Emmeran Le Moal

Gaëtan Juban

Anne-Sophie Sophie Bernard

Tamas Varga

Clotilde Policar

Bénédicte Chazaud

Rémi Mounier
email: remi.mounier@univ-lyon1.fr

Macrophage-derived superoxide production and antioxidant response following skeletal muscle injury

Keywords: oxidative stress, tissue repair, sterile inflammation. Highlights

INTRODUCTION

Macrophages play various important roles in tissue homeostasis [START_REF] Chazaud | Macrophages: supportive cells for tissue repair and regeneration[END_REF].

They exhibit a wide spectrum of phenotypes or inflammatory status, which are associated with a diversity of functions [START_REF] Chazaud | Macrophages: supportive cells for tissue repair and regeneration[END_REF]. Upon an immune challenge, macrophages invading the infected area exhibit a pro-inflammatory phenotype, that support microbicidal activity, mainly through the activation of Toll-like receptors (TLRs) upon recognition of Pathogen Associated Molecular Patterns (PAMPs). Once the pathogens phagocytozed and inactivated, resolution of inflammation takes place to avoid too much tissue degradation [START_REF] Chazaud | Macrophages: supportive cells for tissue repair and regeneration[END_REF]. Inflammation is often associated with an increase of the production of reactive oxygen species (ROS) (Le [START_REF] Moal | Redox Control of Skeletal Muscle Regeneration[END_REF]. It is largely accepted that macrophages produce and release ROS to support bactericidal activity and counteract with invader pathogens [START_REF] West | TLR signalling augments macrophage bactericidal activity through mitochondrial ROS[END_REF]. Thus, ROS are important mediators of macrophage functions during immune challenges [START_REF] Brune | Redox control of inflammation in macrophages[END_REF][START_REF] Forman | Redox signaling in macrophages[END_REF].

After a tissue damage that does not involve bacteria or virus (i.e. sterile inflammation), the activation of macrophages infiltrating the injured area was evidenced via the Damage Associated Molecular Patterns (DAMPs) that also activate TLRs. However, the extent and quality of the pro-inflammatory status of these proinflammatory macrophages have been poorly addressed in vivo. In this context, we recently showed that soon after skeletal muscle injury, monocyte-derived macrophages exhibit a pro-inflammatory status, which is different from that of classical M1 (IFN driven) activation and is likely related to tissue damage [START_REF] Varga | Highly Dynamic Transcriptional Signature of Distinct Macrophage Subsets during Sterile Inflammation, Resolution, and Tissue Repair[END_REF]. Few days after the injury, the resolution of inflammation takes place and macrophages switch their phenotype towards an anti-inflammatory/recovery status to exert repairing function in the tissue [START_REF] Arnold | Inflammatory monocytes recruited after skeletal muscle injury switch into antiinflammatory macrophages to support myogenesis[END_REF][START_REF] Mounier | AMPKalpha1 Regulates Macrophage Skewing at the Time of Resolution of Inflammation during Skeletal Muscle Regeneration[END_REF][START_REF] Perdiguero | p38/MKP-1-regulated AKT coordinates macrophage transitions and resolution of inflammation during tissue repair[END_REF][START_REF] Varga | Tissue LyC6-macrophages are generated in the absence of circulating LyC6monocytes and Nur77 in a model of muscle regeneration[END_REF] .

Whether pro-oxidant characteristics of macrophages that were described in the context of immune challenge does occur in the context of sterile inflammation is still uncertain. iNOS, which participates in the formation of Reactive Nitrogen Species (RNS) and ROS, is expressed by inflammatory macrophages during the first days of muscle regeneration, then its expression declines with time, notably when macrophages acquire the recovery inflammatory status [START_REF] Mounier | AMPKalpha1 Regulates Macrophage Skewing at the Time of Resolution of Inflammation during Skeletal Muscle Regeneration[END_REF]. Mice deficient for inos fail to regenerate correctly because of a defect of myogenesis [START_REF] Rigamonti | Requirement of Inducible Nitric Oxide Synthase for Skeletal Muscle Re= after Acute Damage[END_REF] suggesting, because inos is mainly expressed by inflammatory macrophages infiltrating the injured area, that macrophage-derived iNOS plays a role in skeletal muscle regeneration. However, despite the assumption that inflammatory macrophages generally produce ROS that are detrimental for the tissue, no study has characterized macrophage ROS production in the context of skeletal muscle regeneration or tissue repair.

The aim of the present study was to evaluate the production of ROS by macrophage populations during skeletal muscle regeneration. To address this question, we combined in vitro and ex vivo approaches of direct biochemical measurements of ROS entities, ex vivo analysis of the "oxidative" gene profile of macrophages isolated from regenerating muscle, and in vivo pharmacological antioxidant supplementation during skeletal muscle regeneration.

MATERIALS AND METHODS

Macrophage culture

Macrophages were obtained from bone marrow (BM) precursor cells as previously described [START_REF] Mounier | AMPKalpha1 Regulates Macrophage Skewing at the Time of Resolution of Inflammation during Skeletal Muscle Regeneration[END_REF]. Briefly, total BM was cultured in DMEM (Gibco) containing 20% fetal bovine serum (FBS) and 30% conditioned medium of L929 cell line. Macrophages were seeded (200 000 cells/cm 2 ) and activated in DMEM medium containing 10% FBS for 3 days as follows: IFN- (50 ng/ml), IL-4 (10 ng/ml), IL-10 (10 ng/ml), Muscle Damage Associated Molecular Patterns (DAMPS) (homogenate from a muscle injured by ischemia/reperfusion, 1 g/ml), lipopolysaccharide (LPS, 10 ng/ml) and Phorbol 12-myristate 13 acetate (PMA, 50 ng/ml). After several washes, conditioned medium was obtained after 24h of culture of activated macrophages in DMEM serum-free medium.

Superoxide ion production assessment

Superoxide ion production assessment procedure was previously fully detailed [START_REF] Bernard | Evaluation of the anti-oxidant properties of a SOD-mimic Mn-complex in activated macrophages[END_REF]. Ferricytochrome c reduction was used to assess the superoxide ion production by macrophages. Macrophages activated as described above were incubated at 37°C for 2 h with 1 ml of HEPES buffer (53 mM HEPES) pH 7.4 containing 14.6 mM glucose, 358 mM NaCl, 12.7 mM KCl, 3.1 mM KH 2 PO 4 , 6.1 mM MgSO 4 , 3.1 mM CaCl 2 , 13 mM NaHCO 3 , 0.4 mM EDTA and 100 µM ferricytochrome. Absorbance was read at 550 nm with a spectrophotometer (BioSpectrometer Basic Eppendorf). Contribution of ferricytochrome c at 100 μM (absorbance at 800 nm) was substracted. Similar measurements were realized on macrophages isolated by FACs (see below) and subsequently plated for 2 h in the absence or presence of PMA (50 ng/ml) to induce an oxidative burst [START_REF] Abbas | Detection of superoxide production in stimulated and unstimulated living cells using new cyclic nitrone spin traps[END_REF].

Superoxide dismutase expression

Protein concentration was determined in macrophage lysates using BCA protein assay (Uptima-Interchim). Western blot analysis was performed to examine the expression of MnSOD (rabbit polyclonal anti-human SOD2 200 µg/ml, Santa Cruz Biotechnology) and Cu-ZnSOD (rabbit polyclonal anti-human SOD1 200 µg/ml, Santa Cruz Biotechnology) versus that of actin (Goat polyclonal anti-human actin 200 µg/ml, Santa Cruz Biotechnology). Quantification of the band intensity was performed using ImageJ Software.

Macrophage H 2 O 2 release

Extracellular H 2 O 2 was detected using the Amplex Red hydrogen peroxide/peroxidase assay kit (Invitrogen). Macrophage-conditioned medium was collected and incubated with the Amplex Red reagent (50 μl of 10 mM solution) and horseradish peroxidase (100 μl of 10 U/ml solution) in 0.25 M sodium phosphate buffer for 30 min. Absorbance intensity was measured on an Apollo LB 911 plate reader (Berthold Technologies) at 560 nm.

Skeletal muscle injury

Eight week-old C57/Bl6 and CX3CR1 GFP/+ male mice were used for N-acetylcysteine (NAC, Sigma) supplementation protocol and extraction of macrophages from regenerating muscle, respectively. Mice were bred and used in compliance with French and European regulations. Principal investigators are licensed for animal experimentation and the ethical committee validated the protocol. Muscle injury was caused by injection of cardiotoxin (CTX) in the Tibialis Anterior muscle as previously described [START_REF] Mounier | AMPKalpha1 Regulates Macrophage Skewing at the Time of Resolution of Inflammation during Skeletal Muscle Regeneration[END_REF]. In some experiments, LPS (50 μg/ml final) was coinjected with CTX as previously described [START_REF] Varga | Highly Dynamic Transcriptional Signature of Distinct Macrophage Subsets during Sterile Inflammation, Resolution, and Tissue Repair[END_REF]. In some experiments, NAC was administered at 4% in drinking water for 8 days from the day of injury, as previously described [START_REF] Terrill | N-Acetylcysteine treatment of dystrophic mdx mice results in protein thiol modifications and inhibition of exercise induced myofibre necrosis[END_REF].

Isolation of immune cells from regenerating muscle

Immune cells were isolated as described previously [START_REF] Mounier | AMPKalpha1 Regulates Macrophage Skewing at the Time of Resolution of Inflammation during Skeletal Muscle Regeneration[END_REF][START_REF] Varga | Highly Dynamic Transcriptional Signature of Distinct Macrophage Subsets during Sterile Inflammation, Resolution, and Tissue Repair[END_REF]. Briefly, muscles were minced and digested with collagenase B 0.2%. CD45 + cells were isolated by magnetic sorting procedure (Miltenyi Biotec) and further labeled with APC-CD64 and PE-Ly6C antibodies for C57/Bl6 mice and with an APC-Ly6C (all from eBiosciences) antibody for CX3CR1 GFP/+ mice. Cells were sorted using a FACS Aria II cell sorter (BD Biosciences).

