Yvonne Mccarthy

Liang Yang

Kate B Twomey

Andrea Sass

Tim Tolker-Nielsen

Eshwar Mahenthiralingam

J Maxwell Dow

Robert P Ryan
email: r.ryan@ucc.ie

Tim Tolker- Nielsen

A sensor kinase

Keywords: Intraspecies signalling, Diffusible signal factors, Signal transduction, Biofilm formation, Pathogenesis

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Intercellular signalling is known to regulate functions that contribute to the persistence and virulence of a number of human bacterial pathogens. These bacteria use structurally diverse signal molecules to monitor aspects of their environment such as cell density (quorum sensing) or confinement to niches and to modulate their behaviour accordingly [START_REF] Ng | Bacterial Quorum-sensing network architectures[END_REF]Ryan and Dow, 2008;[START_REF] Shank | New developments in microbial interspecies signalling[END_REF][START_REF] Waters | Quorum sensing: Cell-to-cell communication in bacteria[END_REF][START_REF] Williams | Quorum sensing and environmental adaptation in Pseudomonas aeruginosa: a tale of regulatory networks and multifunctional signal molecules[END_REF]). Interference with such signalling affords a rational approach to aid the treatment of infections, but requires an understanding of both the signalling mechanisms and their role in bacterial behaviour. The identification of sensors for bacterial signal molecules is an important element of such a strategy as these proteins can present particularly attractive targets for interference [START_REF] Rasko | Novel approaches to bacterial infection therapy by interfering with cell-to-cell signalling[END_REF]2010). Bacteria of the Burkholderia cepacia complex (Bcc) are opportunistic pathogens that colonise the lungs of cystic fibrosis (CF) sufferers. One of the most problematic of these CF pathogens is Burkholderia cenocepacia, which accounts for between 50-90% of all Bcc infections diagnosed in CF patients. This species is characterized by low responsiveness to antibiotic therapy and the ability to cause significant reductions in lung function and includes strains exhibiting a high virulence potential as well as strains with a well-documented history of patient-to-patient transmission [START_REF] Govan | Evolving epidemiology of Pseudomonas aerugincisa and the Burkholderia cepacia complex in cystic fibrosis lung infection[END_REF][START_REF] Mahenthiralingam | The multifarious, multireplicon Burkholderia cepacia complex[END_REF][START_REF] Saldias | Interactions of Burkholderia cenocepacia and other Burkholderia cepacia complex bacteria with epithelial and phagocytic cells[END_REF].

In common with many Gram-negative bacteria, B. cenocepacia uses N-acyl homoserine lactones (N-AHLs) as signals. These signal molecules are synthesized by proteins of the LuxI family and bind to transcriptional regulators of the LuxR family to regulate gene expression. B. cenocepacia has two pairs of LuxIR systems; the CepIR system is present in all species of the Bcc whereas the CciIR system is only present in isolates of B. cenocepacia containing the cenocepacia island (cci) found in highly transmissible B. cenocepacia strains such as the notorious ET12 lineage [START_REF] Malott | Characterization of the cciIR quorum-sensing system in Burkholderia cenocepacia[END_REF][START_REF] O'grady | Reciprocal regulation by the CepIR and CciIR quorum sensing systems in Burkholderia cenocepacia[END_REF][START_REF] Sokol | Communication systems in the genus Burkholderia: global regulators and targets for novel antipathogenic drugs[END_REF]. CepI is responsible for the synthesis of primarily oxanoyl-homoserine lactone (C8-HSL) with lesser amounts of hexanoyl-HSL (C6-HSL). CciI primarily synthesizes C6-HSL with lesser amounts of C8-HSL produced. Diggle and colleagues [START_REF] Diggle | Functional genetic analysis reveals a 2-alkyl-4-quinolone signalling system in the human pathogen Burkholderia pseudomallei and related bacteria[END_REF] reported that in addition to N-AHLs, B. cenocepacia produces a diverse set of 2-alkyl-4(1H)- BCAM0581, which is related to RpfF, the DSF synthase of Xanthomonas campestris [START_REF] Barber | A novel regulatory system required for pathogenicity of Xanthomonas campestris is mediated by a small diffusible signal molecule[END_REF][START_REF] Wang | A bacterial cell-cell communication signal with cross-kingdom structural analogues[END_REF]. The Burkholderia signal molecule (called BDSF) is able to activate DSF-dependent responses in X. campestris and to inhibit germ tube formation in Candida albicans, suggesting a role in interspecies and interkingdom communication. Furthermore, BDSF also plays a role as an intraspecies signal in B. cenocepacia J2315 [START_REF] Deng | Differential modulation of Burkholderia cenocepacia virulence and energy metabolism by the quorum-sensing signal BDSF and its synthase[END_REF][START_REF] Ryan | Intraspecies signalling involving the diffusible signal factor BDSF (cis-2-dodecenoic acid) influences virulence in Burkholderia cenocepacia[END_REF]. Mutation of BCAM0581 gives rise to multiple phenotypic effects including reduction in motility, altered adherence to mucin, reduced biofilm formation and attenuated virulence in the Galleria mellonella larva and zebrafish infection models. For most of the phenotypes that were tested, addition of exogenous BDSF restored the wild-type phenotype to the BCAM0581 mutant [START_REF] Deng | Differential modulation of Burkholderia cenocepacia virulence and energy metabolism by the quorum-sensing signal BDSF and its synthase[END_REF][START_REF] Ryan | Intraspecies signalling involving the diffusible signal factor BDSF (cis-2-dodecenoic acid) influences virulence in Burkholderia cenocepacia[END_REF].

Although BDSF and DSF are structurally similar molecules, bioinformatic analysis suggests that the signal perception mechanisms associated with BDSF signalling in B. cenocepacia are different from those established for DSF in the xanthomonads. In these latter bacteria, the sensor kinase RpfC is implicated in DSF perception leading to alterations in bacterial physiology and gene expression. The genome of B. cenocepacia J2315 however does not encode a histidine kinase with a sensory input domain related to that of RpfC, suggesting the existence of an alternative sensor for the BDSF signal. Here we have addressed the mechanism of BDSF perception by B. cenocepacia J2315 by screening a transposon mutant library in a BCAM0581 background for mutants unable to respond to exogenous BDSF. This approach identified that the histidine sensor kinase BCAM0227 had a possible role in signal perception. Comparative transcriptome profiling established that the sensor is required for expression of a subset of genes in the BDSF regulon. A chimaeric sensor kinase in which the input domain of BCAM0227 replaced the input domain of RpfC was active in BDSF signal perception when expressed in X. campestris. Notably, homologs of this sensor are found in other Burkholderia species, which include the Bcc. Taken together the findings indicate that BCAM0227 is a novel sensor involved in BDSF perception in a signalling system that regulates virulence.

RESULTS

Identification of transposon mutants that have altered expression of BDSF regulated genes

We have previously shown that mutation of BCAM0581 in B. cenocepacia J2315 leads to reduced expression of the cable pilus gene cblD and that addition of BDSF restores expression to wild-type [START_REF] Ryan | Intraspecies signalling involving the diffusible signal factor BDSF (cis-2-dodecenoic acid) influences virulence in Burkholderia cenocepacia[END_REF]. This observation formed the basis of a screen for genes involved in BDSF perception. A panel of transposon mutants was created in the BCAM0581 background using mariner transposon mutagenesis and was assessed for a loss of responsiveness to exogenous BDSF using a cblD::lux reporter (see Experimental procedures). Approximately 3,000 mutants were screened for altered bioluminescence in the presence of 50 µM BDSF. A total of 74 mutants that demonstrated altered levels or kinetics of expression of the reporter fusion was isolated.

The sites of transposon insertion for non-responsive mutants were mapped to the genome of B. cenocepacia J2315 by inverse PCR and sequencing of amplicons (see Experimental procedures). Southern blot analysis confirmed that only a single copy of the transposon was inserted into the chromosome in these non-responsive mutants (data not shown). Insertion sites, which were all different, were mapped to 20 genetic loci with 9 genes found more than once (Table 1). None of the mutants showed impaired growth in complex or minimal media, suggesting that the altered response to BDSF could not be attributed to differences in growth.

One insertion was mapped to BCAM2758 (cblS), a known positive regulatory gene of cblD (Tomich et al., 2003;Tomich and Mohr, 2003). In addition, mutations in several genes encoding enzymes involved in metabolic processes (such as BCAM1685, BCAL2745 and BCAL2731), in genes encoding hypothetical proteins (BCAL3149,

BDSF regulates the expression of genes associated with a range of cellular functions

The findings outlined above together with our previous work indicated that both BDSF and BCAM0227 regulate expression of the cblD gene. To examine the extent of the overlap between those genes regulated by BDSF and those regulated by BCAM0227, we compared the global gene expression patterns of specific B. cenocepacia J2315 mutants using a custom microarray [START_REF] Drevinek | Diversity of the parB and repA genes of the Burkholderia cepacia complex and their utility for rapid identification of Burkholderia cenocepacia[END_REF]. In the first set of experiments, the transcriptomes of the wild-type B. cenocepacia J2315 and a BCAM0581 deletion mutant (constructed as described in the Experimental procedures) were compared.

Comparison of wild-type and the BCAM0581 deletion mutant (BDSF-minus strain) established that expression of 372 genes was significantly influenced (> 2 fold change) by mutation of BCAM0581 (Fig. 1; Table S1). The BCAM0581-regulated genes could be grouped into 10 functional categories; selected genes in each category are shown in Table 2. The analysis revealed that BCAM0581 controls genes associated with a range of biological functions that include bacterial motility and attachment, stress tolerance, virulence, regulation, transport, multidrug resistance, detoxification and signal transduction. Genes associated with virulence that are regulated by BCAM0581 included rpoN [START_REF] Saldias | Interactions of Burkholderia cenocepacia and other Burkholderia cepacia complex bacteria with epithelial and phagocytic cells[END_REF], hfq [START_REF] Sousa | The hfq gene is required for stress resistance and full virulence of Burkholderia cepacia to the nematode Caenorhabditis elegans[END_REF], shvR [START_REF] Bernier | A LysR-type transcriptional regulator in Burkholderia cenocepacia influences colony morphology and virulence[END_REF] and those encoding the components of the Bcs type III secretion system (Tomich et al., 2003) (Fig. 1B).

Changes in expression of genes involved in synthesis and/or action of cable pilus and virulence genes are consistent with previous findings that bacterial motility, virulence and attachment are regulated by the BDSF signal [START_REF] Boon | A novel DSF-like signal from Burkholderia cenocepacia interferes with Candida albicans morphological transition[END_REF][START_REF] Ryan | Intraspecies signalling involving the diffusible signal factor BDSF (cis-2-dodecenoic acid) influences virulence in Burkholderia cenocepacia[END_REF]. The effects of mutation of BCAM0581 on expression of fliJ, cblS (BCAM2758) and clbD (BCAM2759) as revealed by the microarray experiments confirmed our previous findings on BCAM0581 regulation of these genes, which were based on quantitative Real Time PCR (qRT-PCR) studies [START_REF] Ryan | Intraspecies signalling involving the diffusible signal factor BDSF (cis-2-dodecenoic acid) influences virulence in Burkholderia cenocepacia[END_REF].

For other selected genes, qRT-PCR was used to confirm the alteration in expression as revealed by the microarray. Fifteen genes with altered expression and one control gene with unaltered expression were chosen (Fig. S1A). These genes were selected to be representative of: (i) a range of observed changes in expression (from 2-to 50fold), and (ii) different functional classes of genes with altered expression. For all genes tested, the alteration in expression measured by qRT-PCR mirrored that seen by microarray (Table 3; Fig. S1A).

In trans expression of the wild-type BCAM0581 gene in the BCAM0581 deletion mutant restored the transcript level of all fifteen genes to that of the wild-type, as tested by qRT-PCR (Fig. S1A). Importantly, the transcript level for 11 of the genes was also restored to wild-type by exogenous addition of 5 µM BDSF (Fig. S1A).

However for BCAM0495, BCAS0297, BCAS0353 and BCAS0387, addition of BDSF up to a concentration of 50 µM was unable to restore wild-type levels of transcript (Table 3; Fig. S1A). The findings may suggest that BCAM0581 has other roles in the bacterial cell in addition to BDSF biosynthesis that influences expression of certain genes. Deng and colleagues [START_REF] Deng | Differential modulation of Burkholderia cenocepacia virulence and energy metabolism by the quorum-sensing signal BDSF and its synthase[END_REF] have proposed for example that BCAM0581 has an additional metabolic function.

BCAM0227 regulates the expression of a sub-set of genes in the BDSF regulon

Comparative transcriptome profiling of the BCAM0227 deletion mutant (sensorminus) and the wild-type parent established that expression of 78 genes was significantly influenced (> 2 fold change) by the loss of the sensor (Table 2; Table S2). Again for selected genes, qRT-PCR was used to confirm the alteration in expression as revealed by the microarray (Table 3; Fig. S1B).

Comparisons of transcriptome profiles revealed that the expression of 65 genes was influenced by mutation of both BCAM0581 and BCAM0227 (Fig. 1). Furthermore, the expression of all 65 genes was altered in the same direction compared to wild-type in the two mutants. Also BCAM0227 did not regulate expression of BCAM0495, F o r P e e r R e v i e w 8 BCAS0297, BCAS0353 and BCAS0387 whose transcription was not responsive to addition of exogenous BDSF (see above). These findings are consistent with the hypothesis that BCAM0227 is involved in BDSF perception leading to expression of this subset of genes, although they do not exclude the scenario that BDSF and BCAM0227 have independent actions on gene expression.

