 new submissions in 2019 (up 11% from the previous year). Its success has led to other disciplinecentric preprint servers -including bioRxiv 2 , which was launched in 2013 for biology and has published 79,570 preprints since then.

The chemistry community came relatively late to preprinting, but the chemistry-focused chemRxiv 3 followed in August 2017 and currently hosts 4,391 preprints -a similar adoption rate, for its second anniversary, as was seen at bioRxiv. Additionally, there are other 'open archives' that accept preprints from any discipline -such as OSF Preprints 4 or HAL 5 -as well as institutional repositories. The recent upward trend 6 in preprinting across all fields of research comes at a time when the academic publishing system is witnessing a rapid acceleration towards more open access when it comes to scientific publishing -and the promotion of open science more generally.

Faster dissemination of ideas

In 2019 a survey of bioRxiv users 7 focused on researchers' motivations for posting preprints and the perceived benefits of doing so. One of the main reasons for preprinting research cited by those surveyed is the rapid dissemination of scientific knowledge to a (potentially) large audience: not only do preprints signal what work is coming out of a particular lab, but they also enable researchers to control when it becomes available. In the context of fast electronic communication being the norm in the modern world -combined with advances in research that can be really quite rapid -it is not surprising that editorial processes that can take several months are considered slow in comparison; colleagues complaining at the coffee machine or on social media about the insufficient pace of peer review is commonplace.

Just like sharing your latest results at a conference, uploading a preprint enables you to share and discuss a draft of your cutting-edge research. Moreover, informal feedback from those who read the preprint can often improve the quality of the final article: 71% of bioRxiv users say they received feedback on their preprints. Although some preprint servers have dedicated comment sections, this feedback most often takes place through other channels including social media, e-mail and discussion at conferences. For example, preprints are regularly publicized on Twitter by the @chemrxiv account, often leading to further discussion on that platform.

Preprinting also has other benefits. One that is frequently cited by researchers 8 is that uploading a preprint also makes it possible to claim priority on new ideas and developments -in a public and indisputable way -without depending on acceptance by the journal to which it will be submitted. This was already cited as one of the motivating factors back when arXiv was launched: some researchers at the time were concerned that unethical journal reviewers would try to delay publication with the intention of publishing the ideas or findings that they were meant to be assessing under their own name [START_REF] Vence | Q&A: 1 Million Preprints and Counting[END_REF] . Although there is often little more than anecdotal evidence for such behaviour, preprinting the submitted version of a paper establishes a clear picture of the state of a piece of work at a specific date. More generally speaking, the use of preprints is a good way to record a timeline of ideas and knowledge as well as the evolution of papers that may change (and hopefully be improved) as a result of the peer-review process: in doing so, the practice of preprinting aligns well with the recent push by some journals to share the content of the peer-review reports and authors' responses to them. This reflects the nonlinear nature of research, where hypotheses are proposed, discussed, proven or refuted over time.

Increased visibility

Several bibliometric studies have shown that preprints also increase the visibility of the work being done 10 , by combining two distinct advantages: they are open access and they appear online earlier than the final peer-reviewed publication. This typically translates into more views and higher impact than non-preprinted articles in the same field 11,12 : preprinted articles typically have better online metrics, attention score, and number of citations. 13 Preprints for all servers are indexed by Google Scholar, and those deposited on chemRxiv are indexed in CAS and CrossRef (and Scopus is in the pipeline), so their discoverability is essentially the same as peer-reviewed papers. Articles can be cited in their preprint form: many servers now provide a DOI as a unique identifier for each preprint, making it easy to include them in reference lists. Citations specifically to the preprint version of an article are added to those of the final version of a paper that ultimately appears in a journal -an important factor at a time when bibliometrics play a significant role in the evaluation of research and researchers alike.

The use of preprints should be particularly attractive to early-career researchers, who typically have few published articles associated with their independent careers. It makes it possible to share articles that are in the process of being submitted in a more concrete fashion than the dreaded 'submitted for publication' phrase often found on CVs. More and more funding agencies accept (and even recommend) references to preprints in grant applications, as evidence of the applicant's previous work [START_REF] Matthews | New boost for preprints after acceptance by ERC[END_REF] : these include, amongst others, the European Research Council, US National Institutes of Health and the UK's Wellcome Trust. Hiring committees can also look favourably on this practice.

