Abdel Lisser

Jia Liu

Tian Xia

Distributionally Robust Chance Constrained Games under Wasserstein Ball

Keywords: Chance constrained games, Wasserstein ball, Elliptical distribution, Nash equilibrium

come   L'archive ouverte pluridisciplinaire

Introduction

The literature on the existence of an equilibrium in game theory started since the paper by John von Neumann [START_REF] Von | On the theory of games of strategy[END_REF], who showed the existence of a mixed strategy saddle point equilibrium for a two-player zero-sum matrix game. Then in 1950, John Nash [START_REF] Jr | Equilibrium points in n-person games[END_REF] showed the existence of a mixed strategy Nash equilibrium for a finite strategic game, which brought the research on the existence of equilibria in game theory to a new stage. Since John von Neumann and John Nash, the traditional games with deterministic payoffs of the players have been widely studied. However, real world problems are significantly subject to uncertainties. Therefore, games with random payoffs are of increasing concern in game theory. Recently, Singh et al. [START_REF] Vikas | Existence of nash equilibrium for chance-constrained games[END_REF][START_REF] Vikas | Distributionally robust chanceconstrained games: existence and characterization of nash equilibrium[END_REF][START_REF] Vikas | A characterization of nash equilibrium for the games with random payoffs[END_REF][START_REF] Peng | Games with distributionally robust joint chance constraints[END_REF] initiated the studies on chance constrained games. By using chance constrained optimization models, we actually consider the random games where the players' payoffs are obtained with a certain confidence. The chance constrained payoff represents the maximal threshold such that the random return/payoff is not less than the threshold with a large probability, e.g., 95% or 90%. In a traditional chance constrained games, each player should know the exact distribution of the random return/payoff, or specify an a-priori distribution before making the decision. However, due to the imperfectness of the historical data, and incompleteness of the information collection, the estimated a-priori distribution may be biased from the true distribution.

If the player does not consider the ambiguity/impreciseness of the distribution and just use the traditional chance constraints, he might over-estimate the payoff at the equilibrium and makes an inefficient decision in the game [START_REF] James | The optimizer's curse: Skepticism and postdecision surprise in decision analysis[END_REF]. Thus, to reduce the potential loss in extreme cases, it is natural to consider the ambiguity set of the distribution in the decision-making model and use the distributionally robust optimization approach to make a decision against the worst-case distribution. The literature on distributionally robust chance constrained games (DRCCG) mainly focuses on the existence of Nash equilibrium when the information to each player is characterized by different kinds of uncertainty sets. For example, Singh et al. [START_REF] Vikas | Distributionally robust chanceconstrained games: existence and characterization of nash equilibrium[END_REF] considered DRCCG with a moment-type uncertainty set. Xu and Zhang [START_REF] Xu | Stochastic nash equilibrium problems: sample average approximation and applications[END_REF] considered the convergence of the sample average approximation in DRCCG. Peng et al. [START_REF] Peng | Games with distributionally robust joint chance constraints[END_REF] studied DRCCG under the divergence distance based uncertainty set and showed the existence of Nash equilibrium.

Wasserstain ball is also a kind of important uncertainty set widely used in distributionally robust optimization. Most of the studies conducted to date focuses on the data-driven case [START_REF] Mohajerin | Data-driven distributionally robust optimization using the wasserstein metric: performance guarantees and tractable reformulations[END_REF][START_REF] Zhao | Data-driven risk-averse stochastic optimization with wasserstein metric[END_REF][START_REF] Chen | Data-driven chance constrained programs over wasserstein balls[END_REF][START_REF] Xie | On distributionally robust chance constrained programs with wasserstein distance[END_REF][START_REF] Ji | Data-driven distributionally robust chance-constrained optimization with wasserstein metric[END_REF][START_REF] Peng | A data-driven distributionally robust game using wasserstein distance[END_REF], where the reference distribution is a discrete distribution. Xie [START_REF] Xie | On distributionally robust chance constrained programs with wasserstein distance[END_REF] gave an exact reformulation of the distributionally robust chance constraint with a data-driven reference distribution. Liu et al. [START_REF] Liu | Distributionally robust chance constrained geometric optimization[END_REF] studied distributionally robust chance constrained geometric optimization where some of the uncertainty sets are constrained by the Wasserstein distance under discrete, full, or nonnegative real-space support. The discrete distributions based data-driven reformulations were generalized by [START_REF] Blanchet | Robust wasserstein profile inference and applications to machine learning[END_REF][START_REF] Rui | Distributionally robust stochastic optimization with wasserstein distance[END_REF] to Polish spaces and continuous distributions. Shen and Jiang [START_REF] Shen | Convex chance-constrained programs with wasserstein ambiguity[END_REF] considered the distributionally robust chance constraint where the reference distribution in the Wasserstein ball is a Gaussian distribution. Peng et al. [START_REF] Peng | A data-driven distributionally robust game using wasserstein distance[END_REF] studied distributionally robust games with expected utility functions and data-driven Wasserstein ball. To the best of our knowledge, DRCCG under Wasserstein distance has not been studied in the literature.

In this paper, we study DRCCG under the Wasserstein ball. We consider the reference distribution as an elliptical distribution, i.e., a distribution from a large family of continuous distributions. A data driven Wasserstein ball and a continuous reference distribution play different rules in distributionally robust optimization. The former allows to calibrate and evaluate the size of the ambiguity set. The later can be viewed as an adjustment of the over-optimism of the decision maker's a-priori distribution information. The radius of the ball reflects the strength of confidence in her/his a-priori information. In many applications, the decision makers ignore the fact that the a-priori distributions are not Gaussian.For instance, wind power and electric load forecasting errors are generally not Gaussian distributed in power system scheduling problems. The stock return rates are often regarded as high kurtosis and fat-tailed [START_REF] Frank J Fabozzi | Fat-tailed and skewed asset return distributions: implications for risk management, portfolio selection, and option pricing[END_REF]. The elliptical distributions are a broad family of probability distributions that generalize the multivariate normal distribution, which thus play an important role in stochastic games.

As far as we know, this paper provides the first contribution which considers an elliptical reference distribution in a Wasserstein ball-based distributionally robust game.

We propose a new approach which leads to the condition of the convexity of the chance constrained payoff with the Wasserstein ball and derive an efficient solution method to the equilibrium problem of this kind of games.

The paper is organized as follows. We derive the reformulation of the distributionally robust chance constrained payoff function in Section 3.2 and show the existence of a Nash equilibrium for DRCCG in Section 3.3. We propose an optimization approach to find Nash equilibrium of DRCCG in Section 3.4. We carry out numerical tests under some popular distributions from the elliptical distribution family in Section 4.

DRCCG under Wasserstein ball

Introduction to chance constrained games

We consider a n-player strategic game. Let I t1, 2, ...nu denotes the set of players. For each i I, A i represents a finite action set of player i and its generic element is denoted by a i . The vector a pa 1 , a 2 , ...a n q denotes the action profile of the game. The set of all action profiles is denoted by the product set A n i1 A i . We denote A ¡i n j1;j$i A j , and a ¡i pa 1 , a 2 , .., a i¡1 , a i 1 , .., a n q A ¡i . Let X i be the set of mixed strategies of player i which is a subset of all probability distributions over the action set A i . A mixed strategy τ i X i is represented by τ i pτ i pa i qq a i A i , where τ i pa i q ¥ 0 is the probability with which player i chooses an action a i and °ai A i τ i pa i q 1. The set of all mixed strategy profiles is denoted by X n i1 X i and a mixed strategy profile τ pτ i q iI is a generic element of X. Denote X ¡i n j1;j$i X j and τ ¡i X ¡i is a vector of mixed strategies τ j , j $ i. We define pτ i , τ ¡i q to be a strategy profile where player i uses the strategy τi and each other player j, j $ i uses strategy τ j . Denote l i paq R the payoff of player i for an action profile a, and r i R |A| the payoff vector of the game where |A| is the cardinality of set A. For such mixed strategies games, Nash [START_REF] Jr | Equilibrium points in n-person games[END_REF] showed that there always exists a Nash equilibrium.

In some practical applications, the players' payoffs are random due to several uncertainty sources. Thus, it is natural to study stochastic games by using stochastic optimization approach. Let pΩ, F, P q be a probability space where l i : Ω Ñ R for each i I. We can view the random payoff vector r i as a measurable function r i pωq pl i pa, ωqq aA : Ω Ñ R |A| , whose distribution is F. Then for a given strategy τ X and a scenario ω Ω, the random payoff R i of player i is

R i pτ, ωq aA £ l i pa, ωq ¹ jI τ j pa j q . ( 1
)
We denote vector η τ pη τ paqq aA for short, where η τ paq ± iI τ i pa i q. Then R i pr i q t η τ .

Considering the randomness of the payoff R i , Singh et al. [START_REF] Vikas | Existence of nash equilibrium for chance-constrained games[END_REF] study the following chance constrained payoff:

u α i i pτq sup 2 v i |P F ppr i q t η τ ¥ v i q ¥ α i @ , (2)
which is the highest level of the player's payoff that he/she can attain with at least a specified level of confidence α i p0, 1q. The confidence level is given in advance and known to all players. That is, the game we study is non-cooperative with complete information.

When the distribution of r i is known and follows a multivariate normal distribution with a mean vector µ i and a positive definite covariance matrix Σ i , the chance constrained payoff equals

u α i i pτq aA £ µ i paq ¹ jI τ j pa j q }Σ 1 2 i η τ } 2 ϕ ¡1 p1 ¡ α i q,
where ϕ ¡1 p¤q is the quantile function of a standard normal distribution N p0, 1q ([3], formula (3.1)).

Distributionally robust chance constrained games

In many practical situations, the probability distribution of r i is not completely known to the players. Instead, the players know uncertainty sets D i , i I, where D i is the set of all possible distributions of r i . We assume that the uncertainty set of each player is known to all players in the game, and the players consider the worst case of their payoffs.

Thus, player i holds distributionally robust chance constrained payoff function:

u α i i pτq sup 4 v i | inf F D i P F ppr i q t η τ ¥ v i q ¥ α i B , (3)
The game is called a distributionally robust chance constrained game (DRCCG) and was studied in [START_REF] Vikas | Distributionally robust chanceconstrained games: existence and characterization of nash equilibrium[END_REF][START_REF] Peng | Games with distributionally robust joint chance constraints[END_REF]. The set of best response strategies of player i, i I against a given strategy profile τ ¡i is

BR α i i pτ ¡i q tτ ¦ i X i | u α i i pτ ¦ i , τ ¡i q ¥ u α i i pτ i , τ ¡i q, dτ i X i u . Definition 1.
A strategy profile τ ¦ X is said to be a Nash equilibrium of a DRCCG for a given α, if for all i I, the following inequality holds,

u α i i pτ ¦ i , τ ¦ ¡i q ¥ u α i i pτ i , τ ¦ ¡i q, dτ i X i . (4)

Wasserstein distance based uncertainty set

Since the true probability distribution of r i , i I is unknown, we replace the true distribution with a reference one which we might derive from some empirical data.

