Dawen Wu
email: dawen.wu@centralesupelec.fr

Abdel Lisser
email: abdel.lisser@l2s.centralesupelec.fr

A dynamical neural network for solving stochastic two-player zero-sum games

Keywords: Stochastic Two-player Zero-sum Game, Saddle point, dynamical neural network

This paper presents a dynamical neural network approach to solve stochastic two-players zero-sum game problems. The original problem is first transformed into an equivalent convex second-order cone programming problem. We develop a dynamical neural network model to solve the problem, where the model's equilibrium point corresponds to the optimal solution of the game problem. Further, we use a Lyapunov function to show that the equilibrium point of the neural network is globally asymptotically stable. Numerical results are given to show the performance of our approach.

Introduction

In [START_REF] Von Neumann | Zur theorie der gesellschaftsspiele[END_REF][START_REF] Von Neumann | Zur theorie der gesellschaftsspiele[END_REF] studies the equilibrium concept in game theory and showed that there always exists a saddle point equilibrium for a finite action two-player zero-sum games. [START_REF] Nash | Equilibrium points in n-person games[END_REF] generalizes this result to n-player games, and shows that there always exists a Nash equilibrium for a finite action general games with a finite number of players. [START_REF] Charnes | Constrained games and linear programming[END_REF] studies a new type of two-player zero-sum game where the mixed strategies of each player are constrained by linear inequalities. He shows that a saddle point equilibrium of such games can be obtained by solving a primal-dual pair of linear programs. Recently, [START_REF] Singh | A second-order cone programming formulation for two player zero-sum games with chance constraints[END_REF] study a stochastic version of two-player zero-sum game, namely chance constrained two-player zero-sum games. They show that a mixed strategy saddle point exists if the random vectors defining stochastic linear constraints follow elliptically symmetric distributions. As regards the solution of the problem, the saddle point of the two-player zero-sum game can be solved by linear programming. A two-player general sum game can be formulated as a linear complementarity problem (LCP), and [START_REF] Lemke | Equilibrium points of bimatrix games[END_REF] gives method to solve it. For n-player general sum cases, [START_REF] Daskalakis | The complexity of computing a nash equilibrium[END_REF] show that the problem of computing nash equilibriums is PPAD-Hard . [START_REF] Van Der Laan | Simplicial variable dimension algorithms for solving the nonlinear complementarity problem on a product of unit simplices using a general labelling[END_REF]; [START_REF] Govindan | A global newton method to compute nash equilibria[END_REF]; [START_REF] Blum | A continuation method for nash equilibria in structured games[END_REF] provide algorithms to solve such problems. [START_REF] Hopfield | neural" computation of decisions in optimization problems[END_REF] bridge neural networks and optimization problems. Since then, many neural network models are proposed to solve varied optimization problems, e.g., linear programming by [START_REF] Wang | Analysis and design of a recurrent neural network for linear programming[END_REF]; Xia (1996b), quadratic programming by Xia (1996a); [START_REF] Nazemi | A neural network model for solving convex quadratic programming problems with some applications[END_REF], nonlinear programming problems by [START_REF] Kennedy | Neural networks for nonlinear programming[END_REF]; [START_REF] Forti | Generalized neural network for nonsmooth nonlinear programming problems[END_REF]; [START_REF] Nazemi | A high performance neural network model for solving chance constrained optimization problems[END_REF], minimax problem by [START_REF] Gao | A neural network for a class of convex quadratic minimax problems with constraints[END_REF]; [START_REF] Nazemi | A dynamical model for solving degenerate quadratic minimax problems with constraints[END_REF], nonlinear complementarity problems by [START_REF] Liao | Solving nonlinear complementarity problems with neural networks: a reformulation method approach[END_REF]; [START_REF] Dang | Neural networks for nonlinear and mixed complementarity problems and their applications[END_REF]; [START_REF] Nazemi | A new neural network framework for solving convex second-order cone constrained variational inequality problems with an application in multi-finger robot hands[END_REF]; [START_REF] Feizi | Solving the stochastic support vector regression with probabilistic constraints by a high-performance neural network model[END_REF]. The neural network model can be reformulated as a first order ordinary differential equation (ODE).

With regards to the solution of an ODE system with a given initial value, Runge-Kutta and backward differentiation approches are commonly used, see [START_REF] Curtiss | Integration of stiff equations[END_REF]; [START_REF] Gottlieb | Total variation diminishing runge-kutta schemes[END_REF].

Python and Julia provide tools implementing such algorithms, see [START_REF] Virtanen | Scipy 1.0: fundamental algorithms for scientific computing in python[END_REF]; [START_REF] Rackauckas | Differentialequations.jl-a performant and feature-rich ecosystem for solving differential equations in julia[END_REF]. Futhermore, with the rapid development of machine learning, [START_REF] Chen | Neural ordinary differential equations[END_REF]; [START_REF] Raissi | Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[END_REF] study deep neural network to handle ODE systems.

The remaining of the paper is organized as follows. Section 2 shows the stochastic zero-sum game together with its second-order cone programming problem (SOCP, for short) reformulation. Section 3 gives the KKT conditions related to the SOCP problem, and the neural network reformulation. Numerical experiments are given in Section 4.

The following notations are used in the remainder of the paper.

• x and y denote the strategies of player1 and player2 respectively. m and n denote the sizes of the action set of the player1 and the player2, respectively.

• J 1 , J 2 denote the probabilistic constraints sets for player1 and player2, respectively. J 1 , J 2 denote the sets sizes.

• A, µ 1 , µ 2 , Σ 1 , Σ 2 , b, d denotes the data for a stochastic two-players zero-sum game. A denotes the payoff matrix. µ 1 , µ 2 , Σ 1 , Σ 2 are the means and the variances of the probability distributions, respectively. ϕ 1 k t 2 and ϕ 2 l t 2 are the characteristic functions of the probability distributions, respectively.

• α 1 and α 2 are the setting of the confidence level of the players 1 and 2, respectively.

• The Ψ -1 ξ 2 l α 2 l , Ψ -1 ξ 1 k α 1
k are quantile functions of 1-dimensional distribution functions induced by characteristic functions ϕ 1 k t 2 and ϕ 2 l t 2 , respectively.

• s = (y, v, δ, λ) are the decision variables of the optimization problem. u is the dual variable of the optimization problem. r = (s, u) = (y, v, δ, λ, u) are the variables of the neural network.

• nr, ns, nu are the number of variables r, s, u, respectively. Moreover, nu also denotes the number of constraints of the optimization problem.

• f (s) = f (y, v, δ, λ) and g(s) = g(y, v, δ, λ) are the objective function and the constraints of the optimization problem. f (s), g(s), ∇f (s), ∇g(s), ∇ 2 g(s) are abbreviated as f, g, ∇f, ∇g, ∇ 2 g.

• Φ(r) = dr dt denotes the dynamical neural network.

Problem formulation

In this section, we present the stochastic two-player zero sum game with a chance constraint. A two player zero-sum game involves two persons called player 1 and player 2. These games are described by a matrix A with m rows and n columns. Matrix A represents the payoffs of player 1 and matrix -A represents the payoffs of player 2. Let I = {1, 2, . . . , m} be the action set of player 1 and J = {1, 2, . . . , n} be the action sets of player 2. We call pure strategies the actions in sets I and J for the players 1 and 2, respectively. We call mixed strategies of a given player, a probability distribution defined over his action set. Let

X = {x ∈ R m | i x i = 1, x i ≥ 0, ∀i ∈ I} and Y = {y ∈ R m | j y j = 1
, y j ≥ 0, ∀j ∈ J} the sets of mixed strategies of player 1 and player 2, respectively. The payoffs of player 1 and player 2 are defined by x T Ay and x T (-A)y respectively, for a given strategy pair (x, y) ∈ X × Y . von Neumann (1928) showed that there exists a saddle point equilibrium in mixed strategies in zero-sum games. [START_REF] Dantzig | A proof of the equivalence of the programming problem and the game problem. Activity analysis of production and allocation[END_REF] showed that the saddle point equilibrium is a solution of primal-dual pair of linear programs. [START_REF] Charnes | Constrained games and linear programming[END_REF] studied a linear constrained two-player zero-sum game problem. For a given a strategy y of player 2, the objective of player 1 is to choose a strategy x which solves the linear programming problem (1). Similarly, the aim of player 2 is to choose a strategy y that solves problem (2) for given a strategy x of player 1.

max x x T Ay s.t. Bx ≤ b 1 T x = 1 x ≥ 0, (1)
min

y x T Ay s.t. Dy ≥ d 1 T y = 1 y ≥ 0, (2)
where

B ∈ R J1×m , D ∈ R J2,n , b ∈ R J1 , d ∈ R J2 .
A strategy pair (x, y) is said to be a saddle point equilibrium for the above constrained zero-sum game if x is an optimal solution of (1) for a given y, and y is an optimal solution of (2) for the given x.

Singh and Lisser (2019) consider the problem where each row vector B k and D l of B and D , respectively, follows an elliptical distribution i.e.

B w k ∼ Ellip m µ 1 k , Σ 1 k , ϕ 1 k and D w l ∼ Ellip n µ 2 l , Σ 2 l , ϕ 2 l . The 55 Ψ -1 ξ 2 l α 2 l , Ψ -1 ξ 1 k α 1
k are the quantile functions of 1-dimensional distribution functions induced by characteristic functions ϕ 1 k t 2 and ϕ 2 l t 2 , respectively. The chance constrained optimization problem can be written as max x x T Ay s.t.

P {B w k x ≤ b k } ≥ α 1 k , ∀k ∈ J 1 1 T x = 1 x ≥ 0, (3) min y
x T Ay s.t.

P {D w l y ≥ d l } ≥ α 2 l , ∀l ∈ J 2 1 T y = 1 y ≥ 0. ( 4
)
We use the SOCP reformulation from [START_REF] Henrion | Structural properties of linear probabilistic constraints[END_REF] to rewrite the probabilistic constraints in (3) and

(4) as follows

x T µ 1 k + Ψ -1 ξ 1 k (α 1 k ) (Σ 1 k ) 1 2 x ≤ b k , ∀k ∈ J 1 , (5)
-y T µ 2 l + Ψ -1 ξ 2 l (α 2 l ) (Σ 2 l ) 1 2 y ≤ -d l , ∀l ∈ J 2 . ( 6
)
We denote the stochastic two-players zero-sum game by G(α) and the feasible strategy sets of the two players by S 1 α 1 and S 2 α 2 ,

S 1 α 1 = x ∈ R m | 1 T x = 1, x ≥ 0, x T µ 1 k + Ψ -1 ξ 1 k (α 1 k ) (Σ 1 k ) 1 2 x ≤ b k , ∀k ∈ J 1 , (7)
S 2 α 2 = y ∈ R n | 1 T y = 1, y ≥ 0, -y T µ 2 l + Ψ -1 ξ 2 l (α 2 l ) (Σ 2 l ) 1 2 y ≤ -d l , ∀l ∈ J 2 . ( 8
)
Assumption 1.

1. The set S 1 (α 1 ) is strictly feasible, i.e., there exists an x ∈ R m which is a feasible point of S 1 (α 1 ) and 60 the inequality constraints of S 1 (α 1 ) are strictly satisfied by x.

2. The set S 2 (α 2 ) is strictly feasible, i.e., there exists an x ∈ R n which is a feasible point of S 2 (α 2 ) and the inequality constraints of S 2 (α 2 ) are strictly satisfied by y.

(x * , y * ) is called a saddle point equilibrium of G(α), if the following inequality holds:

x T Ay * ≤ x * T Ay * ≤ x * T Ay, ∀x ∈ S 1 α 1 , y ∈ S 2 α 2 . ( 9
)
The following theorem shows the saddle point existence of the stochastic two-player zero-sum game problem.

Theorem 1 [START_REF] Singh | A second-order cone programming formulation for two player zero-sum games with chance constraints[END_REF], Theorem 3.5). Consider a constrained zero-sum matrix game where the matrices B w and D w defining the constraints of both the players, respectively, are random. Let the row

vectors B w k ∼ Ellip m µ 1 k , Σ 1 k , ϕ 1 k , k ∈ J 1 , and D w l ∼ Ellip n µ 2 l , Σ 2 l , ϕ 2 l , l ∈ J 2 . For all k and l, Σ 1 k 0 and Σ 2 l 0.
Then, there exists a saddle point equilibrium for the game G(α) for all α ∈ (0.5, 1] J1 × (0.5, 1] J2 .

We refer the reader to [START_REF] Singh | A second-order cone programming formulation for two player zero-sum games with chance constraints[END_REF] for more details about the proof of this theorem and the related results.

