Marie-Cécile Cécile Faure

Jean-Claude Sulpice

Maud Delattre

Marc Lavielle

Magali Prigent

Marie-Hélène Cuif

Chantal Melchiort

Eric Tschirhart

Oliver Nusse

Sophie Dupre-Crochet
email: sophie.dupre@u-psud.fr

Chantal Melchior

Oliver Nüβe

Sophie Dupré-Crochet

The recruitment of p47phox and Rac2G12V at the phagosome is transient and phosphatidylserine-dependent

Keywords: ROS production, live imaging, anionic phospholipids, NADPH oxidase, phagocytosis succinimidyl ester; lactC2, C2

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

During bacterial and fungal infections, phagocytes play a major role in host defence. Neutrophils are the first cells that arrive at the site of infection, ingest the pathogen into a compartment called phagosome and kill it inside this phagosome. Phagocytes express the enzyme NADPH oxidase that generates superoxide anion (O 2 •-) by transferring electrons from NADPH to molecular oxygen. O 2

•-and the other derivative reactive oxygen species (ROS) produced in the phagosome, participate in the killing of the pathogenic agent. Genetic deficits in NADPH oxidase lead to chronic granulomatous disease (CGD) characterized by severe recurrent infections [START_REF] Stasia | Genetics and immunopathology of chronic granulomatous disease[END_REF]. NADPH oxidase is a multicomponent enzyme composed of membrane associated proteins, gp91 phox and p22 phox , cytosolic proteins, p67 phox , p47 phox , p40 phox and the small G protein Rac. Upon stimulation via soluble or particulate stimuli, the cytosolic subunits translocate to the plasma or phagosomal membrane and assemble with the membrane subunits to form an active enzyme [START_REF] Nauseef | Assembly of the phagocyte NADPH oxidase[END_REF]. This translocation induces the activation of the NADPH oxidase. In the resting state, the three cytosolic subunits, p67 phox , p47 phox , p40 phox form a heterotrimeric complex in the cytosol [START_REF] Groemping | Activation and assembly of the NADPH oxidase: a structural perspective[END_REF]. It is unclear, whether these subunits translocate together or sequentially and whether they remain together to form the active enzyme for the entire time of ROS production. Upon stimulation, p47 phox attaches to the membrane via its PX domain and to p22 phox via its SH3 domains and thereby translocates the complex to the membrane [START_REF] Kanai | The PX domains of p47phox and p40phox bind to lipid products of PI(3)K[END_REF][START_REF] Ago | Phosphorylation of p47phox directs phox homology domain from SH3 domain toward phosphoinositides, leading to phagocyte NADPH oxidase activation[END_REF][START_REF] Groemping | Activation and assembly of the NADPH oxidase: a structural perspective[END_REF]. Rac1 and Rac2 are involved in phagocytosis and O 2 •generation [START_REF] Quinn | Translocation of Rac correlates with NADPH oxidase activation. Evidence for equimolar translocation of oxidase components[END_REF][START_REF] Caron | Identification of two distinct mechanisms of phagocytosis controlled by different Rho GTPases[END_REF][START_REF] Mizrahi | A prenylated p47phoxp67phox-Rac1 chimera is a Quintessential NADPH oxidase activator: membrane association and functional capacity[END_REF] . However they may have different roles and it has been proposed that Rac2 is more specific for ROS production [START_REF] Filippi | Localization of Rac2 via the C terminus and aspartic acid 150 specifies superoxide generation, actin polarity and chemotaxis in neutrophils[END_REF]. It has been shown in vitro that phospholipids and especially phosphatidylserine (PS) are essential for oxidase activation [START_REF] Tamura | The superoxidegenerating respiratory burst oxidase of human neutrophil plasma membrane. Phosphatidylserine as an effector of the activated enzyme[END_REF][START_REF] Shpungin | Activation of the superoxide forming NADPH oxidase in a cell-free system by sodium dodecyl sulfate. Absolute lipid dependence of the solubilized enzyme[END_REF] . The translocation of Rac, p47 phox and p40 phox to the membrane involve direct interaction with anionic lipids mainly PPI, phosphatidic acid (PA) and PS. In fact, p47 phox and p40 phox bear a lipid binding domain: the phox homology (PX) domain. The PX domain of p47 phox has two binding pockets: one preferentially binds to phosphatidylinositol-(3,4)-bisphosphate (PI(3,4)P 2 ) and the other binds to PA and PS whereas the PX domain of p40 phox binds to phosphatidylinositol-3-phosphate (PI(3)P) [START_REF] Kanai | The PX domains of p47phox and p40phox bind to lipid products of PI(3)K[END_REF][START_REF] Karathanassis | Binding of the PX domain of p47(phox) to phosphatidylinositol 3,4bisphosphate and phosphatidic acid is masked by an intramolecular interaction[END_REF]. The role of the PX domain in p47 phox may depend on the type of stimulation. The disruption of PI(3,4)P2 binding in this domain does not prevent phagosomal translocation [START_REF] Li | p47phox Phox homology domain regulates plasma membrane but not phagosome neutrophil NADPH oxidase activation[END_REF]. Rac proteins, like many other small GTPases, contain a polybasic domain [START_REF] Williams | The polybasic region of Ras and Rho family small GTPases: a regulator of protein interactions and membrane association and a site of nuclear localization signal sequences[END_REF] that has been shown to act as a targeting signal to localize proteins to membranes [START_REF] Mclaughlin | The myristoyl-electrostatic switch: a modulator of reversible protein-membrane interactions[END_REF][START_REF] Yeung | Receptor activation alters inner surface potential during phagocytosis[END_REF][START_REF] Yeung | Membrane phosphatidylserine regulates surface charge and protein localization[END_REF]. The negative charge of the cytosolic leaflet of cellular membranes depends on their anionic phospholipid content. It creates an electrostatic field that attracts proteins with cationic residues such as the small GTPases [START_REF] Yeung | Lipid signaling and the modulation of surface charge during phagocytosis[END_REF]. Using cationic probes [START_REF] Roy | Mutational and biochemical analysis of plasma membrane targeting mediated by the farnesylated, polybasic carboxy terminus of Kras4B[END_REF], Yeung et al. showed that the surface charge of the cytosolic leaflet of the phagosome decreased during phagocytosis [START_REF] Yeung | Receptor activation alters inner surface potential during phagocytosis[END_REF][START_REF] Yeung | Contribution of phosphatidylserine to membrane surface charge and protein targeting during phagosome maturation[END_REF]. This decrease was attributed mainly to the drop in phosphatidylinositol-(4,5)-bisphosphate that occurred upon closure of the phagosome [START_REF] Botelho | Localized biphasic changes in phosphatidylinositol-4,5bisphosphate at sites of phagocytosis[END_REF][START_REF] Steinberg | Pathogen destruction versus intracellular survival: the role of lipids as phagosomal fate determinants[END_REF] . PS, which represents about 15% of the lipids in the inner leaflet of the plasma membrane [START_REF] Yeung | Lipid signaling and the modulation of surface charge during phagocytosis[END_REF], persists on the phagosome after sealing and contributes to the phagosomal surface charge [START_REF] Yeung | Contribution of phosphatidylserine to membrane surface charge and protein targeting during phagosome maturation[END_REF][START_REF] Magalhaes | Pivotal Advance: Phospholipids determine net membrane surface charge resulting in differential localization of active Rac1 and Rac2[END_REF][START_REF] Minakami | Membrane phospholipid metabolism during phagocytosis in human neutrophils[END_REF] . In the previous studies, the phagosomal PS content was monitored using the C2 domain of the lactadherin fused to a fluorescent protein as a probe. The lactadherin and its C2 domain (lactC2) were shown to bind phosphatidylserine specifically and with a high affinity (Kd around 2 nM for both), in a calcium independent manner [START_REF] Andersen | Functional analyses of two cellular binding domains of bovine lactadherin[END_REF][START_REF] Shi | Lactadherin binds selectively to membranes containing phosphatidyl-L-serine and increased curvature[END_REF]. By contributing to the phagosomal surface charge as well as binding to specific protein domains, PS could play a role in NADPH oxidase activation. Rac and p47 phox are good candidates for a PS dependent recruitment to the membrane as explained above. In this study, we investigated the kinetics of Rac and p47 phox recruitment at the phagosome and the role of PS in their recruitment. To examine this latter point we used the lactC2 probe. Indeed, in vitro studies demonstrated that lactadherin inhibits enzyme complexes of blood coagulation by blocking PS binding sites [START_REF] Shi | Lactadherin inhibits enzyme complexes of blood coagulation by competing for phospholipid-binding sites[END_REF]. We then reasoned that lactC2 may have a dual role in the cell serving as a probe to monitor PS but also masking PS. We observed that p47 phox and a dominant positive form of GFP-Rac2 (GFP-Rac2G12V) were transiently recruited at the phagosomal membrane, at the beginning of phagocytosis and left the phagosome far before the termination of ROS production. In the presence of mCherry-lactC2 the phagosomal recruitment of GFP-Rac2G12V was blocked and p47-GFP translocation slowed down. We then looked at NADPH oxidase activity in cells expressing mcherry-lactC2. The phagosomal ROS production was reduced at the commencement of phagocytosis. These results advocate that lactC2, by masking PS, interfered with the phagosomal recruitment of p47 phox and Rac2 and disturbed NADPH oxidase activity.

Results

Generation of stable PLB-985 cell lines expressing p47-GFP and CFP-Rac2

To examine the accumulation of p47phox and Rac2 at the phagososome we generated neutrophillike cells (differenciated myeloid leukemia cell line, PLB-985 cells) with stable expression of p47-GFP and CFP-Rac2. The level of transgenic protein expression was assessed by flow cytometry (data not shown) and by immunoblot analysis (Fig. 1A). The immunot blot with anti-GFP reveals that only the tagged proteins are expressed in the cells and not the GFP alone. The immunoblot with anti-p47 phox shows that the level of endogenous p47 phox was 5 fold higher than the p47-GFP level in the p47-GFP transgenic cell line (Fig. 1B). In the CFP-Rac2 transgenic cell line similar level of CFP-Rac2 and endogenous Rac2 were observed (Fig. 1C). We further measured the ROS production in the two transgenic cells lines in response to phorbol 12myristate acetate (PMA) by using a luminol assay. ROS production of both lines was similar to non-transfected differentiated cells (data not shown). Immunoprecipitation experiments were then performed to check the interactions of p47-GFP and CFP-Rac2 with the other endogenous NADPH oxidase subunits. Both transgenic PLB-985 cell lines were stimulated by PMA for 10 min and the oxidase membrane subunit gp91 phox was immunoprecipitated from the membrane fraction. As shown in Figure 1D, gp91 phox co-immunoprecipitated with p47-GFP as well as endogenous p47 phox and p67 phox in the p47-GFP transgenic PLB-985 cell line. The same result was obtained with CFP-Rac2 in the transgenic CFP-Rac2 PLB-985 cell line (Fig. 1E). In the control, agarose beads alone without gp91 phox antibodies collected tenfold less gp91 phox than with the specific gp91 phox antibodies.

P47 phox is transiently recruited to the phagosome

The recruitment of p47-GFP was then followed during phagocytosis of opsonized zymosan by wide field videomicroscopy. The cytosolic marker calcein blue was measured in the same cells in order to quantify p47-GFP fluorescence by ratio analysis. The mean time from phagosomal cup formation to phagosome closure, was 48 +/-5 s (n=12). Accumulation of P47-GFP was first visible at the phagosomal cup on average 14 +/-5 s (n=14) before sealing (Fig. 2B). Then p47-GFP fluorescence increased rapidly at the phagosome, with a maximal fluorescence 20s after the phagosome closure, and then decreased (Fig. 2A, C and video S1). Quantification of p47-GFP phagosomal recruitment was performed by measuring the ratio of phagosomal to cytosolic fluorescence. P47-GFP was present at the phagosomal membrane for only 2 min after the phagosome closure (Fig. 2C). To confirm that p47-GFP reflected the properties of endogenous p47 phox , we performed immunofluorescence experiments using a protocol of synchronized phagocytosis [START_REF] Quinn | Neutrophil methods and protocols[END_REF]. Cells were incubated with opsonized texas-red zymosan (time 0); then the number of phagosomes and the presence of phagosomal p47 phox were determined at different time points after incubation. Most of the phagocytosis occurred during the first five minutes (Fig. 2E). In these conditions, we observed a decrease in p47 phox positive phagosomes from 50% to 25% during the first five minutes and then p47 phox presence diminished to 11% after 10 minutes (Fig. 2D,E). The remaining p47 phox positive phagosomes at 10 min were most likely formed during the preceding 5 min. This result confirms that endogenous p47 phox , like p47-GFP, is transiently recruited at the phagosome and leaves the phagosome within a few minutes.

