Michaela Keuper

Matthias Blüher

Michael R Schön

Peter Möller

Anna Dzyakanchuk

Kurt Amrein

Klaus-Michael Debatin

Dr Martin Wabitsch
email: martin.wabitsch@uniklinik-ulm.de

Dr Pamela Fischer-Posovszky

An inflammatory micro-environment promotes human adipocyte apoptosis

Keywords: inflammation, fat cells, macrophages, apoptosis, insulin resistance

Obesity-associated macrophage infiltration into adipose tissue is responsible for both local and systemic inflammation. Recent findings suggest fat cell apoptosis as an initiator of macrophage recruitment.

Here, we investigated the effects of an inflammatory micro-environment on fat cells by using human THP-1 macrophages and SGBS adipocytes. Macrophage-secreted factors induced insulin resistance, inhibited insulin-stimulated Akt phosphorylation, and induced apoptosis of adipocytes. The apoptosisinducing effect was even more pronounced in direct co-cultures of adipocytes and macrophages. Our data suggest a link between insulin resistance and apoptosis sensitivity. Accordingly, pharmacological and genetic inhibition of insulin signaling at the level of Akt2 sensitized adipocytes to apoptosis induction by macrophage-secreted factors.

In conclusion, we describe here a novel interaction of macrophages and fat cells, i.e. induction of apoptosis. Our data suggest a feed-forward cycle in which macrophages further drive the inflammatory process by inducing insulin resistance and concomitant apoptosis of adipocytes.

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

Introduction

Obesity is one of the major health problems in the western world. The excessive accumulation of adipose tissue gives rise to a state of chronic, low-grade inflammation and is associated with severe comorbidities including type 2 diabetes mellitus, liver steatosis and atherosclerosis [START_REF] Shoelson | Obesity, inflammation, and insulin resistance[END_REF][START_REF] Baker | Childhood body-mass index and the risk of coronary heart disease in adulthood[END_REF].

It is now well accepted that adipose tissue inflammation is an important early event in the pathogenesis of obesity complications, especially insulin resistance [START_REF] Xu | Chronic inflammation in fat plays a crucial role in the development of obesity-related insulin resistance[END_REF]. In both humans and rodents, macrophages accumulate in adipose tissue with increasing body weight [START_REF] Xu | Chronic inflammation in fat plays a crucial role in the development of obesity-related insulin resistance[END_REF][START_REF] Weisberg | Obesity is associated with macrophage accumulation in adipose tissue[END_REF]. Several rodent studies demonstrate that an increase in adipose tissue macrophages worsens insulin sensitivity [START_REF] Hirasaka | Deficiency of Cbl-b gene enhances infiltration and activation of macrophages in adipose tissue and causes peripheral insulin resistance in mice[END_REF][START_REF] Kamei | Overexpression of monocyte chemoattractant protein-1 in adipose tissues causes macrophage recruitment and insulin resistance[END_REF]. Likewise, a decrease in adipose tissue macrophages (ATM) results in a decrease in adipose tissue inflammation and inhibits obesity-induced insulin resistance [START_REF] Patsouris | Ablation of CD11cpositive cells normalizes insulin sensitivity in obese insulin resistant animals[END_REF]. Furthermore, genetic ablation of macrophage signaling by knockout of IB kinase-beta (IKK-) or c-Jun N-terminal kinase 1 (JNK1) protects mice from dietinduced insulin resistance [START_REF] Arkan | IKK-beta links inflammation to obesity-induced insulin resistance[END_REF][START_REF] Solinas | JNK1 in hematopoietically derived cells contributes to diet-induced inflammation and insulin resistance without affecting obesity[END_REF].

