Giulia Chinetti-Gbaguidi

Bart Staels
email: <bart.staels@pasteur-lille.fr

Macrophage polarization in metabolic disorders: functions and regulation

Keywords: Adipose Tissue, immunology, Animals, Humans, Inflammation, immunology, Macrophages, cytology, metabolism, Metabolic Diseases, immunology macrophages, obesity, atherosclerosis, nuclear receptors

Purpose of review

To discuss recent findings on the role and regulation of macrophage polarization in obesity and atherosclerosis.

Recent findings

Macrophages

infiltrate the vascular wall during atherosclerosis and adipose tissue during obesity. At least two distinct sub-populations with different functions, the classically (M1) and the alternatively (M2) activated macrophages, have been found in these tissues. Reciprocal skewing of macrophage polarization between the M1 and M2 states is a process modulated by diet, humoral and transcription factors, such as the nuclear receptor Peroxisome Proliferator-Activated Receptor gamma (PPAR ).

γ

INTRODUCTION

Monocyte/macrophages are heterogeneous, versatile cells which functionally adapt to specific microenvironmental signals.

Differential expression of selected surface markers is used to identify monocyte and macrophage sub-populations. Many of the used markers differ, however, between mice and humans.

In humans, the presence of CD16 distinguishes two monocyte subsets. CD14 /CD16 monocytes (representing approximately 90 of

+ - %
circulating monocytes), which express high levels of the chemokine receptor CCR2 (MCP-1 receptor) and low levels of CX3CR1 (fractalkine receptor). CD14 /CD16 monocytes display high CX3CR1 expression and a pro-inflammatory profile . Mouse monocytes

+ + [1 ]
are distinguished on the basis of Ly6C antigen expression . The short-lived CXCR1 CCR2( )Ly6C inflammatory monocytes are Diverse macrophage populations have also been described, but the relationship between the monocyte subsets and macrophage phenotype is unclear. Classically activated M1 macrophages, driven by Th1 cytokines (IFN , TNF) or by LPS recognition, produce high γ levels of IL-12 and IL-23 (in humans) and low levels of IL-10, secrete pro-inflammatory cytokines (TNF, IL-6, IL1 ) and exhibit macrophages, which are functionally anti-inflammatory, but still produce high amounts of inflammatory cytokines .

[5 ]

Moreover, human monocytes differentiated in the presence of granulocyte macrophage colony stimulating factor (GM-CSF) or -M-CSF, respectively, display M1 and M2 properties, and have been referred to as M 1 and M 2 . More recently, the platelet factor-4

ø ø [3 ]
(CXCL4) was shown to induce a unique transcriptional profile generating M4 macrophages, characterized by reduced CD163 and other scavenger receptor expression as well as phagocytic capacity, while some M1 and M2 markers are over-represented , . This [6 • 7 ] classification system represents the extremes of macrophage activation states and probably simplifies the situation where the full in vivo phenotypic spectrum between M1 and M2 can exist . Moreover, an alternative classification of macrophage populations has been

[8 ]
proposed based on three different macrophage homeostatic activities: host defence, wound healing and immune regulation . This

[8 ]

classification distinguishes classically activated macrophages, wound-healing macrophages (corresponding to the alternatively activated macrophages) and regulatory macrophages . This latter polarization, induced by TLR agonists in combination with other stimuli

[8 ]
(amongst which IgG immune complexes, prostaglandins, apoptotic cells), produces high levels of IL-10, thus resembling the M2b phenotype.

Although general properties of macrophages are conserved between species, there are some significant differences between mice and humans. For example markers such as Ym1 (also called chitinase 3-like 3), the transcription factor Found in Inflammatory Zone 1 (FIZZ1)

and arginase 1 (Arg1), and more generally arginine metabolism, are characteristic of the M2 phenotype in mouse but not in human macrophages (see ).

[9 ]

table 1

Macrophage sub-populations in pathophysiology

The role of alternative macrophages in tumorigenesis is well established. Tumour-associated macrophages (TAM), display an alternative-like activation phenotype and play a detrimental pro-tumoural role . In this review we will focus on the involvement of

[10 ]
alternative macrophages in cardio-vascular and metabolic disorders such as atherosclerosis and obesity ( ). figure 1

Macrophage sub-populations and atherosclerosis

Only few data report the presence/characterization of alternative macrophages within atherosclerotic plaques in mice. In apoE-deficient mice on chow diet, early lesions (20 weeks) contain Arg1-positive M2 macrophages, whereas Arg2-positive M1 macrophages appear and prevail in lesions of aged mice (55 weeks). Lesion progression is thus associated with the predominance of the M1 over the M2 phenotype . M2 to M1 phenotype switching of pre-existing cells rather a direct recruitment of M1 macrophages likely [11 ] occurs in this model. High-fat diet feeding of 10 week-old apoE-deficient mice induced the expression of the M2-associated genes, selenoprotein-1 (SEPP1), stabilin-1 (STAB1) and CD163 molecule-like-1 (CD163L1) in parallel with atherosclerotic plaque development.

In contrast, expression of the M1-related marker, pro-platelet basic protein (PPBP), increased only in very early lesions (6 weeks of diet) returning to basal levels in late stages (12 weeks of diet) . A distinct macrophage phenotype, strikingly different from the conventional

[12 ]
M1 and M2 macrophages, has been reported in atherosclerotic lesions of LDL receptor-deficient mice . This phenotype, called Mox,

[13 •• ]
is induced by oxydized phospholipids and is characterized by high expression of heme oxygenase 1 (HO-1), sulforedoxin-1 (Srnx-1) and thioredoxin-1 reductase (txnrd-1), all redox-regulatory genes under the control of the Nrf2 transcription factor . Mox macrophages

[13 •• ]
represent approximately 30 of all macrophages in murine advanced atherosclerotic lesions.