The accumulation of Rac2G12V at the phagosome is also transient

We next investigated the CFP-Rac2 translocation at the phagosome during zymosan phagocytosis by videomicroscopy. However, we could not observe any consistent accumulation of CFP-Rac2 at the phagosome (data not shown). The same result was obtained with PLB-985 cells stably expressing GFP-Rac2 (data not shown). This may be due to abundant endogenous Rac2 which might compete with CFP-Rac2 (or GFP-Rac2) for activation and phagosomal membrane targeting. Therefore, we used the constitutively active form Rac2G12V which displaced the equilibrium Rac2GDP/Rac2GTP in favor of the GTP bound Rac2G12V form. Differentiated PLB-985 cells were transiently transfected with GFP-Rac2G12V and the fusion protein was followed during phagocytosis of zymosan by spinning-disk confocal videomicroscopy (Fig. 3A, video S2). In these cells expressing GFP-Rac2G12V, the formation of the phagosome, from the visible cup until phagosome sealing, lasted 48 s +/-5 s (mean +/-SD, n=9) and GFP-Rac2G12V was first visible at the cup 15 s +/-6 s (mean +/-SD, n=11) prior to phagosome closure (Fig. 3B). As for p47-GFP, the GFP-Rac2G12V fluorescence increased rapidly at the phagosome, with a maximal fluorescence 20 s after the phagosome closure, and then decreased (Fig. 3A,C and video S2). The fluorescence ratio between phagosome and cytosol was calculated to evaluate the protein accumulation at the phagosome. The mean time of GFP-Rac2G12V accumulation at the phagosomal membrane was 3 min (Fig. 3C).

Phosphatidylserine is localized at the phagosomal membrane and its presence decreases slowly after the phagosomal closure As the accumulation of p47 and Rac2G12V at the phagosome may require attachment to anionic phospholipids, we wanted to investigate the role of PS in this process. We first monitored PS accumulation at the phagosome of PLB-985 cells during phagocytosis with the C2 domain of lactadherin that binds specifically to PS (lactC2). The lactC2 domain was fused to the fluorescent protein mCherry, this fusion protein was transiently transfected into differentiated PLB-985 cells, and the distribution of the mCherry-lactC2 probe was visualized during yeast phagocytosis by 3D deconvolution microscopy. At time 0, when the phagosome closed, the lactC2 probe marked the inner leaflet of the plasma membrane as well as the cytosolic leaflet of the phagosomal membrane (Fig. 4A), as observed by Minakami et al. in human neutrophils (Minakami et al., 2010). We also witnessed a cytosolic fluorescence due to the presence of free probe in the cytosol. Following yeast internalization, the lactC2 fluorescence at the phagosome decreased slowly. Quantification of mcherry-lactC2 phagosomal accumulation was performed by measuring the ratio of phagosomal to cytosolic fluorescence. As shown in Figure 4B, the fluorescence ratio decreased from 1.61 at time 0 to 1.26 within 20 min. Statistical analysis shows that this ratio was significantly reduced 15 and 20 min after the phagosome closure, revealing a slow decrease of the phosphatidylserine content in the cytosolic leaflet of the phagosomal membrane. These results indicate that, in PLB-985 cells, the phagosomal accumulation of PS was slowly reduced to reach nearly 50% reduction 20 min after the phagosome closure.

LactC2 expression disturbs p47-GFP accumulation at the phagosome

As the phagosome was enriched in the lactC2 probe especially during the first 20 min after closure, lactC2 could mask phagosomal PS similar to the effects seen with whole lactadherin [START_REF] Hanayama | Identification of a factor that links apoptotic cells to phagocytes[END_REF][START_REF] Shi | Lactadherin inhibits enzyme complexes of blood coagulation by competing for phospholipid-binding sites[END_REF][START_REF] Asano | Masking of phosphatidylserine inhibits apoptotic cell engulfment and induces autoantibody production in mice[END_REF]. Therefore lactC2 may interfere with p47 phox and Rac2 phagosomal membrane targeting. The effect of the lactC2 probe on p47 phox phagosomal recruitment was assessed by transiently transfecting the stable cell line p47-GFP with mCherry-lactC2. Then the accumulation of p47-GFP at the phagosome during phagocytosis of opsonized zymosan was monitored by wide field videomicroscopy as before. P47-GFP was still recruited in the cells that expressed mCherry-lactC2 as revealed by the p47-GFP fluorescent ring around the phagosome (Fig. 5A). However the kinetics of p47-GFP accumulation was modified in these cells compared to control cells. In mCherry-lactC2 coexpressing cells p47 phox -GFP was still present at the phagosomal membrane 240 s after the phagosome closure whereas it had already disappeared at 120 s in cells that did not express the mCherry-lactC2 probe (Fig. 5B). To highlight the impact of lactC2 on p47-GFP translocation, the phagosomal recruitment of p47-GFP was modeled after a monocompartmental pharmacokinetics equation (see materials and methods section for detailed explanation of the method). Comparison of the constant of recruitment k2 between the two cell populations indicates that the recruitment of p47-GFP was significantly decelerated in cells expressing the lactC2 probe compared to control cells: k2 = 2.1 min -1 for control cells and 1.0 min -1 for lactC2 expressing cells (statistical Wald test, p = 0.029). Interestingly, the detachment of p47-GFP was also hindered in lactC2 expressing cells, as revealed by the diminution of the constant of detachment k1: 0.4 min -1 for control cells and 0.2 min -1 in lactC2 cells (statistical Wald test, p = 0.004). Thus, the lactC2 probe interfered with p47-GFP phagosomal accumulation. This is further illustrated by the increase in time needed to reach the maximal phagosomal recruitment of p47-GFP, Tmax, which was 0.6 min in control cells and 1.4 min in lactC2 expressing cells. However the maximal quantity of p47-GFP recruited at the phagosome (Rmax) appeared unaffected. Actually Rmax was 1.15 in control cells and 1.13 in lactC2 cells. These results indicate that the presence of the lactC2 probe results in a delay of p47-GFP dynamics.

LactC2 expression prevents Rac2 but not Rac1 accumulation at the phagosome

As the surface charge of membrane cytosolic leaflet is important of Rac recruitment [START_REF] Mclaughlin | The myristoyl-electrostatic switch: a modulator of reversible protein-membrane interactions[END_REF][START_REF] Yeung | Receptor activation alters inner surface potential during phagocytosis[END_REF][START_REF] Yeung | Membrane phosphatidylserine regulates surface charge and protein localization[END_REF], the presence of lactC2 at the phagosomal membrane may disturb its recruitment. To investigate the effect of lactC2 expression on GFP-Rac2G12V translocation to the phagosomal membrane, mCherry-lactC2 and GFP-Rac2G12V were transiently co-expressed in the cells. GFP-Rac2G12V accumulation at the phagosome was then monitored during phagocytosis of opsonized zymosan by spinning disk confocal microscopy. In the presence of mCherry-lactC2, GFP-Rac2G12V was no more recruited at the phagosomal membrane (Fig. 5C and video S3). The fluorescence ratio of phagosome against cytosol was lower than one, emphasizing the absence of GFP-Rac2G12V recruitment in mCherry-lactC2 expressing cells (Figure 5D). As Rac 2 is no more recruited at the phagosome in the presence of the lactC2 probe we wondered whether this was also the case for Rac1. We repeated the same experiment with GFP-Rac1G12V. As previously described [START_REF] Magalhaes | Pivotal Advance: Phospholipids determine net membrane surface charge resulting in differential localization of active Rac1 and Rac2[END_REF], GFP-Rac1G12V was present at the plasma membrane and decreased from the phagosome after sealing (Fig. 6A,C). In some videos, however, we could observe a very brief increase of the phagosomal fluorescence after the phagosome closure (Fig. 6A). A similar phenomenon was observed in the cells expressing mcherry-lactC2 (Fig. 6B, C and video S4). In both cases, Rac1G12V was present at the phagosome for at least 10 min (Fig. 6C). Thus Rac1G12V is present on the phagosome from the beginning and its presence is not affected by lactC2.

Binding of the lactC2 probe on phosphatidylserine induces a delay in the ROS production

As the recruitment of p47-GFP and GFP-Rac2G12V was modified, we wondered if the lactC2 probe could have an effect in ROS production. We have previously shown that DCFH 2 -yeast is a valuable tool to measure ROS production inside the phagosome by video-microscopy until 30 minutes [START_REF] Tlili | Stable accumulation of p67phox at the phagosomal membrane and ROS production within the phagosome[END_REF] with no photooxydation [START_REF] Tlili | Kinetic analysis of phagosomal production of reactive oxygen species[END_REF]. We could not use DCFH 2zymosan because due to its small size less DCFH2 is attached to the zymosan as compared to yeast and the saturation of the probe in the phagosome is reached after five minutes of ROS production (unpublished observations). Thus the effect of the lactC2 probe on the amplitude and kinetics of the ROS production was assessed with DCFH 2 -yeast by videomicroscopy. Phagocytosis of DCFH 2 -yeasts was filmed and the measure of the yeast fluorescence allowed quantifying cellular ROS production inside the phagosome (Fig. 7A,B). During the first minutes after the phagosome closure, the oxidative burst was reduced in mCherry-lactC2 expressing cells compared to mCherry control cells. This initial delay that occurred during the first 7 min following the phagosome closure was compensated at 10 min (Fig 7C). To analyze more precisely the kinetics of the ROS production in the two cell populations, expressing or not the mCherry-lactC2 probe, we took advantage of the Monolix software that is widely used for pharmacokinetics studies [START_REF] Lavielle | Estimation of population pharmacokinetic parameters of saquinavir in HIV patients with the MONOLIX software[END_REF]. This approach is based on the non linear mixed effects modeling of data (see materials and methods section for detailed explanation of the method). Because we measured an enzymatic activity, we selected a sigmoid model to describe the kinetics of ROS production. The time course of the median DCFH 2 -yeast fluorescence is well correlated between the experimental data and the values predicted by the model (Fig. S5). The time needed to reach half of the maximal fluorescence (Tc) for the control mCherry expressing cells and for the mCherry-lactC2 cells is significantly increased in cells that express the lactC2 probe : 6.3 min for mCherry expressing cells and 10.7 min for mCherry-lactC2 cells (statistical Wald test, p = 0.008). The initial reaction rate is also decreased more than 2 fold for the mCherry-lactC2 cells expressing cells (31 units/min) as compared to mcherry cells( 68 units/min) . These results indicate that the initial ROS production is significantly reduced in lactC2 expressing cells, probably by masking PS on the phagosomal membrane.