The pathogenic mechanisms leading to ATM accumulation in the obese state are still not completely understood. There is increasing evidence that adipocyte hypertrophy and local hypoxia are implicated in macrophage recruitment since both conditions cause an increased production of inflammatory cytokines and chemoattractants [START_REF] Skurk | Relationship between adipocyte size and adipokine expression and secretion[END_REF][START_REF] Wood | Cellular hypoxia and adipose tissue dysfunction in obesity[END_REF]. In general, macrophages play an important role in the immune system and tissue homeostasis by sequestering and clearing pathogens, and old and dead cells [START_REF] Weigert | The liaison between apoptotic cells and macrophages--the end programs the beginning[END_REF]. Measuring the integration of 14 C derived from nuclear bomb tests in genomic DNA by accelerator mass spectrometry, Spalding et al. recently showed that there is a continuous turnover of adipocytes with approximately 10 % of fat cells being renewed annually [START_REF] Spalding | Dynamics of fat cell turnover in humans[END_REF]. Tissue homeostasis includes supply of new cells from progenitors followed by differentiation into specialists. Indeed, it has long been known that a large pool of preadipocytes exists in human adipose tissue [START_REF] Fischer-Posovszky | Endocrinology of adipose tissue -an update[END_REF]. The finding of a pool of early committed white adipocyte precursors in the mural cell compartment of the adipose vasculature supplies an additional putative source for new adipocytes [START_REF] Tang | White fat progenitor cells reside in the adipose vasculature[END_REF].

A c c e p t e d M a n u s c r i p t 4 On the other hand, old cells must be eliminated to keep the number of adipocytes in a certain regulated range. There are several distinct types of cell death [START_REF] Galluzzi | Guidelines for the use and interpretation of assays for monitoring cell death in higher eukaryotes[END_REF], and necrosis of adipocytes has been described [START_REF] Cinti | Adipocyte death defines macrophage localization and function in adipose tissue of obese mice and humans[END_REF]. However, apoptosis is considered the predominant way cells die in multicellular organisms, including humans [START_REF] Weigert | The liaison between apoptotic cells and macrophages--the end programs the beginning[END_REF]. Apoptosis of adipocytes was shown in several disease states accompanied by loss of adipose tissue including tumor cachexia [START_REF] Prins | Human adipocyte apoptosis occurs in malignancy[END_REF], autoimmune lipodystrophy [START_REF] Fischer-Posovszky | Role of CD95-mediated adipocyte loss in autoimmune lipodystrophy[END_REF] and in HIV patients under highly active anti-retroviral therapy [START_REF] Domingo | Subcutaneous adipocyte apoptosis in HIV-1 protease inhibitor-associated lipodystrophy[END_REF]. Most interestingly, not only a pro-apoptotic gene expression profile, but also an increase in apoptosis was detected in mice under a high fat diet [START_REF] Alkhouri | Adipocyte apoptosis, a link between obesity, insulin resistance, and hepatic steatosis[END_REF]. Blocking the function of pro-apoptotic Bid resulted in a significant decrease of fat cell apoptosis thereby reducing obesity-associated macrophage infiltration into adipose tissue and preventing insulin resistance [START_REF] Alkhouri | Adipocyte apoptosis, a link between obesity, insulin resistance, and hepatic steatosis[END_REF]. This suggests an important role of fat cell apoptosis in the progression of obesity-induced disorders.

Taken together, recent findings strongly imply a local interplay between ATM and fat cells. We hypothesize that the local pro-inflammatory micro-environment which is accompanied by insulin resistance enhances adipocyte apoptosis. We developed an in vitro model system of inflamed adipose tissue to study macrophage-mediated apoptosis induction in human fat cells. To this end, we either incubated SGBS adipocytes with conditioned media from THP-1 macrophages or directly co-cultured SGBS adipocytes with macrophages.

A c c e p t e d M a n u s c r i p t 5

Material and Methods

Cell culture, adipocyte and macrophage differentiation

SGBS preadipocytes were cultured and differentiated into adipocytes as described elsewhere [START_REF] Fischer-Posovszky | Role of CD95-mediated adipocyte loss in autoimmune lipodystrophy[END_REF]. THP-1 cells purchased from ATCC (Wesel, Germany) were cultured in RPMI containing 100 µM non-essential amino acids, 2 mM L-glutamine, 1 mM sodium pyruvate, antibiotics (basal medium), and 10% FCS. Differentiation of THP-1 cells into macrophages was induced by incubation with 125 ng/ml phorbol myristate acetate (PMA) in serum-free basal medium containing 0.5% BSA for 48 h. Expression of surface molecules was analyzed by flow cytometry using the following antibodies: CD11c-PE (clone B-Ly6), and isotype-PE (all from BD Pharmingen, Heidelberg, Germany). Data were analyzed using CellQuest software (BD Biosciences, Heidelberg, Germany) and ratio of mean fluorescent intensity (MFI) of CD11c-PE to MFI of isotype-PE was calculated.