The sensor kinase protein BCAM0227 influences a sub-set of BDSF-regulated phenotypes

We have previously shown that mutation of BCAM0581 and associated loss of BDSF synthesis has a pleiotropic phenotypic influence in B. cenocepacia J2315, affecting biofilm formation, motility and adhesion. To further examine the possible role of BCAM0227 in BDSF signalling, we assessed the effect of deletion of BCAM0227 on the same phenotypes. Although mutation of the BCAM0581 leads to a reduction in motility and adherence to mucin, mutation of BCAM0227 did not significantly alter these traits (Fig. 3).

In contrast, mutations in both genes influenced biofilm formation by bacteria growing in flow cells in minimal medium (Fig. 3; see Experimental procedures). Under these conditions, the wild-type and mutant strains formed characteristic microcolonies after initial surface attachment. With the wild-type, biofilms rapidly developed and covered most of the available surface space within a period of 4 days (Fig. 3A). At this time, the biofilms showed substantial staining with propidium iodide. However, biofilm formation by the mutants never appeared to pass the microcolony stage, bacteria did not colonize the entire surface during the course of the experiment and little staining by propidium iodide was evident (Fig. 3B,E). The introduction of cloned BCAM0227 or BCAM0581 genes into the appropriate mutant strains restored biofilm formation and architecture to that seen of the wild-type in each case (Fig. 3,C,F). Addition of exogenous BDSF at 100 nM to cultures of the BCAM0581 mutant also restored the formation of biofilm structures similar to those of the wild-type (Fig. 3D). In contrast, biofilm architecture and formation by the BCAM0227 mutant was not altered by addition of exogenous BDSF (data not shown). Staining with propidium iodide will reflect not only the occurrence of dead bacteria but also the presence of extracellular DNA, a component that is an important part of the biofilm matrix for most bacteria, and which may be produced via pathways controlled by cell-cell signalling [START_REF] Allesen-Holm | A characterization of DNA release in Pseudomonas aeruginosa cultures and biofilms[END_REF][START_REF] Whitchurch | Extracellular DNA required for bacterial biofilm formation[END_REF].

The different effects of mutation of BCAM0581 and BCAM0227 on motility, and adherence to mucin but similar effects on biofilm formation suggest that the sensor kinase BCAM0227 is involved in regulation of a subset of BDSF-regulated phenotypes. This conclusion was further supported by measurement of the susceptibility of BCAM0581 and BCAM0227 mutant strains to different antibiotics. B. cenocepacia is known to exhibit high levels of intrinsic resistance to many antimicrobials [START_REF] Nzula | Influence of taxonomic status on the in vitro antimicrobial susceptibility of the Burkholderia cepacia complex[END_REF]. Mutation of BCAM0581gene led to an increased susceptibility to aminoglycosides, erythromycin, chloramphenicol and ampicillin but had no effect on susceptibility to other antibiotics tested (Table S3). In contrast, mutation of BCAM0227 did not alter resistance to any of the antibiotics tested. These findings correlate with the observation that mutation of BCAM0581 but not BCAM0227, led to down-regulation of a number of genes involved in multidrug efflux, resistance and detoxification (Fig. 1B). Whether these particular genes play a functional role in antibiotic resistance to remains to be determined.

Assessment of the role of BCAM0227 in BDSF perception by domain swapping

Using the Simple Modular Architecture Research Tool (SMART), BCAM0227 was predicted to be a hybrid two-component sensor of 1011 amino acids with a histidine kinase phosphoacceptor (HisKA) domain (residues 529-594), histidine kinase domain (residues 641-757), CheY-like receiver domain (residues 781-892) and a C-terminal Histidine phosphotransfer (HPt) domain (residues 920-996) (Fig. 4A). The sensory input domain of BCAM0227 has predicted to have two transmembrane helices and a large loop located in the periplasm (Fig. 4B). This organization differs from that of RpfC, the DSF sensor from X. campestris, which has five predicted transmembrane helices with very limited periplasmic loops (Fig. 4A,B).

We hypothesized that the sensory input of BCAM0227 was responsible for the perception of BDSF. In order to test this proposed function, domain-swapping experiments were performed in which the portion of RpfC comprising the sensory input domain was replaced with the sensory input domain of BCAM0227. A hybrid construct was made (see Experimental procedures) and the functional response to introduced into an rpfFrpfC double mutant of X.campestris. This mutant strain is defective in the production of extracellular endoglucanase and is insensitive to addition of BDSF or DSF (Ryan et al., 2008). Expression of RpfC in the rpfFrpfC mutant conferred sensitivity to added DSF at concentrations as low as 100 nM and added BDSF at concentrations as low as 10 µM as measured by the induction of extracellular endoglucanase activity. Importantly, expression of the chimeric BCAM0227-RpfC protein also allowed endoglucanase production in the presence of 10 µM BDSF or 10 µM DSF, and activation of the system was seen below these concentrations (Fig. 4C). Similar effects are also seen after expression of a chimeric sensor protein in which the input domain and HisKA domain of BCAM0227 replaces the input domain and HisKA domain of RpfC generating chimeric sensor protein BCAM0227 (residues 1-594)-RpfC (residues 253-726) (Fig. S2).

This induction of endoglucanase was specific to BDSF or DSF. A range of compounds of related structure, including trans-11-methyl dodecenoic acid, cis-2decenoic acid, 11-methyl dodecanoic acid and farnesoic acid showed little or no activity when tested at concentrations of 50 µM, although some activity was seen at 1 mM (Fig. 4C). Importantly we have shown previously that chimaeric proteins in which the input domain of unrelated sensors are fused to RpfC have no activity in sensing either DSF (Ryan et al., 2008) or BDSF (data not shown). Taken together, these observations support the contention that BCAM0227 has a direct role in BDSF perception.

BDSF signalling contributes to cytotoxicity and virulence in B. cenocepacia

We have previously reported that mutation of BCAM0581 led to a reduction of virulence of B. cenocepacia in the wax moth (Galleria mellonella) larvae model, indicating a role for BDSF signalling in virulence. Transcriptome analysis discussed above identified that genes encoding the Bcs type III secretion system and a number of other virulence-associated factors were down regulated in both the BCAM0581 and BCAM0227 mutant strains. These observations prompted us to examine the role of the BCAM0227 sensor in both virulence in the insect model and cytotoxicity to mammalian cells. The BCAM0227 mutant showed reduced ability to cause disease in Galleria mellonella. The minimum inoculum of the wild-type required to kill all of 10 larvae challenged was 10± 5 CFU (n = 10). With the BCAM0227 mutant, a higher level of bacteria (100,000± 1000 CFU) was required to kill all 10 larvae, indicating a reduction in virulence. A similar alteration in virulence is seen with the BCAM0581 mutant. Complementation of the BCAM0581 and BCAM0227 mutant strains with the cognate wild-type gene restored virulence to wild-type levels of virulence; the minimum inoculum level to kill all larvae was between 10 and 15 CFU. (All values are the means of measurements on quadruplicate sets of ten larvae; the whole experiment was repeated twice with essentially the same result). A reduction in the rate of killing of larvae by BCAM0581 and BCAM0227 mutants was also seen when comparisons were made using an inoculum of 10 6 CFU of each strain (Fig. 5A).

Complementation of the BCAM0581 and BCAM0227 mutant strains with the cognate wild-type gene restored wild-type levels of virulence (Fig. 5A). Importantly, addition of BDSF to the inoculum at 100 nM restored virulence to the BCAM0581 mutants, but had little effect on the BCAM0227 strain (Fig. 5A).

Previous studies have established a correlation between the ability of strains of B. cenocepacia and B. cepacia to induce cytotoxicity in cultured mammalian cells and the virulence of these strains to mice [START_REF] Bylund | Exopolysaccharides from Burkholderia cenocepacia inhibit neutrophil chemotaxis and scavenge reactive oxygen species[END_REF][START_REF] Bylund | Burkholderia cenocepacia induces neutrophil necrosis in chronic granulomatous disease[END_REF][START_REF] Cieri | Correlation between an in vitro invasion assay and a murine model of Burkholderia cepacia lung infection[END_REF]Tomich et al., 2003). As a first step towards testing the role of BDSF signalling in mammalian virulence, we examined the effects of wild-type, BCAM0227 and BCAM0581 mutant strains on cytotoxicity to CHO cells (see Experimental procedures). Cytotoxicity was measured by quantifying the release of lactate dehydrogenase (LDH) into the supernatants of cultured CHO cells after a short cocultivation period (6 h) with bacteria. At a multiplicity of infection (MOI) of 50:1, the wild-type B. cenocepacia J2315 strain induced a stronger cytotoxic effect than either the BCAM0227 or BCAM0581 mutant (Fig. 5B). The reduction in cytotoxic phenotype in these mutants could be restored to the level of the wild-type strain by complementation (Fig. 5B). Addition of 100 nM BDSF had a significant effect on increasing the cytotoxicity of the BCAM0581 strain although no significant effect on cytotoxicity of the BCAM0227 mutant strain was evident (Fig. 5B). On the basis on these findings, we extended these studies to utilise the agar bead mouse model of pulmonary infection. C57Bl/6 mice were infected intra-tracheally with B. cenocepacia wild-type, BCAM0227 and BCAM0581 mutant strains embedded in agar beads and tested for virulence. Mice infected with the wild-type strain displayed 50% mortality after 8 days post-infection. In contrast, none of mice infected with the mutants appeared to decline during the experiment, and all survived to 8 days post-infection (Table 4). The lethality wild-type was restored to the mutant strains by complementation in trans with a wild-type copy of the appropriate gene (Table 4).

The effects of mutation of BCAM0581 and BCAM0227 on virulence were associated with a substantial reduction in the number of bacteria that could be isolated from the lungs and spleen of the sacrificed mice (Table 4).

These findings from all three infection models indicate that both the BDSF-synthase BCAM0581 and sensor kinase BCAM0227 significantly contribute to virulence of B. cenocepacia J2315. Addition of BDSF to the BCAM0581 mutant restores the cytotoxity and virulence to G. mellonella. In contrast, addition of BDSF to the BCAM0227 mutant gave no alteration in phenotype, suggesting that the regulatory effect of BDSF on virulence is exerted through the sensor kinase.

DISCUSSION

Previous work has established that BDSF acts as an intraspecies or quorum sensing signal in B. cenocepacia and has identified BCAM0581 as the BDSF synthase [START_REF] Boon | A novel DSF-like signal from Burkholderia cenocepacia interferes with Candida albicans morphological transition[END_REF][START_REF] Deng | Differential modulation of Burkholderia cenocepacia virulence and energy metabolism by the quorum-sensing signal BDSF and its synthase[END_REF][START_REF] Ryan | Intraspecies signalling involving the diffusible signal factor BDSF (cis-2-dodecenoic acid) influences virulence in Burkholderia cenocepacia[END_REF]. The mechanism of signal perception and the full range of genes controlled by signalling have however 13 these other gene products in regulation of cblD is currently unclear; we do not know if they are elements within the BDSF signalling cascade or whether their regulatory action is independent of BDSF signalling. Intriguingly expression of some of these genes including BCAM1009, BCAL3149, BCAM2758 and BCAS0021 was influenced by BDSF (Table S1).

The extent of the BDSF regulon

Comparative transcriptome profiling identified that 390 genes were under the control of BCAM0581 under the conditions tested (Table S1). Genes under BCAM0581 regulation include those involved in bacterial motility and attachment, stress tolerance, regulation, transport, multidrug resistance, detoxification and signal transduction. Our qRT-PCR analysis of 23 selected genes indicated that the expression of 19 of these genes could be restored to wild-type by addition of exogenous BDSF. Complementation with BCAM0581 in trans however restored expression of all of these genes to wild-type. The findings suggest that the regulatory role of BCAM0581 in B. cenocepacia is not restricted to the synthesis of BDSF. Deng and colleagues [START_REF] Deng | Differential modulation of Burkholderia cenocepacia virulence and energy metabolism by the quorum-sensing signal BDSF and its synthase[END_REF] have proposed that BCAM0581, which is a member of the crotonase/enoyl CoA hydratase superfamily of proteins, may have a metabolic function. Another possibility is that accumulation of a precursor of BDSF within the cell of the BCAM0581 mutant may have a regulatory impact that cannot be reversed by addition of exogenous BDSF. Currently nothing is known about the substrate(s) or the enzymatic action of BCAM0581. Nevertheless, addition of exogenous BDSF to the BCAM0581 mutant restores expression of the majority of genes and all of the phenotypes that we have tested (including virulence) to wild-type levels. Overall these findings indicate that BDSF signalling has a broad regulatory influence in B. cenocepacia, which reflects the regulatory impact of DSF in X. campestris [START_REF] He | Genome scale analysis of diffusible signal factor regulon in Xanthomonas campestris pv. campestris: identification of novel cellcell communication-dependent genes and functions[END_REF].

A number of laboratories have proposed criteria to define signalling molecules involved in intraspecies communication [START_REF] Monds | Metabolites as intercellular signals for regulation of community-level traits[END_REF]Winzer et al., 2002). Such signals should be produced during specific stages of growth, under certain physiological conditions, or in response to changes in the environment; the signal should accumulate extracellularly and be recognised by a specific receptor; accumulation of the signal should generate a concerted response, once a critical [START_REF] Deng | Differential modulation of Burkholderia cenocepacia virulence and energy metabolism by the quorum-sensing signal BDSF and its synthase[END_REF][START_REF] Ryan | Intraspecies signalling involving the diffusible signal factor BDSF (cis-2-dodecenoic acid) influences virulence in Burkholderia cenocepacia[END_REF] indicates that BDSF satisfies all of these criteria and thus can be considered a true cell-cell signal molecule.