Discussion

Our results show that p47 phox and a tagged version of the dominant positive form of Rac2 are recruited to the phagosome for only 2-3 min after phagosome sealing. The presence of lactC2 at the phagosome disturbs p47-GFP accumulation and prevents GFP-Rac2G12V recruitment at the phagosome. In PLB-985 cells expressing lactC2 the ROS production is also reduced during the first seven minutes following phagosome sealing. As previously observed [START_REF] Ueyama | Sequential binding of cytosolic Phox complex to phagosomes through regulated adaptor proteins: evaluation using the novel monomeric Kusabira-Green System and live imaging of phagocytosis[END_REF][START_REF] Li | p47phox Phox homology domain regulates plasma membrane but not phagosome neutrophil NADPH oxidase activation[END_REF], p47-GFP was recruited at the phagosomal membrane just before phagosome closure and then accumulated at the phagosome. However, we observed that p47-GFP remained accumulated for only 2 min. This was unexpected since we had previously observed a stable association of p67-GFP at the phagosome for 45min under similar conditions of phagocytosis [START_REF] Tlili | Stable accumulation of p67phox at the phagosomal membrane and ROS production within the phagosome[END_REF]. Immunofluorescence experiments confirmed the transient translocation of endogenous p47 phox to phagosomes containing opsonized zymosan. Other immunofluorescence studies do not agree with this transient translocation of p47 phox however this may depend on the amount of zymosan in the experiment which is a key element for the synchronization of phagocytosis [START_REF] Allen | Transient association of the nicotinamide adenine dinucleotide phosphate oxidase subunits p47phox and p67phox with phagosomes in neutrophils from patients with X-linked chronic granulomatous disease[END_REF]. We could not detect any reproducible accumulation of wild type CFP-Rac2 or GFP-Rac2 at the phagosomal membrane. A discrete and brief enrichment of tagged Rac2 on the phagosome has been observed in Raw 264.7 cells [START_REF] Hoppe | Cdc42, Rac1, and Rac2 display distinct patterns of activation during phagocytosis[END_REF]. The visualisation of a phagosomal localisation of a tagged Rac2 depends on the amount of protein that is recruited and on its time of presence at the phagosome. Our fractionation experiments indicate that less than 5% of the CFP protein is translocated to the plasma membrane upon PMA stimulation (data not shown). The recruitment and the time of presence of Rac2 at the phagosome depend on the regulation of this small GTPase by RhoGDI, GEFs and GAPs and others factors such as the surface charge of the membrane. Rac proteins have a polybasic domain: the Rac2 polybasic domain is moderately charged (+3) whereas the Rac1 domain carries more positive charges (+6). We have shown here that a dominant positive mutant of Rac2, Rac2G12V, is recruited at the phagosomal cup just before sealing whereas the corresponding mutant of Rac1 accumulated at the plasma membrane and started to detach upon phagosome sealing. Similar results were obtained for the other dominant positive mutants Rac1Q61L and Rac2Q61L [START_REF] Yeung | Receptor activation alters inner surface potential during phagocytosis[END_REF][START_REF] Magalhaes | Pivotal Advance: Phospholipids determine net membrane surface charge resulting in differential localization of active Rac1 and Rac2[END_REF]. The accumulation of constitutively active Rac2 and the detachment of constitutively active Rac1 correlated with the decrease of the surface charge of the phagosome [START_REF] Yeung | Lipid signaling and the modulation of surface charge during phagocytosis[END_REF][START_REF] Magalhaes | Pivotal Advance: Phospholipids determine net membrane surface charge resulting in differential localization of active Rac1 and Rac2[END_REF]. Thus, the charges in the polybasic domain appeared to be important for the phagosomal accumulation of GTP bound Rac. We showed that Rac2G12V is only accumulated at the phagosome for 3 min after phagosome closure and leaves the phagosome well before termination of ROS production that lasted more than 10 min (Fig. 7C). Magalhaes et al. observed that Rac2Q61L was still at the phagosomal membrane 30 min after particle internalization [START_REF] Magalhaes | Pivotal Advance: Phospholipids determine net membrane surface charge resulting in differential localization of active Rac1 and Rac2[END_REF]. This difference may be explained by the differential regulation of the two dominant positive mutant proteins. Both mutations inhibit the intrinsic and GAP-stimulated GTP hydrolysis [START_REF] Xu | Differing structural requirements for GTPase-activating protein responsiveness and NADPH oxidase activation by Rac[END_REF]. However RacG12V, like wild type Rac and contrary to RacQ61L, can interact with GEF and with RhoGDI [START_REF] Xu | Guanine nucleotide binding properties of Rac2 mutant proteins and analysis of the responsiveness to guanine nucleotide dissociation stimulator[END_REF][START_REF] Ugolev | Liposomes comprising anionic but not neutral phospholipids cause dissociation of Rac(1 or 2) x RhoGDI complexes and support amphiphile-independent NADPH oxidase activation by such complexes[END_REF]. Accordingly, the regulation of Rac2G12V would be closer to that of wild type Rac2. We were unable to identify an anti-Rac2 antibody suitable for immunostaining to examine the translocation of endogenous Rac2. To monitor PS localization and decrease its accessibility, we used the lactC2 probe that interacts specifically with PS [START_REF] Yeung | Membrane phosphatidylserine regulates surface charge and protein localization[END_REF]. As previously observed [START_REF] Yeung | Contribution of phosphatidylserine to membrane surface charge and protein targeting during phagosome maturation[END_REF][START_REF] Magalhaes | Pivotal Advance: Phospholipids determine net membrane surface charge resulting in differential localization of active Rac1 and Rac2[END_REF][START_REF] Minakami | Membrane phospholipid metabolism during phagocytosis in human neutrophils[END_REF], the lactC2 probe marked the cytosolic leaflet of the phagosome. In the neutrophil-like PLB-985 cells, we observed that lactC2 remained accumulated at the phagosome after particle internalization and that the ratio phagosome/cytosol slowly decreased over 20min. The PS content of the phagosome is probably modulated by PS metabolism and vesicular traffic. The removal of membrane material enriched in PS, from macrophage phagosomes has been described [START_REF] Yeung | Contribution of phosphatidylserine to membrane surface charge and protein targeting during phagosome maturation[END_REF]. Lactadherin and its C2 domain bind PS with high affinity and selectivity [START_REF] Andersen | Functional analyses of two cellular binding domains of bovine lactadherin[END_REF][START_REF] Shi | Lactadherin binds selectively to membranes containing phosphatidyl-L-serine and increased curvature[END_REF][START_REF] Shao | Crystal structure of lactadherin C2 domain at 1.7A resolution with mutational and computational analyses of its membrane-binding motif[END_REF] and an extracellular mutant of lactadherin could mask PS on the outer leaflet of the plasma membrane [START_REF] Hanayama | Identification of a factor that links apoptotic cells to phagocytes[END_REF][START_REF] Shi | Lactadherin inhibits enzyme complexes of blood coagulation by competing for phospholipid-binding sites[END_REF]. Therefore, lactC2 inside the cell could have a dual role serving as a probe to monitor PS but also masking PS. LactC2 masks probably a few percent of the PS, which has an estimated cellular concentration of 100µM [START_REF] Kay | Phosphatidylserine dynamics in cellular membranes[END_REF], however a substantial part of the PS may be associated with protein complexes [START_REF] Kay | Phosphatidylserine dynamics in cellular membranes[END_REF] and consequently not be accessible to the lactC2 probe and other endogenous proteins. So, the presence of the lactC2 probe decreases the fraction of "accessible PS". Indeed, we could show that the lactC2 probe interferes with cellular functions, it modifies Rac2G12V and p47-GFP recruitment and reduces ROS production during the first seven minutes after the phagosome closure. The reduced PS accessibility could explain these defects. Our results suggest that lactC2 prevents Rac2 recruitment at the phagosome. Absence of phagosomal Rac2 may explain the reduced initial ROS production. One might expect that the absence of Rac2 leads to a stronger defect. However it has been shown that neutrophils expressing a dominant negative form of Rac2 have only a modest defect in O2 .-production in response to opsonized zymosan [START_REF] Ambruso | Human neutrophil immunodeficiency syndrome is associated with an inhibitory Rac2 mutation[END_REF][START_REF] Williams | Dominant negative mutation of the hematopoietic-specific Rho GTPase, Rac2, is associated with a human phagocyte immunodeficiency[END_REF]. Furthermore, Rac1 is present in our cells. We observed that, in cells expressing or not mcherry-lactC2, GFP-Rac1G12V is present at the phagosome during its formation and then starts to detach from the phagosome after sealing. Thus Rac1 can probably partly support Rac2 function, when Rac2 is absent. Rac1 localization at the plasma membrane is not affected by lactC2 since Rac1 has been shown to interact preferentially with PS [START_REF] Finkielstein | Cell migration and signaling specificity is determined by the phosphatidylserine recognition motif of Rac1[END_REF] and may have a better affinity than lactC2 probe for the negative cytosolic leaflet of the plasma membrane due to its larger polybasic domain. . P47 phox , via its PX domain, binds PS [START_REF] Karathanassis | Binding of the PX domain of p47(phox) to phosphatidylinositol 3,4bisphosphate and phosphatidic acid is masked by an intramolecular interaction[END_REF][START_REF] Stahelin | Membrane binding mechanisms of the PX domains of NADPH oxidase p40phox and p47phox[END_REF]. The specific role of PS in the phagosomal translocation of p47 phox has not been established. In the presence of mCherry-lactC2 at the phagosome, the recruitment of p47-GFP is slowed down and its time of presence at the phagosome is increased two fold. The mathematical model suggests that attachment as well as removal of p47 phox from the phagosome is slowed down. The PX domain alone does not accumulate at the phagosomal membrane ( [START_REF] Johnson | Gene transfer and expression in human neutrophils. The phox homology domain of p47phox translocates to the plasma membrane but not to the membrane of mature phagosomes[END_REF] and unpublished data). However, our data suggest that this domain is important for the kinetics of p47 phox accumulation at the phagosome: by reducing the accessibility to PS, we slowed down p47 phox recruitment. Masking PS with lactC2 may also have indirect effects on the accumulation of p47-GFP at the phagosomal membrane by interfering with other proteins. For example, the phosphorylation of p47 phox by PKC may be reduced because the PS binding domain (C2 domain) of PKC isoforms is important for its localization and activation [START_REF] Conesa-Zamora | Identification of the phosphatidylserine binding site in the C2 domain that is important for PKC alpha activation and in vivo cell localization[END_REF]. The mutant p47R90A, which has a reduced affinity for PI(3,4)P 2 but binds to PA and PS, translocated to the phagosomal membrane and restored ROS production of p47 phox KO mouse neutrophils [START_REF] Li | p47phox Phox homology domain regulates plasma membrane but not phagosome neutrophil NADPH oxidase activation[END_REF]. The translocation of this mutant presumably relies on a direct interaction with PA, PS and p22 phox . Taken together, these observations confirm the concept that membrane translocation of p47 phox is based on a multicomponent interaction system [START_REF] Steinberg | Pathogen destruction versus intracellular survival: the role of lipids as phagosomal fate determinants[END_REF] to insure maximal flexibility. PS would be one of the targeting motifs of this system.

To summarize, we propose the following model (Fig. 8): Rac2 and p47 phox would be recruited at the same time at the phagosomal cup just before phagosome sealing (Fig. 8A). The surface charge of the phagosome conferred by PPI, PA and PS is important for Rac2 recruitment. The Rac2 protein would serve as an adapter to correctly position p67 phox toward gp91 phox ; (Fig. 8B) then it detaches from the phagosome a few minutes after sealing (Fig. 8C). P47 phox , via its PX domain binding to PS and PI(3,4)P 2 and its SH3 domain binding to p22 phox , would act as a carrier protein, targeting p67 phox and p40 phox to the NADPH oxidase complex (Fig. 8A,B). Once the complex is assembled p47 phox detaches a few minutes after phagosomal closure (Fig. 8C). PI(3,4)P 2 accumulates at the phagosome during its formation and remains present during the first minute after closure, whereas PI(3)P starts to accumulate 1 min after phagosome sealing (Fig8. B, C).. P67 phox would remain assembled with the membrane subunits via its interaction with gp91 phox and p40 phox that binds to PI(3)P (Fig. 8C). A reduced PS level on the phagosomal membrane should affect the ROS production inside the phagosome and may contribute to intraphagosomal survival of pathogens. Indeed, certain pathogens such as Listeria pneumophilia are able to decrease the PS content of the phagosome within a few hours [START_REF] Yeung | Contribution of phosphatidylserine to membrane surface charge and protein targeting during phagosome maturation[END_REF].

Materials and methods

Cell culture

The human myeloid leukemia cell line PLB-985 was a generous gift from Marie-José Stasia (Grenoble, France). Cells were cultured as previously described [START_REF] Tlili | Kinetic analysis of phagosomal production of reactive oxygen species[END_REF]. The differentiation towards neutrophil-like cells was induced by addition of 1.25% dimethylsulfoxide (DMSO) to exponentially growing cells for 5 or 6 days. For all experiments, differentiated PLB-985 cells were centrifuged 3 min at 2000 rpm and resuspended in Hepes buffer containing 140 mM NaCl, 5 mM KCl, 1 mM MgCl 2 , 2 mM CaCl 2 , 10 mM Hepes (pH 7.4), 1,8 mg/ml glucose and heat inactivated FBS (1% for microscopy experiments, otherwise 5%).

Plasmid constructs

The human p47 phox gene (kind gift from Marie-Claire Dagher, Grenoble, France) was amplified by PCR by using primers designed to discard the stop codon of this gene and to introduce HindIII and AgeI restriction sites. By using the Quickchange II Site Directed Mutagenesis kit (Agilent technologies, Santa Clara, CA), the start codon of the pEGFP-N1 vector (Clontech, Palo Alto, CA) was deleted in order to avoid expression of the GFP alone. The p47-GFP vector was then constructed by cloning the p47 phox fragment into the mutated pEGFP-N1 plasmid. The cDNA encoding human Rac2 was amplified from pGEX2T-Rac2 (gift from Marie-Claire Dagher, Grenoble, France) by PCR and the PCR fragment was then cloned in the pECFP-C1 vector using the restriction sites HindIII-SacII. The GFP-Rac2G12V and GFP-Rac1G12V vectors were obtained by cloning Rac2G12V or Rac1G12V into the pEGFP-C1 vector (Clontech). The human lactadherin C2 domain (475bp) was amplified by PCR from a vector pBluescript II KS+BA46 (kind gift from C. Théry, Paris, France). The primers used contained restriction sites (Hind III and Sac II) to allow the cloning into the pECFP-C1 vector (Clontech). MCherry was amplified from p413GPDmCherry (kind gift from R. Tsapis, Paris, France) by PCR and the pmCherryC1 vector was obtained by replacing EGFP in the pEGFP-C1 vector by mCherry, using the restriction site AgeI-HindII. The mCherry-lactC2 vector was constructed by inserting the lactC2 (Photometrics) driven by the Metamorph 7 software (Universal imaging). The temperature was maintained at 37°C using a stage incubator. GFP and mCherry were excited at 491 nm and 561 nm respectively by using a laser illumination for 0.3 and 0.2 s and the fluorescence was detected with a quad band beamsplitter, a 525 nm emission filter for GFP and a 607 nm emission filter for mCherry. Sequential stacks of images were collected for several minutes with a time lapse of 10 s or 30s and an optical spacing of 1 µm.