Macrophage-conditioned media (MacCM) were collected after additional 48 h in serum-free basal medium containing 0.5% BSA and cleared by centrifugation. Cytokine concentrations in conditioned media were assessed using Bio-Plex System (Bio-Rad Laboratories, Hercules, CA). Serum-free basal medium containing 0.5% BSA in corresponding concentrations was used as vehicle control in all further treatments. In vitro differentiated THP-1 macrophages were added to cultures of SGBS preadipocytes or adipocytes at increasing rates (THP-1:SGBS 1:10, 1:2, 1:1, 2:1).

Adipogenic differentiation assay

SGBS preadipocytes were incubated with MacCM (10, 20, 50 %) or vehicle during adipogenic differentiation. At day 10 cells were analyzed. Rate of adipogenic differentiation was assessed by direct counting the number of differentiated cells in the monolayers using a net micrometer.

Basal and insulin-stimulated glucose uptake and de novo lipogenesis

After washing twice with PBS, in vitro differentiated SGBS adipocytes (differentiation rate > 80 %) were incubated in serum-free medium supplemented with indicated concentrations of MacCM or vehicle for 24 h. For glucose uptake, cells were stimulated with indicated doses of insulin (0 -1000 nM) for 15 min and then [ 14 C]-2-deoxy-D-glucose (0.2 µCi/well) was added for another 15 min. For lipogenesis, insulin and [ 14 C]-D-glucose (0.1 µCi/well) were added for 24 h. After incubation with glucose, cells were

Preparation of RNA and quantitative RT-PCR

Total RNA was prepared using RNeasy kit or RNeasy Lipid tissue kit (Qiagen, Hilden, Germany). cDNA synthesis was performed as described previously [START_REF] Fischer-Posovszky | Role of CD95-mediated adipocyte loss in autoimmune lipodystrophy[END_REF]. qPCR was carried out on a LightCycler ® (Roche Diagnostics, Mannheim, Germany). The mRNA levels of target genes were normalized to those of Glycerinaldehyd-3-phosphat-Dehydrogenase (GAPDH) and succinate dehydrogenase complex subunit A (SDHA) using the Ct method. The sequences of gene-specific primers can be found in Table A.2.

Western blot analysis

Western blot analysis was performed as described previously [START_REF] Fischer-Posovszky | Role of CD95-mediated adipocyte loss in autoimmune lipodystrophy[END_REF]. Enhanced chemiluminescence was used for detection (Amersham Bioscience, Freiburg, Germany). Following antibodies were used: rabbit anti-phospho Akt, rabbit anti-phospho p38 MAPK, mouse anti-phospho ERK1/2, rabbit anti-panAkt, mouse anti-p38 MAPK (all from Cell Signaling, Beverly, MA, USA), rabbit anti-Akt2 (R&D Systems Inc, Minneapolis, USA), rabbit anti-ERK1/2 (Sigma, Munich, Germany), mouse anti--actin (Sigma). HRP-conjugated goat anti-mouse IgG and goat anti-rabbit IgG were obtained from Santa Cruz Biotechnology, INC., Europe.

Induction and quantification of apoptosis after treatment with MacCM

Preadipocytes and adipocytes were stimulated with MacCM (10, 20, 50 %) or corresponding vehicle control. Quantitative determination of apoptosis according to Nicoletti was performed as described elsewhere [START_REF] Nicoletti | A rapid and simple method for measuring thymocyte apoptosis by propidium iodide staining and flow cytometry[END_REF][START_REF] Fischer-Posovszky | Inhibition of death-receptor mediated apoptosis in human adipocytes by the insulin-like growth factor I (IGF-I)/IGF-I receptor autocrine circuit[END_REF]. In all experiments the percentage of specific apoptosis was calculated as follows to normalize data: 100 x [experimental apoptosis (%) -