Proposition 1. The chance constrained optimization problems (3) and (4) can be reformulated as the following SOCP problems (P) and (D).

min y,v 1 ,(δ 1 k ) k∈J 1 ,λ 1 v 1 + k∈J1 λ 1 k b k s.t. (i)Ay -k∈J1 λ 1 k µ 1 k -k∈J1 Σ 1 k 1 2 δ 1 k ≤ v 1 1 m (ii) -y T µ 2 l + Ψ -1 ξ 2 l α 2 l Σ 2 l 1 2 y ≤ -d l , ∀l ∈ J 2 (iii) δ 1 k ≤ λ 1 k Ψ -1 ξ 1 k α 1 k , ∀k ∈ J 1 (iv)1 T y = 1 (v)y ≥ 0 (vi)λ 1 k ≥ 0, ∀k ∈ J 1                                                  (P) max x,v 2 ,(δ 2 l ) l∈J 2 ,λ 2 v 2 + l∈J2 λ 2 l d l s.t. (i)A T x -l∈J2 λ 2 l µ 2 l -l∈J2 Σ 2 l 1 2 δ 2 l ≥ v 2 1 n (ii)x T µ 1 k + Ψ -1 ξ 1 k α 1 k Σ 1 k 1 2 x ≤ b k , ∀k ∈ J 1 (iii) δ 2 l ≤ λ 2 l Ψ -1 ξ 2 l α 2 l , ∀l ∈ J 2 (iv)1 T x = 1 (v)x ≥ 0 (vi)λ 2 l ≥ 0, ∀l ∈ J 2                                                  (D)
Proof. We show the process that generate (P) from (3).

The chance constrained optimization problem (3) with the second-order cone reformulation (5) is max

x x T Ay s.t. (i)x T µ 1 k + Ψ -1 ξ 1 k (α 1 k ) (Σ 1 k ) 1 2 x ≤ b k , ∀k ∈ J 1 (iii)1 T x = 1 (v)x ≥ 0. ( 10
)
Given a strategy y of player 2, the problem can be written as the following SOCP problem, where (t 1 k ) k∈J1 are auxiliary variables, max

x,(t 1 k ) k∈J 1 x T Ay s.t. (i) -x T µ 1 k -Ψ -1 ξ 1 k (α 1 k ) t k + b k ≥ 0, ∀k ∈ J 1 (ii)t 1 k -(Σ 1 k ) 1 2 x = 0, ∀k ∈ J 1 (iii)1 T x = 1 (iv)x ≥ 0 (11)
The saddle point of the lagrangian of ( 11) is min

v 1 ,(δ k ) 1 k∈J 1 ,λ 1 ≥0 max x,(t 1 k ) k∈J 1 x T Ay + v 1 1 T x + k∈J1 δ 1 k T t 1 k -(Σ 1 k ) 1 2 x + k∈J1 λ 1 k -x T µ 1 k -Ψ -1 ξ 1 k α 1 k t k + b k (12) For the fixed v 1 , (δ k ) 1 k∈J1 , λ 1 , we have max x,(t 1 k ) k∈J 1 x T Ay + v 1 1 T x + k∈J1 δ 1 k T t 1 k -(Σ 1 k ) 1 2 x + k∈J1 λ 1 k -x T µ 1 k -Ψ -1 ξ 1 k α 1 k t k + b k = max x x T Ay - k∈J1 λ 1 k µ 1 k - k∈J1 Σ 1 k 1 2 δ 1 k -v 1 1 + max (t 1 k ) k∈J 1 k∈J1 δ 1 k T t 1 k -λ 1 k Ψ -1 ξ 1 k α 1 k t 1 k + v 1 + k∈J1 λ 1 k b k (13)
The first and second max problem are unbound unless the following conditions hold,

Ay - k∈J1 λ 1 k µ 1 k - k∈J1 Σ 1 k 1 2 δ 1 k ≤ v 1 1 (14) δ 1 k ≤ λ 1 k Ψ -1 ξ 1 k α 1 k , ∀k ∈ J 1 (15)
The lagrangian dual of the inner maximum problem is given by the following SOCP, min

v 1 ,(δ 1 k ),λ 1 v 1 + k∈J1 λ 1 k b k s.t. (i)Ay - k∈J1 λ 1 k µ 1 k - k∈J1 Σ 1 k 1 2 δ 1 k ≤ v 1 1 m (ii) δ 1 k ≤ λ 1 k Ψ -1 ξ 1 k α 1 k , ∀k ∈ J 1 (iii)λ 1 k ≥ 0, ∀k ∈ J 1 (16)
With the second-order cone constraint (6) for y, we finally get the first SOCP (P).

min y,v 1 ,(δ 1 k ) k∈J 1 ,λ 1 v 1 + k∈J1 λ 1 k b k s.t. (i)Ay -k∈J1 λ 1 k µ 1 k -k∈J1 Σ 1 k 1 2 δ 1 k ≤ v 1 1 m (ii) -y T µ 2 l + Ψ -1 ξ 2 l α 2 l Σ 2 l 1 2 y ≤ -d l , ∀l ∈ J 2 (iii) δ 1 k ≤ λ 1 k Ψ -1 ξ 1 k α 1 k , ∀k ∈ J 1 (iv)1 T y = 1 (v)y ≥ 0 (vi)λ 1 k ≥ 0, ∀k ∈ J 1                                                  (P)
The dual problem (D) can be generated similarly.

The following theorem shows the existence of the saddle point for the chance constrained zero-sum game G(α).

Theorem 2 [START_REF] Singh | A second-order cone programming formulation for two player zero-sum games with chance constraints[END_REF], Theorem 3.7). Consider a constrained zero-sum game where the matrices B w and D w defining the constraints of player 1 and player 2, respectively, are random. Let the row

vector B w ∼ Ellip m µ 1 k , Σ 1 k , ϕ 1 k , k ∈ J 1 ,
where Σ 1 k 0, and the row vector

D w l ∼ Ellip n µ 2 l , Σ 2 l , ϕ 2 l , l ∈ J 2 where Σ 2 l 0.
Let Assumption 1 holds. Then, for a given α ∈ (0.5, 1] p × (0.5, 1] q , (x * , y * ) is a saddle point equilibrium of the game G(α) if and only if there exist v 1 * , δ 1 * k k∈J1 , λ 1 * and v 2 * , δ 2 * l l∈J2 , λ 2 * such that y * , v 1 * , δ 1 * k k∈J1 , λ 1 * and x * , v 2 * , δ 2 * l l∈J2 , λ 2 * are optimal solutions of primal-dual pair of SOCPs (P) and (D), respectively.

We refer the reader to [START_REF] Singh | A second-order cone programming formulation for two player zero-sum games with chance constraints[END_REF] for more details about the proof of this theorem and the related results.

Methodology

In this section, we study a neural network approach to solve the second-order cone programming problem given in section 2. We provide the necessary and sufficient KKT conditions of problem (P). We use a neural network to solve the KKT conditions that the equilibrium point of the neural network is the satisfied KKT point. Then, we study the stability of the equilibrium point by analyzing a Lyapunov function.

90

We transform the equality constraint 1 T y = 1 in (P) into inequality 1 T y -1 ≤ 0, 1 -1 T y ≤ 0. For sake of simplicity, we consider only the primal problem. Denote s = (y, v, δ, λ) = (y, v 1 , (δ 1 k ) k∈J1 , λ 1 ), where

δ = (δ 1 k ) k∈J1 = [δ 1 1 T , . . . , δ 1 J1 T ] T and λ = λ 1 = [λ 1 1 , . . . , λ 1 J1 ] T .
Such that, the optimization problem (P) can be simplified as

min s f (s) s.t. g(s) ≤ 0, ( 17
)
where the objective function f : R ns → R, and the constraints g : R ns → R nu .

KKT conditions

Since the SOCP constraints of g(s) are not differentiable, we introduce the following perturbation = 10 -6 , i.e., s 2 + 2 . Thanks to this smoothness technique, the KKT conditions of the SOCP are necessary and sufficient optimality conditions.

95

The KKT conditions of the SOCP problem (P) are

∇f (s) + ∇g(s) T u = 0 g(s) ≤ 0, u T ≥ 0, u T g(s) = 0 ( 18
)
where the ∇f, u, g, ∇g are as follows

∇f (s) =         ∂f ∂y ∂f ∂v ∂f ∂δ ∂f ∂λ         =         0 1 0 b         (19) u =                  u 1 u 2 u 3 u 41 u 42 u 5 u 6                  (20) g(s) =                  g 1 g 2 g 3 g 41 g 42 g 5 g 6                  =                  Ay -v1 -k∈J1 Σ 1 k 1 2 δ k -k∈J1 λ k µ 1 k (-y T µ 2 l + Ψ -1 ξ 2 l α 2 l Σ 2 l 1 2 y + d l ) l∈J2 ( δ k -λ k Ψ -1 ξ 1 k α 1 k ) k∈J1 1 T y -1 -1 T y + 1 -y -λ                  (21) ∇g(s) =                 
                 =                   A -1 (-(Σ 1 k ) 1 2 ) k∈J1 (-µ 1 k ) k∈J1 (-(µ 2 l ) T + Ψ -1 ξ 2 l (α 2 l ) Σ 2 l 1 2 T Σ 2 l 1 2 y Σ 2 l 1 2 y ) l∈J2 0 0 0 0 0 ( δ k δ k ) k∈J1 (-Ψ -1 ξ 1 k α 1 k ) k∈J1 1 T 0 0 0 -1 T 0 0 0 -I 0 0 0 0 0 0 -I                   (22)
The stationarity, primal feasibility, dual feasibility, and complementary slackness can be written as follows

        0 1 0 b         +                   A -1 (-(Σ 1 k ) 1 2 ) k∈J1 (-µ 1 k ) k∈J1 (-(µ 2 l ) T + Ψ -1 ξ 2 l Σ 2 l 1 2 T Σ 2 l 1 2 y Σ 2 l 1 2 y ) l∈J2 0 0 0 0 0 ( δ k δ k ) k∈J1 (-Ψ -1 ξ 1 k α 1 k ) k∈J1 1 T 0 0 0 -1 T 0 0 0 -I 0 0 0 0 0 0 -I                   T                  u 1 u 2 u 3 u 41 u 42 u 5 u 6                  = 0 (23)                  Ay -v1 -k∈J1 Σ 1 k 1 2 δ k -k∈J1 λ k µ 1 k (-y T µ 2 l + Ψ -1 ξ 2 l α 2 l Σ 2 l 1 2 y + d l ) l∈J2 ( δ k -λ k Ψ -1 ξ 1 k α 1 k ) k∈J1 1 T y -1 -1 T y + 1 -y -λ                  ≤ 0 (24)                  u 1 u 2 u 3 u 41 u 42 u 5 u 6                  ≥ 0 (25)                  u 1 u 2 u 3 u 41 u 42 u 5 u 6                  T                  Ay -v1 -k∈J1 Σ 1 k 1 2 δ k -k∈J1 λ k µ 1 k (-y T µ 2 l + Ψ -1 ξ 2 l α 2 l Σ 2 l 1 2 y + d l ) l∈J2 ( δ k -λ k Ψ -1 ξ 1 k α 1 k ) k∈J1 1 T y -1 -1 T y + 1 -y -λ                  = 0 (26)
The four decision variables of problem (P), namely y, v, δ, λ, have n, 1, J 1 * n, J 1 components, respectively.

The function g is composed of g 1 , g 2 , g 3 , g 41 , g 42 , g 5 , and g 6 , with m, J 2 , J 1 , 1, 1, n, J 1 components, respectively.

The gradient ∇f is a (J 1 + 1) * (n + 1)-vector. The jacobian ∇g is (2 + m + n + 2 * J 1 + J 2 ) × (J 1 + 1) *

(n + 1)-matrix. 100

Neural network model

We now propose a neural network model with a given initial value. Let r = (y, v, δ, λ, u) T be the variables of the neural network.

dr dt =            dy dt dv dt dδ dt dλ dt du dt            =            -∇f y + ∇g T y (u + g) + -∇f v + ∇g T v (u + g) + -∇f δ + ∇g T δ (u + g) + -∇f λ + ∇g T λ 1 (u + g) + (u + g) + -u            (27) r (t 0 ) = r 0 (28)
The complexity for solving the neural network ( 27) is highly dependent on the number variable. The number of the decision variables y, v, δ, λ is ns = (J 1 + 1) * (n + 1), and the number of the dual variables µ

105 is nu = 2 + m + n + 2 * J 1 + J 2 , leading to a total number of variables nr = 3 + m + 2N + (3 + n) * J 1 + J 2
for the neural network.

Theorem 3. The point r * = (y * , v * , δ * , λ * , u * ) T is the equilibrium point of the neural network (27) if and only if it is also the KKT point of the SOCP problem.

Proof. Let r * = (y * , v * , δ * , λ * , u * ) T be the equilibrium of the neural network ( 27). It follows that dr * dt = 0,

-∇f * y + ∇g T y * (u * + g * ) + = 0 -∇f * v + ∇g T v * (u * + g * ) + = 0 -∇f * δ + ∇g T δ * (u * + g * ) + = 0 -∇f * λ + ∇g T λ * (u * + g * ) + = 0 (u * + g * ) + -u * = 0 (29)
Substituting the first four lines by the last line u * = (u * + g * ) + , we have

-∇f * y + ∇g T y * u * = 0 -∇f * v + ∇g T v * u * = 0 -∇f δ + ∇g T δ * u * = 0 -∇f * λ + ∇g T λ * u * = 0, (30)
where the KKT conditions stationarity holds. Moreover, u * = (u * + g * ) + result in

g * ≤ 0, u * ≥ 0, u * T g * = 0, (31)
where the primal feasibility, the dual feasibility and the complementary slackness hold.