%

Using the expression levels of different macrophage markers, as well as morphology and phenotypic analysis, several macrophage sub-populations have been identified in human atherosclerosis. In human coronary arteries, macrophages expressing both CD68 and CD14 appear, while a CD68 CD14 sub-population appear predominant in areas devoid of disease . While the CD68 CD14 cells express

+ - [14 ] + +
many pro-inflammatory genes, the CD68 CD14 cells express high levels of genes involved in reverse cholesterol transport (PPAR ,

+ - γ
LXR , ABCG1) and macrophage emigration (CCR7) from the vessel wall . These two populations are similar to monocytes

α [14 ]
in vitro differentiated with M-CSF or GM-CSF, respectively . Similarly, using the M-CSF (M2) GM-CSF (M1) macrophage model, the M2

[14 ] vs phenotype-associated genes, SEPP1, STAB-1 and CD163L1, were found to be enriched in human atherosclerotic plaques compared to fatty streaks or normal arteries .

[12 ]

We identified CD68 MR (M1) macrophages in the lipid core of human carotid atherosclerotic lesions, while CD68 MR (M2)

+ - + +
macrophages prevail in the shoulder region as well as in the periphery of the plaque , . CD68 MR macrophages appear smaller

[15 16 ] + +
and contain several small lipid droplets in their cytoplasm, while CD68 MR macrophages contain fewer, but bigger lipid droplets.

+ -

In IL4-polarized M2 macrophages are less competent to captate native and oxidized lipoproteins. Whereas CD68 MR macrophages vitro

+ +
display lower cholesterol-handling capacity, they are competent for phagocytosis, since the expression of opsonins and receptors involved in phagocytosis is high in these cells . Moreover, hemorrhaged atherosclerotic plaques contain hemorrhage-associated macrophages

[16 •• ]
(HA-mac) , which contain more iron, express high levels of CD163 and thus highly scavenge the haemoglobin/haptoglobin complex

[17 ]
which induces IL-10 secretion and monocyte differentiation to M2 macrophages. Differentiation to HA-mac was prevented by neutralizing IL-10 antibodies, indicating that IL-10 mediates an autocrine feedback mechanism. Whether these different populations represent completely different polarization states or whether they display overlapping phenotypes, is a challenging question.

Macrophage sub-populations and obesity

Substantial evidences at the cellular and molecular level indicate that obesity is a chronic low-grade inflammatory disease .

[18 ]

Monocytes infiltrate adipose tissue during obesity and differentiate in adipose tissue macrophages (ATM) . ATM from lean mice

[19 ]
express many genes characteristic of M2 macrophages, which may protect adipocytes from inflammation, while diet-induced obesity led to a shift in the activation state to an M1 pro-inflammatory state that contributes to insulin resistance . Comparative studies revealed

[20 -22 ]
that the majority of adipose tissue-produced cytokines (TNF, IL-6), with the exception of leptin and adiponectin, are secreted by non-adipocyte cells and in particular by M1 polarized macrophages. These pro-inflammatory cytokines may contribute to the low-grade inflammatory state. Furthermore, while M2 ATM, which express N-acetyl-galactosamine specific lectin 1 (MGL1), are localized in the interstitial space, MGL1 /CD11c M1 ATM rather surround death adipocytes thus forming the crown-like structures (CLS) . The

- + [23 ]
CD11c surface molecule is considered as an M1 marker and its expression in ATM is considerably increased upon high-fat diet feeding [20 , . The obesity-induced switch from the M2 to M1 phenotype was attributed to a CCR2-dependent monocyte recruitment rather than to 24 ] the conversion of pre-existing M2 macrophages .

[23 ]

However, this strict spatiotemporal polarization concept has been challenged more recently. Indeed, mouse epidydimal ATM recruited in response to a high fat-diet display a mixed M1/M2 phenotype and their transcription profile became more M2-like upon diet duration extension . Using MR and CD11c as markers, three distinct ATM populations have been described . Obesity promotes a shift CCR5, CCR3 and CX3CR1 to be expressed on both MR CD11c and MR CD11c , whereas CCR7 and CCR9 were selectively

- + - -
expressed in MR CD11c and MR CD11c , respectively . These data reveal previously unappreciated similarities between the

- + - - [26 ]
murine and human ATM phenotypes.

In humans, the amount of ATM correlates with BMI, adipocyte size and total body fat mass . Fat mass expansion is associated [START_REF] Curat | Macrophages in human visceral adipose tissue: increased accumulation in obesity and a source of resistin and visfatin[END_REF] with the accumulation of anti-inflammatory or mixed M1/M2 polarized ATM , characterized by increased MMP activities, indicating [START_REF] Zeyda | Human adipose tissue macrophages are of an anti-inflammatory phenotype but capable of excessive pro-inflammatory mediator production[END_REF] a role in tissue remodelling . Human ATM produce the anti-inflammatory cytokines IL-10 and IL-1Ra, but can also secrete [START_REF] Bourlier | Remodeling phenotype of human subcutaneous adipose tissue macrophages[END_REF] pro-inflammatory cytokines such as TNF, IL-6 and IL-1 supporting a role of ATMs in obesity-induced adipocyte dysfunction and β metabolic disorders . Together with increased macrophage numbers, adipose tissue from obese subjects contain increased areas of [START_REF] Zeyda | Human adipose tissue macrophages are of an anti-inflammatory phenotype but capable of excessive pro-inflammatory mediator production[END_REF] fibrosis, when compared to lean subjects . The majority of macrophages appear rather associated with fibrosis, than in CLS. CLS [START_REF] Spencer | Adipose tissue macrophages in insulin-resistant subjects are associated with collagen VI and fibrosis and demonstrate alternative activation[END_REF] macrophages are predominantly M1, whereas other macrophages, particularly those in fibrotic areas, are CD150 , a M2c marker .

+ [30 ]
Regulation of macrophage polarization in metabolic disorders: lessons from humans and mice

Regulation by lifestyle and diet

In obese mice, progressive lipid accumulation in macrophages with age provokes a M2 to M1 polarization . Nearly 50 of the [START_REF] Prieur | Differential lipid partitioning between adipocytes and tissue macrophages modulates macrophage lipotoxicity and M2/M1 polarization in obese mice[END_REF] %

ATM of obese mice accumulate lipids and resemble atherosclerotic foam cells. Lipid accumulation was associated with increased CD11c + ATM (M1) and decreased CD209a ATM (M2). However, a unique subclass of lipid loaded CD209a /CD11c has been identified.