Thus, we consider the uncertainty set D i , i I as the neighbourhood of the reference distribution Fi :

D i tF : d w pF, Fi q ¤ δ i u
, where δ i ¡ 0 is a pre-specified radius of the uncertainty set D i . d w p¤, ¤q is the Wasserstein distance and the set D i is known as a Wasserstein ball. Definition 2. The Wasserstein distance d w is defined by

d w pF, Fi q inf πΠ 4» Ω¢Ω }ξ 1 ¡ ξ 2 }πpdξ 1 , dξ 2 q B
, where Π is the space of all joint distributions of ξ 1 and ξ 2 with marginals F and Fi respectively, } ¤ } is the norm defined as } ¤ } : }Σ ¡ 1 2 i p¤q} 2 where Σ i is a positive definite covariance matrix of the reference distribution Fi . We denote } ¤ } ¦ the dual norm of } ¤ }, i.e., } ¤ } ¦ }Σ 1 2 i p¤q} 2 for each i.

Reformulation of DRCCG under elliptical reference distributions

Reformulation of DRCCG

In this section, we concentrate on the reformulation of the payoff function u α i i pτq, i I for given τ and α, when the uncertainty set D i is a Wasserstein ball centered at an elliptical distribution.

We introduce the definitions of Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) of a random variable, VaR α,F pXq : inf tx :

P F rX ¤ xs ¥ αu , CVaR α,F pXq : min γ 4 γ 1 1 ¡ α E F rpX ¡ γq s B .
It is well known that both VaR and CVaR are translation invariant, monotone and positive homogeneous. Additionally, CVaR is subadditive, which is a coherent risk measure.

Next, we focus on the reformulation of u α i i pτq in (3). Proposition 1. For each player i I, δ i ¡ 0 and α i p0, 1q, suppose that the reference distribution Fi is continuous, then the distributional robust optimal (DRO) payoff (3) is equal to the optimal value of the following optimization problem:

max αi ,v i R v i (5a) s.t. δ i p1 ¡ α i qCVaR α i , Fi p¡f i pτ, r i qq ¤ sup αi r0,1s 3 p1 ¡ αi qCVaR αi , Fi p¡f i pτ, r i qq A ,(5b) 0 ¥ VaR α i , Fi p¡f i pτ, r i qq (5c)
where f i pτ, r i q ¡v i pr i q t η τ c pη τ qtΣ i η τ , and Σ i is the covariance matrix of the reference distribution Fi .

Proof. By Corollary 1 of [START_REF] Shen | Convex chance-constrained programs with wasserstein ambiguity[END_REF], the distributionally robust chance constraint inf

F D i P F ppr i q t η τ ¥ v i q ¥ α i is equivalent to the following group of inequalities, δ i 1¡α i CVaR α i , Fi p¡f i pτ, r i qq ¤ 1 1¡α i E Fi rp¡f i pτ, r i qq s , (6a) 0 ¥ VaR α i , Fi p¡f i pτ, r i qq. ( 6b
)
where f i pτ, r i q ¡v i pr i q t η τ }η τ }¦ ¡v i pr i q t η τ c pη τ qtΣ i η τ , Σ i is the covariance matrix of the reference distribution Fi . Therefore the payoff (3) is equal to the maximal v i subjecting to (6a) and (6b). Meanwhile,

(6a) ô δ i 1¡α i CVaR α i , Fi p¡f i pτ, r i qq ¤ 1 1¡α i E Fi r¡f i pτ, r i q ¤ 1t¡f i pτ, r i q ¥ 0us, ô δ i 1¡α i CVaR α i , Fi p¡f i pτ, r i qq ¤ 1 1¡α i sup t i R E Fi r¡f i pτ, r i q ¤ 1t¡f i pτ, r i q ¥ t i us, ô δ i p1 ¡ α i qCVaR α i , Fi p¡f i pτ, r i qq ¤ sup t i R E Fi r¡f i pτ, r i q ¤ 1t¡f i pτ, r i q ¥ t i us, ô δ i p1 ¡ α i qCVaR α i , Fi p¡f i pτ, r i qq ¤ sup tR 3 p1 ¡ g i ptqqCVaR g i ptq, Fi p¡f i pτ, r i qq A , , ô δ i p1 ¡ α i qCVaR α i , Fi p¡f i pτ, r i qq ¤ sup αi r0,1s
3 p1 ¡ αi qCVaR αi , Fi p¡f i pτ, r i qq A

, where g i ptq 1 ¡ P Fi r¡f i pτ, r i q ¥ ts. The first equivalence is by the definition of r¤s . The second equivalence comes from the fact that t i reaches its optimal value at 0. The third equivalence is by multiplying p1 ¡ α i q to both sides of the inequality. The forth equivalence is a reformulation by the definition of CVaR. The last equivalence is due to the replacement of variable t by its quantile αi g i ptq, which works for the assumption that Fi is continuous such that g i ptq is a continuous function.

Reformulation of DRCCG under elliptical reference distributions

In this section, each player i uses an elliptical reference distribution and considers a worst-case payoff over a Wasserstein ball centered at an elliptical distribution.

Definition 3 ([20]

). A d-dimensional vector X R d follows an elliptical distribution E d pµ, Σ, ψq if the probability density function (PDF) is f pxq |Σ| ¡ 1 2 gppx¡µq t Σ ¡1 px¡µqq, where µ R d is the location parameter, Σ R d¢d is the dispersion matrix, ψ is the characteristic generator and g : R Ñ 0 is the density generator such that the Fourier transform of gp|x| 2 q, as a generalized function, is equal to ψp|ξ| 2 q.

By [START_REF] Alexander | Quantitative risk management: concepts, techniques and tools-revised edition[END_REF], for any matrix A R N ¢d and any vector b R N , we have AX b E N pAµ b, A t ΣA, ψq.

Lemma 1 ([21, Theorem 1]). For an elliptical distributed vector ζ E d pµ, Σ, ψq, a real vector w R d , and α p0, 1q, CVaR α pw t ζq w t µ c

w t ΣwT α , (7)
where

T α π d¡1 2 2αΓp d 1 2 q » V q 2 α pu ¡ q 2 α q d¡1 2 gpuqdu, q α ¡w t µ VaR α pw t ζq c w t Σw ,
and T α denotes the value of CVaR α pw t ζq when ζ E d p0, Σ, ψq and w t Σw 1.

Remark 1. There exists a slight difference between [START_REF] James | The optimizer's curse: Skepticism and postdecision surprise in decision analysis[END_REF] and the definition in [START_REF] Dobrev | Accurate evaluation of expected shortfall for linear portfolios with elliptically distributed risk factors[END_REF] because here CVaR is defined with respect to the loss function rather than the reward function in [START_REF] Dobrev | Accurate evaluation of expected shortfall for linear portfolios with elliptically distributed risk factors[END_REF].

Assumption 1. For each player i I, the confidence level α i p0, 1q and the uncertainty set D i is a Wasserstein ball centered at an elliptical distribution E |A| pµ i , Σ i , ψ i q with a radius δ i ¡ 0.

Theorem 1. Given Assumption 1, the DRO payoff function (3) is equal to the following reformulation:

u α i i pτq pη τ q t µ i β i pη τ q t Σ i η τ , (8)
where

β i sup αi pα i ,1q 1
pα i ¡ α i q rp1 ¡ αi qT αi ¡ p1 ¡ α i qT α i ¡ δ i s . (9)
Proof. By Proposition 1, we know that the DRO payoff function ( 3) is equal to the optimal value of the optimization problem (5a)-(5c). We consider the reformulation of (5b).

In (5b), the CVaR value is evaluated under the reference distribution Fi of r i . By the translation invariance property of CVaR and Lemma 1, we have

CVaR α i , Fi p¡f i pτ, r i qq CVaR α i , Fi £ ¡ ¡v i pr i q t η τ pη τ q t Σ i η τ CVaR α i , Fi £ ¡pr i ¡ µ i q t η τ pη τ q t Σ i η τ v i ¡ µ t i η τ pη τ q t Σ i η τ T α i v i ¡ µ t i η τ pη τ q t Σ i η τ (10)
where ¡pr i ¡µ i q t η τ c pη τ q t Σ i η τ follows a uni-variate elliptical distribution E 1 p0, 1, ψ i q.

Taking ( 10) into (5b), we have

(5b) ô sup αi 3 pα i ¡ α i qµ t i η τ pη τ q t Σ i η τ rp1 ¡ αi qT αi ¡ p1 ¡ α i qT α i ¡ δ i s ¡ pα i ¡ α i qv i A ¥ 0. (11)
Also we have αi ¥ α i by (5c), because otherwise p1 ¡ α i qCVaR α i , Fi p¡f i pτ, r i qq ¥ p1 ¡ αi qCVaR αi , Fi p¡f i pτ, r i qq.

Thus combined with (5c), we have

(5b) ¡ (5c) ñ v i ¤ µ t i η τ pη τ q t Σ i η τ sup αi ¥α i 4 rp1 ¡ αi qT αi ¡ p1 ¡ α i qT α i ¡ δ i s αi ¡ α i B , (12)
that is, (5b) ¡ (5c) ñ [START_REF] Xu | Stochastic nash equilibrium problems: sample average approximation and applications[END_REF].

By Lemma 1 and the translation invariance of VaR, we have

VaR α i , Fi p¡f i pτ, r i qq VaR α i , Fi £ ¡pr i ¡ µ i q t η τ pη τ q t Σ i η τ v i ¡ µ t i η τ pη τ q t Σ i η τ q α i v i ¡ µ t i η τ pη τ q t Σ i η τ , and (5c) ô v i ¤ µ t i η τ ¡ pη τ q t Σ i η τ q α i , (13)
where by Lemma 1, q α i here is just the quantile at α i of the elliptical distribution.

p1¡α i q c 2π ¡Φ ¡1 pα i q standard Laplace 1 c 2 p1 ¡ ln p2 ¡ 2α i qq 1 c 2 ln p2 ¡ 2α i q standard Logistic c 3 π ln α i ¡ α i 1¡α i 1¡α i ¡ c 3 π ln p α i 1¡α i q
Suppose that (8) holds. Let Y ¡pr i ¡µ i q t η τ c pη τ q t Σ i η τ . Then, by the definition of CVaR, we have

p1¡α i qCVaR α i , Fi pY q¡p1¡α i qCVaR αi , Fi pY q E Fi Y ¤ 1tY rVaR αi , Fi pY q, VaR α i , Fi pY qsu % » VaR αi , Fi pY q VaR α i , Fi pY q yd Fi pyq » αi α i VaR s, Fi pY qds ¥ pα i ¡ αi qVaR α i , Fi pY q.
The last equality is obtained by the change of variable s Fi pyq, which induces β i ¤ ¡q α i . Hence (8) ñ (13), and ( 8) ñ (5c). With the combination of (11), we have ( 8) ñ (5b). Thus we get (8) ñ (5b), (5c) which completes the proof.

From Theorem 1 we know that given a confidence level α i for each player i I, and any random variable r i elliptically distributed, we can solve the DRO payoff function u α i i analytically according to [START_REF] Xu | Stochastic nash equilibrium problems: sample average approximation and applications[END_REF].

For some specific distributions in the elliptical distribution family, we have particular values of the coefficient β i defined in Theorem 1. By [START_REF] Khokhlov | Conditional value-at-risk for elliptical distributions[END_REF], for each player i, we have Table 1 as for the value T α i and ¡q α i of three widely used elliptical distributions, where Φp¤q is the cdf of the standard Gaussian distribution.