110

Conversely, let r * = (y * , v * , δ * , λ * , u * ) be the kkt point of the problem (P), then we have

∇f * y + ∇g T y * u * = 0 ∇f * v + ∇g T v * u * = 0 ∇f δ * + ∇g T δ * u * = 0 ∇f * λ + ∇g T λ * u * = 0, (32)
g * ≤ 0, u * ≥ 0, u * T g * = 0. (33)
Conditions (33) lead to u * = (u * + g * ) + . By substituting this into (32), we obtain

-∇f * y + ∇g T y * (u * + g * ) + = 0 -∇f * v + ∇g T v * (u * + g * ) + = 0 -∇f * δ + ∇g T δ * (u * + g * ) + = 0 -∇f * λ + ∇g T λ * (u * + g * ) + = 0 (u * + g * ) + -u * = 0 (34)
which is the equilibrium point of the neural network.

Stability analysis

In this subsection, we study the uniqueness and the stability of the equilibrium point.

Lemma 4. The equilibrium point of the proposed neural network (27) is unique.

Proof. Since the problem (P) has the unique optimal solution (y * , v * , δ * , λ * ), the necessary and sufficient KKT conditions ( 18) have the corresponding unique solution. From Theorem 3, we see that the equilibrium point of the proposed neural network is a necessary and sufficient condition for being a KKT point (18).

Thus the equilibrium point of the neural network is unique.

Lemma 5. For an initial value problem ( 27) and (28), there exists a unique continuous solution r(t).

Proof. Since g, ∇f and ∇g are locally Lipschitz continuous, and the operations +, •, (•) + would not change the locally Lipschitz property, such that ∇f + ∇g T (u + g) + and (u + g) + -u are locally Lipschitz continuous.

According to the Cauchy-Lipschitz theorem, the neural network ( 27) with an initial point r(t 0 ) = r 0 has a unique solution r(t), t ∈ [t 0 , T ), for some T > t 0 . Additionally, if r(t) is globally bound, the interval [t 0 , T ] expand to [t 0 , +∞).

Lemma 6. The Jacobian matrix ∇Φ(r), ∀r ∈ R nr is a negative semidefinite matrix.

Proof. We separate the situations into three cases, depending on the different status of (u + g) + ∈ R nu + , and show under all three situations ∇Φ(r) is negative semidefinite matrix.

For the case where (u + g) + has zero and non-zero components, such that 0 < p < nu

(u + g) + = (u 1 + g 1 , . . . , u p + g p , p 0, . . . , 0 nu-p ), (35)
the jacobian matrix ∇Φ(r), ∀r ∈ R nr is

∇Φ(r) =   -∇ 2 f + p k=1 (u k + g k ) ∇ 2 g p k + ∇g pT ∇g p -∇g pT ∇g p S   , (36)
where

S =   0 p×p 0 p×(nu-p) 0 (nu-p)×p -I (nu-p)×(nu-p)   . ( 37
)
∇g and ∇ 2 g k denote the Jacobian matrix of function g and the Hessian matrix of function g k . ∇g p and ∇ 2 g p k are the same as ∇g and ∇ 2 g k for first p row but the remaining nu -p row are all zero.

Matrix ∇g pT ∇g p is positive semidefinite. Since the functions f and g are assumed to be convex and twice differentiable, the Hessian matrices ∇ 2 f and ∇ 2 g k , k = 1, 2, . . . , p, are positive semidefinite matrices. Furthermore, the positive semidefiniteness of ∇ 2 g k implies that ∇ 2 g p k is positive semidefinite matrix. Moreover, it is clear that matrix S is negative semidefinite matrix. Putting those all together, the Jacobian matrix ∇Φ 135 is a negative semidefinite matrix.

For the case where (u + g) + has all non-zero components, such that p = nu

(u + g) + = (u 1 + g 1 , . . . , u nu + g nu ), (38)
the jacobian matrix ∇Φ(r) is

∇Φ(r) =   -∇ 2 f + m k=1 (u k + g k ) ∇ 2 g k + ∇g T ∇g -∇g T ∇g 0 nu×nu   , (39)
Similar to the previous case, it is to see that ∇Φ(r) is a (ns + nu) × (ns + nu) negative semidefinite matrix.

For the case where (u + g) + has all zero components, such that p = 0

(u + g) + = (0, . . . , 0), (40)
the jacobian matrix ∇Φ(r) is

∇Φ(r) =   -∇ 2 f 0 ns×nu 0 nu×ns -I nu×nu   . (41)
In this case also, it is easy to see that ∇Φ(r) is a negative semidefinite matrix. This completes the proof.

Definition 1. A mapping F : R n → R n is said to be monotonic if:

(x -y) T (F (x) -F (y)) ≥ 0, ∀x, y ∈ R n (42)
Lemma 7 [START_REF] Ortega | Iterative solution of nonlinear equations in several variables[END_REF]). A differentiable mapping F : R n → R n is monotonic, if and only if the jacobian matrix ∇F (x), ∀x ∈ R n is positive semidefinite.

By lemma 7, we have

(r -r * ) T (Φ(r) -Φ (r * )) = (r -r * ) T Φ(r) ≤ 0, ∀r = r * . ( 47
)
This means that Ė(r(t)) ≤ 0.

According to Lyapunov globally stable theorem, the equilibrium r * of the neural network ( 27) is globally stable. Moreover, it follows from ( 27), ( 28), ( 46) and ( 47), that Φ(r) = 0 ⇔ Ė(r) = 0, which means Ė(r) = 0 is true only for the equilibrium point, such that Ė(r) is a negative definite function. Therefore, the equilibrium point of the neural network is globally asymptotically stable.

Numerical Experiments

In this section, we provide numerical results for using the neural network to solve the stochastic twoplayer zero-sum game problem. We use Python3.8, Scipy1.6, Numpy1.20 softwares run on i7-10610U, 1.8GHz

processor with 16 GB RAM to solve our different game problems. We use the following method to study the quality of the KKT point.

Definition 2. The point (s, u) is an approximate KKT point with error if it satisfies

∇f (s) + ∇g (s) T µ ≤ |min {-g i (s) , u i }| ≤ ∀i = 1, . . . , nu g (s) + ≤ (48)
We uniformly generate the data of the game, namely, A, b, d, µ 1 , µ 2 , i.e., A ∼ U (0, 10), b ∼ U (7, 10), d ∼ U (3, 6), µ 1 ∼ U (0, 10), µ 2 ∼ U (0, 10). For the sake of simplicty, we only consider uniformly distributed diagonal matrices Σ 1 and Σ 2 , i.e., Σ 1 , Σ 2 ∼ U (0, 3). By way of illustration, the following is a randomly generated game instance with 4 × 4 payoff matrix. Table 1 shows the experiment results for different game sizes, confidence level α, and probability distributions. Each game size represents the action set size for both players, and each case has linear probability constraints. The data of the games are generated by the above method. We consider the Normal and Laplace distributions, and the confidence levels α are 0.8 or 0,9. The number of iterations of the neural network algorithm is 100.

A =         5
     Σ 1 1 =         1.05 0 0 0 0 0.86 0 0 0 0 2.80 0 0 0 0 1.59         Σ 1 2 =         2.11 0 0 0 0 1.66 0 0 0 0 0.10 0 0 0 0 2.59         Σ 1 3 =         0.24 0 0 0 0 0.55 0 0 0 0 2.32 0 0 0 0 2.38         Σ 2 1 =         0.02 0 0 0 0 2.10 0 0 0 0 2.85 0 0 0 0 1.02         Σ 2 2 =         0
From table 1, we can see that the CPU time depends on the game size. When the game size increases, the number of variables in the neural network increases, which determines the CPU time. Further, the epsilon errors in our experiment are all below 0.06 which shows the high performances of our approach in terms of the quality of the neural network solutions.

Conclusion

In this paper, we studied a neural network model to solve the two-player zero-sum game with stochastic linear constraints problem. We reformulated our problem as an SOCP problem. We show that the equilibrium point of the neural network model is the optimal solution for the original problem. By using the Lyapunov stability theory, we prove the globally asymptotic stability and the uniqueness of the equilibrium point of the proposed neural network. Our numerical experiments show the performance of the neural network to solve

140Theorem 8 .

 8 The equilibrium point r * = (y * , v * , δ * , λ * , u * ) of the proposed neural network (27) is a positive definite function because E(r * ) = 0 and E(r) > 0, ∀r = r * . -r * ) T dr(t) dt = Φ T ∇Φ(r) T + ∇Φ(r) Φ + (r -r * ) r) ∇Φ(r) T + ∇Φ(r) Φ(r) ≤ 0, ∀r = r * .

Table 1 :

 1 Solving stochastic zero sum games by the neural network model large size two-player zero-sum games with up to 200 × 200 payoff matrix within reasonable CPU time.

	Game size	α α 1 α 2	Probability distribution	Neural network model Neural network size CPU time Value epsilon
	4	0.8 0.8 0.9 0.9 0.8 0.8 0.9 0.9	Normal Laplace	39	1.19 0.87 1.14 0.86	3.80 4.00 3.82 4.11	0.01 0.01 0.01 0.00
	10	0.8 0.8 0.9 0.9 0.8 0.8 0.9 0.9	Normal Laplace	75	2.08 2.36 2.11 1.95	4.36 4.46 4.36 4.55	0.06 0.03 0.05 0.01
	50	0.8 0.8 0.9 0.9 0.8 0.8 0.9 0.9	Normal Laplace	315	35.69 36.67 37.14 44.29	5.24 5.29 5.24 5.34	0.05 0.04 0.06 0.02
	100	0.8 0.8 0.9 0.9 0.8 0.8 0.9 0.9	Normal Laplace	615	249.40 345.70 356.80 352.70	5.06 5.00 5.04 5.04	0.02 0.05 0.05 0.04
	200	0.8 0.8 0.9 0.9 0.8 0.8 0.9 0.9	Normal Laplace	1215	8182.92 9202.68 10836.52 7726.93	4.98 5.03 4.99 5.04	0.03 0.03 0.03 0.02


Marie Gilardoni

Daniel Léonço

Fanny Caffin

Fanny Gros-Désormeaux

Camille Eldin

David Béal

Sadia Ouzia

Christophe Junot

François Fenaille

Christophe Piérard

Thierry Douki
email: thierry.douki@cea.fr

Evidence for the systemic diffusion of (2-chloroethyl)-ethyl-sulfide, a sulfur mustard analog, and its deleterious effects in brain

Keywords: 2-Chloroethyl ethyl sulfide, biomarkers, plasma, brain

Introduction

Sulfur mustard (SM), or bis(2-chloroethyl) sulfide, is a chemical warfare agent used on a large scale during World War I [START_REF] Romano | Chemical warfare agents: chemistry, pharmacology, toxicology, and therapeutics[END_REF], and on other battlefields until very recently [START_REF] Sezigen | Recent sulfur mustard attacks in Middle East and experience of health professionals[END_REF]. In addition to these military concerns, SM is a threat for civilians that can be exposed to old ammunitions [START_REF] Missiaen | Evaluation of a chemical munition dumpsite in the Baltic Sea based on geophysical and chemical investigations[END_REF][START_REF] Vanninen | Exposure status of sea-dumped chemical warfare agents in the Baltic Sea[END_REF]. More importantly, SM is thought to be a persistent threat by non-state actors, such as terrorist groups, because of its easy synthesis [START_REF] Wattana | Mustard gas or sulfur mustard: an old chemical agent as a new terrorist threat[END_REF]. Neurological and psychological impacts have also been observed in exposed patients [START_REF] Balali-Mood | Long-term complications of sulphur mustard poisoning in severely intoxicated Iranian veterans[END_REF][START_REF] Darchini-Maragheh | Delayed Neurological Complications of Sulphur Mustard and Tabun Poisoning in 43 Iranian Veterans[END_REF][START_REF] Rowell | The chronic effects of sulfur mustard exposure[END_REF][START_REF] Talabani | Long-term health effects of chemical warfare agents on children following a single heavy exposure[END_REF][START_REF] Thomsen | Chronic neuropathic symptoms after exposure to mustard gas: A long-term investigation[END_REF], suggesting the occurrence of SM-mediated damage to the brain. In vitro experiments have confirmed than neuron were highly sensitive to SM [START_REF] Lanks | Sulfur mustards induce neurite extension and acetylcholinesterase synthesis in cultured neuroblastoma-cells[END_REF][START_REF] Sawyer | Toxicity of sulfur mustard in primary neuron culture[END_REF][START_REF] Turnbull | Induction of axon growth and increased acetylcholinesterase production in mouse neuroblastoma cells by sulfur mustard[END_REF]. Because handling of SM is strictly regulated and banned from most toxicology laboratories by the Organization for the Prohibition of Chemical Weapons (OPCW), other experimental works involved the analog 2-chloroethyl ethyl sulfide (CEES). Neurological effects of CEES confirmed those found for SM in vivo [START_REF] Gadsden-Gray | Induction of neuronal damage in guinea pig brain by intratracheal infusion of 2-chloroethyl ethyl sulfide, a mustard gas analog[END_REF][START_REF] Gros-Desormeaux | Cognitive and emotional impairments after cutaneous intoxication by CEES (a sulfur mustard analog) in mice[END_REF][START_REF] Sharma | Neurobehavioral impairments, generation of oxidative stress and release of proapoptotic factors after chronic exposure to sulphur mustard in mouse brain[END_REF].