Histological analysis

At different time-points after injury (day 4, 8 and 28) regenerating muscles were harvested, snap-frozen in nitrogen-chilled isopentane and kept at -80°C until analysis. 10 m-thick muscle cryosections were prepared and used for H/E staining and immunofluorescent staining. Cryosections were treated with antibodies directed against F4/80 (#ab6640, Abcam), iNOS (#ab15323, Abcam), and CD206 (#sc-58987, Santa Cruz) revealed with FITC-or Cy3-conjugated antibodies (Jackson Immunoresearch Inc). Pictures were recorded with Axioscop microscope (Zeiss) at 20X of magnification connected to AxioCam ICc5 (Zeiss). For each condition in each experiment, at least 8-10 randomly chosen fields were counted. The number of labeled macrophages was calculated using ImageJ software and was expressed as a percentage of total macrophages. For Cross Section Area (CSA) analysis, cryosections were labeled with antibodies directed against Laminin (#L9393, Sigma-Aldrich) and revealed with FITC-conjugated secondary antibody. Pictures covering the whole cryosection were recorded with an Axio Observer.Z1 (Zeiss) at 10X of magnification connected to a CoolSNAP HQ2 CCD Camera (Photometrics). CSA was determined manually using ImageJ software.

A mean of 2345 +/-542 fibers were analyzed per TA muscle.

For satellite cells analysis, fresh cryosections were labeled with antibodies directed against Pax7 (clone 4ea, DHSB) and Laminin (#L9393, Sigma-Aldrich) revealed with Cy3-and FITC-conjugated antibodies, respectively. Pictures were recorded with an Axio Imager.Z1 (Zeiss) at 20X of magnification connected to a CoolSNAP Myo CCD Camera (Photometrics). For each condition, 10 randomly chosen fields were counted. The number of satellite cells was determined manually using ImageJ software and expressed as the number of Pax7+ cells per 100 fibers.

Statistical analyses

Results are expressed as means ± SEM. All experiments were carried out with at least 3 different primary cultures or 3 mice for in vivo analyses. Isolation of macrophages from regenerating muscle required 3 to 5 mice that were pooled.

Mann-Whitney and Kruskal-Wallis tests were done with Prism software.

RESULTS

Pro-and anti-inflammatory macrophages release similar levels of ROS in vitro

To investigate ROS production by differentially activated macrophages, BM-derived macrophages were treated with a (IFN-), with IL-4 or with IL-10 to trigger proinflammatory, alternative and anti-inflammatory activation, respectively [START_REF] Murray | Macrophage activation and polarization: nomenclature and experimental guidelines[END_REF]. Superoxide production in conditioned medium was assessed using ferricytochrome c reduction assay [START_REF] Bernard | Evaluation of the anti-oxidant properties of a SOD-mimic Mn-complex in activated macrophages[END_REF]. This method, based on the reduction of cytochrome c by superoxide to form detectable ferrocytochrome, reflects the extracellular production of superoxide. No significative superoxide production was observed upon the 3 treatments, despite a slight increase by macrophages activated with IFN (Figure 1A). To investigate ROS production by pro-inflammatory macrophages in vitro, cells were treated with various effectors known to induce proinflammatory states including a ROS-inducer (PMA), a bacterial mimetic (LPS) and a lysate from injured skeletal muscle (DAMPS). Because the measurement of anion superoxide release has never been performed in these conditions, we added two positive controls known as superoxide generators: a strong chemical inducer (PMA) [START_REF] Nathan | Hydrogen peroxide release from mouse peritoneal macrophages: dependence on sequential activation and triggering[END_REF] and LPS. LPS activates superoxide production and hydrogen peroxide-derived superoxide. We also tested DAMPS as a putative superoxide activator. Hence, we validated our measurement setup and were able to underline the differences between classical models of activation based on cytokine treatment and the above-mentioned treatments. Among the 3 treatments, only PMA (+139±9%) (p<0.05) treatment induced superoxide detection as compared with untreated macrophages (NT) (Figure 1A). To further explore ROS metabolism in activated macrophages, the level of H 2 O 2 , which is the product of superoxide dismutation by superoxide dismutases (SODs) [START_REF] Holmstrom | Cellular mechanisms and physiological consequences of redox-dependent signalling[END_REF], was measured in conditioned media. Surprisingly, hydrogen peroxide production was similar in all macrophage populations (Figure 1B). The important lability of H 2 O 2 can explain these results. Overall, these data indicate that macrophages released similar levels of superoxide and H 2 O 2 irrespective of their activation state.

We further examined the expression of superoxide dismutase (SOD) enzymes by activated macrophages. MnSOD (or SOD2) is a mitochondrial enzyme responsible of the dismutation of primary formed superoxide into hydrogen peroxide, the most common reaction of ROS metabolism cascade [START_REF] Holmstrom | Cellular mechanisms and physiological consequences of redox-dependent signalling[END_REF].

MnSOD expression was higher in IFN macrophages as compared with (IL-10) macrophages (p<0.05) (Figure 1C). Additionally, cytosolic SOD1 (or CuZn-SOD) expression was not significantly different in the various activated macrophages (Figure 1D). These data indicate that mitochondrial SOD2 was overexpressed only in pro-inflammatory macrophages, leading to an improvement of the dismutation capacity of pro-inflammatory macrophages that may result in a similar ROS release as compared with other activated macrophages.

Macrophages produce low levels of superoxide during post-injury skeletal muscle regeneration in vivo

To investigate ROS production by macrophages during muscle regeneration, macrophages subsets and neutrophils were sorted from regenerating muscle at day 2 after injury and assayed for the evaluation of superoxide production (Figure 2A). At day 2, both LyC6 pos and Ly6C neg populations are equally present [START_REF] Mounier | AMPKalpha1 Regulates Macrophage Skewing at the Time of Resolution of Inflammation during Skeletal Muscle Regeneration[END_REF][START_REF] Varga | Tissue LyC6-macrophages are generated in the absence of circulating LyC6monocytes and Nur77 in a model of muscle regeneration[END_REF]. No significant superoxide production was detected in both

Gene expression profiling does not reveal antioxidant-related response in macrophages after skeletal muscle injury

The oxidative stress signature of macrophages present in regenerating muscles was investigated, using a microarray analysis published in [START_REF] Varga | Highly Dynamic Transcriptional Signature of Distinct Macrophage Subsets during Sterile Inflammation, Resolution, and Tissue Repair[END_REF]. As expected, Ly6C pos macrophages expressed higher levels of a series of genes known to be associated with pro-inflammatory macrophages in skeletal muscle regeneration, including selectin L, CSF1, IL6 and CCR1 (Figure 3, upper panel). A set of antioxidant genes that would identify an antioxidant-related transcriptional signature was analyzed in both macrophage subsets. There was no significant differential expression of 30 genes associated with response to oxidative stress between inflammatory Ly6C pos and restorative Ly6C neg macrophage subsets (Figure 3, lower panel).

NAC treatment impairs skeletal muscle regeneration after injury

To further assess a potential functional role of ROS in macrophages during skeletal muscle regeneration, we used a pharmacological approach using antioxidant NAC oral treatment, given from the time of muscle injury. Indeed, NAC should counteract the ROS released at early time points by pro-inflammatory macrophages. Inversely, if no ROS are present in the injured muscle, NAC should not affect skeletal muscle regeneration.

First, macrophage inflammatory status was investigated by performing double immunofluorescent staining for F4/80 and iNOS or CD206, to identify pro-and antiinflammatory macrophages, respectively [START_REF] Mounier | AMPKalpha1 Regulates Macrophage Skewing at the Time of Resolution of Inflammation during Skeletal Muscle Regeneration[END_REF]. At day 4 after injury, there was no difference in the percentage of macrophages expressing iNOS or CD206 between control and NAC-treated muscles (Figure 4A). The number of necrotic and phagocyted myofibers was calculated 8 days after injury, a time point when most of necrosis and phagocytosis of injured myofibers is over [START_REF] Mounier | AMPKalpha1 Regulates Macrophage Skewing at the Time of Resolution of Inflammation during Skeletal Muscle Regeneration[END_REF]. Although exhibiting high variations between individuals, NAC-treated group did not significantly differ from the control group (Figure 4B), indicating no significant difference in the time course of the regeneration process. Four weeks after injury, NAC treatment was associated with a reduction of body weight (-9%, p<0.01), Tibialis Anterior muscle mass (-15%, p<0.001) and Tibialis Anterior mass/body mass ratio (-8%, p<0.05) (Figure 5B). This was correlated with a decrease in mean Cross Section Area (-15%, p<0.05) and a shift of the myofiber size towards the smaller ones (Figure 5C). Additionally, muscles from NAC-treated group displayed a significant lower number of nuclei per fiber as compared with the control group (-22%, p<0.001) (Figure 5D). Finally, NAC-treated muscles did not show a significant difference in the number of satellite cells (Figure 5E). Altogether, these results suggest that NAC treatment primarily affected myofiber growth than early events related to inflammation during skeletal muscle regeneration.

DISCUSSION

We investigated ROS release from macrophages during regenerating skeletal muscle and provided evidence that after a sterile injury, infiltrated macrophages did not release ROS and that antioxidant supplementation impaired long term skeletal muscle regeneration. To our knowledge, only one study investigated whole skeletal muscle redox changes following skeletal muscle injury induced by venom injection. It was described that skeletal muscle injury is accompanied by an increase of ROS production within the whole tissue from 1 to 7 days after injury [START_REF] Vezzoli | High-mobility group box 1 release and redox regulation accompany regeneration and remodeling of skeletal muscle[END_REF].

However, the pro/antioxidant-related measurements were not specific to macrophage populations. These results, combined to those of the present study, suggest that ROS present in the early injured muscle are mainly derived from the myofiber than from the inflammatory compartment.

ROS are short lived and direct detection is difficult, notably in vivo, despite recent technological advances [START_REF] Murphy | Unraveling the biological roles of reactive oxygen species[END_REF][START_REF] Winterbourn | Reconciling the chemistry and biology of reactive oxygen species[END_REF]. Here, we used biochemical assays such as ferricytochrome c reduction and Amplex Red technology, which are robust, specific and adequate methods to detect the major ROS that are superoxide and hydrogen peroxide produced by the cells and released in their microenvironment [START_REF] Bernard | Evaluation of the anti-oxidant properties of a SOD-mimic Mn-complex in activated macrophages[END_REF][START_REF] Wang | Nanoelectrodes for determination of reactive oxygen and nitrogen species inside murine macrophages[END_REF].