+ + +

Diphtheria toxin induced death of CD11c cells reduces CLS formation and improves insulin sensitivity , emphasizing the pathogenic

- + [32 ]
role of CD11c M1 macrophages. Stimulation of obese db/db mice with resolvin D1, a docosahexaenoic acid-derived anti-inflammatory + mediator, reduced the number of M1 macrophages resident in CLS and increased the percentage of MGL1 macrophages .

+ [33 ]
Gastric surgery of obese subjects modulated the M1/M2 balance by increasing the number of M2 and decreasing the number of M1 macrophages in sub-cutaneous adipose tissue . Weight stabilization by 6-month of very low calorie diet decreased the amount of [START_REF] Tordjman | Human adipose tissue macrophages: m1 and m2 cell surface markers in subcutaneous and omental depots and after weight loss[END_REF] ATM, without changing the relative proportion of M1/M2 macrophages . The discrepancies between these two studies can be due to [START_REF] Kovacikova | Dietary intervention-induced weight loss decreases macrophage content in adipose tissue of obese women[END_REF] differences in the type of patient population and/or cell surface markers used to identify ATM sub-populations.

Exercise training promoted M1 to M2 phenotype switching of adipose tissue macrophages in high fat diet-induced obese mice [START_REF] Kawanishi | Exercise training inhibits inflammation in adipose tissue via both suppression of macrophage infiltration and acceleration of phenotypic switching from M1 to M2 macrophages in high-fat-diet-induced obese mice[END_REF] and enhanced the expression of M2 markers of circulating leukocytes in humans, an effect which may be related to an increased expression of PPAR and its cofactors .

γ [37 ]

Regulation by humoral factors

Macrophage polarization can be modulated by different factors. Among these, C-reactive protein , angiotensin type 1 receptor angiotensin converting enzyme (ACE) favour an M2 differentiation. IL-4 is one of the most potent M2-drivers. In adipose tissue, [START_REF] Kohlstedt | Adipocyte-derived lipids increase angiotensin-converting enzyme (ACE) expression and modulate macrophage phenotype[END_REF] eosinophils are the major IL-4 producing cells and in their absence the macrophage M2 phenotype is greatly attenuated .

[

•• ]

Deficiency in CD40/tumor necrosis factor receptor-associated factor (TRAF)6 signalling skewed the immune response toward an anti-inflammatory M2 profile, thus preventing atherosclerosis . In addition to immune-inflammatory factors, high density lipoproteins

[47 ]

(HDL) also decrease the expression of pro-inflammatory factors and increase M2 markers (Arg1, MR, CD163) in mice, in an apoAI-dependent manner .

[48 enhances PPAR activity through an interaction between PPAR and the Signal Transducer and Activators of Transcription (STAT)6 on γ γ promoters of PPAR -target genes . In response to IL-4, STAT6 and the PPAR -coactivator-1beta (PGC-1 ) induce macrophage

γ [51 ] γ β
programs for fatty acid oxidation and mitochondrial biogenesis ( ). IL-4 and IL-13 also induce PPAR / expression through a [START_REF] Vats | Oxidative metabolism and PGC-1beta attenuate macrophage-mediated inflammation[END_REF] figure 2 β δ STAT6 site in its promoter to activate alternative differentiation .

[53 ]

The first evidence of modulation of macrophage polarization by nuclear receptors came from observations related to obesity.

Macrophage-specific PPAR deletion in Th2-oriented Balbc mice identified a role for PPAR in the polarization of macrophages toward upon bone marrow transplantation . This suggests that macrophage alternative polarization is not completely dependent on nuclear [START_REF] Marathe | Preserved glucose tolerance in high fat diet-fed C57BL/6 mice transplanted with PPARgamma / , PPARdelta / , PPARgamma ---delta / or LXRalpha beta / bone marrow[END_REF] receptor signalling and the effects can be modulated by the genetic background. per se, In obese mice, treatment with the PPAR ligand rosiglitazone promotes lipid redistribution from macrophages towards adipocytes thus γ restoring a M2 phenotype . In line, treatment with PPAR ligands, rosiglitazone and pioglitazone, as well as telmisartan, an [START_REF] Prieur | Differential lipid partitioning between adipocytes and tissue macrophages modulates macrophage lipotoxicity and M2/M1 polarization in obese mice[END_REF] γ

angiotensin II type 1 receptor blocker with PPAR agonist activity , decreased the number of M1 macrophages in visceral adipose γ [START_REF] Matsumura | Telmisartan Exerts Antiatherosclerotic Effects by Activating Peroxisome Proliferator-Activated Receptor-gamma in Macrophages { }[END_REF] tissue and enhanced the expression of M2 markers thus improving insulin sensitivity , .

[ regression significantly enhanced the expression of M2 markers in parallel with a reduction of plaque macrophage and lipid content .

[60 ]

PPAR activation also promotes IL-4 induced M2 differentiation of primary human monocytes resulting in a more pronounced M2 macrophage differentiation of human monocytes .

[61 ]

Despite their important role in the inflammatory response, whether LXRs influence macrophage polarization has not yet been reported.

However, LXR expression and signalling is altered in human IL-4 polarized M2 macrophages . These cells display a reduced

α [16 •• ]
capacity to handle and efflux cellular cholesterol due to low expression levels of LXR and its target genes, ABCA1 and ApoE . The

α [16 •• ]
decreased LXR expression and activity in M2 macrophages is likely related to an enhanced 15-lipoxygenase activity. As a consequence, α PPAR activation has no effect on the cholesterol efflux pathways. By contrast, PPAR activation induces the expression of

γ γ
thrombospondin-1 in M2 macrophages, thus enhancing their phagocytic activity . with dexamethasone prevented ATM accumulation, an effect most pronounced for the F4/80 /CD11b /CD11c M1 subset .

+ + + [64 ]
Other nuclear receptors can also control macrophage polarization. As an example, the mineralocorticoid receptor (MR) is necessary for efficient macrophage polarization by inflammatory cytokines . Macrophages differentiated from MR-deficient myeloid cells [START_REF] Usher | Myeloid mineralocorticoid receptor controls macrophage polarization and cardiovascular hypertrophy and remodeling in mice[END_REF] display a transcription profile of alternative activation and protect against cardiac hypertrophy and fibrosis.