It is established that B. cenocepacia also utilises N-AHLs as intracellular signals to regulate gene expression through the two LuxR family proteins CepR and CciR [START_REF] O'grady | Reciprocal regulation by the CepIR and CciIR quorum sensing systems in Burkholderia cenocepacia[END_REF]. This raises the issue as to the extent to which the regulatory influences of the two classes of signal overlap. Comparison of the BDSF regulon described here with those genes whose expression is regulated by CepR and CciR [START_REF] O'grady | Reciprocal regulation by the CepIR and CciIR quorum sensing systems in Burkholderia cenocepacia[END_REF] indicates only a partial regulatory overlap (Fig. S3).

Furthermore, we have previously shown that mutation of BCAM0581 does not influence the levels of N-AHLs produced by B. cenocepacia J2315 [START_REF] Ryan | Intraspecies signalling involving the diffusible signal factor BDSF (cis-2-dodecenoic acid) influences virulence in Burkholderia cenocepacia[END_REF] and the transcriptome profiling of O'Grady and colleagues [START_REF] O'grady | Reciprocal regulation by the CepIR and CciIR quorum sensing systems in Burkholderia cenocepacia[END_REF], which was done in the K56-2 strain, does not indicate any influence of CepR and CciR on expression of BCAM0581.

BCAM0227 is a novel sensor for DSF-family signals

The findings from the domain-swapping experiments provide direct evidence for a role for BCAM0227 in the perception of BDSF. This contention is supported by the observation that BCAM0227 is required for the induction of cblD gene expression in response to exogenously added BDSF. Furthermore BCAM0227 regulates a sub-set of genes under BDSF control but does not regulate expression of BCAM0581 and shows no alteration in BDSF levels (data not shown), consistent with a role in signal perception rather than signal generation. The sensory domain of BCAM0227 is however quite different from that of RpfC, the DSF sensor in xanthomonads.

Bioinformatics analysis predicts that RpfC has 5 transmembrane helices with periplasmic and cytoplasmic loops of less than 20 amino acids. In contrast, BCAM0227 is predicted to have 2 transmembrane helices and a large periplasmic loop of 300 amino acids (Fig. 4). The precise mechanism by which either sensor recognises BDSF or DSF remains to be determined. Transduction of the DSF signal in xanthomonads involves modulation of levels of the second messenger cyclic di-GMP through the action of RpfG, an HD-GYP domain regulator with cyclic di-GMP phosphodiesterase activity [START_REF] Ryan | Cell-cell signalling in Xanthomonas campestris involves an HD-GYP domain protein that functions in cyclic di-GMP turnover[END_REF][START_REF] Ryan | Cell-cell signal-dependent dynamic interactions between HD-GYP and GGDEF domain proteins mediate virulence in Xanthomonas campestris[END_REF]. In the genome of X. campestris, rpfG is part of the rpfGHC operon that is convergently transcribed with rpfF, which encodes the DSF synthase. Although nothing is known of the BDSF signal transduction mechanism beyond BCAM0227, we speculate that this may involve BCAM0228, which encodes an OmpR-like two component response regulator with a DNA-binding domain, rather than an RpfG-like regulator.

Notably, mutation of BCAM0227 affects expression of only a subset of the genes within the BDSF regulon. Furthermore BCAM0227 does not influence all phenotypes that are regulated by BCAM0581. For example, although BCAM0227 is involved in biofilm formation and virulence, it does not affect motility or adherence to mucin. This partial overlap of regulatory influence may suggest the existence of further sensing mechanisms for BDSF in B. cenocepacia. Although bioinformatic studies of B. cenocepacia identified no homolog of RpfC, a sensor kinase related to BCAM0227 (BCAM0218) was identified. However RT-PCR experiments showed that the putative BCAM0218 sensor did not regulate any of a number of genes under BDSF regulation and domain-swapping experiments using the input domain of BCAM0218 provided no evidence for a role in BDSF perception (data not shown).

Hence the nature of any additional sensory mechanisms for BDSF remains obscure.

Furthermore we cannot rule out that BDSF may have biosurfactant properties that play a biological role. For example other quorum sensing molecules from Gramnegative organisms have been shown to influence surface movement and attachment by their surfactant properties [START_REF] Daniels | Quorum sensing and swarming migration in bacteria[END_REF]2006).

The broader relevance of findings

Bioinformatic studies suggest that BDSF signalling is widespread in the genus organism may be able to sense a signal that it does not make. Such "eavesdropping" on bacterial signalling has been described for a number of bacteria and a range of signals. A pertinent example is the sensing of DSF by Pseudomonas aeruginosa, an organism that has no close homolog of RpfF (the DSF synthase of xanthomonads) or BCAM0581. DSF perception in P. aeruginosa is mediated by PA1396, a sensor kinase with an input domain of related amino acid sequence to the sensory input domain of RpfC. Signal perception leads to alteration of expression of genes involved in increased tolerance to cationic antimicrobial peptides (Ryan et al., 2008). P. aeruginosa can also recognize BDSF through the action of PA1396 suggesting a possible role for BDSF in interspecies signalling with B. cenocepacia that counterbalances the role for N-AHLs in (unidirectional) signalling from P. aeruginosa to B.cenocepacia. Intriguingly, [START_REF] Davies | A fatty acid messenger is responsible for inducing dispersion in microbial biofilms[END_REF] have recently shown that P. aeruginosa does make a signal of the DSF family (cis-2-decenoic acid) but this molecule is not recognised by BCAM0227 or PA1396 (unpublished data) and the synthase for cis-2-decenoic acid has not yet been identified.

In conclusion, we have shown that BCAM0227 is a sensor kinase that is an element of the BDSF mediated cell-cell signalling pathway of B. cenocepacia that contributes to the virulence of this organism. In the context of the global rise of anti-microbial resistance and the slower development of novel antibiotics, cell-cell signalling pathways are being evaluated as potential drug targets. Our findings make a contribution towards the greater understanding of the mechanisms of signal generation and perception that will be important for the development of novel therapeutic strategies to combat bacterial infections.

EXPERIMENTAL PROCEDURES

Bacterial strains, growth conditions and media.

The bacterial strains used in this study are listed in Supporting Table S4. Cultures were routinely grown at 37°C, in Luria broth (LB) with shaking or on 1.5% LB agar plates. For promoter::lux assays, strains were grown in 0.25% trypticase soy broth with 5% Bacto-Peptone (Difco™). When appropriate, the following concentrations of antibiotics were used: 50 µg/ml of gentamicin (Gm) and 100 µg/ml of tetracycline (Tc). Antibiotics were purchased from Sigma-Aldrich. Methods used for assay of virulence factors, adherence to mucin and motility are detailed in [START_REF] Ryan | Intraspecies signalling involving the diffusible signal factor BDSF (cis-2-dodecenoic acid) influences virulence in Burkholderia cenocepacia[END_REF]. Molecular biological methods such as isolation of plasmid and chromosomal DNA, PCR, plasmid transformation as well as restriction digestion were carried out using standard protocols [START_REF] Sambrook | Molecular cloning: a laboratory manual[END_REF]. PCR products were cleaned using the PCR purification kit (Qiagen™) and DNA fragments were recovered from agarose gels using Qiaquick minielute gel purification kit (Qiagen™). Oligonucleotide primers were purchased from Sigma-Genosys™.

Construction of chromosomal reporter fusions.

The protocol used for the construction of the lux reporter fusion for cblD was a modification of the RED recombination protocols previously described by [START_REF] Gerlach | Rapid engineering of bacterial reporter gene fusions by using red recombination[END_REF][START_REF] Lesic | Use of the lambda Red recombinase system to rapidly generate mutants in Pseudomonas aeruginosa[END_REF]. The cblD gene was amplified by PCR using forward and reverse primers cblD-F (5'GGATCCAAGCGCCAGCAGGACGTGCTGATG-3') and cblD-R (5'GGATCC GAGACTGGTGGACGACGGC-3') to include the region 250 bp upstream of the gene and was cloned into pGLuc. The cblD::lux fusion removed by restriction digest using HindIII and XbaI and cloned into pME6032-RedS cut with the same enzymes.

This construct was electroporated into B. cenocepacia wild-type or BCAM0581 mutant. Transconjugants were grown in LB to an optical density at 600 nm of 0.4 at 37°C and recombination was induced by addition of 0.2% L-arabinose for 1.5 h. The correct integration of the reporter cassette was confirmed by colony PCR using a reverse primer specific to the reporter and a forward primer specific to the upstream region of the target gene.

Transposon mutagenesis and screening.

A library of mutants in the B. cenocepacia BCAM0581 strain carrying the cblD::lux fusion was constructed using Mariner transposon vector pBT20 as previously described [START_REF] Kulasekara | A novel two-component system controls the expression of Pseudomonas aeruginosa fimbrial cup genes[END_REF] Qpix robot (Genetix™). The luciferase activity lysates of bacteria from each well were quantified as described previously [START_REF] Deiwick | Environmental regulation of Salmonella pathogenicity island 2 gene expression[END_REF]. Briefly, equal amounts of bacterial cells as adjusted by determination of optical density at 600 nm (OD600) of the cultures were harvested by centrifugation, and lysed. Luciferase activity in bacterial cell lysates was assayed using Luciferase Assay Kit (Promega™) and quantified in black microtiter plates by a Wallac Victor 2 1420 Multilabel Counter plate reader (Perkin Elmer™).

Identification of transposon insertion sites.

Genomic DNA was extracted from transposon mutants of interest as previously described [START_REF] Sambrook | Molecular cloning: a laboratory manual[END_REF]. The nucleotide sequence flanking the transposon insertion site was amplified by PCR as previously described [START_REF] Friedman | Genes involved in matrix formation in Pseudomonas aeruginosa PA14 biofilms[END_REF]) using the primers: Rnd1-TnM (5'-GTGAGCGGATAACAATTTCACACAG-3'), and Rnd2-TNm (5'-ACAGGAAACAGGACTCTAGAGG-3'). Amplicons were sequenced using the TnM specific primer: TnMseq (5'-CACCCAGCTTTCTTGTACAC-3'). Sequence data generated was examined with BLASTn, BLASTx (www.ncbi.nlm.nih.gov/BLAST/), and VGE BLAST (www.vge.ac.uk/BLAST/) software. Bioinformatic analysis of the BCAM0227 sensor kinase was performed using SMART (http://smart.embl-heidelberg.de/).

Cloning and expression of input-BCAM0227-RpfC chimeric protein.

For the construction of BCAM0227-RpfC chimeric sensor protein, the coding sequences for the input domain of BCAM0227 modules (amino acid residues 1 -383) and N-terminal of RpfC (residues 249 -726) were amplified using primers InBCAM0227-F (5'-GGATCCATGGGGGCACCCTGGCCGACCGC-3') and BCAM0227-R (5'-AAGGCTTGCGCACCAGCAGCCACCACAC-3'); RpfCCTerm-F (5'-AAGGCTTGCGATGACCCGTGCAGTGCGC-3') and RpfCCterm-R (5'-GGATCCCTATTCGTTGCTACGGGGGGA-3') respectively. These primers contain restriction sites for HindIII and BamHI (underlined). These amplified products were diluted, mixed and subjected to further PCR amplification using the primers F (5'-GGATCCATGGGGGCACCCTGGCCGACCGC-3') and R (5'-AAGCTTCTATTCGTTGCTACGGGGGGA-3'). The final amplicon was cloned into TOPO (Invitrogen™) and verified by sequencing. The product was excised from 19 same enzymes. This construct was conjugated into X. campestris strains 8004 (wildtype), the rpfC mutant and rpfCrpfF double mutant using triparental mating as described previously (Ryan et al., 2008).

Construction of complementing clone.

To complement the B. cenocepacia BCAM0227 mutant, the wild-type gene BCAM0227 was amplified by PCR from genomic DNA of B. cenocepacia with the primers (see Supporting Table S4) with the following thermal cycling conditions: 95°C for 5 min followed by 30 cycles of 95°C for 45 s, 60°C for 45 s, and 72°C for 2 min, with a final extension at 72°C for 10 min. The resulting amplicon was digested with the restriction enzymes HindIII and BamHI and ligated into pBBR1MCS cut with the same enzymes. The resulting plasmid, pBCAM0227, was introduced into the B. cenocepacia BCAM0227 mutant by conjugation.

RNA manipulations.

B. cenocepacia strains was subcultured to obtain an initial optical density at 600 nm of 0.8 in LB and grown without selection. Total RNA was isolated using a RiboPure bacterial RNA isolation kit (Ambion™) and subjected to DNase treatment. Samples were confirmed to be free of DNA prior to cDNA synthesis by PCR using Taq polymerase (Invitrogen™).

Microarray sample preparation and hybridization.

Three independent RNA samples from B. cenocepacia strains grown for 24 h were used in microarray experiments. Gene expression profiles were generated using custom B. cenocepacia J2315 microarrays (Agilent™) [START_REF] Drevinek | Diversity of the parB and repA genes of the Burkholderia cepacia complex and their utility for rapid identification of Burkholderia cenocepacia[END_REF]. A reference pool of J2315 cDNA was fluorescently labelled with Cy3 while the test cDNA samples (WT J2315, BCAM0581 and BCAM0227) were fluorescently labelled with Cy5. cDNA generation and labelling was performed using the CyScribe Post-Labelling kit (GE Healthcare™) according to the manufacturer's protocol. Spike-in controls (Agilent™) were included into the labelling procedure for quality control purposes. cDNA purification was performed by ethanol precipitation and the labelled cDNA was purified using a CYSCRIBE GFX purification kit and eluted with water.