We report that among a set of in vitro activation, only macrophages treated with PMA, which induces the oxidative burst in macrophages [START_REF] Abbas | Detection of superoxide production in stimulated and unstimulated living cells using new cyclic nitrone spin traps[END_REF], present a significant ROS production. Additionally, no difference was observed in the release of H 2 O 2 , which is the product of superoxide dismutation, between the different activated macrophage populations. Protein expression of MnSOD, responsible of the dismutation of superoxide into hydrogen peroxide, the most common reaction of ROS metabolism cascade at this step [START_REF] Holmstrom | Cellular mechanisms and physiological consequences of redox-dependent signalling[END_REF], is higher in IFN treated macrophages as compared with other activated macrophages. This distinct pattern of MnSOD is a hallmark of IFN-activated macrophages [START_REF] Lawrence | Transcriptional regulation of macrophage polarization: enabling diversity with identity[END_REF]. This result indicates that a higher MnSOD protein expression may counteract a higher production of superoxide (not detectable in our experimental conditions) as a stress response to restore cellular homeostasis.

Ex vivo isolation of pure populations of pro-inflammatory and anti-inflammatory macrophages (Ly6C pos and Ly6C neg , respectively) two days after muscle injury revealed no significant superoxide release by these cells. Technically, sorted cells, here neutrophils, were shown to secrete superoxide after PMA treatment. Similar results were observed after muscle injury upon LPS co-injection, that induces a stronger pro-inflammatory profile in Ly6C pos macrophages [START_REF] Varga | Highly Dynamic Transcriptional Signature of Distinct Macrophage Subsets during Sterile Inflammation, Resolution, and Tissue Repair[END_REF]. ROS production is an important feature of macrophages, notably following activation with pathogens [START_REF] Fang | Antimicrobial reactive oxygen and nitrogen species: concepts and controversies[END_REF][START_REF] Laskin | Macrophages and tissue injury: agents of defense or destruction?[END_REF][START_REF] West | TLR signalling augments macrophage bactericidal activity through mitochondrial ROS[END_REF], that constitutes a different context from sterile tissue injury. The absence of a high oxidative response in macrophages upon muscle injury, notably in Ly6C pos macrophages at early time points after injury, was confirmed by gene expression analysis. The mRNA expression of about 30 genes related to oxidative stress did not differ between inflammatory Ly6C pos and restorative/Ly6C neg macrophage from injured muscle. More broadly, gene profiling analysis of these two populations indicated that following a sterile muscle injury, inflammatory Ly6C pos macrophages exhibit a specific inflammatory profile which is different from the classical M1 (IFN driven) status and which is strongly different from that of pathogen-challenged macrophages [START_REF] Varga | Highly Dynamic Transcriptional Signature of Distinct Macrophage Subsets during Sterile Inflammation, Resolution, and Tissue Repair[END_REF].

Finally, our in vivo pharmacological approach aimed at targeting ROS production early after injury, with the antioxidant NAC, a molecule commonly used as therapeutics in a large number of pathophysiological contexts including skeletal muscle diseases and in exercise and sport performance context (Le [START_REF] Moal | Redox Control of Skeletal Muscle Regeneration[END_REF][START_REF] Michailidis | Thiol-based antioxidant supplementation alters human skeletal muscle signaling and attenuates its inflammatory response and recovery after intense eccentric exercise[END_REF]. We showed that NAC treatment did not alter macrophage phenotype at early time points after injury, in accordance with the above results indicating the absence or ROS release from macrophages in this context.

While the number of necrotic myofibers was not significantly different from that observed in the control muscle, NAC-treated muscles showed a highly heteregenous, somewhat increased level of necrosis 8 days after the injury, deserving further investigation on the phagocytic capacities of macrophages in these conditions.

Finally, parameters of myofiber growth and fusion were altered in NAC-treated muscle 28 days after injury, suggesting and impact of anti-oxidant treatment on ROSunrelated properties of restorative macrophages or directly on the myogenesis process. However, the number of satellite cells was unchanged after NAC treatment, suggestive of an impact of NAC on the late stage of myogenesis, that deserves further exploration. Indeed, several pathways involved in myogenesis processes may be regulated by oxidative stress, for instance IGF-I signaling or Akt-mTOR pathways (Le [START_REF] Moal | Redox Control of Skeletal Muscle Regeneration[END_REF]. This is also suggested by a study showing that NAC supplementation after an exercise-induced muscle damage (EIMD) in human blunts the circulating inflammatory response (assessed by indirect measurements in blood) and decreases the EIMD-induced increase in the phosphorylation of a series or proteins involved in myofiber growth (PKB, mTOR, p70S6K, S6, MAPK P38).

Consequently, performance of the NAC-treated group was decreased [START_REF] Michailidis | Thiol-based antioxidant supplementation alters human skeletal muscle signaling and attenuates its inflammatory response and recovery after intense eccentric exercise[END_REF].

Our results appear to be in sharp contrast with what is observed during muscular dystrophies. Indeed, the increase in oxidative stress in muscular dystrophies is a well-known process and correlates with the severity of the pathology (Le [START_REF] Moal | Redox Control of Skeletal Muscle Regeneration[END_REF]. Accordingly, anti-oxidant treatments ameliorate the morphological, biochemical, and functional phenotype of the diseased muscle, including a decrease in inflammatory cells (Le [START_REF] Moal | Redox Control of Skeletal Muscle Regeneration[END_REF]. However, the redox status of inflammatory cells, including macrophages, has not been addressed in this pathological context yet. Nevertheless, as the inflammatory status of macrophages is different during skeletal muscle regeneration and during dystrophies [START_REF] Lemos | Nilotinib reduces muscle fibrosis in chronic muscle injury by promoting TNF-mediated apoptosis of fibro/adipogenic progenitors[END_REF][START_REF] Vidal | Fibrinogen drives dystrophic muscle fibrosis via a TGFbeta/alternative macrophage activation pathway[END_REF], it is likely that their redox status will also differ significantly between acute and chronic injuries.

In conclusion, this study indicates that, contrary to an immune challenge implicating bacteria or virus, macrophage-derived ROS release is absent or weak after a sterile tissue damage, and does not impact the skeletal muscle regeneration process.

Additionally, we report that antioxidant supplementation from the time of injury does not impact macrophage inflammatory status but may conversely be deleterious in the later restorative phase of skeletal muscle regeneration. Two days after injury, Ly6C pos and Ly6C neg macrophages were sorted from regenerating muscle and processed for microarray as in [START_REF] Varga | Highly Dynamic Transcriptional Signature of Distinct Macrophage Subsets during Sterile Inflammation, Resolution, and Tissue Repair[END_REF]. Upper panel: heatmap of genes referred as highly expressed in Ly6C pos macrophage subset. Lower panel: heatmap of 30 genes associated with oxidative stress. Results represent the mean of 3 independent experiments.

  Ly6C pos and Ly6C neg macrophage subsets and neutrophils, as compared with the control solution (Figure 2A, right panel). To test the experimental conditions and the ability of myeloid cells isolated from regenerating muscles to produce superoxide, sorted macrophages and neutrophils were immediately treated with PMA, which induces an oxidative burst and is widely used to study ROS production in myeloid cells. In this condition, PMA-stimulated neutrophils significantly produced ROS, with an upregulation of superoxide release of 162.6±9.4% (p<0.001) (Figure 2A, right panel). Additionally, a skeletal muscle injury model with concomitant bacterial contamination with LPS was used to induce a stronger pro-inflammatory profile of Ly6C pos macrophages (Varga et al., 2016) (Figure 2B, left panel). No significant superoxide production was detected in both macrophage subsets and neutrophils, as compared with the control solution (Figure 2B, right panel). Again, treatment with PMA stimulated only neutrophils to produce superoxide (144.1±13.2%, p<0.05) (Figure 2B, right panel). Taken together, these results indicate that after an injury, macrophages invading skeletal muscle release, if any, very low levels of superoxide.

  FIGURES LEGENDS

Figure 2 .

 2 Figure 2. Superoxide detection in conditioned medium of macrophages sorted

Figure 3 .

 3 Figure 3. Gene expression profiling of oxidative stress-associated network in

Figure 4 :

 4 Figure 4: NAC supplementation effect on early steps of muscle regeneration:

Figure 5 .

 5 Figure 5. NAC supplementation effect on late steps of muscle regeneration:

ACKNOWLEDGEMENTS

This work was funded by CNRS, INSERM and University of Lyon.

DISCLOSURE STATEMENT

Authors declare no conflict of interest.


Achour Mostéfaoui
email: achour.mostefaoui@univ-nantes.fr

Michel Raynal
email: michel.raynal@irisa.fr

Matthieu Roy May

M Roy
email: roy@laas.fr

Time-Efficient Read/Write Register in Crash-prone Asynchronous Message-Passing Systems

Keywords: Asynchronous message-passing system, Atomic read/write register, Concurrency, Fast operation, Process crash failure, Synchronous behavior, Time-efficient operation

The atomic register is one of the most basic and useful object of computing science, and its simple read-write semantics is appealing when programming distributed systems. Hence, its implementation on top of crash-prone asynchronous message-passing systems has received a lot of attention. It was shown that having a strict minority of processes that may crash is a necessary and sufficient requirement to build an atomic register on top of a crash-prone asynchronous message-passing system.

This paper visits the notion of a fast implementation of an atomic register, and presents a new time-efficient asynchronous algorithm that reduces latency in many cases: a write operation always costs a round-trip delay, while a read operation costs a round-trip delay in favorable circumstances (intuitively, when it is not concurrent with a write). When designing this algorithm, the design spirit was to be as close as possible to the original algorithm proposed by Attiya, Bar-Noy, and Dolev.

object, usually called a register, provides its users (processes) with a write operation which defines the new value of the register, and a read operation which returns the value of the register. When considering sequential computing, registers are universal in the sense that they allow us to solve any problem that can be solved [START_REF] Turing | On computable numbers with an application to the Entscheidungsproblem[END_REF]. Although named differently in recent architectures such as multi-core systems, the read/write semantics of a register is the most clean and easy to understand abstraction of shared memory and is extensively used in multi-threaded programs.

Register in message-passing systems. In a message-passing system, the computing entities communicate only by sending and receiving messages transmitted through a communication network. Hence, in such a system, a register is not a communication object given for free, but constitutes a communication abstraction which must be built with the help of the communication network and the local memories of the processes.