Existence of the Nash equilibrium

Assumption 2. For each player i I, we assume that the radius of the uncertainty set D i satisfies δ i ¥ max αi pα i ,1q rp1 ¡ αi qT αi ¡ p1 ¡ α i qT α i s such that the parameter β i in Theorem 1 meets the condition β i ¤ 0. Remark 2. For each i I, as αi pα i , 1q in β i , the condition δ i ¥ max αi pα i ,1q rp1 ¡ αi qT αi ¡ p1 ¡ α i qT α i s in Assumption 2 is necessary and sufficient for β i ¤ 0. Theorem 2. Given Assumptions 1,2, the DRO payoff function u α i i p¤, τ ¡i q is a concave function of τ i for every τ ¡i X ¡i .

Proof. From the definition of η τ , we have

η τ pη τ paqq aA £ ¹ jI τ j pa j q aA £ τ i pa i q ¹ j$i τ j pa j q aA .
For a given τ ¡i X ¡i and for each a A, each element of η τ is the form expressed as τ i pa i qK a , where K a ± j$i τ j pa j q R is a known coefficient. Therefore, every element of η τ is both convex and concave of τ i for every τ ¡i X ¡i .

Moreover, we know that pη τ q t Σ i η τ is convex of each element of η τ , and each element of η τ is convex of τ i for every τ ¡i X ¡i . Then, pη τ q t Σ i η τ is convex of each element of η τ . By Assumption 2, β i ¤ 0, and thus β i pη τ q t Σ i η τ is concave of each element of η τ . Based on the above results, pη τ q t µ β i pη τ q t Σ i η τ is concave of τ i for every τ ¡i X ¡i .

Therefore, the payoff function u α i i p¤, τ ¡i q is a concave function of τ i for every τ ¡i X ¡i . Lemma 2 ([4, Theorem 1]). For a given confidence level vector α r0, 1s n , assume that for each i I, 1. the payoff function of player i, u α i i : X i ¢ X ¡i Ñ R defined by (3) is a continuous function.

2. the payoff function u α i i p¤, τ ¡i q is a concave function of τ i for every τ ¡i X ¡i .

Then, there always exists a mixed strategy Nash equilibrium of a DRCCG at a confidence level α. Theorem 3. Consider an n-player finite strategic game where the payoff vector r i pr i paqq aA of each player i, i I is a random vector, given Assumptions 1,2, then there always exists a mixed strategy Nash equilibrium.

Proof. Given Assumptions 1,2, by Theorem 2, u α i i p¤, τ ¡i q, i I is a concave function of τ i for every τ ¡i X ¡i . From (8), u α i i p¤, τ ¡i q, i I is a continuous function of τ . That is, both conditions of Lemma 2 are satisfied. Thus, there exists a mixed strategy Nash equilibrium.

Mathematical programming formulation to compute the Nash equilibrium

To compute the Nash equilibrium of a DRCCG, we solve the following convex program, whose global maximizer is the Nash equilibrium of the DRCCG.

Remark 3. Consider the DRCCG whose reference distribution is defined in Theorem 1, for each player i I, and any action profile a i A i ,. The expected payoff u α i i for a fixed strategy τ ¡i can be reformulated by the following optimization program

min τ i ¡β i pη τ q t Σ i η τ ¡ µ t i η τ (14a) s.t. a i A i τ i pa i q 1, (14b)
τ i pa i q ¥ 0. ( 14c
)
The dual problem of ( 14) is

max λ i ,v i λ i (15a) s.t. λ i ¤ a¡i A ¡i ¹ jI;j$i τ j pa j q ¡β i pv i q t Σ i v i ¡ µ i pa i , a ¡i q % , (15b)
}v i } ¤ 1. (15c)
As ( 14) is an SOCP problem, strong duality holds if [START_REF] Peng | A data-driven distributionally robust game using wasserstein distance[END_REF] has a finite optimal value, i.e., the optimal value of ( 14) is equal to the optimal value of (15). Thus, the optimal policy τ ¦ i of the i-th player can be found by solving the following set of equations

5 ¡β i pη τ q t Σ i η τ ¡ µ t i η τ λ i , p16aq (14b) ¡ (14c), (15b) ¡ (15c). p16bq (16)
Alternatively, we can solve an optimization problem instead of solving the equilibrium equations by penalizing the violation in the objective function ,i.e., max λ i ,v i ,τ i , i1,...,n iI

λ i β i pη τ q t Σ i η τ µ t i η τ % (17a) s.t. (14b) ¡ (14c), (15b) ¡ (15c), i 1, . . . , n. (17b)
u α 1 1 , u α 2 2 both increase as δ i converges to 0. In fact, when the radius of Wasserstein ball increases, the model covers more possible situations and thereby presents more robust results. The reduction of the payoffs can be seen as the trade off for the robustness. For the radius δ i 0.5, 0.1 and 0, we choose the results of Nash equilibrium τ 1 , τ 2 and the corresponding payoffs u α 1 1 , u α 2 2 under the standard Gaussian distribution when α i 0.95. We randomly generate 100 groups of Gaussian distributions and take τ 1 , τ 2 , u α 1 1 , u α 2 2 for which we compute the satisfaction probability P K i ppr i q t η τ ¥ u α i i q, where K i is the random Gaussian distribution. Figure 1 shows the values of the satisfaction probability P K 1 under 100 randomly generated Gaussian distributions for player 1 when δ 1 0.5, 0.1, and 0. The results of the players 1 and 2 are similar, thus we present only the player 1 results. From Figure 1, when δ 1 0.5, 0.1, we see that P K 1 for all 100 distributions are beyond 0.95. When δ 1 0, i.e., the distributionally robust model is reduced to the non robust model, there are only 3 random distributions' satisfaction probabilities beyond 0.95. Through comparing the satisfaction probabilities of these three radii, we see the performance of robustness for the solution of Nash equilibrium u α 1 1 . Therefore, it is clear that the robustness of the non robust model is by far less than the distributionally robust one.

Figure 1 :

 1 Figure 1: Values of Satisfaction probability P K1 under 100 randomly generated Gaussian distributions

Table 1 :

 1 The value T α i , ¡q α i of different elliptical distributions

	Distribution	T α i	¡q α i
	standard Gaussian	e ¡	q 2 1¡α i 2

Table 2 :

 2 Nash equilibrium solution and payoff under different radius and tolerance probability

	Distribution	pα 1 , α 2 q δ i	β i	τ 1	τ 2	u α1 1	u α2 2
			0.5 -12.0627 p 2731 10000 , 6166 10000 , 1104 10000 q p 3943 10000 , 2949 10000 , 3108 10000 q -12.4130 -12.8836
	Robust-Gaussian	(0.95,0.95)	10 ¡1 -4.0627 p 3922 10000 , 5257 10000 , 821 10000 q p 4178 10000 , 2252 10000 , 3570 10000 q 2.2079 1.5976
			10 ¡5 -1.6588	(1,0,0)	p 7199 10000 , 0, 2801 10000 q 6.5906 5.0894
	Non robust-Gaussian (0.95,0.95)	0	-1.6449	(1,0,0)	p 7219 10000 , 0, 2781 10000 q 6.6184 5.1198
			0.5	-4.8894 p 3428 10000 , 5759 10000 , 812 10000 q p 3987 10000 , 2522 10000 , 3491 10000 q 0.7406 0.1460
	Robust-Laplace	(0.85,0.85)	10 ¡1 -2.1050	(1,0,0)	p 6712 10000 , 0, 3288 10000 q 5.6730 4.1194
			10 ¡5 -0.8611	(1,0,0)	p 9667 10000 , 0, 333 10000 q 7.9579 6.8917
	Non robust-Laplace (0.85,0.85)	0	-0.8513	(1,0,0)	p 9733 10000 , 0, 267 10000 q 7.9685 6.9153
			0.5	-2.7774 p 7456 10000 , 2058 10000 , 486 10000 q p 5776 10000 , 669 10000 , 3556 10000 q 4.3319 3.3341
	Robust-Logistic	(0.7,0.7)	10 ¡1 -1.2856	(1,0,0)	p 7900 10000 , 0, 2100 10000 q 7.3088 5.9140
			10 ¡5 -0.4744	(0,1,0)	p0, 5010 10000 , 4990 10000 q 10.0532 8.9392
	Non robust-Logistic	(0.7,0.7)	0	-0.4671	(0,1,0)	p0, 5031 10000 , 4969 10000 q 10.0720 8.9555

Numerical experiments

In this section, we carry out a series of numerical tests under three different kinds of reference distributions. We compute the Nash equilibrium and the corresponding payoff of DRCCG by solving the mathematical program [START_REF] Rui | Distributionally robust stochastic optimization with wasserstein distance[END_REF]. Through 100 randomly generated groups of Gaussian distributions, we compare the performances of the Nash equilibrium of different radius δ i and observe the robustness of our model.

We consider a two-players DRCCG example introduced in [START_REF] Vikas | Distributionally robust chanceconstrained games: existence and characterization of nash equilibrium[END_REF], where I t1, 2u, A 1 t1, 2, 3u, A 2 t1, 2, 3u. The mean vectors for both players are µ 1 p10, 9, 11, 8, 12, 10, 7, 8, 13q t , µ 2 p9, 7, 8, 9, 10, 10, 10, 9, 8q t and the covariance matrices for both players are , which are both positive definite. We consider three kinds of reference distributions in elliptical distributions family as the center of the Wasserstein ball in DRCCG, namely Gaussian, Logistic and Laplace distributions respectively. We find the coefficient β i , i I by solving the optimization problem (9) with the function "globalsearch" in MATLAB when the radius δ i 0.5, 10 ¡1 , 10 ¡5 , 0. When δ i 0, the robust model is reduced to the non-robust one, and β i ¡q α i according to the results in Section 3.2 of [START_REF] Vikas | Existence of nash equilibrium for chance-constrained games[END_REF]. We find the Nash equilibrium by solving the optimization problem [START_REF] Rui | Distributionally robust stochastic optimization with wasserstein distance[END_REF] with BARON solver in YALMIP tool box of MATLAB.

In Table 2, We list our computational results of the Nash equilibrium τ i and the corresponding payoffs u α i i for three different elliptical distributions under different radius δ i and confidence probability coefficients α i respectively. From Table 2, we observe that the value of β i converges to ¡q α i when the radius goes to zero, which means the distributionally robust model gradually recovers the non robust one. Correspondingly, the payoff


Mojgan Allahyari

Majid Golkar
email: golkar@pasteur.ac.ir

Pezhman Fard-Esfahani
email: fard-esfahani@pasteur.ac.ir

Isabelle Dimier-Poisson

Marie-Nöelle Mévélec
email: mevelec@univ-tours.fr

Marie- Noëlle Mévélec

Marie-Noëlle Mévélec

Co-delivery of PLGA nanoparticles loaded with rSAG1 antigen and TLR ligands: An

Keywords: Toxoplasma gondii, rSAG1, Poly (D, L-lactide-co-glycolide) PLGA, Monophosphoryl lipid A, Imiquimod. PDI, Poly Dispersity Index, A, adsorption, E, encapsulation. * Encapsulation efficiency, ** Adsorption efficiency

Introduction

Toxoplasma gondii is an obligatory intracellular protozoan parasite with a considerably worldwide distribution which estimates to infect more than one-third of human population.