Fluorescence quantification: intraphagosomal ROS production and recruitment of p47 phox and Rac2G12V

All images were analyzed with the ImageJ software. To measure DCF fluorescence, ROIs that cover the whole yeast were manually drawn and the mean fluorescence intensity of these ROIs was calculated for each time point. The accumulation of lactC2 at the phagosomal membrane was measured by calculating a fluorescence ratio between the mean fluorescence intensity of the phagosomal ring and the mean cytoplasmic fluorescence intensity. The fluorescence intensity of non phagocyting cells remained stable indicating that there was no photobleaching during the experiments. To quantify p47-GFP fluorescence without artifacts due to volume variations when using wide field microscopy, ratiometric films were obtained with the cytosolic marker calcein blue. The ratio between the GFP and the calcein blue fluorescence was determined after background subtraction. These ratiometric films were then used to analyze p47 phox translocation to the phagosome. P47-GFP, GFP-Rac2G12V or GFP-Rac1G12V phagosomal membrane recruitment were determined by evaluating the accumulation of the fusion proteins at the phagosomal membrane relative to the cytoplasm, as described above for the lactC2 probe. A fluorescence ratio higher than 1 indicates that the protein is accumulated at the phagosomal membrane.

Immunofluorescence

The immunofluorescence experiments were performed as described before [START_REF] Tlili | Stable accumulation of p67phox at the phagosomal membrane and ROS production within the phagosome[END_REF]. Briefly, the cells were mixed with opzonised Texas Red zymosan, spun down at 13°C to synchronize phagocytosis and then incubated at 37°C for 0 to 10 min. The cells were fixed with 4% paraformaldehyde, permeabilized with 0,1% tritonX100 in PBS and blocked with 10% decomplemented human serum for 1h at 37°C. The cells were then immunostained with anti-p47 phox antibody (BD Bioscience, diluted 1:100) followed by Alexa 488 goat anti-mouse (Life Technologies). Cells were imaged on a spinning disk microscope described above.

Subcellular fractionation, immunoprecipitation and western blotting

To analyze the expression of the fusion protein in the stable p47-GFP or CFP-Rac2 expressing cell line, cells were disrupted in lysis buffer (40000 cells/µl) containing Hepes-Na 25 mM, NaCl 150 mM, MgCl 2 5 mM, EGTA 0.5 mM, TritonX100 0.5%, NaF 10 mM and supplemented with a protease inhibitor cocktail (Roche, Basel, Switzerland). Protein concentration was determined using the BioRad Protein assay (BioRad) and 25 µg of the samples were diluted in 6x Laemmli buffer and boiled for 5 min. The proteins were separated on a 12% SDS-PAGE gel and transferred onto nitrocellulose membrane (BioRad, Marnes La Coquette, France). The membranes were blotted with anti-GFP monoclonal antibody (Roche, diluted 1:500) or anti-p47 phox monoclonal antibody (BD Bioscience, diluted 1:1000) or anti-Rac2 rabbit polyclonal antibody (diluted 1:1000, Millipore) followed by incubation with horseradish peroxidase conjugated secondary antibodies (GE Healthcare). Proteins were detected by using the ECL reagent (GE Healthcare Europe, Orsay, France) and the image acquisition system Fusion Fx7 (Fisher Scientific, Illkirch, France). Relative protein quantification on western blot was done using BioD1 software (Fisher Scientific). Subcellular fractionation was adapted from Dang et al. [START_REF] Dang | Priming of human neutrophil respiratory burst by granulocyte/macrophage colony-stimulating factor (GM-CSF) involves partial phosphorylation of p47(phox)[END_REF]. The two differentiated cells lines CFP-Rac2 and p47-GFP were incubated at 37°C first with 5 µM of cytochalasinB for 10 min and then stimulated with 200 nM PMA for 10 min. The stimulation was stopped on ice by addition of ice cold PBS and cells were resuspended in relaxation buffer (100 mM KCl, 3 mM NaCl, 3.5 mM MgCl 2 , 10 mM Pipes KOH pH7.3, and 1mM EGTA) supplemented with protease inhibitors and NaF(10mM). Cells were disrupted by sonication for 3 x 10 s on ice, and the unbroken cells and nuclei were removed by a 400 g centrifugation for 8 min. The supernatants were transferred to a sucrose gradient (16%-32%) and ultracentrifuged at 200000 g for 30 min (Centrikon T-2050, Kontron Medical, S t Germain en Laye, France). The plasma membrane fraction was collected and ultracentrifuged at 200000g for 30 min. The membrane pellet was resuspended in relaxation buffer with 1% triton and incubated 30 min at 4°C to solubilize the proteins. Then 5 µg of mouse monoclonal anti-gp91 phox (Abcam, Cambrige, UK) were added and the mixture was incubated at 4°C for 1 h. For negative controls, antibodies were omitted. Protein G-agarose beads were added and the mixture was again incubated for 1 h at 4°C. The beads were collected, washed with relaxation buffer with 1% triton, then resuspended in 50 µl of 2x Laemmli sample buffer, boiled and centrifuged at 14 000 g for 5 min. The supernatants were subjected to SDS-PAGE followed by immunoblotting with anti-gp91 phox (diluted 1:1000), anti-p47 phox , anti-GFP, anti-p67 phox (Millipore, Bedford, MA) and anti-Rac2.

Statistical analysis and mathematical models

All data are presented as mean  SEM. Statistical analysis of the quantification of the LactC2 probe at the phagosomal membrane was performed using a Mann Whitney test with GraphPad Prism 5 software. Values of p < 0.05 were considered statistically significant. The ROS production kinetics over 30 min measured by videomicroscopy was analyzed using nonlinear mixed effects models. We assumed that the production of ROS by NADPH oxidase corresponds to an enzymatic kinetics. Thus we have tested several sigmoid models to describe the ROS production and selected the function: F(t) = Fmin + (Fmax-Fmin)/(1+e -α(t-Tc) ), where F(t) is the fluorescence at time t, Fmin is the fluorescence base line, Fmax is the maximal fluorescence, Tc is the time when 50% of the maximal fluorescence increase is reached, and α is proportional to the slope of the tangent at Tc. A lactC2 effect was assumed to occur on parameters α and Tc, and this effect was modelled as α lactC2 = α control exp(β α ), and Tc lactC2 = Tc control exp(β Tc ). Statistical analysis was performed using the Monolix software (http://www.monolix.org/, user guide) that is mainly dedicated to pharmacokinetics-pharmacodynamics applications in a population approach. The parameters of the structural model (i.e. Fmin, Fmax, α and Tc) are assumed to be random variables in the population. We assumed that Fmin is constant in each condition, and Fmax, α and Tc are random variables that have a log-normal distribution. The Wald nullity test was used to test if βα=0 or βTc=0, i.e. to compare the logTc and logα values in the two conditions and to statistically determine whether the kinetics of the two cell populations is different or not. Here, p-values < 0.05 indicate that the observed differences for Tc were statistically significantThen we have simulated data (N=100000) and calculated the two simulated cell population parameters that are representative of experimental data, by using Monte-Carlo methods. In order to illustrate the statistical difference observed between the two cell populations, the initial reaction rate at the time of phagosome closure was calculated in each cell population by deriving the function F(t) at time 0 min.

To analyze the effect of the lactC2 probe on p47 phox phagosomal recruitment, the same strategy described above was applied. The statistical model used to describe p47 phox recruitment derives from a monocompartmental pharmacokinetics model: R(t) = R inf + A*e (-k 1 t) -(R inf + A -R 0 )*e (-k 2 t) , where R inf is the fluorescence ratio base line, A is a factor that determines the amplitudes of the two exponentials, R 0 is the ratio at time of phagosome closure, k 1 is a constant of detachment of p47 phox and k 2 is a constant of its recruitment. We assumed that a lactC2 effect occurred on parameters k 1 and k 2 . The parameters R inf , A, R 0 , k 1 and k 2 are random variables that have a log-normal distribution. By using the Monolix software, the parameters were estimated. Data simulation was performed by Monte Carlo method to calculate the time when maximal fluorescence ratio (R max ) was reached (T max ). This time can be defined as the point when the derivative of the function is null.

Figure legends

Figure 1 Generation of stable PLB-985 cell lines expressing p47-GFP and CFP-Rac2 (A, B, C) Generation of p47-GFP and CFP-Rac2 expressing stable cell lines: (A) Western blot analysis of p47-GFP (left panel) and CFP-Rac2 (right panel) using an anti-GFP antibody. (B) Western blot analysis of p47-GFP using an anti-p47 phox antibody. (C) Western blot analysis of CFP-Rac2 using an anti-Rac2 antibody. (D, E): Differentiated PLB-985 cells expressing p47-GFP (D) or CFP-Rac2 (E) were stimulated for 10 min with PMA and then lysed by sonication. Membrane and cytosol were separated by ultracentrifugation on sucrose gradient. Membrane fractions were immunoprecipitated (IP) with antibodies specific for gp91 phox . The negative control contained no antibody during precipitation. Immunoprecipitates were resolved by SDS-PAGE followed by immunoblotting with anti-gp91 phox , anti-p47 phox , anti-Rac2, anti-GFP and anti-p67 phox antibodies. The star marks the heavy chain of Immunoglobulin G anti-gp91 phox (A) Differentiated PLB-985 cells were transiently transfected with mCherry-lactC2, and phagocytosis of yeasts (*) by the transfected cells was imaged by wide field videomicroscopy, followed by a 3D-deconvolution analysis. Time 0 was set when the phagosome closure was observed. Scale bar, 5 µm. (B) Quantification of lactC2 accumulation at the phagosomal membrane was evaluated by measuring the phagosomal fluorescence relative to the cytosolic fluorescence. Data are mean ± SEM from at least seven phagosomes for each time point, * p < 0.05.

Figure 2 .

 2 Figure 2. Recruitment of p47 during phagocytosis of opsonized zymosan (A, B, C) Wide-field videomicroscopy was used to monitor phagocytosis of zymosan (*) by p47-GFP expressing cells. (A) Representative micrographs of ratiometric films between p47-GFP and the cytosolic marker calcein blue. Scale bar, 5 µm. (B) The time of the first appearance of p47-GFP on the phagosome membrane was analyzed for 13 positive phagosomes. The time of phagosome closure is defined as time zero. The mean is represented by a black line. (C) P47-GFP accumulation at the phagosomal membrane was quantified by calculating the relative fluorescence intensity at the phagosomal membrane compared to the cytosol on the ratiometric films. At least ten phagosomes per condition were analyzed and are shown as mean ± SEM. Time 0 (black arrow) corresponds to the phagosome closure. (D, E) Immunofluorescence experiments were performed to detect endogenous p47 phox -positive phagosomes at 0, 2, 5 and 10 min after incubation of the cells with texas red opsonized zymosan at 37°C. (D) Representative micrographs at time 2 and 10 min are shown. The phagosomes are indicated by arrows. The arrow in the left micrograph shows a p47 phox -positive phagosome. (E, left panel) Percentage of phagosomes ((number of phagosomes/number of cells)*100) at the different time points. For each condition, in each experiment, at least 150 cells have been observed. (E, right panel) Percentage of p47 phox -positive phagosomes at the different time points. The results represent the mean of at least three independent experiments.

Figure 3 .

 3 Figure 3. Recruitment of GFP-Rac2G12V during phagocytosis of opsonized zymosan Cells were transiently transfected with GFP-Rac2G12V. (A) Representative micrographs of zymosan (*) phagocytosis by transfected cells acquired by spinning-disk confocal videomicroscopy. Time 0 was set when the phagosome closure was observed. Scale bar, 5 µm. (B) The time of the first appearance of GFP-Rac2G12V on phagosome membrane was analyzed for 11 positive phagosomes. The time of phagosome closure is defined at time zero. The mean is represented by a black line. (C) GFP-Rac2G12V accumulation at the phagosomal membrane was quantified by calculating the relative fluorescence intensity at the phagosomal membrane compared to the cytosol. Data are mean ± SEM from at least seven phagosomes per condition. Time 0 (black arrow) corresponds to the phagosome closure.

Figure 4 .

 4 Figure 4. Localization of lactC2 decorated phosphatidylserine during phagocytosis.(A) Differentiated PLB-985 cells were transiently transfected with mCherry-lactC2, and phagocytosis of yeasts (*) by the transfected cells was imaged by wide field videomicroscopy, followed by a 3D-deconvolution analysis. Time 0 was set when the phagosome closure was observed. Scale bar, 5 µm. (B) Quantification of lactC2 accumulation at the phagosomal membrane was evaluated by measuring the phagosomal fluorescence relative to the cytosolic fluorescence. Data are mean ± SEM from at least seven phagosomes for each time point, * p < 0.05.

Figure 5 .

 5 Figure 5. Recruitment of p47-GFP and CFP-Rac2 in the presence of mcherry-lactC2 (A, B) The p47-GFP transgenic cell line was transiently transfected with mCherry-lactC2. Widefield videomicroscopy was used to monitor phagocytosis of zymosan (*). (A) Representative micrographs of ratiometric films between p47-GFP and the cytosolic marker calcein blue. Scale bar, 5 µm. (B) Quantification of p47 accumulation at the phagosome in the absence (black line) or presence of mcherry-lactC2 (red line). At least ten phagosomes per condition were analyzed and are shown as mean ± SEM. Time 0 (black arrow) corresponds to the phagosome closure. (C, D) Cells were transfected with GFP-Rac2G12V alone and mCherry-lactC2. (C) Representative micrographs of zymosan (*) phagocytosis by transfected cells were acquired by spinning-disk confocal videomicroscopy. Time 0 was set when the phagosome closure was observed. Scale bar, 5 µm. (C) Quantification of GFP-Rac1G12V at the phagosomal membrane in the absence (black line) or presence of mcherry-lactC2 (red line). Data are mean ± SEM from 8 phagosomes per condition. Time 0 corresponds to the phagosome closure.

