Giovanna D'onofrio

Filomena Tramontano

Annalisa Susanna Dorio

Alessia Muzi

Valeria Maselli

Domenico Fulgione

Grazia Graziani

Maria Malanga

Piera Quesada

POLY(ADP-RIBOSE) POLYMERASE SIGNALING OF TOPOISOMERASE 1-DEPENDENT DNA DAMAGE IN CARCINOMA CELLS

Keywords: Topotecan (TPT), PARP-1 and -2, PJ34 inhibitor, p53, carcinoma cells Classification: Antibiotics and Chemotherapeutics

A c c e p t e d M a n u s c r i p t 2 ABSTRACT A molecular approach to enhance the antitumour activity of Topoisomerase 1 (TOP1) inhibitors relies on the use of chemical inhibitors of poly(ADP-ribose)polymerases (PARP). Poly(ADP-ribosyl)ation is involved in the regulation of many cellular processes such as DNA repair, cell cycle progression and cell death. Recent findings showed that poly(ADP-ribosyl)ated PARP-1 and PARP-2 counteract camptothecin action facilitating resealing of DNA strand breaks. Moreover, repair of DNA strand breaks induced by poisoned TOP1 is slower in the presence of PARP inhibitors, leading to increased toxicity.

In the present study we compared the effects of the camptothecin derivative Topotecan (TPT), and the PARP inhibitor PJ34, in breast (MCF7) and cervix (HeLa) carcinoma cells either PARP-1 proficient or silenced, both BRCA1/2 +/+ and p53 +/+ . HeLa and MCF7 cell lines gave similar results: i) TPT-dependent cell growth inhibition and cell cycle perturbation were incremented by the presence of PJ34 and a 2 fold increase in toxicity was observed in PARP-1 stably silenced HeLa cells; ii) higher levels of DNA strand breaks were found in cells subjected to TPT+PJ34 combined treatment; iii) PARP-1 and -2 modification was evident in TPT-treated cells and was reduced by TPT+PJ34 combined treatment; iv) concomitantly, a reduction of soluble/active TOP1 was observed. Furthermore, TPT-dependent induction of p53, p21 and apoptosis were found 24-72 h after treatment and were increased by PJ34 both in PARP-1 proficient and silenced cells. The characterization of such signaling network can be relevant to a strategy aimed at overcoming acquired chemoresistance to TOP1 inhibitors.

INTRODUCTION

The camptothecin derivative Topotecan (TPT) is a DNA Topoisomerase 1 (TOP1) inhibitor approved for the treatment of ovarian cancer, non small-cell lung cancer and under clinical investigation for a number of advanced solid tumours and haematological malignancies [START_REF] Pommier | Topoisomerase I inhibitors: camptothecins and beyond[END_REF]. The drug reversibly abolishes the DNA religation activity of TOP1 generating single strand breaks (SSBs) to which the protein is covalently linked. Double strand breaks (DSBs) arise when replication forks collide with the SSBs and run off.

Thus, TPT-induced DSBs are largely replication dependent or S phase specific [START_REF] Staker | The mechanism of topoisomerase I poisoning by a camptothecin analog[END_REF][START_REF] Pommier | Repair of and checkpoint response to topoisomerase I-mediated DNA damage[END_REF].

Eukaryotes have two pathways for repairing DSBs: homologous recombination (HR) and non homologous end joining (NHEJ). The relative contribution of these two DSB repair pathways seems to differ depending on the cell cycle phase; HR acts mainly in the S and G2 phases, whereas NHEJ mainly in the G1 phase [START_REF] Takata | Homologous recombination and non-homologous end-joining pathways of DNA double strand break repair have overlapping roles in the maintenance of chromosomal integrity in vertebrate cells[END_REF][START_REF] Essers | Homologous and non-homologous recombination differentially affect DNA damage repair in mice[END_REF]. For these reasons, TPTinduced replication-dependent DSBs are usually repaired by the HR pathway [START_REF] Arnaudeau | DNA double-strand breaks associated with replication forks are predominantly repaired by homologous recombination involving an exchange mechanism in mammalian cells[END_REF].

Poly(ADP-ribosyl)ation is a post-translational modification catalyzed by poly(ADPribose)polymerase-1 and -2 (PARP-1 and PARP-2) and is one of the earliest cellular responses to DNA damage. PARP-1 and PARP-2 belong to a family of enzymes that cleave -NAD + in nicotinamide and ADP-ribose to form long and branched (ADPribose) polymers (PAR) on glutamic acid residues within the primary sequence of PARP-1 and PARP-2 (automodification) and of other cellular proteins (heteromodification). This process causes chromatin decondensation around damage sites, recruitment of repair machineries, such as base excision repair complexes, and accelerates DNA damage repair [START_REF] Burkle | ADP-ribosyl)ation[END_REF][START_REF] Schreiber | Poly(ADPribose): novel function for an old molecule[END_REF]. In contrast, when DNA damage exceeds cell repair capacity PARP-1 undergoes cleavage by caspases into two fragments of 89 kDa and of 24 kDa, thereby avoiding futile cycling of PAR that would otherwise deplete the cell of -NAD + required for the A c c e p t e d M a n u s c r i p t 4 onset of apoptosis [START_REF] Scovassi | Poly(ADP-ribosyl)ation and apoptosis[END_REF]. Moreover, interaction of PAR with the p53 oncoprotein is able to modulate its transcriptional activity [START_REF] Malanga | Poly(ADP-ribose) binds to specific domains of p53 and alters its DNA binding functions[END_REF].

PARP-1 also affects DSBs repair as indicated by the increased sensitivity of PARP-1deficient cells to DSBs inducing agents, especially to camptothecin [START_REF] Staker | The mechanism of topoisomerase I poisoning by a camptothecin analog[END_REF]. Furthermore, the molecular mechanisms underlying tumour chemosensitization to TOP1 poisons by PARP inhibitors have been in part clarified by recent findings showing that poly(ADPribos)ylated PARP-1 and PARP-2 counteract camptothecin action facilitating resealing of DNA strand breaks [START_REF] Malanga | The role of poly(ADP-ribose) in the DNA damage signaling network[END_REF]. This occurs through noncovalent yet specific interaction of PAR with particular TOP1 sites which results in inhibition of DNA cleavage and stimulation of the religation reaction [START_REF] Malanga | Poly(ADP-ribose) reactivates stalled DNA topoisomerase I and Induces DNA strand break resealing[END_REF]. Another mechanism proposed to explain the potentiation of campthotecin cytotoxicity by PARP inhibitors, is via the inhibition of base excision repair system, of which PARP-1 and -2 are important components. This model is supported by the association of tyrosyl phosphodiesterase-1, which removes the TOP1 cleavable complex, with base excision repair components that interact with PARP-1 [START_REF] Smith | The novel poly(ADP-Ribose) polymerase inhibitor, AG14361, sensitizes cells to topoisomerase I poisons by increasing the persistence of DNA strand breaks[END_REF].

Indeed, PARP-1 inhibition enhances the cytotoxic effects of TPT [START_REF] Tentori | Chemopotentiation by PARP inhibitors in cancer therapy[END_REF]. The potential of PARP inhibitors to increase the efficacy of chemotherapy has led to the development of a wide range of specific inhibitors -quinazolinone derivates-like NU1025 or PJ34 which display increased potency compared to the prototype 3-aminobenzamide (3-ABA) [START_REF] Sandhu | Poly(ADP-ribose) polymerase inhibitors in cancer treatment: a clinical perspective[END_REF]. In this regard, we previously demonstrated a TPT-dependent PARP-1 activation in glioblastoma cells, while co-treatment with the PARP inhibitor NU1025 increased the TPT-dependent p53 up-regulation [START_REF] Cimmino | Poly(ADPR)polymerase-1 signalling of the DNA damage induced by DNA topoisomerase I poison in D54(p53wt) and U251(p53mut) glioblastoma cell lines[END_REF]. Moreover, we showed PJ34 chemo-potentiation of cisplatin in colon carcinoma cells [START_REF] Gambi | Poly(ADPR)polymerase inhibition and apoptosis induction in cDDP-treated human carcinoma cell lines[END_REF].

It has been reported that PARP inhibitors would be particularly effective in BRCA1/2 mutated breast carcinoma cells [START_REF] Bryant | Specific killing of BRCA2-deficient tumours with inhibitors of poly(ADP-ribose) polymerase[END_REF]. In fact, PARP-1 and PARP-2 are required for the base excision repair pathway, whereas the BRCA proteins are critical for the HR Furthermore, a factor supposed to be involved in determining the sensitivity of cells to TOP1 inhibitors is p53. However, for breast cancer cells the p53 status was not found to be predictive of sensitivity to camptothecins [START_REF] Davis | Complex response of breast epithelial cell lines to topoisomerase inhibitors[END_REF].

On the basis of such evidences, we have investigated the role of PARP-1 in the DNA damage response to TOP1 inhibitors, in human BRCA1/2 +/+ and p53 +/+ mammary (MCF7) and cervix (HeLa) carcinoma cells treated with TPT as single agent or in association with a PARP inhibitor. Furthermore, TPT sensitivity of HeLa cells in which PARP-1 has been knocked down by RNA interference, has been compared to that of HeLa cells treated with the PARP inhibitor. Nicotinamide adenine [adenylate-32 P] dinucleotide-[ 32 P]-NAD + (1,000 Ci/mmole, 10 mCi/ml) was supplied by GE Healthcare (Milano, Italy).

Propidium iodide (PI) and RNAse were from Sigma-Aldrich (Milano, Italy). PVDF (poly-vinylidene-fluoride) membrane was from MILLIPORE S.p.A. (Milano, Italy). Anti-PARP-1 mouse monoclonal antibody (F1-23) was from Alexis Biochemicals (Vinci-Biochem, Firenze, Italy) and anti-DNA TOP1 (Scl-70) human antibody from Topogen (ABCAM, Cambridge, UK). Anti-p53 (DO-1), anti-p21 (C-19), anti-BAX (P-19) and anti-GAPDH (H-2) mouse monoclonal antibodies were from Santa-Cruz Biotechnology (DBA, Milano, Italy); anti-actin (A2066) mouse monoclonal antibody and goat anti-mouse and goat anti-rabbit IgG HRP-conjugated antibodies were from Sigma-Aldrich (Milano, Italy). Anti-H2AX (ser139, 2577) rabbit antibody was from Cell Signaling (Invitrogen Milano, Italy).

All other chemicals were of the highest quality commercially available.

Cell cultures

Cervix (HeLa) and mammary (MCF7) carcinoma cells were maintained in Dulbecco's modified Eagle's medium (DMEM) containing 10% (v/v) heat-inactivated foetal bovine serum (FBS), 100 U/ml penicillin, 100 g/ml streptomycin, 5 mM Lglutamine and incubated at 37°C in a humidified atmosphere, plus 5% CO 2 .

A c c e p t e d M a n u s c r i p t 7 Stably PARP-1 silenced HeLa cells (hereafter referred to as HeLa SiP-1 ) or transfected with the pBabe vector carrying the puromycin resistance gene (hereafter referred to as HeLa Babe ) were obtained as previously described [START_REF] Tentori | Pharmacological Inhibition of Poly(ADP-ribose) Polymerase (PARP) Activityin PARP-1 Silenced Tumour Cells Increases Chemosensitivity to Temozolomide and to a N3-Adenine Selective Methylating. Agent[END_REF].

Cell growth inhibition

MCF7 and HeLa cells were seeded at 1×10 5 cells; after 24 h, cell cultures were treated with graded concentrations of TPT and PJ34 and cell growth inhibition was assessed at different time points [START_REF] Yelamos | Toward specific functions of poly(ADP-ribose) polymerase-2[END_REF]48, 72 h) using the 3-[4,5-dimethylthiazol-2-yl]-2,5diphenyltetrazolium bromide (MTT) assay. All the experiments were performed in triplicate.