Although immunocompetent individuals are rarely affected by toxoplasmosis, it still remains as major health concern and makes serious consequences in immunocompromised patients caused by AIDS or chemotherapy and in developing fetus due to mother's primary infection [START_REF] Tenter | Toxoplasma gondii: from animals to humans[END_REF]. Undoubtedly, vaccination is known as the most promising approach against toxoplasmosis. Firstly, primary infection confers protective immunity against re-infection confirming the potential of vaccination [START_REF] Innes | A one health approach to vaccines against Toxoplasma gondii[END_REF]. On the other hand, available therapeutics do not eradicate tissue cysts and share severe side effects [START_REF] Dunay | Treatment of Toxoplasmosis: Historical Perspective, Animal Models, and Current Clinical Practice[END_REF]. Protective immune response to T. gondii and other intracellular parasites is complicated and involves the collaboration of innate immunity, humoral and cellular acquired immunity, directed against the multi stages of the parasite [START_REF] Mévélec | Key Limitations and New Insights Into the Toxoplasma gondii Parasite Stage Switching for Future Vaccine Development in Human, Livestock, and Cats[END_REF]. Appropriate adaptive immune response depends mainly on the ability of CD4 + and CD8 + T lymphocytes to produce IFN-γ [START_REF] Sasai | Innate, adaptive, and cell-autonomous immunity againstToxoplasma gondii infection[END_REF]. Along with immune T cells, antibodies contribute also to protection [START_REF] Kang | Decreased resistance of B cell-deficient mice to infection with Toxoplasma gondii despite unimpaired expression of IFN-gamma, TNFalpha, and inducible nitric oxide synthase[END_REF]. Therefore, if vaccination enhances potent Th1 T cell responses, it can successfully provide protective immunity to Toxoplasma infection. Among numerous T. gondii antigens, surface antigen 1 (SAG1) has attracted many attentions in vaccine development due to remarkable characteristics [START_REF] Wang | Research progress on surface antigen 1 (SAG1) of Toxoplasma gondii[END_REF]. Subunit vaccine candidates based on native or recombinant SAG1 proteins have been evaluated in animal models against, acute, chronic and congenital toxoplasmosis, using various adjuvant formulations to overcome the low intrinsic immunogenicity of the protein [START_REF] Pagheh | Toxoplasma gondii surface antigen 1 (SAG1) as a potential candidate to develop vaccine against toxoplasmosis: A systematic review[END_REF]. Immune responses and significant protections were obtained in most of these studies.

With the continuous improvement of knowledge and awareness regarding the immune system, the application of vaccine delivery strategies such as Poly (D, L-lactic-co-glycolic acid) (PLGA) particles could bring hope to improve vaccine efficacy [START_REF] Allahyari | Peptide/protein vaccine delivery system based on PLGA particles[END_REF]. Indeed, we previously showed that rSAG1 adsorbed on the surface of PLGA nanoparticles or rSAG1 encapsulated in PLGA nanoparticles elicited higher systemic IFN-γ and specific anti-T. gondii IgG antibodies than rSAG1 alone and conferred significant protection against acute toxoplasmosis [START_REF] Allahyari | In-vitroandin-vivocomparison of rSAG1-loaded PLGA prepared byencapsulation and adsorption methods as an efficient vaccine against Toxoplasma gondii[END_REF]. Similar results were also obtained with rSAG1 protein [START_REF] Chuang | Induction of long-lasting protective immunity against Toxoplasma gondii in BALB/c mice by recombinant surface antigen 1 protein encapsulated in poly (lactide-co-glycolide) microparticles[END_REF] and a multimeric recombinant T. gondii vaccine including SAG1 epitopes [START_REF] Roozbehani | Characterization of a multi-epitope peptide with selective MHC-binding capabilities encapsulated in PLGA nanoparticles as a novel vaccine candidate against Toxoplasma gondii infection[END_REF] or a recombinant chimeric protein rSAG1/2 [START_REF] Chuang | Encapsulation of chimeric protein rSAG1/2 into poly(lactide-co-glycolide) microparticles induces long-term protective immunity against Toxoplasma gondii in mice[END_REF] combined with PLGA nano-or microparticles.

TLRs agonists belong to a class of adjuvant known as immune stimulating agents. TLRs not only mediate the activation of innate immune cells, but also directly modulate vaccine specific response [START_REF] Dowling | Toll-like receptors: the swiss army knife of immunity and vaccine development[END_REF]. MPL, a portion of Salmonella minnesota lipopolysaccharide peptidoglycan, as specific agonist of TLR4 has been applied in some vaccination studies against T. gondii infection. Golkar et al. [START_REF] Golkar | Evaluation of protective effect of recombinant dense granule antigens GRA2 and GRA6 formulated in monophosphoryl lipid A (MPL) adjuvant against Toxoplasma chronic infection in mice[END_REF] demonstrated the protective efficacy of recombinant GRA2 with MPL against chronic infection by T. gondii and confirmed the role of MPL as an efficient immunostimulator to induce a protective Th1 immune response. The MPL adjuvant with toxofilin DNA vaccine alsoinduced significantly enhanced humoral and Th1-biased immune responses compared to the non-adjuvant toxofilin DNA vaccine [START_REF] Song | Vaccination with toxofilin DNA in combination with an alum-monophosphoryl lipid A mixed adjuvant induces significant protective immunity against Toxoplasma gondii[END_REF]. IMQ, a synthetic analog of imidazoquinoline family, as an agonist of TRL7 has been approved by US Food and Drug Administration (FDA) as an immunopotentiator agent for local topical administration [START_REF] Stanley | Imiquimod and the imidazoquinolones: mechanism of action and therapeutic potential[END_REF]. Recently, IMQ has been considered as a candidate for use against Toxoplasmosis both therapeutically and prophylactically [START_REF] Zaki | Effect of Imiquimod on Tachyzoites of Toxoplasma gondii and Infected Macrophages in vitro and in BALB/c Mice[END_REF][START_REF] Hamie | Imiquimod Targets Toxoplasmosis Through Modulating Host Toll-Like Receptor-MyD88 Signaling[END_REF], but there is not any research on vaccine design against T. gondii. Although TLR agonists are powerful immune stimulators, induced unwanted cytokine release syndrome leading to potential adverse events and safety concerns is a major factor limiting their usage. To minimize these side effects and to enhance their efficacy, TLR agonists are increasingly combined with delivery systems including PLGA particles [START_REF] Hamdy | Enhanced antigen-specific primary CD4+ and CD8+ responses by codelivery of ovalbumin and toll-like receptor ligand monophosphoryl lipid A in poly (D,L-lactic-co-glycolic acid) nanoparticles[END_REF][START_REF] Ilyinskii | Adjuvantcarrying synthetic vaccine particles augment the immune response to encapsulated antigen and exhibit strong local immune activation without inducing systemic cytokine release[END_REF][START_REF] Kim | Poly (d,l-lactide-co-glycolide) Nanoparticles as Delivery Platforms for TLR7/8 Agonist-Based Cancer Vaccine[END_REF].

As PLGA nanoparticles can serve for dual-delivery of antigens and immunomodulators, in present study, mice vaccination was performed with rSAG1-PLGA nanoparticles in combination with IMQ or MPL encapsulated into PLGA nanoparticles in order to assess their impacts on eliciting immune responses and protection against T. gondii infection. Since, we previously proved that adsorption is a more suitable approach than encapsulation in antigen loading on PLGA nanoparticles [START_REF] Allahyari | In-vitroandin-vivocomparison of rSAG1-loaded PLGA prepared byencapsulation and adsorption methods as an efficient vaccine against Toxoplasma gondii[END_REF], rSAG1-PLGA was prepared by adsorption method. It is worthy to note that during encapsulation process, protein exposure to water-oil interface, harsh mechanical, thermal and chemical stresses could affect protein integrity and consequently its immunogenicity [START_REF] Van De Weert | Protein instability in poly (lactic-co-glycolic acid) microparticles[END_REF]. Protection was evaluated against chronic infection in CBA/J mice susceptible to cyst formation and development of toxoplasmosis encephalitis. In addition, mice were challenged bythe oral route, the major natural route of infection, with tissue cysts.

Materials and Method

Materials

Poly (D, L-lactide-co-glycolide) polymer (PLGA), Resomer®RG503 (50:50, lactide: glycolide ratio) (viscosity 0.32-0.44 dl/g) was purchased from Boehringer Ingelheim, Germany. PVA [poly vinyl alcohol; molecular weight (MW) 30,000-70,000 Da, 88% hydrolyzed] was obtained from Sigma Chemical Company. The materials applied for SDS PAGE gel electrophoresis and protein molecular weight marker were supplied from Roche Applied Sciences (Mannheim, Germany) and Fermentas, Vilnius, Lithuania, respectively. Dichloromethane (DCM) (analytical grade) was obtained from Merck Ltd. Cell culture reagents including RPMI-1640, Fetal Calf Sera (FCS), HEPES, L-glutamine, sodium pyruvate, penicillin, and streptomycin, were obtained from Gibco (Life Technologies GmbH, Karlsruhe, Germany). Concanavalin A (conA), lyophilized powder of MPL (monophosphoryl from Salmonella enterica serotype minnesota Re 595) were purchased from Sigma-Aldrich (Darmstadt, Germany). Imiquimod Vacci Grade TM was supplied from InvivoGen (CA, USA).

All other chemicals used were of analytical reagent grade. All solutions were prepared by MilliQ™ ultrapure (Milli-QSystem, Millipore, Molsheim, France).

Purification and characterization of rSAG1 protein

rSAG1 protein was cloned, expressed and purified according to our previous studies and its purity, antigenicity, and immunogenicity were confirmed [START_REF] Allahyari | Production of in-vitro refolded and highly antigenic SAG1 for development of a sensitive and specific Toxoplasma IgG ELISA[END_REF]. In order to remove of contaminants including LPS, and E.coli DNA, Sartobind® Q strong 0.08 mL (Sartorius) was used. Briefly, rSAG1 buffer was exchanged to binding buffer of Sartobind® Q strong (Tris-HCl 20 mM, NaCl 50 mM, pH 8) by MicroSpin G-25 column. After column equilibration by mentioned buffer, rSAG1 was applied to Sartobind® Q strong. The Sartobind® Q strong operated as negative chromatography, so DNA attached to the column, and rSAG1 passed through Sartobind® Q strong. The concentration of bacterial endotoxin in purified rSAG1 was measured by limulus amboebocyte lysate assay using LAL chromogenic endotoxin quantitation kit (Pierce®, Thermo scientific, USA).