A step toward open science

The use of preprints also ensures a wider circulation of the article's content -especially if the final published version appears in a subscription-based journal -because the preprint remains freely accessible after formal publication. Preprints can also be linked to the final published version of the article and can even be updated to the latest 'author accepted' version if the policy of the journal in question allows it. The dissemination of research results to the widest possible audience is considered to be a moral obligation of academic research 15 , especially when the majority of published research is funded from public monies 16 . Although a transition of the publishing system is under way, with a considerable push toward open access, preprints represent a cost-free way to achieve open access: it is a form of self-archiving by the authors, often referred to as 'green open access'. Preprints therefore provide a guarantee of access to those beyond well-funded research institutions, such as NGOs, patient organizations, journalists, independent researchers, researchers from developing countries and educators at non-research institutions. They also ensure the long-term sustainability of access to such files: preprint servers have an archiving and mirror system in place -and in many cases institutional guarantees. This is a marked difference with some commercial article-sharing platforms which sometimes require readers to create an account (so the content is not open access) and offer no longterm guarantee for the content that they host. Moreover, the rapid sharing of research as it happens is in line with the idea of open science: preprints can help shape the future research in the field by making ideas spread faster. They contribute by filling a role that was traditionally served by oral and poster communications at conferences and workshops.

Although conferences have increased in both size and number, preprints share none of the ecological, financial or personal costs of travel associated with those -and they also scale better by being available to all. This is particularly important with the current wave of travel restrictions related to the COVID-19 outbreak. Finally, it should also be noted that preprint servers also allow data associated with the research to be uploaded alongside the article itself, just as regular journals do. This is not limited to traditional supporting materials in PDF format, but also datasets of experimental and computational data, in machine-readable formats, enabling future researchers to build upon the work.

There are, of course, also some risks associated with the use of preprints. An important one is a potential blurring of the lines between peer-reviewed articles and preprints, meaning that preprints could create confusion in the eyes of journalists and the general public 17 . In particular, in topics with public-health implications it is important that preprints, if they are covered in the media, are not overblown -especially if, after review, the science turns out to be flawed in some way. Here it is important to note that, unlike published papers, preprints are typically viewed as part of the ongoing scientific discussion taking place in the community: since there is no formal peer review or quality control (beyond some basic moderation), there is no formal process for retraction or expression of concern on preprints posted online. However, we know that even peer-reviewed publications can later turn out to be incorrect, so perhaps this is an opportunity for a larger debate about the scientific method and how research works.

The COVID-19 outbreak and ensuing public health crisis is quite revealing of both the importance and limitations of preprint servers. While scientific journals provide accelerated peer review for work related to the current crisis, immediate and open access to the latest research data truly accelerates the pace of research, with analysis and critique often provided mere hours after preprints are posted. At the same time, the multitude of studies on a hot topic of such importance to public health has led preprint servers to add banners clearly reminding readers that, such as in the case of medRxiv, 'these are preliminary reports that have not been peer-reviewed. They should not be regarded as conclusive, guide clinical practice/health-related behavior, or be reported in news media as established information.'

Preprints in chemistry

As noted earlier in this article, the chemistry community as a whole came relatively late to preprinting.

In some subdisciplines, researchers have been uploading preprints for a while now, especially in areas

Figure 1 |

 1 Figure 1 | How preprints fit into the ecosystem of academic publishing. The process in the top part of the figure shows the traditional route to publication in which a manuscript is submitted to a journal and then undergoes editorial and peer evaluation before finally being accepted for publication (and not necessarily in the first journal it was submitted to). The lower-half shows how preprints fit into this model, comparing the availability of preprints and journal articles to be read (and cited). Preprints are available immediately and any community feedback can be used to help in revising the manuscript during the formal evaluation at a journal.

Figure 2 |

 2 Figure 2 | Distribution of broad subject areas of 4,391 preprints posted on chemRxiv as of April 2020. There is a wide spread of subject areas, but the largest one is in the subdiscipline of chemistry (theory and computation) that has traditionally embraced preprinting and open-science principles more so than other areas.

such as physical chemistry, computational chemistry and, more recently, biochemistry, but this did not become a common occurrence in our field until very recently 18 . In 2017, not one but two preprint servers for the chemical sciences were launched: chemRxiv, backed by a consortium of the American Chemical Society, the Royal Society of Chemistry, the German GDCh, the Chinese Chemical Society and the Chemical Society of Japan; and ChemRN, launched by Elsevier after its acquisition of SSRN (a preprint archive that previously focused on social sciences and law). These servers have met with success and the number of preprints in chemistry is now rapidly increasing. ChemRxiv published 2,314 new preprints in 2019 -compared to 1,031 in 2018 -and has seen more than 3 million article reads in the last year. This is still a drop in the ocean when it comes to chemistry publishing, with roughly 500,000 academic papers in chemistry and chemistry-related fields appearing each year.