One specific feature of SM and CEES is their ability to induce neuropathological disorders and brain damage after dermal exposure on small skin areas. This is partly explained by the fact that vesicant agents readily diffuse through skin, reach the blood stream and then diffuse to internal organs. Presence of SM in blood has been shown by using labeled compounds and quantifying the parent molecule as well as its degradation products [START_REF] Chilcott | Human skin absorption of bis-2-(chloroethyl)sulphide (sulphur mustard) in vitro[END_REF][START_REF] Cullumbine | Medical aspects of mustard gas poisoning[END_REF][START_REF] Meng | Visualization of sulfur mustard in living cells and whole animals with a selective and sensitive turn-on fluorescent probe[END_REF]. Evidence was also provided by the detection of adducts to circulating proteins [START_REF] Chen | A sensitive quantification approach for detection of HETE-CP adduct after benzyl chloroformate derivatization using ultra-high-pressure liquid chromatography tandem mass spectrometry[END_REF][START_REF] Nie | Improvements in monitoring the N-terminal valine adduct in human globin after exposure to sulfur mustard and synthesis of reference chemicals[END_REF][START_REF] Noort | Retrospective detection of sulfur mustard exposure by mass spectrometric analysis of adducts to albumin and hemoglobin: an in vivo study[END_REF][START_REF] Pantazides | A quantitative method to detect human exposure to sulfur and nitrogen mustards via protein adducts[END_REF]. The fact that SM then reaches internal organs, and in particular the brain, has been documented in human by post-mortem quantification of the toxic [START_REF] Drasch | Concentrations of mustard gas [bis(2-chloroethyl)sulfide] in the tissues of a victim of a vesicant exposure[END_REF].

Evidence was also provided by the quantification of SM adducts in nuclear DNA [START_REF] Batal | DNA damage in internal organs after cutaneous exposure to sulphur mustard[END_REF][START_REF] Batal | Temporal and spatial features of the formation of DNA adducts in sulfur mustard-exposed skin[END_REF][START_REF] Brookes | The reaction of mustard gas with nucleic acids in vitro and in vivo[END_REF][START_REF] Fidder | Synthesis, characterization and quantitation of the major adducts formed between sulfur mustard and DNA of calf thymus and human blood[END_REF][START_REF] Yue | Distribution of DNA Adducts and Corresponding Tissue Damage of Sprague-Dawley Rats with Percutaneous Exposure to Sulfur Mustard[END_REF].

Using the latter biomarker, brain was found to be a major target, possibly because of its large fat content. The present work, performed in a hairless mice model, aimed at further understanding the chain of events leading form dermal exposure by vesicant agents to brain damage. We wished to use CEES for which numerous other toxicological data are available.

In particular, we reported that acute dermal exposure to CEES induced a significant increase in anxiety-like reactivity and an alteration of working memory [START_REF] Gros-Desormeaux | Cognitive and emotional impairments after cutaneous intoxication by CEES (a sulfur mustard analog) in mice[END_REF].

We thus wanted to gather novel information on the diffusion step in blood and on the final damage to brain.

These two goals first relied on the quantification of novel biomarkers. SM and CEES are both strong alkylating agents reacting with the nucleophilic sites present in biomolecules. This leads, as already mentioned, to the formation of DNA adducts. Another important target of SM is glutathione (GSH), the most abundant intracellular non-proteic thiol that plays major roles in antioxidant defense and detoxification. SM and CEES are electrophilic agents, which under their episulfonium form, exhibit strong alkylating properties. Like numerous electrophilic compounds, detoxification of SM and CEES occurs via conjugation to GSH. A widely proposed explanation to the toxicity of SM and CEES is their ability to deplete the GSH pool [START_REF] Abel | Protection against 2-chloroethyl ethyl sulfide (CEES) -induced cytotoxicity in human keratinocytes by an inducer of the glutathione detoxification pathway[END_REF]Black et al., 1992a;Black et al., 1992b;[START_REF] Davison | Metabolism of bis-beta-chloroethyl sulfide (sulfur mustard gas)[END_REF][START_REF] Kinsey | Action of mustard gas and other poisons on yeast cells: study of the relationship between inhibition of carbohydrate metabolism and inhibition of growth by various poisons, and effects of other toxic agents on yeast[END_REF] and thereby inducing oxidative stress by reducing cellular antioxidant capacities. This effect is reflected by the modulation of a wide variety of markers of oxidative stress [START_REF] Husain | Effect of topically applied sulphur mustard on antioxidant enzymes in blood cells and body tissues of rats[END_REF][START_REF] Jafari | Dose-and time-dependent effects of sulfur mustard on antioxidant system in liver and brain of rat[END_REF][START_REF] Mukhopadhyay | Modulation of the expression of superoxide dismutase gene in lung injury by 2-chloroethyl ethyl sulfide, a mustard analog[END_REF][START_REF] Paromov | Sulfur mustard toxicity following dermal exposure: role of oxidative stress, and antioxidant therapy[END_REF][START_REF] Pohanka | Sulfur mustard induced oxidative stress and its alteration by epigallocatechin gallate[END_REF][START_REF] Pohanka | Sulfur mustard causes oxidative stress and depletion of antioxidants in muscles, livers, and kidneys of Wistar rats[END_REF][START_REF] Varmazyar | Time course study of oxidative stress in sulfur mustard analog 2-chloroethyl ethyl sulfide-induced toxicity[END_REF]. Evidence of the role of oxidative stress is confirmed by the ability of antioxidants at counteracting the toxic effects of SM [START_REF] Mukherjee | Protection of half sulfur mustard gas-induced lung injury in guinea pigs by antioxidant liposomes[END_REF][START_REF] Paromov | Protective effect of liposome-encapsulated glutathione in a human epidermal model exposed to a mustard gas analog[END_REF][START_REF] Paromov | The influence of N-acetyl-Lcysteine on oxidative stress and nitric oxide synthesis in stimulated macrophages treated with a mustard gas analogue[END_REF][START_REF] Rappeneau | Efficient protection of human bronchial epithelial cells against sulfur and nitrogen mustard cytotoxicity using drug combinations[END_REF][START_REF] Shohrati | Therapeutics effect of Nacetyl cysteine on mustard gas exposed patients: evaluating clinical aspect in patients with impaired pulmonary function test[END_REF][START_REF] Tewari-Singh | Efficacy of Glutathione in Ameliorating Sulfur Mustard Analog-Induced Toxicity in Cultured Skin Epidermal Cells and in SKH-1 Mouse Skin In Vivo[END_REF]. These toxic effects impact glutathione content and are likely to affect the function of many organs exposed to vesicant agents and participate to their systemic effects. The conjugate of CEES to GSH (GSH-CEES) appear thus as an attractive effect biomarker that we quantified in plasma and brain. We also quantified two of its metabolites: the cysteine conjugate (Cys-CEES) and the final product of the mercapturate pathway Nacetyl-cysteine conjugate (NAC-CEES). Data were also gathered on the DNA adduct 2-ethylthioethyl-N7-guanine (N7Gua-CEES), both in nuclei and in plasma.

As mentioned above, although previous studies have shown that toxicity of SM and CEES are associated with inflammation, oxidative stress, and DNA damages, the underlying biochemical mechanisms remain largely ignored. Therefore, we intended to reinforce current hypotheses and shed new light on CEES action mechanisms and effects through novel biomarkers that strongly. First, we targeted molecular biomarkers that like NAC-CEES had never been quantified nor investigated in brain. Second, we applied an extensive and state-of-the art untargeted metabolomics approach for getting deeper insight into underlying biochemical mechanisms. It should be stressed that few metabolomics studies have been published and involved NMR and/or GC-MS analysis of plasma samples from humans or rats exposed to SM [START_REF] Liu | Nuclear magnetic resonance-based plasma metabolomics revealed the protective effect of tea polyphenols on sulfur mustard-induced injury in rats[END_REF][START_REF] Nobakht | NMR-and GC/MS-based metabolomics of sulfur mustard exposed individuals: a pilot study[END_REF][START_REF] Nobakht | NMR spectroscopy-based metabolomic study of serum in sulfur mustard exposed patients with lung disease[END_REF]. Although relevant, those studies provided partially overlapping results in terms of SM-altered metabolic pathways, probably because of the low number of metabolites monitored (less than 50). In the present work, we used an untargeted metabolomics approach based on liquid chromatography coupled to highresolution mass spectrometry that routinely enables the detection of about 200 metabolites in plasma samples [START_REF] Boudah | Annotation of the human serum metabolome by coupling three liquid chromatography methods to high-resolution mass spectrometry[END_REF][START_REF] Moreau | Blood metabolomics uncovers inflammation-associated mitochondrial dysfunction as a potential mechanism underlying ACLF[END_REF][START_REF] Tabone | The effect of acute moderate-intensity exercise on the serum and fecal metabolomes and the gut microbiota of cross-country endurance athletes[END_REF] for getting a more precise view of CEES-associated systemic damages. In addition to support the ability of vesicant at diffusing in blood and reaching brain, which is at the origin of neurological and psychological disorders, an aim of our study was also to validate in animals biomarkers of interest to develop for human exposure.

Materials and methods

Animal experiment

The protocol was validated by the ethic committee of Health Services of French Armed Forces.

SKH-1 hairless mice were purchased from Charles River and used at 7 weeks. Mice were anesthetized by intraperitoneal injection of a mixture of ketamine (90 mg/kg) and xylazine (10 mg/kg). Under a fume hood, mice were then locally exposed through the skin to the vapors emitted by 10 µL of neat CEES using cap method (screw cap for HPLC vial, PTFE septum and polypropylene cap, total volume 300 µ). Based on the known vapor pressure of 3.4 mm Hg for CEES, it can be calculated that this exposure corresponds at the most to 0.4 mg/kg if the totality of the vapors is absorbed by the skin. Four sites on the mice back were exposed to CEES vapors during 30 minutes. After 4h, the skin was decontaminated by soap and water.

To relieve pain induced by CEES, buprenorphine (0.05 mg/kg) was subcutaneously injected after skin decontamination and codeine was daily administrated in drinking water (4 mg/100 mL). One, 3, 7 and 14 days after exposure, the animals (n=16 to 20) were sacrificed by an injection of pentobarbital (150 mg/kg, ip route). The blood was collected by heart puncture.

Plasma was separated by centrifugation (2000 g, 10 minutes, 4°C) and samples were kept frozen until quantification of biomarkers. For this purpose, an aliquot fraction of 100 µl was collected from each sample. The brain of the sacrificed animals was removed by surgery and kept frozen at -80°C until use.

Chemicals and enzymes

Reduced L-glutathione (GSH), (2-chloroethyl)-ethyl sulfide (CEES) and L-cysteine (Cys) were purchased from Sigma Aldrich (Saint Quentin Falavier, France). N-Acetyl-L-cysteine (NAC) was purchased from Roche (Boehringen Mannheim) (Mannheim, Germany).

2'-Deoxyguanosine (dGuo) was purchased from Pharma Waldhof (Düsseldorf, Germany).

Formic acid (LC-MS grade) was purchased from Thermo Fisher Scientific (Rockford, USA), acetonitrile (HPLC-MS grade) from VWR (Fontenay-sous-Bois, France), methanol (HPLC grade) and ammonium formate from Sigma Aldrich (Saint Quentin Falavier, France). Ribonuclease T1, ribonuclease A, phosphodiesterase II, deoxyribonuclease II, alkaline phosphatase and nuclease P1 were obtained from Sigma. Protease was purchased from Qiagen and phosphodiesterase I from Worthington (Lakewood, NJ, USA). Isotopically labelled molecules Cys* ( 13 C3 99%, 15 N 99%), NAC* ( 13 C3 97-99% 15 N 97-99%), GSH* ( 13 C2 98% 15 N 96-99%) and dGuo* ( 15 N5 98%) were purchased from Cambridge Isotope Laboratories (Andover, USA).