The preparation of different nanoparticles

Encapsulation of IMQ and MPLwas carried out completely according to the preparation of blank PLGA mentioned in our previous study [START_REF] Allahyari | In-vitroandin-vivocomparison of rSAG1-loaded PLGA prepared byencapsulation and adsorption methods as an efficient vaccine against Toxoplasma gondii[END_REF] using double emulsion solvent evaporation technique. The only distinction was the substitution of IMQ or MPL ligands for PBS. The IMQ and MPL were usedin concentration of 5 mg/mL and 2.5 mg/mL, respectively. rSAG1 was adsorbed on blank PLGA as stated by Allahyari et al. [START_REF] Allahyari | Synergistic effect of rSAG1 and rGRA2 antigens formulated in PLGA microspheres in eliciting immune protection against Toxoplasama gondii[END_REF].

Various batches of individual PLGA nanoparticles loaded by IMQ, MPL, and rSAG1 were prepared to provide sufficient quantity of homogenous nanoparticles. rSAG1-PLGA nanoparticles were mixed with either IMQ-PLGA or MPL-PLGA nanoparticles to make different formulations just before each immunization.

Nanoparticles characterization

Particle size, zeta potential and polydispersity index

The particle size (Z-average mean), particle size distribution (PSD), polydispersity index (PDI) and surface charge of all nanoparticles were determined based on Allahyari et al. [START_REF] Allahyari | Synergistic effect of rSAG1 and rGRA2 antigens formulated in PLGA microspheres in eliciting immune protection against Toxoplasama gondii[END_REF].

Both types of measurements were performed at 25°C using a ZetasizerNano ZS (Malvern Instruments, Worcestershire, UK). All measurements were performed in triplicate.

Process yield determination

Process yield for both blank PLGA nanoparticles and PLGA nanoparticles loaded by rSAG1, IMQ, and MPL were assessed in three preparation batches as mentioned in Allahyari et al. [START_REF] Allahyari | In-vitroandin-vivocomparison of rSAG1-loaded PLGA prepared byencapsulation and adsorption methods as an efficient vaccine against Toxoplasma gondii[END_REF].

Evaluation of encapsulation efficiency

Encapsulation efficiency of IMQ, and MPL into PLGA nanoparticles was determined by direct methods. Therefore, after hydrolysis of individual encapsulated PLGA nanoparticles, the quantity of encapsulated IMQ and MPL was calculated in definite amount of PLGA nanoparticles. The quantification of IMQ was done using spectrophotometry. Standard calibration curve of IMQ was drawn by ascertaining adsorption of definite concentration of IMQ at 320 nm. Afterwards, IMQ encapsulated in PLGA nanoparticles was quantified against standard curve. The amount of encapsulated MPL into PLGA nanoparticles was calculated by reverse phase (RP) HPLC (Kenuver, AZURA) through the area under curve (AUC), in comparison with defined amount of MPL. Encapsulated MPL was extracted by methanol, choloroform solvents (2:1 v/v). After centrifugation (at 20000 rpm, 20 mins), 50 µl of supernatant was loaded on TSKgel Octadecyl-4PW column (2.0 mm ID × 15.0 cm length, particle size 7.0 μm) (TOSOH Bioscience). The elution was carried out with a linear gradient at a flow rate of 0.5 mL/min (Sykam delivery system S2100 and at 210 nmusing UV detector (Sykam UV/Vis detector S3210) according to Kazzaz et al. [START_REF] Kazzaz | Encapsulation of the immune potentiators MPL and RC529 in PLG microparticles enhances their potency[END_REF]. Evaluation of all encapsulation efficiencies was evaluated by direct method according to following equation in triplicate:

Encapsulation efficiencies % = "Amount of TLR ligand encapsulated into PLGA nanoparticles"/"Total amount of TLR ligand used for encapsulation" × 100. Blank PLGA nanoparticles were used as negative control.The amount of rSAG1 adsorbed on PLGA nanoparticles was calculated as described by Allahyari et al. [START_REF] Allahyari | Synergistic effect of rSAG1 and rGRA2 antigens formulated in PLGA microspheres in eliciting immune protection against Toxoplasama gondii[END_REF].

Vaccination studies

Mice immunization and Vaccination schedules

The 6-week-old female CBA/J (H-2k) mice (Janvier, Le Genest St. Isle, France) resistant to acute toxoplasmosis infection and susceptible to cyst formation in chronic infection were applied in present study. Mice were kept under pathogen-free conditions in the animal house of University of Tours. Experiments were performed in accordance with the guideline for animal experimentation (EU Directive 2010/63/EU) and the protocol was approved by the local ethics committee (number 00807-02, Comité d'Ethique en Expérimentation Animale Val de Loire). In-vivo studies were divided in two experiments. Vaccination groups in first experiment and second experiment were shown in table 1 and table 2, respectively. CBA/J (H-2k) mice were randomly divided into groups of 12 mice, except for IMQ-PLGA control group in second experiment: only four mice for cellular analysis. All groups were immunized subcutaneously (s.c.) two times at 3-week intervals with different formulations containing 20 μg rSAG1, IMQ (64.5μg), and MPL (34 μg), as represented in Table 1and 2, respectively.

T. gondii extract (TE) preparation

Preparation of TE containing both cytoplasmic and membrane antigens was prepared from tachyzoites of the RH strain obtained by serial passaging in human foreskin fibroblast (HFF) cell monolayers, as previously described [START_REF] Ismael | The MIC3 gene of Toxoplasma gondii is a novel potent vaccine candidate against toxoplasmosis[END_REF]. Briefly, the obtained tachyzoites were washed in PBS and sonicated for three 10-min periods at 60 W/s. The Toxoplasma sonicate was centrifuged at 2,000 g for 30 min. The protein concentration was determined in the supernatant by the Micro BCA protein assay reagent kit using bovine serum albumin (BSA) as the standard (Pierce, Rockford, III.). The TE was stored at -20°C until use.

Humoral response

Anti SAG1-specific IgG antibodies were measured by ELISA on sera collected three weeks after second immunization as previously described [START_REF] Ismael | The MIC3 gene of Toxoplasma gondii is a novel potent vaccine candidate against toxoplasmosis[END_REF]. Briefly, Flat-bottomed 96-well plates (Nunc) were coated with 10 μg/mL TE. Serial two-fold dilutions of serum were performed (starting at a 1:100 dilution) and added to the wells. Sample of naive mice (untreated) served as negative controls. Bound antibodies were detected with Goat anti-Mouse IgG alkaline phosphatase (1:5,000, Sigma). The optical density of each sample was read at 405 nm. The endpoint antibody titer for each sample is given as the reciprocal of the highest dilution producing an OD that was 2.5-fold greater than that of the serum of naïve mice (serum of naïve mice gave optical density readings of less 0.1).

The levels of anti-SAG1 IgG subclasses were measured by ELISA as described above, except that sera were added to the plates at a single dilution (1:100 for sera of mice immunized with rSAG1, rSAG1-PLGA, rSAG1-PLGA and IMQ-PLGA, PLGA, IMQ-PLGA or MPL-PLGA nanoparticles and 1:800 for sera of mice immunized with rSAG1-PLGA and MPL-PLGA nanoparticles). The alkaline phosphatase-conjugated Rat anti-Mouse IgG1 and IgG2a were used at 1:1,000 (BD Pharmingen). IgG subclasses were evaluated using optical density.

Western Blotting

Western blottings were performed as previously described with either rSAG1 or T. gondii extractas the source of antigenand pooled sera obtained from mice in each group three weeks after second immunizations [START_REF] Ismael | The MIC3 gene of Toxoplasma gondii is a novel potent vaccine candidate against toxoplasmosis[END_REF]. rSAG1 (6 µg/1-cm-wide slot) or TE (60 µg/1-cm-wide slot) were electrophoresed in a 12 % SDS-polyacrylamide gel (SDS-PAGE) under non-reducing conditions, transferred to nitrocellulose, and probed with pooled sera from untreated mice (T-), T. gondii infected mice (inf), mice immunized with soluble rSAG1 in PBS (G1), rSAG1-PLGA (G2), rSAG1-PLGA and IMQ-PLGA (G3), PLGA (G4) or IMQ-PLGA (G5). Sera were diluted at 1:100. A mouse monoclonal antibody (mAb), anti-SAG1 MAb 1E5 (diluted at 1:100) isused as positive control [START_REF] Couvreur | Surface antigens of Toxoplasma gondii[END_REF]. Anti-SAG1 mAB 1E5 was kindly provided by Jean-François Dubremetz.

Cellular response (cytokine assay)

Four mice in each group were sacrificed three weeks after the last immunization. Single spleen cell suspensions were individually obtained by filtration through nylon mesh.

Erythrocytes were removed by lysis (hypotonicshock) and the remaining cells were washed and suspended in RPMI 1640 medium supplemented with 5% FBS, 25 mM HEPES, 2 mM Lglutamine, 1 mM sodium pyruvate, 50 µM β-mercaptoethanol, 100 U/mL penicillin, and 100 µg/mL streptomycin. Cells (5 × 10 5 cells/well) were stimulated in triplicate with 4 µg/mL endotoxin depleted rSAG1, or with medium alone (negative control). Concanavalin A (5 µg/mL) was used as a positive control for proliferation. Supernatants were harvested and assayed for IL-2 after 24 h and for IFN-γ, IL-5, IL-10 after 72 h. The concentrations of cytokines were determined using ELISA kits (eBioscience, San Diego, CA) according to the manufacturer's protocol. Cytokine concentrations were determined by reference to standard curves constructed with known amounts of cytokines provided by the kits.

Challenge

Mice (8/group) were orally challenged with 15 cysts of the 76K T. gondii strain, three weeks after the last immunization. To evaluate the protection, brain cyst loads were evaluated one month after challenge. Mouse brains were homogenized in 5 mL of RPMI medium, and the number of tissue cysts per brain was determined microscopically by counting 8 samples (10 μl) of each mouse brain by light microscope.

Statistical analyses

Statistical significance was analyzed using GraphPad Prism software. Statistical analysis was done by one-way ANOVA followed by a Tukey's multiple comparison test or using a Kruskall-Wallis test followed by Dunn's multiple comparison test. p < 0.05 was considered to be statistically significant.

Results

Production of rSAG1

rSAG1 was expressed and purified, its antigenicity and immunogenicity were confirmed as described in our previous study [START_REF] Allahyari | Production of in-vitro refolded and highly antigenic SAG1 for development of a sensitive and specific Toxoplasma IgG ELISA[END_REF]. Endotoxin concentration in purified protein was determined less than 0.05 EU/μg by limulus amboebocyte lysate assay.

Nanoparticles characterization

All different nanoparticles were identified regarding zeta average size, PDI, zeta potential, encapsulation efficiency and process yield. The results were summarized in Table 3.

The sizes of IMQ-PLGA and MPL-PLGA were about 451.3 ± 27and 403 ± 25 nm, respectively. There was significant difference (p < 0.05) between mean sizes of MPL-PLGA and IMQ-PLGA or rSAG1-PLGA nanoparticles. The difference between mean sizes of IMQ-PLGA and rSAG1-PLGA was not significant. All prepared nanoparticles share a PDI less than 0.2. Surface charges of IMQ-PLGA, MPL-PLGA, nanoparticleswere negative, -4.9 ± 0.26 and -5.6 ± 0.75 mV,respectively. Among all nanoparticles, rSAG1-PLGA showed the lowest negative charge about -2.37 ± 0.3 mV. Zeta potential in rSAG1-PLGA was significantly (p < 0.001) lower than all other PLGA nanoparticles.