Hybridization and washing of arrays were performed according to the two-colour

Microarray data analysis.

Microarray data analysis was performed using GeneSpring GX 11.1 software (Agilent™). The enhanced Agilent FE import method was used to preprocess the initial data before per-spot and per-chip normalizations were performed for all arrays.

After filtering, genes were selected on the basis of changes, for which a 2-fold cutoff was used for comparison. The entire microarray data set has been deposited in the ArrayExpress database http://www.ebi.ac.uk/arrayexpress under accession number (waiting accession number).

Quantitative RT-PCR.

RT-PCR was performed using a cDNA synthesis kit (Invitrogen™). For quantitative RT-PCR, quantification and melting curve analyses were performed with a LightCycler ® (Roche™) and QuantiFast SYBR Green (Qiagen™) according to manufacturers' instructions. RT-PCR reactions were performed in triplicate, and the data shown are from at least two independent experiments. Expression levels were normalized to levels of the 16S RNA gene transcript in each experiment.

Antimicrobial susceptibility testing.

Susceptibility experiments were carried out as previously described by [START_REF] Hassett | Quorum sensing in Pseudomonas aeruginosa controls expression of catalase and superoxide dismutase genes and mediates biofilm susceptibility to hydrogen peroxide[END_REF] with the following modifications. Selected bacterial strains were grown shaking aerobically over night at 37°C. For sensitivity experiments for most compounds were carried out by adding a 0.1 ml suspension of stationary phase bacteria spread over the entire surface of LB agar plates. After 17 h of growth at 37°C, the stationary phase 21 organisms were removed from the agar surface with 10 ml of 0.9% saline and a sterile glass rod. Filter paper disks impregnated with various concentrations of selected compounds were placed on the agar surface. For H 2 O 2 and NaOCl sensitivity experiments, suspensions (0.1 ml) were diluted in 3 ml of molten soft agar (0.6%), poured onto the surface of L agar plates, and the agar was allowed to solidify. Filter paper disks (7 mm, Whatmann) saturated with 10 ml of 30% H 2 O 2 were placed on the agar overlay. Sensitivity to each compound was then recorded as the mean ± SE (n = 9) of the diameter of growth inhibition after 24 h of growth at 37°C. The results represent the average of three independent replicas.

Cultivation and imaging of biofilms in flow chambers.

Biofilms were grown in flow chambers with individual channel dimensions of 1 × 4 × 40 mm. The flow system was assembled and prepared as described by [START_REF] Sternberg | Growing and analyzing biofilms in flow cells[END_REF]. The flow chambers were inoculated by injecting 350 µl of overnight culture diluted to an OD600 of 0.01 into each flow channel with a syringe. After inoculation, flow channels were left without flow for 1 h, after which medium flow at a linear rate of 0.2 mm/s was started using a Watson Marlow 205S peristaltic pump. All microscopic observations and image acquisitions were performed on an LSM510 confocal laser scanning microscope (CLSM) (Carl Zeiss™) equipped with detectors and filter sets for monitoring Syto9 fluorescence. Simulated three-dimensional images, shadow projections and sections were generated using the Imaris™ software package.

Cytotoxicity assay.

Cytotoxicity was assessed by measuring release of lactate dehydrogenase (LDH) from CHO (Chinese hamster ovary) cells. The CHO cells were routinely grown in F-12 medium supplemented with 10% fetal bovine serum (FBS) and 2 mM glutamine.

Prior to infection, confluent CHO cells were washed and incubated with F-12 containing 1% FBS and 2 mM glutamine. B. cenocepacia was grown overnight in LB, subcultured into fresh LB, and grown to mid-log phase.

G. mellonella killing assays.

Larvae were stored in wood chips at 4°C. A 10-µl Hamilton syringe was used to inject 5-µl aliquots of bacterial suspensions into G. mellonella via the hindmost left proleg. 10-fold serial dilutions of bacteria starting at 2 x 10 8 CFU per ml were made in PBS supplemented with 100 µg/ml of ampicillin. 5 µl of each dilution was injected into each of ten larvae. Control larvae were injected with 5 µl of PBS plus 100 µg/ml of ampicillin. Following injection, larvae were placed in a static incubator in the dark at 30°C, the optimum temperature for insect growth and development. Larvae were scored as dead or alive at 48 and 72 h post-infection. The minimum inoculum required to kill all 10 larvae was determined for each strain.

Mouse infection model.

C57Bl 6 mice were infected with B. cenocepacia strains by using the agar bead model of lung infection [START_REF] Bylund | Exopolysaccharides from Burkholderia cenocepacia inhibit neutrophil chemotaxis and scavenge reactive oxygen species[END_REF][START_REF] Bylund | Burkholderia cenocepacia induces neutrophil necrosis in chronic granulomatous disease[END_REF][START_REF] Cieri | Correlation between an in vitro invasion assay and a murine model of Burkholderia cepacia lung infection[END_REF]. Bacteria were incorporated into agar beads, and 1 x 10 9 CFU were inoculated into each mouse intra-tracheally. The animals were sacrificed after 8 days post-infection, and the lungs and spleens were harvested, homogenized, and cultured on LB agar to assess the severity of infection.

FIGURES AND TABLES LEDGENDS

Table 1. Location of mariner transposon insertions in the Burkholderia cenocepacia J2315 genome leading to lack of responsiveness to exogenous BDSF. Table 2. Mutation of BCAM0581 and BCAM0227 in Burkholderia cenocepacia affects expression of genes in a number of functional groups, as revealed by microarray analysis. The sensor kinase BCAM0227 regulates a sub-set of the genes regulated by the BDSF synthase BCAM0581. Table 3. Comparison of effects of mutation of BCAM0581 and BCAM0227 on expression of selected genes in Burkholderia cenocepacia as measured by microarray and quantitative RT-PCR. Table 4. BDSF signalling is associated with lethality in an agar bead mouse model of infection. Mice were infected intra-tracheally using agar beads with 1 x 10 9 CFU of B. cenocepacia J2315 wild-type or strains with mutations in BCAM0581 or BCAM0227.

Survival of mice was monitored over a period of 8 days. Animals were sacrificed 8 days post-infection, and the bacterial loads in the lungs and spleen were determined.

F o r P e e r R e v i e w

  ), including 2-heptyl-4(1H)-quinolone (HHQ), but does not synthesize 2-heptyl-3-hydroxy-4(1H)-quinolone (PQS), which is a signal molecule in Pseudomonas aeruginosa. The proposed role of HHQ in signalling in B. cenocepacia has yet to be confirmed. Until recently, these molecules (N-AHLs and AHQs) were the only cell-cell signals defined in B. cenocepacia. However, work by Boon and colleagues demonstrated that B. cenocepacia J2315 also produces cis-2-dodecenoic acid, a molecule structurally related to the diffusible signal factor (DSF; 11-methyl-cis-2-dodecenoic acid) found in xanthomonads. Synthesis of cis-2-dodecenoic acid by B. cenocepacia requires

  , BCAM1242) and in genes encoding regulatory proteins including a putative diguanylate cyclase (BCAM2836), a putative two-component sensor kinase (BCAM0227), methyl-accepting chemotaxis protein (BCAM1572) were identified. Since a histidine kinase (RpfC) is implicated in the perception of BDSFlike molecules in X. campestris, subsequent work focused on the potential role of the putative sensor kinase BCAM0227 in BDSF perception.

  tested. Constructs expressing either full-length RpfC or the chimaeric protein BCAM0227 (residues 1-367)-RpfC (residues 172-726) were

  remained obscure. Here we have utilized a transposon mutagenesis screen, transcriptome profiling and domain swapping experiments to identify BCAM0227 as a sensor for BDSF and to establish the extent of the BDSF regulation in B. cenocepacia. Our findings indicate that BCAM0227 acts in regulation of a subset of genes of the BDSF regulon and has a role in virulence of B. cenocepacia in both insect and mammalian models.BCAM0227 was one of a number of genes in which transposon insertion led to a loss of responsiveness to exogenous BDSF, as measured by cblD expression. The role of

  been reached; most importantly, the cellular response to the signal should extend beyond physiological changes required to metabolize or detoxify it. The work shown here and previously

Burkholderia.

  Homologues of BCAM0227 occur widely within bacteria of the Bcc, including B. pseudomallei and B. ambifaria, as well as in other species that are plant associated. Homologs of BCAM0581, which encodes the BDSF synthase, are also found in a number of Burkholderia spp. but not in B. pseudomallei. The occurrence of the sensor in the absence of a signal synthase in B. pseudomallei suggests that this

  . Briefly, 100 ng of transposon-containing plasmid was electroporated into newly prepared electrocompetent B. cenocepacia BCAM0581 mutant cells. Strains carrying the transposon insertion were identified by selection on Gm. The library of mutants was screened for loss of responsiveness of cblD::lux expression to exogenous BDSF. Strains were inoculated into microtiter wells containing LB medium with 50nmol BDSF (Allesen-Holm et al., 2006a) by using a

  expression analysis protocol (Agilent™) with minor modifications where the 25× fragmentation buffer was omitted and the cDNA mix and 10× blocking agent were heat-denatured for 3 min at 98°C and cooled to room temperature before adding the hybridization buffer. Washing of microarrays was performed including acetonitrile as well as stabilization and drying solution (Agilent™). A G2565 BA microarray scanner and the scan control software (Agilent™) were used. Scanning resolution was set to 5 µm and the scan region was adjusted to 61 × 21.6 mm. The extended dynamic range function was switched on with 100% and 10% photomultiplier gain settings. Images were analysed with the feature extraction software (Agilent™).

  2.5×10 5 CHO cells per well were infected with B. cenocepacia at an initial multiplicity of infection (MOI) of 50:1 in triplicate experiments. Culture supernatants were collected after 6 hours of infection and centrifuged for 10 min at 3,220×g to sediment bacteria and CHO cells.LDH in the supernatant was measured with a Roche LDH kit in accordance with the instructions. Percentage LDH release was calculated relative to that of the uninfected control, which was set at 0% LDH release, and that of cells lysed by Triton X-100, which was set at 100% LDH release.

  ., Hardie, K.R., andWilliams, P. (2002). Bacterial cell-to-cell communication: Sorry, can't talk now -gone to lunch! Curr Opin Microbiol 5, 216-222.

Fig. 1 .

 1 Fig. 1. Genes differentially expressed in BCAM0581 or BCAM0227 mutants compared to the wild-type B. cenocepacia J2315 as revealed by microarray analysis. (A) Venn diagrams showing the overlap of genes whose expression is influenced by both BCAM0227 and BCAM0581: (i) up-regulated; (ii) down-regulated gene subsets.

  Detailed analysis of expression of (i) genes involved in the regulation of stress response and detoxification and (ii) genes involved in the regulation of virulence and pathogenesis from microarray analysis representing the effects of mutation of BCAM0227 and BCAM0581 on global gene expression with decreased (green), increased (red) or no significant change (grey) in expression in each mutant compared to J2315 according to microarray analysis. Only changes in transcript level >2 fold were deemed significant.

Fig. 2 .

 2 Fig. 2. Mutation of genes encoding BCAM0581 and BCAM0227 does not have the same effect on motility and adherence to porcine mucin. (A) Although the BCAM0581 mutant showed reduced motility in 0.6% Eiken agar, no alteration was seen in the BCAM0227 mutant. Motility of the BCAM0581 mutant was restored to wild-type level by complementation or addition of 50 µM BDSF to the plates. Each experiment was performed at least three times in triplicate. (B). The BCAM0581 mutant showed a reduction in adherence to porcine mucin while mutation of BCAM0227 mutant had no effect. Adherence of the BCAM0581 mutant was restored to wild-type level by complementation. Each experiment was performed at least three times in triplicate. Error bars represent the standard errors of the mean.

Fig. 3 .

 3 Fig. 3. Mutation of genes encoding BCAM0581 and BCAM0227 has similar effects on biofilm architecture within biofilms of B. cenocepacia J2135. Images are of 4-dayold biofilms in flow cells that were stained with both Syto9 (green) to stain all bacteria and propidium iodide (red) to stain dead cells and extracellular DNA. Separate images for each fluorescent dye were merged. The central pictures show horizontal CLSM optical sections, and the flanking pictures show vertical CLSM optical sections. Scale bars, 20 µm. The B. cenocepacia J2135 strains were (A) wildtype; (B) BCAM0581 mutant; (C) BCAM0581 complemented mutant; (D) BCAM0581 mutant plus 100 nM BDSF. (E) BCAM0227 mutant; (F) BCAM0227 complemented mutant.

Fig. 4 .

 4 Fig. 4. BCAM0227 is a two-component sensor with a role in BDSF perception. (A) Domain architecture of BCAM0227 and of RpfC, the DSF/BDSF sensor of xanthomonads as revealed by the SMART algorithm. Key: Bars: transmembrane helices; HisKA: histidine kinase phosphoacceptor domain: HATPase_c: histidine

Fig. S2 .