Cytofluorimetric analysis

Control and treated cells were detached by enzymatic treatment (Trypsin/EDTA 0.02%), washed in PBS w/o Ca ++ /Mg ++ pooled with floating cells and recovered by centrifugation at 1,200 rpm for 15 min at 4°C. Cells were fixed in 70% ethanol and stored at -20°C until analysis. After washing in PBS w/o Ca ++ /Mg ++ , cells were stained in 2 ml of propidium iodide (PI) staining solution [50 g/ml of PI, 1 mg/ml of RNAse A in PBS w/o Ca ++ /Mg ++ , pH 7.4] overnight at 4°C and DNA flow cytometry was performed in duplicate by a FACScan flow cytometer (Becton Dickinson Franklin Lakes, NJ USA) coupled with a CICERO work station (Cytomation). Cell cycle analysis was performed by the ModFit LT software (Verity Software House Inc. Topsham, ME USA). FL2 area versus FL2 width gating was done to exclude doublets from the G2/M region. For each sample 15,000 events were stored in list mode file.

Alkaline Comet Assay

A c c e p t e d M a n u s c r i p t 8 Cells were suspended in PBS at a density of 10 4 cells/ml and mixed with an equal volume of fresh low-melting agarose (LMA, 1% in PBS); 80 μl of agarose cell suspension was spread on normal-melting agarose (NMA, 1% in PBS) slides and covered with a cover-slip. Two slides were prepared per sample. After gelling for 5 min on an ice bed, the cover-slip was gently removed and another layer was added, cover-slipped and allowed to solidify for 5 min on ice before gently removing the cover-slip. The slides were then immersed in a freshly prepared ice-cold lysis solution (2.5 M NaCl, 0.1 M Na 2 EDTA, 0.01 M Tris, 1% Triton X-100, 10% DMSO, pH 10) for 1 h. The slides were drained and placed in a horizontal electrophoresis tank filled with freshly prepared alkaline buffer (0.3 M NaOH, 1 mM Na 2 EDTA, pH 13). Electrophoresis was carried out in this buffer for 20 min at 300 mA. Finally, the slides were gently washed twice in a neutralization buffer (Tris-HCl 0.4 M, pH 7.5) for 5 minutes to remove alkali and detergent, and stained with 50 μl/ml DAPI (3 h). Images of a minimum of hundred cells from each sample were analysed on a fluorescence microscope (Nikon Instruments S.p.A. Firenze, Italy); overlapping figures were avoided from each slide. Quantitative assessment of DNA damage was performed using Comet Score 1.5 Image Analysis (TriTek Corporation, Sumerduck VA, USA) software which computes the integrated intensity profile for each cell. DNA damage was measured as olive tail moment [(Tail mean -Head mean) x % of DNA in the tail/100]. The results were analysed by Student's t-test and were considered statistically significant at P<0.008.

Analysis of [ 32 P]-PAR synthesis

Following treatment with 10 µM TPT +/-5 µM PJ34 of intact cell (5x10 6 cells/plate), [ 32 P]-PAR synthesis was determined by substituting the culture medium with 1 ml of 56 mM HEPES buffer pH 7.5, containing 28 mM KCl, 28 mM NaCl, 2 mM MgCl 2 , 0.01% PJ34 efficiency as PARP inhibitor, was determined in an in vitro enzymatic activity assay using permeabilized cells: cell pellets were resuspended in 40 mM Tris-HCl pH 7.8, 0.6 mM EDTA, 30 mM MgCl 2 , 0.05% Triton X-100, 1 mM -mercaptoethanol, 20% glycerol, 1 mM PMSF and a 1:25 dilution of the cocktail of protease. To maximally stimulate PAR synthesis, DNA strand breaks were induced by sonication for 30 sec at medium intensity; finally, samples were incubated at 30°C for 1 h with 5 Ci/ml [ 32 P]-NAD + and 50 M unlabeled -NAD + , in the presence or absence of 5 M PJ34.

Reactions were stopped by TCA addition (20% final concentration) and the samples were processed and analyzed as described above.

Isolation of nuclear and post-nuclear fractions

To isolate sub-cellular fractions, cells were suspended in a buffer containing 30 mM Tris-HCl pH 7.5, 1.5 mM MgCl 2 , 10 mM KCl, 1% (v/v) Triton X-100, 20% glycerol, 2 mM PMSF and the protease inhibitors cocktail solution. After 30 min of incubation on Nuclear fractions were resuspended in 20 mM HEPES pH 7.9, containing 20 mM KCl, 0.2 mM EDTA, 1.5 mM MgCl 2 , 25% glycerol and the protease inhibitors cocktail solution. Protein concentration was determined using the Bradford protein assay reagent (BIO-RAD Milano, Italy) with bovine serum albumin as a standard.

Autoradiographic and immunological analyses

Aliquots of 120 g of cellular proteins were separated by SDS-PAGE (5-15% gradient gels) and transferred onto a PVDF membrane using an electroblotting apparatus (BIO-RAD). The membrane was subjected to autoradiographic analysis by the PhosphorImager (BIO-RAD) and/or to immunodetection after blocking with 5% non-fat milk in TBST 1 h, with anti-PARP-1 (F1-23; diluted 1:5,000), anti-TOP1 (Scl-70; diluted 1:2,500), anti-p53 (DO-1; diluted 1:5,000), anti-p21 (C-19; diluted 1:1,000), anti-Bax (P-19; diluted 1:500), anti-GAPDH (H2; diluted 1:5,000), anti-H2AX (2577; diluted 1:1,000) and anti-actin (A2066; diluted 1:1,000).

As secondary antibodies goat-anti-mouse or goat-anti-rabbit IgG HRP-conjugate (diluted 1:10,000-1:20,000) in 3% (w/v) non-fat milk in TBST were used. Peroxidase activity was detected using the ECL Advance Western Blotting Kit of GE Healthcare (Milano, Italy) and quantified using the Immuno-Star Chemiluminescent detection system GS710 (BIO-RAD) and the Arbitrary Densitometric Units normalised on those of the GAPDH loading control.

Effect of PJ34 on TPT-induced growth inhibition in human carcinoma cells

In preliminary experiments human cervical (HeLa) and mammary (MCF7) carcinoma cell lines showed comparable TPT-dependent growth inhibition, as measured by the MTT assay (data not shown). Furthermore, PARP-1 silencing by stable shRNA expression in HeLa cells (HeLa SiP-1 ) rendered these cells more sensitive to the cytotoxic effects of the drug. In particular, while in a 72 h assay, 10 M TPT for 1 h exerted mainly cytostatic effects in control cells (Hela Babe ), the same treatment caused 45% (+/-5) of PARP-1 silenced cells (HeLa SiP-1 ) to die. In the presence of 5 M PJ34 30% (+/-6) of PARP-1 proficient and 60% (+/-9) of PARP-1 deficient cells underwent cell death (data not shown).

To gain insight into the mechanism of enhanced TPT toxicity as a consequence of alteration of the cellular poly(ADP-ribosyl)ation status, we analysed cell cycle distribution at different recovery times after 1 h exposure to increasing concentrations of TPT, in the presence or absence of a functional PARP-1 (i.e., PARP-1 wild type HeLa or MCF7 cells versus HeLa SiP-1 cells). In another set of experiments, the PARP inhibitor PJ34 was used in combination with TPT, at a fixed concentration of 5 µM, maintained in the culture medium all over the recovery time. As shown in Figure 1 HeLa SiP-1 cells treated with TPT concentrations comprised between 0.2 and 0.4 M underwent a more pronounced increase of G2/M cell accumulation with respect to Hela Babe cells exposed to the same concentrations of the TOP1 poison. Interestingly, 0.4 M TPT caused in HeLa SiP-1 cells effects comparable to those observed in HeLa Babe cells treated with 0.4 M TPT plus the PARP inhibitor. However, PARP-1 silenced cells retained sensitivity to PJ34 and the combination 1.25 M TPT+PJ34 caused S phase accumulation at a higher extent in HeLa SiP-1 than in Hela Babe cells (Figure 1).

Cytofluorimetric analyses at a longer recovery time (i.e., 72 h after treatment), revealed that alterations of the poly(ADP-ribosyl)ation system caused TPT to be cytotoxic at a concentration (1.25 µM ) that was primarily cytostatic in control cells, as indicated by the appearance of a sub-diploid peak (apoptotic cells) both in PARP-1 silenced (HeLa SiP-1 ) and PJ34-treated PARP-1 wild type cells (Hela Babe ) (Figure 2). In this regard, the lack of PARP-1 appeared to be more effective than PARP activity inhibition as the fraction of apoptotic cells was 62% in TPT-treated HeLa SiP-1 versus 38% in HeLa Babe , subjected to a combined TPT+PJ34 treatment (Figure 2).

Analysis of TPT and/or PJ34 dependent DNA damage in carcinoma cells

By alkaline comet assay, we analysed the level of both SSBs and DSBs [START_REF] Tice | Single cell gel/comet assay: guidelines for in vitro and in vivo genetic toxicology testing[END_REF] induced by 10 M TPT+/-PJ34 treatments. Figure 3A shows that the olive tail moment determined for both HeLa (Babe and SiP-1) and MCF7 cells 24 h after 1 h treatment with TPT was increased in the cells left to recover in the presence of PJ34. The definition of a DSBs level was obtained by looking at the H2AX phoshorylation in isolated nuclei from Babe and SiP-1 cells. Figure 3B shows that 72 h after 1 h treatment TPT induced a A c c e p t e d M a n u s c r i p t 13 higher level of histone phosphorylation in HeLa SiP-1 than in Hela Babe cells. H2AX phoshorylation was further incremented by PJ34 addition in both PARP-1 proficient and silenced cells.

Analysis of PAR synthesis in carcinoma cells after treatment with TPT+/-PJ34

First, PJ34 efficacy as a PARP inhibitor at the concentration used in this study was assessed in an in vitro enzyme activity assay by incubating permeabilized and sonicated HeLa cells with exogenous 50 M [ 32 P]-NAD + in the presence or not of 5 M PJ34.

Sonication was performed to induce DNA strand breaks and thus maximally stimulate endogenous PARP activities. PAR synthesis on protein acceptors was analyzed by SDS-PAGE followed by electroblotting onto PVDF membrane and autoradiography. As shown in Figure 4A, a high amount of protein-bound PAR was produced in HeLa cells and such an activity was completely inhibited by 5 M PJ34. Although a wide range of modified proteins could be visualized, the main PAR acceptor was most likely PARP-1 as suggested by the strong radioactivity signal at the top of the gel and by the concomitant reduction of the PARP-1 immunoreactive band in the sample incubated with -NAD + alone compared to that incubated with -NAD + and PJ34 (Figure 4B). Such a difference is explained by a band depletion due to the automodification-related electrophoretic mobility shift of a fraction of heavily poly(ADP-ribosylated) PARP-1.

After quantification of immunoreactive bands by scanning densitometry and normalization of PARP-1 to GAPDH content it could be estimated that about 50% of PARP-1 underwent automodification.

The same kind of analysis carried out in HeLa SiP-1 cells, revealed a strongly reduced ADP-ribosylation capacity of these cells as a consequence of PARP-1 silencing (Figure As no PARP-1 could be detected in these cells by western blotting (Figure 4B) the modest ADP-ribosylation activity detected by the in vitro assay may be due to PARP-2 and/or other PARP.

Then, we used a different experimental setting to determine whether or not TPT could induce PARP(s) activation in intact cells. To this purpose, growing MCF7 cells were first exposed to the drugs and then PAR synthesis was measured in situ by incubation in the presence of 0.01% digitonin and 0.125 M [ 32 P]-NAD + . By autoradiography (Figure 5A) we observed a main signal slightly up to PARP-1 molecular weight (113 kDa), indicating that DNA damage induced by TPT caused PARP-1 activation and automodification that was apparent already after 1 h treatment and further increased in the following 24 h recovery time. Such a trend was confirmed by scanning densitometry and normalization of data from autoradiography (Figure 5A) to those relative to PARP-1 immunoreactive band (Figure 5B). Minor autoradiographic bands were evident in the 90-50 kDa MWs range (Figure 5A) indicating other PAR acceptors, possibly including other PARP. PARP-2 was detectable in this region as a 62 kDa protein band; a modification-related electrophoretic mobility shift could explain the lack of correspondence between the autoradiographic signal (Figure 5A) and the PARP-2 immunoreactive band (Figure 5B).