Several types of registers can be defined according to which processes are allowed to read or write it, and the quality (semantics) of the value returned by each read operation. We consider here registers which are single-writer multi-reader (SWMR) and atomic. Atomicity means that (a) each read or write operation appears as if it had been executed instantaneously at a single point of the time line, between is start event and its end event, (b) no two operations appear at the same point of the time line, and (c) a read returns the value written by the closest preceding write operation (or the initial value of the register if there is no preceding write) [START_REF] Lamport | On interprocess communication, Part I: basic formalism[END_REF]. Algorithms building multi-writer multi-reader (MWMR) atomic registers from single-writer single-reader (SWSR) registers with a weaker semantics (safe or regular registers) are described in several textbooks (e.g., [START_REF] Attiya | Distributed computing: fundamentals, simulations and advanced topics[END_REF][START_REF] Lynch | Distributed algorithms[END_REF][START_REF] Raynal | Concurrent programming: algorithms, principles and foundations[END_REF]).

Many distributed algorithms have been proposed to build a register on top of a message-passing system, be it failure-free or failure-prone. In the failure-prone case, the addressed failure models are the process crash failure model, or the Byzantine process failure model (see, the textbooks [START_REF] Attiya | Distributed computing: fundamentals, simulations and advanced topics[END_REF][START_REF] Lynch | Distributed algorithms[END_REF][START_REF] Raynal | Distributed algorithms for message-passing systems[END_REF][START_REF] Raynal | Fault-tolerant message-passing distributed systems: an algorithmic approach[END_REF]). The most famous of these algorithms was proposed by H. Attiya, A. Bar-Noy, and D. Dolev in [START_REF] Attiya | Sharing memory robustly in message passing systems[END_REF]. This algorithm, which is usually called ABD according to the names its authors, considers an n-process asynchronous system in which up to t < n/2 processes may crash (it is also shown in [START_REF] Attiya | Sharing memory robustly in message passing systems[END_REF] that t < n/2 is an upper bound of the number of process crashes which can be tolerated). This simple and elegant algorithm, relies on (a) quorums [START_REF] Vukolic | Quorum systems, with applications to storage and consensus[END_REF], and (b) a simple broadcast/reply communication pattern. ABD uses this pattern once in a write operation, and twice in a read operation implementing an SWMR register.

Fast operation. To our knowledge, the notion of a fast implementation of an atomic register operation, in failure-prone asynchronous message-passing systems, was introduced in [START_REF] Dutta | How fast can a distributed atomic read be?[END_REF] for process crash failures, and in [START_REF] Dutta | Fast access to distributed atomic memory[END_REF] for Byzantine process failures. These papers consider a three-component model, namely there are three different types of processes: a set of writers W , a set of readers R, and a set of servers S which implements the register. Moreover, a client (a writer or a reader) can communicate only with the servers, and the servers do not communicate among themselves.

In these papers, fast means that a read or write operation must entail exactly one communication round-trip delay between a client (the writer or a reader) and the servers. When considering the process crash failure model (the one we are interested in in this paper), it is shown in [START_REF] Dutta | How fast can a distributed atomic read be?[END_REF] that, when

(|W | = 1) ∧ (t ≥ 1) ∧ (|R| ≥ 2), the condition (|R| < |S| t -2
) is necessary and sufficient to have fast read and write operations (as defined above), which implement an atomic register. It is also shown in [START_REF] Dutta | How fast can a distributed atomic read be?[END_REF] that there is no fast implementation of an MWMR atomic register if

(|W | ≥ 2) ∧ (|R| ≥ 2) ∧ (t ≥ 1) .
Algorithms suited to this three-tier model (writers, readers, and servers) providing very fast read operations are described in [START_REF] Hadjistasi | Oh-RAM! One and a half round atomic memory[END_REF].

Another notion of efficiency is related to the size of the control information carried by messages. This issue is mainly addressed in [START_REF] Mostéfaoui | Two-bit messages are sufficient to implement atomic read/write registers in crash-prone systems[END_REF], where it is shown that two bits are sufficient.

Content of the paper: bounded delay assumption and time-efficiency. The work described in [START_REF] Dutta | How fast can a distributed atomic read be?[END_REF][START_REF] Dutta | Fast access to distributed atomic memory[END_REF] is mainly on the limits of the three-component model (writers, readers, and servers constitute three independent sets of processes) in the presence of process crash failures, or Byzantine process failures. These limits are captured by predicates involving the set of writers (W ), the set of readers (R), the set of servers (S), and the maximal number of servers that can be faulty (t). Both the underlying model used in this paper and its aim are different from this previous work.

While keeping the spirit (basic principles and simplicity) of ABD, our aim is to design a time-efficient implementation of an atomic register in the classical model used in many articles and textbooks (see, e.g., [START_REF] Attiya | Sharing memory robustly in message passing systems[END_REF][START_REF] Attiya | Distributed computing: fundamentals, simulations and advanced topics[END_REF][START_REF] Lynch | Distributed algorithms[END_REF][START_REF] Raynal | Concurrent programming: algorithms, principles and foundations[END_REF]). This model, where any process can communicate with any process, can be seen as a peer-to-peer model in which each process is both a client (it can invoke operations) and a server (it manages a local copy of the register that is built). 1Adopting the usual distributed computing assumption that (a) local processing times are negligible and assumed consequently to have zero duration, and (b) only communication takes time, this paper focuses on the communication time needed to complete a read or write operation. For this reason the term time-efficiency is defined here in terms on message transfer delays, namely, the cost of a read or write operation is measured by the number of "consecutive" message transfer delays they require to terminate. Let us notice that this includes transfer delays due to causally related messages (for example round trip delays generated by request/acknowledgment messages), but also (as we will see in the proposed algorithm) message transfer delays which occur sequentially without being necessarily causally related. Let us notice that this notion of a time-efficient operation does not involve the model parameter t.

In order to give a precise meaning to the notion of a "time-efficient implementation" of a register operation, this paper considers the duration of read and write operations based on a specific additional synchrony assumption, usually named "bounded delay" assumption. This assumptions and the associated time-efficiency of the proposed algorithm are the following.

Let us consider the case where every message takes at most ∆ time units to be transmitted from its sender to any of its receivers. In such a context, the algorithm presented in the paper has the following time-efficiency properties:

-A write operation takes at most 2∆ time units.

-A read operation which is write-latency-free takes at most 2∆ time units. (The notion of write-latency-freedom is defined in Section 3. Intuitively, it captures the fact that the behavior of the read does not depend on a concurrent or faulty write operation, which is the usual case in read-dominated applications.) Otherwise, it takes at most 3∆ time units, except in the case where the read operation is concurrent with a write operation and the writer crashes during this write, where it can take up to 4∆ time units. (Let us remark that a process can experience at most once the 4∆ read operation scenario.)

Hence, while it remains correct in the presence of any asynchronous message pattern (e.g., when each message takes one more time unit than any previous message), the proposed algorithm is particularly time-efficient when "good" scenarios occur. Those are the ones defined by the previous synchrony patterns where the duration of a read or a write operation is a single round-trip delay. Moreover, in the other synchrony scenarios, where a read operation is concurrent with a write, the maximal duration of the read operation is precisely quantified. A concurrent write adds uncertainty whose resolution by a read operation requires one more message transfer delay (and two if the concurrent write crashes).

Roadmap. The paper consists of 7 sections. Section 2 presents the system model. Section 3 defines the atomic register abstraction, and the notion of a time-efficient implementation. Then, Section 4 presents an asynchronous algorithm providing an implementation of an atomic register with time-efficient operations, as previously defined. Section 5 proves its properties. Section 6 provides insights on the practical impact of the latency-efficiency in real systems. Finally, Section 7 concludes the paper.

System Model

Processes. The computing model is composed of a set of n sequential processes denoted p 1 , ..., p n . Each process is asynchronous which means that it proceeds at its own speed, which can be arbitrary and remains always unknown to the other processes.

A process may halt prematurely (crash failure), but executes correctly its local algorithm until it possibly crashes. The model parameter t denotes the maximal number of processes that may crash in a run. A process that crashes in a run is said to be faulty. Otherwise, it is correct or non-faulty.

Communication. The processes cooperate by sending and receiving messages through bi-directional channels. The communication network is a complete network, which means that any process p i can directly send a message to any process p j (including itself). Each channel is reliable (no loss, corruption, nor creation of messages), not necessarily first-in/first-out, and asynchronous (while the transit time of each message is finite, there is no upper bound on message transit times).

A process p i invokes the operation "send TAG(m) to p j " to send p j the message tagged TAG and carrying the value m. It receives a message tagged TAG by invoking the operation "receive TAG()". The macro-operation "broadcast TAG(m)" is a shortcut for "for each j ∈ {1, . . . , n} send TAG(m) to p j end for". (The sending order is arbitrary, which means that, if the sender crashes while executing this statement, an arbitrary -possibly empty-subset of processes will receive the message.)

Let us notice that, due to process and message asynchrony, no process can know if an other process crashed or is only very slow.

Notations. In the following, the previous computation model, restricted to the case where

t < n/2, is denoted CAMP n,t [t < n/2] (Crash Asynchronous Message- Passing).
It is important to notice that, in this model, all processes are a priori "equal". As we will see, this allows each process to be at the same time a "client" and a "server". In this sense, and as noticed in the Introduction, this model is the "fully connected peer-to-peer" model (whose structure is different from other computing models such as the client/server model, where processes are partitioned into clients and servers, playing different roles).

3 Atomic Register and Time-efficient Implementation

Atomic read/write register

Read/write register. A concurrent object is an object that can be accessed by several processes (possibly simultaneously). An SWMR register (say REG) is a concurrent object which provides exactly one process (called the writer) with an operation denoted REG.write(), and all processes with an operation denoted REG.read(). When the writer invokes REG.write(v) it defines v as being the new value of REG.

Effective operation. The notion of an effective read or write operation is from [START_REF] Raynal | Fault-tolerant message-passing distributed systems: an algorithmic approach[END_REF]. When a process crashes while executing an operation, this operation may take effect or not. The notion of an effective operation captures the fact that, due to process crashes, some operations "participate" in the computation, while others do not. More precisely, we have the following [START_REF] Raynal | Fault-tolerant message-passing distributed systems: an algorithmic approach[END_REF].

-A read operation is effective if the invoking process does not crash during its execution. -A write operation is effective if the invoking process does not crash during its execution, or, despite the fact that it crashes during its execution, the value it writes is returned by an effective read operation.