Nevertheless, the acceptance and adoption of the practice of preprinting is rapidly growing.

Parallel to this uptake, chemistry journals have clarified or revisited their positions regarding preprints.

While some editors previously considered that uploading preprints compromised the novelty of the findings reported in a submitted manuscript, preprints are now universally welcomed by chemistry journals, from all different communities -and the vast majority of journals have a well-documented preprint policy, which can be found with online tools such as SHERPA/RoMEO 19 . The distribution of preprints among the different sub-fields of chemistry is currently biased towards fields that already practiced preprinting before the launch of the chemistry-specific servers, such as theoretical chemistry and materials science (Fig. 2). However, there is now a rapid rise in the areas of organic chemistry and chemical biology.

Preprints have made significant inroads into chemistry, but worries are occasionally expressed suggesting that the act of preprinting an article will, despite the assurances on journal websites, prejudice its chances of appearing in what are considered by some researchers to be high-profile venues for publication. The counter-argument to this is to highlight examples of published research papers that were first shared as preprints. There are many examples, but to select just a few, you can look to the 16-step synthesis of isoryanodane diterpene (+)-perseanol by Han and co-workers 20,21 , the CryoEM methodological paper by Jones and colleagues 22,23 , or the study of modular radical crosscoupling with sulfones by the Baran group and collaborators 24,25 .

The field of chemistry has seen a rapid increase in the adoption of preprints in the past couple of years, linked to the development of community preprint servers and the shift to preprint-friendly editorial policies by the journals in the field. Preprints, coupled with comment platforms and social media, allow for quick and broad dissemination of a research team's latest results as well as feedback from the community and the overall process replicates some aspects typically associated with conference talks, posters and even informal social interactions such as coffee breaks. They are an established part of the scientific conversation in many disciplines and there is no reason why chemists should not take advantage of the benefits they offer, particularly when it comes to giving credit to early career researchers. The way in which scientific results are disseminated has changed significantly with the advent of the internet and preprints represent an important step in the evolution of scholarly communication.

François-Xavier Coudert

Chimie ParisTech, PSL University, CNRS, Institut de Recherche de Chimie Paris, 75005 Paris, France.

Competing interests

François-Xavier Coudert is a member of ChemRxiv Scientific Advisory Board, and Editorial Board Member for Communications Chemistry.


Sylvie Lortal,'" Jean Van Heijenoort

Karin Gruber3

E B Sleytr3 Znm

Slayer of Lactobacillus helveticus ATCC 12046: isolation, chemical characterization and re-formation after extraction with lithium chloride

In a previous study, electron microscopic examinations of thin sections of ~?obacilIus helveticus ATCC 12046 revealed a three-layered structure of the cell wall. The outermost component was identilied as a layer of a nonglycosylated 52 kDa protein. Freeze-etched preparations of intact cells have now demonstrated that this protein layer is an oblique surface layer (Slayer) lattice (a = 4.5 nm, 6 = 9.6 nm, y = 77 ") which completely covers the cell surface. Treatment with 5M-LiCl extracted the S-layer protein from intact cells efficiently and selectively. Viability did not decrease signi6cantly. Moreover, the Slayer reappeared when treated cells were allowed to grow again. I n vitm self-assembly products obtained upon aggregation of isolated S-layer subunits exhibited the same oblique S-layer symmetry as observed on intact cells in viva The purified Slayer protein had a high content (44 %) of hydrophobic amino acids. The N-terminal sequence was mainly composed of alanine, threonine, asparagine and aspartic acid.

Introduction

Crystalline surface layers (S-layers) are a common feature among eubacteria and archaeobacteria (Kandler & Konig, 1985;[START_REF] Sleytr | Crystalline surface layer in procaryotes[END_REF][START_REF] Messner | Crystalline bacterial cell surface layers (S-layers)[END_REF]. They are composed of identical subunits consisting of a single protein species, which may be glycosylated, and completely cover the cell surface [START_REF] Beveridge | Ultrastructure, chemistry and functions of the bacterial cell wall[END_REF] ; [START_REF] Sleytr | Crystalline surface layers on bacteria[END_REF][START_REF] Koval | Paracrystalline protein surface arrays on bacteria[END_REF]. The biological functions of S-layers in eubacteria include protection, cell adhesion and surface recognition [START_REF] Beveridge | Ultrastructure, chemistry and functions of the bacterial cell wall[END_REF][START_REF] Smit | Protein surface layers of bacteria[END_REF][START_REF] Gruber | Influence of an additional proteinaceous layer (S-layer) on surface properties of Bacillus stearothermophilus[END_REF].

