Walid Saad
email: saad@unik.no

Zhu Han

Mérouane Debbah
email: merouane.debbah@supelec.fr

Are Hjørungnes

Tamer Bas ¸ar

A Game-Based Self-Organizing Uplink Tree for VoIP Services in IEEE 802.16j Networks

In this paper, we propose a game theoretical approach to tackle the problem of the distributed formation of the uplink tree structure among the relay stations (RSs) and their serving base station (BS) in an IEEE 802.16j WiMAX network. Unlike existing literature which focused on the performance assessment of the network in the presence of the RSs, we investigate the topology and dynamics of the tree structure in the uplink of an 802.16j network. We model the problem as a network formation game, with the RSs being the players. In the proposed game, each RS aims to maximize its utility that accounts for the gains from cooperation in terms of bit error rate (BER) and the costs resulting from multi-hop transmission in terms of delay. The proposed utility model is based on the concept of the R-factor which is a parameter suitable for assessing the performance of VoIP services. For forming the tree structure, we propose a distributed myopic best response dynamics in which each RS can autonomously choose the path that connects it to the BS through other relays while optimizing its utility. Using the proposed dynamics, the RSs can self-organize into the tree structure, and adapt this topology to environmental changes such as mobility while converging to a Nash tree network. Simulation results show that the proposed algorithm presents significant gains in terms of average achieved MS utility reaching up to 40.3% compared to the star topology where all RSs are directly connected to the BS, and up to 42.75% compared to the case with no RSs. The results also show that the average number of hops in the tree does not exceed 4 even for a network with a large number of RSs.

I. INTRODUCTION

Distributed spatial diversity using cooperative relaying has been recently proposed as an effective technique for mitigating the fading effects of the wireless channel. By cooperative relaying, several nodes can cooperate with a given source node in the transmission of its data to a far away destination, consequently, providing spatial diversity gains for the source node without the burden of having several antennas physically present on the node. This class of cooperative relaying is known as cooperative communications [1]. It has been shown that by using one or multiple relays [1-3] a significant improvement can be witnessed in terms of bit error rate, throughput or other QoS parameters. Moreover, different aspects of cooperative transmission have been discussed in the literature such as performance analysis [1-3] or resource allocation [START_REF] Han | Resource allocation for wireless networks: basics, techniques, and applications[END_REF]. Due to this proven advantage of cooperative communications, incorporating relaying into wireless network standards has been under thorough investigation recently. For instance, the IEEE 802.16j mobile multi-hop relay (MMR) task group introduced the concept of multi-hop relaying in the IEEE 802.16 WirelessMAN (WiMAX) family of broadband networks [START_REF]The Relay Task Group of IEEE[END_REF]. The main novelty in the standard is the introduction of a This work was supported by the Research Council of Norway through the projects 183311/S10 and 18778/V11. new node, the relay station (RS), for improving the capacity and coverage of 802.16 networks.

Deploying RSs in an 802.16j network faces several challenging issues. For instance, the authors in [START_REF] Lin | Optimal relay station placement in IEEE 802.16j networks[END_REF] maximize the total downlink rate in an 802.16j network by finding the optimal position of one RS. This optimal RS placement is further studied in [START_REF]Relay station placement in IEEE 802.16j dual-relay MMR networks[END_REF] for multiple RSs, aiming to maximize the throughput by the usage of the concept of dual relaying. Moreover, the work in [START_REF] Yu | A clustering approach to planning base station and relay station locations in IEEE 802.16j multi-hop relay networks[END_REF] provides an algorithm to find the optimal locations of the BS and the RSs minimizing the cost for deployment of a full scale 802.16j network. Nonetheless, one challenging area in the context of 802.16j networks is the design and formation of the tree structure connecting the BS to the RSs in its coverage area as dictated by the standard [START_REF]The Relay Task Group of IEEE[END_REF]. Clearly, existing literature focused on the performance of the 802.16j network and the optimal placement of the RSs, with few work tackling the problem of the formation of the tree structure. The 802.16j standard states that the tree can be formed either at the BS (centralized approach) or by the sole decisions of the RSs (distributed approach) [START_REF]The Relay Task Group of IEEE[END_REF], [START_REF] Lee | Link adaptive multihop path management for IEEE 802.16j[END_REF]. The main existing work is in [START_REF] Saad | Network formation games for distributed uplink tree construction in IEEE 802.16j networks[END_REF] where we presented a distributed network formation game model for forming the uplink tree in an 802.16j network. However, the model in [START_REF] Saad | Network formation games for distributed uplink tree construction in IEEE 802.16j networks[END_REF] does not account for the costs in terms of the delay incurred by multihop transmission. In order to take into account the delay in the network due to the traffic flow (queueing and transmission delay), a new model and algorithm must be proposed. Another contribution tackling the tree formation in 802.16j is given in [START_REF] Lee | Link adaptive multihop path management for IEEE 802.16j[END_REF] through a centralized approach. However, the work in [START_REF] Lee | Link adaptive multihop path management for IEEE 802.16j[END_REF] does not provide a clear algorithm for the tree formation and it does not account for the gains by using cooperative transmission nor for the resulting delay in the network. Finally, a centralized approach can yield some significant overhead and complexity, namely in networks with a rapidly changing environment due to RS mobility or incoming traffic load.

The major contribution of this paper is to provide a distributed algorithm that allows the RSs in an 802.16j network to autonomously form the uplink tree structure, adapting it to environmental changes as the network evolves. Another main contribution is to model the gains in terms of cooperative transmission, while accounting for the costs in terms of the delay incurred by the multi-hop transmission. For this purpose, we adopt a utility function based on the concept of R-factor, which is a suitable criterion for evaluating the performance of VoIP services [START_REF]The emodel, a computational model for use in transmission planning[END_REF]. The R-factor provides a utility that ties the packet success rate (PSR) and the delay over a transmission path. For forming the tree, we model the problem as a network formation game and we propose a myopic dynamics in which each RS plays its best response strategy that maximizes its utility. The proposed dynamics allows the RSs to self-organize into a Nash network tree structure rooted at the BS. Finally, we illustrate the performance gains of the proposed algorithm through simulations, and we show how the proposed algorithm allows the RSs to autonomously adapt the topology to changes in the environment due to RS mobility or the presence of high traffic.

The rest of this paper is organized as follows: Section II presents the system model and the game formulation. Section III exposes the proposed utility model and presents the proposed dynamics. Simulation results are presented and analyzed in Section IV. Finally, conclusions are drawn in Section V.

II. SYSTEM MODEL AND GAME FORMULATION A. System Model

Consider the uplink of an 802.16j network with M RSs (fixed, mobile or nomadic) and one BS. The 802.16j standard imposes a tree structure to form between the RSs and the BS [START_REF]The Relay Task Group of IEEE[END_REF]. Once the uplink tree forms, mobile stations (MSs) can hook to the network by selecting a serving RS (or directly connecting to the BS). Each MS is assigned to the closest RS. We assume that the MSs deposit their packets to their serving RSs using direct transmission. In their turn, the serving RSs act as source nodes transmitting the received MS packets to the BS through one or more hops in the formed tree, using cooperative transmission. This assumption of direct transmission between MS and RS allows to provide a tree formation algorithm that can be easily incorporated in a new or existing 802.16j network without relying on external entities such as the MSs.

For cooperative transmission between the RSs and the BS, we use the decoded relaying multi-hop diversity channel of [START_REF] Boyer | Multihop diversity in wireless relaying channels[END_REF] whereby each intermediate node on the path between a transmitting RS and the BS combines, encodes and re-encodes the received signal from all preceding terminals before relaying (decode-and-forward). Formally, each MS k in the network is considered as a source of traffic following a Poisson distribution with an average arrival rate λ k . With such Poisson streams at the entry points of the network (the MSs), we assume that for every RS incoming packets are stored and transmitted in a first in first out (FIFO) fashion and that we have the Kleinrock independence approximation [START_REF] Bertsekas | Data networks[END_REF]Chap. 3] with each RS being an M/D/1 queueing system. With this approximation, the total traffic that a RS i receives from the MS that it is serving is a Poisson process with an average arrival rate of T i = Li l=1 λ l where L i is the number of MSs served by RS i. Moreover, RS i also receives packets from RSs that are connected to it with a total average rate R i . For these R i packets, the sole role of RS i is to relay them to the next hop. In addition, any RS i that has no assigned MSs (L i = 0 and T i = 0), transmits "HELLO" packets, generated with a Poisson arrival rate of η 0 in order to maintain its link to the BS active during periods of no actual MS traffic. An illustrative example of this model is shown in Fig. 1.