Atomic read/write register. An SWMR atomic register (we also say the register is linearizable [START_REF] Herlihy | Linearizability: a correctness condition for concurrent objects[END_REF]) is defined by the following set of properties [START_REF] Lamport | On interprocess communication, Part I: basic formalism[END_REF].

-Liveness. The invocation of an effective operation by a correct process terminates.

-Consistency (safety). All the effective operations appear as if they have been executed sequentially and this sequence S of operations is such that:

each effective read returns the value written by the closest write that precedes it in S (or the initial value of REG if there is no preceding write), if an effective operation op1 terminated before an effective operation op2 started, then op1 appears before op2 in the sequence S.

This set of properties states that, from an external observer point of view, the object appears as if it was accessed sequentially by the processes, this sequence i) respecting the real time access order, and ii) belonging to the sequential specification of a read/write register.

Bounded delay-based time-efficient implementation

The notion of a time-efficient operation is not related to its correctness, but is a property of its implementation. It is sometimes called non-functional property. In the present case, it captures the time efficiency of operations 2 . Time-efficiency is a property that is closely related to the notion of fast [START_REF] Dutta | How fast can a distributed atomic read be?[END_REF] implementation. While a fast implementation focus on the minimum number of round-trip transfers that are necessary, a time-efficient operation, as we consider here, provides us with time guarantees when favorable synchrony conditions hold on the system execution.

Let us remember that it is assumed that the local processing times needed to implement these high level read and write operations are negligible, and consider a scenario where all message transfer delays are upper bounded by ∆. Notice that we consider the synchrony assumption applies to parts of the execution only.

Write-latency-free read operation and interfering write. Intuitively, a read operation is write-latency-free if its execution does "not interleave" with the execution of a write operation. More precisely, let τ r be the starting time of a read operation. This read operation is write-latency-free if (a) it is not concurrent with a write operation, and (b) the closest preceding write did not crash and started at a time τ w < τ r -∆.

Let op r be a read operation, which started at time τ r . Let op w be the closest write preceding op r . If op w started at time τ w ≥ τ r -∆, it is said to be interfering with op r . A graphical representation of write-latency-freedom and interfering operations is depicted in Fig. 1. 2 Another example of a non-functional property is quiescence. This property is on algorithms implementing reliable communication on top of unreliable networks [START_REF] Aguilera | On quiescent reliable communication[END_REF]. It states that the number of underlying implementation messages generated by an application message must be finite. Hence, if there is a time after which no application process sends messages, there is a time after which the system is quiescent. -A write operation takes at most 2∆ time units.

-A read operation which is write-latency-free takes at most 2∆ time units.

-A read operation which is not write-latency-free takes at most -3∆ time units if the writer does not crash while executing the interfering write, -4∆ time units if the writer crashes while executing the interfering write (this scenario can appear at most once for each process).

An Algorithm with Time-efficient Operations

The algorithm described in Algorithm 1 implements an SWMR atomic register in the asynchronous system model CAMP n,t [t < n/2], and is time-efficient with respect to the "bounded delay"-based definition. This algorithm is voluntarily formulated to be as close as possible to ABD, for facilitating its understanding by ABD-aware readers.

Local variables. Each process p i manages the following local variables.

reg i contains the value of the constructed register REG, as currently known by p i . It is initialized to the initial value of REG (e.g., the default value ⊥). -wsn i is the sequence number associated with the value in reg i .

rsn i is the sequence number of the last read operation invoked by p i .

swsn i is a synchronization local variable. It contains the sequence number of the most recent value of REG that, to p i 's knowledge, is known by at least (nt) processes. This variable (which is new with respect to other algorithms) is at the heart of the time-efficient implementation of the read operation. -res i is the value of REG whose sequence number is swsn i .

Client side: operation write() invoked by the writer. Let p i be the writer. When it invokes REG.write(v), it increases wsn i , updates reg i , and broadcasts the message when READ(rsn) is received from p j do (12) send STATE(rsn, wsn i ) to p j . WRITE(wsn i , v) (line 1). Then, it waits until it has received an acknowledgment message from (nt) processes (line 2). When this occurs, the operation terminates (line 3). Let us notice that the acknowledgment message is a copy of the very same message as the one it broadcast.

Server side: reception of a message WRITE(wsn, v). When a process p i receives a WRITE(wsn, v) message, if this message carries a more recent value than the one currently stored in reg i , then p i updates accordingly wsn i and reg i (line 7). Moreover, if this message is the first message carrying the sequence number wsn, p i forwards to all the processes the message WRITE(wsn, v) it has received (line 8). This broadcast has two aims: to be an acknowledgment for the writer, and to inform the other processes that p i "knows" this value. 3Moreover, when p i has received the message WRITE(wsn, v) from (nt) different processes, and swsn i is smaller than wsn, it updates its local synchronization variable swsn i and accordingly assigns v to res i (lines 9-11).

Server side: reception of a message READ(rsn). When a process p i receives such a message from a process p j , it sends by return to p j the message STATE(rsn, wsn i ), thereby informing it on the freshness of the last value of REG it knows (line 12). The parameter rsn allows the sender p j to associate the messages STATE(rsn, -) it will receive with the corresponding request identified by rsn.

Client side: operation read(). When a process p i invokes REG.read(), it first broadcasts the message READ(rsn i ) with a new sequence number. Then, it waits until "some" predicate is satisfied (line 5), and finally returns the current value of res i . Let us notice that the value res i that is returned is the one whose sequence number is swsn i .

The waiting predicate is the heart of the algorithm. Its first part states that p i must have received a message STATE(rsn, -) from (nt) processes. Its second part, namely (swsn i ≥ maxwsn), states that the value in p i 's local variable res i is as recent or more recent than the value associated with the greatest write sequence number wsn received by p i in a message STATE(rsn, -). Combined with the broadcast of messages WRITE(wsn, -) issued by each process at line 8, this waiting predicate ensures both the correctness of the returned value (atomicity), and the fact that the read implementation is time-efficient.

Proof of the Algorithm

Termination and atomicity

The properties proved in this section are independent of the message transfer delays (provided they are finite).

Lemma 1 If the writer is correct, all its write invocations terminate. If a reader is correct, all its read invocations terminate.

Proof Let us first consider the writer process. As by assumption it is correct, it broadcasts the message WRITE(sn, -) (line 1). Each correct process broadcasts WRITE(sn, -) when it receives it for the first time (line 8). As there are at least (nt) correct processes, the writer eventually receives WRITE(sn, -) from these processes, and stops waiting at line 2.

Let us now consider a correct reader process p i . It follows from the same reasoning as before that the reader receives the message STATE(rsn, -) from at least (nt) processes (lines 5 and 12). Hence, it remains to prove that the second part of the waiting predicate, namely swsn i ≥ maxwsn (line 5) becomes eventually true, where maxwsn is the greatest write sequence number received by p i in a message STATE(rsn, -). Let p j be the sender of this message. The following list of items is such that item x =⇒ item (x + 1), from which follows that swsn i ≥ maxwsn (line 5) is eventually satisfied.

1. p j updated wsn j to maxwsn (line 7) before sending STATE(rsn, maxwsn) (line 12). 2. Hence, p j received previously the message WRITE(maxwsn, -), and broadcast it the first time it received it (line 8). 3. It follows that any correct process receives the message WRITE(maxwsn, -) (at least from p j ), and broadcasts it the first time it receives it (line 8). 4. Consequently, p i eventually receives the message WRITE(maxwsn, -) from (nt) processes. When this occurs, it updates swsn i (line 10), which is then ≥ maxwsn, which concludes the proof of the termination of a read operation.

2 Lemma 1

Lemma 2 The register REG is atomic.

Proof Let read[i, x] be a read operation issued by a process p i which returns the value with sequence number x, and write[y] be the write operation which writes the value with sequence number y. The proof of the lemma is the consequence of the three following claims.

-Claim 1. If read[i, x] terminates before write[y] starts, then x < y.

-Claim 2. If write[x] terminates before read[i, y] starts, then x ≤ y.

-Claim 3. If read[i, x] terminates before read[j, y] starts, then x ≤ y.

Claim 1 states that no process can read from the future. Claim 2 states that no process can read overwritten values. Claim 3 states that there is no new/old read inversions [START_REF] Attiya | Distributed computing: fundamentals, simulations and advanced topics[END_REF][START_REF] Raynal | Distributed algorithms for message-passing systems[END_REF].

Proof of Claim 1. This claim follows from the following simple observation. When the writer executes write[y], it first increases its local variable wsn which becomes greater than any sequence number associated with its previous write operations (line 1). Hence if read[i, x] terminates before write[y] starts, we necessarily have x < y.

Proof of Claim 2. It follows from line 2 and lines 7-8 that, when write[x] terminates, there is a set Q w of at least (nt) processes p k such that wsn k ≥ x. On another side, due to lines 4-5 and line 12, read[i, y] obtains a message STATE() from a set Q r of at least (nt) processes.

As

|Q w | ≥ n -t, |Q r | ≥ n -t,
and n > 2t, it follows that Q w ∩ Q r is not empty. There is consequently a process p k ∈ Q w ∩ Q r , such that that wsn k ≥ x. Hence, p k sent to p i the message STATE(-, z), where z ≥ x.

Due to (a) the definition of maxwsn ≥ z, (b) the predicate swsn i ≥ maxwsn ≥ z (line 5), and (c) the value of swsn i = y, it follows that y = swsn i ≥ z when read[i, y] stops waiting at line 5. As, z ≥ x, it follows y ≥ x, which proves the claim.

Proof of Claim 3. When read[i, x] stops waiting at line 5, it returns the value res i associated with the sequence number swsn i = x. Process p i previously received the message WRITE(x, -) from a set Q r1 of at least (nt) processes. The same occurs for p j , which, before returning, received the message WRITE(y, -) from a set Q r2 of at least (nt) processes.

As

|Q r1 | ≥ n -t, |Q r2 | ≥ n -t,
and n > 2t, it follows that Q r1 ∩ Q r2 is not empty. Hence, there is a process p k which sent STATE(, x) to p i , and later sent STATE(-, y) to p j . As swsn k never decreases, it follows that x ≤ y, which completes the proof of the lemma.

2 Lemma 2

Theorem 1 Algorithm 1 implements an SWMR atomic register in the distributed system model CAMP n,t [t < n/2].