Figure 6 .

 6 Figure 6. Transient accumulation of GFP-Rac1G12V at the phagosome in the absence or presence of mCherry-lactC2 (A, B) Cells were transfected with GFP-Rac1G12V alone (A) or together with mCherry-lactC2 (B). Representative micrographs of zymosan (*) phagocytosis by transfected cells were acquired by spinning-disk confocal videomicroscopy. Time 0 was set when the phagosome closure was observed. Scale bar, 5 µm. (C) Quantification of GFP-Rac1G12V at the phagosomal membrane. Data are mean ± SEM from eight phagosomes per condition. Time 0 corresponds to the phagosome closure.

Figure 7 .

 7 Figure 7. Effect of lactC2 expression on kinetics of ROS production. (A) Cells were transfected with mcherry (upper panel) or mcherry-lactC2 (lower panel). Images of phagocytosis by transfected cells at 0 to 10 min after phagosome closure showing mcherry or mcherry-lactC2 (red) and DCFH 2 yeast (green). In the lower panel some DCFH 2 -yeasts outside the mcherry positive cell present a strong fluorescence since they have been phagocytosed by neighboring cells that are negative for mcherry-lactC2. The phagosomes are indicated by an arrow. Time 0 was set when the phagosome closure was observed. Scale bar, 5 µm. (C) The kinetics of the ROS production was assessed by videomicroscopy by quantifying DCFH 2 -labelled yeast fluorescence in cells with mcherry-lactC2 or mcherry. Time course of the average phagosomal fluorescence (± SEM) during the first 12 min from at least ten DCFH 2 labeled yeasts for each time point of both conditions.

Figure 8 .

 8 Figure 8. Model of NADPH oxidase assembly and phosphatidylserine involvement during phagocytosisA model for NADPH oxidase activation on the phagosomal membrane based on the literature and the data in this article is shown in 3 steps from left to right: The distribution of NADPH oxidase subunits prior to activation (A), at the beginning of NADPH oxidase activity (B) and during continuous activity beyond 3 to 5 min after phagosome closure (C). The presence of 3 different anionic phospholipids is shown by colored head groups.

Acknowledgement

cDNA for the human p47 phox gene and for the human lactadherin C2 domain were kind gifts from Marie-Claire Dagher, Grenoble, France and C. Théry, Paris, France. MCF was a recipient of an AFR PhD Grant from the Fonds National de la Recherche -Luxembourg. We wish to thank Sebastien Plançon for help. This work has benefited from the facilities and expertise of IMAGIF (Centre de recherche de Gif); we thank Marie Noelle Soler for her help on the spinning disk confocal. We also thank Marie Erard for her help and her critical reading of the manuscript.

Funding

This work was supported by grants from the French research ministry, programs ACI DRAB and Microbiology, the French League against cancer and the BQR-program of University Paris-Sud.

domain from pECFP-C1lactC2 into the pmCherryC1 vector at the restriction site Hind III, Sac II. All constructs were controlled by sequencing.

Transient and stable transfection

Differentiated PLB-985 cells were transiently transfected using the Nucleofector device or the 4D-Nucleofector (Lonza, Basel, Switzerland) according to the manufacturer's protocol. For each condition, using the Nucleofector kit for human dendritic cells and the U-015 program or the SF Cell Line Kit and the EH-100 program, 2.10 6 cells were nucleofected with 1 to 4 µg of vector and incubated in culture medium without antibiotics. Four hours post-transfection, cells were processed for experiments. CFP-Rac2 and p47-GFP expressing PLB-985 cell lines were generated as previously described [START_REF] Tlili | Stable accumulation of p67phox at the phagosomal membrane and ROS production within the phagosome[END_REF]. These cell lines were cultured continuously in the presence of 0.5 mg/ml of G418 to maintain the selection.

Preparation and opsonization of yeasts, DCFH2-yeasts and zymosan

Yeasts (Saccharomyces cerevisiae) and zymosan particles from Saccharomyces cerevisiae (Sigma, Saint Louis, MO) were prepared as described previously [START_REF] Tlili | Kinetic analysis of phagosomal production of reactive oxygen species[END_REF]. To measure the intraphagosomal ROS production, yeasts and zymosan were covalently labeled with 2', 7'dichlorodihydrofluorescein diacetate, succinimidyl ester (OxyBURST Green H2DCFDA, succinimidyl ester from Life Technologies, Saint Aubin, France) as described [START_REF] Tlili | Kinetic analysis of phagosomal production of reactive oxygen species[END_REF]. Yeasts (10 8 /ml), DCFH 2 -yeasts were washed and resuspended in PBS (same concentration) before opsonization by incubation with polyclonal rabbit anti-yeast serum (50% diluted, 1 hour, 37°C). Opsonized yeasts were washed twice with PBS, resuspended in Hepes buffer (5x10 7 /ml). Zymosan or DCFH 2 -zymosan were opsonized with total human serum following the same protocol as for yeasts and resuspended in Hepes buffer (20 mg/ml).

Cell preparation for microscopy

For some experiments, cells (2x10 6 /ml) were loaded with 2 µM of the cytoplasmic cell tracer calcein blue-AM (Life Technologies) during 10 min at room temperature and then washed once. For in vivo imaging, 50 µl of cells (4x10 6 /ml in saline media) were allowed to adhere on glass coverslips for 5 min at 37°C and 450 µl of warm Hepes buffer were added.

Live cell imaging

The localization of the mCherry-lactC2 probe was assessed at 37°C by using a DM IRE2 microscope (Leica, Nanterres, France) with a 100x plan Apo 1.4 oil immersion objective, a stage incubator and a Cool SNAP HQ2 camera (Photometrics, Huntington Beach, CA) driven by the Metamorph 7 software (Universal imaging,Westchester, PA). Stacks of 9 frames with a z-optical spacing of 1 µm were collected every minute for 20 min. Images were deconvoluted with Metamorph 7 software and single z planes were extracted. To monitor the intraphagosomal ROS production and p47-GFP phagosomal recruitment in cells co-expressing or not mCherry-lactC2, images were acquired by using a wide field fluorescence microscope described elsewhere [START_REF] Tlili | Kinetic analysis of phagosomal production of reactive oxygen species[END_REF]. Serial images were recorded every 5 s for about 20 min, except for the DCF fluorescence which was recorded every 2 s with binning 2 for up to 40 min. Phagocytosis of opsonized zymosan by GFP-Rac2G12V or GFP Rac1G12V cells cotransfected or not with mCherry-lactC2, was filmed at 37°C by using a spinning disk confocal system (Yokogawa CSU-X1-A1, Yokogawa Electrics, Japan) mounted on a Nikon Eclipse Ti E inverted microscope equipped with a 100x plan Apo 1,4 oil immersion objective and an EM-CCD eVolve camera

Author contribution

Online Supplementary Material

Video S1 : P47-GFP is transiently recruited to the phagosome Ratiometric film between p47-GFP and the cytosolic marker calcein blue during zymosan (*) uptake. Scale bar 5µm.


Mourad Choulli
email: mourad.choulli@univ-lorraine.fr

Laurent Kayser ¶
email: laurent.kayser@univ-lorraine.fr

Yavar Kian
email: yavar.kian@univ-amu.fr

AND Eric Soccorsi
email: eric.soccorsi@univ-amu.fr

§institut Élie

¶institut Élie

HEAT TRACE ASYMPTOTICS AND COMPACTNESS OF ISOSPECTRAL POTENTIALS FOR THE DIRICHLET LAPLACIAN

Keywords:

Let Ω be a C ∞ -smooth bounded domain of R n , n ≥ 1, and let the matrix a ∈ C ∞ (Ω; R n 2 ) be symmetric and uniformly elliptic. We consider the L 2 (Ω)-realization A of the operator -div(a∇•) with Dirichlet boundary conditions. We perturb A by some real valued potential V ∈ C ∞ 0 (Ω) and note A V = A + V . We compute the asymptotic expansion of tr e -tA V -e -tA as t ↓ 0 for any matrix a whose coefficients are homogeneous of degree 0. In the particular case where A is the Dirichlet Laplacian in Ω, that is when a is the identity of R n 2 , we make the four main terms appearing in the asymptotic expansion formula explicit and prove that L ∞ -bounded sets of isospectral potentials of A are H s -compact for s < 2.

Introduction

In the present paper we investigate the compactness issue for isospectral potentials sets of the Dirichlet Laplacian by means of heat kernels asymptotics.

1.1. Second order strongly elliptic operator. Let a = (a ij ) 1≤i,j≤n be a symmetric matrix of R n 2 , n ≥ 1, with coefficients in C ∞ (R n ). We assume that a is uniformly elliptic, in the sense that there is a constant µ ≥ 1 such that the estimate (1.1)

µ -1 ≤ a(x) ≤ µ, holds for all x ∈ R n in the sense of quadratic forms on R n .

1

We consider a bounded domain Ω ⊂ R n , with C ∞ boundary ∂Ω and introduce the selfadjoint operator A generated in L 2 (Ω) by the closed quadratic form

(1.2) a[u] = Ω a(x)|∇u(x)| 2 dx, u ∈ D(a) = H 1 0 (Ω),
where H 1 0 (Ω) is the closure of C ∞ 0 (Ω) in the topology of the standard first-order Sobolev space H 1 (Ω). Here ∇ stands for the gradient operator on R n . By straightforward computations we find out that A acts on its domain D(A) = H 2 (Ω) ∩ H 1 0 (Ω), as

(1.3) A = -div(a(x)∇•) = - n i,j=1
∂ j (a ij (x)∂ i ).

Let V ∈ C ∞ 0 (R n ) be real-valued. We define the perturbed operator A V = A + V as a sum in the sense of quadratic forms. Then we have D(A V ) = D(A) by [START_REF] Reed | Methods of Modern Mathematical Physics IV: Analysis of Operators[END_REF][Theorem X.12].

1.2. Main results. Put (1.4) Z V Ω (t) = tr e -tAV -e -tA , t > 0. Much of the technical work developped in this paper is devoted to proving the existence of real coefficients c k (V ), k ≥ 2, such that following symptotic expansion (1.5) Z V Ω (t) = t -n/2 tc 2 (V ) + t 3/2 c 3 (V ) + . . . + t k/2 c k (V ) + O t k/2+1/2 , t ↓ 0, holds for a homogeneous of degree 0. In the peculiar case where a is the identity matrix then (1.5) may be refined, providing (1.6) Z V Ω (t) = t -n/2 td 1 (V ) + t 2 d 1 (V ) + . . . + t p d p (V ) + O t p+1 , t ↓ 0, where d k (V ), k ≥ 1, is a real number depending only on V . Moreover we shall see that (1.4)-(1.5) remain valid upon substituting R n for Ω in the definition of A (and subsequently H 1 (R n ) for H 1 0 (Ω) in (1.2)). Since Ω is bounded then the injection H 1 0 (Ω) ֒→ L 2 (Ω) is compact. Thus the resolvent of A V is a compact operator and the spectrum of A V is pure point. Let {λ V j , j ∈ N * } be the non-decreasing sequence of the eigenvalues of A V , repeated according to their multiplicities. We define the isospectral set associated to the potential V ∈ C ∞ 0 (Ω) by Is(V ) = {W ∈ C ∞ 0 (Ω); λ V k = λ W k , k ∈ N * }. The computation carried out in §5.2 of the coefficients d j (V ) appearing in (1.6), for j = 1, 2, 3, 4, leads to the following compactness result.

Theorem 1.1. Let a be the identity of R n 2 . Then for all V ∈ C ∞ 0 (Ω) and any bounded subset

B ⊂ L ∞ (Ω) such that V ∈ B, the set Is(V ) ∩ B is compact in H s (Ω) for each s ∈ (-∞, 2).
1.3. What is known so far. It turns out that the famous problem addressed by M. Kac in [Ka], as whether one can hear the shape of drum, is closely related to the following asymptotic expansion formula for the trace of e t∆g on a compact Riemannian manifold (M, g):

(1.7) tr e t∆g = t -n/2 e 0 + te 1 + t 2 e 2 + . . . + t k e k + O t k+1 .
Here ∆ g is the Laplace-Beltrami operator associated to the metric g and the coefficients e k , k ≥ 0, are Riemannian invariants depending on the curvature tensor and its covariants derivatives. There is a wide mathematical literature about (1.7), with many authors focusing more specifically on the explicit calculation of e k , k ≥ 0. This is due to the fact that these coefficients actually provide useful information on g and consequently on the geometry of the manifold M . The key point in the proof of (1.7)is the construction of a parametrix for the heat equation ∂ t -∆ g , which was initiated by S. Minakshisudaram and Å. Pleijel in [MP].