B. Game Formulation and Strategies

For modeling the interactions among the RSs seeking to form the uplink tree structure, as imposed by the 802.16j standard, network formation games provide a suitable framework [START_REF] Demange | Group formation in economics: networks, clubs and coalitions[END_REF][START_REF] Johari | A contract based model for directed network formation[END_REF][START_REF] Derks | Local dynamics in network formation[END_REF]. In such games, several independent decision makers (players) interact in order to form a network graph. Depending on the goals of each player, a final network graph G, resulting from individual players' decisions, forms. We model the uplink tree formation problem in 802.16j as a network formation game with the RSs being the players. The result of the interactions among the RSs is a directed graph G(V, E) with V = {1, . . . , M + 1} denoting the set of all vertices (M RSs and the BS) and E denoting the set of all edges (links) between pairs of RSs. Each link between two RSs i and j, denoted ij ∈ E, corresponds to an uplink traffic flow between RS i and RS j. First and foremost, we define the notion of a path:

Definition 1: A path between two nodes i and j in the graph G is defined as a sequence of nodes i 1 , . . . , i K such that i 1 = i, i K = j and each directed link i k i k+1 ∈ G for each k ∈ {1, . . . , K -1}. We denote the set Q i as the set of all paths from node i to the BS, and thus |Q i | represents the number of paths from node i to the BS.

As the 802.16j standard imposes a tree structure between the RSs and the BS, we adopt the following convention throughout the remainder of this paper Convention 1: Each RS i is connected to the BS through at most one path, and thus 0 ≤ |Q i | ≤ 1, ∀i ∈ V . Hence, we denote by q i ∈ Q i the path between any RS i and the BS.

Finally, we delineate the possible actions or strategies that each RS can take in the proposed network formation game. The strategy space of each RS i consists of the RSs (or the BS) that i wants to connect to. Consequently, the strategy of a RS i is to select the link that it wants to form from the available strategy space. We note that, a RS i cannot connect to a RS j which is already connected to i, in the sense that if ji ∈ G, then ij / ∈ G. Formally, for a current network graph G, let A i = {j ∈ V \ {i}|ji ∈ G} be the set of RSs from which RS i accepted a link ji, and S i = {ij|j ∈ V \ ({i} A i )} as the set of links corresponding to the nodes (RSs or the BS) with whom i wants to connect (note that i cannot connect to RSs that are already connected to it, i.e., RSs in A i .). In consequence, the strategy of a RS i is to select the link s i ∈ S i that it wants to form, i.e., choose the RS that it will connect to. Based on Convention 1, a RS can be connected to at most one other node in our game so selecting to form a link s i implicitly implies that RS i will replace its previously connected link (if any) with the new link s i .

III. NETWORK FORMATION GAME: UTILITY AND DYNAMICS

In this section, we derive suitable utilities for modeling the PSR-delay trade off in an 802.16j network and we construct a dynamics for distributed network tree formation.

A. Utility

Having modeled the tree formation problem as a network formation game, we propose a utility function that takes into account the QoS in terms of PSR as well as the delay incurred by multi-hop transmission. Consider an actual network tree graph G where each RS extracts a positive utility from the packets successfully transmitted to the BS out of the packets received from the MSs. Each transmitted packet is subject to a bit error rate (BER) due to the transmission over the wireless channel using one or more hops. For instance, for a transmission between a RS V 1 ∈ V to a destination V n+1 (the destination is always the BS) going through n -1 intermediate relays {V 2 , . . . , V n } ⊂ V , let N r be the set of all receiving terminals, i.e., N r = {V 2 . . . V n+1 } and N r(i) be the set of terminals that transmit a signal received by a node V i . Hence, for a relay V i on the path from the source V 1 to the destination V n+1 , we have N r(i) = {V 1 , . . . , V i-1 }. Given this notation, the BER between a source RS V 1 ∈ V and the destination V n+1 = BS is computed along the path q V1 = {V 1 , . . . , BS} ∈ Q V1 using the tight upper bound given in [3, Eq. ( 10)] for the decoded relaying multi-hop diversity channel with Rayleigh fading and BPSK modulation as follows

P e q V 1 ≤ Ni∈Nr 1 2     N k ∈N r(i)     Nj ∈N r(i) N j =N k γ k,i γ k,i -γ j,i × 1 - γ k,i γ k,i + 2
.

(1)

Here, γ i,j = Pi•hi,j σ 2 the average received SNR at node j from node i where P i is the transmit power of node i, σ 2 the noise variance and h i,j = 1

d α i,j
is the path loss with d i,j the distance between i and j and α the path loss exponent. Without loss of generality, we assume that all the RSs transmit with equal power P i = P , ∀i. Finally, for a RS i which is connected to the BS through a direct transmission path q d i ∈ Q i with no intermediate hops, the BER can be given by P e

q d i = 1 2 1 - γi,BS 2+γi,BS
[2], [START_REF] Boyer | Multihop diversity in wireless relaying channels[END_REF]; where γ i,BS is the SNR received at the BS from RS i. Using the BER expression in (1) and with no channel coding, the packet success rate (PSR) ρ i,qi perceived by a RS i over a path q i is defined as follows

ρ i,qi (G) = (1 -P e qi ) B , ( 2
)
where B is the number of bits per packet. The PSR is a function of the network graph G as the path q i varies depending on how RS i is connected to the BS in the formed network tree structure.

Transmitting over multi-hop links incurs a significant delay due to buffering and multiple transmission. For this purpose, we consider the average delay τ i,qi along the path q i = {i 1 , . . . , i k } ∈ Q i from RS i 1 = i to the BS. A measure of the average delay over q i in a network with Poisson arrivals at the entry points and considering the Kleinrock approximation as in the previous section (each RS is an M/D/1 queueing system) is given by [START_REF] Bertsekas | Data networks[END_REF]Chap.

τ i,qi (G) = i k i k+1 ∈qi Λ i k ,i k+1 2µ i k ,i k+1 (µ i k ,i k+1 -Λ i k ,i k+1 ) + 1 µ i k ,i k+1 .
(3) where Λ i k ,i k+1 = T i k + R i k is the total traffic (packets/s) originating from MSs (T i k ) and from RSs (R i k ) traversing link i k i k+1 ∈ q i between RS i k and RS i k+1 . The ratio

1 µi k ,i k+1
represents the average transmission time (service time) on link i k i k+1 ∈ q i with µ i k ,i k+1 being the service rate on link i k i k+1 .

This service rate is given by

µ i k ,i k+1 = Ci k ,i k+1 B
where

C i k ,i k+1 = W log (1 + ν i k ,i k+1 ) (4)
is the capacity of the direct transmission between RS i k and RS

i k+1 , ν i k ,i k+1 = P hi k ,i k+1 σ 2
is the received SNR from RS i k at RS i k+1 , and W is the bandwidth available for RS i k which is assumed the same for all RSs in the set of vertices V , without loss of generality. Similar to the PSR, the delay depends on the paths from the RSs to the BS, hence is a function of the network graph G.

For VoIP services, given the delay and the PSR, an appropriate utility function can be defined through the concept of the Rfactor [START_REF]The emodel, a computational model for use in transmission planning[END_REF]. The R-factor is an expression that links the delay and packet loss to the voice quality [START_REF]The emodel, a computational model for use in transmission planning[END_REF] as follows

U i (G) = Ω a -ǫ 1 τ i,qi (G) -ǫ 2 (τ i,qi (G) -α 3 )H -υ 1 -υ 2 ln (1 + 100υ 3 (1 -ρ i,qi (G))) (5)
where τ i,qi is the delay given by (3) expressed in milliseconds, 100υ 3 (1ρ i,qi ) represents the packet loss percentage (we consider only the packet loss due to errors through ρ i,qi given in (2), ignoring packet loss due to overloaded links). The remaining parameters are constants defined as: Ω a = 94.2, ǫ 1 = 0.024, ǫ 2 = 0.11, ǫ 3 = 177.3, H = 0 if τ i,qi < ǫ 3 , H = 1 otherwise. υ 1 , υ 2 and υ 3 are parameters dependent on the voice speech codec. The relationship between the R-factor and the speech quality is such that as the R-factor increases, the voice quality improves. For different voice codecs, different R-factor ranges provide an indication on the voice quality varying from poor, low, medium, high to best as the R-factor increases [START_REF]The emodel, a computational model for use in transmission planning[END_REF].

Although the RSs are the players of the game, the final results must assess the performance of the MSs in terms of the R-factor achieved (considered as MS utility). For computing the R-factor of the MSs, the PSR and the delay for the whole transmission from MS to BS must be considered. For instance, based on the proposed network model in Section II, the PSR perceived by each MS i served by a RS j is given by

ζ i,j (G) = ρ i,ij • ρ j,qj (G), (6)
where ρ i,ij is the PSR on the direct transmission between MS i and RS j (independent of network graph G) and ρ j,qj (G) is the PSR from RS j to the BS along path q j given by (2) (which can be either a multi-hop transmission or a direct transmission depending on how RS j is connected in the tree graph G).