Proof The proof follows from Lemma 1 (termination) and Lemma 2 (atomicity). 2 T heorem 1

5.2 Time-efficiency with respect to the bounded delay assumption

As already indicated, this underlying synchrony assumption considers that every message takes at most ∆ time units. Moreover, let us remind that a read (which started at time τ r ) is write-latency-free if it is not concurrent with a write, and the last preceding write did not crash and started at time τ w < τ r -∆.

Lemma 3 A write operation takes at most 2∆ time units.

Proof The case of the writer is trivial. The message WRITE() broadcast by the writer takes at most ∆ time units, as do the acknowledgment messagesWRITE() sent by each process to the writer. In this case 2∆ correspond to a causality-related maximal round-trip delay (the reception of a message triggers the sending of an associated acknowledgment).

2 Lemma 3

When the writer does not crash while executing a write operation The cases where the writer does not crash while executing a write operation are captured by the next two lemmas.

Lemma 4 A write-latency-free read operation takes at most 2∆ time units.

Proof Let p i be a process that issues a write-latency-free read operation, and τ r be its starting time. Moreover, Let τ w the starting time of the last preceding write. As the read is write latency-free, we have τ w + ∆ < τ r . Moreover, as messages take at most ∆ time units, and the writer did not crash when executing the write, each non-crashed process p k received the message WRITE(x, -) (sent by the preceding write at time τ w + ∆ < τ r ), broadcast it (line 8), and updated its local variables such that we have wsn k = x (lines 7-11) at time τ w + ∆ < τ r . Hence, all the messages STATE() received by the reader p i carry the write sequence number x. Moreover, due to the broadcast of line 8 executed by each correct process, we have swsn i = x at some time τ w + 2∆ < τ r + ∆. It follows that the predicate of line 5 is satisfied at p i within 2∆ time units after it invoked the read operation.

2 Lemma 4

Lemma 5 A read operation which is not write-latency-free, and during which the writer does not crash during the interfering write operation, takes at most 3∆.

Proof Let us consider a read operation that starts at time τ r , concurrent with a write operation that starts at time τ w and during which the writer does not crash. From the read operation point of view, the worst case occurs when the read operation is invoked just after time τ w -∆, let us say at time τ r = τ w -∆ + . As a message STATE(rsn, -) is sent by return when a message READ(rsn) is received, the messages STATE(rsn, -) received by p i by time τ r + 2∆ can be such that some carry the sequence number x (due to last previous write) while others carry the sequence number x + 1 (due to the concurrent write) 4 . Hence, maxwsn = x or maxwsn = x + 1 (predicate of line 5). If maxwsn = x, we also have swsn i = x and p i terminates its read. If maxwsn = x + 1, p i must wait until swsn i = x + 1, which occurs at the latest at τ w + 2∆ (when p i receives the last message of the (nt) messages WRITE(y, -) which makes true the predicates of lines 9-10, thereby allowing the predicate of line 5 to be satisfied). When this occurs, p i terminates its read operation.

As τ w = τ r + ∆ -, p i returns at the latest τ r + 3∆time units after it invoked the read operation.

2 Lemma 4

When the writer crashes while executing a write operation. The problem raised by the crash of the writer while executing the write operation is when it crashes while broadcasting the message WRITE(x, -) (line 1): some processes receive this message by ∆ time units, while other processes do not. This issue is solved by the propagation of the message WRITE(x, -) by the non-crashed processes that receive it (line 8). This means that, in the worst case (as in synchronous systems), the message WRITE(x, -) must be forwarded by (t + 1) processes before being received by all correct processes. This worst scenario may entail a cost of (t + 1)∆ time units. Algorithm 2 presents a simple modification of Algorithm 1, which allows a fast implementation of read operations whose executions are concurrent with a write operation during which the writer crashes. The modifications are underlined.

When a process p i receives a message READ(), it now returns a message STATE() containing an additional field, namely the current value of reg i , its local copy of REG (line 12).

When a process p i receives from a process p j a message STATE(-, wsn, v), it uses it in the waiting predicate of line 5, but executes before the lines 7-11, as if this message was WRITE(wsn, v). According to the values of the predicates of lines 7, 9, While the notion of a time-efficient algorithm has a theoretical interest -the lower bounds and the worst cases are not the "end of the road"-its practical implication is much more important.

In all possible scenarios that may happen in an actual execution, the worst case scenarios in terms of message transfer delays and asynchrony patterns are usually rare. From a practical viewpoint, it is more important to optimize cases that occur regularly than just focus on those worst case scenarios. What we show with this algorithm is that it possible to match the lower bound, whatever asynchrony pattern happens in the system, and still provide a fast and time-efficient solution when the system is more synchronous. The price to pay for optimizing those good cases is an increase of the message complexity of a write operation, making the algorithm well suited for read-dominated applications that run on almost always synchronous systems.

Indeed, many real-life systems experience lots of "stability" periods during which the results described in this paper apply -cloud systems deployed by Google, Apple, Facebook, Amazon and Microsoft run on dedicated in-house hardware solutions, and are not purely asynchronous. With regards to IoT applications, sensor networks usually have a synchronous mode of operation in order to reduce energy consumption: a Time-Division Multiple Access (TDMA) communication scheme allows sensors to sleep most of the time and wake up at predefined time slots to communicate.

Conclusion

This work presented a new distributed algorithm that implements an atomic read/write register on top of an asynchronous n-process message-passing system in which up to t < n/2 processes may crash. When designing it, the constraints we imposed on this algorithm were (a) from an efficiency point of view: provide time-efficient implementations for read and write operations, (b) and from a design principle point of view: remain "as close as possible" to the flagship ABD algorithm introduced by Attiya, Bar-Noy and Dolev [START_REF] Attiya | Sharing memory robustly in message passing systems[END_REF]. Table 1 recapitulates the "time-efficiency" property of the proposed algorithm. Under the "upper bound ∆ on message transfer delays", any write operation takes then at most 2∆ time units, and a read operation takes at most 2∆ time units when executed in good circumstances (i.e., when there is no write operation concurrent with the read operation). Hence, the inherent cost of an operation is a round-trip delay, always for a write and in favorable circumstances for a read. A read operation concurrent with a write operation during which the writer does not crash, may require an additional cost of ∆, which means that it takes at most 3∆ time units. Finally, if the writer crashes during a write concurrent with a read, the read may take at most 4∆ time units. This shows clearly the incremental cost imposed by the adversaries (concurrency of write operations, and failure of the writer).

It is important to remind that the proposed algorithm remains correct in the presence of any asynchrony pattern. Its time-efficiency features are particularly interesting when the system has long synchrony periods.

Fig. 1

 1 Fig.1Left: A write-latency-free read operation opr. Right: a write operation opw interfering with a read.

  Algorithm 1 Time-efficient SWMR atomic register in CAMP n,t [t < n/2] local variables initialization: reg i ← ⊥; wsn i ← 0; swsn i ← 0; rsn i ← 0. operation write(v) is (1) wsn i ← wsn i + 1; reg i ← v; broadcast WRITE(wsn i , v); (2) wait WRITE(wsn i , -) received from (n -t) different processes ; (3) return() end operation. operation read() is % the writer may directly return reg i % (4) rsn i ← rsn i + 1; broadcast READ(rsn i ); (5) wait (msgs STATE(rsn, -) rec. from (n -t) different proc.) ∧ (swsn i ≥ maxwsn) where maxwsn is the greatest seq. nb in the previous STATE(rsn, -) msgs ; (6) return(res i ) end operation. when WRITE(wsn, v) is received do (7) if (wsn > wsn i ) then reg i ← v; wsn i ← wsn end if; (8) if (not yet done) then broadcast WRITE(wsn, v) end if; (9) if WRITE(wsn, -) received from (n -t) different processes (10) then if (wsn > swsn

i ) ∧ (not already done) then swsn i ← wsn; res i ← v end if

[START_REF] Mostéfaoui | Two-bit messages are sufficient to implement atomic read/write registers in crash-prone systems[END_REF]

end if.

  READ(rsn) is received from p j do (12) send STATE(rsn, wsn i , reg i ) to p j .

Algorithm 2 Time-efficient read in case of concurrent writer crash

when WRITE(wsn, v) or STATE(rsn, wsn, v) is received do (7) if (wsn > wsn i ) then reg i ← v; wsn i ← wsn; broadcast WRITE(wsn, v) end if; (8) if (not yet done) then broadcast WRITE(wsn, v) end if; (9) if WRITE(wsn, -) received from (n -t) different processes

[START_REF] Lynch | Distributed algorithms[END_REF]

then if (wsn > swsn i ) ∧ (not already done) then swsn i ← wsn; res i ← v end if (11) end if. when

Considering the three-component model where each reader is also a server (i.e., R = S), we obtain a two-component model with one writer and reader-server processes. In this model, the necessary and sufficient condition (|R| < |S| t -2) can never be satisfied, which means that, it is impossible to design a fast implementation of a SWMR atomic register in such a two-component model.

Let us observe that, due to asynchrony, it is possible that wsn i > wsn when p i receives a message WRITE(wsn, v) for the first time.

Messages STATE(rsn, x) are sent by the processes that received READ(rsn) before τw, while the messages STATE(rsn, x+1) are sent by the processes that received READ(rsn) between τw and τr +∆ = τw + .

Acknowledgments

The authors want to thank the referees for their constructive comments. This work has been partially supported by the Franco-German DFG-ANR Project DISCMAT (DFG-ANR 40300781) devoted to connections between mathematics and distributed computing, and the French ANR project DESCARTES (ANR-16-CE40-0023-03) devoted to distributed software engineering.

and 10, this allows p i to expedite the update of its local variables wsn i , reg i , swsn i , and res i , thereby favoring fast termination.

The reader can check that these modifications do not alter the proofs of Lemma 1 (termination) and Lemma 2 (atomicity). Hence, the proof of Theorem 1 is still correct. Lemma 6 A read operation which is not write-latency-free, and during which the writer crashes during the interfering write operation, takes at most 4∆ time units.

Proof Let τ r be the time at which the read operation starts. As in the proof of Lemma 4, the messages STATE(rsn, -, -) received p i by time τ r + 2∆ can be such that some carry the sequence number wsn = x (due to last previous write) while some others carry the sequence number wsn = x + 1 (due to the concurrent write during which the writer crashes). If all these messages carry wsn = x, the read terminates by time τ r + 2∆. If at least one of these messages is STATE(rsn, x + 1, -), we have maxwsn = x+1, and p i waits until the predicate swsn i ≥ maxwsn (= x+1) becomes true (line 5).