Untargeted metabolomics experiments by liquid chromatography coupled to high-resolution mass spectrometry (LC-HRMS)

The metabolomic study was performed according to a protocol previously developed and routinely used in our team, which relies on reverse phase chromatography, hydrophilic interaction liquid chromatography and high resolution mass spectrometry [START_REF] Boudah | Annotation of the human serum metabolome by coupling three liquid chromatography methods to high-resolution mass spectrometry[END_REF][START_REF] Moreau | Blood metabolomics uncovers inflammation-associated mitochondrial dysfunction as a potential mechanism underlying ACLF[END_REF][START_REF] Tabone | The effect of acute moderate-intensity exercise on the serum and fecal metabolomes and the gut microbiota of cross-country endurance athletes[END_REF]. Two 50-μL aliquots were withdrawn from each plasma sample. Each aliquot was then mixed with 200 μL of methanol and incubated on ice for 90 min to allow complete protein precipitation. After centrifugation, the supernatants were withdrawn and evaporated to dryness under a nitrogen stream. Finally, the extracts were resuspended in 100 µL of starting mobile phases of the two chromatographic conditions. Quality control samples were obtained by pooling aliquots of each sample and these were injected every 10 samples throughout the analysis for further data normalization/standardization. Metabolic extracts were analyzed by LC-HRMS using an Ultimate 3000 chromatographic system (Thermo Fisher Scientific, Courtaboeuf, France) coupled to an Exactive mass spectrometer from Thermo Fisher Scientific fitted with an electrospray (ESI) source and operating in the positive and negative ion modes for metabolite separations on C18 and ZIC-pHILIC columns, respectively. LC-HRMS conditions were exactly those previously described by our group [START_REF] Boudah | Annotation of the human serum metabolome by coupling three liquid chromatography methods to high-resolution mass spectrometry[END_REF]. Data processing and statistical analysis were achieved using the Workflow4Metabolomics (W4M) platform [START_REF] Giacomoni | Workflow4Metabolomics: a collaborative research infrastructure for computational metabolomics[END_REF]. Annotation of metabolite features was performed by using our spectral database first according to accurate measured masses and chromatographic retention times [START_REF] Boudah | Annotation of the human serum metabolome by coupling three liquid chromatography methods to high-resolution mass spectrometry[END_REF]. Metabolite identification was further confirmed for discriminant metabolites by running additional LC-MS/MS experiments using a Dionex Ultimate chromatographic system combined with a Q-Exactive mass spectrometer (Thermo Fisher Scientific, San Jose, CA, USA) under non-resonant collision-induced dissociation conditions using higher-energy C-trap dissociation (HCD). To be identified, metabolites had to match at least two orthogonal criteria (among accurate measured mass, retention time, and MS/MS spectrum) to those of an authentic chemical standard analyzed under the same analytical conditions, as proposed by the Metabolomics Standards Initiative [START_REF] Sumner | Proposed minimum reporting standards for chemical analysis Chemical Analysis Working Group (CAWG) Metabolomics Standards Initiative (MSI)[END_REF].

Brain sample preparation

DNA extraction from brain

DNA extraction from the brain was achieved as previously described [START_REF] Ravanat | Cellular background level of 8-oxo-7,8-dihydro-2'deoxyguanosine: an isotope based method to evaluate artefactual oxidation of DNA during its extraction and subsequent work-up[END_REF].

Then, DNA was hydrolyzed in two steps as previously developed in the group [START_REF] Douki | Predominance of the 1,N-2-propano 2 '-deoxyguanosine adduct among 4-hydroxy-2-nonenal-induced DNA lesions[END_REF]. Each step involved a specific enzymatic cocktail: nuclease P1, phosphodiesterase II and DNase II at pH 6 for the first; phosphodiesterase I and alkaline phosphatase at pH 8 for the second. Each step was carried out at 37°C for 2 h. The samples were then left at 90°C for 20 min in a heating block in order to depurinate DNA adducts. The final samples thus contain normal DNA bases under the form of nucleosides, which are quantified by a UV detector placed before the inlet of the mass spectrometer, and adducts as modified bases quantified by mass spectrometry. Samples were filtered prior to analysis at 0.2 µm on filtration column (VWR) centrifuged at 8000xg for 5 min.

Metabolites extraction from brain

The used protocol is based on the work of Malik et al [START_REF] Malik | Extraction and Analysis of Pan-metabolome Polar Metabolites by Ultra Performance Liquid Chromatography-Tandem Mass Spectrometry (UPLC-MS/MS)[END_REF]. In brief, samples were thawed in an ice bath and spiked with isotopically labelled internal standards. A stainless steel bead and 300µL of a cold methanol:chloroform solution were added to each brain sample.

Samples were then homogenised by shaking at 25Hz for 4 min in a TissueLyser (Qiagen).

Then, samples are placed back on ice. Then, 100 µL of cold water was added before vortexing.

Sample were then centrifuged at 18,787xg for 7 min at 4 °C. The upper layer, which contains the polar metabolites of interest, was carefully collected and placed in a new tube to be dried in a SPD111V speed vac (Thermo Scientific). The resulting residue was dissolved in 50 µl of water and filtered at 0.2 µm on filtration column (VWR) centrifuged at 8000xg for 5 min.

On-line solid phase extraction of plasma

An improved version of the previously reported assay for plasma samples [START_REF] Roser | Glutathione conjugates of the mercapturic acid pathway and guanine adduct as biomarkers of exposure to CEES, a sulfur mustard analog[END_REF] was used. Classical solid phase extraction was replaced by on line SPE. Prior to analysis, the samples (90 µL of plasma) were spiked with internal standards (10 µL, 0.2 µM) and loaded onto a 0.2 µm filtration column (VWR). Then, they were centrifuged at 8000 x g for 5 min. The filtrate was collected and loaded onto a 30 kDa Nanosep® tube. This second filtration step involved a centrifugation for 30 min at 8000 x g. Additional sample clean-up was performed directly on-line within the UHPLC system. The samples were injected (50 µL) first in a Macherey-Nagel Nucleodur® PFP column (50 mm x 2.0 mm ID, 5 µm particle size) used as an on-line SPE column. Mobile phase A consisted of 5mM ammonium formate (AmF) in MilliQ water and mobile phase B was HPLC-LC/MS grade methanol with 5mM AmF. Samples were cleaned by a gradient from 0 to 50% of B in 1 minute. The flow rate was set to 350 µL/min, the column temperature maintained at 50°C and the sample storage compartment set to 15°C.

After that, samples are injected by backflush into the analytical column.

UHPLC-MS/MS Quantification of the biomarkers

Quantification by ultra high performance liquid chromatography associates with tandem mass spectrometry (UHPLC-MS/MS) of GSH-CEES, Cys-CEES, NAC-CEES and N7Gua-CEES was carried out using a recently described approach [START_REF] Roser | Glutathione conjugates of the mercapturic acid pathway and guanine adduct as biomarkers of exposure to CEES, a sulfur mustard analog[END_REF]. Chromatographic separations were carried out on an ExionLC system (SCIEX, Framingham, MA) equipped. The on-line SPE part consisted in an autosampler, two pumps and a 2×50 mm ID pentafluorophenyl column (5µm particle size, Nucleodur PFP, Macherey-Nagel). The analytical part involved two pumps, a heated column compartment and the detectors. Analytical separation was performed using a Macherey-Nagel Nucleodur® C18 column (100 mm x 2.0 mm ID, 1.8 µm particle size).

Connection between the on-line SPE and the analytical sub-units was controlled by a 6-ways valve. The mobile phase consisted in a gradient of 2mM AmF and LC/MS grade acetonitrile.

Detection was afforded by a SCIEX QTRAP® 6500+ triple quadrupolar mass spectrometer (Framingham, MA) equipped with an Ion Drive™ Turbo V source operated in positive ionization mode. Data were collected in multiple reaction monitoring (MRM) scan mode using previously reported transitions [START_REF] Roser | Glutathione conjugates of the mercapturic acid pathway and guanine adduct as biomarkers of exposure to CEES, a sulfur mustard analog[END_REF]. Quantification of N7Gua-CEES in DNA also involved on-line UV quantification on unmodified nucleosides in order to determine the amount of analyzed DNA.

Statistical analysis

Statistical analysis were performed on pools of replicates originating from different animals.

Data were statistically analyzed in GraphPad PRISM using one-way ANOVA followed by the non-parametric Kruskal-Wallis test for multiple comparisons. Univariate and multivariate analyses of metabolomics data were also performed using the W4M platform [START_REF] Giacomoni | Workflow4Metabolomics: a collaborative research infrastructure for computational metabolomics[END_REF], while the online MetaboAnalyst platform was used for heatmap generation and metabolic pathway enrichment [START_REF] Chong | MetaboAnalyst 4.0: towards more transparent and integrative metabolomics analysis[END_REF].

Results

Untargeted metabolomics analysis of plasma samples from mice exposed to CEES

Metabolomics analyses of mouse plasma samples were performed using two complementary LC-HRMS platforms involving reversed-phase chromatography [C18(+)]) and Hydrophilic

Interaction Liquid Chromatography [HILIC(-)], for the analysis of hydrophobic and polar metabolites in the positive and the negative ionization modes, respectively. Under these conditions, 57 metabolite features from the C18(+) and 165 from the HILIC(-) analysis matched the accurate masses and retention times of the metabolites included in our chemical database, thus yielding 199 unique annotated metabolites in total. Although principal component analysis (PCA) did not reveal any obvious differences between the 4 post-exposure times (data not shown), a clustered analysis heatmap was generated to group the related metabolites (top 50 ones by ANOVA p-values), and highlighted a time-dependent evolution of metabolite patterns through the 14 days of exposure (Fig. 1A). A small portion of the top 50 metabolites was more abundant at day 1 while most of them accumulated at day 3 and plateauing afterwards, before returning to "baseline" levels at day 14 (i.e. similar profile than non-exposed mice). The metabolic profile at day 1 post-exposure proved significantly different from those at days 3 and 7 (Fig. 1A). Tables S1,S2 and S3 present the metabolites with p-values < 0.05 and fold changes > 1.5 when comparing non-exposed mice to day 1, day 1 to day 3, and day 3 to day 7 post-exposure, respectively. Differences between day 1 and day 3 were the more striking and thus the more precisely studied (Table S2). Under these conditions, 65 and 26 statistically different metabolites were retrieved from HILIC(-) and C18(+) datasets, respectively (Table S1). Metabolic pathway analysis showed that arginine and proline metabolism, arginine biosynthesis, histidine metabolism, purine metabolism and primary bile acid synthesis were among the most affected metabolic pathways (Fig. S1). Figure 2

Targeted CEES biomarkers in blood plasma

Analysis of plasma samples benefited from an efficient and reproducible on-line SPE approach [START_REF] Kuklenyik | Design of online solid phase extraction-liquid chromatography-tandem mass spectrometry (SPE-LC-MS/MS) hyphenated systems for quantitative analysis of small organic compounds in biological matrices[END_REF] that was used to improve our UHPLC-MS/MS method for CEES biomarkers [START_REF] Roser | Glutathione conjugates of the mercapturic acid pathway and guanine adduct as biomarkers of exposure to CEES, a sulfur mustard analog[END_REF]. None of the analytes could be detected in the plasma of untreated animals. The time-course variation of the level of N7Gua-CEES in plasma exhibited a fast decrease following a maximal concentration found at the first investigated time of 1 day post exposure (Table 1). At day 14, the level of adduct was low but still above the limit of detection (LOD). GSH-CEES was detected in most samples but its level was always closed to the LOD and hardly varied with time. In contrast, Cys-CEES and NAC-CEES were found to be present in large concentrations. Their amounts were maximal at day 1 and then readily decreased. The rate of this second phase was slightly lower for Cys-CEES than NAC-CEES.

Detection of NAC-CEES was yet more sensitive and its concentration after 14 days was still 12 times higher than its limit of detection (LOD).

< Table 1 >

Quantification of CEES biomarkers in brain

The molecular biomarkers were then quantified in the brain of exposed mice sacrificed just before the exposure or 1, 3, 7, and 14 days after topical application of CEES. A first analyte of interest was N7Gua-CEES. Its occurrence was first measured in the nuclear DNA extracted from the whole brain (Fig. 3a). No trace of adduct was found before exposure. Over time, the amount of N7Gua-CEES was maximal at day 1 and day 3. The difference between the two values was not statistically significant. After this peak, the value decreases. N7Gua-CEES is though still detectable after 14 days (18.0 ± 6.7 adducts per 10 6 bases) with a sensitivity corresponding to 28 times the value of its LOD. N7Gua-CEES was also detected in the organ's extracts prepared by homogenisation and solvents extraction (Fig. 3a). The N7Gua-CEES concentration was highest at day 1 and then decreased to 10% of this value at day 14.

< Figure 3 >

The concentrations of the three CEES conjugates GSH-CEES, Cys-CEES and NAC-CEES arising from the mercapturate pathway were also determined in the brain extracts (Fig. 3b).