The autoradiographic signals were drastically reduced (up to 75% reduction) in cells co-treated with TPT and the PARP inhibitor with respect to cells treated with TPT as single agent.

Similar results were obtained in HeLa Babe cells, while a [ 32 P]-PAR signal was undetectable in HeLa SiP-1 (data not shown).

A c c e p t e d M a n u s c r i p t 15

Immunological analysis of PARP-1, TOP1, p53, p21 level in TPT+/-PJ34 treated cells

By western blotting we analysed changes in the endogenous levels of PARP-1, TOP1 and p53 in HeLa and MCF7 cells at different times (24, 48 and 72 h) after treatment with TPT+/-PJ34.

Figure 6 shows a comparable amount of PARP-1 in MCF7 cell samples at all time points, whereas the amount of soluble/active TOP1 was lowered (~50%) till 72 h after treatment with TPT alone or in combination with PJ34. Conversely, an up-regulation of p53 endogenous levels was evident until 72 h after treatment with TPT+/-PJ34.

Furthermore, the p53-dependent p21 induction was evidenced starting from 24 h after TPT treatment.

Figure 7 shows that the amount of soluble/active TOP1 was drastically lowered also in HeLa cells (up to 70%-80% reduction both in PARP-1 proficient and silenced cells) as a consequence of the treatments. Interestingly, such a decrease was sustained till 72 h after 1 h treatment. Again, we observed a TPT-dependent p53 up-regulation in both PARP-1 proficient and silenced cells, which appeared further increased by the use of PARP inhibitor (Figure 7).

By densitometric scanning of immunoreactive bands we quantified the changes in p53 levels at different times after single and combined treatments. As shown in Figure 8, the p53 level was 2-4 fold increased in HeLa Babe cells 72 h after 1 h TPT-/+PJ34 treatment. In HeLa SiP-1 cells a 10 fold increase was induced by TPT alone and this value increased (13 fold) in the presence of PJ34 during the recovery time.

Finally, 72 h after TPT treatment we analysed the expression of apoptotic markers. A c c e p t e d M a n u s c r i p t

DISCUSSION

The evaluation of PARP inhibitors as chemosensitizers is based on evidences linking PARP-1 and recently PARP-2, to the cellular DNA damage response [START_REF] Smith | The novel poly(ADP-Ribose) polymerase inhibitor, AG14361, sensitizes cells to topoisomerase I poisons by increasing the persistence of DNA strand breaks[END_REF]. This has led to the development of a multitude of potent inhibitors with various bioavailability and pharmacokinetic characteristics whose efficacy in the treatment of cancer in vivo has been evaluated in animal models [START_REF] Tentori | Chemopotentiation by PARP inhibitors in cancer therapy[END_REF][START_REF] Giansanti | PARP inhibitors: New tools to protect from inflammation[END_REF]; several PARP inhibitors are currently under investigation in clinical trials [START_REF] Sandhu | Poly(ADP-ribose) polymerase inhibitors in cancer treatment: a clinical perspective[END_REF][START_REF] Rouleau | PARP inhibition: PARP1 and beyond[END_REF]. However, a clear understanding of the mechanism(s) whereby PARP inhibitors potentiate the activity of antineoplastic agents is still lacking. Moreover, isoform specific PARP inhibitors are still missing while it is known that PARP-2 accounts for 10-20% of the total PARP activity in response to DNA damage [24 and references therein].

In our studies we used the hydrophilic PARP inhibitor PJ34 that has been recently reported to synergize with cisplatin in triple-negative breast cancer cells [START_REF] Hastak | Synergistic chemosensitivity of triple-negative breast cancer cell lines to PARP inhibition, gemcitabine and cisplatin[END_REF], in combination with the DNA TOP1 inhibitor, TPT. For our experiments we performed 1 h treatment with up to 10 M TPT that was already reported to be sufficient for trapping TOP1 in MCF7 cells [START_REF] Feeney | Tracking the cell cycle origins for escape from topotecan action by breast cancer cells[END_REF]. PJ34 was used at a concentration (5 M) that was capable of inhibiting PARP activity but devoid of cytotoxic effects We found that TPT toxicity was higher when PAR synthesis was strongly reduced by either PARP-1 silencing (HeLa Sip-1 cells) or PJ34 administration (both in HeLa and MCF7 cells). MCF7 and HeLa cells, according with their comparable PARP-1 +/+ BRCA1/2 +/+ and p53 +/+ status showed the same sensitivity to TPT, which determined a cell cycle arrest until 72 h after treatment. However, in combination with PJ34, TPT was cytotoxic even at a very low concentration (1.25 µM). Accordingly, 1.25 µM TPT alone was cytotoxic in PARP-1 silenced cells (HeLa SiP-1 ). Nevertheless, the PARP inhibitor further increased the sensitivity of SiP-1 cells with respect to PARP-1 proficient cells treated with the drug combination, suggesting a PARP-2 involvement in the signaling of TPT-dependent DNA damage.

Consistently with the idea that poly(ADP-ribosyl)ation plays a role in the response to TPT-induced DNA damage, we found increased PAR synthesis following cell exposure to 10 µM TPT. The PARP inhibitor PJ34 prevented PARP activity and concomitantly caused intensification of cell cycle perturbations and increased DNA damage.

In particular, we observed distinct cell cycle perturbation effects depending on the concentration of the TOP1 inhibitor and on the association with the PARP inhibitor: in the low TPT dose range, PJ34 in combination with 0.2-0.4 µM TPT caused more cells to be arrested in the G2/M phase, whereas combined with 1.25 M TPT it arrested at the S phase cells that escaped TPT action. Furthermore, the G2/M block induced by 0.4 M TPT in PARP-1 wild type cells was magnified in PARP-1 silenced HeLa cells. These evidences agree with the concept that after 1 h pulse (whatever the dose) of TPT not all the cells are prevented from entry in mitosis and then G2 cell lineages could survive TPT-mediated cytotoxicity [START_REF] Tuguri | Topoisomerase I suppresses genomic instability by preventing interference between replication and transcription[END_REF]. Therefore, accumulation at the G2/M phase of tumour cells that escaped TPT action, provoked by PARP inhibition or by PARP-1 silencing, can be seen as a mechanism to overcome resistance to camptothecin derivatives. Interestingly, in PARP-1 silenced HeLa cells PJ34 increased the TPT S phase arrest as a further indication of PARP-2 implication.

Consistently, the TPT-dependent DNA damage level was increased by co-treatment with PJ34 either in PARP-1 proficient and PARP-1 silenced cells 24 h after treatment. In nuclei of such cells, differences in H2AX levels deriving from TPT+/-PJ34, also support PARP-1 and -2 stimulation of TPT-dependent DSBs repair. Moreover, we found a sustained PAR synthesis from 1 to 24 h after treatment and most of the newly synthesized polymer was linked to PARP-1 itself. Two other PAR acceptors in the 55-95 kDa MW's range appeared to be TPTand PJ34-dependent.

Accordingly with the magnified effects of TPT+PJ34 treatment in PARP-1 silenced cells the PARP-2 modification could represent the mechanism of its participation in DSBs signaling and HR repair [START_REF] Yelamos | Toward specific functions of poly(ADP-ribose) polymerase-2[END_REF]. Indeed, these evidences suggest that the lack of PAR synthesis, by interfering with the repair of TOP1-induced DNA damage, causes DNA strand breaks accumulation and further delays cell cycle progression. Moreover, we found that TPT-treated cells entered the apoptotic program as a consequence of PARP-1 silencing and/or PARP inhibition.

The last set of results was based on mechanistic investigations addressed to show the long-term response to TPT action: after 1 h TPT pulse TOP1 soluble/active fraction was drastically reduced for at least 3 cell duplication cycles and p53/p21 levels increased within the same time frame. Such an up-regulation was even higher in cells lacking PARP-1 and further increased by TPT+PJ34 treatment, supporting again the involvement of PARP-2 in the signaling of TPT-dependent DNA damage.

These results are in agreement with those previously reported in the same cells treated with the methylating agent temozolomide in combination with the PARP inhibitor GPI 15427, suggesting the involvement of PARP-2 (or other PARP) in the repair of DNA damage provoked by temozolomide [START_REF] Tentori | Pharmacological Inhibition of Poly(ADP-ribose) Polymerase (PARP) Activityin PARP-1 Silenced Tumour Cells Increases Chemosensitivity to Temozolomide and to a N3-Adenine Selective Methylating. Agent[END_REF].

Our data also suggest a synergistic interaction of PARP-1 and PARP-2 with p53 in tumour suppression through their role in DNA damage response and genome integrity surveillance. Another study showed that in MCF7 cells inhibition of endogenous PARP-1 function suppresses the transactivation function of p53 in response to ionizing radiation [START_REF] Wieler | Poly(ADP-ribose) Polymerase-1 Is a Positive Regulator of the p53-mediated G1 Arrest Response following Ionizing Radiation[END_REF]. We also observed that p53-dependent BAX expression and caspase-dependent PARP-1 proteolysis were sustained by the PARP inhibitor as a result of apoptosis induction.

A c c e p t e d M a n u s c r i p t 20 By the all of such evidences we envisaged a TPT-dependent DNA damage signaling network, involving PARP. Indeed, the DNA damage arising from the trapping of TOP1 was signaled by PARP-1 and -2 and gathered by effectors like p53 and p21. Previous results suggest that p53 causes resistance of cells to TPT [START_REF] Tomicic | Topotecan-triggered degradation of topoisomerase I is p53-dependent and impacts cell survival[END_REF]. Our findings suggest a PARP modification induced by TPT-dependent DNA damage, while PARP-1 and -2 inactivation switches on p53/p21 pro-apoptotic role. Indeed, caspase-dependent PARP-1 proteolysis contributes to restoring the apoptotic program in neoplastic cells. Nuclear caspases-mediated PARP-1 cleavage has been described in camptothecin-induced apoptosis as an early event that precedes the release of cytochrome c and AIF, generally thought to activate the chemotherapy-induced apoptosis by DNA-damaging drugs [START_REF] Rodrıguez-Hernandez | Nuclear caspase-3 and capase-7 activation, and Poly(ADP-ribose) polymerase cleavage are early events in camptothecin-induced apoptosis[END_REF].

In conclusion, our findings contribute to the understanding of the molecular events triggered by TOP1 poison-dependent genomic damage and provide a rationale for the development of new approaches to sensitize cancer cells to chemotherapy. Fifty ng of human recombinant PARP-2 (hrPARP-2) was also loaded as a standard. Cells were treated with 10 M TPT for 1 h and allowed to recover in fresh medium in the presence or not of 5 M PJ34 for the indicated times. GAPDH was used as loading control.

  can survive when one repair system breaks down, but they start to die when both DNA repair mechanisms stop functioning.

  , antibodies and chemicals TPT was from Glaxo Smith-Kline (Verona, Italy) and PJ34 [N-(6-oxo-5,6,dihydrophenanthridin-2-yl) -(N,Ndimethylamino) Acetamide] from Alexis Biochemicals (Vinci-Biochem, Firenze, Italy). The cocktail of protease inhibitors was from ROCHE-Diagnostic (Milano, Italy).

  1 mM PMSF, 1:25 dilution of a cocktail of protease inhibitors, 0.125 M NAD + and 5 Ci [ 32 P]-NAD + (1,000 Ci/mmole). After incubation at 37°C for 30 min, cells were scraped off the plates, transferred to eppendorf tubes and mixed with TCA at 20% (w:v) final concentration. After 15 min standing on ice, samples were collected by centrifugation at 1,200 rpm for 15 min, washed twice with 5% TCA and three times with ethanol. [ 32 P]-PAR incorporated in the TCA-insoluble fraction was measured by Cerenkov counting using a LS8100 liquid scintillation spectrometer (Beckman Coulter S.p.A. Milano, Italy). Finally, TCA protein pellets were resuspended in Laemmli buffer; proteins were separated by 5-15% SDS-PAGE and after electroblotting on PVDF membrane, [ 32 P]-PAR acceptors were visualized by autoradiography. Immunodetection of specific proteins was accomplished on the same blots after autoradiography.