CONCLUSIONS

Although significant progress has been made in characterizing the phenotype and functions of macrophage sub-populations during obesity and atherosclerosis development, much remains to be discovered. It appears of crucial importance to define a panel of markers in mice and humans allowing identification of the macrophage sub-types in different microenvironments. Investigations on the role of these different macrophages in adipose tissue and vascular wall patho-physiology and identification of endogenous as well as novel synthetic molecules controlling the M1/M2 macrophage balance, among which potential nuclear receptor ligands, represent exciting areas of future research which may lead to possible therapeutic applications.

KEY POINTS

Macrophages are a heterogeneous cell population exhibiting a wide spectrum of phenotypes;

Classically M1 and alternatively M2 activated macrophages are found in adipose tissue and atherosclerotic plaques of the vascular wall;

M1 and M2 macrophages are phenotypically and functionally different;

Macrophage M1 to M2 skewing is transcriptionally regulated by nuclear receptors including PPARγ

Acknowledgements:

Grants REFERENCES AND RECOMMENDED READING Papers of particular interest, published within the annual period of review, have been highlighted as:

• of special interest

Table 1

Mouse and human macrophage sub-population markers Some markers are common amongst different macrophage subtypes, suggesting that overlapping phenotypes can exist. Caution is required when a novel macrophage sub-type is identified solely on the basis of the expression of one or two markers.

Markers Mouse Human

  inflamed tissues, while the CX3CR1 CCR2( )Ly6C subset are characterized by a CX3CR1-dependent high low recruitment to non-inflamed tissues.

  , thus conferring a Th1-like response. During the inflammatory response, inflammation is usually spatially and temporally counteracted by protective mechanisms operated by the alternatively activated M2 macrophages. The M2 alternative phenotype is typically driven by Th2 cytokines. 3 different sub-classes have been identified: IL-4 and IL-13 lead to M2a macrophages, immune complexes in combination with IL-1 or LPS drive the M2b subtype, while IL-10, TGF or glucocorticoids induce M2c β β macrophages. M2 macrophages are characterized by high IL-10 and TGF (especially prominent in M2c) and low IL-12 production, high β endocytic clearance capacities, protecting surrounding tissues from a detrimental immune response . An exception are the M2b [4 ]

  from a predominant MR CD11c population (expressing marginally M1 and M2 markers, but high levels of MCP-1 and CCL7) to two + -MR populations: MR CD11c cells exhibiting a M1 inflammatory phenotype and MR CD11c cells expressing low levels of --+ -inflammatory markers and high levels of M2 markers such as Arg1 and Ym1 . Analysis of chemokine receptors identified CCR2, [26 ]

[25 59 ]

 59 Information on the role of nuclear receptors in macrophage polarization in the context of atherosclerosis is scarce. In the Reversa mouse model (Ldlr Apob Mttp Mx1-Cre ), treatment with the PPAR ligand pioglitazone during the phase of atherosclerosis /

  Glucocorticoid receptor (GR) activation induces a unique M2 phenotype, characterized by the high expression levels of CD36, CD163, IL-10 and IL-1 receptor type II and enhanced antioxidative homeostasis and iron recycling , . Treatment of high-fat diet-fed mice[62 63 ]

Figure 1

 1 Figure 1Circulating monocytes can infiltrate the vascular wall during atherosclerosis and adipose tissue during obesity and can differentiate in different macrophage sub-populations, exemplified by the extreme pro-inflammatory M1 and anti-inflammatory M2 macrophage subtypes

γ

  IL-4 induces PPAR expression and activity by generating natural PPARγ γ ligands via the 12/15-lipoxygenase (12/15 LOX) pathway PPAR activation by its ligands promotes monocyte differentiation toward M2 γ macrophages in mice and humans.

  Transcription factor networks are important regulators of macrophage polarization. Among these, nuclear receptor family members, including the Peroxisome Proliferator-Activated Receptors (PPAR , PPAR and PPAR / ) and the Liver X Receptors (LXR and LXR )

	α	γ	β δ		α	β
	are highly expressed in macrophages where they control the inflammatory response	[49 ]	. Glass et al. reported that IL-4 induces PPAR	γ
	expression and activity by generating natural PPAR ligands via the 12/15-lipoxygenase pathway γ	[50 ]	. Furthermore, IL-4 signalling

•• ]

Regulation by transcription factors

  However, there is some controversy, since Marathe et al. reported that Th1-oriented C57BL/6 mouse macrophages lacking PPAR , PPAR / or LXR / display unaltered expression profiles of alternative macrophage markers and presented normal glucose tolerance

		γ		γ
	an alternative phenotype. Disruption of PPAR in myeloid cells impairs the alternative differentiation, an effect associated with the γ
	development of diet-induced obesity, insulin resistance and glucose intolerance	[54 ]	. Myeloid PPAR / -deficiency rendered macrophages β δ
	unable to shift toward an alternative phenotype, causing inflammation and metabolic perturbations in adipocytes and hepatocytes leading
	to insulin resistance, increased adipocyte lipolysis and hepatosteatosis	, [53 55 ]	.
	β δ	α β

γ

  from the R gion Nord-Pas de Calais/FEDER (CPER N. 1449), COST actions BM0602 and BM0904, the Agence Nationale de la Recherche (AlMHA project), the transatlantic Leducq HDL Network, the Fondation Coeur et Art res and the European Community s 7th

	è	'
	Framework Programme (FP7/2007-2013, grant agreement n 201608) are acknowledged.

é °References:

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Benoît Darties

Fabrice Theoleyre

Andrzej Duda

A restricted-Weakly Connected Dominating Set for Role Assignment in a Multichannel MAC for Wireless Mesh Network

Keywords: wireless mesh networks, Molecular MAC, WCDS, 2-approximation

We propose an efficient way of constructing the wireless mesh structure associated with Molecular MAC, a multichannel access method designed for efficient packet forwarding. We base our role assignment on a restricted Weakly Connected Dominating Set structure. After presenting a formal definition of the role assignment problem, we prove its NP-completeness. Then, we propose a centralized 2-approximation algorithm that maximizes the sum of radio link capacities in the molecular structure. Finally, we extend this protocol so that it can operate in a distributed way still providing the same guarantee. This distributed protocol is self-stabilizing thus robust to topology changes. Our simulation results show that the 2-approximation distributed protocol provides an improvement in throughput with respect to other protocols.