Moreover, the delay perceived by a MS i served by RS j is given by ( 3) by taking into account, in addition to the delay on the path q j , the traffic on the link ij between the MS and the RS, i.e., the buffering and transmission delay at the MS level.

Having the PSR given by ( 6) and the delay, the utilities of the MSs can be computed for performance assessment.

B. Dynamics of Network Formation

Having formulated a network formation game between the RSs, the next step is to define an algorithm of interaction between the RSs in order to form the required network graph. Network formation algorithms are referred to in the literature as dynamics of network formation. For our game, prior to providing a dynamics for network formation, we first show that any dynamics will result in a connected tree structure, as follows Property 1: Any network graph G resulting from a dynamics applied to the proposed network formation game is a connected directed tree structure rooted at the BS.

Proof: By inspecting the utility in ( 5), we notice that if a RS i is not connected to the BS through direct transmission or other multi-hop paths, the BER will be P e qi = 1 since Q i = ∅ (no path from the RS i to the BS). Consequently, for the utility in (5) the last term will be maximized since ρ i,qi = 0 in this case and thus the whole utility will be minimized. In other words, a disconnection by any RS drastically decreases its utility. Hence, there is no incentive for any RS to disconnect from the BS and any graph G formed using the proposed game will be a connected graph. Moreover, due to Convention 1, the connected graph is thus a tree rooted at the BS.

Due to the high disconnection cost, if a RS is unable to find any partner suitable for forming a link, it will connect to the BS through direct transmission. Thus, our network initially starts with all the RSs connected to the BS (star topology), before engaging in the network formation game.

Denote G si,s-i as the graph G formed when RS i plays a strategy s i ∈ S i while all other RSs maintain their vector of strategies s -i = [s 1 , . . . , s i-1 , s i+1 , . . . , s M ]. We define the best response for a RS as follows [START_REF] Derks | Local dynamics in network formation[END_REF].

Definition 2: A strategy s * i ∈ S i is a best response for a RS i ∈ V if U i (G s * i ,s-i ) ≥ U i (G si,s-i ), ∀s i ∈ S i .
Thus, the best response for RS i is to make the selection of the link that maximizes its utility given that the other RSs maintain their vector of strategies.

By using the different properties of the RS network formation game, we propose best response dynamics that allow a distributed formation of the network graph. The proposed dynamics assume that each RS is myopic, in the sense that the RSs aim at improving their payoff considering only the current state of the network without taking into account the future evolution of the network. Finding an optimal dynamics is a very complex problem, and there exist no strict rules for the network formation dynamics in the literature [START_REF] Demange | Group formation in economics: networks, clubs and coalitions[END_REF]. Therefore for each network formation game model, different dynamics rules must be applied suited to the considered model. Network formation literature encompasses several myopic dynamics for various game models with directed and undirected graphs [START_REF] Demange | Group formation in economics: networks, clubs and coalitions[END_REF][START_REF] Johari | A contract based model for directed network formation[END_REF][START_REF] Derks | Local dynamics in network formation[END_REF]. In this paper, we propose a myopic dynamics inspired from [START_REF] Demange | Group formation in economics: networks, clubs and coalitions[END_REF] and [START_REF] Derks | Local dynamics in network formation[END_REF]. The All the RSs start by directly connecting to the BS in a star topology.

Two phases in each round of the dynamics

Phase 1 -Fair Prioritization:

Prioritize the RSs from the highest to the lowest current BER.

Phase 2 -Myopic Dynamics:

The RSs take action sequentially by priority. Each RS i plays its best response s * i , maximizing its utility (R-factor). The best response s * i of each RS is a replace link operation through which a RS i splits from its current parent RS and replaces it with a new RS that maximizes its utility. Multiple rounds are run until convergence to the final Nash tree G † where no RS can improve its utility by a unilateral change of strategy.

proposed algorithm is composed of several rounds where each round consists mainly of two phases: a fair prioritization phase and a dynamics phase. In the fair prioritization phase, we propose a priority function that assigns a priority to each RS. In the dynamics phase, by increasing priority order, the RSs are allowed to interact.

Therefore, each round of the proposed algorithm begins with the fair prioritization phase where each RS is assigned a priority depending on its actual perceived BER: RSs with a higher BER are assigned a higher priority. The motivation behind this procedure is to fairly allow RSs that are perceiving a bad channel to possess an advantage in selecting their partners; for the purpose of improving their BER. Thus, the RSs experiencing a high BER can select their partners out of a larger space of strategies during the dynamics phase. Other priority functions can also be used, and in a general case, a random priority function can be defined. Following prioritization, the RSs start selecting their strategies sequentially by order of priority. During its turn, each RS i chooses to play its best response s * i ∈ S i in order to maximize its utility at each round given the current network graph resulting from the strategies of the other RSs. The best response of each RS can be seen as a replace operation, whereby the RS will replace its current link to the BS with another link that maximizes its utility (if such a link is available).

Multiple rounds consisting of the above two phases will be run until convergence to the final tree structure G † where the RSs can no longer improve their utility through best responses. The proposed algorithm is summarized in Table I.

The stability of the final graph G † is given using the concept of Nash equilibrium applied to network formation games [START_REF] Derks | Local dynamics in network formation[END_REF] Definition 3: A network graph G in which no node i can improve its utility by a unilateral change in its strategy s i ∈ S i is a Nash network.

Therefore, a Nash network is a network where the links chosen by each node are the best responses. The Nash network is, thus, a network where the nodes are in a Nash equilibrium with no node able to improve its utility by unilaterally changing its current strategy. When our dynamics converge, and as an immediate consequence of playing a best response dynamics, we have Lemma 1: The final tree structure G † resulting from the proposed dynamics is a Nash network.

IV. SIMULATION RESULTS AND ANALYSIS

For simulations, we consider a square area of 4 km × 4 km with the BS at the center. We deploy the RSs and the MSs within this area. The transmit power is P = 50 mW (RSs and MSs), the noise level is -110 dBm, and the bandwidth per RS is set to W = 100 kHz. For path loss, we set the propagation loss to α = 3. For the VoIP traffic, we consider a traffic of 64 kbps, divided into packets of length B = 200 bits with an arrival rate of 320 packets/s. For the HELLO packets we set η 0 = 1 packet/s with the same packet length of B = 200 bits. Finally, without loss of generality, we select the G.729 codec for voice, hence υ 1 = 12, υ 2 = 15 and υ 3 = 0.6 [START_REF]The emodel, a computational model for use in transmission planning[END_REF].

In Fig. 2 we randomly deploy M = 10 RSs within the BS area. For clarity of the figure, we show only the area around the BS which contains RSs. The network formation game starts with the star topology where all RSs are connected directly to the BS. Prior to the presence of the MSs in the network (only HELLO packets present), the RSs interact and converge to a final Nash tree structure shown by the solid lines in the figure. This figure clearly shows how the RSs connect to their nearby partners, forming the tree structure. Furthermore, we randomly deploy 50 MSs in the area, and show how the RSs autonomously adapt the topology to this incoming traffic. The new resulting network structure upon deployment of MSs is shown in dashed lines. The RSs autonomously self-organize and adapt to the deployed traffic. For instance, RS 3 can no longer accommodate the traffic generated by RS 2 as it drastically decreases its utility. As a result, RS 2 takes the decision to disconnect from RS 3 and improve its R-factor by directly connecting to the BS. Similarly, RS 7 disconnects from RS 10 and connects directly to the BS. Moreover, RS 9 finds it beneficial to replace its link with RS 5 with a link with the less loaded RS 6. In a nutshell, Fig. 2 summarizes the operation of the proposed adaptive dynamics with and without the presence of external MS data traffic.

Furthermore, we assess the effect of mobility on the network structure. We consider the network of Fig. 2 prior to the deployment of the MSs and we assume that RS 5 is moving upwards on the positive y-axis while the other RSs remain fixed. The changes in utility of the concerned RSs during the movement of RS 5 are shown in Fig. 3. As RS 5 moves upwards, its utility starts by dropping since the distance to its serving RS (RS 6) increases. Upon moving 0.3 km, RS 5 finds it beneficial to replace its link with RS 6 with a direct link to the BS, adapting the topology. Meanwhile, RS 9 decides to remain connected to RS 5 since it cannot improve its utility otherwise. However, when RS 5 has moved 0.4 km, RS 9 decides to disconnect from RS 5 and connect directly to RS 6 improving its utility. This decreases the traffic on RS 5 which is now closer to RS 3 and can maximize its utility by connecting to RS 3. As RS 5 gets closer to RS 3, its utility continues to improve. After moving 1.2 km, RS 5 approaches RS 8 which finds it beneficial to disconnect its link with RS 3 and connect to RS 5. At this point the utility of RS 5 drops due to the newly accepted traffic; moreover, this utility drops further as RS 5 distances itself from its serving RS (RS 3). The utility of RS 8 continues to improve as RS 5 gets closer to it. Through these results, we clearly illustrate how the RSs can autonomously self-organize adapting the topology to mobility. Similar results can be shown when new RSs enter the network or RSs leave the network with or without the presence of MSs, but these results are omitted due to space limitation.