  A c c e p t e d M a n u s c r i p t 10 ice, cellular suspensions were centrifuged at 960 x g for 90 sec at 4°C and the nuclear fractions recovered in the pellet. The supernatant represents the post-nuclear fraction.

  , as early as 24 h after treatment, graded concentrations of TPT induced a progressive increase of cell accumulation in the G2/M phase starting from 0.2 M up to 1.25 M. Higher TPT concentrations, instead, promptly arrested the cells in S phase.The addition of the PARP inhibitor PJ34 to TPT concentrations <1.25 M significantly increased G2/M cell accumulation, whereas when combined with ≥1.25 M TPT concentrations, PJ34 induced S phase cell accumulation. As also shown in Figure1

A

  c c e p t e d M a n u s c r i p t 12 cell cycle kinetics was unaffected by treatment of HeLa Babe cells with PJ34 used as single agent.

  on the autoradiography only a light smear at the top of the gel could be visualized.

Figure 9

 9 Figure9shows in HeLa Babe cells the caspase-dependent PARP-1 cleavage. In MCF7

Figure 1 -

 1 Figure 1 -Cell cycle analysis of HeLa Babe and HeLa SiP-1 cells treated with TPT and PJ34 as

Figure 2 -

 2 Figure 2 -Cell death analysis of HeLa cells subjected to TPT and PJ34 single and combined

Figure 3 -A:

 3 Figure 3 -DNA damage in HeLa Babe , HeLa SiP-1 and MCF7 cells subjected to TPT+/-PJ34

A c c e p t e d M a n u s c r i p t 26 B:

 26 Western Blot analysis of H2AX levels in HeLa Babe and HeLa SiP-1 cell nuclei treated 1 h with 10 M TPT and allowed to recover in fresh medium in the presence or not of 5 M PJ34 for 72 h. Actin was used as loading control.

Figure 4 -

 4 Figure 4 -PJ34-dependent inhibition of PAR synthesis in HeLa Babe and HeLa SiP-1 cells.Cells were resuspended in lysis buffer, sonicated and incubated with 50 µM [ 32 P]-NAD + +/-5 M PJ34 as described in Materials and Methods.

Figure 5 -

 5 Figure 5 -TPT-dependent PARP activation in MCF7 cells.

Figure 6 -

 6 Figure 6 -Western Blot analysis of PARP-1, TOP1, p53 and p21 in MCF7 cells.

Figure 7 -

 7 Figure 7 -Western blot analysis of PARP-1, TOP1, p53 and p21 in HeLa Babe and HeLa SiP-1

Figure 8 -

 8 Figure 8 -Densitometric analysis of p53 levels in HeLa cell samples.

Figure 9 -

 9 Figure 9 -Western Blot analysis of PARP-1 and BAX in HeLa and MCF7 cells.

  Figure1

Figure

  Figure

FigureFigure

  Figure

ACKNOWLEDGEMENTS:

This work was supported by funding from the Italian Ministery of Education and Research "Programmi di Ricerca Scientifica di Rilevante Interesse Nazionale" (PRIN 2007(PRIN -2009)).

*Graphical Abstract


Francesco Amoroso

Sinnou David

Umberto Zannier

On fields with the Property (B)

Keywords:

Let K be a number field and let L/K be an infinite Galois extension with Galois group G. Let us assume that G/Z(G) has finite exponent. We show that L has the Property (B) of Bombieri and Zannier: the absolute and logarithmic Weil height on L * (outside the set of roots of unity) is bounded from below by an absolute constant. We discuss some feature of Property (B): stability by algebraic extensions, relations with field arithmetic. As a as a side result, we prove that the Galois group over Q of the compositum of all totally real fields is torsion free.

Introduction

Let h be the absolute and logarithmic Weil height on Q. Following Bombieri and Zannieri [Bo-Za], we say that a set A of algebraic numbers has the Bogomolov property (B) if there exists a real number T 0 = T 0 (A) > 0 such that the set of non-zero α ∈ A of height < T 0 consists of all roots of unity in A.

There are several interesting examples of subfields of Q with Property (B). For instance, the field Q tr of all totally real algebraic numbers has this property (see [Sc] and [Sm]). Also the abelian closure Q ab satisfies (B) (see [Am-Dv]) and, more generally, the abelian closure K ab of a number field K satisfies (B) (see [Am-Za]). In the latter case, Property (B) holds uniformly in [K : Q]; the height on (K ab ) * outside roots of unity is bounded from below by a positive constant depending only on [K : Q] (see ).

Another family of fields with Property (B) is provided by fields with bounded local degrees at some finite place. Let K be a number field and L/K be an infinite extension.

1 Fix a non-archimedean valuation v of K. We say that L/K has bounded local degree at v if there exists an integer d 0 such that for every extension w of v to L we have [L w : K v ] ≤ d 0 . Bombieri and Zannier prove (see [Bo-Za], Theorem 2) that a Galois extension L/Q with bounded local degree at some rational prime satisfies the Property (B).

A third family has been recently exhibited by Habegger [Hab]. Let E be an elliptic curve defined over Q. Then the field Q(E tors ) obtained by adjoining all torsion points of E has the Bogomolov property. Observe that if E does not have complex multiplication then this field is an infinite non-abelian extension of Q.

The first two examples above suggest the following problem.

Problem 1.1 Let K/Q be an extension with bounded local degree at some rational prime. Is it true that K ab has the Property (B)?

The first result of this article is a partial answer to this problem. In section 4 we prove the following generalization of both [Bo-Za], Theorem 2 and [Am-Za2], Theorem 1.2.

Theorem 1.2 Let K be a number field and let L/K be an infinite Galois extension with Galois group G. Let E ⊆ L be the subfield fixed by Z(G) and assume that E/K has local degree at some non-archimedean valuation v of K bounded by d 0 . Then L has the Property (B), uniformly in v, d 0 and [K : Q].

More explicitely, there exists a positive function c which depends effectively only on v, d 0 and [K : Q] such that for any α ∈ L * which is not a root of unity we have h(α) ≥ c.

As a special case of the quoted result of Bombieri and Zannier, a Galois extension L/Q with Galois group of finite exponent has Property (B). Indeed Gal(L/K) has finite exponent if and only if L/K has uniformly bounded local degrees at every prime of K (see [Ch], Theorem 2.2.2, for a more precise statement). Similarly, our Theorem 1.2 implies:

Corollary 1.3 Let K be a number field and let L/K be an infinite Galois extension with Galois group G. Let us assume that G/Z(G) has finite exponent b. Then L has the Property (B), uniformly in b and [K : Q].

Our second result deals with finite extensions F/L of a field L with Property (B). Assume that L/K is an abelian extension of a number field K. By the main theorem of [Am-Za], F has the Property (B). Similarly, if L/Q has bounded local degree at some rational prime, then F/Q has the same property and thus, by [Bo-Za], it has the Property (B). Therefore the following question arises naturally.

Problem 1.4 Let L be a field with Property (B) and let F/L be a finite extension. Is it true that F has necessarily Property (B)?

In section 5 we give a negative answer to this question and we provide some related remarks.

In this respect, it may not be out of place to recall another definition of [Bo-Za] (see also [Dv-Za]). A set A of algebraic numbers has the Northcott property (N) if there exists a positive real number T such that the set of α ∈ A of height < T is finite. By a celebrated theorem of Northcott, the set of algebraic numbers of degree at most d has Property (N). Moreover, it is easy to see that if an extension L/Q has Property (N), then every finite extension F/L has again (N). Thus Property (B) and Property (N) behave in radically different ways under finite extensions.

Section 6 explores some speculative relations between Property (B) and field arithmetic. Two central definitions in this area are Pseudo Algebraically Closed and Hilbertian fields. We recall that a field K is Pseudo Algebraically Closed (PAC) if each absolutely irreducible variety defined over K has a K-rational point (see [Fr-Ja], chapter 11, for more details). A field K is Hilbertian if it satisfies Hilbert's Irreduciblity Theorem: for every irreducible f ∈ K[x, y] which is separable in x there exists a ∈ K such that f (x, a) is irreducible over K (see [Fr-Ja], chapter 12, for more details).

We consider the following problems. Does there exist a PAC field K ⊆ Q which satisfies (B)? What are the relations between (B) and hilbertianity ?

We give some evidence for a negative answer to the first question and we provide examples of a Hilbertian (resp. non Hilbertian) field which does not satisfy (resp. which satisfies) Property (B).

Finally, in section 7 we prove that the Galois group over Q of the compositum of all totally real fields is torsion free. This results is needed in section 5, and it is apparently not known.

Aknowledgements. The authors are indebted with P. Dèbes, B. Deschamps and M. Fried for useful discussions on the subject of sections 6 and 7. We also thank D. Simon who provides us with the reference [Ko].

Notations and auxiliary results

Let K be a number field. Given a place v of K we denote by | • | v the corresponding absolute value normalized so to induce on Q one of the standard absolute values.

We shall use the followings couple of lemmas.

Lemma 2.1 Let K be a number field, v be a finite place of K over a rational prime p and let ρ > 0 s(p,ρ,λ) with s(p, ρ, λ) → +∞ for λ → +∞. More precisely, let us define an integer k = k(p, ρ) by k = 0 if (p -1)ρ > 1 and by

. Let γ 1 , γ 2 ∈ O K such that |γ 1 -γ 2 | v ≤ p -ρ . Then for any non-negative integer λ we have |γ p λ 1 -γ p λ 2 | v ≤ p -
p k-1 (p -1)ρ ≤ 1 < p k (p -1)ρ otherwise. Then s(p, ρ, λ) = p k ρ + max(0, λ -k) .
Proof. Let us denote by the same letter v the only valuation of K(ζ p λ ) extending v. We write

γ p λ 1 -γ p λ 2 = (γ 1 -γ 2 ) λ j=1 ζ p j (γ 1 -ζ p j γ 2 )
where the inner product is on the roots of unity ζ p j of order p j . The ultrametric inequality shows that

|γ 1 -ζ p j γ 2 | v = |γ 1 -γ 2 + (1 -ζ p j )γ 2 | v ≤ max(p -ρ , p -1/p j-1 (p-1) ) . Then |γ p λ 1 -γ p λ 2 | v ≤ p -s with s = ρ + λ j=1 min(p j-1 (p -1)ρ, 1) = p k ρ + max(0, λ -k) .
Lemma 2.2 Let L/K be a Galois extension of number fields and let σ ∈ Gal(L/K). Let ℘ be a prime of O K over the rational prime p. Let also a, b ∈ N and ρ > 0. Let us assume that

∀γ ∈ O L , ∀v | ℘, |γ a -σ(γ) b | v ≤ p -ρ .
Then for every α ∈ L such that α a = σ(α) b we have

h(α) ≥ 1 a + b [K ℘ : Q p ] [K : Q]
ρ log p -log 2 ; .

Proof.

We have already used implicitly this lemma in the proofs of [Am-Dv], Proposition 1, and of [Am-Za2], Proposition 3.2. We briefly recall the demonstration.

Let v be a place of L, normalized so to induce on Q one of the standard places. We shall estimate |α a -σ(α) b | v . Suppose to start with that v|℘.

By the Strong Approximation Theorem, there exists an integer β ∈ L such that αβ is integer and

|β| v = max{1, |α| v } -1 .
(see [Am-Dv], lemma 1, for details). Then we have

|(αβ) a -σ(αβ) b | v ≤ p -ρ and |β a -σ(β) b | v ≤ p -ρ .
Using the ultrametric inequality, we deduce that

|α a -σ(α) b | v = |β| -a v |(αβ) a -σ(αβ) b + (σ(β) b -β a )σ(α) b | v ≤ c(v) max(1, |α| v ) a max(1, |σ(α)| v ) b
with c(v) = p -ρ . This last inequality plainly holds for an arbitrary place v of L with

c(v) = 1 if v ∞ ; 2 if v | ∞ .
Applying the Product Formula as in [Am-Za2], Proposition 3.2, to α a -σ(α) b we get, after a standard computation,

1 ≤ 2 [K:Q] p ρ[K℘:Qp] H(α) a[K:Q] H(σ(α)) b[K:Q] .
The conclusion follows.

We now fix some notations which we follow in the next two sections.