 S2 Fig. S2. Domain swapping experiments between BCAM0227 and RpfC indicate that BCAM0227 is a sensor of BDSF. These experiments were conducted by expression of chimeric sensors in the rpfFrpfC double mutant of Xcc and measurement of the effects of addition of exogenous BDSF on the DSF-regulated phenotype of endoglucanase production. Expression of RpfC in this mutant background has no effect on endoglucanase activity (which is very low), but endoglucanase activity was increased upon addition of exogenous BDSF at 50 µM. Similar effects are also seen after expression of a chimeric sensor protein in which the input domain and HisKA domain of BCAM0227 replaces the input domain and HisKA domain of RpfCgenerating chimeric sensor protein BCAM0227 (residues 1-594)-RpfC (residues 253-726); the rpfFrpfC strain expressing this chimeric protein can recognize synthetic BDSF at 50 µM to induce endoglucanase activity. In contrast, addition of trans-11-methyl-2-dodecenoic acid and cis-2-decenoic acid (cis-DA) at 50 µM did not induce endoglucanase production in this strain, although some effects were seen at the higher concentration of 1 mM.

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 5 .

 1235 Fig. 1. Genes differentially expressed in BCAM0581 or BCAM0227 mutants compared to the wildtype B. cenocepacia J2315 as revealed by microarray analysis. 105x127mm (300 x 300 DPI)

Table 1 .

 1 BCAM0581, BCAM0227 mutants (as reported here) and cepR or cciR mutants as reported by O'Grady et al., (2009) are compared. Location of transposon insertions found in the Burkholderia cenocepacia 2 J2315 genome leading to lack of responsiveness to exogenous BDSF. 3

	1
	Disrupted	Polar effect	Number of different	Gene product
	gene a	possible? b	transposon mutants c
	BCAL0223	N			1	50S ribosomal protein L1
	BCAL1543	N			2	major facilitator superfamily protein
	BCAL1740	N			2	major facilitator superfamily protein
	BCAL1791 BCAL1957	F N o r Y		1 1	hypothetical protein putative sugar transferase
	BCAL2276	N	P	1	putative FAD dependent oxidoreductase
	BCAL2731 BCAL2745	N N	3 1 r e e	ATP-dependent Clp protease adaptor protein putative CoA transferase family protein
	BCAL3149	N			R 1	hypothetical protein
	BCAM0227 BCAM0580	N Y			2 2	hybrid two-component system kinase-response regulator protein v e i e RNase II stability modulator
	BCAM0700	N			2	w TetR family regulatory protein
	BCAM1009	N			2	putative acyltransferase
	BCAM1242	N			1	hypothetical protein
	BCAM1572	N			1	methyl-accepting chemotaxis protein
	BCAM1685	N			1	nirB; putative nitrite reductase
						32

Table 2 .

 2 Classification of a selection of BDSF-regulated genes by functional groups. Gene numbers are from http://www.sanger.ac.uk/Projects/B_cenocepacia. Annotations and functional assignments are from B. cenocepacia J2315 (Holden et al., 2009). b. A complete outline of expression changes in both BCAM0581 and BCAM0227 mutant backgrounds is detailed in TableS1 and S2.c. NC: No significant (< 2 fold) change in gene expression compared to wild-type.

	Fold changes b

Table 3 .

 3 Summary of microarray and quantitative RT-PCR analysis of selected genes.

	Microarray gene expression
	relative to wild-type

b

RT-

qPCR gene expression analysis relative to wild- type c Gene name a Gene product and function

  Gene numbers are from http://www.sanger.ac.uk/Projects/B_cenocepacia. Annotations and functional assignments are from B. cenocepacia J2315. b. Change (fold) in BCAM0581 or BCAM0227 mutant compared to wild-type determined by microarray analysis.

		a
			BCAM0581	BCAM0227	BCAM0581	BCAM0227	BCAM0581
							(+BDSF)
	BCAM0495	putative DNA-binding protein	-12.1	-5	-9.5 (1.5)	-6.1 (0.4)	ND d
	BCAM1265	putative amino acid permease	-2.5	NC d	-2.0 (0.3)	ND	-3.1 (0.4)
	BCAM1619	putative DNA-binding cold-shock protein	-3.1	-2.5	-2.9 (0.9)	-3.1 (0.3)	ND
	BCAM2039	putative transcriptional regulator	-42.5	-33.5	-7.3 (1.1)	-5.2 (0.4)	ND
	BCAM2041	type III secretion system protein (BcscR)	-2.2	NC	-2.2 (0.4)	-2.0 (0.5)	ND
	BCAM2045 BCAM2051 BCAM2758 BCAM2759 BCAL0114 BCAL0132 BCAL0813 BCAL2820 BCAS0297 BCAS0299 BCAS0353 BCAS0387 a. F type III secretion system protein (BcscU) type III secretion system protein (BcscU) cable pilus (CblS) cable pilus (CblD) flagellin (FliC) chemotaxis protein methyltransferase (CheR) RNA polymerase factor sigma-54, (RpoN) efflux system outer membrane protein (OprM) hypothetical protein flp type pilus subunit (PilA) putative aldose 1-epimerase putative haloacid dehalogenase-like hydrolase o r P e e r R e	-2.8 -2.7 -2.1 -2 5.5 2.7 -2.5 -125 -5.3 -2.1 -2.6 -4.3	-2.3 -2.1 -2.1 -2 5.4 2.2 NC NC -2.1 NC -3 NC	-2.5 (0.9) -2.6 (0.8) -2.2 (0.5) -2.5 (0.6) 6.3 (1.2) 3.1 (0.9) -2.9 (0.4) -15.2 (1.6) -2.0 (1.2) -2.5 -4.0 (0.2) -3.1 (0.2)	-2.1 (0.7) -2.1 (0.9) -2.1 (0.3) -2.4 (0.9) 6.9 (1.8) 2.2 (1.2) ND -2 (1.1) -2.3 (0.9) ND -3.5 (0.3) ND	ND ND ND ND ND ND ND ND -3.8 (0.4) ND -2.2 (0.5) -2.5 (1.1)
		v i e
			w

Table 4 .

 4 BDSF signalling is associated with lethality in an agar bead mouse model of infection. Mice were infected intra-tracheally using agar beads with 1 x 10 9 CFU of B. cenocepacia J2315 wild-type or strains with mutations in BCAM0581 or BCAM0227. Survival of mice was monitored over a period of 8 days. Animals were sacrificed 8 days post-infection, and bacterial loads in the lungs and spleen were determined.

	Strain	No. of	No. of	Log CFU per (mean +/-S.D.)
		mice per	surviving
		group	mice 8 days
			post-infection
				Lung	Spleen
	J2315	6	3	5.3 (+/-0.01)	3.3 (+/-0.05)
	BCAM0581	6	6	0.8 (+/-0.01)	0.2 (+/-0.01)
	BCAM0227	6	6	1.1(+/-0.01)	0.7 (+/-0.01)
	BCAM0581	6	3	6.3 (+/-0.01)	2.3 (+/-0.05)
	(pBAM0581)
	BCAM0227	6	3	5.9 (+/-0.01)	2.9 (+/-0.05)
	(pBCAM0227)

Molecular Microbiology

ACKNOWLEDGMENTS

The authors are indebted to Michael Hensel and George O'Toole for providing them with strains and plasmids to assist in this study. Y.M., K.T., J.M.D. and R.P.R. were supported in part by grants awarded by the Science Foundation of Ireland (SFI 07/IN.1/B955 to J.M.D. and SFI 09/SIRG/B1654 to R.P.R.). A.S and E.M. acknowledge funding from the US Cystic Fibrosis Foundation Therapeutics program for the B. cenocepacia microarray program (grant MAHENT06V0)

SUPPORTING INFORMATION

Table S1. A complete list of genes with altered expression in BCAM0581 mutant compared to wild-type J2315. Table S2. A complete list of genes with altered expression in BCAM0227 mutant compared to wild-type J2315. The genes that are also regulated by BCAM0581 (BDSF synthase) are indicated. Table S3. Antimicrobial susceptibilities of B. cenocepacia J2315, BCAM0227 and BCAM0581 mutants Table S4. Bacterial strains, plasmids and primers used in this study.


A Athanassoulis
email: agis.athanassoulis@math.polytechnique.fr

T Paul

F Pezzotti

M Pulvirenti
email: pulvirenti@mat.uniroma1.it

SEMICLASSICAL PROPAGATION OF COHERENT STATES FOR THE HARTREE EQUATION

Introduction

Let us consider the Hartree equation in R d :

iε∂ t Ψ ε (x, t) = -ε 2
2 ∆Ψ ε (x, t) + (V (x, t) + U (x, t)) Ψ ε (x, t),

Ψ ε (x, 0) = Ψ ε 0 (x), (1)
where

V (x, t) = φ(|x -y|)|Ψ ε (y, t)| 2 dy (2)
is a self-consistent potential given by a smooth two-body interaction, φ : R R, even, and U (•, t) : R d R for all t ≥ 0, is a smooth external potential (see the next section for the precise assumptions on φ and U ).

In a recent paper [START_REF] Athanassoulis | Strong semiclassical approximation of Wigner functions for the Hartree dynamics[END_REF] the authors of the present one considered the semiclassical limit of the version of the Hartree equation corresponding to mixed states, for initial data whose Wigner functions do not concentrate at the classical limit.

The problem we deal with in the present paper is the semiclassical asymptotics for [START_REF] Athanassoulis | Strong semiclassical approximation of Wigner functions for the Hartree dynamics[END_REF] when the initial state is a coherent state centered around the point q, p of the classical phase space, namely:

Ψ ε 0 (x) = ε -d 4 a 0 x -q √ ε e i p•(x-q) ε := ψ a 0 qp (x). (3)
This problem was studied in [START_REF] Lions | Sur les mesures de Wigner[END_REF] in the kinetic (Wigner) picture, see Théorème IV.2 therein. There it is shown that, under appropriate conditions, the solution W ε of the Wigner equation corresponding to the dynamics (1) namely

∂ t W ε + k • ∂ x W ε = i ε(2π) d e iξy V (x + εy 2 , t) -V (x - εy 2 , t) dy W ε (x, k -ξ)dξ + + i ε(2π) d e iξy U (x + εy 2 , t) -U (x - εy 2 , t) dy W ε (x, k -ξ)dξ, (4)
where V (x, t) is the same as in (1) equivalently written as

V (x, t) = φ(|x -y|)W ε (y, k, t)dkdy, (5)
converges, in weak * -sense, to the solution of the (classical) Vlasov equation

∂ t f + k • ∂ x f -∂ x V 0 (x, t) • ∂ k f -∂ x U (x, t) • ∂ k f = 0, f (x, k, t)| t=0 = f 0 (x, k), (6)
where V 0 (x, t) = φ(|x -y|)f (y, k, t)dkdy, and U (x, t) is the same as in [START_REF] Athanassoulis | Strong semiclassical approximation of Wigner functions for the Hartree dynamics[END_REF] . The initial condition for [START_REF] Hagedorn | Semiclassical quantum mechanics. I. The → 0 limit for coherent states[END_REF] is given by f 0 = w - * lim ε→0 W ε 0 . It is easy to check that the conditions of Théorème IV.2 in [START_REF] Lions | Sur les mesures de Wigner[END_REF] are satisfied for W ε 0 (x, v) = W ε [Ψ ε 0 ](x, v), Ψ ε 0 as in equation [START_REF] Carles | Nonlinear coherent states and Eherenfest time for Schrödinger equation[END_REF]. In that case (under appropriate assumptions on the pairinteraction potential φ and the external potential U ) it can be seen that the Wigner measure of the wave function verifies

W ε [Ψ ε ](x, k, t) δ(x -X(t))δ(k -K(t)), as ε → 0,
where Ẋ(t) = K(t), K(t) = -∇U (X(t), t), X(0) = q, K(0) = p. In that sense, the semiclassical limit of the problem (1) is known to be the Vlasov dynamics [START_REF] Hagedorn | Semiclassical quantum mechanics. I. The → 0 limit for coherent states[END_REF], since it is easy to recognize that, due to the smoothness of the potentials, the limiting measure δ(x -X(t))δ(k -K(t)) is the unique (weak) solution of the Vlasov equation with initial datum δ(x -q)δ(k -p).

The goal of the present work is to strengthen this approximation. First of all, we construct L 2 approximations, as opposed to the with weak * -limit, and this yields an explicit control of the error in ε which allows to recover the shape with which W ε concentrates to a δ in phase-space.

Main result

We will consider the Hartree equation in R d :

iε∂ t Ψ ε (x, t) = -ε 2 2 ∆Ψ ε (x, t) + (V (x, t) + U (x, t)) Ψ ε (x, t), Ψ ε (x, 0) = Ψ ε 0 (x), (7)
where

V (x, t) = φ(|x -y|)|Ψ ε (y, t)| 2 dy (8)
The initial condition will be of the form

Ψ ε 0 (x) = ε -d 4 a 0 x -q √ ε e i p•(x-q) ε := ψ a 0
qp and we will make the following assumptions on a 0 , φ and U :

Assumption 1. a 0 L 2 = Ψ ε 0 L 2 = 1, x A ∂ B x a 0 (x) ∈ L 2 for any pair A, B ∈ N d with |A| + |B| 3, x i |a 0 (x)| 2 dx = 0, ∀ i = 1 . . . d. ( 9
)
k i | a 0 (k)| 2 dk = 0, ∀ i = 1 . . . d. ( 10
) Assumption 2. C 3 b (R) φ even Assumption 3. U ∈ C 1 R + t , C 3 b (R d x ) .
Here and henceforth we denote by C k b (R m ) the space of continuous and uniformly bounded functions on R m whose all derivatives up the order k are also continuous and uniformly bounded.