I. INTRODUCTION

We consider wireless mesh networks that use IEEE 802.11 wireless LANs for interconnecting mesh routers. When mesh routers use the legacy IEEE 802.11 networks with a single interface, the performance of packet forwarding quickly degrades with the number of hops due to channel contention and spatial problems such as hidden, exposed, masked, and blocked nodes [START_REF] Chaudet | Performance Issues with IEEE 802.11 in Ad Hoc Networking[END_REF]. We can observe that the capacity of a wireless mesh network strongly depends on the ability of nearby mesh routers to communicate in parallel, which is only possible if neighbor routers that may interfere use different channels.

One way of improving performance is to use multiple nonoverlapping channels so that stations can transmit in parallel and without collisions. When a mesh router has several radio interfaces, it can tune them to different channels and use them simultaneously. The main problem in this approach is thus to assign static channels to interfaces in a way that maximizes network capacity, either with a centralized [START_REF] Nguyen | Channel Assignment for Multicast in Multi-channel Multi-radio Wireless Mesh Networks[END_REF] or a distributed algorithm [START_REF] Avallone | A Channel Assignment Algorithm for Multi-Radio Wireless Mesh Networks[END_REF].

A more dynamic approach consists of using a MAC layer allowing each node to switch to a different channel on demand. In this way, the MAC layer can distribute load more uniformly over channels thus achieving better throughput. Moreover, This work was partially supported by the European Commission project WIP under contract 2740, the French Ministry of Research project AIRNET under contract ANR-05-RNRT-012-01 and ARESA under contract ANR-05-RNRT-01703.

the approach may benefit to stations with single or multiple interfaces. An example of such a method is MMAC (Multi-Channel MAC) [START_REF] So | Multi-channel mac for ad hoc networks: handling multi-channel hidden terminals using a single transceiver[END_REF] that proposed periodic Rendezvous Points during which nodes reserve channels to use later for data transmission. Molecular MAC [START_REF] Nassiri | Molecular architecture for autonomic wireless mesh networks[END_REF] defined a molecular architecture based on allocating roles and channels to mesh nodes according to the analogy with atoms containing a nucleus and electrons. This approach requires a specific MAC mechanism to support multichannel multiplexing for efficient packet forwarding and results in significant performance improvement over MMAC.

In this paper, we address the problem of constructing the structure associated with Molecular MAC: it requires allocating different roles (nucleus or electron) to mesh nodes. Although such a structure is related to the problem of clustering, Molecular MAC imposes some specific constraints: we must guarantee global connectivity along with the requirement of alternating the roles of nodes-neighboring nodes that directly communicate must not have the same role so they do not experience the deafness problem when a node tries to send a frame on one channel while the intended receiver is listening or sending on another one. In addition to that, we need to maximize network capacity.

The contributions of this paper are threefold:

• we formally define the problem of role assignment and show its NP-completeness in a general case, • we propose a centralized near-optimal algorithm for finding the required structure for Molecular MAC that guarantees the number of links closed to the optimal by the factor of 2 (such a guarantee is called 2-approximation); the algorithm has running time O(m), where m is the number of links; • we also propose a distributed version of this 2approximation that cope with node and/or link failures.

II. MOLECULAR MAC OVERVIEW & MOTIVATION

The present paper aims at defining protocols for constructing a suitable network structure required for Molecular MAC. Molecular MAC proposes to extend IEEE 802.11 to efficiently use multichannel nodes in multi-hop networks. It achieves efficient dynamic multichannel multiplexing along with solving the deafness problem.

The idea of Molecular MAC is to partition a mesh network into spatially distinct atoms that are similar to traditional IEEE 802.11 infrastructure mode cells: a nucleus node in each cell always uses a channel different from interfering nuclei. Nodes between two nuclei, which we call electrons, dynamically switch channels to communicate with neighbor nuclei. There is no direct link between two electrons so to communicate, they use a nucleus like an access point in a 802.11 WLAN. Two nuclei do not directly communicate neither, because they use different channels. If they need to communicate, they can do it through neighboring electrons.

Molecular MAC uses a nucleus as a kind of a virtual access point: it is either a receiver or a transmitter of any frame in an atom. Packet forwarding relies on electrons that switch between channels of all neighboring atoms. This scheme assigns one channel to each nucleus and all electrons use it for communication with the nucleus (for both reception and transmission). By choosing different channels, two neighboring atoms that would have interfered, if they had used the same channel, can limit interference problems, which contributes to achieving higher capacity.

In Molecular MAC, a nucleus chooses its channel according to the load measured locally, announces it to all electrons, and stays tuned to it. An electron belonging to several atoms knows the channels of its nuclei and can switch to a given channel when it wants to communicate.

For efficient packet forwarding an electron explicitly requests a data frame from a nucleus by sending a special control frame, a pull that acts a little bit like a Clear To Send (CTS) reservation frame. Each nucleus piggybacks the list of pending destinations in each data frame so that the electrons know when sending a pull is relevant. An electron periodically scans each atom it belongs to for pending packets. Moreover, if some packets are pending, it randomly chooses to send or receive one. A nucleus maintains activity by sending an empty data frame to implement a quick notification. When a node wants to join the mesh, it performs multichannel neighborhood discovery to detect eventual neighboring nuclei and become an electron. Otherwise, it takes the role of a nucleus.

Figure 1 illustrates the structure of a molecular mesh. The identifiers of nuclei are numbers while they are letters for electrons. We can note that there is no link between two electrons nor between two nuclei. However, we need to keep the number of unused links small to allow for redundant paths in the network for better connectivity and fault-tolerance.

Note that in this paper, we limit its scope to mesh nodes with a single radio interface, however we can easily generalize the results to the case of multiple interfaces.