In Fig. 4, we show the average achieved utility per MS as the number of RSs, M , in the network increases. The results are averaged over random positions of the MSs and the RSs in a network having 40 MSs. We compare the performance of the proposed algorithm against the star topology whereby each RS is directly connected to the BS as well as the scenario where no RSs exist in the network. In this figure, we clearly see that as the number of RSs in the network increases the performance of the proposed algorithm as well as for the star topology improves. However, for the star topology the slope of increase is much slower than that of the proposed algorithm. In addition, the proposed algorithm presents a clear performance advantage, increasing with the number of the RSs, and reaching up to 40.3% and 42.75% (at M = 25 RSs) relative to the star topology and the no RSs case, respectively.

In Fig. 5, we plot the average M av and maximum M max number of hops achieved in the network tree as the number of RSs increases for a network with 40 MSs (results are averaged over random positions of MSs and RSs). It must be noted that the plotted number of hops corresponds to the RS-RS or RS-BS links in the tree, without accounting for the entry point MS-RS hop. First and foremost, as the number of RSs M in the network increases both the average and the maximum number of hops increase. However, we note that, almost 5 additional RSs must be deployed before the average and the maximum number of hops increase of 1. This result shows that, due to the delay cost for multi-hop transmission, the number of hops in the tree do not increase drastically with the number of RSs in the network. In general, we can easily remark that the increase slope of M av and M max with respect to the number of RSs M is small. Finally, we note that the maximum number of hops reaches 7 hops while the average number of hops does not exceed 4 hops, even for a network with M = 25 RSs.

V. CONCLUSIONS This paper proposed a novel distributed approach to the uplink tree formation problem in 802.16j networks through a network formation game model. In the proposed game, the RSs engage in pairwise interactions for forming directed links that result in a final uplink tree structure rooted at the BS. Each RS aims to maximize its utility which accounts for the gains in terms of PSR while taking into account the delay costs incurred by multi-hop transmission. For VoIP services, a suitable utility is derived through the concept of the R-factor which is a parameter that assesses VoIP quality through an expression that ties the PSR and the delay. We proposed a best response dynamics for network formation that converges to a Nash network. Using the proposed dynamics, the RSs are able to autonomously form the tree topology and adapt it to environmental changes such as mobility or incoming traffic. Simulation results show that the proposed algorithm presents significant gains in terms of average achieved MS utility reaching up to 40.3% improvement compared to the star topology (all RSs directly connected to the BS), and up to 41.75% compared to the case with no RSs. The results also show that the average number of hops in the tree does not exceed 4 even for networks having a large number of RSs.

Fig. 1 .

 1 Fig. 1. A prototype of the uplink tree model.

  3, Eqs. (3.42), (3.45) and (3.93)]

Fig. 2 .

 2 Fig.2. Snapshot of a tree topology formed using the proposed algorithm with 10 RSs before (solid line) and after (dashed line) the random deployment of 50 MSs (MS positions not shown for clarity).

  by connecting to RS 3 RS 9 disconnects from RS 5 and connects to RS 6. RS 5 (with RS 9 connected to it) splits from RS 6 and connects directly to the BS. RS 8 disconnects from RS 3 and connects to RS 5.

Fig. 3 .

 3 Fig.3. Self-adaptation of the network's tree topology to mobility shown through the variation of the utility of RS 5 as it moves upwards on the y-axis prior to any MS presence.

Fig. 4 .

 4 Fig. 4. Performance assessment of the proposed distributed tree algorithm for a network with 40 MSs shown through the average achieved MS utility vs. number of RSs M in the network (average over random positions of MSs and RSs).

Fig. 5 .

 5 Fig. 5. Average and maximum number of hops in the final tree structure for a network with 40 MSs vs. number of RSs M in the network (average over random positions of MSs and RSs).


Béatrice Bonafos

Antonio J Cortés-Espinar

Laurence Balas

Laurence Pessemesse

Karen Lambert

Melha Benlebna

Sylvie Gaillet

François Pelletier

Pierre Delobel

Javier Ávila-Román

Miquel Mulero Abellán

Christelle Bertrand-Gaday

Thierry Durand

Charles Coudray

François Casas
email: francois.casas@inrae.fr

Christine Feillet-Coudray

9-PAHPA long term intake in DIO and db/db mice ameliorates insulin sensitivity but has few effects on obesity and associated metabolic disorders

1.Introduction

Obesity is a worldwide major public health issue [START_REF] Jaacks | The obesity transition: stages of the global epidemic[END_REF] . It generally results from an imbalance between body energy intake and expenditure from a metabolic point of view and is associated with numerous metabolic dysfunctions including insulin resistance and inflammation [START_REF] Lingvay | Obesity management as a primary treatment goal for type 2 diabetes: time to reframe the conversation[END_REF] . Obesity affects key metabolic organs, including liver and skeletal muscle. While liver is a key organ that performs a wide range of biochemical functions necessary for the metabolic homeostasis of the entire body, skeletal muscle is essential for energy metabolism and is the predominant site of insulin-mediated glucose uptake in the post prandial state [START_REF] Defronzo | Skeletal muscle insulin resistance is the primary de-fect in type 2 diabetes[END_REF] .

Fatty acid esters of hydroxy fatty acids (FAHFAs) are a vast family of endogenous lipids [START_REF] Yore | Discovery of a class of endogenous mammalian lipids with anti-diabetic and antiinflammatory effects[END_REF][START_REF] Brejchova | Understanding FAHFAs: from structure to metabolic regulation[END_REF][START_REF] Benlebna | Potential physiopathological effects of branched fatty acid esters of hydroxy fatty acids[END_REF]. Some of them, in particular PAHSAs (palmitic acid esters of hydroxystearic acid), were proven to favorably modulate insulin sensitivity and glucose metabolism [ 4 , 7-11 ]. The anti-inflammatory ability of some FAHFAs, notably from omega-3 fatty acid but also from the PAHSA family, has also been demonstrated [ 4 , 12-14 ]. In our laboratory we showed that 9-PAHPA (9-palmitic acid esters of hydroxypalmitic acid) and 9-OAHPA (9-oleic acid esters of hydroxypalmitic acid) increased insulin sensitivity in mice when incorporated to a chow diet [START_REF] Benlebna | Long-term high intake of 9-PAHPA or 9-OAHPA increases basal metabolism and insulin sensitivity but disrupts liver homeostasis in healthy mice[END_REF] or lowered insulin resistance in mice when incorporated to a high fat high sucrose (HFHS) diet [START_REF] Benlebna | Long-term intake of 9-PAHPA or 9-OAHPA modulates favorably the basal metabolism and exerts an insulin sensitizing effect in obesogenic diet-fed mice[END_REF] . Moreover, 9-PAHPA and 9-OAHPA induced hepatic steatosis and fibrosis in some healthy mice but not in HFHS mice [ 15 , 16 ], with few effects on inflammation. How-ever, preventive supplementation with 9-PAHPA and 9-OAHPA in HFHS mice did not prevent significant weight gain or the development of hyperglycemia. Our assays performed on C2C12 muscle cells suggested a muscular origin of the increase in insulin sensitivity, as 9-PAHPA and 9-OAHPA induced a switch toward a more oxidative contractile phenotype of skeletal muscle [START_REF] Benlebna | FAH-FAs Regulate the Proliferation of C2C12 Myoblasts and Induce a Shift toward a More Oxidative Phenotype in Mouse Skeletal Muscle[END_REF] . However, the ability of FAHFAs to lower obesity and improved major associated metabolic dysfunction has not been addressed yet The purpose of this work was therefore to study whether curative supplementation with 9-PAHPA may have beneficial effects in liver and skeletal muscle associated metabolic dysfunction in two animal models of obesity, namely the classical diet-induced obesity (DIO) and the db/db mice [ 18 , 19 ]. For this purpose, C57Bl6/J mice were fed for 8 weeks with HFHS diet and then were supplemented with 9-PAHPA for 8 weeks. Moreover, db/db mice were fed for 7 weeks with a chow diet supplemented with 9-PAHPA. Whole-body metabolism was explored with a particular focus on the liver and skeletal muscle functions.

2.Materials and methods

FAHFAs synthesis

The synthesis of 9-PAHPA was performed using our previously reported procedure [START_REF] Balas | Region controlled syntheses of FAHFAs and LC-MS/MS differentiation of regioisomers[END_REF] .

Animals and diets

DIO mice: Thirty 6-weeks old male C57BL/6J mice (Charles River, L'Arbresle, France), weighing about 22 g, were housed (five per cage) under conditions of constant temperature (23-24 °C), humidity (35-50%) and a standard dark cycle (19.30-07.30 h). The mice were randomly separated into three groups of 10 animals; a control group (control) was fed for 16 weeks with a control diet, a HFHS group was fed for 16 weeks with a HFHS diet and a 9-PAHPA group (HFHS + 9-PAHPA) was fed for 8 weeks with a HFHS diet and subsequently for eight other weeks with a HFHS diet enriched with 9-PAHPA.