The asymptotic expansion formula (1.6) was proved by Y. Colin de Verdière in [Co] by adaptating (1.7). An alternative proof, based on the Fourier transform, was given in [BB] by R. Bañuelos and A. Sá Barreto. The approach developped in this text is rather different in the sense that (1.5) is obtained by linking the heat kernel of e -tAV to the one of e -tA through Duhamel's principle. The asymptotic expansion formulae (1.6) and (1.7) are nevertheless quite similar, but, here, the coefficients d k , k ≥ 1, are stated as integrals over Ω of polynomial functions in V and its derivatives. This situation is reminiscent of [BB][Theorem 2.1] where the same coefficients are expressed in terms of the tensor products V ⊗ . . . ⊗ V , where V is the Fourier transform of the potential V . Since the present work is not directly related to the analysis of the asymptotic expansion formula (1.7), we shall not go into that matter further and we refer to [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF][START_REF] Chavel | Eigenvalues in riemannian geometry[END_REF][START_REF] Gilkey | Asymptotic formulae in spectral geometry[END_REF][START_REF] Kac | Can one hear the shape of drum ?[END_REF][START_REF] Mckean | Curvature and the eigenvalues of the laplacian[END_REF] for more details.

As will appear in section 5, the proof of the compactness Theorem 1.1 boils down to the calculation of the four main terms in the asymptotic expansion formula (1.5). This follows from the basic identity k≥1 e -λ V k t = tr e -tAV = tr e -tA + Z V Ω (t), linking the isospectral sets of A V to the heat trace of A. Compactness results for isospectral potentials associated to the operator ∆ g + V were already obtained by Brüning in [Br][Theorem 3] for a compact Riemannian manifold with dimension no gretaer that 3, and further improved by Donnelly in [Don]. Their approach is based on trace asymptotics borrowed to [Gi1][Theorem 4.3]. Our strategy is rather similar but the heat kernels asymptotics needed in this text are explicitly computed in the first part of the article.

1.4. Outline. Section 2 gathers several definitions and auxiliary results on heat kernels and trace asymptotics neeeded in the remaining part of the article. The asymptotic formulae (1.5)-(1.6) areestablished in Section 3. Finally section 5 contains the proof of Theorem 1.1.

Preliminaries

In this section we introduce some notations used throughout this text and derive auxiliary results needed in the remaining part of this paper.

2.1. Heat kernels and trace asymptotics. With reference to the definitions and notations introduced in §1 we first recall from [Ou] that the operator (-A V ), where V ∈ C ∞ 0 (Ω), generates an analytic semi-group e -tAV on L 2 (Ω). We note K V the heat kernel associated to e -tAV , in such a way that the identity (2.1) e -tAV f (x) = Ω K V (t, x, y)f (y)dy, t > 0, x ∈ Ω, holds for every f ∈ L 2 (Ω). Let M V be the multiplier by V . Then we have e -tAV = e -tAt 0 e -(t-s)A M V e -sAV ds, t > 0, from Duhamel's formula. From this and (2.1) then follows that

(2.2) K V (t, x, y) = K(t, x, y) - t 0 Ω K(t -s, x, z)V (z)K V (s, z, y)dzds, t > 0, x, y ∈ Ω,
where K denotes the heat kernel of e -tA . Upon solving the integral equation (2.2) with unknown function K V by the successive approximation method, we obtain that

(2.3) K V (t, x, y) = j≥0 K V j (t, x, y), t > 0, x, y ∈ Ω, with (2.4) K V 0 (t, x, y) = K(t, x, y) and K V j+1 (t, x, y) = - t 0 Ω K(t -s, x, z)V (z)K V j (s, z, y)dsdz for all j ∈ N.
Thus, for each t > 0 and x, y ∈ Ω, we get by induction on j ∈ N * that

K V j (t, x, y) = (-1) j Ω n t 0 t1 0 . . . tj-1 0 j i=1 K(t i-1 -t i , z i-1 , z i )V (z i ) K(t j , z j , y)dz j dt j ,
where t 0 = t, z 0 = x, and du j = du 1 . . . du j for u = z, t. From this, the following reproducing property (2.5) Ω K(t -s, x, z)K(s, z, y)dz = K(t, x, y), t > 0, s ∈ (0, t), x, y ∈ Ω, and the estimate K ≥ 0, arising from [Fr], then follows that

(2.6) |K V j (t, x, y)| ≤ V j ∞ t j j! K(t, x, y), t > 0, x, y ∈ Ω, j ∈ N.
Therefore, for any fixed x, y ∈ Ω, the series in the rhs of (2.3) converges uniformly in t > 0.

Having said that we consider the fundamental solution Γ to the equation

∂ t -div(a(x)∇ •) = ∂ t - n i,j=1 ∂ j (a ij (x)∂ i •) = 0 in R n .
Then there is a constant c > 0, depending only on n and µ, such that we have

(2.7) Γ(t, x, y) ≤ (ct) -n/2 e -c|x-y| 2 /t , t > 0, x, y ∈ R n ,
according to [FS]. Further, arguing as in the proof of Lemma 2.1 below, it follows from the maximum principle that

(2.8) 0 ≤ K(t, x, y) ≤ Γ(t, x, y), t > 0, x, y ∈ Ω.
Thus, for all fixed t > 0, the series in the rhs of (2.3) converges uniformly with respect to x and y in Ω according to (2.6) and (2.7), and we have (2.9)

Ω K V (t, x, x)dx = j≥0 A V j (t) where A V j (t) = Ω K V j (t, x, x)dx, j ∈ N.
Since σ(e -tAV ) = {e -tλ V k , k ≥ 1} from the spectral theorem then e -tAV is trace class by [Kat]. On the other hand, e -tAV being an integral operator with smooth kernel (see e.g. [Da]), we have

(2.10) tr e -tAV = Ω K V (t, x, x)dx = k≥1 e -tλ V k , t > 0.
Notice that the right identity in (2.10) is a direct consequence of Mercer's theorem (see e.g. [Ho]), entailing Finally,putting (1.4) and (2.9)-(2.10) together, we find out that

K V (t, x, y) = k≥1 e -tλ V k φ V k (x) × φ V k (y), t > 0, x, y ∈ Ω, where {φ V k , k ∈ N * } is an orthonormal basis of eigenfunctions φ V k of A V , associated to the eigenvalue λ V k .
(2.11) Z V Ω (t) = j≥1 A V j (t), t > 0.
2.2. Estimation of Green functions. We start with the following useful comparison result:

Lemma 2.1. For δ > 0 put Ω δ = {x ∈ Ω; dist(x, ∂Ω) > δ}. Then we have 0 ≤ Γ(t, x, y) -K(t, x, y) ≤ (ct) -n/2 e -cδ 2 /t , 0 < t ≤ 2cδ 2 n , x ∈ Ω, y ∈ Ω δ ,
where c is the constant appearing in the rhs of (2.7).

Proof. Fix y ∈ Ω δ . Then u y (t, x) = Γ(t, x, y) -K(t, x, y) being the solution to the following initial boundary value problem

   ∂ t u y (t, x) -n i,j=1 ∂ j (a ij ∂ i u y (t, x)) = 0, t > 0, x ∈ Ω, u y (0, x) = 0, x ∈ Ω, u y (t, x) = Γ(t, x, y), t > 0, x ∈ ∂Ω,
we get from the parabolic maximum principle (see e.g. [Fr]) that u y (t, x) ≤ max z∈∂Ω 0<s≤t Γ(s, z, y) for ae x ∈ Ω.

Therefore we have

u y (t, x) ≤ max z∈∂Ω 0<s≤t (cs) -n/2 e -c|z-y| 2 /s ≤ max 0<s≤t (cs) -n/2 e -cδ 2 /s , t > 0, x ∈ Ω,
by (2.7). Now the desired result follows readily from this and (2.8) upon noticing that s → (cs) -n/2 e -cδ 2 /s is non-decreasing on (0, 2cδ 2 /n).

Remark 2.1. a) The functions K(t, •, •) and Γ(t, •, •) being symmetric for all t > 0, the statement of Lemma 2.1 remains valid for x ∈ Ω δ and y ∈ Ω as well. b) A result similar to Lemma 2.1 can be found in [Mi] for the Dirichlet Laplacian, which corresponds to the operator A in the peculiar case where a is the identity matrix. This claim, which was actually first proved by H. Weyl in [We], is a cornerstone in the derivation of the classical Weyl's asymptotic formula for the eigenvalues counting function (see e.g. [Dod]). c) We refer to [Co] for an alternative proof of Lemma 2.1 that is based on the classical Feynman-Kac formula (see e.g. [SV]) instead of the maximum principle.

Let us extend V ∈ C ∞ 0 (Ω) to R n by setting V (x) = 0 for all x ∈ R n \ Ω, and, with reference to (2.3)-(2.4), put

(2.12) Γ V 0 (t, x, y) = Γ(t, x, y) and Γ V j+1 (t, x, y) = - t 0 R n Γ(t -s, x, z)V (z)Γ V j (s, z, y)dsdz, j ∈ N,
for all t > 0 and x, y ∈ R n . Armed with Lemma 2.1 we may now relate the asymptotic behavior of A V j (t) as t ↓ 0 to the one of (2.13)

B V j (t) = Ω Γ V j (t, x, x)dx, t > 0, j ∈ N. Proposition 2.1. Let j ∈ N * . Then for each k ∈ N we have A V j (t) = B V j (t) + O(t k ) as t ↓ 0. Proof.
Choose δ > 0 so small that supp(V ) ⊂ Ω δ , where Ω δ is the same as in Lemma 2.1, and pick t ∈ (0, 2cδ 2 /n). Then, for all x, y ∈ Ω, we have

|Γ V 1 (t, x, y) -K V 1 (t, x, y)| ≤ t 0 Ω δ Γ(t -s, x, z)|V (z)|[Γ(s, z, y) -K(s, z, y)]dzds + t 0 Ω δ [Γ(t -s, x, z) -K(t -s, x, z)]|V (z)|K(s, z, y)dzds,
by (2.4) and (2.12). This, together with Lemma 2.1 and part a) in Remark 2.1, yields

(2.14) |Γ V 1 (t, x, y) -K V 1 (t, x, y)| ≤ V ∞ (ct) -n/2 e -cδ 2 /t t 0 R n Γ(s, x, z)dz + t 0 R n Γ(s, z, y)dz ,
for all t > 0 and a.e. x, y ∈ Ω. Here we used the estimae 0 ≤ K ≤ Γ and the fact that the function s → (cs) -n/2 e -cδ 2 /s is non-decreasing on (-∞, 2cδ 2 /n]. Further, due to (2.7), there is a positive constant C, independent of t, such that

t 0 R n Γ(s, x, z)dz + t 0 R n Γ(s, z, y)dz ≤ Ct, t > 0, x, y ∈ Ω, so we obtain |Γ V 1 (t, x, y) -K V 1 (t, x, y)| ≤ (2C V ∞ )t(ct) -n/2 e -cδ 2 /t , t > 0,
x, y ∈ Ω, by (2.14). Similarly, using (2.6) and arguing as above, we get

|Γ V j (t, x, y) -K V j (t, x, y)| ≤ (2C V ∞ ) j t j j! (ct) -n/2 e -cδ 2 /t , t > 0, x, y ∈ Ω,
by induction on j ∈ N * . Now the result follows from this, (2.9) and (2.13).

2.3. The case of a homogeneous metric with degree 0. We now express the function (t, x) ∈ R * + ×R n → Γ V j (t, x, x), j ∈ N * , defined by (2.12), in terms of the heat kernel Γ and the perturbation V , in the particular case where a is homogeneous of degree 0. The result is as follows.

Lemma 2.2. Assume that a is homogeneous of degree 0. Then for every j ∈ N * , t > 0 and x ∈ R n , we have

Γ V j (t, x, x) = (-1) j t j-n/2 (R n ) n 1 0 s1 0 . . . sj-1 0 j i=1 Γ(s i-1 -s i , x + w i-1 , x + w i )V (x + √ tw i )
×Γ(s j , x + w j , x)ds j dw j , with s 0 = 1, w 0 = 0, and dβ j = dβ 1 . . . dβ j for β = s, w.

Proof. The main benefit of dealing with a homogeneous function a of degree 0 is the following property:

Γ(ts, x, y) = t -n/2 Γ s, x √ t , y √ t , t, s > 0, x, y ∈ R n .
From this and the following the identity arising from (2.12) for all t > 0 and x, y ∈ R n ,

Γ V j (t, x, y) = (-1) j t j (R n ) 2 1 0 s1 0 . . . sj-1 0 j i=1 Γ(t(s i-1 -s i ), z i-1 , z i )V (z i ) Γ(ts j , z j , y)dz j ds j , with z 0 = x, then follows that Γ V j (t, x, y) = (-1) j t j-(j+1)n/2 (R n ) n 1 0 s1 0 . . . sj-1 0 j i=1 Γ s i-1 -s i , z i-1 √ t , z i √ t V (z i ) Γ s j , z j √ t , y √ t dz j ds j .
Thus, by performing the change of variables (z 1 , . . . z j ) = √ t(w 1 , . . . w j ) + (x, . . . , x) in the above integral, we find out that Γ V j (t, x, y) = (-1) j t j-n/2

(R n ) n 1 0 s1 0 . . . sj-1 0 j i=1 Γ s i-1 -s i , x √ t + w i-1 , x √ t + w i V x + √ tw i ×Γ s j , x √ t + w j , y √ t dw j ds j .
Finally, we obtain the desired result upon taking y = x in the above identity and recalling that Γ = Γ a verifies

Γ a t, x √ t + z, x √ t + w = Γ a • -x √ t (t, z, w) = Γ a( √ t • -x) (t, z, w) = Γ a( √ t • ) (t, z+x, w+x) = Γ a (t, z+x, w+x),
for all t > 0 and x, z, w in R n .