Theorem 2.1. Under Assumptions 1, 2 and 3 there exists a constant C such that, ∀ t ≥ 0,

Ψ ε (•, t) -e i L(t) ε +iγ(t) ψ βt q(t)p(t) L 2 Ce Cte C e C t • √ ε. ( 11
)
where β t is the solution of

i∂ t β t (x) = - ∆ 2 + φ (0)x 2 2 + < x, ∇ 2 U (q(t), t)x > 2 β t (x), (12)
β 0 (x) = a 0 (x), (13)
γ(t) = - φ (0) 2 t 0 η 2 |β s (η)| 2 dηds, (14)
(q(t), p(t)) is the Hamiltonian flow associated with p 2 2 + U (q, t) + φ(0) issued from (q, p),

L(t) := t 0 p(s) 2 /2 -U (q(s), s) -φ(0) ds
(the Lagrangian action along such Hamiltonian flow).

Remarks:

• As shown in the proof of the Theorem, the constant C depends only on d,

||U || W 3,∞ , ||φ|| W 3,∞ and sup |A|+|B| 3 ||x B ∂ A x a 0 || L 2 .
• Note that in the classical flow the nonlinear potential enters only via the inessential constant φ(0). Indeed, due to the symmetry and smoothness of φ, we have φ (0) = 0 so that, in case of concentration as ε → 0, the self-consistent field ∇V vanishes. • A similar problem for φ (0) ≥ 0 has been faced in [START_REF] Belov | Semiclassical Soliton-Type Solutions of the Hartree Equation[END_REF] in a semirigorous way. Here we treat the case φ (0) ≤ 0 as well and present an explicit control of momenta and derivatives of the solutions (see Lemma 2.3 below) which allow us to estimate the error in L 2 . • For a related result (Gross-Pitaevskii equation with a different scaling) see [START_REF] Carles | Nonlinear coherent states and Eherenfest time for Schrödinger equation[END_REF].

• Assumption 1 can be relaxed by dismissing equation [START_REF] Lions | Sur les mesures de Wigner[END_REF]. Indeed even if [START_REF] Lions | Sur les mesures de Wigner[END_REF] does not hold one can always make a change of variables x → x -x|a 0 (x)| 2 dx. However in that case one would have to adjust appropriately the external potential, which of course is not translation invariant.

3. Proofs 3.1. A Lemma. We first prove the following Lemma 3.1. b t (x) := e iγ(t) β t (x) as defined by [START_REF] Paul | Échelles de temps pour l'évolution quantique à petite constante de Planck[END_REF][START_REF] Reed | Methods of modern mathematical physics[END_REF]14) is the unique solution of the equation:

i∂ t + 1 2 ∆ b t (x) = φ (0) 2 |x -η| 2 |b t (η)| 2 dη b t (x) + <x,∇ 2 U (q(t),t)x> 2 b t (x), b 0 (x) = a 0 (x). ( 15
)
Proof. i∂ t b t (x) = -γ (t)b t (x) + e iγ(t) i∂ t β t (x).

(16) By virtue of equations ( 12), ( 13) and (14) we find

i∂ t b t (x) = φ (0) 2 η 2 |β t (η)| 2 dη b t (x) + e iγ(t) -∆ 2 β t (x) + φ (0) 2 x 2 β t (x) + <x,∇ 2 U (q(t),t)x> 2 β t (x) , b 0 (x) = a 0 (x), (17)
namely

i∂ t b t (x) = -∆ 2 b t (x) + φ (0) 2 x 2 b t (x) + φ (0) 2 η 2 |β t (η)| 2 dη b t (x) + <x,∇ 2 U (q(t),t)x> 2 b t (x), b 0 (x) = a 0 (x). ( 18
)
We first notice that the equation ( 12) for β t (x) is a linear Schrödinger equation with an harmonic potential; therefore the solution β t (x) of the initial value problem ( 12)-( 13) is uniquely determined in L 2 (R d ) and

β t L 2 = a 0 L 2 = 1, ∀ t ∈ R. ( 19
) As a consequence of that, it turns out that equation (18) can be rewritten as

i∂ t b t (x) = -∆ 2 b t (x) + φ (0) 2 x 2 |β t (η)| 2 dη b t (x) + φ (0) 2 η 2 |β t (η)| 2 dη b t (x) + <x,∇ 2 U (q(t),t)x> 2 b t (x), b 0 (x) = a 0 (x).
(20) Furthermore, it is easy to check that if

xa 0 (x), ∂ x a 0 (x) ∈ L 2 (R d ), (21)
then xβ t (x), ∂ x β t (x) ∈ L 2 (R d ), for all t, (22)
(see Observation 4.3 below). Condition (21) is satisfied under Assumption 1, so the property (22) holds and, in particular, there exists a constant C finite for any time t such that

|η| 2 |β t (η)| 2 dη < C, ∀ t ∈ R. (23)
Thus, by virtue of (23) and of Assumptions 1, 2 and 3, it follows that the initial value problem (20) is guaranteed to have a unique solution in L 2 and, clearly, b t L 2 = a 0 L 2 = 1, ∀ t. In fact, the equation for b t (x) has turned to be a linear Schrödinger equation with an harmonic potential (and all constants appearing in the potential terms are finite thanks to Assumptions 2 and 3 and to (23)). Now, it remains only to recognize that (20) is exactly the same as (15). To this end it is sufficient to observe that, since the equation ( 12) for β t (x) is a linear Schrödinger equation with an harmonic potential and conditions ( 9) and ( 10) are satisfied at time t = 0, we are guaranteed that

η|β t (η)| 2 dη = 0, ∀ t. (24)
Thus, by virtue of (24), it follows straightforwardly that (20) can be rewritten as

i∂ t b t (x) = -∆ 2 b t (x) + φ (0) 2 |x -η| 2 |β t (η)| 2 dη b t (x) + <x,∇ 2 U (q(t),t)x> 2 b t (x), b 0 (x) = a 0 (x). (25)
Finally, it is clear, by the definition of b t (x), that |β t (x)| = |b t (x)| for any x and t. Therefore (25) turns to be exactly the same as (15).

3.2.

Proof of Theorem 2.1. We seek an approximate solution to equation (1) of the form as e.g. in [START_REF] Hagedorn | Semiclassical quantum mechanics. I. The → 0 limit for coherent states[END_REF][START_REF] Hagedorn | Raising and Lowering Operators for Semiclassical Wave Packets[END_REF][START_REF] Hepp | The classical limit for quantum mechanical correlation functions[END_REF][START_REF] Paul | Semiclassical methods with an emphasis on coherent states[END_REF][START_REF] Paul | Échelles de temps pour l'évolution quantique à petite constante de Planck[END_REF]]

Ψ ε (x, t) = ε -d 4 a( x -q(t) √ ε , t)e i p(t)•(x-q(t)) ε e i L(t) ε ( 26
)
where

q(t) = p(t), ṗ(t) = -∇U (q(t), t). ( 27
)
By inserting the ansatz (26) in equation ( 1) we get

iε∂ t Ψ ε (x, t) = ε -d 4 iε∂ t a( x -q(t) √ ε , t) -i √ ε∇a( x -q(t) √ ε , t) • q(t)+ -L (t)a( x -q(t) √ ε , t) -( ṗ(t)(x -q(t)) -p(t) q(t)) a( x -q(t) √ ε , t) × ×e i p(t)•(x-q(t)) ε e i L(t) ε , (28)
and

- ε 2 2 ∆Ψ ε (x, t) = ε -d 4 - ε 2 ∆a( x -q(t) √ ε , t) + p 2 (t) 2 a( x -q(t) √ ε , t)+ -i √ ε∇a( x -q(t) √ ε , t) • p(t) e i p(t)•(x-q(t)) ε e i L(t) ε , (29)
while, with regard to the potential terms in (1), we find

(V (x, t) + U (x, t)) Ψ ε (x, t) = ε -d/4 φ(|x -y|)ε -d/2 |a( y -q(t) √ ε , t)| 2 dy + U (x, t) × × a( x -q(t) √ ε , t)e i p(t)•(x-q(t)) ε e i L(t) ε . (30)
By ( 28), ( 29) and (30) we get that the amplitude a solves the following initial value problem:

i∂ t + 1 2 ∆ a(µ, t) = 1 ε V ε (µ, t)a(µ, t)+ + 1 ε [U (q(t) + √ εµ, t) -U (q(t), t) - √ ε∇U (q(t), t) • µ] a(µ, t), a(µ, 0) = a 0 (µ), (31)
where

V ε (µ, t) = φ( √ ε|µ -η|) -φ(0) |a(η, t)| 2 dη, (32)
q(t), p(t) are as in the claim of Theorem 2.1 and we have used the rescaling µ = x-q(t) √ ε . Note that we should have

V ε (µ, t) = φ √ ε|µ -η| |a(η, t)| 2 dη -φ(0), (33)
instead of (32) in equation ( 31). However equation (31) with potential (33) is an Hartree equation which preserves the L 2 norm so that we can replace (33) by (32). Since φ ∈ C 3 b (R) is even φ (0) = 0 and the Taylor expansion yields

φ( √ ε|µ -η|) -φ(0) = ε|µ-η| 2 2 φ (0) + ε 3 2 R(|µ -η|), |R(|µ -η|)| C||φ || L ∞ |µ -η| 3 , (34)
while for the terms involving U we find:

U (q(t) + √ εµ, t) -U (q(t), t) - √ ε∇U (q(t), t) • µ = ε < µ, ∇ 2 U (q(t),t) 2 µ > +ε 3 2 R U (µ, t), |R U (µ, t)| C sup α∈N d :|α|=3 |∇ α U (q(t), t)||µ| 3 , (35)
where

∇ 2 := ∇ ⊗ ∇.
The core of the proof is to estimates the two remainders ε

3 2 R(|µ -η|) and ε 3 2 R U (µ, t) so that we can substitute (φ( √ ε|µ -η|) -φ(0)) by ε|µ-η| 2 2 φ (0) (as in (34)) and U (q(t) + √ εµ, t) - U (q(t), t) - √ ε∇U (q(t), t) • µ by ε < µ, ∇ 2 U (q(t),t) 2 µ > (as in (35)).
In the framework of semiclassical approximation for the linear Schrödinger equation using coherent states the method is standard (see e.g. [START_REF] Hagedorn | Semiclassical quantum mechanics. I. The → 0 limit for coherent states[END_REF][START_REF] Hagedorn | Raising and Lowering Operators for Semiclassical Wave Packets[END_REF][START_REF] Hepp | The classical limit for quantum mechanical correlation functions[END_REF][START_REF] Paul | Semiclassical methods with an emphasis on coherent states[END_REF][START_REF] Paul | Échelles de temps pour l'évolution quantique à petite constante de Planck[END_REF]), however we establish these estimates again for completeness.

Denote a t (µ) := a(µ, t) and

h t (µ) = b t (µ) -a t (µ). ( 36
)
By straightforward substitution we get that h 0 (µ) = 0 (see (15)) and

     i∂ t + 1 2 ∆ - φ (0) 2 |µ -η| 2 |b t (η)| 2 dη+ < µ, ∇ 2 U (q(t), t) 2 µ > V Q (µ,t)      h t (µ) = = φ (0) 2 |µ -η| 2 |b t (η)| 2 -|a t (η)| 2 dη a t (µ) I 1 (µ,t) + - √ ε R(|µ -η|)|a t (η)| 2 dη a t (µ) I 2 (µ,t) - √ εR U (µ, t)a t (µ). ( 37
)
By standard manipulations it turns out that

h t L 2 d dt h t L 2 |φ (0)| 2 | I 1 , h t | + √ ε| I 2 , h t | + √ ε| R U (•, t)a t , h t |. (38)
Moreover, the term involving I 1 can be estimated as follows:

| I 1 , h | ≤ µ η |µ -η| 2 |b t (η)| 2 -|a t (η)| 2 dη a t (µ)h t (µ)dµ = = µ η |µ -η| 2 (|b t (η)| -|a t (η)|) (|b t (η)| + |a t (η)|) dη a t (µ)h t (µ)dµ ≤ ≤ µ η |µ -η| 2 |h t (η)| (|b t (η)| + |a t (η)|) dη a t (µ)h t (µ)dµ ≤ ≤ 2 h t 2 L 2 (1 + |µ| 2 ) 2 [|a t (µ)| + |b t (µ)|] 2 dµ, (39)
while, thanks to (34), the term involving I 2 is estimated by:

| I 2 , h | ≤ C φ L ∞ µ η |µ -η| 3 |a t (η)| 2 dη a t (µ)h(µ, t)dµ ≤ ≤ C φ L ∞ dη |η| 3 |a t (η)| 2 ||a t || L 2 ||h t || L 2 + +3 dη |η| 2 |a t (η)| 2 3/2 ||h t || L 2 + 3 dµ |µ| 4 |a t (µ)| 2 1/2 dη |η| |a t (η)| 2 ||h t || L 2 + + dµ |µ| 6 |a t (µ)| 2 1/2 ||a t || 2 L 2 ||h t || L 2 . ( 40
)
One should observe here that dη

|η| |a t (η)| 2 ≤ dη (1 + |η| 2 ) |a t (η)| 2 .
Finally, due to (35), the term involving R U (µ, t) is controlled as follows:

| R U (•, t)a t , h t | ≤ C sup α:|α|=3 |∇ α U (q(t), t)| dµ |µ| 3 |a t (µ)| |h t (µ)| ≤ ≤ C sup t sup α:|α|=3 |∇ α U (q(t), t)| dµ |µ| 6 |a t (µ)| 2 1/2 ||h t || L 2 . ( 41
)
Making use of Lemma 4.2 and equation (70) below to estimate terms of the form

|| | • | m a t || L 2 = dη|η| 2m |a t (η)| 2 1/2 , for m ≤ 3, and || | • | m b t || L 2 = dη|η| 2m |b t (η)| 2 1/2
, for m ≤ 2, in terms of the same quantities evaluated at time t = 0, we easily show, by summing up the previous estimates, that there exist three ε-independent functions C 1 (t), C 2 (t) such that:

d dt ||h t || L 2 ≤ √ εC 1 (t) + C 2 (t)||h t || L 2 . ( 42
)
In particular C 1 (t), C 2 (t) depend on the potentials φ and U and on the L 2 -norm of moments and derivatives of a 0 (up to the order 3 ). Therefore it follows (see e.g. [START_REF] Reed | Methods of modern mathematical physics[END_REF]) that the corresponding time-dependent linear problem

i∂ t + 1 2 ∆ u(µ, t) = 1 ε (φ( √ ε|µ -η|) -φ(0)) |a(η, t)| 2 dη u(µ, t)+ + 1 ε (U (q(t) + √ εµ, t) -U (q(t), t) - √ ε∇U (q(t), t) • µ) u(µ, t), u(x, 0) = u 0 (x), u 0 ∈ L 2 (R d ) u 0 L 2 = 1, (43)
has a unique and well-defined L 2 propagator.