Db/db mice: Ten 5-weeks old male db/ + mice weighing about 22 g and twenty 5-weeks old male db/db mice weighing about 30 g (Janvier, Saint-Berthevin Cedex, France), were housed (two per cage) as previously mentioned. The db/ + mice were fed for 7 weeks a control diet (db/ + group). The db/db mice were randomly separated into two groups of 10 animals; a group (db/db) was fed for 7 week a control diet and a 9-PAHPA group (db/db + 9-PAHPA) was fed for 7 weeks a control diet plus 9-PAHPA.

The detailed composition has been published previously [ 15 , 16 ]. The control diet contains 5% lipids as a mixture of rapeseed oil, high oleic sunflower oil, sun-flower oil, and linseed oil (oil mixture of Carrefour) and the HFHS diet contains 25% lipids (5% of a mixture of rapeseed oil, high oleic sunflower oil, sunflower oil, and linseed oil and 20% of lard) and 30% sucrose. The lipid fraction of the control diet was composed of 12.2% saturated fatty acids, 60.6% monounsaturated fatty acids and 27.3% polyunsaturated fatty acids and that of the HFHS diet was com-posed of 30.4% saturated fatty acids, 53.4% monounsaturated fatty acids, and 16.3% polyunsaturated fatty acids. The 9-PAHPA was incorporated into the diet after dis-solution in the oil mixture of Carrefour and the final content of PAHPA was set at 300 μmol/kg diet. Throughout the study, mice were given free access to food and tap water. Mice body weight was assessed weekly and food consumption was determined every 2 d (week) or 3 d (week-end). A detailed scheme of the study design is provided in Figure 1 .

Body composition and metabolic analyses

Mice whole-body composition (fat and lean mass) was measured every 2 weeks throughout the study using an EchoMRI-700 whole-body composition analyzer (Echo Medical Systems, Houston, TX, USA), according to the manufacturer's instructions. Oxygen consumption and carbon dioxide production were measured using a Comprehensive Lab Animal Monitoring System (Oxymax-CLAMS) (Columbus Instruments, Columbus, OH, USA). Mice were housed in individual cage inside a con-trolled cabinet. The environmental enclosure allows precise control over the temperature and light/dark cycle. Mice were acclimatized individually in metabolic cages at 24 °C with ad libitum access to food and water for 24-h, prior to a 24-h period of automated recordings. Sampled air from individual cages was passed through sensors to determine O 2 and CO 2 content. Sensors were calibrated before each experiment with commercial gas mixtures of accurately determined composition (20.5% O2, 0.5% CO2, and 79% N2). For each mouse, volume of O 2 (VO 2 ) and VCO 2 were measured 111 times in 24 h. The respiratory exchange ratio (RER) was calculated as the volume of CO 2 versus volume of O 2 (VCO 2 /VO 2 ) ratio.

Oral Glucose Tolerance Test (OGTT) and Insulin Tolerance Test (ITT)

OGTT was performed only in DIO mice and ITT was performed in DIO and db/db mice. Following an overnight fasting (16 h), mice were administrated glucose solution (2 g/kg) by oral gavage, and blood were collected from the tail vein at the indicated times for glucose and insulin determination. Insulin tolerance was assessed after 2 h of fasting by administration of human insulin (0.75 U/kg for DIO mice and 1.5 U/kg for db/db mice) and blood glucose monitoring. Glycemia was measured using a OneTouch Verio glucometer (Lifescan).

Sampling and routine biochemical analyses

Four to five days after the OGTT or the ITT, blood from 12 h-fasted mice was collected from the retro-orbital sinus and distributed into heparinized tubes. Blood tubes were centrifuged at 1,0 0 0 g for 10 min at 4 °C, plasma was collected and stored at -80 °C until analysis. After cervical dislocation of mice, liver and muscles were removed, rinsed with 0.9% NaCl, weighed, cut into several parts, when necessary, plunged into liquid nitrogen and then kept at -80 °C until analysis. In addition, liver and muscle fresh samples were frozen in Tissue-Tek (Microm Microtech, Brignais, France) for histological analysis and stored at -80 °C. Plasma levels of glucose, total cholesterol, triglycerides and free fatty acids as well as enzymatic activity of alanine aminotransferase were measured at the ANEX-PLO/CREFRE analysis platform (CHU RANGUEIL-BP 84225, France). Plasma levels of insulin, leptin and IL-6 were quantified with ELISA kits (Merck Millipore, Darmstadt, Germany; Crystal Chem, Zaandam, Netherlands; Abcam, Paris, France, respectively).

Tissue neutral lipids measurement

Liver and muscle samples were homogenized in aqueous NaCl solution (9 g/L) and Triton X-100 (0.1%) and free fatty acids, triglycerides and total cholesterol levels were quantified on the tissue homogenate by enzymatic methods (Wako-NEFA-C kit, Oxoid, Dardilly, France; Cholesterol CHOD-PAP SOBIODA kit and triglycerides LQ SOBIODA kit, Sobioda 38330 Montbonnot-Saint-Martin, France) [START_REF] Coudray | A mitochondrial-targeted ubiquinone modulates muscle lipid profile and improves mitochondrial respiration in obesogenic diet-fed rats[END_REF] .

Histological study

After euthanasia by cervical dislocation, liver samples and the tibialis anterior muscles were collected. Liver were immediately fixed in 4% Paraformaldehyde (PFA) at 4 °C for 24 h then embedded in paraffin. Tissue sections were cut at a 4 μm thickness and stained with Hematoxylin & Eosin. Tibialis anterior muscle were freshly frozen in Tissue-Tek (Microm Microtech, Brignais, France), and then stored at -80 °C. For immunostaining, the sections were fixed in Phosphate Buffered Saline (PBS) (Euromedex, Souffelweyersheim, France), 4% PFA at room temperature for 5 min, permeabilized 30 min in PBS, 20% horse serum, and 0.1% triton at room temperature, and incubated with the anti-laminin (1/200, rabbit polyclonal, Sigma L9393) in a solution of PBS, 20% goat serum, and 1% Bovine Serum Albumin (BSA) for 24 h at 4 °C. Sections were washed in PBS 3X for 10 min and incubated with the secondary antibody in PBS for 1 h at 37 °C. Sections were washed in PBS 2X for 10 min, incubated 30 s with DAPI, and washed once in PBS for 10 min and mounted. For the morphometric analysis, muscle sections were scanned using a NanoZoomer (Hamamatsu Photonics, Massy, France) with a 20X objective. Image J free software was used to analyze and quantify the pictures for each entire area.

Tissue oxidative stress status and mitochondrial enzymatic activities

Long-established oxidative stress parameters (TBARS, -SH, GSH, catalase, SOD and MnSOD, GPX), citrate synthase (CS) and different mitochondrial respiratory complex (CI, CII, CII + III and COX) activities were measured in liver and muscle as previously reported [ 15 , 16 ].

Western blotting

Protein levels were assessed by Western blotting. Total proteins were measured using the Bio-Rad protein assay (Hercules, CA, USA) according to the manufacturer's instructions. Total proteins were lysed in RIPA buffer (20 mM Tris, 150 mM NaCl, 1 mM EDTA, 1% NP40, pH 7.5 + complete cocktail protease inhibitors + Pierce phosphatase inhibitors tablet). 40 μg of cell extracts were loaded and blotted onto a nitrocellulose membrane. The membranes were washed, blocked with 5% nonfat milk and incubated in primary antibody overnight at 4 °C. The membranes were then washed in TBS-Tween (10 mM Tris, 140 mM NaCl, and 0.2% Tween-20), incubated with the appropriate secondary antibody coupled to horseradish peroxidase, and washed again in TBS-Tween. Signals were revealed using a Clarity Western ECL Substrate Kit (Hercules, CA, USA), and proteins were visualized by enhanced chemiluminescence using the ChemiDoc Touch Imaging System (Hercules, CA, USA) and quantified with Image Lab. Touch Software (version 5.2.1) (Hercules, CA, USA). Primary antibodies used include: anti-PPARalpha (1/1,0 0 0, Braissant et al, 1996); anti-chREBP (1/1,0 0 0, ABCam ab92809) and anti-Nrf2 (1/1,0 0 0, ABCam ab137550).

Statistical analysis

Results were expressed as means ±SD. DIO mice: HFHS diet versus control diet and HFHS + 9-PAHPA versus HFHS diet were tested by a student t test. Db/db mice: db/db versus db/ + and db/db + 9-PAHPA versus db/db were tested by a student t test. Statistical analyses were performed using the StatView program (SAS Institute, Cary, NC, USA).