If a is the identity matrix I, then Γ(t, x, y) is explicitly known and coincides with the following Gaussian kernel

(2.15) G(t, x -y) = (4πt) -n/2 e -|x-y| 2 /(4t) , t > 0, x, y ∈ R n .
This and Lemma 2.2 entails the following:

Lemma 2.3. Assume that a = I. Then, using the same notations as in Lemma 2.2, we have

Γ V j (t, x, x) = (-1) j t j-n/2 (R n ) n 1 0 s1 0 . . . sj-1 0 j i=1 G(s i-1 -s i , w i-1 -w i )V (x + √ tw i )
×G(s j , w j )dw j ds j , for all t > 0 and x ∈ R n , where G is defined by (2.15).

Asymptotic expansion formulae

In this section we establish the asymptotic expansion formulae (1.5)-(1.6). The strategy of the proof is, first, to establish (1.5)-(1.6) where (3.1) Z V (t) = tr(e -tHV -e -tH ), t > 0, is substituted for Z V Ω (t), and, second, to relate the asymptotics of Z V Ω (t) as t ↓ 0 to the one of Z V (t). Here H is the selfadjoint operator generated in L 2 (R n ) by the closed quadratic form

h[u] = R n a(x)|∇u(x)| 2 dx, u ∈ D(h) = H 1 (R n ),
and H V = H + V as a sum in the sense of quadratic forms. It is easy to check that H acts on its domain D(H) = H 2 (R n ), the second-order Sobolev space on R n , as the rhs of (1.3). Moreover we have

D(H V ) = D(H) since V ∈ L ∞ (R n ).
In other words H (resp., H V ) may be seen as the extension of the operator A (resp., A V ) acting in L 2 (R n ), and, due to (2.12) and (3.1), we have

(3.2) Z V (t) = j≥1 H V j (t), t > 0, where H j (t) = R n Γ V j (t, x, x)dx, j ∈ N.
In light of this and Lemma 2.2, we apply Taylor's formula to V ∈ C ∞ 0 (Ω), getting for all j ≥ 1 and p ≥ 1,

(3.3) j k=1 V (x + tw k ) = p-1 ℓ=0 t ℓ   |α1|+...|αj |=ℓ 1 α 1 ! . . . α j ! j k=1 ∂ α k V (x)w α k k   + t p R p j (t, x, w 1 . . . , w j ),
where

(3.4) R p j (t, x, w 1 . . . , w j ) = |α1|+...|αj |=p p α 1 ! . . . α j ! 1 0 (1 -s) p-1 j k=1 ∂ α k V (x + stw k )w α k k ds.
For the sake of notational simplicity we note (3.5)

α j = (α j 1 , . . . , α j j ) ∈ (N n ) j , α j ! = j k=1 α j k ! and W α j j = j k=1 w α j k k ,
so that (3.3)-(3.4) may be reformulated as

(3.6) j k=1 V (x + tw k ) = p-1 ℓ=0 t ℓ   |α j |=ℓ W α j j α j ! j k=1 ∂ α j k V (x)   + t p R p j (t, x, w 1 . . . , w j ), with (3.7) R p j (t, x, w 1 . . . , w j ) = |α j |=p pW α j j α j ! 1 0 (1 -s) p-1 j k=1 ∂ α k V (x + stw k )ds, j, p ∈ N * .
Next, with reference to (3.5) we define for further use (3.8)

c α j (x) = 1 α j ! (R n ) n 1 0 s1 0 . . . sj-1 0 W α j j j i=1 Γ(s i-1 -s i , x + w i-1 , x + w i ) Γ(s j , x + w j , x)dw j ds j ,
where, as usual, s 0 = 1, w 0 = 0, and du j stands for du 1 . . . du j with u = s, w, and we put (3.9)

P α j (V ) = Ω c α j (x) j k=1 ∂ α j k V (x)dx, j ∈ N * .
We now state the main result of this section.

Proposition 3.1. Let p ∈ N \ {0, 1, 2}. Then, under the assumption (1.1), Z V (t) and Z V Ω (t) have the following asymptotic expansion p-1 ℓ=2 t ℓ/2 P ℓ (V ) + O(t p/2 ) as t ↓ 0, where

P ℓ (V ) = 1≤j≤ℓ/2 (-1) j |α j |=ℓ-2j P α j (V ),
the coefficients P α j (V ) being given by (3.8)-(3.9).

Proof. In view of (2.13) and Lemma 2.2, we have

t n B V j (t 2 ) = (-1) j p-1 ℓ=0 t ℓ+2j |α j |=ℓ P α j (V ) + O(t p+2j ), t > 0, j ∈ N * , hence t n B V j (t 2 ) = (-1) j p-1 ℓ=2j t ℓ |α j |=ℓ-2j P α j (V ) + O(t p ), t > 0, j ∈ N * .
Summing up the above identity over all integers j between 1 and (p -1)/2, we find that

t n 1≤j≤(p-1)/2 B V j (t 2 ) = 1≤j≤(p-1)/2 (-1) j p-1 ℓ=2j t ℓ |α j |=ℓ-2j P α j (V ) + O(t p ) = p-1 ℓ=2 t ℓ 1≤j≤ℓ/2 (-1) j |α j |=ℓ-2j P α j (V ) + O(t p ).
As a consequence we have

t n 1≤j≤(p-1)/2 B V j (t 2 ) = p-1 ℓ=2 t ℓ P ℓ (V ) + O(t p ), hence (3.10) t n j≥1 B V j (t 2 ) = p-1 ℓ=2 t ℓ P ℓ (V ) + O(t p ).
Next, bearing in mind that V is supported in Ω, we see that

P α j (V ) = R n c α j (x) j k=1 ∂ α j k V (x)dx for all j ∈ N * . This entails (3.11) t n j≥1 H V j (t 2 ) = p-1 ℓ=2 t ℓ P ℓ (V ) + O(t p ),
upon substituting (3.2) (resp., H V j ) for (2.13) (resp., B V j ) in the above reasonning. Finally, putting (2.11), (3.10) and Proposition 2.1 (resp., (3.2) and (3.11)) together we obtain the result for Z V Ω (resp., Z V ).

Proposition 3.1 immediately entails the:

Corollary 3.1. Let V 0 ∈ C ∞ 0 (Ω). Then, under the conditions of Proposition 3.1, each V ∈ Is(V 0 ) verifies P ℓ (V ) = P ℓ (V 0 ), ℓ ≥ 2.
In the particular case where a = I, (3.8) may be rewritten as

(3.12) c α j (x) = c α j = 1 α j ! (R n ) n 1 0 s1 0 . . . sj-1 0 W α j j G(1 -s 1 , w 1 ) j k=1 G(s k -s k+1 , w k -w k+1 )dw j ds j ,
with s j+1 = w j+1 = 0. Here G is defined by (3.5) and the notations α j ! and W α j j are the same as in (3.5). Thus we have (3.13)

P α j (V ) = c α j P α j (V ), P α j (V ) = Ω j k=1 ∂ α j k V (x)dx,
from (3.9), hence Proposition 3.1 entails the: Proposition 3.2. Assume that a = I. Then, for any p ∈ N * , the asymptotics of Z V (t) and Z Ω (t) as t ↓ 0 have the expression

p ℓ=1 t ℓ P 2ℓ (V ) + O(t p+1 ),
where

(3.14) P ℓ (V ) = 1≤j≤ℓ/2 (-1) j |α j |=ℓ-2j P αj (V ) = 1≤j≤ℓ/2 (-1) j |α j |=ℓ-2j c α j P α j (V ),
the coefficients P αj (V ), c α j and P α j (V ) being defined by (3.12)-(3.13).

Proof. Upon performing the change of variables (w 1 , . . . , w j ) -→ (-w 1 , . . . , -w j ) in the rhs of (3.12) we get that c α j = (-1) |α j | c α * . Therefore c α j = 0 hence P αj (V ) = 0 by (3.13), for |α j | odd. As a consequence we have

P 2ℓ+1 (V ) = 1≤j≤ℓ (-1) j |α j |=2(ℓ-j)+1 P αj (V ) = 0.
Thus, applying (3.10) where 2(p + 1) is substituted for p, we find out that

t n j≥1 B V j (t 2 ) = 2p+1 ℓ=2 t ℓ P ℓ (V ) + O(t 2(p+1) ) = p ℓ=1 t 2ℓ P 2ℓ (V ) + O(t 2(p+1) ),
which, in turns, yields

t n/2 j≥1 B V j (t 2 ) = p ℓ=1 t ℓ P 2ℓ (V ) + O(t p+1 ).
Now the result follows from this by arguing as in the proof of Proposition 3.1.

Remark 3.1. It is clear that the asymptotic formula stated in Proposition 3.1 (resp., Proposition 3.2) for Z V remains valid upon substituting

R n c α j (x) j k=1 ∂ α j k V (x)dx (resp., c α j R n j k=1 ∂ α j k V (x)dx) for P α j (V ), if V is taken in the Schwartz class S (R n ).

Two parameter integrals

In this section we collect useful properties of two parameter integrals appearing in the proof of Theorem 1.1, presented in section 5. As a preamble we consider the integral (4.1)

I n (f ) = R n R n 1 0 s1 0 f (w 1 , w 2 )G(1 -s 1 , w 1 )G(s 1 -s 2 , w 1 -w 2 )G(s 2 , w 2 )dw 1 dw 2 ds 1 ds 2 , where f ∈ C ∞ (R n × R n
) and G is defined by (2.15). For all σ ∈ σ n , the set of permutations of {1, . . . , n}, and all z = (z 1 , . . . , z n ) ∈ R n , we write σz = (z σ(1) , . . . , z σ(n) ). Similarly, for every

w 1 , w 2 ∈ R n , we note σ(w 1 , w 2 ) = (σw 1 , σw 2 ) and f • σ(w 1 , w 2 ) = f (σ(w 1 , w 2 )).
The following result gathers several properties of I n that are required in the remaining part of this section.

Lemma 4.1. Let f ∈ C ∞ (R n × R n ).
Then it holds true that: i) I n (f ) = I n (Sf ), where S denotes the "mirror symmetry" operator acting as

Sf (w 1 , w 2 ) = f (w 2 , w 1 ); ii) I n (f ) = I n (f • σ) for all σ ∈ σ n ; iii) If there are f k ∈ C ∞ (R × R), k = 1, . . . , n, such that f (w 1 , w 2 ) = n k=1 f k (w k 1 , w k 2 ), w i = (w 1 i , . . . , w n i ), i = 1, 2,
and if any of the f k is an odd function of (w k 1 , w k 2 ), then we have I n (f ) = 0. Proof. i) Upon performing successively the two changes of variables τ 1 = 1 -s 1 and τ 2 = 1 -s 2 in the rhs of (4.1), we get that

I n (f ) = R n R n 1 0 1 τ1 f (w 1 , w 2 )G(τ 1 , w 1 )G(τ 2 -τ 1 , w 1 -w 2 )G(1 -τ 2 , w 2 )dw 1 dw 2 dτ 2 dτ 1 = R n R n 1 0 τ2 0 f (w 1 , w 2 )G(τ 1 , w 1 )G(τ 2 -τ 1 , w 1 -w 2 )G(1 -τ 2 , w 2 )dw 1 dw 2 dτ 1 dτ 2 ,
so the result follows by relabelling (w 1 , w 2 ) as (w 2 , w 1 ). ii) In light of (2.15) we have G(t, w) = G(t, σ -1 w) for all t > 0, w ∈ R n and σ ∈ σ n , hence I n (f ) is equal to

R n R n 1 0 s1 0 f (w 1 , w 2 )G(1 -s 1 , σ -1 w 1 )G(s 1 -s 2 , σ -1 w 1 -σ -1 w 2 )G(s 2 , σ -1 w 2 )dw 1 dw 2 ds 1 ds 2 ,
according to (4.1). The result follows readily from this upon performing the change of variable ( w 1 , w 2 ) = σ -1 (w 1 , w 2 ). iii) This point is a direct consequence of the obvious identity I n (f ) = n k=1 I 1 (f k ), arising from (2.15) and (4.1).

We turn now to evaluating integrals of the form (4.2)

I α,β = I α,β (s 1 , s 2 ) = R R x α y β g(1 -s 1 , x)g(s 1 -s 2 , x -y)g(s 2 , y)dxdy, α, β ∈ N, s 1 , s 2 ∈ R,
for α + β even, where g denotes the one-dimensional Gaussian kernel defined by (2.15) in the particular case where n = 1. This can be achieved upon using the following result.