Let K be a number field of degree d over Q. We consider a finite Galois extension L/K of Galois group G. Let N be a normal subgroup of G contained in Z(G). We let E = L N be the fixed field of N . We remark that N is abelian (since it is contained in Z(G)). Thus L/E is an abelian extension of Galois group N .

We fix a prime ideal q of O E . Let ℘ = q ∩ O K and let (p) = ℘ ∩ Z. We define d 0 as the local degree [E q : K ℘ ].

We also denote by µ the group of roots of unity and by µ p ∞ the group of roots of unity of order a p-th power.

A conditional result

The main result of this section is the following generalization of Proposition 3.2 of [Am-Za2].

Proposition 3.1 Let α ∈ L * \µ. Assume further that for any τ ∈ Gal(L/E) τ (α)/α ∈ µ p ∞ \{1} . (3.1) Then h(α) ≥ c
for some c > 0 depending only on p, d 0 and d.

Proof. We can assume L = E(α). Indeed Gal(E(α)/E) is a normal subgroup of Gal(E(α)/K) contained in the center. Thus, condition (3.1) can be rephrased by saying that for any non-trivial τ ∈ Gal(L/E) we have τ (α)/α ∈ µ p ∞ . A first case occurs when q does not ramify in L. Let φ be the Frobenius automorphism of Q/q, where Q is any prime of O L over q. Since L/E is abelian, φ does not depend on the choice of Q. Thus for any γ ∈ O L γ q ≡ φ(γ) mod qO L , where q is the norm of q. Let now q be an other prime ideal of O E over ℘, fix a prime Q of O L over q and let φ be the Frobenius of Q /q . Then φ and φ are both in N and are conjugate in G. Since N is contained in Z(G) we deduce that φ = φ. This shows that for any γ ∈ O L and for any place v of L with v | ℘ we have

|γ q -φ(γ)| v ≤ p -1/e0
, where e 0 is the ramification index of q over ℘.

Lemma 2.1 shows that there exists λ depending only on p, on e 0 and on d such that

|γ qp λ -φ(γ) p λ | v ≤ p -2d .
Since α is not a root of unity, α qp λ -φ(α) p λ = 0. By lemma 2.2

h(α) ≥ 1 p λ (q + 1) (2[K ℘ : Q p ] log p -log 2) ≥ log 2 p λ (q + 1) ≥ c 1
for some c 1 > 0 depending only on p, d 0 and d.

Assume now that ℘ is ramified in L and let, as in [Am-Za2] Proposition 2.3,

H q := {σ ∈ N such that ∀γ ∈ O L , σγ q ≡ γ q mod qO L }
where q is the norm of q. By the quoted proposition, H q is non trivial. As in the nonramified case, let q be an other prime ideal of O E over ℘. Then H q and H q are both subgroups of N and are conjugate in G. Since N is contained in Z(G) we deduce that H q = H q . Let σ be a non trivial automorphism of this subgroup. Then, for any γ ∈ O L and for any place v of L with v | ℘ we have

|γ q -σ(γ) q | v ≤ p -1/e0
, where e 0 is the ramification index of q over ℘. We use Lemma 2.1 and Lemma 2.2 as in the first part of the proof. We remark that α qp λ -σ(α) qp λ = 0 thanks to (3.1) and to the remark at the beginning of the proof. We get h(α) ≥ c 2 for some c 2 > 0 depending only on p, d 0 and d. It is now enough to choose c = min(c 1 , c 2 ).

4 An unconditional result and the proof of Theorem 1.2

The radical reduction of [Am-Za2], section 4, does not apply in the present situation. Indeed, following the beginning of the proof of Proposition 4.3 in op. cit., k is not necessarily a power of a prime and so we cannot bound the degree of E(ζ k )/E in terms of d 0 and d.

Fortunately, we can modify the argument of op.cit. in a such a way that it applies in the present situation. As an extra-bonus the proof becomes simpler.

We first state a simplified and slightly precise version of a special case of [Am-Za2], Lemma 4.2. The present statement applies only to subgroups of (Z/kZ) * for k a prime power, but this is enough for our purposes.

Lemma 4.1 Let p be a rational prime, k be a p-th power and B be a positive integer. Then, for every subgroup H of (Z/kZ) * of index < B, there are integers x, y such that x mod k ∈ H, y mod k ∈ H and

2 < y -x < 6B .
Proof. The proof is a standard application of the box principle. Let Λ = {x ∈ N | x mod k ∈ H} and define, for j ∈ N, the real interval

I j = [6(j -1)B, 6jB) . Assume by contradiction ∀j ∈ N, |I j ∩ Λ| ≤ 3 .
Let J be a large integer and put r = [6JB/k]. Then

r|H| = |Λ ∩ [0, rk)| ≤ |Λ ∩ [0, 6JB)| ≤ 3J
which implies 2B|H| ≤ 6JB/r ≤ k(r + 1)/r. Letting J → +∞ we get a contradiction:

2B|H| ≤ k ≤ 2(1 -1/p)k = 2|(Z/kZ) * | < 2B|H| .
Thus there exist integers x = x 1 < x 2 < x 3 < x 4 = y such that x i mod k ∈ H and y -x < 6B.

We can now prove an unconditional version of Proposition 3.1.

Proposition 4.2 Let α ∈ L * \µ. Then h(α) ≥ c
for some c > 0 depending only on p, d 0 and d.

Proof. There exists a p-th power k and a primitive k-root of unity

ζ k ∈ L such that L ∩ Q(µ p ∞ ) = L ∩ Q(ζ k ) .
We identify Gal(E(ζ k )/E) to a subgroup of (Z/kZ) * of index, say, B -1. By Galois theory,

B -1 = [E ∩ Q(ζ k ) : Q]. Since k is a p-th power, the prime p is totally ramified in E ∩ Q(ζ k ). This shows that B -1 ≤ e(q|p) ≤ d 0 d. By Lemma 4.1 there exist σ 1 , σ 2 ∈ Gal(E(ζ k )/E) such that σ i ζ k = ζ gi k with g = g 2 -g 1 satisfiying 2 < g < 6(d 0 d + 1) . (4.1)
Let σi ∈ Gal(L/E) extending σ i . We want to apply Proposition 3.1 with α ← β, where

β = σ2 (α) α g σ1 (α) . (4.2)
To do this we need to prove that β ∈ µ and that τ (β)/β ∈ µ p ∞ \{1} for any τ ∈ Gal(L/E). Let us verify these requirements. We argue by contradiction.

• β ∈ µ.

Then, by (4.2),

gh(α) = h(α g ) = h(σ 2 (α)/σ 1 (α)) ≤ 2h(α) .
Since g > 2 by (4.1) we get α ∈ µ. Contradiction.

• There exists τ ∈ Gal(L/E) such that θ := τ (β)/β ∈ µ p ∞ \{1}.

Let η = τ (α)/α. Apply (4.2) and its conjugate by τ , taking into account that we are working in an abelian extension of E. We obtain

θ = τ (β) β = τ σ2 (α) τ (α g )τ σ1 (α) σ2 (α) α g σ1 (α) -1 = σ2 (η) η g σ1 (η)
.

Hence gh(η) ≤ 2h(η) which implies h(η) = 0 by (4.1). Thus η ∈ µ. Write η as η = η 1 η 2 with η 1 ∈ µ p ∞ and with η 2 of order not divisible by p. By Bezout's identity,

η 1 ∈ Q(η) ⊆ L.
Thus there exists an integer a such that η 1 = ζ a k . By the choice of σi we see that

σ2 (η 1 ) η g 1 σ1 (η 1 ) = 1 . Thus θ = σ2 (η 2 ) η g
2 σ1 (η 2 ) has order not divisible by p. But θ ∈ µ p ∞ and θ = 1. Contradiction. Applying (3.1) with α ← β we get h(β) ≥ c. By (4.1) and (4.2),

h(β) ≤ (g + 2)h(α) ≤ (6d 0 d + 7)h(α) . Thus h(α) ≥ c
with c = c/(6d 0 d + 7).

We are now in position to prove Theorem 1.2. Let K be a number field and let L/K be an infinite Galois extension with Galois group G. Let E ⊆ L be the subfield fixed by Z(G) and assume that E/K has local degree at some prime ℘ bounded by d 0 . Let α ∈ L * not a root of unity. We choose a subfield L containing α and such that L /K is Galois. We put E = L ∩ E. Then it is easy to see that L /E is Galois and Gal(L /E ) is contained in the center of Gal(L /K). Moreover E /K has local degree at ℘ bounded by d 0 . By Proposition 4.2, the height of α is bounded from below by a positive constant depending only on ℘, d 0 and on [K : Q]. Theorem 1.2 follows.

Property (B) and field extensions

In this section we show that the Property (B) is not generally preserved by taking a finite extension. As remarked in the introduction, this on the contrary holds for the Property (N).

Let Q tr be the compositum of all totally real extensions. Thus Q tr is a Galois extension of Q and α ∈ Q tr if and only if α is totally real. We denote by i a square root of -1 in Q. Note that Q tr (i)/Q is also Galois, as the composite of the Galois extensions Q tr /Q and Q(i)/Q. Let τ be the generator of Gal(Q tr (i)/Q tr ). Then for any Q-embedding σ : Q tr (i) → C we have σ = στ . This implies that if an archimedean absolute value of α ∈ Q tr (i) is 1 then all its archimedean absolute values are equal to 1. The following lemma shows that the converse is also true.

Lemma 5.1 Let α ∈ Q be such that all its archimedean absolute values are equal to 1.

Then α ∈ Q tr (i). Proof. We define a = 1 2 (α + α -1 ), b = 1 2i (α -α -1 ) . Then α = a + bi and a, b ∈ Q tr . Indeed, let σ : Q tr (i) → C. Since 1 = |σα| 2 = σα • σα , we have σa = 1 2 (σα + σα) = Re(σ(α)), σb = 1 2σ(i) (σα -σα) = Im(σ(α)) .
Remark 5.2 We recall that a number field L is a CM field if it is a totally complex quadratic extension of a totally real number field. It is well-known (see for instance [Wa], p.38) that in a CM field L the complex conjugation of C defines an involution τ of L which is independent of the embedding into C., i.e. for any Q-embedding σ : L → C we have σ = στ . Let α ∈ L. The argument of the proof of Lemma 5.1 shows that

a = 1 2 (α + τ (α)), b = 1 2i (α -τ (α)) .
are totally reals. Thus α = a + ib ∈ Q tr (i). This proves that any CM field is contained in Q tr (i) (and actually Q tr (i) is the compositum of all CM fields).

We are now in position to give the promised example.

Example 5.3 The field Q tr satisfies (B), but its quadratic extension Q tr (i) does not.

Proof. For the first assertion, see [Sc] and [Sm]. For the second one, [Am-Nu], Theorem 1.3, shows that there exists an infinite sequence (α k ) of algebraic numbers such that the fields Q(α k ) are CM-fields, α k is not a root of unity, and h(α k ) → 0. By remark 5.2,

Q(α k ) ⊆ Q tr (i).
Thus the field Q tr (i) does not satisfy (B). A more direct example is the following. For k ∈ N let

α k = 2 -i 2 + i 1/k .
Then all the archimedean absolute values of α k are equal to 1. Thus, by Lemma 5.1, α k ∈ Q tr (i). Obviously α k is not a root of unity and h(α k ) → 0 (note, however, that extracting roots is not the only manner to construct number of small height in Q tr (i). See again [Am-Nu], sections 4 and 5 for details).

In view of this example, it may not be out of place to study the Galois group of Q tr /Q. The absolute Galois group of Q tr is known. By a result of Freid, Haran and Völklein (see [Fr-Ha-Vo]) Gal(Q/Q tr ) is freely generated by a subset of involutions, homeomorphic to the Cantor set. Nevertheless, nothing is apparently known on Gal(Q tr /Q).

By a well known theorem of Artin-Schreier-Baer (see [Ar-Sc] and [Ba]) the only non trivial elements of finite order in the absolute Galois group Gal(Q/Q) are the complex conjugations. In the present situation we have: Theorem 5.4 There is no automorphism of finite order > 1 in Gal(Q tr /Q).