3.

Results

In DIO mice 9-PAHPA increased body weight and fat mass and decreased energy expenditure

As expected, energy intake, final body weight and fat mass were increased with 16 weeks of HFHS diet by comparison to con-trols ( Fig. 2 A). Surprisingly, 9-PAHPA supplementation (HFHS + 9-PAHPA) expanded weight gain and fat mass ( Fig. 2 A). However, no specific effect of 9-PAHPA was found on energy intake or on IL6 plasma levels ( Fig 2 A). In addition, while plasma cholesterol levels were significantly increased with HFHS diet compared to controls, no specific effect of 9-PAHPA was observed on plasma triglycerides, cholesterol and free fatty acids levels ( Fig. 2 A).

We next investigate the basal metabolism, oxygen consumption and carbon dioxide production using Oxymax-CLAMS system. We found that oxygen consumption reflecting metabolic expenditure was decreased in animals fed a HFHS diet compared to controls ( Fig. 2 B). In addition, we observed that 9-PAHPA supplementation worsened this decrease of energy expenditure ( Fig. 2 B). This result probably explains the increase in body weight and adipose tissue induced by 9-PAHPA ( Fig. 2 A). Moreover, analysis of RER revealed that HFHS diet favored the oxidation of lipids, which was reflected by a lower RER value compared to controls ( Fig. 2 C). 9-PAHPA supplementation further significantly decreased RER com-pared to HFHS group ( Fig. 2 C).

In DIO mice 9-PAHPA improved insulin sensitivity

Fasted plasma glucose was increase with HFHS diet compared to controls ( Fig. 2 D). 9-PAHPA supplementation had no effect on glycemia compared to HFHS group ( Fig. 2 D). We found that 9-PAHPA supplementation increased insulin sensitivity in comparison to HFHS diet alone as attested by ITT experiments ( Fig. 2 E). However, OGGT indicated that 9-PAHPA had no effect on glucose tolerance compared to HFHS group ( Fig. 2 F).

In DIO mice 9-PAHPA had no effect on liver phenotype nor on mitochondrial function and oxidative stress

Liver weight and the measure of liver triglycerides, total cholesterol and free fatty acid revealed no difference whatever the group of diets ( Fig. 3 A-B). We also showed that 9-PAHPA was able to abolish the induction of TNF αinduced by HFHS diet ( Fig. 3 C). Furthermore, histologic analysis indicated the presence of some lipid micro droplets in liver of DIO mice compared to controls (Fig. 3 D). Finally, we did not observe any evidence of fibrosis (Fig. 3 D). In line with this observation, plasma ALAT activity (Alanine aminotransferase), a biochemical marker of hepatic fibrosis, was not modified regardless of the diet (Fig. 3 E).

In addition, liver gene expression of peroxisome proliferator-activated receptors alpha (a major metabolic regulator of lipids homeostasis), ChREBP (a transcription factor coupling hepatic glucose utilization and lipid synthesis) and Nrf2 (a master regulator of cellular redox homeostasis) was studied by western-blot. We found that none of these proteins were modified by the diet (Fig. 3 E). However, we observed that 9-PAHPA induced a slight increase in ChREBP and a slight decrease in Nrf2. Moreover, we observed that liver 3hydroxyacyl CoA dehydrogenase activity ( β-HAD), involved in β-oxidation was increase by HFHS diet ( Fig. 3 G).

We next investigated the mitochondrial function and oxidative stress parameters in liver. We found a slight increase of the activities of mitochondrial respiratory chain complexes II + III and IV with the HFHS diet compared to control whereas the activities of the complexes activity I and II remained unchanged (Fig. 4 A). However, 9-PAHPA supplementation had no impact ( Fig. 4 A). CS, a marker of mitochondrial activity, was not modified whatever the diet (Fig. 4 B). Moreover, we observed that 9-PAHPA induced a slight decrease level of TBARS (a lipid peroxidation marker) compared to HFHS diet whereas the other biochemical markers of oxidized proteins remained unchanged (SH groups, GSH and GS = SG) (Fig. 4 C-D). In addition, we found that the activities of the SOD (superoxide dismutase) antioxidants enzymes were slightly increased by HFHS diet whereas no change was ob-served for catalase and GPx (glutathion peroxidase) (Fig. 4 E). Last, 9-PAHPA supplementation induced an increase of catalase activity compared to HFHS group (Fig. 4 E).

In DIO mice 9-PAHPA had no effect on muscle phenotype nor on mitochondrial function and oxidative stress

We found that the weight of the various muscles analyzed (quadriceps, gastrocnemius and tibialis) was not modified either by the HFHS diet or by the HFHS + 9-PAHPA diet (Fig. 5 A). In addition, muscle triglycerides were increased by HFHS diet compared to controls and 9-PAHPA further increased the level of triglycerides in DIO mice (Fig. 5 B). However, cholesterol and FFA remained un-changed whatever the diet (Fig. 5 B). Next, we examined the cross-sectional area (CSA) of the tibialis muscle from our mice. Quantification of tibialis fiber number and area indicated that neither the HFHS diet nor 9-PAHPA supplementation had any impact on total fiber number and mean CSA (Fig. 5 C-D). Furthermore, analysis of the number of fibers in each range of area also showed no difference with or without 9-PAHPA (Fig. 5 E).

Then, we measured the mitochondrial respiratory chain activities that determine the metabolic properties of skeletal muscle. We found a slight increase of the activities of mitochondrial respiratory chain complexes IV with the HFHS diet compared to control whereas the activities of other complexes remained unchanged (Fig. 6 A). In addition, CS and β-HAD activities were also increased in DIO mice (Fig. 6 B-C). Regarding oxidative stress, HFHS diet increased TBARs levels and GSH content but had no impact on -SH content and antioxidant enzyme activities. 9-PAHPA supplementation slightly increased CS compared to HFHS diet but had no im-pact on mitochondrial respiratory chain and β-HAD activities and on oxidative stress markers ( Fig. 6 A-C).

In db/db mice, 9-PAHPA had no influence on body weight and energy expenditure

We next investigate the influence of 9-PAHPA supplementation in db/db mice, a genetical model of obesity and type 2 diabetes. Db/ + mice were used as control. We show that energy intake and final body weight of db/db mice were greatly increased after 7 weeks of diet by comparison to controls (Fig. 7 A). However, no specific effect of 9-PAHPA was found in energy intake and body weight (Fig 7 A). As expected, the lack of leptin receptor leaded to the over-production of leptin compared to db/ + mice (Fig. 7 A). In addition, no changes were observed on IL-6, plasma triglycerides and free fatty acids levels whatever the group (Fig. 7 A).

We next investigate the basal metabolism, oxygen consumption and carbon dioxide production using Oxymax-CLAMS system. We found that oxygen consumption reflecting metabolic expenditure was strongly decreased in db/db mice compared to db/ + (Fig. 7 B). Moreover, analysis of RER revealed that oxidation of carbohydrates is increased in db/db mice, which was reflected by a higher RER value compared to db/ + (Fig. 7 C). This increase in RER reflects the fact that db/db mice were eating during the diurnal phase while db/ + animals were mainly sleeping. 9-PAHPA supplementation had no effect on either VO2 or RER compared with the db/db mice (Fig. 7 B-C). Insulin tolerance test (ITT). For each mouse, VO2 and VCO2 were measured 111 times in about 24 h. The respiratory exchange ratio (RER) was calculated as the volume of CO2 versus volume of oxygen (VCO2/VO2) ratio. Mean VO2, VCO2 and RER values are mean of 8 mice for each group considering all the cycles. Results were expressed as means ±SD, n = 10 animals per group except for VO2, CO2 and RER data. db/ + versus db/db and db/db versus db/db + 9-PAHPA were tested by a student t test; * P < .05 db/db versus db/ + and $ P < .05 db/db + 9-PAHPA versus db/db.

In db/db mice, 9-PAHPA improved insulin sensitivity but had no effect on hyperglycemia

In fasted state we showed that plasma glucose, insulin levels and HOMA-IR index were dramatically increased in db/db mice compared to controls (Fig. 7 D). Unfortunately, 9-PAHPA supple-mentation did not improve these parameters compared to db/db mice (Fig. 7 D). However, we found that 9-PAHPA supplementation increased insulin sensitivity in comparison to db/db mice as at-tested by ITT experiments (Fig. 7 E).

In db/db mice, 9-PAHPA supplementation did not improve liver steatosis nor mitochondrial function and oxidative stress

In db/db mice liver weight and ALAT activity were strongly in-creased compared to db/ + (Fig. 8 A). In line with these observations, in db/db mice, histologic analysis revealed a macrovesicular steatosis with large lipid droplet that filled up the hepatocyte and displaced the nucleus to the periphery (Fig. 8 B). Moreover, we did not observe any evidence of fibrosis. 9-PAHPA supplementation did not correct the hepatomegaly and hepatic steatosis observed in db/db mice (Fig. 8 A-B).