All statistical analysis was done by one way ANOVA. The efficacy of IMQ encapsulation into IMQ-PLGA nanoparticles measured by spectrophotometry method was about 70 ± 3.1 % (Table 3), which was calculated through interpolating of absorbance with the concentration.

In addition, as shown in Table 3, the encapsulation efficacy of MPL into MPL-PLGA nanoparticles was quantified by RP-HPLC through comparison of AUC of encapsulated MPL and AUC related to defined amount of MPL, nearly 73.1 ± 1.2 (Fig. 1).

The adsorption efficacy of rSAG1-PLGA nanoparticles was 69.01 ± 1.8 %. Blank PLGA and rSAG1-adsorbed PLGA were prepared as mentioned by Allahyari et al. [START_REF] Allahyari | Synergistic effect of rSAG1 and rGRA2 antigens formulated in PLGA microspheres in eliciting immune protection against Toxoplasama gondii[END_REF]. The results represent mean ± SD of 5 independent PLGA nanoparticle preparations.

3.3Vaccination with rSAG1-PLGA in combination with IMQ-PLGA

The effect of IMQ adjuvant, when encapsulated into PLGA and co-administrated with rSAG1-PLGA was first investigated. CBA/J mice (12 mice/group) were immunized subcutaneously (s.c.) two times at 3 weeks interval with rSAG1-PLGA nanoparticles or with the mixture of rSAG1-PLGA and IMQ-PLGA nanoparticles. Control mice were immunized with rSAG1 in PBS, blank PLGA nanoparticles or IMQ-PLGA nanoparticles (Table 1).The induced humoral and cellular immune responses and the protective efficacy against chronic toxoplasmosis following oral challenge, were evaluated.

Humoral immune responses

The humoral response was analyzed three weeks after the second immunization.

Immunoblot analysis of mouse sera was first performed against rSAG1 (Fig. 2A). Sera from mice immunized either with rSAG1, rSAG1-PLGA or rSAG1-PLGA and IMQ-PLGA reacted mainly with two protein bands corresponding to rSAG1 monomers at the expected size (around 30 kDa). These sera reacted also with dimers (around 70 kDa) and probably multimers of rSAG1 (> 70 kDa). These bands are also recognized by serum IgG antibodies from T. gondii infected mice (inf). Monoclonal antibodie 1E5 reacted only with the two protein bands corresponding to rSAG1 monomers. Monoclonal antibodie 1E5 recognizes a conformational epitope which may not be accessible (or problem of conformation) in the multimeric forms of rSAG1. Importantly, immunoblot analysis of mouse sera performed against T. gondii extract, showed that sera from mice immunized either with rSAG1, rSAG1-PLGA or rSAG1-PLGA and IMQ-PLGA reacted strongly with native SAG1 (Fig. 2B). As expected, sera from mice immunized with PLGA (G4) and IMQ-PLGA (G5), and untreated mice (T-) identified no protein in rSAG1 or T. gondii extract. ELISA using T. gondii extract containing native SAG1 as coating antigen was used to determine endpoint anti-SAG1 IgG antibodies titers (Fig. 2C). In our experimental conditions, anti-SAG1-specific IgG antibodies were detected only in 6/12 mice immunized with rSAG1 in PBS (for 6 mice, OD value failed to reach 2.5 times the background value at the initial dilution tested), whereas all mice immunized with rSAG1-PLGA nanoparticles produced detectable anti-SAG1-specific IgG antibodies. These results indicated that a stronger humoral immune response was induced in mice immunized with rSAG1-PLGA nanoparticles compared to mice immunized with rSAG1 alone, however, the difference between the two immunized groups did not reach statistical significance. In mice immunized with rSAG1-PLGA and IMQ-PLGA nanoparticles, higher antibodies titers were found compared to mice immunized either with rSAG1 in PBS or rSAG1-PLGA nanoparticles, however, a significant statistical difference was found only between the rSAG1 and rSAG1-PLGA+IMQ-PLGA groups (p < 0.01). Mice vaccinated with PLGA nanoparticles or IMQ-PLGA nanoparticles did not produce any anti-SAG1 antibodies. To find out whether a Th1 and/or a Th2 humoral response was induced by immunization, the IgG subclasses were analyzed (Fig 2D). The levels of IgG1 exceeded those of IgG2a in sera of mice immunized with PLGA nanoparticles with or whithout IMQ (IgG1/IgG2a OD ratio > 2), suggesting mixed Th1/Th2 humoral-type response with a bias towards a Th2 response. In conclusion, vaccination with the mixture of rSAG1-PLGA and IMQ-PLGA enhanced a mixed Th1/Th2 humoral response witha bias towards a Th2 response.

Cellular immune responses

The potency of the various formulations to induce T cell immune responses was investigated by measuring the specific cytokine responses in spleen three weeks after second immunization. Splenocytes from mice immunized with rSAG1-PLGA or the mixture of rSAG1-PLGA and IMQ-PLGA nanoparticles elicited significant (p < 0.05) amounts of IFN-γ, IL-2, TNF-α (Th1 cytokines) and IL-5 (Th2 cytokine) in response to rSAG1 stimulation compared to their respective control (PLGA and IMQ-PLGA, respectively). However, any significant difference in cytokine production was shown between the two above-mentioned immunized groups. Furthermore, splenocytes from mice immunized with rSAG1 elicited very low levels of IFN-γ, IL-2, TNF-α and IL-5 as compared to mice immunized with rSAG1-PLGA or the mixture of rSAG1-PLGA and IMQ-PLGA. In addition, the amounts of IFN-γ, TNF-α, IL-5 (p < 0.05) in mice immunized with rSAG1-PLGA and the level of IL-2 (p < 0.01) in mice immunized with the mixture of rSAG1-PLGA and IMQ-PLGA were significantly higher than those in mice immunized by rSAG1. Although the level of IL-5 significantly increased in groups immunized with rSAG1-PLGA and the mixture of rSAG1-PLGA and IMQ-PLGA nanoparticles in comparison with control groups, all immunized groups and control groups share a considerably low amount of induced IL-5.

Fig. 3. Cellular immune response in mice immunized with rSAG1-PLGA nanoparticles

or the mixture of rSAG1-PLGA and IMQ-PLGA nanoparticles. Three weeks after the second immunization, splenocytes were collected from four mice in each group, and cultured with 4 µg/mL rSAG1 or with medium alone (no stimulation). Cell-free supernatants were harvested after 24 (IL-2) or 72 h (IFN-γ, IL-5, and TNF-α) for cytokine assays. Results are expressed as the median and interquartile range. Kruskall-Wallis, Dunn's multiple comparisons test. *p < 0.05; **p < 0.01 significant differences between the immunized groups.

Challenge studies

In order to evaluate the protective effect of these vaccine formulations against chronic toxoplasmosis, mice were orally challenged with15 cysts of the 76K strain 3 weeks after the second immunization and sacrificed one month after challenge.

The numbers of brain cyst after challenge were significantly decreased in mice immunized with the mixture of rSAG1-PLGA and IMQ-PLGA compared to both control groups (p < 0.05 compared to PLGA control group; p < 0.01 compared to IMQ-PLGA control group, respectively). In addition, the reduction in brain cyst load was attained in rSAG1-PLGA immunized mice compared to IMQ-PLGA control group (p < 0.05), but not compared to PLGA control group. Mice immunized with the mixture of rSAG1-PLGA and IMQ-PLGA and mice immunized with rSAG1-PLGA shared the least brain cyst loads about 51% and 42% brain cyst reduction, respectively, versus control IMQ-PLGA group (p < 0.01). However, the difference between these two groups did not meet any statistical significance. Moreover, mice immunized with rSAG1 alone did not show any significant reduction in brain cyst compared to control groups.

Vaccination with rSAG1-PLGA in combination with IMQ-PLGA or MPL-PLGA

After the investigation of the effect of IMQ, effect of MPL adjuvant, when encapsulated into PLGA and co-administrated with rSAG1-PLGA, was evaluated. CBA/J mice were immunized subcutaneously (s.c.) two times at 3 week intervals with the mixture of rSAG1-PLGA and MPL-PLGA nanoparticles (12 mice) or with the mixture of rSAG1-PLGA and IMQ-PLGA nanoparticles (12 mice). Control mice were immunized with MPL-PLGA ( 12mice) or IMQ-PLGA nanoparticles (4 mice) for the analysis of the cellular immune response (Table2).

Humoral immune responses

Higher endpoint anti-SAG1 IgG titers were found in mice immunized with rSAG1-PLGA and MPL-PLGA nanoparticles (Fig 5A ), compared to mice immunized with rSAG1-PLGA and IMQ-PLGA nanoparticles (p < 0.01). In both groups, the levels of IgG1 exceeded those of IgG2a (Fig 5B ) with a IgG1/IgG2a OD ratio slightly different (rSAG1-PLGA and IMQ-PLGA nanoparticles, ratio > 2; rSAG1-PLGA and MPL-PLGA nanoparticles, ratio < 2).

Compared to IMQ, MPL potentiated the humoral response and induced a more balanced Th1/Th2 humoral response (IgG1/IgG2a OD around 1.5).

Cellular immune responses

Splenocytes from mice immunized with the mixture of rSAG1-PLGA and IMQ-PLGA nanoparticles or rSAG1-PLGA and MPL-PLGA nanoparticles elicited significant (p < 0.05) levels of IFN-γ, IL-2, TNF-α (Th1 cytokines) and IL-5 (Th2 cytokine) in response to rSAG1 stimulation compared to their respective control (IMQ-PLGA and MPL-PLGA, respectively).

More IL-2 and IFN-γ were found in the supernatants of restimulated splenocytes from mice immunized with the mixture of rSAG1-PLGA and MPL-PLGA nanoparticles compared to those from mice immunized with rSAG1-PLGA and IMQ-PLGA. However the differences did not reach statistical significance. Three weeks after the second immunization, splenocytes were collected from four mice in each group, and cultured with 4 µg/mL rSAG1or medium alone (no stimulation). Cell-free supernatants were harvested after 24 (IL-2) or 72 h (IFN-γ, IL-5, and TNF-α) for cytokine assays. Results were expressed as the median and interquartile range. Mann Whitney test was done. "ns", no significant difference between the two immunized groups.

Challenge studies

Significant protection was displayed in the group of mice immunized with the mixture of rSAG1-PLGA and MPL-PLGA compared to its control group (MPL-PLGA) (p < 0.01), demonstrating 41% brain cyst reduction. The brain cyst number of mice immunized with the mixture of rSAG1-PLGA and MPL-PLGA was similar to that of mice immunized with the mixture of rSAG1-PLGA and IMQ-PLGA (1583 +/-427 versus 1698 +/-519). These results suggest that in our experimental conditions, MPL and IMQ have similar protective effects against chronic toxoplasmosis when encapsulated into PLGA nanoparticles and coadministrated with rSAG1-PLGA nanoparticles.

Fig. 7. Evaluation of the protection against chronic toxoplasmosis in mice immunized

with the mixture of rSAG1-PLGA and MPL-PLGA or with the mixture of rSAG1-PLGA and IMQ-PLGA nanoparticles. Mice in all groups were orally challenged with15 cysts of the 76 K strain, 3 weeks after second immunization. One month after challenge, all mice were sacrificed and the number of brain cysts in each mouse was calculated individually. Data represent the mean ± SEM of measurements from eight mice. ANOVA, Tukey's multiple comparison test. **p < 0.01; "ns", no significant difference.