Lactobacillus helveticus has great technological significance, particularly in cheese making [START_REF] Accolas | Les levains lactiques thermophiles : propriktks et comportement en technologie laitibre[END_REF][START_REF] Turner | Swiss type cheese. 11. The role of thermophilic lactobacilli in sugar fermentation[END_REF]. In previous work [START_REF] Lortal | Cell wall and autolytic system of Lactobacillus helveticus ATCC 12046[END_REF] the cell wall of L. helveticus ATCC 12046 was found to have a three-layered structure. The external layer was shown to be composed of a 52 kDa protein, non-covalently linked to the internal layers. These results strongly suggested the presence of an S-layer on L.

Author for correspondence. Tel. 9928 5334; fax 9928 5350.

Abbreviations: CE, crude extract; GHCI, guanidinium hydrochloride; PCF, polycationic ferritin; S-layer, surface layer. helveticus ATCC 12046, especially since an S-layer has been detected on other strains of L. helveticus [START_REF] Masuda | Distribution and chemical characterization of regular arrays in the cell wall of strains of the genus Lactobacillus[END_REF].

In this paper the presence of an S-layer on L. helveticus ATCC 12046 was confirmed by analysis of the cell surface topography using freeze-etc hing techniques. Isolation procedures and reassembly experiments in vitro are described as well as re-formation of the S-layer on intact cells after extraction. The amino acid composition and the N-terminal sequence of the purified S-layer protein were determined.

Methods

Bacterial strain and growth conditions. Lactobacillus helveticus

Extraction of the S-layer protein with LiCl and S-layer re-formation on growing cells.

Exponential-phase cells were harvested and washed once with distilled water. Ten-fifteen mg of the moist pellet were suspended per ml of 5 M-LiCl and kept at 0 "C for 15 min, followed by centrifugation (30000 g, 15 min). The supernatant was designated the crude extract (CE). The pellet of LiC1-extracted cells was washed once in distilled water, resuspended in prewarmed (43 "C) MRS broth to an OD650 of 0.5 (corresponding to 0.2 mg dry wt ml-I) and incubated at 43 "C. An aliquot was kept at 4 "C.

PuriJication of the S-layer protein.

The CE was dialysed against distilled water overnight at 4°C. For analysis of the amino acid composition, a more extensive purification of the S-layer protein was carried out by gel permeation chromatography : the CE was applied to a column (2.5 x 50 cm) of Sephacryl S200 (Pharmacia) equilibrated in 5 M-Licl. Upward elution was done at a flow rate of 50 ml h-l and fractions of 5 ml were collected. Protein was determined by the Lowry method using BSA as standard. Amino acid analysis was performed with a Pharmacia LKB Alpha Plus instrument as described by Lortal et al. (1991).

Preparation of cell walls. Cell walls were prepared by mechanical disruption of whole cells using a French press as described by Lortal et a/. (1991).

Electron microscopy and labelling with plycationized ferritin. Negative staining was obtained by treating the grids for 2 min with a drop of uranyl acetate (2%, w/v, in distilled water). Thin sectioning was performed as described by Lortal et al. (1991). Freeze-fracturing, freeze-etching and optical diffraction procedures for determining lattice constants are described by Messner et al. (1984) and Pum et al.

(1 989). For freeze-drying preparations, the pioloform-coated gold grids (400 mesh) were glow-discharged (CTA 010, Balzers) and treated with poly-L-lysine (M, 56000, Sigma, 0.1 % in distilled water) for 10 min to produce a positively charged surface. After floating upside down on the bacterial suspension for 10 min, the grids were washed three times in distilled water and moved onto a drop of polycationic ferritin (PCF, Sigma), incubated for 10min and washed three times in 0 . 0 5 ~-Tris/HCl buffer, pH 7.2. The grids were blotted onto wet filter paper for 45 s [START_REF] Kistler | Freeze-drying and shadowing a two-dimensional periodic specimen[END_REF] after washing in distilled water to remove residual buffer. The grids were immediately frozen by plunging into liquid nitrogen and were mounted onto a precooled freeze-drying table of a Balzers BAF 400 freeze-etching machine. After freeze-drying at -80 "C for 1 h the samples were shadowed unidirectionally with 2 nm platinum at an angle of 45". The heavy metal shadow was reinforced with a 15 nm carbon layer deposited at an angle of 90 " (Pum et al., 1989). The preparations were first rinsed in acetone and then incubated in chloroform (both from Merck) at room temperature for 3 h followed by an overnight treatment with 30% (v/v) chromic acid at 37 "C. After four washes in distilled water the grids were air-dried. Electron microscopy was performed with a Philips EM 301 microscope using an acceleration voltage of 80 kV.