We shall give a proof of this theorem in section 7. For the moment, let us pause for some consequences of this statement.

Let G be a profinite group such that any Galois extension L/Q with Galois group G satisfy (B). Let L/Q be any of such extensions. We could ask if any finite extension of L satisfy again (B).

The group G = Gal(Q tr /Q) provide a counterexample to this assertion. Indeed, if L/Q has Galois group G, then L ⊆ Q tr . Otherwise we could find an involution in G, contradicting Theorem 5.4. By the quoted result of [Sc] and [Sm], any Galois extension L/Q with Galois group G satisfy (B).

Thus G is a profinite group such that any Galois extension L/Q with Galois group G satisfy (B). However, as proved in example 5.3, Q tr (i) is a quadratic extension of L = Q tr which does not satisfy (B) and Gal(L/Q) = G.

The situation seems to be different if we allow base change.

Definition 5.5 Let G be a profinite group. We say that G has the Property (B) if for any number field K and for any Galois extension L/K with Galois group G, the field L satisfies (B).

By Galois theory, a profinite group G satisfies (B) if and only at least one of its subgroups of finite index satisfies (B). Indeed, let H be a subgroup of finite index of G which satisfies (B). Let L/K be a Galois extension of a number field K such that Gal(L/K) = G. Then L H is a finite extension of K, hence a number field. Since Gal(L/L H ) = H satisfies (B), the field L satisfies (B).

We remark that Gal(Q tr /Q) does not satisfy (B). Indeed, Gal(Q tr (i)/Q(i) Gal(Q tr /Q) and the field Q tr (i) does not satisfy (B).

We also remark that abelian groups and groups with finite exponent satisfy (B). Moreover, let L/K be a Galois extension of a number field with Galois group abelian or of finite exponent. Then any finite extension of L satisfies again (B), as we have already seen in the introduction in the special case K = Q.

More generally, let G be a profinite group such that all its subgroups of finite index satisfy (B) (this is the case for abelian groups and for groups of finite exponent). Then any finite extension E of a Galois extension L/K of a number field satisfies (B), provided that Gal(L/K) = G. To see this, select a primitive element α ∈ E over L. Thus E = L(α) and, by Galois theory L(α)/K(α) is a Galois extension with Galois group isomorphic to

H = Gal(L/L ∩ K(α)) ⊆ G of index [L ∩ K(α) : K] ≤ [K(α) : K] < ∞. Since K(α) is a number field, if H satisfies (B) then L(α) satisfies (B).
These remarks suggest the following questions:

Problem 5.6 Let G be a profinite group which satisfy (B). i) Is it true that any subgroup of G of finite index satisfies (B)?

ii) Let K be a number field and let L/K be a Galois extension with Galois group G. Is it true that any finite extension of L satisfies (B)? By the remarks above, i) implies ii).

Let 1 → H → G → G → 1 be a group extension of profinite groups. A positive answer to problem 5.6 ii) would imply that if H is finite and G satisfies (B), then G satisfies (B). We remark that we cannot replace "H finite" by "H satisfies (B)" in this last statement. Indeed, for p prime the field

L = Q(µ p ∞ , 2 1/p , 2 1/p 2 , . . .)
obviously does not satisfy (B). However, its Galois group G over Q is an extension of a profinite abelian group by an other profinite abelian group:

H = Gal(Q(µ p ∞ )/Q) Z * p , and
G = Gal(L/Q(µ p ∞ )) Z p .
6 Relations with field arithmetic

In this section we explore some speculative relations between Property (B) and field arithmetic.

We recall that a field K is Pseudo Algebraically Closed (PAC) if each absolutely irreducible variety defined over K has a K-rational point (see [Fr-Ja], chapter 11, for more details). Obviously an algebraically closed field is PAC. We give some evidences for a negative answer to the following problem: Problem 6.1 Does there exist a PAC field K ⊆ Q which satisfies (B)?

First we remark that algebraic extensions of PAC fields are again PAC fields by a theorem of Ax-Roquette ([Fr-Ja], corollary 11.2.5). Similarly, if K does not satisfy (B) and if E/K is an algebraic extension, than E does not satisfy (B).

Only few examples of non-trivial PAC subfields of Q are known. For instance, as a consequence of a deep result of Pop, Q tr (i) is a PAC field (see [Pop], Theorem S, p.21 and [Ja-Ra], section 7 before Lemma 7.1). It is an open problem if the maximal solvable extension Q solve of Q is pseudo algebraically closed (see [Fr-Ja], problem 11.5.9 (a)). Observe that both Q tr (i) and Q solve do not have Property (B) (this is obvious for the second field and it is true for the first one by example 5.3).

An other example of PAC subfield of Q is provided by the compositum Q symm of all symmetric extensions of Q (i.e. of all Galois extensions over Q with Galois group a symmetric group), see [Fr-Ja], Th. 18.10.4. Again, this field does not satisfy (B). To prove this statement, it is enough to find a family L 1 , L 2 , . . . of symmetric extensions of Q such that lim n→+∞ inf{h(α) | α ∈ L n , α not a root of unity} = 0 . (6.1)

We can choose for {L n } the set of splitting fields of x p +x+1 for p > 3 prime, p ≡ 1 mod 4. Indeed for p prime x p +x+1 is irreducible and, if p satisfies the said condition, its splitting field is a symmetric extension (see [Ko]). Moreover, let α be a root of x p + x + 1. Then α is not a root of unity and ph(α) = h(α + 1) ≤ h(α) + log 2 by well known properties of Weil's height. Thus h(α) ≤ (log 2)/(p -1). The absolute Galois group Gal(Q/Q) is a compact group, thus admits a translation invariant Haar mesure. Let e ∈ N. By a theorem of Jarden (PAC Nullstellensatz, see [Fr-Ja], Th. 18.6.1), for almost all σ ∈ Gal(Q/Q) e the field Q σ fixed by σ 1 , . . . , σ e is PAC. It is again quite simple to prove that for almost all σ ∈ Gal(Q/Q) e the field Q σ does not satisfy (B). For this purpose we quote the following immediate consequence of [Fr-Ja], Lemma 18.5.3. Lemma 6.2 Let e ∈ N and let L 1 , L 2 , . . . be linearly disjoint number fields. Let us assume that L 1 , L 2 , . . . satisfy (6.1) and in addition

∞ n=1 [L n : Q] -e = ∞ .
Then for almost all σ ∈ Gal(Q/Q) e the field Q σ contains infinitely many L n and hence it does not satisfy (B).

Proof. We apply the quoted lemma of [Fr-Ja] with K = Q and

A n = {Gal(Q/L n )}.
Then, the set of σ ∈ Gal(Q/Q) e such that σ 1 , . . . , σ e ∈ Gal(Q/L n ) for infinitely many n has Haar measure 1. Thus for almost all σ ∈ Gal(Q/Q) e the field Q σ contains infinitely many L n . Since L 1 , L 2 , . . . satisfy (6.1), Q σ does not have (B).

It is easy to exhibit L 1 , L 2 , . . . which satisfy the requirements of the lemma. To simplify, let us assume e = 1. Then we can simply choose for {L n } the set of Q(2 1/p ) for p prime (indeed 1/p = ∞).

An other central definition in field arithmetic is the hilbertianity. A field K is Hilbertian if it satisfies Hilbert's Irreduciblity Theorem: for every irreducible f ∈ K[x, y] which is separable in x there exists a ∈ K such that f (x, a) is irreducible over K (see [Fr-Ja], chapter 12, for more details). As for PAC fields, we could ask for relations between hilbertianity and (B). But now, we do not have any direct implication. Indeed Q tr is not hilbertian (choose f (x, y) = x 2 -y 2 -1) and satisfies (B); on the contrary Q tr (i) is hilbertian (by Weissauer's Theorem, [Fr-Ja], capter 13, Theorem 13.9.1) and does not satisfy (B). An other example of field with these properties is Q symm which is hilbertian (see [Fr-Ja], Theorem 18.10.4) and does not satisfy (B), as already remarked.

7 On the Galois group Gal(Q tr /Q)

The field Q tr has a subset of totally positive elements, i.e., those all of whose conjugates are nonnegative. We shall repeatedly use the easy observation that:

A square root of a totally positive element lies in Q tr . Indeed, if α is totally positive, and β 2 = α, we have (σβ) 2 = σα, which is real and ≥ 0 for every Q-embedding σ : Q → C; hence σβ ∈ R for each such σ, as asserted.

We shall now prove Theorem 5.4. Let σ be a non trivial Q-automorphism of finite order of the field of totally real algebraic numbers. By replacing σ with a power of it, we may suppose that its order is a prime l. We shall derive a contradiction. Let us start with the Case l = 2. Since σ is supposed to have order 2, its fixed field F is such that [Q tr : F ] = 2. We have Q tr = F (β) for a β such that α := β 2 ∈ F .

We shall use the following Lemma 7.1 The number α is a sum of two squares of elements of F .

Proof. We shall use a result of Hilbert and Landau (see [La], Exercise 1, p. 461, or [Ra], Theorem. 15.11,p. 224): a totally positive α is a sum of squares in Q(α). In the Appendix below we shall give a self-contained proof of this result. Note that since β is totally real, α is totally positive. By the quoted result, α is then a sum of squares in Q(α), hence in F . Let α = a 2 1 + . . . + a 2 m , a 1 , . . . , a m ∈ F , be such a representation with minimal m. If m ≥ 2, let ξ := a 2 1 + . . . + a 2 m-1 . Since the a i lie in Q tr , we have that ξ is totally positive. Hence √ ξ is totally real, i.e. there exists µ ∈ Q tr

The c i are surely in Q(α 1 , . . . , α d ), but must actually be in Q, as can be seen by uniqueness and taking conjugates of these relations, or also directly by noting that the independence of the v i amounts to the fact that g 2 1 (α), . . . , g 2 d (α) is a basis of Q(α)/Q. We contend that c i ≥ 0 for all i. But this contradicts our choice of .

Then the c i are nonnegative rationals, and therefore each of them is a sum of squares of rational numbers: for a, b positive integers, the fraction a b is the sum of ab equal squares 1 b 2 . Then, relation (8.2) for i = 1 proves the sought conclusion.

Remark 8.2 Note that the proof shows that only d distinct squares (each repeated a suitable number of times) suffice to represent α in the sought shape.

  Indeed, let M := max |c i | and suppose that M = |c r |. Evaluating (8.2) at i = r and recalling (8.1) we haveM ≤ (d -1)M + |α r | ≤ M/2 + max |α i |,proving that M ≤ 2 max |α i |. Now, suppose by contradiction that c s < 0, and evaluate (8.1) at i = s. We obtain0 < α s ≤ j =s c j g 2 j (α s ) ≤ M (d -1) ≤ 2(d -1)(max |α i |) .

with µ 2 = ξ. We can write µ = p + βq with p, q ∈ F , and then ξ = p 2 + αq 2 + 2pqβ. Since ξ ∈ F we must have pq = 0, but of course we may assume that p, q do not both vanish. If p = 0, then q = 0 and we obtain a representation of α as a sum of m -1 squares, a contradiction. Then q = 0. But then ξ = p 2 and α = p 2 + a 2 m , as required.

By the lemma, we may find x, y ∈ F * such that x 2 -αy 2 = -1. Set γ := x + βy, so N (γ) = -1, where N is the norm from Q tr to F .

We have N (γ 2 ) = 1. Set η := 1+γ 2 , and denote with an accent the said automorphism (i.e. the conjugation of Q tr over F ). We have γ 2 (γ

Since γ ∈ Q tr , the element η is totally positive, whence η = δ 2 for some δ ∈ (Q tr ) * . Hence γ 2 (δ ) 2 = δ 2 , leading to γ = ±δ/δ . But then N (γ) = 1 and we have a contradiction, concluding the proof in the case l = 2.

Case l > 2.

We recall that Q tr (i)/Q is also Galois, as the composite of the Galois extensions Q tr /Q and Q(i)/Q. Note that Q tr (i) contains all roots of unity (as an immediate consequence of lemma 5.1). We also recall that the complex conjugation of C defines an automorphism τ of Q tr (i) which is independent of the embedding into C. By abuse of notation, we shall denote this automorphism by the usual symbol α := τ (α).