Regarding mitochondrial respiratory chain activities and oxidative stress, we found a slight increase of the activities of mitochondrial respiratory chain complexes I and II + III in db/db mice com-pared to db/ + animals (Fig. 8 C). In contrast, the activities of mitochondrial respiratory chain complexes II and IV were decreased in db/db mice (Fig. 8 C). In addition, CS activity was also decreased in db/db mice (Fig. 8 D). However, 9-PAHPA supplementation had no impact on mitochondrial respiratory chain complexes and CS activities compared to db/db mice alone (Fig. 8 C-D). Moreover, we recorded a modest decrease level of TBARS, a modest increase of SH groups, an increase of GSH and a decrease of GS = SG content in db/db mice compared to controls (Fig. 8 E-F). In addition, we observed an increase of catalase activity and a decrease of Mn-SOD and GPx activities in db/db mice compared to db/ + whereas SOD activity remained unchanged (Fig. 8 G). However, 9-PAHPA supple-mentation had no impact on oxidative stress parameters (Fig. 8 E-G).

In db/db mice, 9-PAHPA had no effect on muscle phenotype nor on mitochondrial function and oxidative stress

In db/db mice, a strong muscle atrophy was recorded compared to db/ + animals as attested by the fall of the muscle weights analyzed (Fig. 9 A). Furthermore, during tissue collection we observed that the muscles of the db/db mice appeared to be very fatty. This observation was confirmed by the measurement of the intramuscular triglyceride content which was strongly increased in db/db mice (Fig. 9 B). As in DIO model, 9-PAHPA supplementation was not able to compensate for this muscle loss or this increase of triglyceride content (Fig. 9 A-B).

Histological analyses of the tibialis muscle (Fig. 9 C) revealed that this drop in muscle mass recorded in db/db mice was not related to a reduction in fibers number but to a significant decrease in mean CSA (Fig. 9 D-E). In addition, analysis of the number of fibers in each range of area also showed that in db/db mice there were more small-sized fibers and fewer large-sized fibers (Fig. 9 F). These analyses confirmed that 9-PAHPA supplementation had no impact on the total number of fibers, CSAs, and fiber distribution compared with db/db mice alone (Fig. 9 D-F).

Next, we measured the mitochondrial respiratory chain activities that determine the metabolic properties of skeletal muscle. In db/db mice, we found a slight decrease of the activities of mitochondrial respiratory chain complexes IV and II + III compared to db/ + animals ( Fig. 10 A). CS and β-HAD activities remained un-changed (Fig. 10 B-C). However, 9-PAHPA supplementation had no impact on mitochondrial respiratory chain, CS and β-HAD activities compared to db/db mice. In addition, analysis of antioxidant enzymes (SOD, Mn-SOD, catalase, and glutathione peroxidase), GSH and markers of oxidative stress (TBARS and SHgroup) revealed no difference with or without 9-PAHPA in db/db mice (Fig. 10 D-F).

Discussion

We previously showed that 9-PAHPA and 9-OAHPA increased insulin sensitivity in healthy mice when incorporated into a chow diet [START_REF] Benlebna | Long-term high intake of 9-PAHPA or 9-OAHPA increases basal metabolism and insulin sensitivity but disrupts liver homeostasis in healthy mice[END_REF] or lowered insulin resistance in obese mice when incorporated into an HFHS diet [START_REF] Benlebna | Long-term intake of 9-PAHPA or 9-OAHPA modulates favorably the basal metabolism and exerts an insulin sensitizing effect in obesogenic diet-fed mice[END_REF] . However, in C57Bl6/J mice fed an HFHS diet, preventive supplementation with 9-PAHPA and 9-OAHPA failed to prevent weight gain and hyperglycemia [START_REF] Benlebna | Long-term intake of 9-PAHPA or 9-OAHPA modulates favorably the basal metabolism and exerts an insulin sensitizing effect in obesogenic diet-fed mice[END_REF] . The aim of this work was therefore to investigate whether curative supplementation with 9-PAHPA may have beneficial effects against obesity and metabolic dysfunction when given curatively in al-ready obese animals. For this purpose, we chose to use two animal models of obesity, the conventional DIO and db/db mice. Surprisingly, in DIO mice, we observed that 9-PAHPA did not improve the phenotype but rather worsened it, as evidenced by increased body weight and fat mass. In line with this observation, we found that 9-PAHPA supplementation decreased energy expenditure compared to HFHS diet alone. We also recorded a more marked hyperglycemia with 9-PAHPA supplementation. This lack of positive effect on weight gain is in agreement with recent reports indicating that PAHSA [ 10 , 11 ] or 12-OAHSA treatment [START_REF] Aryal | Distinct biological activities of isomers from several families of branched fatty acid esters of hydroxy fatty acids (FAHFAs)[END_REF] did not reduce body weight gain and fat mass in DIO mice. In the db/db model of obesity, 9-PAHPA supplementation had no effect on the dramatic weight gain and hyperglycemia observed in db/db mice after few weeks on chow diet. In the same db/db mouse model, Wang et al. also showed that 30 d of treatment with 5-PAHSA was not able to prevent significant weight gain in these mice [START_REF] Wang | High glucose concentration impairs 5-PAHSA activity by inhibiting AMP-activated protein kinase activation and promoting nu-clear factor-Kappa-B-mediated inflammation[END_REF] .

Obesity is a major risk factor for nonalcoholic fatty liver dis-ease (NAFLD) [START_REF] Byrne | NAFLD: a multisystem disease[END_REF] . Although C57BL6/J mice are widely used in obe-sity and diabetes research [ 24 , 25 ], their ability to develop NAFLD is quite heterogeneous according to studies [ 26 , 27 ]. In this work, de-spite the obesogenic diet, we did not observe any increase in liver weight in HFHS mice and only the histological analysis revealed the presence of some lipid micro-droplets in the liver. In line with this limited impact on the liver, the different metabolic parame-ters analyzed (plasma cholesterol, triglycerides, ALAT, and hepatic mitochondrial activities and oxidative stress parameters) were not modified either by the diet or by 9-PAHPA supplementation. However, we observed that 9-PAHPA increased in the expression of the transcription factor ChREBP. Interestingly, several studies reported that ChREBP acts as a key modulator of hepatic fatty acid com-position and insulin sensitivity [ 28 , 29 ]. Our result suggests that 9-PAHPA, by upregulating ChREBP, may participate in the previously described increase in insulin sensitivity. In the db/db obesity model, as expected liver weight and plasma ALAT activity were strongly increase compared to db/ + . In line with these observations, in db/db mice, histologic analysis revealed a macrovesicular steatosis with large lipid droplet. However, 9-PAHPA supplementation did not correct the hepatomegaly and hepatic steatosis ob-served in db/db mice. 9-PAHPA had also no impact on the different metabolic parameters analyzed (plasma cholesterol, triglycerides, ALAT, and hepatic mitochondrial activities and oxidative stress parameters). A study from Wang et al. demonstrated that 5-PAHSA treatment tends to promote promoted fatty liver and inflammation in db/db mice [START_REF] Wang | High glucose concentration impairs 5-PAHSA activity by inhibiting AMP-activated protein kinase activation and promoting nu-clear factor-Kappa-B-mediated inflammation[END_REF] . These set of results indicate no beneficial impact of 9-PAHPA supplementation on liver metabolism on both models of obesity studied.

Skeletal muscle is essential for whole metabolism. It is one of the main regulators of glucose homeostasis, responsible for 80% of postprandial glucose uptake from the circulation [START_REF] Defronzo | Skeletal muscle insulin resistance is the primary de-fect in type 2 diabetes[END_REF] . Recently, we showed that 12 weeks of 9-PAHPA or 9-OAHPA supplementation had no effect on the skeletal muscle mass, mitochondrial activity, and oxidative stress parameters of mice fed a chow diet. How-ever, we observed that 9-PAHPA or 9-OAHPA induces a switch toward a more oxidative contractile phenotype in these mice [START_REF] Benlebna | FAH-FAs Regulate the Proliferation of C2C12 Myoblasts and Induce a Shift toward a More Oxidative Phenotype in Mouse Skeletal Muscle[END_REF] . In this work, we found that in DIO mice, 9-PAHPA had no effect on muscle phenotype. In db/db mice, strong muscle atrophy was recorded compared to db/ + animals. However, 9-PAHPA supple-mentation was not able to compensate for this muscle loss, nor did it induce any change in the overall parameters studied.