They were found to be in the same range at day 1. This value was the highest of the four investigated days for GSH-CEES and NAC-CEES. A higher concentration was found for Cys-CEES at day 3 but the difference was not statistically different from day 1. A concentration decrease was then observed for the three biomarkers. It was faster for GSH-CEES. Because this analyte exhibited the lowest sensitivity in the analytical method, it was below the limit of quantification in all samples at day 3 and in a large fraction of those at day 7 and 14. In contrast to GSH-CEES, the concentrations of Cys-CEES and NAC-CEES decreased slowly over the 2 weeks of follow-up. They were still detectable at day 14, with a good sensitivity for NAC-CEES corresponding to a factor of 9 with respect to the LOD. The quantification of Cys-CEES at day 14 was hampered by the presence of traces of interfering peaks.

Discussion

In addition to directly observable effects on skin, eyes and lungs, SM exhibits neurological and psychological effects [START_REF] Balali-Mood | Comparison of early and late toxic effects of sulfur mustard in Iranian veterans[END_REF][START_REF] Isono | Long-term neurological and neuropsychological complications of sulfur mustard and Lewisite mixture poisoning in Chinese victims exposed to chemical warfare agents abandoned at the end of WWII[END_REF][START_REF] Sharma | Neurobehavioral impairments, generation of oxidative stress and release of proapoptotic factors after chronic exposure to sulphur mustard in mouse brain[END_REF].

However, most of the related observational studies were performed on veterans of Iran-Irak war (1980-88), many years after their initial exposure. It is thus not possible to distinguish between the toxic effects of SM itself on the one hand, and war stress on the other hand.

Similarly, two non-exclusive origins for the neurological and pathological effects of SM and vesicants can be proposed. The first is a response to the exposure stress while the second is a diffusion of SM and CEES through the exposed skin followed by direct and indirect diffusion to the brain with resulting local effects. Thus, our group carried out an in vivo study on hairless SKH-1 mice topically exposed to CEES, the SM analog used in the present study, in order to clarify the origin of the behavioral effects observed in humans [START_REF] Gros-Desormeaux | Cognitive and emotional impairments after cutaneous intoxication by CEES (a sulfur mustard analog) in mice[END_REF].

In this study, CEES-treated mice were compared with a control group burnt with hydrochloric acid. Our results evidenced that the anxiety and long-lasting working memory impairments observed in mice were due to the vesicant agent itself, and not to the burn effect or to the war stress. This conclusion is actually strongly supported by the observation that brain was one of the organs exhibiting the largest concentration of SM in a post-mortem study on an exposed Iranian soldier [START_REF] Drasch | Concentrations of mustard gas [bis(2-chloroethyl)sulfide] in the tissues of a victim of a vesicant exposure[END_REF].

Explanation to the effects of vesicants in brain can be found in a few studies showing the induction of apoptosis and oxidative stress in the brain of mice chronically exposed to SM by percutaneous route [START_REF] Sharma | Neurobehavioral impairments, generation of oxidative stress and release of proapoptotic factors after chronic exposure to sulphur mustard in mouse brain[END_REF]. Histological and immunohistochemical analyses of brains of guinea pigs exposed to CEES by intratracheal injection showed a loss of neural cells in all brain regions, likely resulting from an activation of microglial cells by -synuclein [START_REF] Gadsden-Gray | Induction of neuronal damage in guinea pig brain by intratracheal infusion of 2-chloroethyl ethyl sulfide, a mustard gas analog[END_REF]. In spite of these biochemical modifications, no major tissue damage was reported in the brain of rats topically exposed to large doses of SM (up to 45 mg/kg) with the exception of haemorrhage [START_REF] Yue | Distribution of DNA Adducts and Corresponding Tissue Damage of Sprague-Dawley Rats with Percutaneous Exposure to Sulfur Mustard[END_REF]. These results unambiguously

show the presence of SM or CEES in brain of treated animals. However, they do not provide quantitative information on the kinetic of diffusion from the exposed site or on the distribution between internal organs. This is the reason why quantification of exposure and effect biomarkers represent an interesting complementary approach. Evidence that SM and its analogs diffuse in the blood stream is shown by the detection of its degradation products [START_REF] Li | Simultaneous quantification of seven plasma metabolites of sulfur mustard by ultra high performance liquid chromatography-tandem mass spectrometry[END_REF][START_REF] Manandhar | Rapid analysis of sulfur mustard oxide in plasma using gas chromatography-chemical ionization-mass spectrometry for diagnosis of sulfur mustard exposure[END_REF][START_REF] Qi | Simultaneous determination of sulfur mustard and related oxidation products by isotope-dilution LC-MS/MS method coupled with a chemical conversion[END_REF] and of protein adducts in plasma [START_REF] John | Forensic evidence of sulfur mustard exposure in real cases of human poisoning by detection of diverse albumin-derived protein adducts[END_REF][START_REF] Liu | An improved method for retrospective quantification of sulfur mustard exposure by detection of its albumin adduct using ultra-high pressure liquid chromatography-tandem mass spectrometry[END_REF][START_REF] Noort | Retrospective detection of sulfur mustard exposure by mass spectrometric analysis of adducts to albumin and hemoglobin: an in vivo study[END_REF][START_REF] Xu | Four sulfur mustard exposure cases: Overall analysis of four types of biomarkers in clinical samples provides positive implication for early diagnosis and treatment monitoring[END_REF]. SM or CEES may also react in plasma with small molecules such as glutathione, which leads to the presence of conjugates in blood. The GSH conjugates may also be produced within cells by detoxification enzymes glutathione-S-transferases [START_REF] Sheehan | Structure, function and evolution of glutathione transferases: implications for classification of non-mammalian members of an ancient enzyme superfamily[END_REF] and further excreted into biological fluids.

Quantification of such biomarkers in plasma would thus reflect a systemic effect of vesicant agents because their presence in blood would originate from all organs.

Therefore, our result confirm both the diffusion of CEES from skin to blood and the subsequent exposure of internal organs. We obtained data not only on GSH-CEES, but also on Cys-CEES and NAC-CEES, two metabolites of GSH-CEES resulting from the action of several enzymes [START_REF] Hanna | The mercapturic acid pathway[END_REF][START_REF] Wu | Glutathione metabolism and its implications for health[END_REF]. We recently developed a method for their quantification and validated their biological relevance [START_REF] Roser | Glutathione conjugates of the mercapturic acid pathway and guanine adduct as biomarkers of exposure to CEES, a sulfur mustard analog[END_REF]. Using an improved version of this approach, we detected large amounts of GSH-CEES metabolites in the plasma transporters [START_REF] Griffith | The Role of Glutathione Turnover in the Apparent Renal Secretioonf Cystine[END_REF][START_REF] Hanigan | γ-Glutamyl transpeptidase, a glutathionase: Its expression and function in carcinogenesis[END_REF][START_REF] Hughey | Specificity of a particulate rat renal peptidase and its localization along with other enzymes of mercapturic acid synthesis[END_REF]. There, the Cys-S-conjugates are acetylated by the N-acetyl-transferase NAT8 (Chambers et al., 2010).

Another interesting observation was the confirmation of the presence of the DNA adduct N7Gua-CEES in plasma samples [START_REF] Roser | Glutathione conjugates of the mercapturic acid pathway and guanine adduct as biomarkers of exposure to CEES, a sulfur mustard analog[END_REF]. This modified nucleic acid base may have several origins. The most likely is the depurination from DNA, which a characteristic of N7-alkylated guanine adducts [START_REF] Boysen | The formation and biological significance of N7-guanine adducts[END_REF]. Likewise, N7Gua-CEES could originate from RNA and the pool of nucleotides. These results on N7Gua-CEES are in line with those obtained with SM [START_REF] Zhang | Simultaneous determination of four sulfur mustard-DNA adducts in rabbit urine after dermal exposure by isotope-dilution liquid chromatography-tandem mass spectrometry[END_REF] and further validate the use of CEES as a relevant analog of SM. In terms of time course, all the detected biomarkers were at their maximal plasma concentration at day 1 and then rapidly decreased. At day 14, the respective level of NAC-CEES and Cys-CEES was 9 and 20% of that at day 1, while the corresponding value was 1 % for N7Gua-CEES. These data confirm results from the literature that SM readily diffuses into blood after dermal exposure [START_REF] Nagy | The penetration of vesicant vapors into human skin[END_REF]. These data also reflect the formation of CEES-mediated damage to biomolecules all-over the mouse organism.

To further study the latter point, we performed an untargeted metabolomics study that enabled the detection of almost 200 unique metabolites in the plasma of exposed mice. These results revealed a systemic alteration of metabolism, involving various metabolic pathways (arginine and proline metabolism, arginine biosynthesis, histidine metabolism, purine metabolism and primary bile acid synthesis,…), with the most striking differences occurring between days 1 and 3 post-exposure (Fig. 1, Fig. S1). Noteworthy is the fact that a few particular metabolites accumulate at day 1, which might reflect a self-protection effect to alleviate the injury of CEES exposure and/or inflammatory response. Among those metabolites, arginine and histidinerelated metabolites (e.g., urocanic acid, anserine, carnosine) are of special interest. Arginine analogues have been demonstrated to exhibit protective characteristics against SM poisoning [START_REF] Rappeneau | Efficient protection of human bronchial epithelial cells against sulfur and nitrogen mustard cytotoxicity using drug combinations[END_REF], while SM analogues were demonstrated to form covalent adducts with histidine and histidine-containing peptides [START_REF] Hemme | Mass spectrometric analysis of adducts of sulfur mustard analogues to human plasma proteins: approach towards chemical provenancing in biomedical samples[END_REF]. Carnosine, a dipeptide composed of β-alanine and L-histidine, and its methylated analog anserine are major constituents within mammalian skeletal muscle that play key roles in muscle contraction/excitation as intracellular buffer and antioxidant [START_REF] Boldyrev | Physiology and pathophysiology of carnosine[END_REF]. The antioxidant activity of carnosine and related peptides has been demonstrated as well as their ability to prevent the formation of advanced lipoxidation end-products and advanced glycoxidation end-products, for instance by forming adducts with reactive lipid oxidation products [START_REF] Boldyrev | Physiology and pathophysiology of carnosine[END_REF]. Urocanic acid, existing as trans and cis isomers, is a breakdown (deamination) product of histidine produced by the catalytic action of histidine ammonia-lyase [START_REF] Walterscheid | Cis-urocanic acid, a sunlight-induced immunosuppressive factor, activates immune suppression via the 5-HT2A receptor[END_REF]. In the epidermis, it accumulates and may be both a UV protectant and an immunoregulator. Trans-urocanic acid upon UV light absorption is isomerized to the cis-isomer in the stratum corneum. Cis-urocanic acid can suppress cellular immunity in mice [START_REF] Hug | The potential role for urocanic acid and sunlight in the immune suppression associated with protein malnutrition[END_REF], and has been identified as a serotonin receptor ligand [START_REF] Walterscheid | Cis-urocanic acid, a sunlight-induced immunosuppressive factor, activates immune suppression via the 5-HT2A receptor[END_REF]). In addition to potential anti-inflammatory effects, recent data demonstrate that urocanic acid also promotes glutamate biosynthesis and release in various brain regions which can impact recognition memory and motor learning in mice [START_REF] Zhu | Moderate UV Exposure Enhances Learning and Memory by Promoting a Novel Glutamate Biosynthetic Pathway in the Brain[END_REF].

Beside, we also observed accumulation of hydroxyglutaric acid and glutaconic acid in the plasma of exposed mice at day 1. This is a feature commonly observed in glutaric aciduria type I caused by inherited deficiency of glutaryl-CoA dehydrogenase which is involved in the catabolic pathways of L-lysine, L-hydroxylysine and L-tryptophan [START_REF] Kolker | Diagnosis and management of glutaric aciduria type I--revised recommendations[END_REF]. In this case, high levels of those organic acids can cause damage to the brain (and also other organs), but particularly the basal ganglia, which are regions that help regulate movement [START_REF] Kolker | Diagnosis and management of glutaric aciduria type I--revised recommendations[END_REF]. Such organic acids can be detoxified by carnitine [START_REF] Strauss | Glutaric acidemia type 1: Treatment and outcome of 168 patients over three decades[END_REF]. Thus, these observations might also be linked with the modulated levels of carnitine species observed upon CEES exposure (Table S1). Last, many other metabolites present at low levels at day 1 proved up-regulated at day 3. Among those are metabolites belonging to purine and pyrimidine metabolism, pentose phosphate pathway, beta-oxidation, etc…, which underlines the broad systemic effect caused by CEES poisoning. Also the strong dysregulation of bile acids (taurocholic and taurodeoxycholic acid) can indicate liver dysfunction and/or an impact on gut microbiota (Łuczykowski et al., 2021). Of note, untargeted metabolomics does not detect CEES adducts due to insufficient detection sensitivity.

Overall, our metabolomics approach provided an unprecedented view on the impact of CEES on mouse metabolism. Metabolite profiling until 14 days post-exposure highlighted a two-wave organism response to CEES poisoning. An early phase response at day 1 that might reflect a self-protection effect through the accumulation of anti-inflammatory nucleophilic compounds precedes a more systematic effect from day 3 involving many different metabolic pathways.