When it received STATE(rsn, x + 1, -), if not yet done, p i broadcast the message WRITE(rsn, x + 1, -), (line 8 of Algorithm 2), which is received by the other processes within ∆ time units.

If not yet done, this entails the broadcast by each correct process of the same message WRITE(rsn, x + 1, -). Hence, at most ∆ time units later, p i has received the message WRITE(rsn, x + 1) from (nt) processes, which entails the update of swsn i to (x + 1). Consequently the predicate of line 5 becomes satisfied, and p i terminates its read operation.

When counting the number of consecutive communication steps, we have: The message READ(rsn) by p i , followed by a message STATE(rsn, x+1, -) sent by some process and received by p i , followed by the message WRITE(rsn, x + 1) broadcast by p i , followed by the message WRITE(rsn, x + 1) broadcast by each non-crashed process (if not yet done). Hence, when the writer crashes during a concurrent read, the read returns within at most τ r + 4∆ time units.

2 The algorithm presented in this paper shares many similarities with the well-known ABD algorithm. ABD is at the basis of real-life implementations of scalable replicated storage services used in today's large data centers and clouds, and distributed shared memory systems. Keeping as close as possible to the design ideas of the original algorithm eases the integration of the proposed ideas in implemented systems.

Cost of the operations according to upper bound ∆ the underlying synchrony assumption on transfer delays write operation 2∆ read operation (no concurrent write) 2∆ read operation (concurrent write) 3∆ read operation (conc. crashing write) 4∆

Table 1 Summary for the upper bound delay assumption Differently from the proposed algorithm, the ABD algorithm does not display different behaviors in different concurrency and failure patterns. In ABD, the duration of all write operations is upper bounded by 2∆ time units, and the duration of all read operations is upper bounded by 4∆ time units. The trade-off between ABD and our algorithm lies the message complexity, which is O(n) in ABD for both read and write operations, while it is O(n 2 ) for a write operation and O(n) for a read operation in the proposed algorithm. Hence our algorithm is particularly interesting for registers used in read-dominated applications run on systems that are often synchronous.


Fatima Ibersiene

Camille Latouche

Claudine Katan

Abdou Boucekkine

On the contribution of f electrons to the quadratic hyperpolarizability: the case of Lanthanide Terpyridyl Complexes

Over the last decades, trivalent lanthanide ions (Ln 3+ ) have gained much attention due to their peculiar luminescence, which opened the way to a broad range of applications, from medical diagnostic to lasers. Their impact on nonlinear optical (NLO) properties also attracted interest, especially in the framework of lanthanide complexes. Several experimental studies demonstrated that the quadratic hyperpolarizability varies with the number of 4felectrons, with a stronger effect on dipolar than octupolar components. The main interpretation put forward to explain the observed trends relied on the polarizable character of the 4felectrons. We report here a first step towards understanding the role of 4f-electrons in NLO responses, considering a series of dipolar terpyridyl-trinitro lanthanide complexes LLn(NO 3 ) 3 (Ln = Gd, Dy,Yb, Lu as well as La and Y; L=terpyridil-like ligand). Using DFT and TD-DFT we investigate their linear and non-linear optical properties. Consistently with earlier experimental findings, simulated UV-visible spectra show minor changes by varying Ln. The same holds for dipole moments and polarizabilities, whereas the nature of the lanthanide affects hyperpolarizabilities. It is shown that the observed changes are not a direct effect of the 4felectrons that behave like core electrons.

Introduction

In the past decades, transition metal and lanthanide complexes have been widely investigated, especially for their strong ability to be luminescent 1 but also as candidates for new nonlinear optical (NLO) active materials.

2-14

In 2006, Le Bozec, Maury and coll. 7 who synthesized and characterized the first dipolar lanthanide terpyridyl complex family LLn(NO 3 ) 3 (L= 4,4''dimethyl,4'-(N,Ndimethylaminophenyl)2,2',6',2''terpyridine Ln = La, Gd, Dy and Yb) found that their first order hyper-polarizabilities (β), measured using Harmonic Light Scattering (HLS) experiments, increase with the number of 4f electrons, whereas their UVvisible absorption spectra remain almost unchanged. Such behavior is rather surprising. A similar trend of the secondorder NLO complexes (Ln= La, Ce, Sm, Eu and Er, L= N,N′-1,3-propylenbis(salicylideniminato). 9 In these studies, both HLS and EFISH techniques were used to measure β. This helped to establish the respective influence of dipolar and octupolar contributions to the quadratic hyperpolarizability, which was shown to be dominated by the octupolar part whereas the dipolar one is significantly influenced by the number of 4f electrons. From these studies, it was concluded that β undergoes a systematic increase with the number of f electrons, and the main explanation put forward by several authors was an unexpected polarizable character of the 4f electrons in the Ln complexes.

10 Furthermore, Tanner, Wong and coll. 11 first examined an extended series of dipolar Ln complexes of trans-cinnamic acid, followed by Parker and coll. 12 with a series of ninecoordinate chiral triazacyclononane complexes having both dipolar and octupolar contributions. Whereas both these extensive investigations are almost fully consistent with previously reported trends, they reveal that β reaches a maximum around the center of the whole Ln series. More recently, a series of octupolar phtalocyanine-based Ln complexes (Ln = Nd, Eu, Dy, Lu compared to Y) also revealed a strong dependence of β HLS on the number of 4f electrons with similar trends for neutral, reduced and oxidized species.

14

These results call into question the interpretation of the firstorder hyperpolarizability dependence on the number of 4f electrons in terms of polarization of the f electrons. This study aims to provide a first step towards a better understanding of the structure-property relationship between the 4f-electron shell and quadratic hyperpolarizability of Ln complexes. We start with a series of dipolar complexes to restrict complexity and chose the terpyridyl-trinitro lanthanide complexes LLn(NO 3 ) 3 for which experimental data are available (Ln = La, Gd, Dy, Yb as well as Y; L=terpyridyl-like ligand) 7 also including Ln = Lu. Based on DFT and TD-DFT computations with the PBE0 functional (see computational details), [15][16][17][18][19][20][21][22][23][24] we shall successively examine their structural, electronic and linear optical properties and finally their NLO responses.

Results and discussion

Molecular structure

The general structure of the complexes under investigation is displayed in Figure 1. All complexes exhibit a typical dipolar push-pull structure, with N-dimethyl aniline as the donating group and the terpyridine moiety as the main electron withdrawing group. Even so X-ray diffraction data are available for two complexes, we optimized the structure of each complex in the solvent (CH 2 Cl 2 ) to afford computed data relevant to available experimental photophysical data. 7 Table 1 reports selected structural data derived from these optimized geometries that are compared with X-ray diffraction data.

7

Since we compare optimized geometries in solution (CH 2 Cl 2 ) to solid state ones, the overall agreement between the X-ray data and the computed geometrical parameters is good. In particular, the computed dihedral angles and the X-ray ones are expected to be different. Indeed, (i) the dihedral angle Θ between the terpyridyl and the N-dimethyl aniline moieties is computed to be ca. 20°-30° lower in solution than in the solid state, (ii) the NCCN dihedral angles are larger in solution than in solid state for LGd(NO 3 ) 3 whereas the reversed situation is observed in LY(NO 3 ) 3 . This clearly evidences steric effects and intermolecular interactions induced by the molecular packing in the crystalline solid state. Besides, the Gd-O and Gd-N bond lengths are higher than the Y-O and Y-N, consistently with the larger ionic radius of Gd 3+ as compared to Y 3+ .

Electronic structure and UV-visible spectra

Table 2 summarizes computed (Figure S2) and measured 7 UVvis data, namely the wavelengths of the two first main absorption bands and corresponding calculated oscillator strengths f as well as experimental molar extinction coefficients ε. The main transitions leading to the computed absorption bands are also given with relevant molecular orbitals (MO) plotted in Figure 2. Overall, the computed photophysical data show good agreement with experimental findings. We find a small systematic overestimation of the highest wavelength absorption band corresponding to ca. 0.15 eV for all complexes. The second very intense absorption band at shorter wavelengths is well reproduced by the simulations, with a systematic small underestimation of the wavelengths corresponding to ca. 0.20 eV. Noteworthy, such differences are usual given the level of theory used to compute excited states, in this instance the solvation model and the lack of vibronic contributions. Besides, when using other DFT exchange-correlation functionals, i.e. B3LYP, 21 CAM-B3LYP 22 and M062X, 23 comparison between the simulated and experimental UV-visible spectra worsened, in particular with CAM-B3LYP and M062X that led to large underestimation of the absorption energies of ca. 1 eV (Supplementary Table S2).

The MOs involved in the transitions reported in Table 2 are depicted in Figure 2 for LY(NO 3 ) 3 and LGd(NO 3 ) 3 . The frontier MOs are similar for all complexes. It is interesting to note that the α and β frontier MOs are identical. The lowest energy transitions are dominated by an electronic transition from the highest occupied MO (HOMO) to the lowest unoccupied MO (LUMO), whereas the second absorption band mainly involves HOMO-1 and LUMO. For the two LY(NO 3 ) 3 and LGd(NO 3 ) 3 complexes, the atomic orbital composition of these MOs as well as their energies are almost identical since they do not involve the metal ion. As expected for the ligands used in these complexes, the HOMO is dominated by the Ndimethylaminophenyl donating moiety whereas the LUMO is mainly localized on the terpyridyl ligand.

Table 1 Selected structural data of LGd(NO 3 ) 3 and LY(NO 3 ) 3 complexes with L= terpyridyl-like ligand, from computed optimized geometries and X-ray data. 7 Distances are given in Å and angles in degrees. Please do not adjust margins

Ln 3+ /M 3+ ion Gd 3+ Gd 3+ X-ray Y 3+ Y 3+ X-ray M-O O40-M39 2

Please do not adjust margins

A more detailed frontier MO diagram of this complex is given in Figure S1. Interestingly the SOMO and SOMO-1 of the complexes are localized on the terpyridine ligand as well as the LUMO+1, so that the absorption bands observed at ca. 390 nm correspond to intra-ligand excitation and not to an inter-ligand charge transfer one, like the band at ca. 450 nm. The LUMO+2, which contributes to the higher lying band also visible in the experimental spectra (ca. 325 nm) is different from the other unoccupied MOs since it is localized on the electron withdrawing nitro groups.