SDS-PAGE. (i)

For determination of the apparent molecular mass of the S-layer protein and for electroblotting experiments, SDS-PAGE was performed using a 15% (w/v) acrylamide separating gel (15 x 15 cm) as described by Lortal et al. (1991). Molecular mass standard proteins were obtained from Pharmacia.

(ii) For all other experiments, SDS-PAGE was performed using a 10% acrylamide separating gel (8 x 8 cm) and a 4% acrylamide stacking gel. Samples of whole cells (equal weights when comparison between lanes was required), cell walls or isolated S-layer protein were suspended in a 3% (w/v) SDS solution in 0.05 M-Tris/HCl buffer, pH 7.2, and treated for 10 min in a boiling water bath; then the samples were centrifuged and 25 pl of the supernatant was mixed with 40 pl of the following solution: 0.1 % SDS, 0.1 % 2-mercaptoethanol, 30% (w/v) glycerol, 0.2% Orange G in 0.01 M-Tris/HCl buffer, pH 7-2. Gels were stained with Coomassie blue G250.

Electroblotting and N-terminal sequence. Immediately following SDS-PAGE, the proteins were electrohoretically (50 V, 1.25 h) transferred to an Immobilon sheet (PVDF 0.45 pm, Millipore). The membrane was stained for 30 s with a solution composed of 0.1 % (w/v) amido black, 45 % (v/v) methanol and 7 % (v/v) ethanol in distilled water, and washed once in distilled water. The area of the membrane containing the 52 kDa protein was cut out with a sterile scalpel and kept at -20 "C until microsequencing was performed with a gas-phase analyser at the Institute of Neurophysiology at the CNRS, Gif sur Yvette, France.

Treatment of intact cells with Iytic enzymes.

Lysozyme and rnutanolysin (both from Sigma) were dissolved before use in 0.05 M- Tris/HCl buffer (pH 7-2) and kept at 4 "C. The effect on the bacteria was followed by the decrease in OD650 of a cell suspension in Tris/HCl buffer, pH 7.2, at 37 "C. The concentration of the buffer was 0.05 M for mutanolysin and 0.1 M for lysozyme; concentrations of the enzymes are indicated in the Figure legends or in the text.

Morphological characterization of the S-layer

Freeze-etched preparations of whole cells of L. helveticus ATCC 12046 (Fig. la,b) revealed the presence of a crystalline array which completely covers the cell surface. A clear fracture plane between the peptidoglycan-containing layer and the S-layer could frequently be observed in oblique fracture faces along the cell envelope (Fig. lb). In such areas [indicated by the arrowheads in Fig. l(b)] the thin outermost S-layer can clearly be recognized. In thin sections (Fig. lc) the S-layer appeared only approximately 9 nm thick, whereas the underlying layer was three-to four-times as thick. Optical diffraction experiments (Fig. 46) with selfassembly products of the 52 kDa S-layer protein showed that the length of the lattice base vectors were 4.5 and 9.6 nm, respectively, with an interaxial angle y of 77 O.

Isolation, purification and chemical characterization of the S-layer protein

Previous work [START_REF] Lortal | Cell wall and autolytic system of Lactobacillus helveticus ATCC 12046[END_REF] showed that the outermost layer of the cell wall was composed of a nonglycosylated 52 kDa protein. In this investigation, several procedures for extracting the S-layer protein from whole cells and isolated cell walls were tested (Table 1).

Since the S-layer protein represented approximately 14 % of the total protein content of the bacteria and gave a dominant band on SDS electrophoretograms (Fig. 2, lane A), the efficiency of the various extraction procedures could be followed by SDS-PAGE (data not shown). Tween 80, varying concentrations of urea and heating for 15 min in 0.1 M-sodium phosphate buffers at 50 "C (pH 5, 6, 7.5, 9) were inefficient. HC1 (0.1 M) had an extractive effect but also led to hydrolytic damage of the S-layer protein. SDS, GHCl and LiCl were found to be efficient extraction agents. Among them, LiCl (5 M, 15 min, 0 "C) was the most selective (Fig. 2, lane B) and was therefore chosen for further assays. Lower concentrations of LiCl (1 and 2.5 M) extracted less protein. A linear relationship was observed between the concentration of LiCl and the quantity of protein solubilized. Repeated LiCl treatment had an insignificant effect on the increase in efficiency of extraction (Fig. 2, lane C).