III. RELATED WORK

The problem of constructing a molecular mesh relates to the problem of clustering for which many efficient distributed algorithms exist. We can elect clusterheads with a Maximum Independent Set [START_REF] Lin | Adaptive clustering for mobile wireless networks[END_REF], however in some cases, clusterheads can be exactly three hops apart, which means that under Molecular MAC the network becomes disconnected.

Many authors studied the Weakly Connected Dominating Sets (WCDS) and applied them to various network problems such as routing or pre-shared key distribution [START_REF] Pathan | A key-predistribution-based weakly connected dominating set for secure clustering in dsn[END_REF]. Construction of a WCDS consists of electing dominators so that each node is a neighbor of at least one dominator and the set of edges for which one endpoint is a dominator forms a connected set. The problem of finding a WCDS with minimum cardinality is NP-hard [START_REF] Dunbar | On weakly connected domination in graphs[END_REF]. Chen et al. [START_REF] Chen | Approximating minimum size weaklyconnected dominating sets for clustering mobile ad hoc networks[END_REF] extended the centralized algorithm of Guha et al. [START_REF] Guha | Approximation Algorithms for Connected Dominating Sets[END_REF]: in each step, they choose the best node to color, the component forming the WCDS being larger for each round. Other authors proposed a divide-and-conquer approach to reduce the time complexity of the algorithm [START_REF] Chen | Maintaining weakly-connected dominating sets for clustering ad hoc networks[END_REF], [START_REF] Han | Clustering wireless ad hoc networks with weakly connected dominating set[END_REF]. The concept of WCDS appears as a suitable structure for electing electrons and nuclei, however, we cannot use it as is since two nuclei cannot directly communicate in Molecular MAC to avoid deafness. Thus, Molecular MAC leads to a structure similar to a WCDS in which we remove edges between dominators.

In our previous work, we have first focused on functional MAC aspects of the molecular approach [START_REF] Nassiri | Molecular architecture for autonomic wireless mesh networks[END_REF]. In particular, we have evaluated the performance of the proposed Molecular MAC and shown its efficiency for multi-hop packet forwarding. Then, we have considered the problem of constructing a suitable mesh structure and proposed a centralized optimal algorithm through MILP (Mixed Integer Linear Programming) formulation and two heuristic protocols [START_REF] Theoleyre | Assignment of roles and channels for a multichannel mac in wireless mesh networks[END_REF]. In this paper, we extend the last work by proposing a protocol that achieves 2approximation: in any case, it guarantees at least 50% performance of the optimal centralized algorithm, the performance objective being the sum of link capacities in the constructed molecular mesh network.

IV. PROBLEM DEFINITION

We consider the problem of constructing a mesh network suitable for Molecular MAC. First, we introduce our notations:

• we model the network as an undirected graph G = (V, E)

of n vertices and m edges, where vertices V are nodes and edges E are communication links between nodes. • N (u) denotes the neighbors of u (excluding u).

• A weight function w : E → R assigns the capacity of a link to each edge in the graph.

In Molecular MAC, a link may exist between two nodes if and only if its endpoints have different roles (nucleus or electron). Maintaining network connectivity implies the existence of at least one path between each pair of nodes that alternates electrons and nuclei, i.e. the union of edges (nucleus, electron) must form a bipartite connected graph.

We aim at maximizing the sum of link capacities while ensuring network connectivity. Thus, we should assign the roles so that the resulting graph maximizes the sum of radio link capacities between nuclei and electrons. However, simultaneously assigning roles and channels in a distributed way and attaining this objective is difficult. Thus, a good trade-off between performance and complexity would be to split the problem into two parts solved sequentially:

1) first define node roles regardless of channel assignment that would minimize interference; 2) then allocate channels to nuclei in a way that minimizes interference. This allocation represents the familiar coloring problem widely studied in the literature [START_REF] Kosowski | On Greedy Graph Coloring in the Distributed Model[END_REF].

The first part corresponds to our main objective in this paper. Formally speaking, finding the optimal role assignment that maximizes the sum of edge capacities regardless of interference corresponds to finding the following subgraph that we call a Maximum Weighted Edge Connected Bipartite Subgraph (Max-WECBS) defined as follows: Input: a connected graph G = (V, E) and a weight function

w : E ⇒ R Output: a bipartite connected graph G B = (X, Y, E ′ ), where X ∪ Y = V , E ′ ⊆ E and E ′ = {(x, y)}/x ∈ X and y ∈ Y (set X contains electrons while set Y contains nuclei). Objective: maximize e∈E ′ w(e).
The considered problem also relates to that of Maximum Weighted Cut (Max-WC), the weighted version of the Maximum Cut [START_REF] Alimonti | Hardness of Approximating Problems on Cubic Graphs[END_REF] that partitions the vertices of a graph into two subsets by maximizing the number of edges with one endpoint in each subset. If we only consider instances with strictly positive weights, then Max-WC and Max-WECBS lead to the same objective: indeed one can show that the edges of any maximum weighted cut necessarily form a bipartite connected graph. However, they differ with respect to the fact that Max-WC does not have the connectivity constraint whereas it appears in Max-WECBS. Thus, approximation protocols for Max-WC do not necessarily lead to valid solutions for Max-WECBS. Note also that fully local 2-approximation protocols exist for Max-WC. On the contrary, we have mentioned in the previous work [START_REF] Theoleyre | Assignment of roles and channels for a multichannel mac in wireless mesh networks[END_REF] that there is no fully localized protocol for constructing a Max-WECBS. If we accept negative or zero weights, both problems are clearly different since the optimal solution for Max-WC does not necessarily results in a connected component.

We conclude this section with the following theorem: Theorem 1: Max-WECBS is NP-complete.

Proof: The proof follows from a reduction from the Maximum Cut problem. Recall that Maximum Cut is NPcomplete and hard to approximate [START_REF] Alimonti | Hardness of Approximating Problems on Cubic Graphs[END_REF] (the weighted version is as hard to approximate as the original problem [START_REF] Crescenzi | To Weight or Not To Weight: Where is the Question?[END_REF]).

Let a connected graph G = (V, E) be an instance of Max-Cut and let E ′ ⊆ E be the edges belonging to the optimal solution. Then G ′ = (V, E ′ ) is bipartite. Moreover, G ′ is necessary connected, otherwise E ′ will not be maximum. Thus, G ′ is also a solution of Max-WECBS on the instance given by G and by the weight function that assigns the same strictly positive weight to all edges. Reciprocal is also straightforward.