Having confirmed the systemic diffusion of CEES in topically exposed SKH-1 mice, we then focussed on the brain. First, in agreement with previous works on SM [START_REF] Batal | DNA damage in internal organs after cutaneous exposure to sulphur mustard[END_REF][START_REF] Yue | Distribution of DNA Adducts and Corresponding Tissue Damage of Sprague-Dawley Rats with Percutaneous Exposure to Sulfur Mustard[END_REF], we detected large amounts of N7Gua-CEES adducts in the nuclear DNA of this organ. Interestingly, the same adduct was detected in brain extracts aimed at isolating metabolites. Like in plasma, N7Gua-CEES found in these samples likely results from depurination of DNA, RNA or nucleotide pool. The formation DNA adducts is an interesting observation that could partly explain some of the neurological effects of vesicant agents.

Indeed, the formation of large amounts of DNA damage has been proposed as one of the mechanisms leading to cell death [START_REF] Debiak | Role of poly(ADP-ribose) polymerase in sulfur mustard toxicity[END_REF][START_REF] Kehe | Acute effects of sulfur mustard injury-Munich experiences[END_REF][START_REF] Papirmeister | Molecular basis for mustard-induced vesication[END_REF].

We also observed that brain extracts contained the three investigated metabolites resulting from the reaction of CEES with GSH. Like in plasma, the parent molecule in this pathway, GSH-CEES, was present in the lowest amount after the first day of exposure. This observation is reminiscent of those made in in vitro experiment using CEES on human skin cells and skin explants [START_REF] Roser | Glutathione conjugates of the mercapturic acid pathway and guanine adduct as biomarkers of exposure to CEES, a sulfur mustard analog[END_REF]. In contrast, Cys-CEES was the most frequent and more persistent. This clearly illustrates the efficiency of the metabolism of GSH conjugates that aims at maintaining a sufficient cellular content in this essential tripeptide [START_REF] Forman | Glutathione: overview of its protective roles, measurement, and biosynthesis[END_REF][START_REF] Griffith | Biologic and pharmacologic regulation of mammalian glutathione synthesis[END_REF][START_REF] Wu | Glutathione metabolism and its implications for health[END_REF]. NAC-CEES was also unambiguously detected, in amounts representing roughly 75% of that of Cys-CEES. This is slightly unexpected since N-acetyltransferases are believed to be mostly renal and hepatic enzymes (Chambers et al., 2010;[START_REF] Hinchman | Efficient hepatic uptake and concentrative biliary excretion of a mercapturic acid[END_REF]. Evidence from the literature yet suggest that brain also expresses N-acetyltransferases [START_REF] Hanna | The mercapturic acid pathway[END_REF]. Accordingly, Miller et al. [START_REF] Miller | Metabolism of 5-(Glutathione-S-Yl)-Alpha-Methyldopamine Following Intracerebroventricular Administration to Male Sprague-Dawley Rats[END_REF] observed the formation of a mercapturic acid derivative in all brain regions. More recently, the mercapturate pathway activity was also detected in the brain although with a lower efficiency than in liver [START_REF] Sidell | Mercapturate metabolism of 4-hydroxy-2-nonenal in rat and human cerebrum[END_REF]. An additional explanation to the observation of large amounts of NAC-CEES in brain is that this organ receives 15-20% of the cardiac output, attesting to the high vascular demand of the brain. Thus, a part of the detected NAC-CEES could originate from the blood where it is present in large concentration as shown by the measurements in the plasma. A last observation regarding biomarkers in brain is that their decreases more slowly than in plasma, with the exception of GSH-CEES. This may be explained by a longer persistence of CEES in brain than in the rest of the organism because of a larger fat content and the formation of a reservoir compartment.

Conclusion

The present work, based on novel experimental approaches involving untargeted metabolomics and quantification of recently validated specific molecular biomarkers, provides observations relevant to a better understanding of the occurrence of neurological and psychological disorders after exposure to vesicant agents like SM. Our work confirms that, following diffusion through skin, vesicant agents enter the blood stream and cross the hematoencephalic barrier. Then, the compounds enter the brain and induce a series of damaging processes. Our data reveals that the formation and clearance of the studied biomarkers spreads over two weeks, reflecting the persistence of the chemical in the brain.

Our work also shed some light on the systemic effects of CEES, potential underlying biochemical mechanisms and especially the interest of some biomarkers. NAC-CEES and Gua-CEES were found to be those detected with the best LOD in our UHPLC-MS/MS and to persist for more than two weeks in brain and plasma. The last point is important for applications of the UHPLC-MS/MS assay to human. NAC-CEES 41.0±2.9 †, ‡ ‡, § § 17.3±1.1 **, ‡ ‡, § § 3.8±0.5 **, † † 5.7±2. **, † † N7Gua-CEES 34.7±2.9 † †, ‡ ‡, § § 4.2±0.4 *, ‡, § § 0.8±0.1 **, † 0.3±0.1 **, † †

  presents the post-exposure evolution of some key metabolites showing markedly increased or decreased concentration levels between day 1 and day 3, also by comparison to other time points post-exposure and non-exposed animals.

  of the CEES-treated mice in the following decreasing order of concentration: Cys-CEES > NAC-CEES >> GSH-CEES. The observation of a very low concentration in GSH-CEES is in agreement with well identified features of the mercapturate pathway involved in the detoxification of electrophilic species such as CEES or SM. GSH-CEES is converted into CysGly-CEES and Cys-CEES through the consecutive actions of membrane-bound-enzymes  glutamyl-transferase (GGT) and dipeptidase or aminopeptidase-M, respectively[START_REF] Gonçalves-Dias | Mercapturate Pathway in the Tubulocentric Perspective of Diabetic Kidney Disease[END_REF][START_REF] Hanna | The mercapturic acid pathway[END_REF]. The last step of the mercapturate pathway involves the entrance of Cys-S-conjugates into the renal tubular cells and hepatocytes via various transporters including organic anion transport polypeptides and cystine/cysteine

FIGURESFigure 1 :

 1 FIGURES

Figure 2 :

 2 Figure 2: variation of the concentration of a series of plasma metabolites following exposure to CEES. The metabolites underwent either a) decrease or b) increase in their concentration levels. NE: non exposed mice.

Figure 3 :

 3 Figure 3: Levels of biomarkers of exposure to CEES in brains of exposed mice. Brains were collected after sacrifice after different periods. Half of the organ was used for DNA extraction and the other was homogenized in chloroform:methanol and extracted with water. Samples were then analysed by UHPLC-MS/MS with on-line SPE. A) Amount of N7Gua-CEES adducts measured either in the DNA of the organ (expressed in adduct per 10 6 bases) or detected in extract of the whole organ (expressed in fmol/mg tissue) b) Concentration of GSH, Cys and NAC conjugates to CEES in brain extracts (expressed in fmol/mg). The number on animals varied from 16 to 20 per group. The reported values are the means ± SEM. The symbols for statistical significance of the differences at p<0.05 or p<0.01 are: * and ** for comparison with day 1; † and † † for comparison with day 3; ‡ and ‡ ‡ for comparison with day 7; and § and § § for comparison with day 14.

TABLES Table 1 :

 TABLES1 Concentration (in nM) of four biomarkers of exposure to CEES determined in the plasma of mice topically exposed to CEES. Plasma was collected after sacrifice after different periods and analysed by UHPLC-MS/MS with on-line SPE. The number on animals varied from 16 to 20 per group. The reported values are the means ± SEM. The symbols for statistical significance of the differences at p<0.05 or p<0.01 are: * and ** for comparison with day 1; † and † † for comparison with day 3; ‡ and ‡ ‡ for comparison with day 7; and § and § § for comparison

	with day 14.
	day	1	3	7	14
	GSH-CEES	0.8±0.1 §	1.1±0.4	1.3±0.5	0.5±0.3 *
	Cys-CEES	42.9±3.3 ‡ ‡, § §	37.6±3.2 ‡, § §	14.8±4.3 **, †	3.9±2.0 **, † †

Table S1 . Annotated and significant metabolites between non-exposed mice and day 1 after CEES exposure.

 S1 Fold changes > 1.5, p-values < 0.05 (Wilcoxon with Benjamini-Hochberg correction).

	HILIC(-) conditions	Fold change D1/NE	log2(FC)	p.ajusted Identification status	C18(+) conditions	Fold change D1/NE	log2(FC)	p.ajusted	Identification status
	Cyclic AMP		0.1013	-3.3034	0.014461	a, b	Nalpha-Acetyl-L-asparagine	0.01481	-6.0773	0.024113	a, b
	Phosphoenolpyruvic acid		0.1523	-2.715	0.00025799	a, b	Taurocholic-acid	0.05335	-4.2284	0.024113	a, b, c
	GMP		0.15345	-2.7042	0.00041656	a, b, c	Indolelactic-acid	0.34417	-1.5388	0.029765	a, b, c
	Medroxyprogesterone		0.17092	-2.5486	0.00075628	a, b	Nicotinamide	0.42485	-1.235	0.017882	a, b, c
	UMP		0.19061	-2.3913	0.028242	a, b, c	5-hydroxytryptophan	0.44083	-1.1817	0.02428	a, b
	L-Histidinol		0.19801	-2.3364	0.00075628	a, b	Propionylcarnitine	0.44768	-1.1595	0.017882	a, b, c
	Hexose-Phosphate		0.20741	-2.2695	0.00012959	a, b	Riboflavin	0.45643	-1.1315	0.024113	a, b, c
	D-Sedoheptulose 7-phosphate		0.20951	-2.2549	0.00075628	a, b	Hexanoylcarnitine	0.47746	-1.0665	0.024113	a, b, c
	Taurocholic acid		0.2287	-2.1285	0.0077006	a, b, c	Allantoic-acid	0.59578	-0.74716	0.024113	a, b
	Dihydroxyacetone phosphate/Diglycolic acid /Malic acid 0.23164	-2.1101	0.0055025	a, b	Acetyl-L-carnitin	0.64446	-0.63384	0.024113	a, b, c
	Prostaglandin E1/Prostaglandin B1/Prostaglandin A1 0.247	-2.0174	0.0031558	a, b
	D-Sphingosine /Elaidic acid/trans-Vaccenic acid	0.27309	-1.8726	1.5483E-05	a, b	Metabolite identification status
	Disaccharide		0.30244	-1.7253	0.033416	a, b	a. Based on accurate mass matching
	Cholic acid		0.33332	-1.585	0.037612	a, b, c	b. Based on chromatographic retention time similarity with that of an authentic standard
	cis-8,11,14-Eicosatrienoic acid		0.35115	-1.5098	4.2134E-05	a, b	c. Based on MS2 spectral similarity with that of an authentic standard
	Myristic acid		0.36217	-1.4653	4.2134E-05	a, b
	13(S)-Hydroxyoctadeca-9Z,11E-dienoic acid		0.38152	-1.3902	3.0085E-05	a, b, c
	gamma-Linolenic acid		0.38596	-1.3735	0.00020346	a, b
	L-Aspartic acid		0.46681	-1.0991	2.1541E-05	a, b, c
	Nicotinamide		0.48102	-1.0558	0.0077006	a, b
	O-Phosphorylethanolamine		0.5122	-0.96523	0.0022392	a, b, c
	Diethyladipate/Nonanoic acid		0.51913	-0.94582	0.00020346	a, b
	N-Acetyl-a-D-glucosamine 1-phosphate		0.51941	-0.94505	0.0040478	a, b, c
	Isobutyric acid		0.53237	-0.90949	0.0047267	a, b, c
	4-Methyl-2-oxovaleric acid/2-Ketohexanoic acid/3-Methyl-2-oxovaleric acid 0.5418 -0.88417	0.00020346	a, b
	cis-5,8,11,14,17-Eicosapentaenoic acid		0.54188	-0.88395	0.00236	a, b, c
	N-Acetylglycine		0.55797	-0.84175	0.0024695	a, b, c
	Guanidinosuccinic acid		0.56244	-0.83023	0.02684	a, b
	Azelaic acid		0.56474	-0.82435	0.0055025	a, b, c
	N-Acetyl-L-asparagine		0.56491	-0.8239	0.0027245	a, b, c
	D-erythro-Dihydrosphingosine (Sphinganine)		0.59419	-0.751	0.00075628	a, b
	N-acetyl-DL-Valine		0.62656	-0.67449	0.0079669	a, b
	Methyl acetoacetate		0.64059	-0.64252	0.00075628	a, b, c
	Isoleucine/Leucine		0.64379	-0.63534	0.0077006	a, b, c
	N-Isobutyrylglycine		0.65029	-0.62084	0.011653	a, b, c
	Methylhistidine		1.5362	0.61935	0.00066837	a, b, c
	Hexose		1.5709	0.65159	0.00020346	a, b, c
	5-Hydroxylysine		1.5772	0.65735	0.025891	a, b, c
	2,6-Dihydroxybenzoic acid		1.5874	0.66665	0.022888	a, b
	L-Anserine		1.5944	0.67305	0.0055286	a, b, c
	4-Hydroxy-3-methoxyphenylglycol sulfate		1.637	0.71105	0.0055286	a, b, c
	Citric acid/Isocitric acid		1.6997	0.76529	0.0047375	a, b, c
	L(+)arabinose/D-Xylulose/D-Arabinose		1.7574	0.8134	0.042033	a, b, c
	D-Arabinose /L(+)arabinose		1.814	0.8592	0.015196	a, b, c
	Glutaconic acid		1.9477	0.9618	0.024794	a, b, c
	Hydroxyglutaric acid		2.215	1.1473	0.0075293	a, b, c
	Indole-3-propionic acid /Methyl indole-3-acetate 2.2311	1.1577	0.00011977	a, b, c
	D-(-)-Quinic acid		2.2457	1.1672	0.015196	a, b, c
	L-Glutathione oxidized		2.3602	1.2389	0.037612	a, b, c
	Arginine		3.5531	1.8291	0.0002166	a, b, c

Table S2 . Annotated and significant metabolites between day 1 and day 3 after CEES exposure.