During dialysis of the LiCl CE against distilled water at 4"C, a white precipitate appeared which could be recovered by centrifugation. The supernatant (Fig. 3, lane A) revealed some contaminant proteins whereas the white pellet (Fig. 3, lane B) contained mainly the 52 kDa protein (Fig. 4a). Optical diffraction analysis of electron micrographs (Fig. 4b) revealed that the sheet-like selfobserved in freeze-etched preparations of intact cells (Fig. 1 a). Purification of the S-layer protein was carried out on a Sephacryl S-200 column. The first Deak eluted consisted of the -52 kDa protein, as verified by SDS-PAGE (data not shown). The combined fractions were dialysed against distilled water and the precipitated protein was dried overnight using a speed vacuum centrifuge (Savant Instruments). After hydrolysis, the amino acid composition of the protein was determined.

As summarized in Table 2, the hydrophobic amino acids accounted for 44% of the amino acid residues present. protein and was therefore regarded as pure enough for analysis of its N-terminal sequence. In the sequence obtained -ATTINADSAINANTNAKYDVDVTalanine, threonine, asparagine and aspartic acid accounted for 70% of the amino acids. Negative staining revealed that the precipitated material obtained during dialysis of the LiCl extract was reassembled S-layer Treatment of bacteria with 5 M-Licl led to a limited loss of viability (80%, i.e. < 1 log unit) whereas the extraction with 5 M-GHCl led to a drastic loss of viability (6.2 log units). On repeated treatment with 5 M-LiCl, viability decreased significantly (3.2 log units). For all treatments, results were the same whether the procedures were performed at 0 "C or at room temperature.

Eflect of LiCl and GHCl treatment on viability

Reformation of the S-layer after extraction with LiCl

LiC1-treated cells (sample B, Fig. 5a) incubated at 43 "C in MRS broth, grew again as shown in Fig. 5

(a). Sample

A (untreated cells) and samples B-K harvested at different times, as shown in Fig. 5(a), were analysed by SDS-PAGE (Fig. 5 b), and labelled with PCF and freezedried for electron microscopical observations (Fig. 6).

Hardly any PCF absorbed to S-layer-carrying cells (Fig. 6a). This implies that the surface of the S-layer is not negatively charged. However, PCF formed a dense layer on the surface of cells from which the S-layer had been removed (Fig. 6b). This confirmed the complete extraction of the S-layer by LiC1. As PCF does not absorb onto the S-layer, Fig. 6(c) and (d) suggested that the binding of PCF molecules was prevented by the progressive reformation of the S-layer on LiCl-treated cells of L. helveticus ATCC 12046.

Sensitivity of cells to lytic enzymes before and after removal of the S-layer

To investigate a potential protective role for the S-layer against muramidases, whole cells before and after LiCl treatment were exposed to lysozyme and mutanolysi.l. LiCl treatment did not increase sensitivity to lysozyme, whereas mutanolysin had a more rapid lytic effect on cells devoid of an S-layer (Fig. 7).

l o 0 i 80

Discussion

The presence of an oblique S-layer on the external surface of L. helveticus ATCC 12046 was clearly established by freeze-etching and negative staining. This has extended our previous work [START_REF] Lortal | Cell wall and autolytic system of Lactobacillus helveticus ATCC 12046[END_REF] where the more external layer of the cell wall was shown to consist of a 52 kDa non-glycosylated protein, noncovalently bound to other cell wall constituents. Other strains of L. helveticus have also been found to contain an S-layer (M. Sara, personal communication). Masuda & Kawata (1983) found an S-layer in other L. helveticus strains.

Although several studies of the structure and chemical composition of the cell walls of lactobacilli have been performed or LiClextracted (------), were tested for their susceptibility to the lytic action of mutanolysin at the h a 1 concentrations indicated (pg ml-l) in 0-05 ~-Tris/HCl buffer, pH 7.2.

1960; [START_REF] Williams | Cell wall composition and enzymology of lactobacilli[END_REF], only two strains have been investigated using freeze-etching techniques : Lacto- bacillus fermenti [START_REF] Kawata | Regular array in the cell wall of Lactobacillus fermenti as revealed by freeze-etching and negative staining[END_REF] and Lactobacillus brevis [START_REF] Masuda | Ultrastructure and partial characterization of a regular array (RA) in the cell wall of Lactobacillus brevis[END_REF]. These displayed regular arrays with oblique symmetry like t . helveticus.