Taking into account this fact we propose an algorithm that approximates the optimal solution for constructing a Max-WECBS.

V. 2-APPROXIMATION FOR MAX-WECBS

This section describes a centralized algorithm that provides 2-approximation for Max-WECBS with the computational time of O(m), where m is the number of edges.

Given input graph G = (V, E) and weight function w, we construct partition {X, Y } of V such that bipartite subgraph

G ′ = (X, Y, E ′ = {u, v}|u ∈ X, v ∈ Y ) is connected (set X contains
electrons while set Y contains nuclei). We initially place nodes in set R = V while both X and Y are empty. The algorithm starts with choosing a random node from R and placing it in X. During the iterative part, we successively take nodes from R and place them in one of the sets X or Y according to the criteria described below. To maintain the connectivity of the resulting bipartite subgraph, the protocol only chooses nodes from R incident to at least one node from X ∪ Y .

To decide about the set (X or Y ) for placement of node u, the algorithm computes two values: w X (u) and w Y (u) corresponding to the sum of weights of the edges between u and nodes belonging to X for w X (u), respectively to Y for w Y (u). The algorithm decides to place node u in X if w X (u) < w Y (u) and in Y otherwise. Thus, at each iteration, the sum of weights of the new edges included in the solution is greater or equal to the sum of weights of forbidden edges (i.e. edges between the newly included node and nodes in the same set).

Obviously, the resulting bipartite graph will be connected. Indeed, let assume by induction that the component of nodes already in X ∪ Y is connected. The algorithm chooses a neighbor of a node already either in X or in Y (i.e. w Y (u) > 0 or w X (u) > 0). If node u is placed in X (respectively in Y ), at least one neighbor of the component is in Y (respectively in X). This edge will be kept in the bipartite graph and the new component remains connected.

At the end of the execution, the sum of weights of edges added to the solution is at least the half of the sum of weights of all the edges of the graph, which gives us a 2approximation. Algorithm 1 presents a formal specification.

VI. PROTOCOL VERSION OF THE 2-APPROXIMATION

While Algorithm 1 is efficient and finds a 2-approximation, it is centralized so we cannot directly apply it to multi-hop networks. Moreover, it does not take into account topology changes that would require an entire re-execution. In this section we present a distributed version of the algorithma protocol achieving the 2-approximation in a mesh network: nodes periodically broadcast hello packets and determine their own role from hello packets transmitted by adjacent nodes. Since in a realistic radio environment packets may be lost due to bad channel conditions, fading, or collisions, and the network topology may vary in time, we propose a selfstabilizing version so that the protocol converges to a valid solution in spite of lost packets or added as well as removed nodes.

A. Description

Centralized Algorithm 1 requires selecting a node at random at the execution startup (line 6). In the distributed version, the node with the highest id becomes the startup node called also a "leader". Then, we construct an order in the network by considering the instant at which a node decides on its role. This instant is a variable local to each node so that nodes do not require global synchronization. A node decides on its role in function of the role decided by its neighbors.

1) Election of the leader: As multi-hop networks usually use hellos for neighbor discovery, we add the fields leader-id and seq-num required by our protocol to existing hellos. A node includes in the leader-id field the highest id received so far or its id, if it is greater than the received values (in this case the node considers itself as a leader). The leader increments seq-num sent in each hello packet so that nodes can detect its departure: if seq-num does not increase for e.g. 3 hello periods, nodes consider leader-id as invalid.

2) Creation of an order: To guarantee the 2-approximation, nodes have to make consistent decisions, even if some hellos are lost. Consequently, we create an order through the decision-date variable that represents the instant at which a node decides on its role. It is only a local variable so nodes do not need to synchronize.

3) Decision on a role: When a node decides on its role, it takes into account the role of all neighbors that have already taken a role, i.e. their decision-date is lower. It chooses the role that maximizes the capacity of links with neighbors with a different role.

4) Protocol operation: The leader node chooses the role of a nucleus, sets its decision-date to 0, and broadcasts its decision in a hello packet. Each node maintains a neighborhood table called originator-table fed by received hellos. When a node receives a hello with a higher leader-id, it proceeds as follows:

1) it activates a timer set to a random value t backof f (t backof f ∈ (0..∆ decision-date ], where ∆ decision-date is a configurable parameter), 2) it sets its own decision-date to the decision--date of the sender +t backof f , 3) after t backof f , the node decides on its role: it must maximize the sum of the weights of all radio links to a neighbor with a different role and a lower decision--date, 4) it broadcasts its new role and decision-date in a hello, 5) the node stores the list of all neighbors with an earlier decision-date than itself at the instant of its decision in originator-table . It is important to note that only these nodes infuence its decision.

B. Dealing with inconsistent decisions

Nodes may make inconsistent decisions because of packet losses, if a node does not know that a neighbor has already chosen a role. However, the decision-date variable helps to detect such situations: a node detects that a neighbor is not in originator-table, but it has an earlier decision--date than itself. If decision-date values are equal, the node will compare the ids to break the tie. This means that the list of nodes that influenced its decision changed and the node needs to reconsider its role by following the rules below:

1) it adds the conflicting neighbor to originator--table, 2) it reconsiders its role by taking into account all the nodes in the updated originator-table, 3) if its role has changed, it immediately sends a hello.

Since such situations may lead to cascading changes, a node has to react to role changes. In particular, if a neighbor of a given node present in the originator-table has changed its decision, the node must reconsider its own decision in the same way by applying the rules above. Note that the rules result in a self-stabilizing behavior of the distributed protocol.

C. Illustration

Let us illustrate the execution of the protocol on a simple topology presented in Figure 2. For the sake of simplicity, we assume that all radio links have the same weight and we name each node by its decision-date. We only show how nodes decide on their roles so we assume that the node at the top is the leader.