Let L denote the fixed field of σ in Q tr ; we may extend σ to an automorphism, denoted again σ, of Q tr (i), fixing i, so L(i) is the fixed field of σ in Q tr (i) and [Q tr (i) : L(i)] = [Q tr : L] = l. Note that Q tr (i) contains the l-th roots of unity, whose degree over L(i) divides l -1; but this degree also divides l, whence L(i) contains the l-th roots of unity. By Kummer's theory we then have Q tr (i) = L(i)(β) where β l =: α ∈ L(i), and σ(β) = θβ for some primitive l-th root of unity θ. Since L ⊂ Q tr , L(i) is sent into itself by complex conjugation, so also β generates Q tr (i) over L(i) and β l = α ∈ L(i). By Kummer's theory again, we have

for some r coprime to l (only its residue class mod l matters in (7.1)).

Applying to (7.1) complex conjugation we get β = γβ r , and using (7.1) in this last equation we obtain

, and then we may suppose r = ±1 in (7.1), so also γγ r = 1.

If r = 1 we have γγ = 1 whence (by Hilbert 90 for L(i)/L) γ = u/u for some u ∈ L(i). Then uβ = uβ, so uβ is totally real, and then uβ ∈ Q tr . But since σ fixes L(i) pointwise and stabilizes Q tr , this contradicts that σ(β)/β is a primitive l-th root of unity, so not totally real.

Therefore we have r = -1, and ββ = γ ∈ L(i) (actually γ ∈ L since this shows it is totally real).

Note now that Q tr (i) = L(i)(ρβ s ) for every ρ ∈ L(i) * and every s coprime to l, hence we may replace β with such ρβ s to generate Q tr (i)/L(i). We choose s = 2 and ρ = (ββ) -1 , so ρβ s = β/β =: ξ, say, and Q tr (i) = L(i)(ξ), where µ := ξ l = β 2l /γ l ∈ L(i).

All conjugates of ξ have absolute value 1, so all the conjugates of ξ 1 l have absolute value 1, and therefore, by lemma 5.1 ξ 1 l lies in Q tr (i), for every choice of the l-th root. Note that (ξ

At this point the proof mimics an argument of Artin, proving that C has no automorphisms of finite odd order (see Lang's quoted book, p. 299, Cor. 9.3). The polynomial x l 2 -µ = x l 2 -ξ l lies in L(i)[x] and has a root (actually all roots) in Q tr (i). But [Q tr (i) : L(i)] = l, so the polynomial is reducible. By Capelli's Theorem (see Lang's quoted book, Ch. VIII, Theorem 16), we have µ = a l for some a ∈ L(i). This yields ξ = ζa for an l-th root of unity ζ, and since ζ lies in L(i) as noted above, we deduce that ξ ∈ L(i), a contradiction which proves the theorem.

Appendix

In this Appendix we provide a self-contained proof of the following result, used in the proof of Lemma 7.1: Theorem 8.1 Let α be a totally real algebraic number. Then an element of Q(α) is a sum of squares in Q(α) if and only if it is totally positive.

This theorem is due to Hilbert and Landau, in a more general form when α is not necessarily totally real and it is required that all embeddings in R of the relevant number are positive. See Lang's or Rajwade's above-quoted references for a proof which depends on Artin-Schreier's theory of real fields. The simple proof below is independent of this theory and would seemingly work with small modifications also for the more general assertion.

To prove Theorem 8.1 note that one half of the conclusion is clear and so it suffices to work on the assumption that α is totally positive, and to prove that α is a sum of squares in Q(α).

We let d := [Q(α) : Q] and we denote by α 1 = α, α 2 , . . . , α d the (positive) distinct conjugates of α over Q.

There are polynomials f 1 , . . . , f d ∈ R[x] of degree at most d -1, and such that f i (α j ) equals 2 if i = j and 0 otherwise. (Just solve a Vandermonde linear system, or else take

Let us choose > 0 as a real positive number < min αi 2d max αi . By approximating the coefficients of f i with rational numbers, and squaring, we may then find polynomials g 1 , . . . , g d ∈ Q[x] (also of degree ≤ d -1) such that, for i = 1, . . . , d,

Note that v 1 , . . . , v d are linearly independent: if t 1 v 1 + . . . + t d v d = 0 with real t i not all 0, we may assume on dividing by the coefficient of maximal absolute value, that t r = 1 while |t i | ≤ 1 for i = 1, . . . , d. The relation implies t 1 g 2 r (α 1 ) + • • • + t d g 2 r (α d ) = 0, whence 1 < g 2 r (α r ) ≤ (d -1) , a contradiction with the said choice of . Then we may find (uniquely !) real numbers c 1 , . . . , c d such that . . . , d. (8.2)


Jose Luis

Patino Vilchis

Murray Evans

François James Ferryman

François Bremond

Monique Thonnat

Unsupervised Activity Extraction on Long-Term Video Recordings employing Soft Computing Relations

Keywords: Video data mining, vision understanding, discovery in multimedia database, soft computing

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

Introduction

Scene understanding corresponds to the real time process of perceiving, analysing and elaborating an interpretation of a 3D dynamic scene observed through a network of sensors (including cameras and microphones). This process consists mainly in matching signal information coming from sensors observing the scene with a large variety of models which humans are using to understand the scene. Although activity models can be built by experts of the domain, this might be a hard and time-consuming task depending on the application and the spectrum of activities that may be observed. The challenge thus consists of discovering, in an unsupervised manner, the significant activities observed from a video sequence. Knowledge discovery systems (KDS) aim at helping the human operator on this aspect. KDS systems have become a central part on many domains where data is stored in a database, but little research has been only done in the field of video data-mining. It must be said the task is particularly challenging because of the difficulty in identifying the interesting patterns of activity in the video due to noise, incomplete or uncertain information inherently present in the data. Soft computing methodologies are particularly suitable for these tasks because they provide the capability to process uncertain or vague information, as well as a more natural framework to cope with linguistic terms and produce natural language-like interpretable results. Fuzzy sets [START_REF] Zadeh | Fuzzy sets[END_REF] are the corner stone of soft computing together with other techniques such as neural networks and genetic algorithms. The relation between different existing fuzzy sets can be graded with the use of fuzzy relations [START_REF] Zadeh | Similarity relations and fuzzy ordering[END_REF]. Various fuzzy-based soft computing systems have been developed for different applied fields of data mining; but only a few systems employ soft computing techniques to partially characterize video activity patterns [START_REF] Doulamis | A fuzzy video content representation for video summarization and content-based retrieval[END_REF][START_REF] Lee | Activity and Location Recognition Using Wearable Sensors[END_REF]; In this paper we present a fully unsupervised system exploiting the use of fuzzy relations for the discovery of activities from video. First we model spatial and temporal properties from detected mobile objects employing fuzzy relations. We employ typical soft-computing algebra to aggregate these relations. A clustering algorithm based on the transitive closure calculation of the fuzzy relations allows finding spatio-temporal patterns of activity with different granularities. We have applied the proposed technique to different video sequences of apron monitoring in the Toulouse airport in France.

The reminder of this paper is as follows. In the next section, we give a short overview of the related work. We give a general overview of the system architecture and present our global approach in section 3 . The object detection and tracking process is given in section 4, then the data preprocessing steps previous to activity extraction are explained (section 5). We give the activity clustering methodology in section 6. Section 7 gives the main results and evaluation. Finally, Section 8 draws the main conclusions and describes our future work.

Related Work

Extraction of the activity contained in the video by applying data-mining techniques represents a field that has only started to be addressed. Although the general problem of unsupervised learning has been broadly studied in the last couple of decades, there are only a few systems which apply them in the domain of behaviour analysis. A few systems employ soft computing techniques to characterize video activity patterns [START_REF] Doulamis | A fuzzy video content representation for video summarization and content-based retrieval[END_REF][START_REF] Lee | Activity and Location Recognition Using Wearable Sensors[END_REF] but the methodology to self-discover new activities is still missing. Because of the complexity to tune parameters or to acquire knowledge, most systems limit themselves to object recognition [START_REF] Dalal | Human detection using oriented histograms of flow and appearance[END_REF]. For behaviour recognition, three main categories of learning techniques have been investigated.

• The first class of techniques learns the parameters of a video understanding program. These techniques have been widely used in case of event recognition methods based on neural networks [START_REF] Foresti | Event classification for automatic visualbased surveillance of parking lots[END_REF], Bayesian classifiers and HMMs [START_REF] Lv | Left luggage detection using bayesian inference[END_REF][START_REF] Wilson | Hidden Markov models for modeling and recognizing gesture under variation[END_REF].

• The second class consists in using unsupervised learning techniques to deduce abnormalities from the occurring events [START_REF] Xiang | Video behaviour profiling and abnormality detection without manual labelling[END_REF].

• The third class of methods focuses on learning behaviour based on trajectory analysis. This class is the most popular learning approach due to its effectiveness in detecting normal/abnormal behaviours; for instance, on abnormal trajectory detection on roads [START_REF] Piciarelli | Trajectory clustering and its applications for video surveillance[END_REF][START_REF] Stauffer | Learning patterns of activity using real-time tracking[END_REF] or pedestrian trajectory characterisation [START_REF] Anjum | Single camera calibration for trajectory-based behavior analysis[END_REF]. Hidden Markov Models (HMM) have also been employed to detect different states of predefined normal behaviour [START_REF] Bashir | Object Trajectory-Based Activity Classification and Recognition Using Hidden Markov Models[END_REF][START_REF] Porikli | Learning object trajectory patterns by spectral clustering[END_REF]. All these techniques are interesting, but little has been said about the semantic interpretability of the results. Indeed, more than trajectory characterisation, we are interested in extracting meaningful activity, where different trajectory types may be involved. This work comes thus into the frame of behaviour extraction from trajectory analysis however we have in addition a higher semantic level that employs spatial and temporal prox-imity relations between detected mobiles to characterise the ongoing different activities of the scene. In a similar framework, Dubba et al [START_REF] Dubba | Event model learning from complex videos using ilp[END_REF] have researched into transforming tracking data into a relational form where the relations are spatio-temporal relations among objects. This is the most comparable work to us, not only because they also look to identify activities as groups with coherent spatio-temporal information, but because they have worked on the same dataset. However, Dubba et al. employ a supervised approach (based on Inductive Logic Programming) and thus requires large quantities of annotated data. Our proposed approach is on the contrary completely unsupervised.

3 General overview of the system Our proposed system is mainly composed of two different processing components. The first one is for the detection and tracking of objects. The second subsystem works off-line and achieves the extraction of activity patterns from the video. This subsystem is composed of two modules: The trajectory speed analysis module, and the activity analysis module. The first is aimed at segmenting the trajectory into tracklets of fairly similar speed. The latter is aimed at extracting complex patterns of activity, which include spatial information (coming from the trajectory analysis) and temporal information related to the interactions of mobiles observed in the scene.

Streams of video are acquired at a speed of 10 frames per second. The on-line (real time) analysis subsystem takes its input directly from the data acquisition component; the video is stored in the DB parallel to the real time processing.

Real-time processing object detection and tracking

The detection and tracking is performed using multiple cameras with an overlapping field of view, and consists of three stages: Detection in the image plane, tracking in the image plane, fusion and tracking in 3D.

Detection

Detection is performed by combining change detection and motion detection. The first detector is the Adaptive Gaussian Mixture Model of Zivkovic [START_REF] Zivkovic | Efficient adaptive density estimation per image pixel for the task of background subtraction[END_REF]. This method builds on the standard Gaussian Mixture Model approach but permits an adaptive number of components per pixel. This generally produces good object silhouettes and runs very fast.

To complement the change detector, a motion detector is employed. In this method, the three most recent frames {I(t), I(t -1), I(t -2)} are used to determine the motion in the most recent frame I(t). A set of corner features is detected in frame I(t + 1) using the method in [START_REF] Shi | Good features to track[END_REF]. These features are then tracked forwards to frame I(t) and backwards to frame I(t + 2) using the sparse optical flow method in [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF]. This results in two direction vectors for each feature, [d 0→1 , d 1→2 ]. Features are clustered based on their motion with a constraint on the maximum distance between any two features. A triangulation of each cluster of features is performed such that the cluster can be rendered to a binary motion mask. The two binary motion masks, from the change detector and the motion detector, are combined through a simple logical AND.