Induction and quantification of apoptosis in macrophage-adipocyte co-cultures

In vitro differentiated THP-1 macrophages were added to cultures of SGBS preadipocytes or adipocytes as described above. THP-1 macrophages are morphologically clearly distinguishable from SGBS cells. This allowed assessing the number of adherent preadipocytes and adipocytes by direct counting the number of cells in the monolayers using a net micrometer after three days of co-culture. Externalization of phosphatidylserine was measured by Annexin V staining after 24 h of co-culture. CD221-PE staining was used to identify preadipocytes in the co-culture system. Adipocytes were separated from macrophages according to their FSC/SSC profile as previously published [START_REF] Lee | Simple flow cytometric method used to assess lipid accumulation in fat cells[END_REF][START_REF] Schaedlich | A simple method to sort ESC-derived adipocytes[END_REF]. For immunofluorescence, SGBS cells were grown and differentiated on chamberslides. At day 8 of adipogenic differentiation and after 24 h co-culture with THP-1 macrophages, monolayers were washed, and fixed with 4% PFA followed by staining with CD11c-PE antibody (BD Pharmingen, Heidelberg, Germany) and BODIPY 493/503 (1 µg/ml, Invitrogen, Karlsruhe, Germany). Finally, cells were mounted with DAKO fluorescence mounting medium (DAKO, Hamburg, Germany) containing DAPI (0.5 µg/ml, Invitrogen). Slides were viewed using an Olympus AX70 fluorescence microscope equipped with appropriate filters (Olympus Optical Co., Hamburg, Germany). Images were captured with a CC12 digital camera (Soft Imaging System, Münster, Germany) using cell^P software (Olympus, Hamburg, Germany).

Experimental subjects

Paired human visceral (visc) and subcutaneous (sc) fat tissue samples were obtained from the Leipzig (Germany) cohort fat tissue bank described in previous studies [START_REF] Harman-Boehm | Macrophage infiltration into omental versus subcutaneous fat across different populations: effect of regional adiposity and the comorbidities of obesity[END_REF]. Fat tissue samples were obtained from 54 Caucasian obese and lean men (n = 30) and women (n = 24) who underwent open abdominal surgery for elective cholecystectomy or explorative laparotomy. Patients with malignant diseases or acute or chronic inflammatory disease have not been included. Mean age of patients was 62 years (range 31 to 84 years) and their mean BMI was 30 kg/m² (range 29 to 42 kg/m²).

All procedures were performed in accordance with the Declaration of Helsinki guidelines and approved by the ethics committee of the University of Leipzig. Patients gave a written informed consent in advance.

A c c e p t e d M a n u s c r i p t 8 2.9. Immunohistochemistry

The sc and visc fat samples were fixed at room temperature in 4% formaldehyde and embedded in paraffin. 3 µm serial sections were mounted on glass slides, deparaffinized in xylol, and stained for CD11c (Clone BU15, mouse IgG 1 , Immunotech, Marseille, France) or cleaved caspase-3 (rabbit polyclonal, Cell Signaling, Beverly, MA, USA) using the Envision detection system (Dako, Glostrup, Denmark). Macrophages were identified in the adipose parenchyma (CD11c + within blood vessels were excluded) when staining for CD11c was present along with an identifiable mononuclear nucleus. Data are presented as the number per 100 adipocytes (% macrophages, % apoptotic cells). Counting was performed blinded to sample data and was confirmed by a certified pathologist (P.M.).

Lentiviral-mediated over expression of shRNA

Specific shRNA sequences targeting the human Akt2 sequence (#1: start 1423: 5'-ggttcttcctcagcatcaact-3'; #2: start 679: 5'-gcaagggaacctttggcaaag-3') and a nonsense shRNA sequence with no corresponding part in the human mRNA (5'-gatcatgtagatacgctca -3') were cloned into the pENTR™/H1/TO entry vector using the BLOCK-iT Inducible H1 RNAi Entry Vector Kit (Invitrogen, Karlsruhe, Germany) and subsequently transferred into the pLenti4/BLOCK-iT™-DEST expression vector by targeted recombination using BLOCK-iT Inducible H1 Lentiviral RNAi system (Invitrogen) according to manufacture's protocol. The pLenti expression constructs were transfected with packaging plasmids by Lipofectamine™ 2000 (Invitrogen) in the 293FT producer cell line using ViraPower™ Lentiviral Expression Systems (Invitrogen) according to the manufacturer's protocols. Three days after transfection the viral supernatant was harvested, filtered through a 0.45 µM PVDF filter, and stored at -80°C. SGBS cells were infected at MOI = 1 in the presence of 8 µg/ml polybrene. Stable bulk cultures were generated by selection with 0.2 mg/ml zeocin for 14 d, and down-regulation of Akt2 protein was controlled by Western blot. Experiments were performed with cells from three independent lentiviral infections.