Release of the S-layer protein from isolated cell walls of L. fermenti was achieved by enzymic hydrolysis of the peptidoglycan and from L. brevis by extraction with GHCl, urea and SDS. Negative staining and electrophoresis also revealed S-layers on L. casei (Barker & Thorne, 1970), L. brevis (Masuda & Kawata, 1980, 1983), L. buchneri, L. acidophilus and L. bulgaricus [START_REF] Masuda | Characterization of a regular array in the wall of Lactobacillus buchneri and its reattachment to the other wall components[END_REF], 1985). For all these lactobacilli the apparent molecular mass of the S-layer protein was in the range 41-52 kDa. We found that it was possible to extract the S-layer protein from whole cells of L. helveticus directly at a remarkably high concentration compared to other proteins using 5 M-LiCl. During the removal of the LiCl by dialysis, the S-layer subunits reassembled spontaneously into sheets with the same p2 symmetry and lattice constants as observed in vim, showing that they were not substantially altered by the LiCl treatment. Reattachment experiments with isolated Slayer subunits on naked peptidoglycan-containing layers of L. helveticus ATCC 12046 have not been performed although studies on homologous or heterologous reattachment with S-layers from other lactobacilli have provided interesting data on the reattachment of S-layers amongst different organisms (Masuda & Kawata, 1985). In agreement with data on S-layers from other bacterial species [START_REF] Masuda | Distribution and chemical characterization of regular arrays in the cell wall of strains of the genus Lactobacillus[END_REF][START_REF] Sleytr | Crystalline surface layer in procaryotes[END_REF], the S-layer protein of L. helveticus had a high content of hydrophobic amino acids and no or very little sulphur-containing amino acids. As observed by [START_REF] Luckevitch | Characterization of a dynamic S-layer on Bacillus thuringiensis[END_REF], no homology between the N-terminal sequence determined here and that from other S-layer proteins was evident.

The LiCl procedure used in this work for extracting the S-layer protein of L. helveticus is a new and interesting alternative to the classical methods with GHCl or SDS. Electron microscopy, electrophoresis and chemical analysis demonstrated the efficiency and selectivity of this method. Moreover, LiCl was less lethal than GHCl to the bacteria. One of the most interesting features was the reformation of the Slayer after its removal. The new S-layer material appeared during the exponential-phase on the cell surface in a random distribution. The change in the mode of distribution of the PCF molecules from Fig. 6(c) to 6(d) could be interpreted as a 'charge drift'. Pushed or partially masked by growing and fusing crystallites of the newly synthesized S-layer protein, the long teichoic acid molecules protruding from the peptidoglycan layer could form areas of high negative charge in the closing S-layer. For two Bacillus species [START_REF] Gruber | Localized insertion of new S-layer during growth of Bacillus stearothermophilus strains[END_REF][START_REF] Howard | Expansion of the tetragonally arrayed cell wall protein layer during growth of Bacillus sphaericur[END_REF], the incorporation of newly synthesized S-layer protein into a coherent S-layer lattice has been shown to be non-random. However, the S-layer reformation described here was a repair process of cells stripped of the S-layer and, thus, probably reflected conditions quite different from the incorporation of S-layer domains in an existing lattice during cell growth.

No specific biological significance of the S-layer lattice from lactobacilli has been detected so far. In L. helveticus ATCC 12046 the S-layer appeared as a thin layer completely covering the thick peptidoglycancontaining layer. Since no morphological changes of the cells could be detected after removal of the S-layer with LiCl and during subsequent cell growth and S-layer reformation, a cell-shape-determining function of the crystalline array [START_REF] Beveridge | The bacterial surface : general considerations towards design and function[END_REF][START_REF] Sleytr | Crystalline surface layer in procaryotes[END_REF]) can be excluded. Previous studies on the permeability properties of S-layers from mesophilic and thermophilic Bacillaceae (Sira & Sleytr, 1987;Siira et al. 1990) have shown that S-layers have the potential to function as selective molecular sieves. S-layers preventing the entry of molecules with molecular masses exceeding 10oo(r15000 kDa could provide the organism with a selective advantage, since most lytic enzymes present in natural habitats have larger molecular masses (Stewart et al., 1986; Glenn, 1976). Since the S-layer of L. helveticus could not protect the peptidoglycan from lysozyme (M, 14600) and mutanolysin (M, 24000) it can be concluded from the molecular dimensions of these enzymes that the pores in the oblique S-layer lattice must be larger than 3.5 nm (Sira et al., 1990). Further studies will show if the S-layer can protect the cells from proteases which generally have molecular masses larger than common muramidases. Only a few molecules of polycationized ferritin were adsorbed on whole cells o f t . helveticus showing that the S-layer surface did not exhibit a net negative charge. The same observations were made with different Clostridium (Ska et al., 1988) and Bacillus (Pum et al., 1989) species. Therefore, the functional role of the S-layer of L. helveticus could be to mask the underlying strongly negatively charged peptidoglycancontaining layer, as has been proposed previously for mesophilic and thermophilic Bacillaceae (Sdra et al., 1988;Sira et al., 1990;[START_REF] Sleytr | Crystalline surface layer in procaryotes[END_REF]. Further study on the interaction of L. helveticus ATCC 12046 with environmental components such as milk proteins will help to elucidate this functional aspect.