Lemma 4.2. For all α, β ∈ N and all s 1 , s 2 ∈ R, we have:

i) I 1,1 (s 1 , s 2 ) = 2(4π) -1/2 (1 -s 1 )s 2 ; ii) I α,β (s 1 , s 2 ) = 2(1 -s 1 )s 2 [2(α -1)(β -1)(s 1 -s 2 )I α-2,β-2 (s 1 , s 2 ) + (α + β -1)I α-1,β-1 (s 1 , s 2 )]; iii) I α,β (s 1 , s 2 ) = 2(1 -s 1 ) [(α -1)s 1 I α-2,β (s 1 , s 2 ) + βs 2 I α-1,β-1 (s 1 , s 2 )]; iv) I α,β (s 1 , s 2 ) = 2(1 -s 1 ) [(α + β -1)s 1 I α-2,β (s 1 , s 2 ) -2β(β -1)s 2 (s 1 -s 2 )I α-2,β-2 (s 1 , s 2 )]; v) I 2α,0 (s 1 , s 2 ) = (4π) -1/2 (2α)!/(α!)s α 1 (1 -s 1 ) α ; vi) I 0,2α (s 1 , s 2 ) = (4π) -1/2 (2α)!/(α!)s α 2 (1 -s 2 ) α . Proof. a) In light of the basic identity (4.3) zg(t, z) = -2t∂ z g(t, z), t > 0, z ∈ R, we have R R xyg(1 -s 1 , x)g(s 1 -s 2 , x -y)g(s 2 , y)dxdy = -2(1 -s 1 ) R R y∂ x g(1 -s 1 , x)g(s 1 -s 2 , x -y)g(s 2 , y)dxdy = 2(1 -s 1 ) R R yg(1 -s 1 , x)∂ x g(s 1 -s 2 , x -y)g(s 2 , y)dxdy,
by integrating by parts. Thus, applying (4.3) once more, we obtain that R R

xyg(1 -s 1 , x)g(s 1 -s 2 , x -y)g(s 2 , y)dxdy (4.4) with the help of the reproducing property. On the other hand, an integration by parts providing

= -2(1 -s 1 ) R R yg(1 -s 1 , x)∂ y g(s 1 -s 2 , x -y)g(s 2 , y)dxdy = 2(1 -s 1 ) R R yg(1 -s 1 , x)g(s 1 -s 2 , x -y)∂ y g(s 2 , y)dxdy + 2(1 -s 1 )(4π) -1/2 = 2(1 -s 1 ) R yg(1 -s 2 , y)∂ y g(s 2 , y)dy + 2(1 -s 1 )(4π) -1/2 ,
R yg(1 -s 2 , y)∂ y g(s 2 , y)dy = - R g(1 -s 2 , y)g(s 2 , y)dy - R y∂ y g(1 -s 2 , y)g(s 2 , y)dy = -(4π) -1/2 - s 2 1 -s 2 R yg(1 -s 2 , y)∂ y g(s 2 , y)dy,
we get that R yg(1 -s 2 , y)∂ y g(s 2 , y)dy = -(4π) -1/2 (1 -s 2 ). Thus Part i) follows from this and (4.4). b) Applying (4.3) with z = x and t = 1 -s 1 we find that

I α,β (s 1 , s 2 ) = -2(1 -s 1 ) R R x α-1 y β ∂ x g(1 -s 1 , x)g(s 1 -s 2 , x -y)g(s 2 , y)dxdy = 2(α -1)(1 -s 1 ) R R x α-2 y β g(1 -s 1 , x)g(s 1 -s 2 , x -y)g(s 2 , y)dxdy - 2(1 -s 1 ) 2(s 1 -s 2 ) R R x α-1 y β (x -y)g(1 -s 1 , x)g(s 1 -s 2 , x -y)g(s 2 , y)dxdy,
by integrating by parts wrt x, so we get

(4.5) (1 -s 2 )I α,β (s 1 , s 2 ) = 2(α -1)(1 -s 1 )(s 1 -s 2 )I α-2,β (s 1 , s 2 ) + (1 -s 1 )I α-1,β+1 (s 1 , s 2 ).
Doing the same with z = y and t = s 2 we obtain that (4.6)

s 1 I α,β (s 1 , s 2 ) = 2(β -1)(s 1 -s 2 )s 2 I α,β-2 (s 1 , s 2 ) + s 2 I α+1,β-1 (s 1 , s 2 ).
Thus, upon successively substituting (α -1, β + 1) and (α -2, β) for (α, β) in (4.6), we find that (4.7)

s 1 I α-1,β+1 (s 1 , s 2 ) = 2βs 2 (s 1 -s 2 )I α-1,β-1 (s 1 , s 2 ) + s 2 I α,β (s 1 , s 2 ) and (4.8) s 1 I α-2,β (s 1 , s 2 ) = 2(β -1)(s 1 -s 2 )I α-2,β-2 (s 1 , s 2 ) + s 2 I α-1,β-1 (s 1 , s 2 ).
Plugging (4.7)-(4.8) in (4.5) we end up getting Part ii). Further we obtain Part iii) by following the same lines as in the derivation of Part ii), and Part iv) is a direct consequence of Parts ii) and iii). c) Arguing as in the derivation of Part i) in a), we establish for any α ≥ 2 that

I α,0 (s 1 , s 2 ) = 2(α -1)s 1 (1 -s 1 )I α-2,0 (s 1 , s 2 ).
This and the obvious identity I 0,0 (s 1 , s 2 ) = (4π) -1/2 yields Part v) upon proceeding by induction on α.

Finally, Part vi) follows from Part v) upon noticing from (4.2) that I 0,α (s 1 , s 2 ) = I α,0 (1 -s 2 , 1 -s 1 ).

Further, for all α = (α k ) 1≤k≤n and β = (β k ) 1≤k≤n in N n , we put (4.9)

I (α, β) = 1 0 s1 0 R n R n x α y β G(1 -s 1 , x)G(s 1 -s 2 , x -y)G(s 2 , y)dxdy ds 1 ds 2 ,
and establish the:

Lemma 4.3. For each α = (α k ) 1≤k≤n and β = (β k ) 1≤k≤n in N n we have: i) I (α, β) = 1 0 s1 0 n k=1 I α k ,β k (s 1 , s 2 )ds 1 ds 2 . ii) J (α, β) = J (β, α). iii) J (α, β) = 0 if any of sums α k + β k for 1 ≤ k ≤ n, is odd.
Proof. Part i) follows readily from the identity G(s, z) = n k=1 g(s, z k ) arising from (2.15) for all s ∈ R * and all z = (z k ) 1≤k≤n ∈ R n , and from the very definitions (4.2) and (4.9). Next, Part ii) is a direct consequence first assertion of Lemma (4.1), while Part iii) follows from the third point of Lemma (4.1).

Proof of Theorem 1.1

We start by establishing two identities which are useful for the proof of Theorem 1.1. 5.1. Two useful identities. They are collected in the following: Proposition 5.1. Let V ∈ C ∞ 0 (Ω) be real-valued and assume that a = I. Then, with reference to the definitions (3.12)-(3.13), we have

(5.1) (4π) n/2 |α 2 |=2 c α 2 P α 2 (V ) = - 1 12 Ω |∇V | 2 dx and (5.2) (4π) n/2 |α 2 |=4 c α 2 P α 2 (V ) = 1 120 k Ω ∂ 2 kk V 2 dx + 13 360 k =ℓ Ω ∂ 2 kℓ V 2 dx. Proof. Since (5.3) c α 2 = I (α 2 1 , α 2 2 ) α 2 ! , α 2 = (α 2 1 , α 2
2 ), by (3.12) and (4.9), we know from the two last points in Lemma 4.3 that (5.4) c α 2 = 0 if the sum (α 2 1 ) k + (α 2 2 ) k is odd for any k ∈ {1, . . . , n}, and In light of (5.4)-(5.5) we deduce from (5.6)-(5.7) that

|α 2 |=2 c α 2 P α 2 (V ) = 1 6 (4π) -n/2 Ω ∆V V dx + 1 12 (4π) -n/2 Ω |∇V | 2 dx.
Taking into account that Ω ∆V V dx = -Ω |∇V | 2 dx, we obtain (5.1) from the above line.

We now compute |α 2 |=4 c α 2 P α 2 (V ). As a preamble we first invoke Lemma 4.2 and get simultaneously (5.8) I 2,2 (s 1 , s 2 ) = 2(1 -s 1 )[s 1 I 0,2 (s 1 , s 2 ) + 2s 2 I 1,1 (s 1 , s 2 )] = 4(4π) -1/2 (1 -s 1 )s 2 [s 1 (1 -s 2 ) + 2(1 -s 1 )s 2 ], and

(5.9) I 3,1 (s 1 , s 2 ) = 2(1 -s 1 )[2s 1 I 1,1 (s 1 , s 2 ) + s 2 I 2,0 (s 1 , s 2 )] = 12(4π) -1/2 (1 -s 1 ) 2 s 1 s 2 , Finally, since Ω ∂ 2 km V (x)V (x) 2 dx = -2 Ω ∂ k V (x)∂ m V (x)V (x)dx for all natural numbers 1 ≤ k, m ≤ n, by integrating by parts, we see that there is a constant C n depending only on n such that we have according to (3.13). This, together with the identity P 8 (V ) = c α 4 =0 P α 4 =0 (V ) -

|α 3 |=2 c α 3 P α 3 (V ) ≤ C n V ∞ Ω |∇V (x)| 2 dx,
|α 3 |=2 c α 3 P α 3 (V ) + |α 2 |=4 c α 2 P α 2 (V ) - |α 1 |=6 c α 1 P α 1 (V ),
arising from (3.14), and (5.2), (5.16), (5.18), then yield (5.21)

|γ|=2 Ω |∂ γ V (x)| 2 dx + Ω V (x) 4 dx ≤ C ′ n |P 8 (V )| + V ∞ Ω |∇V (x)| 2 dx. ,
for some constant C ′ n > 0 depending only on n. In light of (5.20)-(5.21) the set Is(V 0 ) ∩ B is thus bounded in H 2 (Ω) from Corollary 3.1. This entails the desired result since H 2 (Ω) is compactly embedded in H s (Ω) for all s < 2.

  c α 2 = c α 2 for α 2 = (α 2 2 , α 2 1 ), We first compute |α 2 |=4 c α 2 P α 2 (V ).In what follows we note (0, . . . , β k , . . . , 0), 1 ≤ k ≤ n, β ∈ R, the vector (β j ) 1≤j≤n ∈ R n such that β j = 0 for all 1 ≤ j = k ≤ n and β k = β. In view of (5.3) we apply the first point in Lemma 4.3 for α 2 = ((0, . . .2 k , . . . 0), (0, . . . , 0)), 1 ≤ k ≤ n, 0 (s 1 , s 2 ) n-1 I 2,0 (s 1 , s 2 )ds 1 ds 2 = (4π) -(n-10 (s 1 , s 2 )ds 1 ds 2 = (4π) -n/2 12 ,with the aid of Part v) in Lemma 4.2. Similarly, for α 2 = ((0, . . . , 1 k , . . . , 0), (0, . . . , 1 k , . . . , 0)), 1 ≤ k ≤ n, we use the first part of Lemma 4.2 and obtain that (1 (s 1 , s 2 )ds 1 ds 2 = (4π) -n/2 12 .

from Part iii), and

(5.10) I 4,0 (s 1 , s 2 ) = 12(4π) -1/2 s 2 1 (1 -s 1 ) 2 , from Part v). Thus, for all k ∈ {1, . . . , n} it follows from fhe first part of Lemma 4.3 and (5.10) upon taking α 2 = ((0, . . . , 4 k , . . . , 0), (0, . . . , 0)) in (5.3) that (5.11)

Further, choosing α 2 = ((0, . . . , 3 k , . . . , 0), (0, . . . , 1 k , . . . , 0)) we deduce in the same way from (5.9) that,

(5.12)

and with α 2 = ((0, . . . , 2 k , . . . , 0), (0, . . . , 2 k , . . . , 0)), we get from (5.8) that

(5.13)

Finally, upon taking α 2 = ((0, . . . , 2 k , . . . , 0), (0, . . . , 2 ℓ , . . . , 0)) in (5.3), for 1 ≤ k = ℓ ≤ n, we derive from the two last parts of Lemma 4.2 that

(5.14)

with the aid of the first part. Putting (5.11)-(5.15) together and recalling (5.4)-(5.5) we end up getting (5.2).

Armed with Proposition 5.1 we are now in position to prove Theorem 1.1.

5.2.

Completion of the proof. By applying the reproducing property (2.5) to the kernel G, defined in (2.15), we derive from (3.12) for all j ≥ 1 that

where s j+1 = w j+1 = 0. In light of (3.13)-(3.14), (5.16) then yields that (5.17)

Next, bearing in mind that the potential V is compactly supported in Ω, we notice from (3.13) that

(5.18)

As a consequence we have

Further, as P 6 = -c α 3 =0 P α 3 =0 (V )+ |α 2 |=2 c α 2 P α 2 (V )-|α 1 |=4 c α 1 P α 1 (V ), it follows from (5.1) and (5.16) that

(5.20)