Statistical analysis

If not otherwise stated, data from three independent triplicate experiments were expressed as mean ± standard error of the mean (SEM). For statistical comparison, Student's two-tailed paired t-test was used. p<0.05 was considered statistically significant in comparison to the respective control cells. For correlation analysis spearman's correlation was performed using SigmaStat version 3.5 statistical analysis software (Systat Software GmbH, Inc. San Jose, California, US).

Establishment of an in vitro model system for human inflamed adipose tissue

In order to study interactions of macrophages and adipocytes on a cellular level, we designed an experimental setup to mimic adipose tissue inflammation in vitro. To this end, we differentiated human THP-1 monocytes into macrophages. Increased mRNA expression of CD11c (Fig. 1A) and the upregulation of CD11c surface expression (Fig. 1B,C) demonstrated successful macrophage differentiation. Furthermore, the expression of inflammatory cytokines like TNF, IL-1 and IL-6 (Fig. 1D-F) was increased after differentiation into macrophages, and cytokines accumulated in the medium supernatants (TNF ~ 39 ng/ml, IL-1 ~ 390 pg/ml, IL-6 ~ 2100 pg/ml). Macrophage-conditioned media (MacCM) were used in the in vitro setting together with cultures of human preadipocytes and adipocytes (SGBS cells) to mimic the inflammatory micro-environment found in vivo.

Macrophage-secreted factors inhibit insulin-dependent processes in human adipocytes

To examine whether macrophage factors interfere with insulin sensitivity in SGBS adipocytes we performed a panel of experiments related to insulin function. First, we investigated the insulin-dependent differentiation of preadipocytes into mature, functional adipocytes. We found an inhibition of adipogenesis upon stimulation with MacCM as seen by determination of adipogenic differentiation rate (Fig. 2A) and mRNA expression of the adipocyte marker gene FABP4 (fatty acid binding protein 4) (Fig. 2B). Furthermore, we studied if macrophage-secreted factors interfere with insulin-stimulated biochemical functions of mature fat cells, i.e. glucose uptake and de novo lipogenesis. When we preincubated adipocytes with MacCM, the insulin stimulated uptake of [ 14 C]-2-deoxy-D-glucose was robustly decreased (fold change 100 nM insulin vs. basal, 0% MacCM: 3.91 ± 0.47; 50% MacCM: 1.37 ± 0.12) (Fig. 2C and2D). Likewise, the insulin stimulated de novo lipogenesis determined by [ 14 C]-Dglucose incorporation into cellular lipids was decreased when adipocytes were incubated with MacCM (Fig. 2E and2F). Inhibitory effects of macrophage factors on insulin sensitivity were also observed in 3T3-L1 and human primary preadipocytes [START_REF] Constant | Macrophage-conditioned medium inhibits the differentiation of 3T3-L1 and human abdominal preadipocytes[END_REF] demonstrating that our model system is well suitable for studying the effects of macrophage-secreted factors on fat cell function.

A c c e p t e d M a n u s c r i p t 10

Macrophage-secreted factors inhibit insulin signaling in adipocytes

We next aimed at identifying molecular changes induced by macrophage-secreted factors. Knowing that MacCM inhibits insulin-dependent processes (Section 3.2.), we hypothesized that macrophage-secreted factors interfere with insulin signaling pathways of adipocytes. Thus, we pre-treated SGBS adipocytes with MacCM for 24 h, stimulated the cells with 10 -7 M insulin for 15 min and subjected protein samples Additional Western blot analysis (Fig. 3A) to validate these findings demonstrated that macrophagesecreted factors abolished basal phosphorylation of Akt (Fig. 3B) and down-regulated insulin-stimulated phosphorylation by ~ 65 % (Fig. 3C). The up-regulation of p38 phosphorylation by macrophage-secreted factors was not detected in the protein array.

The data on insulin-stimulated fat cell function (Section 3.2.) together with the insights on phosphorylation of insulin-stimulated downstream targets (Section 3.3.) demonstrates that factors secreted from macrophages provoked insulin resistance in human fat cells.