In this study, the major effect that emerged from curative 9-PAHPA supplementation was its ability to decrease insulin resistance in DIO and db/db mice. This positive result of 9-PAHPA on insulin sensitivity is in agreement with our previous results with healthy or diet-induced obese mice [ 15 , 16 ] and those of other groups [ 4 , 7 , 9 -11 ]. Furthermore, a possible effect of 9-PAHPA supplementation on hepatic neoglucogenesis was not addressed in this work. However, the observation during insulin sensitivity tests that the return of blood glucose to normal in the second part of the experiment (between 60 and 120 min) was slower in animals supplemented with 9-PAHPA suggests that it could also decrease hepatic glucose production as it has been shown previously for 9-PAHSA [START_REF] Zhou | PAHSAs enhance hepatic and systemic insulin sensitivity through direct and indirect mechanisms[END_REF] .

In conclusion, we have investigated the impact of curative 9-PAHPA supplementation in two animal models of obesity, the classical DIO and the db/db mice. In these two obesity models, 9-PAHPA was not able to prevent either weight gain or the development of hyperglycemia. Moreover, 9-PAHPA had no impact on the metabolic parameters analyzed at the hepatic and muscular level. In the present study, the only beneficial effect of a curative 9-PAHPA supplementation observed was a decrease in insulin resistance in DIO and db/db mice. These results are comparable with those obtained recently with other FAHFAs and highlight that although these molecules alone are able to improve insulin sensitiviity, they have little effects on obesity and associated metabolic dis-orders. Perhaps their action could be more interesting in the longer term or in combination with other molecules or with physical ac-tivity practice.

Authors Contributions

Author contributions were as follows: study design (CC, FC, CF-C), FAHFAs synthesis (LB, MB), data collection (BB, LP, MB, KL, AJCE, CBG), statistical analysis (KL, FC, CF-C), data interpretation (CC, FC, LP, AJCE, MMA, JAR, SG, TD, KL, LB, CF-C), manuscript preparation (FC, CF-C), literature search (CC, FC, CF-C) and funds collection (CC, FC, CF-C). All the authors have read and approved the final version of this manuscript.

Fig. 1 .

 1 Fig. 1. Design of the study.

Fig. 2 .

 2 Fig. 2. DIO mice characteristics. (A) Body weight, lean and fat mass, energy intake, plasma IL6, and plasma lipids. (B) Oxygen consumption in mice (VO2). (C) Respiratory exchange ratio (RER) in mice. (D) fasting blood glucose. (E) Insulin tolerance test (ITT). (F) Glucose tolerance test (OGTT). For each mouse, VO2 and VCO2 were measured 111 times in about 24 h. The respiratory exchange ratio (RER) was calculated as the volume of CO2 versus volume of oxygen (VCO2/VO2) ratio. Mean VO2, VCO2 and RER values are mean of 8 mice for each group considering all the cycles. For all other parameters, n = 10 animals per group. Results were expressed as means ±SD, Control diet versus HFHS diet and HFHS diet versus HFHS + 9-PAHPA diet were tested by a student t test; * P < .05 HFHS versus Control and §P < .05 HFHS + 9-PAHPA versus HFHS.

Fig. 3 .

 3 Fig. 3. Liver phenotype. (A) Liver weight after 16 weeks of diet. (B) Hepatic content of cholesterol triglycerides and free fatty acids (FFA). (C) Relative liver gene expression of TNF-α. (D) H&E staining of liver, pictures are representative from each group. Scale bar, 10 μm. (E) Plasma ALAT activity. (F) Relative Liver protein expression of key players in lipogenesis, glucose and cellular redox homeostasis. (G) Liver β-HAD activity. Results were expressed as means ±SD, n = 10 animals per group. Control diet versus HFHS diet and HFHS diet versus HFHS + 9-PAHPA diet were tested by a student t test; * P < .05 HFHS versus Control and §P < .05 HFHS + 9-PAHPA versus HFHS.

Fig. 4 .

 4 Fig. 4. Mitochondrial activity and hepatic oxidative stress parameters. (A) Mitochondrial respiratory chain activities. (B) Citrate synthase activity. (C) TBARS and -SH values. (D) GSH and GSSG values. (E) SOD, MnSOD, Catalase and GPx activities. Results were expressed as means ±SD, n = 10 animals per group. Control diet versus HFHS diet and HFHS diet versus HFHS + 9-PAHPA diet were tested by a student t test; * P < .05 HFHS versus Control and §P < .05 HFHS + 9-PAHPA versus HFHS.

Fig. 5 .

 5 Fig. 5. Muscle phenotype . (A) Quadriceps, gastrocnemius, and tibialis muscles weight. (B) Quadriceps muscle content of cholesterol triglycerides and FFA measured on tissue homogenate by enzymatic method. (C) Representative anti-laminin staining on cryosections of tibialis muscles. Scale bar, 200 μm. (D) Total fibers number. (E) Mean fibers area. (F) Fiber size distribution in tibialis muscles. Results were expressed as means ±SD, n = 10 animals per group. Control diet versus HFHS diet and HFHS diet versus HFHS + 9-PAHPA diet were tested by a student t test; * P < .05 HFHS versus Control and §P < .05 HFHS + 9-PAHPA versus HFHS.

Fig. 6 .

 6 Fig. 6. Mitochondrial activity and muscle oxidative stress parameters. (A) Mitochondrial respiratory chain activities. (B) Citrate synthase activity. (C) β-HAD activity. (D) TBARS and -SH values. (E) SOD, MnSOD, Catalase and GPx activities (F) GSH values. Results were expressed as means ±SD, n = 10 animals per group. Control diet versus HFHS diet and HFHS diet versus HFHS + 9-PAHPA diet were tested by a student t test; * P < .05 HFHS versus Control and §P < .05 HFHS + 9-PAHPA versus HFHS.

Fig. 7 .

 7 Fig. 7. db/db mice characteristics. (A) Body weight, energy intake, leptin and IL6 plasma levels, and plasma lipids. (B) Oxygen consumption in mice (VO2). (C) Respiratory exchange ratio (RER) in mice. (D) fasting blood glucose, insulin levels and HOMA-IR index. (E)Insulin tolerance test (ITT). For each mouse, VO2 and VCO2 were measured 111 times in about 24 h. The respiratory exchange ratio (RER) was calculated as the volume of CO2 versus volume of oxygen (VCO2/VO2) ratio. Mean VO2, VCO2 and RER values are mean of 8 mice for each group considering all the cycles. Results were expressed as means ±SD, n = 10 animals per group except for VO2, CO2 and RER data. db/ + versus db/db and db/db versus db/db + 9-PAHPA were tested by a student t test; * P < .05 db/db versus db/ + and $ P < .05 db/db + 9-PAHPA versus db/db.

Fig. 8 .

 8 Fig. 8. db/db liver phenotype. (A) Liver weight and plasma ALAT activity after 7 weeks of diet. (B) H&E staining of liver, pictures are representative from each group. Scale bar, 10 μm. (C) Liver mitochondrial activity. (D) Citrate synthase activity. (E) TBARS and -SH values. (F) GSH and GSSG values. (G) SOD, MnSOD, Catalase and GPx activities. Results were expressed as means ±SD, n = 10 animals per group. db/ + versus db/db and db/db versus db/db + 9-PAHPA diet were tested by a student t test; * P < .05 db/db versus db/ + .

Fig. 9 .

 9 Fig. 9. Muscle phenotype . (A) Quadriceps, gastrocnemius, and tibialis muscles weight. (B) Muscle triglycerides content. (C) Representative anti-laminin staining on cryosections of tibialis muscles. Scale bar, 200 μm. (D) Total fibers number. (E) Mean fibers area. (F) Fiber size distribution in tibialis muscles. Results were expressed as means ±SD, n = 10 animals per group. db/ + versus db/db and db/db versus db/db + 9-PAHPA diet were tested by a student t test; * P < .05 db/db versus db/ + .

Fig. 10 .

 10 Fig. 10. Mitochondrial activity and muscle oxidative stress parameters. (A) Mitochondrial respiratory chain activities. (B) Citrate synthase activity. (C) β-HAD activity. (D) TBARS and -SH values. (E) SOD, MnSOD, Catalase and GPx activities. (F) GSH values. Results were expressed as means ±SD, n = 10 animals per group. db/ + versus db/db and db/db versus db/db + 9-PAHPA diet were tested by a student t test; * P < .05.

Acknowledgments

We thank the animal stafffor daily animal care. The authors also wish to thank the animal from the Metamus DMeM facility ( https://doi.org/10.15454/WYR2-8706 ), which belongs to the Montpellier animal facilities network (RAM), Biocampus for technical support and expertise for metabolism phenotyping, and Metamontp platform supported by the European fund for the regional development of Occitanie (FEDER). The authors also thank the "Réseau d'Histologie Expérimentale de Montpellier"(RHEM) histology facility for processing our liver samples.

Funding

Melha Benlebna thanks the Algerian Ministry of Higher Education and Scientific Research for the financial support of her PhD program. The authors also acknowledge the financial support of the French Lipid Nutrition Group, and the National Research Insti-tute for Agriculture, Food, and Environment (INRAE).

Declaration of competing interests

The authors declare no conflict of interest.