Discussion

The main goal of this study was to investigate the adjuvanticity of IMQ and MPL loaded into PLGA nanoparticles when administrated with rSAG1 antigen adsorbed on PLGA nanoparticles and to compare the protective efficacies against T. gondii chronic infection in CBA/J after challenge with T. gondii cysts by the oral route.It has been well established that vaccine delivery systems can compensate for poor immunogenicity of subunit vaccines.

Hence, both PLGA micro and nanoparticles were applied to serve as delivery system and adjuvant dual role for recombinant SAG1 proteins [START_REF] Chuang | Induction of long-lasting protective immunity against Toxoplasma gondii in BALB/c mice by recombinant surface antigen 1 protein encapsulated in poly (lactide-co-glycolide) microparticles[END_REF][START_REF] Naeem | Induction of Th1 typeoriented humoral response through intranasal immunization of mice with SAG1-Toxoplasma gondii polymeric nanospheres[END_REF], a multimeric recombinant T. gondii vaccine including SAG1 epitopes [START_REF] Roozbehani | Characterization of a multi-epitope peptide with selective MHC-binding capabilities encapsulated in PLGA nanoparticles as a novel vaccine candidate against Toxoplasma gondii infection[END_REF] or a recombinant chimericprotein rSAG1/2 [START_REF] Chuang | Encapsulation of chimeric protein rSAG1/2 into poly(lactide-co-glycolide) microparticles induces long-term protective immunity against Toxoplasma gondii in mice[END_REF]. To further potentiate protective immunity, particulate delivery of rSAG1 can be coadministrated with an immunostimulant molecule.

In our design, IMQ and MPL that target TLR7/8 and TLR4, respectively, were encapsulated in PLGA nanoparticles to be co-administrated with rSAG1 adsorbed on PLGA nanoparticles. Among the properties of PLGA nanoparticles including size, shape, surface charge and hydrophobicity, particle sizes are of high priority and play considerable role in antigen uptake by antigen presenting cells (APCs) [START_REF] Bachmann | Vaccine delivery: a matter of size, geometry, kinetics and molecular patterns[END_REF][START_REF] Danhier | PLGA-based nanoparticles: an overview of biomedical applications[END_REF]. In the present study, all prepared PLGA nanoparticles (rSAG1-PLGA, MPL-PLGA, IMQ-PLGA) shared diameters less than 500 nm which are compatible with desired size mentioned by others. It is worthy to note that PLGA nanoparticles < 500 nm more potently elicit CTLs (cytotoxic T lymphocytes) response than larger ones (> 2 µm) [START_REF] Hamdy | Activation of antigenspecific T cell-responses by mannan-decorated PLGA nanoparticles[END_REF].

One of the most critical parameters affecting the size of PLGA-based nanoparticles is hydrophobicity orwater solubility of their cargo [START_REF] Allahyari | Synergistic effect of rSAG1 and rGRA2 antigens formulated in PLGA microspheres in eliciting immune protection against Toxoplasama gondii[END_REF]. MPL was solved in organic solvents (oil phase, first emulsion) like PLGA, while IMQ and rSAG1 were solved in aqueous solutions.

Hence, the disruption of MPL into PLGA oil phase resulted in smaller nanoparticles than those of IMQ and rSAG1. The above-mentioned reason well justifies why the size of MPL-PLGA was different from the size of IMQ-PLGA and rSAG1-PLGA nanoparticles. All prepared nanoparticles shared a PDI less than 0.2 indicating more or less homogeneity in the size of individual nanoparticles. The surface charge similarities among IMQ-PLGA and MPL-PLGA and blank PLGA nanoparticles were attributed to the method of preparation (encapsulation) and confirmed that IMQ and PLGA completely have been encapsulated into PLGA nanoparticles, while rSAG1-PLGA showed less negative charge because of rSAG1 adsorption on PLGA nanoparticles [START_REF] Allahyari | Synergistic effect of rSAG1 and rGRA2 antigens formulated in PLGA microspheres in eliciting immune protection against Toxoplasama gondii[END_REF]. It is worthy to note that we previously proved that adsorption is a more suitable approach than encapsulation in antigen loading on PLGA nanoparticles [START_REF] Allahyari | In-vitroandin-vivocomparison of rSAG1-loaded PLGA prepared byencapsulation and adsorption methods as an efficient vaccine against Toxoplasma gondii[END_REF]. It should be mentioned that mixing different nanoparticles to prepare two formulations (PLGA-IMQ + rSAG1-PLGA and PLGA-MPL + rSAG1-PLGA) may affect nanoparticles features. Thus, the characterizations of different nanoparticles after mixing and preparing final vaccine formulations are necessary in order to provide better understanding of nanoparticles behavior at least in vitro. Unfortunately, we did not perform these characterizations before vaccination studies in this research. It should be emphasized that one of the most critical consequences of stability issues regarding nanopoarticles is aggregation and particle precipitation. As this problem often occurs during time, different PLGA nanoparticles of each vaccine formulation were mixed just before injection. Fortunately, any precipitation or aggregation was not observed after mixing different PLGA nanoparticles of each vaccine formulation. Absolutely, if aggregation was happened, the needle would be clogged and syringe aspiration and injection to mice would be impossible.

Another possible outcome of mixed formulations would be its influence on rSAG1 adsorption efficiency. In the light of this fact, the mixing of two nanoparticles (rSAG1-PLGA and IMQ-PLGA or rSAG1-PLGA and MPL-PLGA), and formulations preparation were done exactly before injection to mice in order to eliminate or decrease any adverse effect of nanoparticles mixing on rSAG1 adsorption in-vitro. Of course, if rSAG1 adsorption was measured after mixing of different nanoparticles, it could provide satisfying insight in this regard.

In first experiment, mice vaccinated with the mixture of rSAG1-PLGA and IMQ-PLGA developed the highest amount of SAG1-specific IgG antibodies, with amixed Th1/Th2 humoral response biased towards a Th2 response similar to rSAG1-PLGA group (IgG1/IgG2a OD ratio >2). While, SAG1-specific IgG antibodies titers were higher in mice vaccinated with the mixture of rSAG1-PLGA and MPL-PLGA in comparison with those vaccinated with the mixture of rSAG1-PLGAand IMQ-PLGA in second experiment, with a more balanced Th1/Th2-type antibody response (IgG1/IgG2a OD ratio < 2). Therefore, we could deduce that the formulation rSAG1-PLGA, MPL-PLGA resulted in more potent humoral responses than rSAG1-PLGA, IMQ-PLGA against SAG1. These findings clearly confirmed the role of IMQ and MPL in eliciting antibody response against SAG1, although MPL showed higher impact than IMQ. Mice vaccinated with rSAG1-PLGA and those with the mixture of rSAG1-PLGA and IMQ-PLGA did not show any significant differences in eliciting IFN-γ, IL-2 and TNF-α indicating Th1-biased immunity. Moreover, rSAG1-PLGA + MPL-PLGA group elicited higher amounts of mentioned cytokines than rSAG1-PLGA + IMQ-PLGA group, though no significant difference was observed. The less cyst numbers in mice immunized with rSAG1-PLGA, the mixture of rSAG1-PLGA and IMQ-PLGA, and the mixture of rSAG1-PLGA and MPL-PLGA were supported by higher amounts of Th1-associated cytokines and anti-SAG1 IgG antibodies in comparison with control groups.

The protective efficiency against toxoplasmosis depends on the Th1 immune response [START_REF] Sasai | Innate, adaptive, and cell-autonomous immunity againstToxoplasma gondii infection[END_REF].

Although antibodies action is limited to extracellular parasites, B-cell antibody responses play a role in preventing persistent proliferation of tachyzoites in the brain and lung during the chronic phase of infection [START_REF] Kang | Decreased resistance of B cell-deficient mice to infection with Toxoplasma gondii despite unimpaired expression of IFN-gamma, TNFalpha, and inducible nitric oxide synthase[END_REF]. SAG1 contains T epitopes [START_REF] Velge-Roussel | Differences in immunological response to a T. gondii protein (SAG1) derived peptide between two strains of mice: effect on protection in T. gondii infection[END_REF][START_REF] Nielsen | Complete protection against lethal Toxoplasma gondii infection in mice immunized with a plasmid encoding the SAG1 gene[END_REF] and neutralizing B epitopes [START_REF] Nielsen | Complete protection against lethal Toxoplasma gondii infection in mice immunized with a plasmid encoding the SAG1 gene[END_REF][START_REF] Mineo | Antibodies to Toxoplasma gondii major surface protein (SAG-1, P30) inhibit infection of host cells and are produced in murine intestine after peroral infection[END_REF][START_REF] Velge-Roussel | Anti-SAG1 peptide antibodies inhibit the penetration of Toxoplasma gondii tachyzoites into enterocyte cell lines[END_REF] and both the cellular responses [START_REF] Nielsen | Complete protection against lethal Toxoplasma gondii infection in mice immunized with a plasmid encoding the SAG1 gene[END_REF][START_REF] Velge-Roussel | Intranasal immunization with Toxoplasma gondii SAG1 induces protective cells into both NALT and GALT compartments[END_REF][START_REF] Mendes | MyD88dependent protective immunity elicited by adenovirus 5 expressing the surface antigen 1 from Toxoplasma gondii is mediated by CD8(+) T lymphocytes[END_REF] and the humoral responses [START_REF] Fu | Generation of a neutralizing human monoclonal antibody Fab fragment to surface antigen 1 of Toxoplasma gondii tachyzoites[END_REF][START_REF] Siachoque | Toxoplasma gondii: immunogenicity and protection by P30 peptides in a murine model[END_REF] directed against SAG1 were shown to play a role in the protection observed against acute or chronic T. gondii infections. More specifically, in CBA/J or C3H mice (with H-2k haplotype), parenteral immunizations with SAG1 or SAG1 peptides conferred protection against chronic T. gondii infection following oral challenge with T. gondii cysts, when Th1 immune responses were induced [START_REF] Velge-Roussel | Differences in immunological response to a T. gondii protein (SAG1) derived peptide between two strains of mice: effect on protection in T. gondii infection[END_REF][START_REF] Letscher-Bru | Protective effect of vaccination with a combination of recombinant surface antigen 1 and interleukin-12 against toxoplasmosis in mice[END_REF]. The brain cyst number in rSAG1-PLGA + MPL-PLGA group was less than those in rSAG1-PLGA + IMQ-PLGA group. Nevertheless, this difference was no significant. The lack of any significant difference in cyst number reduction between rSAG1-PLGA group and rSAG1-PLGA + IMQ-PLGA group is in accordance with their cytokine profiles and IgG1/IgG2a OD ratio.