Macrophage-secreted factors induce apoptosis in preadipocytes and lipid laden, mature adipocytes

To investigate the susceptibility of preadipocytes to enter into apoptosis induced by macrophage-secreted factors, we treated SGBS preadipocytes with escalating doses of MacCM. We found a time-and dosedependent induction of DNA fragmentation (50% MacCM, 48h: 20.11% ± 1.16 specific apoptosis), which is known as a hallmark of apoptosis induction (Fig. 4A and4B). Externalization of phosphatidylserine to the cell surface, which served as an eat-me-signal for macrophages [START_REF] Weigert | The liaison between apoptotic cells and macrophages--the end programs the beginning[END_REF], was demonstrated by Annexin V staining (Fig. 4C). This method is another gold standard for demonstrating apoptosis. We then extended our experiments to mature, lipid-laden adipocytes. Similar to preadipocytes, macrophage-secreted factors induced apoptosis in a dose-and time-dependent manner in A c c e p t e d M a n u s c r i p t 11 adipocytes (50% MacCM, 48h: 15.43% ± 0.76 specific apoptosis) (Fig. 4D and4E). This set of experiments clearly demonstrates that macrophage-secreted factors induced apoptosis in human preadipocytes and adipocytes.

Macrophages induce apoptosis in human preadipocytes and adipocytes

We next aimed to study the role of direct cell-cell contacts between fat cells and macrophages in apoptosis induction. We therefore extended our studies to a direct co-culture system of both cell types.

After 24 h of co-culture we detected a significant increase in phosphatidylserine externalization in both CD221 + -preadipocytes (Fig. 5A) and mature, lipid-laden adipocytes (Fig. 5B). Most interestingly, we observed a robust decrease in the cell number of adherent preadipocytes after co-culture with macrophages for 72 h (decrease by 43.0% ± 9.7 at ratio THP-1:SGBS 2:1) (Fig. 5C). Also the number of mature adipocytes was significantly decreased upon co-culture with macrophages (decrease by 53.1% ± 13.5% at ratio THP-1:SGBS 2:1) (Fig. 5D). Along with the observation of no dead, detached cells swimming in the culture supernatants, this set of data suggests that macrophages induce apoptosis in preadipocytes and adipocytes and directly phagocytose them. An immunofluorescence approach labeling macrophages by CD11c and intracellular lipid droplets by BODIPY further corroborated this finding (Fig. 5E). CD11c + -macrophages surrounded lipid-filled adipocytes and we even detected BODIPYstained lipid droplets inside of macrophages. Notably, our in vitro observation (Fig. 5E) highly resembles the in vivo situation (Fig. 5F). It has long been proposed that macrophages localize around dead adipocytes [START_REF] Cinti | Adipocyte death defines macrophage localization and function in adipose tissue of obese mice and humans[END_REF]. However, the knowledge on fat cell apoptosis in different human adipose tissue depots is still limited. We therefore performed immunohistochemistry to detect macrophages by CD11c along with apoptotic adipocytes by staining the active fragment of caspase-3 in both the subcutaneous and the visceral adipose tissue depot of 54 subjects. As expected, apoptotic adipocytes were surrounded by CD11c + -macrophages in crown-like structures (Fig. 5F, details about tissue distribution are given in Table A.1). A subsequent correlation analysis performed with the data obtained in the patients having positive cells for both CD11c and active caspase-3 (sc: n = 19; visc: n = 30) revealed that the percentage of apoptotic adipocytes strongly correlated with the percentage of macrophages in both subcutaneous (Fig. 5G) and visceral (Fig. 5H) adipose tissue depots (sc: r = 0.888; p < 0.001; visc: r = 0.560, p = 0.003).

A c c e p t e d M a n u s c r i p t 12 Our data unambiguously show that in vivo macrophages were closely located to apoptotic adipocytes, further supporting our in vitro data on the direct connection between macrophages and fat cell apoptosis.