 S2 Fold changes > 1.5, p-values < 0.05 (Wilcoxon with Benjamini-Hochberg correction).

	HILIC(-) conditions	Fold Change D3/D1	log2(FC)	p.adjusted	Identification status	C18(+) conditions	Fold Change D3/D1	log2(FC)	p.adjusted Identification status
	Arginine		0.066097	-3.9193	3.25E-06	a, b, c	N-alpha-acetyl-L-arginine	0.5843	-0.77521	1.98E-03	a, b
	4-Imidazoleacrylic acid (urocanic acid)		0.44291	-1.1749	8.00E-06	a, b, c	4-Imidazoleacrylic-acid-(urocanic-acid)	0.59193	-0.75651	4.96E-05	a, b, c
	Hydroxyglutaric acid		0.51393	-0.96037	0.00021442	a, b, c	N6-Acetyl-L-lysine	0.65282	-0.61525	3.82E-04	a, b, c
	L-Anserine		0.56527	-0.823	0.00021442	a, b, c	Acetyl-L-carnitine	1.5134	0.59782	7.9642E-05	a, b, c
	Corticosterone/21-Deoxycortisol		0.57514	-0.798	0.015843	a, b	Uracil	1.6647	0.73526	0.022903	a, b, c
	Sphingosine 1-phosphate		0.57697	-0.79343	0.0012463	a, b, c	6-Hydroxypyridine-2-carboxylic-acid/3-Hydroxypicolinic-acid/6-Hydroxypyridine-3-carboxylic-acid-(6-hydroxynicotinic-acid)/Isonicotinic-acid-N-oxide/Nicotinic-acid-N-oxide 1.7467 0.80459 0.0082908 a, b
	Glutaconic acid		0.58865	-0.76451	0.0027181	a, b, c	5-Deoxy-5-(methylthio)adenosine	1.7722	0.82553	0.0016388	a, b, c
	Thymine		0.61656	-0.6977	0.000024844	a, b, c	Allantoic-acid	1.8692	0.90246	2.3845E-06	a, b
	Carnosine		0.62023	-0.68912	0.000039292	a, b	Glutathione	1.9318	0.94992	0.00038243	a, b, c
	D-(-)-Quinic acid		0.62992	-0.66675	4.99E-02	a, b, c	4-Guanidinobutyric-acid	1.9338	0.95141	0.0067105	a, b, c
	5-Hydroxylysine		0.63411	-0.65718	2.00E-03	a, b, c	Octanoylcarnitine	2.0873	1.0616	0.0016388	a, b, c
	Deoxyuridine		0.66361	-0.5916	0.000062734	a, b	Propionylcarnitine	2.0935	1.0659	0.00002314	a, b, c
	N-Acetyl-L-aspartic acid		1.5019	0.58677	0.00033479	a, b, c	alpha-Methylhistamine/3-Methylhistamine	2.2114	1.1449	0.0039612	a, b, c
	Muramic acid		1.5059	0.59062	0.048692	a, b	Nicotinamide	2.2888	1.1946	8.4012E-06	a, b, c
	Phosphoserine		1.5473	0.62978	0.0068168	a, b, c	4-Pyridoxic-acid-	2.5335	1.3412	2.3845E-06	a, b
	D-Saccharic acid		1.5538	0.6358	0.00010908	a, b, c	5-hydroxytryptophan	2.6221	1.3907	0.004747	a, b
	Taurine		1.5574	0.63917	0.00022214	a, b, c	Adenine	2.7396	1.454	7.9642E-05	a, b, c
	Azelaic acid		1.6277	0.70281	0.00015082	a, b, c	Adenosine	2.8914	1.5318	0.00019561	a, b
	Galactonic acid		1.6418	0.71528	0.00014052	a, b, c	Disaccharide	3.5956	1.8462	0.00049871	a, b
	N-Glycolylneuraminic acid		1.6741	0.74339	0.000008001	a, b, c	Indolelactic-acid	3.9772	1.9918	0.012103	a, b, c
	N acetyl L glutamine		1.7886	0.8388	0.00005379	a, b, c	Hexanoylcarnitine	4.9322	2.3022	3.2394E-06	a, b, c
	Cystathionine		1.8062	0.85297	0.019735	a, b, c	Riboflavin	5.9475	2.5723	1.1344E-05	a, b, c
	aminobutyric acid		1.8312	0.8728	0.000053687	a, b	Xanthosine	12.087	3.5954	0.0002588	a, b, c
	L-Glutamic acid		1.8698	0.90286	0.00002877	a, b, c	Nalpha-Acetyl-L-asparagine	83.092	6.3766	1.6401E-06	a, b
	Uridine		1.8791	0.91003	0.00099663	a, b, c	Taurocholic-acid	511.4	8.9983	2.838E-08	a, b, c
	N-Acetyl-L-asparagine		1.9716	0.97934	0.00005379	a, b, c	Taurodeoxycholic-acid	1575	10.621	1.6401E-06	a, b, c
	Pentose-Phosphate		2.0154	1.0111	0.03016	a, b, c
	N-acetyl-DL-Valine		2.0768	1.0544	2.72E-03	a, b
	O-Phosphorylethanolamine		2.3198	1.214	1.50E-05	a, b, c
	N-Acetyl-glutamic acid		2.3435	1.2286	0.00051833	a, b, c	Metabolite identification status
	cis-5,8,11,14,17-Eicosapentaenoic acid		2.4316	1.2819	0.0003671	a, b, c	a. Based on accurate mass matching
	Morphine-6-D-glucuronide		2.5591	1.3556	0.013793	a, b, c	b. Based on chromatographic retention time similarity with that of an authentic standard
	4-Pyridoxic acid		2.5748	1.3645	0.000011774	a, b, c	c. Based on MS2 spectral similarity with that of an authentic standard
	Guanidinosuccinic acid		2.5944	1.3754	0.00049995	a, b
	L-Aspartic acid		2.8371	1.5044	1.6339E-07	a, b, c
	N-Acetyl-a-D-glucosamine 1-phosphate		3.0295	1.5991	0.00002877	a, b, c
	Nicotinamide		3.1556	1.6579	3.25E-06	a, b
	cis-8,11,14-Eicosatrienoic acid		3.159	1.6595	7.8912E-06	a, b
	Succinic acid		3.196	1.6763	0.000077059	a, b, c
	gamma-Linolenic acid		3.2921	1.719	0.000007364	a, b
	Myristic acid		3.6533	1.8692	3.99E-06	a, b
	13(S)-Hydroxyoctadeca-9Z,11E-dienoic acid		3.8278	1.9365	2.41E-06	a, b, c
	D-Sphingosine /Elaidic acid/trans-Vaccenic acid	3.8995	1.9633	3.78E-06	a, b
	Disaccharide		3.9478	1.981	1.24E-04	a, b
	N-Isobutyrylglycine		3.9794	1.9925	1.47E-05	a, b, c
	L-Glutathione oxidized		4.0439	2.0158	1.16E-04	a, b, c
	Medroxyprogesterone		5.0897	2.3476	4.41E-04	a, b
	Isobutyric acid		5.486	2.4558	2.41E-06	a, b, c
	L-Histidinol		5.7465	2.5227	3.03E-05	a, b
	Prostaglandin E1/Prostaglandin B1/Prostaglandin A16.1569	2.6222	0.0027181	a, b
	Dihydroxyacetone phosphate/Diglycolic acid /Malic acid 7.1984	2.8477	1.41E-04	a, b
	D-Sedoheptulose 7-phosphate		8.5937	3.1033	8.8833E-06	a, b
	Xanthosine		10.256	3.3584	0.0008667	a, b
	Hexose-Phosphate		10.742	3.4252	7.2412E-06	a, b
	Cyclic AMP		13.778	3.7842	1.6339E-07	a, b
	Phosphoenolpyruvic acid		14.563	3.8642	0.000021656	a, b
	Stachyose		19.606	4.2932	7.89E-06	a, b, c
	Cholic acid		27.856	4.7999	0.00002877	a, b, c
	UDP-N-acetylglucosamine		30.571	4.9341	7.51E-04	a, b, c
	GMP		32.903	5.0401	2.4073E-06	a, b, c
	5-Methyldeoxycytidine		140.67	7.1362	3.94E-04	a, b
	UMP		235.48	7.8794	9.43E-06	a, b, c
	Uridine 5'-diphosphogalactose		711.04	9.4738	0.00026195	a, b, c
	Taurocholic acid		1704.4	10.735	1.13E-07	a, b, c
	Taurodeoxycholic acid		1837.5	10.844	3.25E-06	a, b

Table S3 . Annotated and significant metabolites between day 3 and day 7 after CEES exposure.

 S3 Fold changes > 1.5, p-values < 0.05 (Wilcoxon with Benjamini-Hochberg correction).

	HILIC(-) conditions	Fold Change D7/D3	log2(FC)	p.adjusted Identification status	C18(+) conditions	Fold Change D7/D3	log2(FC)	p.adjusted	Identification status
	Disaccharides	0.25494	-1.9718	0.0002821	a, b	Disaccharides	0.30902	-1.6942	0.043942	a, b
	Cystathionine	0.41459	-1.2702	1.23E-04	a, b, c	4-Guanidinobutyric-acid	0.43931	-1.1867	0.02458	a, b, c
	D-Raffinose	0.49893	-1.0031	8.21E-03	a, b, c	Adenosine	0.49276	-1.0211	0.02458	a, b
	Indole-3-propionic acid /Methyl indole-3-acetate 0.55325	-0.854	0.0082054	a, b, c	Aminohippuric-acid	0.59494	-0.74918	0.018407	a, b
	5-Hydroxylysine	0.5603	-0.83573	0.019063	a, b, c	N-alpha-acetyl-L-arginine	0.64845	-0.62494	0.041023	a, b
	Methylhistidine	0.56075	-0.83456	4.4617E-05	a, b, c	Lactamide/Sarcosine/beta-Alanine/Alanine	1.6398	0.71349	0.025615	a, b
	L-(-)-Arabitol/Adonitol/D-(+)-Arabitol	0.65967	-0.60019	0.0067463	a, b, c	Benzyl-alcohol	2.5138	1.3298	0.0094991	a, b
	Creatine	1.5653	0.64646	0.0082054	a, b, c	Serotonin-hydrochloride	3.0428	1.6054	0.018407	a, b, c
	L-a-Glycerophosphorylcholine	1.5666	0.64764	0.006656	a, b
	Benzoic acid/4-Hydroxybenzaldehyde	1.593	0.67174	0.0082054	a, b	Metabolite identification status
	Carnosine	2.7684	1.4691	0.00010262	a, b	a. Based on accurate mass matching
	Prostaglandin E1/Prostaglandin B1/Prostaglandin A1 3.5546	1.8297	0.01364	a, b	b. Based on chromatographic retention time similarity with that of an authentic standard
	Xanthine	91.915	6.5222	4.4617E-05	a, b, c	c. Based on MS2 spectral similarity with that of an authentic standard

Acknowledgements

This work was supported by the "Agence de l'Innovation de Défense" (French Defense Ministry) and the "NRBC" program of the French Alternative Energies and Atomic Energy Commission.

Conflict of Interest

The authors declare that they have no conflict of interest.

Evidence for the systemic diffusion of (2-chloroethyl)-ethyl-sulfide, a sulfur mustard analog, and its deleterious effects in brain By Marie Roser, Daniel Léonço, Fanny Caffin, Fanny Gros-Désormeaux, Camille Eldin, David Béal, Sadia Ouzia, Christophe Junot, François Fenaille, Christophe Piérard and Thierry Douki

Supplementary data

Content :

Table S1. Annotated and significant metabolites between non-exposed mice and day 1 after CEES exposure.

Table S2. Annotated and significant metabolites between day 1 and day 3 after CEES exposure.

Table S3. Annotated and significant metabolites between day 3 and day 7 after CEES exposure.

Figure S1. Summary of metabolic pathway enrichment analysis performed in MetaboAnalyst

using the metabolites from Table S1 that are altered between day 1 and day 3 after CEES exposure. S1 that are altered between day 1 and day 3 after CEES exposure. The color and size of each circle are based on p-value and pathway impact value, respectively.