Lemma 4.2 (Propagation of Moments and derivatives for a(x, t)). Let a(x, t) be the solution of the initial value problem (31). Suppose that for some m ∈ N there exists an ε-independent

constant M m > 0 such that ||x A ∂ B x a 0 || L 2 M m (44) for all A, B ∈ N d such that |A| + |B| m. For m ≥ 2, assume φ ∈ C m b (R d ) and U ∈ C 1 (R + t , C m b (R d x )).
Then, there exists a (finite)

ε-independent constant C m such that ||x A ∂ B x a(t)|| L 2 C m e Cme Cmt M m , ( 45
)
for all A, B ∈ N d such that |A| + |B| m. For m = 1 inequality (45) holds by assuming φ ∈ C 2 b (R d ) and U ∈ C 1 (R + t , C 2 b (R d x ))
, while in the case m = 0 formula (45) becomes an equality and holds with unitary constant (for all t) by simply assuming

φ ∈ C 0 b (R d ) and U ∈ C 1 (R + t , C 1 b (R d x )
). Remark:The proof makes no use of an energy conservation argument, and this is the reason why the Lemma can be established for both signs of φ (0).

Proof. Denote ψ A,B (x, t) = x B ∂ A x a(x, t), (46)
e.g. ψ 0,0 (x, t) := a(x, t).

It is straightforward to check that

i∂ t + 1 2 ∆ -1 ε V ε (x, t) -1 ε U (q(t) + √ εx, t) + 1 ε U (q(t), t) + 1 √ ε ∇U (q(t), t) • x ψ A,B (x, t) = = - d k=1 B k (B k -1) 2 ψ A,B-2e k (x, t) + B k ψ A+e k ,B-e k (x, t) + 1 ε 0 l<A d k=1 A k l k ∂ A-l x V ε (x, t)ψ l,B (x, t)+ + 1 ε 0 l<A d k=1 A k l k ∂ A-l x U (q(t) + √ εx, t)ψ l,B (x, t) -1 √ ε 0<l A, |l|=1 d k=1 A k l k ∂ l x (∇U (q(t), t) • x) ψ A-l,B (x, t) (47
) where B = (B 1 , . . . , B k , . . . , B d ), l = (l 1 , . . . , l k , . . . , l d ), A = (A 1 , . . . , A k , . . . , A d ) and 0 ≤ l < A means that 0 ≤ l k < A k for any k = 1, 2, . . . , d. The consistent initial data for (47) are defined by

ψ A,B (x, 0) = x B ∂ A x a 0 (x),
and in particular ψ 0,0 (x, 0) := a 0 (x). Some remarks with regard to our notation are in order; it is clear for example that if B k = 0 or B k = 1, then the first term on the right-hand side yields no contribution. Similarly for B k = 0 in the second term and |A| = 0 for the remaining terms respectively. The derivation of (47) is straightforward by induction.

Denote by P (t, τ ) the propagator associated with the left-hand side of equation (47), which is known to be uniquely well defined in L 2 (see Observation 4.1). As a consequence, for m = 0, the result claimed by Lemma 4.2 follows from the existence of the propagator. We will proceed for m ∈ N by induction.

We will work with vectors including all the moments and derivatives, namely,

- → Ψ = {ψ A,B } A,B:|A|+|B|≤m ∈ X m and || - → Ψ || Xm = 0 |A|+|B| m ||ψ A,B || L 2 , where X 0 := L 2 (R d ).
For m = 1 we have

i∂ t + 1 2 ∆ - 1 ε V ε (x, t) - 1 ε U (q(t) + √ εx, t) + 1 ε U (q(t), t) + 1 √ ε ∇U (q(t), t) • x ψ e j ,0 (x, t) = = 1 ε ∂ x j V ε (x, t)ψ 0,0 (x, t) + 1 ε ∂ x j U (q(t) + √ εx, t)ψ 0,0 (x, t) - 1 √ ε ∂ z j U (z, t)| z=q(t) ψ 0,0 (x, t), (48)
and

i∂ t + 1 2 ∆ - 1 ε V ε (x, t) - 1 ε U (q(t) + √ εx, t) + 1 ε U (q(t), t) + 1 √ ε ∇U (q(t), t) • x ψ 0,e j (x, t) = = ψ e j ,0 (x, t) (49)
By virtue of the Duhamel formula we get: ψ e j ,0 (x, t) = P (t, 0)ψ e j ,0 (x, 0)

+ t 0 dτ P (t, τ ) 1 ε ∂ x j V ε (x, τ )ψ 0,0 (x, τ ) + + t 0 dτ P (t, τ ) 1 ε ∂ x j U (q(τ ) + √ εx, τ )ψ 0,0 (x, τ ) - 1 √ ε ∂ z j U (q(τ ), τ ) ψ 0,0 (x, τ ) (50)
and ψ 0,e j (x, t) = P (t, 0)ψ 0,e j (x, 0) + t 0 dτ P (t, τ ) ψ e j ,0 (x, τ ) .

Then, by recalling that P (t, τ ) is L 2 -norm preserving, we find ψ e j ,0 (t) L 2 ≤ ψ e j ,0 (0

) L 2 + t 0 dτ 1 ε ∂ x j V ε (x, τ )ψ 0,0 (τ ) L 2 + + t 0 dτ 1 ε ∂ x j U (q(τ ) + √ εx, τ ) - 1 √ ε ∂ z j U (q(τ ), τ ) ψ 0,0 (τ ) L 2 (52)
and

ψ 0,e j (t) L 2 ≤ ψ 0,e j (0) L 2 + t 0 dτ ψ e j ,0 (τ ) L 2 . (53)
Now, taking into account the terms involving U in (52), we get

1 ε ∂ x j U (q(τ ) + √ εx, τ ) - 1 √ ε ∂ z j U (q(τ ), τ ) = = 1 √ ε ∂ z j U (z, τ )| z=q(τ )+ √ εx - 1 √ ε ∂ z j U (z, τ )| z=q(τ ) = = ∂ 2 z j U (z, τ )| z=q(t)+ √ δ x x j , for some δ ∈ (0, ε), (54)
therefore

1 ε ∂ x j U (q(τ ) + √ εx, τ ) - 1 √ ε ∂ z j U (q(τ ), τ ) ψ 0,0 (τ ) L 2 = = ∂ 2 z j U (z, τ )| z=q(τ )+ √ δ x x j ψ 0,0 (τ ) L 2 = ∂ 2 z j U (z, τ )| z=q(τ )+ √ δ x ψ 0,e j (τ ) L 2 ≤ ≤ sup τ ∈[0,t] ∂ 2 U (•, τ ) L ∞ ψ 0,e j (τ ) L 2 . ( 55
)
On the other side, with regard to the term involving V ε in (52), we have

1 ε ∂ x j V ε (x, τ ) = dη ∂ x j 1 ε φ( √ ε|x -η|)|ψ 0,0 (η, τ )| 2 ≤ ≤ dη φ ( √ ε|x -η|) √ ε |ψ 0,0 (η, τ )| 2 ≤ L dη |x -η||ψ 0,0 (η, τ )| 2 , ( 56
)
where L is the global Lipschitz constant of φ (i.e., the L ∞ -norm of φ ) that is known to be finite since φ ∈ C 2 b (R d ). Then, by (56) we get that

1 ε ∂ x j V ε (x, τ )ψ 0,0 (τ ) 2 L 2 ≤ L 2 dx dη|x -η||ψ 0,0 (η, τ )| 2 dη |x -η ||ψ 0,0 (η , τ )| 2 |ψ 0,0 (x, τ )| 2 ≤ L 2 |x| 2 |ψ 0,0 (x, τ )| 2 dx + 3L 2 |η||ψ 0,0 (η, τ )| 2 dη 2 ≤ L 2 || |x|ψ 0,0 (τ )|| 2 L 2 + 3L 2 || |x|ψ 0,0 (τ )|| 2 L 2 , ( 57
)
where we made use of the fact that ||ψ 0,0 (τ

)|| L 2 = ||ψ 0,0 (0)|| L 2 = ||a 0 || L 2 = 1, for any time τ . At this point we observe that || |x|ψ 0,0 (τ )|| 2 L 2 = || |x|a(τ )|| 2 L 2 = j ||ψ 0,e j (τ )|| 2 L 2 .
So that, we have just proven that there exists a constant C > 0 depending only on the L ∞ -norm of the second derivative of φ, such that

1 ε ∂ x j V ε (x, τ )ψ 0,0 (τ ) L 2 C j ||ψ 0,e j (τ )|| 2 L 2 = ||ψ 0,1 (τ )|| L 2 . ( 58
)
Now, by using ( 55) and ( 58) in (52), we obtain that

ψ e j ,0 (t) L 2 ≤ ψ e j ,0 (0) L 2 + C t 0 dτ ||ψ 0,1 (τ )|| L 2 + C t 0 dτ ψ 0,e j (τ ) L 2 , ( 59
)
where C is not the same constant of formula (58) -we denoted it by the same symbol just for the sake of simplicity -since here it is depending on φ, as previously, but even on U (through the L ∞ -norm of its second derivative, according to (55)). Now after (55), summing over j = 1, . . . , d in equations ( 59) and ( 53) and then adding them, we get

|| - → Ψ (t)|| X 1 ≤ || - → Ψ (0)|| X 1 + C t 0 dτ || - → Ψ (τ )|| X 1 . ( 60
)
The conclusion follows by applying the Gronwall lemma, i.e.

|| - → Ψ (t)|| X 1 ≤ || - → Ψ (0)|| X 1 e Ct ≤ M 1 e Ct , (61)
where M 1 has been defined in (44).

For m ≥ 2, the previous inductive step from m = 1 applies almost verbatim: first, by virtue of the Duhamel formula, we write the solution of equation ( 47) by using the propagator P (t, τ ) associated with the time evolution on the left-hand side. Then, by using the L 2 -control on P (t, τ ), it only remains to show that the "source terms" appearing on the right-hand side of (47) are suitably uniformly bounded in terms of ||ψ For example, let us look at the term involving the potential V ε on the right-hand side of (47), i.e.

1 ε 0 l<A d k=1 A k l k ∂ A-l x V ε (x, t)ψ l,B (x, t) = 1 ε 0 l<A |A-l|=1 d k=1 A k l k ∂ A-l x V ε (x, t)ψ l,B (x, t) + + 1 ε 0 l<A |A-l|>1 d k=1 A k l k ∂ A-l x V ε (x, t)ψ l,B (x, t). ( 62
)
The estimation for any of the terms in the last sum reads as:

||∂ A-l x 1 ε φ( √ ε|x -η|)|ψ 0,0 (η, t)| 2 dη ψ l,B (x, t)|| 2 L 2 ε |A-l|-2 2 ||φ (A-l) (x)|| L ∞ 2 || |x -η||ψ 0,0 (η, t)| 2 dη ψ l,B (x, t)|| 2 L 2 2D|| |η|ψ 0,0 (t)|| L 2 ||ψ l,B (t)|| L 2 || |x|ψ l,B (t)|| L 2 + D|| |η|ψ 0,0 (t)|| L 2 ||ψ l,B (t)|| 2 L 2 + D|| |x|ψ l,B (t)|| 2 L 2 = = 2D||ψ 0,1 (t)|| L 2 ||ψ l,B (t)|| L 2 ||ψ l,B+1 (t)|| L 2 + D||ψ 0,1 (t)|| 2 L 2 ||ψ l,B (t)|| 2 L 2 + D||ψ l,B+1 (t)|| 2 L 2 , (63)
where D is a constant only depending on ||φ (A-l) (x)|| L ∞ (that is finite under our assumptions since A -l ≤ m). Furthermore, it is clear that, by construction, we are guaranteed that the exponent |A-l|-2 2 for ε is non negative. On the other side, the estimate for any of the terms in the first sum on the right-hand side of (62) is given by

||∂ A-l x 1 ε φ( √ ε|x -η|)|ψ 0,0 (η, t)| 2 dη ψ l,B (x, t)|| 2 L 2 ≤ L|| |x -η||ψ 0,0 (η, t)| 2 dη ψ l,B (x, t)|| 2 L 2 2L|| |η|ψ 0,0 (t)|| L 2 ||ψ l,B (t)|| L 2 || |x|ψ l,B (t)|| L 2 + L|| |η|ψ 0,0 (t)|| L 2 ||ψ l,B (t)|| 2 L 2 + L|| |x|ψ l,B (t)|| 2 L 2 = = 2L||ψ 0,1 (t)|| L 2 ||ψ l,B (t)|| L 2 ||ψ l,B+1 (t)|| L 2 + L||ψ 0,1 (t)|| 2 L 2 ||ψ l,B (t)|| 2 L 2 + L||ψ l,B+1 (t)|| 2 L 2 , (64)
where L is the global Lipshitz constant of φ (see ( 56)), which is guaranteed to be finite since φ ∈ C m b (R d ), with m ≥ 2. Now, by virtue of the estimate we proved for m = 1 (see (60)), from ( 63) and (64) we find that

||∂ A-l x 1 ε φ( √ ε|x -η|)|ψ 0,0 (η, t)| 2 dη ψ l,B (x, t)|| 2 L 2 ≤ KM 1 e Ct ||ψ l,B (t)|| L 2 ||ψ l,B+1 (t)|| L 2 + KM 1 e Ct ||ψ l,B (t)|| 2 L 2 + K||ψ l,B+1 (t)|| 2 L 2 , (65)
where K = max{D, L} and and we recall that |l|+|B| ≤ |A|-1+|B| ≤ m-1 and |l|+|B|+1 ≤ |A| -1 + |B| + 1 ≤ m. Thus:

||∂ A-l x 1 ε φ( √ ε|x -η|)|ψ 0,0 (η, t)| 2 dη ψ l,B (x, t)|| 2 L 2 ≤ K(M 1 e Ct + 1)|| - → Ψ (t)|| 2 Xm . (66)
Concerning the terms involving the potential U on the right-hand side of (47), the idea is quite similar. In fact, we observe that