Pharmacological and genetic inhibition of Akt signaling sensitizes adipocytes for apoptosis induced by macrophage-secreted factors

We have previously shown an anti-apoptotic function of insulin-like growth factor I (IGF-1) in human fat cells [START_REF] Fischer-Posovszky | Inhibition of death-receptor mediated apoptosis in human adipocytes by the insulin-like growth factor I (IGF-I)/IGF-I receptor autocrine circuit[END_REF]. Insulin mediates survival in several cell types [START_REF] Díaz | Apoptotic cell death of proliferating neuroepithelial cells in the embryonic retina is prevented by insulin[END_REF][START_REF] Tanaka | Insulin prevents apoptosis of external granular layer neurons in rat cerebellar slice cultures[END_REF] and has anti-apoptotic effects comparable to IGF-1 in brown adipocytes [START_REF] Navarro | Insulin/IGF-I rescues immortalized brown adipocytes from apoptosis down-regulating Bcl-xS expression, in a PI 3-kinase-and map kinase-dependent manner[END_REF][START_REF] Tseng | Differential roles of insulin receptor substrates in the anti-apoptotic function of insulin-like growth factor-1 and insulin[END_REF]. This leads us to hypothesize that insulin resistant adipocytes, where insulin is no longer capable of exerting its survival effects, are sensitive to apoptosis induction. Thus, insulin resistance induced by macrophage-secreted factors would sensitize primarily apoptosis resistant adipocytes to apoptosis induction.

In our experiments, phosphorylation of Akt, the key molecule in the insulin signaling pathway [START_REF] Kohn | Insulin stimulates the kinase activity of RAC-PK, a pleckstrin homology domain containing ser/thr kinase[END_REF], was most robustly inhibited by MacCM (Section 3.3.) suggesting that inhibition of Akt is involved in insulin-mediated anti-apoptosis. To further support our hypothesis, we blocked insulin signaling upstream of Akt in human SGBS adipocytes by using the pharmacological inhibitor LY294002, a small molecule inhibitor targeting phosphatidylinositol 3-kinase (PI3K). Inhibition of Akt phosphorylation was achieved at 20 µM of LY294002, while phosphorylation of other kinases including p38 and ERK was not affected (Fig. 6A). Inhibition of PI3K/Akt signaling resulted in a significant increase in apoptosis induction by a low concentration of MacCM (10%) in these cells (MacCM + vehicle: 7.05% ± 1.11, MacCM + LY 19.48% ± 2.15) (Fig. 6B). We additionally inhibited Akt by lentiviral-mediated overexpression of shRNA. Out of three existent Akt isoforms, Akt2 (also known as PKBß) is most highly expressed in insulin-sensitive tissues [START_REF] Brazil | Ten years of protein kinase B signalling: a hard Akt to follow[END_REF]. We designed and cloned two different shRNA constructs targeting human Akt2. We achieved a down-regulation of Akt2 expression by > 80 % with both constructs compared to expression of a nonsense shRNA (Fig. 6C).

In line with the pharmacological inhibition, knock-down of Akt2 caused a significant increase in apoptosis induction by macrophage-secreted factors compared to nonsense shRNA control cells (nonsense: 12.23% ± 1.27; Akt2#1: 33.05% ± 3.85; Akt2#2:24.56% ± 3.62) (Fig. 6D). Previous studies of our group showed that human fat cells are protected from death receptor induced apoptosis although expressing all components of the apoptosis signaling pathway [START_REF] Fischer-Posovszky | Inhibition of death-receptor mediated apoptosis in human adipocytes by the insulin-like growth factor I (IGF-I)/IGF-I receptor autocrine circuit[END_REF]. The insulin-like growth factor I (IGF-I)//IGF-I-receptor autocrine circuit is responsible for this apoptosis resistance [START_REF] Fischer-Posovszky | Inhibition of death-receptor mediated apoptosis in human adipocytes by the insulin-like growth factor I (IGF-I)/IGF-I receptor autocrine circuit[END_REF]. It is known that insulin/IGF-I act as survival factors in several cell types, including cancer cells [START_REF] Valentinis | The IGF-I receptor protects tumor cells from apoptosis induced by high concentrations of serum[END_REF] as well as brown adipocytes [START_REF] Navarro | Insulin/IGF-I rescues immortalized brown adipocytes from apoptosis down-regulating Bcl-xS expression, in a PI 3-kinase-and map kinase-dependent manner[END_REF]. Therefore, we hypothesized that white adipocytes are resistant to apoptosis as long as they are insulin sensitive. Upon overweight and obesity, macrophages infiltrate human adipose tissue and promote both local and systemic inflammation as well as the development of obesity-related disorders [START_REF] Weisberg | Obesity is associated with macrophage accumulation in adipose tissue[END_REF][START_REF] Xu | Chronic inflammation in fat plays a crucial role in the development of obesity-related insulin resistance[END_REF]. It is well accepted that the insulin sensitivity of fat cells is decreased upon inflammation [START_REF] Olefsky | Macrophages, inflammation, and insulin resistance[END_REF]. In line with our hypothesis this loss of insulin sensitivity seems to be accompanied by increased apoptosis sensitivity. Fat cell apoptosis has been reported in diseases associated with insulin resistance and inflammation including cancer [START_REF] Prins | Human adipocyte apoptosis occurs in malignancy[END_REF], HIV-associated lipodystrophy [START_REF] Domingo | Subcutaneous adipocyte apoptosis in HIV-1 protease inhibitor-associated lipodystrophy[END_REF] or autoimmune lipodystrophy [START_REF] Fischer-Posovszky | Role of CD95-mediated adipocyte loss in autoimmune lipodystrophy[END_REF].