First, the leader becomes a nucleus at the decision--date of 0 and its neighbors increment their decision--date by random t backof f . The node that has chosen the backoff of 2 decides first: it sets its decision-date to 2, chooses the inverse role of the leader since it has only one neighbor, and notifies its neighbors with a hello. The neighbor below 2 sets its decision-date to 2 and adds a random backoff of 4. Node 3 makes a similar decision to node 2. In the next step node 4 chooses to be a nucleus, because becoming an electron results in eliminating links with 2 and 3 while being a nucleus only eliminates one link to the leader. Finally, the network forms a valid Max-WECBS: the resulting molecular structure contains at least half of links and it is connected.

D. Fault tolerance

If the network topology changes due to link failures as well as leaving or arriving nodes, the protocol converges to a valid state by reacting to the changes in the same way as for inconsistent decisions. A node just has to lookup its neighbors in originator-table and re-apply its decision rule as soon as it detects any change. Besides, a topology change may require reordering decisions: it happens when a node has the local maximum decision-date, but it is not the leader. It would potentially break the connectivity of the bipartite graph. In this case, the node looks for the minimum decision-date among its neighbors and increments it by a random value (rand ∈ (0..∆ decision-date ]). Then, it just has to decide on its new role according to the same rule and updates originator-table. Using the minimum (and not the maximum) decision-date limits the number of decision changes among the neighbors: if the role does not change, only the node with the minimum decision-date has to re-consider its decision.

As mentioned previously, a node detects the departure of the leader: if the sequence number does not increase in the received hellos, a node can safely consider the leader dead. Thus, nodes will re-consider their decision after the propagation of the new maximum id.

We can observe that this protocol is self-stabilizing, but we omit here the proof due to space limitation. However, it is quite straightforward by considering all the cases that may occur and proving the protocol convergence in each situation if no topology or decision change occurs during a sufficiently long time.

VII. PERFORMANCE EVALUATION

We have simulated the 2-approximation protocol in Ws-Net [START_REF] Chelius | Wsnet: a modular event-driven wireless network simulator[END_REF] using the COIN-CBC linear programming library [START_REF]Coin-or branch and cut[END_REF]. We randomly place nodes in a circular simulation area. We assume that nodes use IEEE 802.11a network cards with the radio range of 10 units and the interference range of 30 units. We adjust the simulation area to maintain an average density of 10 neighbors.

The results correspond to statistics averaged over 10 different simulations of 240 seconds. The graphs present averaged values with 95% confidence intervals. We compared the proposed 2-approximation with a centralized optimal (OPT) solution, the Maximum Independent Set protocol (MIS), and a self-stabilizing Spanning Tree (ST) (introduced in the earlier work [START_REF] Theoleyre | Assignment of roles and channels for a multichannel mac in wireless mesh networks[END_REF]). Note that the OPT strategy results in the best solution, however it is intractable for medium to large networks (> 40 nodes).

Since all these protocols use uniquely hellos for their decisions, the reader can note that they present exactly the same overhead.

First, we present in Figure 3 the route stretch factor (i.e. the ratio between the length of the shortest route via the links (edge, nucleus) and the length of the shortest route in the original graph). We can note that the MIS and ST strategies use long routes while 2-approx achieves the lowest route stretch factor since it maximizes the number of radio links to keep in the resulting structure. The OPT strategy uses a little longer routes than the 2-approximation since it directly maximizes the global throughput: although both objectives are correlated, they are not entirely identical.

Then, we consider the minimum guaranteed throughput for an increasing number of nodes (cf. Figure 4). The OPT strategy computes this throughput as the objective of the MILP formulation (cf. [START_REF] Theoleyre | Assignment of roles and channels for a multichannel mac in wireless mesh networks[END_REF]). We directly apply here the same MILP formulation with different roles computed by different strategies and extract the objective value. We can observe that maximizing the number of radio links in the molecular structure really improves the throughput of the structure. The OPT strategy obtains the highest throughput, but it is intractable for larger networks. Our 2-approximation outperforms the ST and MIS protocols. Even if the objective of Max-WECBS is to maximize connectivity, it also results in improved throughput of the molecular structure.

VIII. CONCLUSION AND FUTURE WORK

In this paper, we have formally defined the structure associated with the Molecular MAC and proposed a centralized 2-approximation protocol that maximizes the number of radio links in the molecular structure. We then have extended this protocol so that it can operate in a distributed way still providing the same guarantee. Moreover, it is self-stabilizing thus robust to topology changes.

We are currently working on comparing the performance of this approach with other strategies that do not provide guarantees. Moreover, unlike the present work in which we assign roles first and then allocate channels, we consider assigning jointly channels and roles to develop protocols with the global guarantee of performance.

Fig. 1 .

 1 Fig. 1. Example of a molecular structure.

Algorithm 1 : 1 Initialization of the algorithm; 2 for u ∈ X do 3 w 5 randomly choose u ∈ V ; 6 R 7 X ←-{u}; 8 Y ←-∅; 9 for v ∈ N (u) do 10 w 14 R 20 X

 112356789101420 2-approximation for Max-WECBSinput : Connected graph G = (V, E), weight functionw : E ⇒ R output: Partition {X, Y } of V inducing a connected bipartite subgraph of G begin ←-V -{u}; X (u) = w({u, v}); 11 iterative part; 12 while R = ∅ do 13 choose u ∈ R such that N (u) ∩ {X ∪ Y } = ∅; ←-R -{u}; 15 if w X (u) ≥ w Y (u) then 16 Y = Y ∪ {u}; 17 for v ∈ {R ∩ N (u)} do 18 w Y (v) = w Y (v) + w({u, v}); 19 else = X ∪ {u}; 21 for v ∈ {R ∩ N (u)} do 22 w X (v) = w X (v) + w({u, v}); the computational time of Algorithm 1: it executes the initialization part (lines 2-11) in O(n) time complexity. The iterative part (lines 12-23) runs exactly n times and each part can be executed in O(δ(u)) time complexity, where u is the current selected node and δ(u) is its degree in G. Thus, the protocol achieves the running time of O( u∈V δ(u)), i.e O(m), where m is the number of edges.

Fig. 2 .

 2 Fig. 2. Execution of the protocol on a simple topology.

Fig. 3 .Fig. 4 .

 34 Fig.3. Route stretch factor for a varying number of nodes in a random mesh network.