Image Plane Tracking

Tracking in the image plane is performed using two simple templates and a KLT feature tracker. When the detector returns a detection, it can either be associated to an existing tracked target, or to a new target. When a new target is created, two small images are created. One is a greyscale image of the size of the detection bounding box, while the other is an RGB image of the same size. The greyscale image is the detection mask template D t , and is initialised from the binary motion mask of the current image M t , while the RGB image is the appearance template A t and is initialised from the RGB pixel values of the current image I t . Thus, on initialisation, if the top left corner of the detection bounding box is at image coordinates x, y:

D t (u, v) = 0 if M t (x + u, y + v) = 0 255 otherwise (1) A t (u, v) = I t (x + u, y + v) (2)
When a detection is associated to a new target, the detection and appearance templates are updated as a running average. Should the detection indicate a change in the width or height of the bounding box, the template images can be easily expanded or cropped as required.

Each tracked target maintains a set of KLT features that are tracked between frames. The overall plane tracking method is generally good enough to reliably maintain a track on large objects such as vehicles, which often stop for extended periods in the scene. It is not intended to track objects through occlusions, but rather to detect the presence of objects, and maintain the presence of static objects.

Multi-camera Fusion and 3D Tracking

The final stage of tracking is performed in the 3D coordinate system of the scene (though tracking remains 2D on the ground plane). Camera calibration is used to project the bounding boxes of per-camera tracking targets to each other camera view as four epipolar lines from the four corners of the bounding box. This provides a mechanism for rating the extent to which tracking targets are related between views, by determining the extent to which a bounding box fits between the extremal epipolar lines of a bounding box from another view. Agglomerative clustering is used to determine possible solutions for the correct fusion of targets, and an optimisation process then determines the optimal clustering for a single frame of video. Optimal solutions are retained over a temporal window, and an overall optimal association of per-camera targets to fused targets is determined, and fused tracking targets updated for every new frame.

Data preprocessing

In order to discover meaningful activity clusters, it is of prime importance to have available detailed information allowing to detect the different possible interactions between mobiles. As our system is based on trajectory analysis, the first step to prepare the data for the activity clustering methodology is to extract tracklets of fairly constant speed allowing to characterise the displacements of the mobile or its stationary state.

If the dataset is made up of N objects, the trajectory tr j for object O j in this dataset is defined as the set of points [x j (t), y j (t)] corresponding to their position points; x and y are time series vectors whose length is not equal for all objects as the time they spend in the scene is variable. The instantaneous speed for that mobile at point [x j (t), y j (t)] is then v (t) = ẋ (t)

2 + ẏ (t)

2 1 2
. The objective is then to detect those points of changing speed allowing to segment the trajectory into tracklets of fairly constant speed so that the trajectory can be summarised as a series of displacements at constant speed or in stationary state. The mobile object time series speed vector is analysed in the frame of a multiresolution analysis of a time series function v (k) with a smoothing function, ρ 2 s (k) = ρ (2 s k), to be dilated at different scales s. In this frame, the approximation A of v (k) by ρ is such that A 2 s-1 v is a broader approximation of A 2 s v. By analyzing the time series v at coarse resolutions, it is possible to smooth out small details and select those points associated with important changes.

The speed change points are then employed to segment the original trajectory tr j into a series of i tracklets tk. Each tracklet is defined by two key points, these are the beginning and the end of the tracklet, [x i j (1), y i j (1)] and [x i j (end), y i j (end)] as they define where the object is coming from and where it is going to and also with approximative constant speed. We build a feature vector from these two points. By globally reindexing all tracklets, let m be the number of total tracklets extracted, we obtain the following tracklet feature vector :

tk m = [x m (1), y m (1), x m (end), y m (end)] (3)

Activity clustering methodology

We understand activity as the interactions occurring between mobile objects themselves and those between mobiles and the environment. We propose in this work to model those interactions employing Soft computing techniques. The motivation is that they provide uncertain information processing capability; set a framework to work with symbolic/linguistic terms and thus allows producing natural language-like interpretable results.

Preliminary definitions

A fuzzy set is a set of ordered pairs such as A = {(x, µ A (x)) | xǫX} and the belonging of x to A is given by µ A . Any relation between two sets X and Y is known as a binary relation R:

R = {((x, y) , µ R (x, y)) | (x, y) ǫX × Y }
and the strength of the relation is given by µ R (x, y). Let's consider now two different binary relations, R1 and R2, linking three different fuzzy sets X, Y, and Z : R1 = x is relevant to y; R2 = y is relevant to z. It is then possible to find to which measure x is relevant to z (noted R=R1oR2 ) by employing the extension principle:

µ R=R1•R2 (x, z) = max y min [µ R1 (x, y) , µ R2 (y, z)]
It is interesting to verify whether the resulting relation is symmetric, R (x, y) = R (y, x) , reflexive R (x, x) = 1, which make of R a compatibility relation and occurs in most cases when establishing a relationship between binary sets. Because R was calculated employing the extension principle, R is also a transitive relation.

R (x, y) is a transitive relation if ∃ z ∈ X, z ∈ Y /R (x, y) max z min [R (x, z) , R (z, y)]. R can be made furthermore closure transitive fol- lowing the next steps Step1. R ′ = R ∪ (R • R) Step2. If R ′ = R, makeR = R ′ and go to step1 Step3. R = R ′ Stop. R is the transitive closure where R • R (x, y) = max z min (R (x, z) , R (z, y)) (4)
R is now a transitive similarity relation with R indicating the strength of the similarity. If we define a discrimination level α in the closed interval [0,1], an α -cut can be defined such that

R α (x, y) = 1 ⇔ R (x, y) α; R = ∪ α αR α (5)
It is thus implicit that α 1 > α 2 ⇔ R α1 ⊂ R α2 ; thus, the R α form a nested sequence of equivalence relations, or from the classification point of view, R α induces a partition π α of X × Y (or X 2 ) such that α 1 > α 2 implies π α1 is a partition refinement of π α2 .

Clustering of video data

We now set out to establish the appropriate relations between detected mobiles in the video reflecting spatio-temporal similarities in order to obtain activity patterns. With this aim, we define the following relations: R1 ij : mobile object O(i) meets mobile object O(j). In this case the action 'meets' must be understood spatially and thus gives a degree of spatial closeness between the two mobiles.

R1 ij = min ( tk i (1), tk j (1) , tk i (1), tk j (2) ) ,

( tk i (2), tk j (1) , tk i (2), tk j (2) ) (6)
R2 ij : mobile object O(i) starts equal to mobile object O(j). Here we are attempting to relate mobile objects that share temporal closeness.

R2 ij = 1 -abs(start time(i) -start time(j)) (7)
R3 ij : mobile object O(i) starts after mobile object O(j). Here we are attempting to relate mobile objects that appear in a sequential manner.

R3 ij = 1 -abs(start time(i) -end time(j)) (8)
Obtaining the patterns of activity is achieved by aggregating the above spatio-temporal relations with a typical T-norm operator. R = R1 ∪ R2 ∪ R3 aggregates temporal similarity relations between mobiles. We calculate the transitive closure of this new relation. Analogically to section 6.1 an α -cut can be defined such that

R α (x, y) = 1 ⇔ R (x, y) α and R α induces a new partition π α = {C α 1 , ..., C α i , ..., C α n α }; each C α i represents a discovered spatio-temporal activity pattern.

Results and Evaluation

The algorithm can be applied to any given period monitoring the servicing of an aircraft in the airport docking area. In order to evaluate whether the activity extraction algorithm works properly and to asses the correctness of the results, we took five video datasets (each lasting about 1 hour) with available Groundtruth annotation and containing the start and end time for the most relevant activity events of the sequence.

The procedure to find the activity clusters is applied as given in section 6.2. In this work, the final relation R, which verifies the transitive closure, is thresholded for different α -cut values going from 0.05 to 0.95 and with a step value of 0.10. Low α -cut values produce only a few number of clusters of broad resolution as most of the activity is merged spatially and temporally. α -cut values near to one produce activity clusters of higher resolution with more precisely defined activities. At each resolution it is possible to calculate the temporal overlap between the extracted activity clusters and the ground-truth clusters. The quantitative result of this comparison is given in table 1. For each video sequence and for each ground-truth event only the best overlap across all α -cut values is reported. Remark that specific activities involving one mobile require precise definition obtained with activity clusters of higher resolution while loading operations involving the interaction of several mobiles are defined with mid-resolution activity clusters (α-cut values of 0.75 or 0.85). For instance, figure 1 presents a frontal loading activity cluster obtained for an α -cut value of 0.75. In general, all events are recognized correctly in all video sequences. When the percentage of overlap decreases or even goes to zero, it is mainly due to low-level object occlusion problems, which do not allow extracting all mobile trajectories and disturbs then the analysis of all possible mobile interactions.

Table 1. Percentage of overlap between discovered activities and the reference events contained in the ground-truth. The symbol * indicates that for all video sequences, the ground-truth event matches a discovered activity obtained for an alpha value of 0.95. NA indicates 'does not apply' (does not appear in the video sequence). Our results can partially be compared to those obtained with a supervised approach to learn apron activity models with Inductive Logic Programming (Dubba et al. [START_REF] Dubba | Event model learning from complex videos using ilp[END_REF]). As previously indicated, Dubba et

Conclusions

We have presented in this paper, a novel approach to extract activity patterns from video. The technique is unsupervised and is based on the use of fuzzy relations to model Spatial and temporal properties from detected mobile objects. Fuzzy relations are aggregated employing typical soft-computing algebra.

A clustering algorithm based on the transitive closure calculation of the fuzzy relations allows finding spatio-temporal patterns of activity. Our current results are encouraging as the final patterns of activity are given with coherent spatial and temporal information, which is understandable for the end-user. When comparing our results with explicit ground-truth given by a domain expert, we were able to identify the events in general with a temporal overlap of at least, or near, 50%. Events with small temporal overlap in some video sequences is because of low-level detection problems. The comparison with a supervised method on the same data indicates that our approach is able to extract the interesting activities signalled in the ground-truth with a higher True Positive Rate. More importantly, our approach is completely unsupervised. In our future work we will try to work on improving our technique to determine the meaningfulness (or abnormality) of single activity patterns. We also plan to work on the semantical description of the activity clusters.

Fig. 1 .

 1 Fig. 1. Example of an activity cluster obtained. The figure presents the tracklets of the mobiles participating in the Frontal Loading activity. Red points indicate the beginning of a tracklet. Blue points indicate the end of a tracklet.

Table 2 .

 2 al. have worked on the same apron monitoring video dataset from the Toulouse airport in France. Dubba et al. have concentrated on supervised learning of four apron activities: Aircraft arrival; Aircraft departure; Rear Loading/unloading; Jet Bridge Positionning. Dubba et al. obtained a global True Positive Rate (TPR) of 74%. In our work (from Table 1), we have 80% global True Positive Rate for the recognition of eight apron activities: Aircraft arrival; Aircraft departure; Rear Loading/unloading; Jet Bridge Positionning; Frontal Loading/Unloading; GPU vehicle arrival; Handler deposits chocks; Push-back positioning. Dubba et al. approach works as a hit or miss recognition system and in their paper there is no information on what is the temporal overlap between the recognised activities and the ground-truth activities. In our case such temporal overlap has a global value of 73%. The event-by-event comparison between the two approaches is detailed in table 2. Results comparison between our results and those presented in Dubba et al. at ECAI 2010 [5].

		Dubba et al.		Our Approach
	Reference Event	TPR	TPR	Mean temporal
				overlap with GT
	Rear Loading / Unloading	80 %	100 %	53 %
	Aircraft arrival	100 %	100 %	80 %
	Aircraft departure	57 %	100 %	80 %
	Jet Bridge positionning	57 %	100 %	59 %
	Frontal Loading / Unloading	--	100 %	58 %
	GPU vehicle arrival	--	80 %	86 %
	Handler deposits chocks	--	100 %	66 %
	Push-back positioning	--	60 %	85 %