1 ε 0 l<A d k=1 A k l k ∂ A-l x U (q(t) + √ εx, t)ψ l,B (x, t)- -1 √ ε 0<l A, |l|=1 d k=1 A k l k ∂ l x (∇U (q(t), t) • x) ψ A-l,B (x, t) = = 0<l A, |l|=1 C A,l,B 1 ε ∂ x U (q(t) + √ εx, t)ψ A-l,B (x, t) -1 √ ε ∂ x (∇U (q(t), t) • x) ψ A-l,B (x, t) + + 1 ε 0 l<A, |A-l|>1 d k=1 A k l k ∂ A-l x U (q(t) + √ εx, t)ψ l,B (x, t), (67)
where we made a discrete change of variable l → A -l in the first term of the left-hand side. Now, with regard to first term of the right-hand side, the estimation that has to be used is exactly the one we did in (55), thus one finds, ∀ l : |l| = 1

1 ε ∂ x U (q(t) + √ εx, t)ψ A-l,B (t) - 1 √ ε ∂ x (∇U (q(t), t) • x) ψ A-l,B (t) L 2 ≤ ≤ sup t ∂ 2 U (•, t) L ∞ ψ A,B (t) L 2 ≤ sup t ∂ 2 U (•, t) L ∞ || - → Ψ (t)|| Xm (68)
(the adjustment for l = 0 is obvious). Now, for the last term in (67) we have

1 ε ∂ A-l x U (q(t) + √ εx, t)ψ l,B (x, t) L 2 ≤ ε |A-l|-2 2 sup t ||∂ (A-l) U (•, t)|| L ∞ ψ l,B (t) L 2 ≤ ≤ sup t ||∂ (m) U (•, t)|| L ∞ || - → Ψ (t)|| Xm , (69)
where we used that A -l ≤ A ≤ m, |A -l| -2 ≥ 0 and l + B < A + B ≤ m.

Similar (simpler, in fact) estimates can be shown for the other terms on the right-hand side of (47). Observation 4.3. [Propagation of moments and derivatives for β t (x)] β t (x) was defined in equations ( 12), [START_REF] Reed | Methods of modern mathematical physics[END_REF]. Under the assumptions of Lemma 4.2, regularity estimates for β t (x) analogous to Lemma 4.2 for a(x, t) hold, i.e., for any t > 0

||x B ∂ A x β t || L 2 C m (t) |A |+|B | m ||x B ∂ A x a 0 || L 2 , ∀ A, B ∈ N d : |A| + |B| ≤ m.
Remarks:

• The proof is in fact simpler with respect to the one of Lemma 4.2: it can be checked easily that, due to the fact that we have to deal with harmonic potentials, the terms that arise from the differentiation of the potentials turn to be exactly of the form

xβ t (x) (∼ || - → Ψ β (t)
U ∈ C 1 R + t , C m b (R d x )
and φ (0) is finite, we can get a result for b t (x) e.g. analogous to Lemma 4.2 for a(x, t), i.e.

x B ∂ A x b t ∈ L 2 , ∀ A, B ∈ N d : |A| + |B| ≤ m ∀ t,
under the same assumption (44) on the (common) initial datum a 0 (x).

Note that, in particular, by Assumptions 1, 2, 3 we are guaranteed that there exists a εindependent constant C t > 0 depending on the L ∞ -norm of the second x-derivative of U (x, t), on |φ (0)| and on time t (but finite for any t), such that

dx |x| 2m |b t (x)| 2 < C t , ∀ t, m ≤ 3. ( 70
)
We observe that (70), for m = 2, is exactly what we need to make the proof of Theorem 2.1 work succesfully (see (39), (40) and (41)).

Higher order approximations

On the basis of the above results, it seems natural to ask whether it is possible to go beyond the √ ε-approximation discussed previously (see [START_REF] Paul | Semiclassical methods with an emphasis on coherent states[END_REF]) and to find higher order corrections a (k) t (µ) to the amplitude a (0) t (µ) := b t (µ) so that the right-hand side of (11) gets of size of any power of , as this is the case for the linear Schrödinger equation [START_REF] Paul | Semiclassical methods with an emphasis on coherent states[END_REF][START_REF] Paul | Échelles de temps pour l'évolution quantique à petite constante de Planck[END_REF]. Although we will not present all the (tedious) details of the construction, we claim that one can determine a semiclassical expansion

a ε t (µ) = a (0) t (µ) + √ ε a (1) t (µ) + ε a (2) t (µ) + • • • + ε k/2 a (k) t (µ) + . . . , (71) with a
(k) 0 (µ) = δ k,0 a 0 (µ), ( 72
) such that Ψ ε (x, t) = e i L(t) ε +iγ(t) ψ βt q(t)p(t) + O( ∞ ).
In order to determine the equations governing the evolution for each coefficient a (k) t (µ) we need to look at the expansion for the potential terms appearing in (31). With regard to the nonlinear part involving the pair interaction φ, we get:

1 ε φ( √ ε|µ -η|) -φ(0) = |µ -η| √ ε φ (0) + |µ -η| 2 2 φ (0) + √ ε|µ -η| 3 3! φ (0) + + • • • + ( √ ε) k |µ -η| k k! φ (k) (0) + . . . (73)
In Theorem 2.1 we were assuming φ ∈ C 3 b (R). Clearly, if we want to go to higher orders in the approximation we need more smoothness on φ and on the external potential U . Therefore, here and henceforth we assume:

φ ∈ C ∞ b (R)
, and φ even (74) so that we have

1 ε φ( √ ε|µ -η|) -φ(0) = |µ -η| 2 2 φ (0) + • • • + ( √ ε) 2n-1 |µ -η| 2n (2n)! φ (2n) (0) + . . . n ≥ 2 (75)
Observation 5.1. Assumption (74) is actually too strong if one wants to deal with an approximation up to a certain order k.

With regard to the linear terms in (31) involving the external potential U , we get:

1 ε (U (q(t) + √ εµ, t) -U (q(t), t) - √ ε∇U (q(t), t) • µ) = =< µ, ∇ 2 U (q(t),t) 2 µ > + • • • + ( √ ε) n-1 ∇ n U (q(t),t) n! • µ n + . . .
where of course n 3. Here we are using the notation

∇ n U • µ n = α 1 ...α d : α i =n ∂ n U ∂x α 1 1 . . . ∂x α d d µ α 1 . . . µ α d . (76)
Analogously to what we observed for the pair-interaction φ, we need more smoothness for U , so that here and henceforth we require:

U ∈ C 1 (R + t , C ∞ b (R d x )). (77)
Now, inserting (71), ( 75) and ( 76) in (31) we readily arrive to a sequence of problems for the coefficients a (k) t (µ) of the expansion (71). For k = 0 we obviously find:

     i∂ t + ∆ µ 2 + φ (0) 2 dη |µ -η| 2 |a (0) t (η)| 2 + < µ, ∇ 2 U (q(t), t) 2 µ > a (0) t (µ) = 0, a (0) 0 (µ) = a ε 0 (µ), (78)
namely, the initial value problem that we had for b t (µ) in the previous sections (see (15)). Then, for k = 1, we find:

             i∂ t + ∆ µ 2 + φ (
0 (µ) = 0.

(79) This is a linear initial value problem where the left-hand side is known to have a unique welldefined L 2 -propagator P (0) (t) due to the existence and uniqueness in L 2 of the solution a (0) t (µ) of the zero-order initial value problem (78) and to the L 2 -control on its first moment (see Observation 4.4 and (70)). Then, it is easy to see that, writing the solution a

(1) t (µ) through the Duhamel formula (with "leading" propagator P (0) (t)), the well-posedeness in L 2 for (79) is guaranteed by the L 2 -control on the source term ∇ 3 U (q(t),t) 3!

• µ 3 a (0) t (µ) (which is achieved thanks to the smoothness of U and to the L 2 -control on the third moment of a Here C a (0) t > 0 is a constant that only depends on the moments of a (0) t (µ) up to order 3, depending on the potentials U (trought the L ∞ -norm of its second derivative) and φ (through the quantity |φ (0)|) and on the initial derivatives and moments of a (0) t (µ) up to the order 3 (see Observation 4.4), that, as in the previous sections, we assume to be finite. By virtue of (80) and the L 2 -control on the term involving U in (79), the Duhamel formula and the Gronwall lemma allow to conclude that a

t ∈ L 2 (R d ), ∀ t.

(81) Moreover, following the same lines of the proofs presented and discussed in the previous sections (see Lemma 4.2, Observation 4.4 and subsequent remarks), it can be easily checked that by assuming enough regularity for the (zero-order) "full" initial datum a (0) 0 (µ) = a 0 (µ), in such a way that we control in L 2 a sufficiently high number of moments and derivatives, we can control the derivatives and moments of a

( 1 )

 1 t (µ) up to any fixed order m, i.e:x B ∂ A x a (1) t ∈ L 2 (R d ), ∀ t, ∀ A, B ∈ N d : |A| + |B| ≤ m.(82)

  ). With regard to the time dependence, C 1 (t), C 2 (t) are double exponentials Ce Ce Ct , following Lemma 4.2 and observations 4.3, 4.4.The conclusion follows with application of the Gronwall lemma.

4. Auxiliary results

Observation 4.1. Observe that under our assumptions the nonlinear equation (31) can be shown to have, for any T > 0, a unique solution in C 1 [0, T ], L 2 (R d ) (see e.g.

[START_REF] Cazenave | The Cauchy problem for the nonlinear Schrödinger equation in H 1[END_REF]

  A,B || L 2 or || -→ Ψ || Xm . The way to do that is by using ||ψ A,B || L 2 , |A| + |B| < m as constants now.

  || X 1 if we denote by ψ A,B β (x, t) the quantities x B ∂ A x β t (x) and we define As a consequence of Observation 4.3, by Assumptions 1, 2, 3 we are guaranteed that, in particular, there exists a ε-independent constant C > 0 depending on the L ∞ -norm of the second x-derivative of U (x, t) and on |φ (0)|, such thatdx |x| 2 |β t (x)| 2 < dx |x| 2 |a 0 (x)| 2 e Ct < ∞, ∀ t. Observation 4.4. [Propagation of Moments and derivatives for b t (x)] Although apparently b t (x) solves a nonlinear equation, it can be obtained as the solution of a linear Schrödinger equation with an harmonic potential whose coefficients are determined by the L 2 -norm of the first moment of β t (x), by φ (0) and ∇ 2 U (see (25) and Lemma 3.1). Therefore, as a consequence of Observation 4.3, it follows that, as long as

	We remind that this is exactly what we need to make the proof of Theorem 2.1 work
	succesfully (see (23)).
	-→ Ψ β (t)
	consistently), i.e., precisely the kind of objects we want to recover to apply the Gronwall
	Lemma (see the proof of Lemma 4.2).

•

Acknowledgments. We would like to thank R. Carles for pointing out that an extra hypothesis was needed, and a mistake in the remainder which appears in the main theorem.

This will be crucial to go on with the higher orders dynamics because, for example, the equation for the second coefficient a

(2)

t (η)a

(2)

0 (µ) = 0, (83) So, again, as for the case k = 1, we obtained a linear initial value problem where the propagator associated with the left-hand side is P (0) (t), that is known to be uniquely well-defined in L 2 . Then, as before, the solution a

(2) t (µ) can be written through the Duhamel formula, applying the propagator

t (µ) and to the various source terms in (83). The term which is linear in a

(2) t (µ) is estimated as in (80) while the source terms are controlled in L 2 by virtue of the control on moments and derivatives of a (0) t (µ) and a

(1) t (µ). In the end, by using the Gronwall lemma, we get a

(2)

and, moreover, by assuming a sufficiently high number of moments and derivatives of the (zeroorder) "full" initial datum a (0) 0 (µ) = a 0 (µ) to be controlled in L 2 , we can control as well the derivatives and moments of a (2) t (µ) up to any fixed order m, i.e:

At this point it is clear how to proceed in general. The equation for a