Immune-enhancing effects of MPL have been reported for a number of antigens, including T. gondii antigens in mouse model [START_REF] Golkar | Evaluation of protective effect of recombinant dense granule antigens GRA2 and GRA6 formulated in monophosphoryl lipid A (MPL) adjuvant against Toxoplasma chronic infection in mice[END_REF][START_REF] Song | Vaccination with toxofilin DNA in combination with an alum-monophosphoryl lipid A mixed adjuvant induces significant protective immunity against Toxoplasma gondii[END_REF]. Adjuvant/delivery vehicles containing MPL have been developed and licensed in FDA-approved vaccines for human use, such as AS04 and AS01 [START_REF] Laupèze | Adjuvant Systems for vaccines: 13 years of post-licensure experience in diverse populations have progressed the way adjuvanted vaccine safety is investigated and understood[END_REF]. Compared to Aluminum salts, AS04 (Aluminium salt + MPL) is more efficient in inducing the amplification and differentiation of CD4 + T cells and promotes a Th1-biased response, MPL therefore provides a counterbalance to the Th2-differentiating properties of alum [START_REF] Didierlaurent | AS04, an aluminum salt-and TLR4 agonist-based adjuvant system, induces a transient localized innate immune response leading to enhanced adaptive immunity[END_REF]. In our experimental conditions, co-administration of rSAG1-PLGA and MPL-PLGA resulted in a more potent humoral response than rSAG1-PLGA and rSAG1-PLGA plus IMQ-PLGA, with a more polarized Th1 humoral response, confirming the ability of MPL to promote Th1 bias response. Compared to PLGA, IMQ potentiated the humoral response but did not potentiate the Th1 humoral response and did not improve the magnitude of the Th cell response. In present study, the enhanced humoral response obtained by coadministration of IMQ may be due to B cell-intrinsic TLR7 signaling. This signaling has been shown to promote vigorous memory B cell responses following co-immunization of an antigen with IMQ [START_REF] Castiblanco | Co-Stimulation of BCR and Toll-Like Receptor 7 Increases Somatic Hypermutation, Memory B Cell Formation, and Secondary Antibody Response to Protein Antigen[END_REF]. However, compared to PLGA, IMQ did not potentiate the Th1 humoral response and did not improve the magnitude of the Th cell response. This could be due to the modest Th1-polarizing responses from TLR7 pDC signaling [START_REF] Dowling | Recent Advances in the Discovery and Delivery of TLR7/8 Agonists as Vaccine Adjuvants[END_REF].

Our results are similar to those obtained by Kasturi et al. [START_REF] Kasturi | Programming the magnitude and persistence of antibody responses with innate immunity[END_REF] who compared the humoral responses induced in C57BL/6 mice subcutaneously immunized with hemagglutinin (HA) from avian influenza H5N1 virus encapsulated in PLGA nanoparticles combined or not with either PLGA-MPL or PLGA-IMQ nanoparticles. Similar results were also obtained with IMQ in an anionic liposome formulation, administrated by the intramuscular route to C57BL/6 mice with a recombinant Plasmodium berghei circumsporozoite protein which was compared to the TLR4 agonist Glucopyranosyl lipid adjuvant (GLA) [START_REF] Fox | A nanoliposome delivery system to synergistically trigger TLR4 AND TLR7[END_REF]. Importantly, compared to immunization with a single TLR ligand, either IMQ or MPL in PLGA nanoparticles, immunization with PLGA nanoparticles including both TLR ligands has been shown to induce synergistic responses. It would be of interest to investigate if whether or not, in our experimental conditions, combination of PLGA-MPL and PLGA-IMQ could synergize to enhance the protection. Dendritic cell targeting is also a promising strategy to provide protection against T. gondii. SAG1 targeting to DEC205 + dendritic cells via an antibody fragment single-chain fragment variable (scFv) by intranasal and subcutaneous administration to CBA/J mice has been shown to improve the protection against chronic T. gondii infection [START_REF] Lakhrif | Targeted Delivery of Toxoplasma gondii Antigens to Dendritic Cells Promote Immunogenicity and Protective Efficiency against Toxoplasmosis[END_REF]. This targeting strategyto dendritic cells, applied to ovalbumin as a model antigen encapsulated in targeted PLGA nanoparticles together with TLR agonists, led to strong enhancement of vaccine potency and induction of T cell responses compared to non-specific delivery of nanoparticles to dendritic cells [START_REF] Tacken | Targeted delivery of TLR ligands to human and mouse dendritic cells strongly enhances adjuvanticity[END_REF][START_REF] Cruz | Targeting nanoparticles to CD40, DEC-205 or CD11c molecules on dendritic cells for efficient CD8(+) T cell response: a comparative study[END_REF]. Furthermore, as Toxoplasma infection is mainly acquired by consumption of oocysts in contaminated water or vegetables or by ingestion of tissue cysts contained in infected meat, a vaccine strategy able to induce both systemic and mucosal immune responses would be of great interest to tackle the parasite at the portal of entry. Nasal administration is a suitable route to induce such immune responses and nanoparticles based on PLGA or PLGA derivatives have great potential for thisstrategy.

For example, PLGA nanoparticulate intranasal administration of a combined TLR7/NOD2 agonist with HIV p24 antigen was recently shown to induce high-quality humoral and adaptive responses both in systemicand mucosal compartments [START_REF] Gutjahr | New chimeric TLR7/NOD2 agonist is a potent adjuvant to induce mucosal immune responses[END_REF]. Since T. gondii is an intracellular parasite with various life cycles and antigenic variations, the development of effective vaccine against T. gondii is a challenging endeavour. This study used SAG1, a well known candidate vaccine antigen. SAG1 is a major and stage-specific antigen expressed in Toxoplasma tachyzoite and is highly conserved among virulent strains of T. gondii.

Combining SAG1 with antigens from the different T. gondii stages would undeniably improve vaccine efficacy.

Nowadays, vaccine strategy is implied as a puzzle that all its parts should be accurately designed and selected in order to achieve immunization goal. These pieces in a vaccine consist of antigen, adjuvant, delivery system, route of immunization and model.

Conclusion

Using SAG1 as a potential candidate vaccine and PLGA nanoparticles as delivery system, this study indicated that co-delivery of immunomodulators such as TLR agonists and antigens with PLGA nanoparticles could be appropriate to develop efficacious vaccines against toxoplasmosis.

Fig. 1 .

 1 Fig. 1. Quantification of encapsulated MPL in MPL-PLGA nanoparticles by RP-HPLC (Knauer, AZURA). The numbers presented on graphs demonstrated area under curve (AUC). Std, standard, is referred to defined amount of MPL. The column specification; TSKgel Octadecyl-4PW column (2.0 mm ID × 15.0 cm length, particle size 7.0 μm) (TOSOH Bioscience).

Fig. 2 .

 2 Fig. 2. Specific antibody response in miceimmunized with rSAG1-PLGA nanoparticles or the mixture of rSAG1-PLGA and IMQ-PLGA nanoparticles. Analysis of humoral response was done three weeks after second immunization. For immunoblot analysis, rSAG1 (A) or T. gondii extract (B) electrophoresed and transferred to nitrocellulose, were probed with pooled sera from immunized mice, T. gondii infected mice (inf) and untreated mice (T-). A mouse mAb (anti-SAG1 MAb 1E5) was used as positive control (mAb). The molecular weights are shown on the left side. Sera and and Mab 1E5 were diluted at 1:100. mAb (anti-SAG1 MAb 1E5), inf (T. gondii infected mice), G1 (rSAG1), G2 (rSAG1-PLGA), G3 (rSAG1-PLGA + IMQ-PLGA), G4 (PLGA), and G5 (IMQ-PLGA), T-(untreatedmice).C) Determination of specific anti-SAG1 antibody titers by ELISA using T. gondii extractas coating antigen. Sera from each mouse in each group (n = 12/group) were analyzed individually. The antigen-specific antibody titer (endpoint titer) was given as the reciprocal of the highest dilution making an optical density (OD) that was 2.5-fold greater than that of the serum of non-immunized mice. Dotted line represent the lowest mouse sera dilution tested. Titers below the limit of detection (< 100) were assigned a value of 50 for analysis. Symbols

Fig. 4 .

 4 Fig. 4. Evaluation of the protection against chronic toxoplasmosis in mice immunized with rSAG1-PLGA nanoparticles or the mixture of rSAG1-PLGA and IMQ-PLGA nanoparticles. Mice in all groups were orally challenged with 15 cysts of the 76 K strain 3 weeks after the second immunization. One month after challenge, all mice were sacrificed (n = 8/group) and the number of brain cysts in each mouse was calculated individually. Data represent the mean ± SEM of measurements from eight mice. ANOVA, Tukey's multiple comparison test. *p < 0.05 and **p < 0.01.

Fig. 5 .

 5 Fig. 5. Specific antibody response in mice immunized with the mixture of rSAG1-PLGA and MPL-PLGA or with the mixture of rSAG1-PLGA and IMQ-PLGA nanoparticles. Analysis of humoral response was done three weeks after the second immunization. A). Determination of specific anti-SAG1 antibody titers by ELISA using T. gondii as coating antigen. Sera from each mouse in each group were analyzed individually (n = 12/group, except for IMQ-PLGA nanoparticles group, n = 4). The antigen-specific antibody titer (endpoint titer) was given as the reciprocal of the highest dilution making an optical density (OD) that was 2.5-fold greater than that of the serum of non-immunized mice. Dotted line represent the lowest mouse sera dilution tested. Symbols represent individual animals. Results are presented on scatter plots as geometric mean with the 95% confidence interval. Kruskall-Wallis, Dunn's multiple comparisons test. **p < 0.01. B) Determination of the IgG subclasses profiles by ELISA using T. gondii extract as coating antigen. Sera (n = 6/group, except for IMQ-PLGA nanoparticles group n = 4) were evaluated individually.Results are expressed as the mean of the optical density (OD) ± SEM. Sera were tested at a single dilution (rSAG1-PLGA + IMQ-PLGA group and control groups, dilution 1:100 ; rSAG1-PLGA + IMQ-PLGA group, dilution 1:800 ).

Fig. 6 .

 6 Fig. 6. Cellular immune response in mice immunized with the mixture of rSAG1-PLGA and MPL-PLGA or with the mixture of rSAG1-PLGA and IMQ-PLGA nanoparticles.

Figure 2 mAb

 2 Figure 2

Table 3 .

 3 Characterization of IMQ-PLGA, MPL-PLGA, rSAG1-PLGA and blank PLGA nanoparticles. Results represent mean ± SD of five independent PLGA particle preparation batches.

	Formulation	Size (nm)	PDI	Zeta potential (mV)	A/E efficiency (%)	Yield (%)
	IMQ-PLGA	451.3 ± 27	0.15 ± 0.02	-5.2 ± 0.26	70 ± 3.1*	87.3 ± 2.8
	MPL-PLGA	403 ± 25	0.18 ± 0.04	-5.6± 0.75	73.1± 1.2*	88.4± 2
	rSAG1-PLGA	447 ± 9	0.11 ± 0.05	-2.37 ± 0.3	69.01 ± 1.8**	88.4 ± 5.1
	Blank PLGA	412 ± 13	0.10 ± 0.02	-6.15 ± 0.7	-	90.3 ± 2.7

Acknowledgments

Mojgan Allahyari was supported by a PhD research scholar ship from Pasteur Institute of Iran, Iran (grant N° BD-8831). Also, the recurring funds for this study were supported by University of Tours. Authors thank B. Héraut for skillful technical assistance.

Table1. First experiment