Next, we inspect position of the 4f lanthanide orbitals. As expected they behave as core orbitals in the complexes. For instance, in the case of LGd(NO 3 ) 3 the 4f atomic orbitals (AO) are very low lying in energy (HOMO-73 to HOMO-79), at ca. 8.5 eV below the HOMO. For Dy, Yb and Lu complexes the 4f orbitals are located 7.5, 6.4 and 7.1 eV below the HOMO, respectively. Moreover, these 4f AOs exhibit no mixing with the ligand orbitals. This is confirmed by considering the 4f electrons as core electrons, incorporating them into each pseudopotential (4f in-core calculation) 24 for the Gd 3+ , Dy 3+ , Yb 3+ and Lu 3+ ions that possess a non-empty 4f shell.

Moreover, other orbitals namely the 5d and 6s ones neither contribute to the frontiers MOs of the complexes. The virtual MOs of LGd(NO 3 ) 3 containing a high contribution of the 4f Gd orbitals are rather high in energy. The lowest ones are β MOs, namely LUMO+8 to LUMO+14.

NLO properties

Next, we compute the dynamical hyperpolarizabilities β based on the CPHF method 20 and taking into account the solvent (CH 2 Cl 2 ). We used the following expression to derive  tot from the components of the tensor:

This expression is consistent with the measurement of quadratic hyperpolarizabilities using the HLS method 25 and the X convention for the non-linear tensor. 26 Interestingly, computed β tot show a remarkable agreement with experimental values (Table 3). In fact, computed quadratic hyperpolarizabilities follow almost the same trend as the experimental ones, namely they increase with the atomic number of the Ln 3+ cation for the considered atomic species.

The only discrepancy is that the computed value of the Yb complex is smaller to that of the Dy complex, which runs counter experimental data, but differences range within experimental standard errors (15%)

.

Meanwhile, the smaller β tot for Ln=Yb as compared to Dy is fully in line with the experimental results reported for a larger extent of lanthanides on two different series of different complexes. 11,12 Besides, for the lutetium ion, which has the largest number of 4f electrons, the increase of β also collapses in LLu(NO 3 ) 3. Unfortunately, the quadratic hyperpolarizability of LLu(NO 3 ) 3 has not been measured experimentally, but, our computed value falls between that of the La and Yb complexes, which is again consistent with the trends observed in experimental data obtained on tris(cinnamate) complexes and for triazacyclonoane-based complexes. 11,12 Thus, our results on LLn(NO 3 ) 3 are consistent with previously reported observations, 11,12 namely that the quadratic hyperpolarizability reaches a maximum in the vicinity of the center of the lanthanide series. Then, we further checked the assumption of possible impact of the polarizable character of 4f electrons. To this end, we also computed the dynamic hyperpolarizabilities β using a basis set where the 4f electrons are incorporated in the pseudopotentials (4f in-core calculations) 24 for the Gd 3+ , Dy 3+ , Yb 3+ and Lu 3+ ions, which possess a non-empty 4f shell.

Surprisingly, the trends are the same for both levels of theory with comparable computed β tot values (Table 3). This demonstrates the core-like nature of the 4f electrons (vide

COMMUNICATION

Please do not adjust margins Please do not adjust margins supra) and that these core electrons affect only marginally the NLO response.

Since the hyperpolarizability depends on the nature of the metal/lanthanide ion, we found interesting to investigate if other molecular electrical properties i.e. ground state dipole moments and polarizabilities exhibit a similar behavior. Interestingly, both the dipole moments µ and mean isotropic polarizabilities (α iso ) are almost constant in the series of complexes (Table 4), and the small variations are not in line with changes computed for β tot . Thus, the nature of the lanthanide cation has little influence on these properties. In addition, all complexes exhibit also similar excited state dipole moments. Using all the computed values within the two-level model we further checked if the model is capable of providing the correct trend. This is not the case, highlighting contributions stemming from other excitations. Thus, only the higher order responses to the perturbing applied electromagnetic field seem to depend on the nature of the lanthanide ion and the peculiar trend reported for the quadratic hyperpolarizabilities when varying the lanthanide ion cannot be traced back to a direct impact of the 4f-electron shell.

Computational details

The DFT calculations have been performed using the Gaussian09 package. 16 The geometries of all compounds have been optimized without symmetry constraints using the PBE0 functional 15 and the LANL2DZ ). Spin contamination was found very low for the open-shell systems. Next, TD-DFT calculations have been performed using the optimized geometries to determine the electronic transitions involved in the excitations and to simulate the absorption spectra (Figure S2). The β hyperpolarizabilities have been calculated at the same level of theory using the CPHF time dependent perturbation theory 20 as implemented in the Gaussian09 package.

Conclusions

In summary, considering a series of terpyridyl-trinitro lanthanide complexes LLn(NO 3 ) 3 of dipolar nature we examined the impact of the number of 4f electrons on their linear and nonlinear optical properties. We found that their ground and excited state dipole moments, polarizabilities and absorption spectra change only slightly in the series Ln = Y, La, Gd, Dy, Yb and Lu. Meanwhile, the dynamic first-order hyperpolarizability β undergoes sizeable variations, consistently with earlier experimental findings on these LLn(NO 3 ) 3 complexes and other lanthanide complexes. Given the range of metal and lanthanide ions considered, our results also suggest that β increases with the number of f electrons, but β is likely to reach its maximum around Dysprosium. We further inspected possible effect of polarization of the 4felectrons on computed properties by comparing results obtained when the f-electrons are maintained among the valence electrons with those obtained when they are considered as core electrons. We found out that the 4f electrons have a core-like nature and do affect only marginally the quadratic hyperpolarizability, thus their unexpected polarizable character is hardly responsible for the observed changes and alternative explanations deserve to be explored in the future.

Fig. 1

 1 Fig. 1 Structure of the LLn(NO3)3complex (L=terpyridyl-like ligand) with atomic numbering, Ln is number 39. Color code is turquoise blue for lanthanide, red for oxygen, grey for carbon and royal blue for nitrogen. Hydrogen atoms have been omitted for clarity.

Fig. 2 Frontier

 2 Fig. 2 Frontier MOs of LY(NO3)3 and LGd(NO3)3, L=terpyridyl-like ligand. (a) HOMO and LUMO of LY(NO3)3 (b) LGd(NO3)3: α SOMO and LUMO (left) β SOMO and LUMO (right).

Table 2

 2 Computed and experimental 7 photophysical data of LLn(NO 3 ) 3 complexes, with Ln = La, Gd, Dy, Yb, Lu as well as Y and with L=terpyridyl-like ligand, in CH 2 Cl 2 .

	Ln (4f 3+ /M n ) spin multiplicity 3+ ion	λmax nm calc.	f calc.	Main transitions (weight) a	λmax nm exp. 7	Ε L.mol -1 .cm -1 exp. 7
	Y 3+	480	0.18	H→L(99%)	454	4750
	2S+1=1	372	0.56	H-1→L(92%)	392	28900
		326	0.51	H-1→L+1(77%)
	La (4f 3+ 0 )	475 370	0.17 0.56	H→L(99%) H-1 →L ((93%)	453 391	5400 24500
	2S+1=1	323	0.52	H-1→L+1(74%)
	Gd (4f 3+ 7 )	476	0.19	S β→L β (95%) S α→L α(100%)	452	5800
	2S+1= 8	367	0.56	S-1 α→L α(93%)	393	26900
				S-1 β→L β(91%)
		325	0.20	S-1 β→L+1 β (42%)
				S β→L+2 β(31%)
				S α→L+2 α(23%)
	Dy (4f 9 ) 3+	482	0.19	S β→L β(97%) S α→L α(100%)	450	5400
	2S+1= 6	370	0.51	S-1 α→L α(85%)	394	26700
				S-1 β→L β(84%)
		326	0.39	S α→L+2 α(52%)
				S-1 α→L+1 α(43%)
				S β→L+4 β(32%)
	Yb (4f 3+ 13 )	472	0.22	S-3 α →L α(26%) S β →L+1 β(84%)	452	5150
	2S+1=2			S-3 α →L+1 α(23%)
		376	0.43	S-1β →L+1β(85%)	394	28850
				S-1 α →L α(75.9%)
	Lu (4f 3+ 14 )	373 482	0.56 0.17	H-1→L(92%) H→L(99%)	-	-
	2S+1=1	327	0.51	H-1→L(77%)

a H=HOMO, L=LUMO and S=SOMO; calc. = calculated, exp. = experimental.

Table 3

 3 Computed βtot and experimental 7 βexp (βHLS) first order dynamical hyperpolarizabilities of LLn(NO3)3 complexes, with Ln = La, Gd, Dy, Yb, Lu as well as Y and with L=terpyridyl-like ligand, in CH2Cl2 at λ= 1907 nm. 4f electrons are either treated as valence electrons (4f-valence) or incorporated in the pseudopotentials (4f-in-core).

	Ln 3+ /M 3+ ion	βtot	βtot	βHLS
		4f-valence	4f-in-core	exp. 7
		10 -30 esu	10 -30 esu	10 -30 esu
	Y 3+	232	-	191
	La 3+	212	-	186
	Gd 3+	239	247	246
	Dy 3+	253	256	260
	Yb 3+	242	243	288
	Lu 3+	226	228	-

Table 4

 4 Computed dynamic polarizabilities αiso (at λ=1907 nm) and dipole moments µ of LLn(NO3)3 complexes, with Ln = La, Gd, Dy, Yb, Lu as well as Y and with L=terpyridyl-like ligand, in CH2Cl2 at λ= 1907 nm. 4f electrons are either treated as valence electrons (4f-valence) or incorporated in the pseudopotentials (4fin-core).

	Ln 3+ /M 3+	αiso	αiso	µ	µ
	ion	4f-valence	4f-in-core	4f-valence	4f-in-core
		10 -24 esu	10 -24 esu	(D)	(D)
	Y 3+	126	-	27	-
	La 3+	127	-	29	-
	Gd 3+	127	128	27	28
	Dy 3+	124	125	26	27
	Yb 3+	126	126	27	27
	Lu 3+	126	126	26	26

Acknowledgements

This work was granted access to the HPC resources of CINES and IDRIS under the allocation 2016-080649 and 2017-080649 made by GENCI. A.B. and C.L. acknowledge the hospitality of Prof. Vincenzo Barone at SNS Pisa, where part of the present study has been done. COST CM-1006 action is also acknowledged.

Notes and references

Conflicts of interest

There are no conflicts to declare.