  ATCC 12046 was obtained from the Pasteur Institute, Paris, France, and stored at -30 "C in MRS broth (De Man et al., 1960) containing 15% (v/v) glycerol. The strain was grown unstirred in MRS medium at 43 "C after inoculation with 1 % (v/v) overnight culture. Determination of the effect of LiCl and guanidium hydrochloride (GHCI) treatment on cell viability. Viability was determined as c.f.u. on MRS agar plates after incubation for 48 h at 43 "C under anaerobic conditions. Exponential-phase cells (1.6 mg dry mass ml-' ; 6.5 x lo9 c.f.u. mt1) were incubated for 15 min at several temperatures with LiCl or guanidinium hydrochloride (GHCl) to extract the S-layer OOOI-7105 O 1992 SGM protein. The treated bacteria were recovered by centrifugation, washed once in sterile distilled water and resuspended in the initial volume of MRS broth for the determination of viable bacteria by plate counting.

Fig. 1 .

 1 Fig. 1. (a) Freezeetched preparations of whole cells of L. helueticus ATCC 12046 exhibiting an oblique (p2) S-layer lattice which completely covers the cell surface. (6) Oblique fracture faces of the cell envelope revealing a thick peptidoglycancontaining layer (PG) and a thin S-layer (S); CM, concave fracture face of the cytoplasmic membrane. (c) Thin section of whole cells of L. helveticus ATCC 12046; note the thin S-layer [abbreviations as for (b)]. Bars, 0.1 pm.

  Solubilization of S-layer protein; -, no effect. t The S-layer protein was clearly degraded by this treatment.

Fig. 2 .

 2 Fig. 2. SDS-PAGE. Lanes: A, whole cells; B, whole cells treated with 5 M-Licl once; c, whole cells treated with 5 M-Licl twice; D, molecular mass standards.

  Fig. 3. SDS-PAGE of (A) the supernatant and (B) the pellet obtained upon centrifugation of the dialysed LiCl crude extract. Lane C, molecular mass standards.

Fig. 4 .

 4 Fig. 4. (a) Negative staining of the S-layer self-assembly product obtained upon dialysis of the LiCl cell extract (bar, 0.05 pm). (6) Diffraction pattern from S-layer self-assembly product reflecting the oblique symmetry of the crystalline lattice.

Fig. 5 .

 5 Fig. 5. (a) Growth (OD,,,) of LiCl-treated whole cells (B) after suspension in MRS broth at 43 "C; sample X was kept at 4 "C and did not grow. (b) SDS-PAGE of whole cells harvested at different times (B-K) indicated on the growth curve in (a), showing the reappearance of the 52 kDa S-layer protein. Lane A corresponds to untreated whole cells; lane X corresponds to sample X.

  Not determined.

Fig. 6 .

 6 Fig. 6. Freeze-dried preparations of whole cells labelled with polycationic femtin (PCF). (a) Untreated cells, (b) LiC1-treated cells, and (c, d) stationary phase cells J and K (Fig. 5 4 , respectively. The reappearance of the Slayer is shown by the concomitant decrease of the number of bound PCF molecules. Bars, 0.1 pm.

  (Baddiley & Davison, 1961; Ikawa & Snell, L. helveticus ATCC 12046 cells, untreated ( -)

  Part of this work was done during a visit by S. L. to the Zentrum fiir Ultrastrukturforschung, Vienna, Austria, which was supported by grants from INRA and from the Fonds zur Fiirderung der Wissenschaftlichen Forschung and Bundesministerium fiir Wissenschaft und Forschung in Osterreich. The kindness and excellent technical assistance of Ms Andrea Sheberl and Mrs Jutta Frank were particularly appreciated by S. L., as well as the numerous interesting discussions about cheese, bacteria and humans with Seta Kii#.

Table 2 . Amino acid composition of the S-layer protein

 2

	Amino acid	Molar ratio (%)
	Asx Ala	14-15 11.4
	Thr	11.1
	LYS Ser GlY Val	10.9 9.5 8-75 7.9
	Glx	7.35
	Ile
	Pro
	Leu
	TYr Phe
	A% His
	Met
	CYS TW