To test our hypothesis on a cellular level, we established an in vitro model of adipose tissue inflammation. Macrophage-secreted factors inhibited insulin-stimulated processes such as adipogenic differentiation, glucose uptake and de novo lipogenesis accompanied by decreased phosphorylation of insulin-stimulated intracellular kinases. Kinases important for the insulin signaling including Akt were robustly inhibited by the macrophage-secreted factors assumable by cytokines such as IL-6 and/or IL-1.

Both cytokines inhibit phosphorylation of Akt in human primary adipocytes and 3T3-L1 cells [START_REF] Lagathu | Long-term treatment with interleukin-1beta induces insulin resistance in murine and human adipocytes[END_REF][START_REF] Xie | Interactive changes between macrophages and adipocytes[END_REF]. Under normal physiological conditions human fat cells are protected from apoptosis induction by death ligands [START_REF] Fischer-Posovszky | Inhibition of death-receptor mediated apoptosis in human adipocytes by the insulin-like growth factor I (IGF-I)/IGF-I receptor autocrine circuit[END_REF]. Notably, macrophage-secreted factors induced apoptosis in both preadipocytes and adipocytes in a dose-dependent manner. This effect A c c e p t e d M a n u s c r i p t 14 was even more pronounced in macrophage-adipocyte co-cultures. We conclude that the apoptosis inducing effect of macrophages is mediated in a paracrine manner and is not dependent on direct cell-cell interaction, but cell contact ameliorates the process. Our findings demonstrate that macrophage-secreted factors increased apoptosis sensitivity in human adipocytes by blocking insulin signaling. Obviously, an inflammatory micro-environment abrogates the cellular survival system. In line, we demonstrated earlier that the loss of adipose tissue in autoimmune lipodystrophy is due to adipocyte apoptosis induced by a combination of cytokines together with death ligands, i.e. INF, TNF and CD95L (Fischer-Posovszky et al., 2006).

Pharmacological or genetic inhibition of insulin signaling at the level of Akt sensitized human fat cells for apoptosis induction further supporting a causal connection between insulin resistance and apoptosis sensitivity. Akt is a key molecule in the insulin signaling pathway [START_REF] Kohn | Insulin stimulates the kinase activity of RAC-PK, a pleckstrin homology domain containing ser/thr kinase[END_REF]. Among the three isoforms of Akt, Akt2 is the most abundant isoform in insulin sensitive tissues [START_REF] Brazil | Ten years of protein kinase B signalling: a hard Akt to follow[END_REF] and it is the major isoform responsible for insulin action [START_REF] Brazil | Ten years of protein kinase B signalling: a hard Akt to follow[END_REF]. Most interestingly, Akt2 has already been linked to loss of adipose tissue, most likely by apoptosis of fat cells in mice and in humans. Along with insulin resistance, Akt2 knockout mice display an age-dependent loss of adipose tissue [START_REF] Garofalo | Severe diabetes, agedependent loss of adipose tissue, and mild growth deficiency in mice lacking Akt2/PKB beta[END_REF]. Likewise, a family with a mutation in the Akt2-gene shows an autosomal dominant inheritance of severe insulin resistance and lipodystrophy [START_REF] George | A family with severe insulin resistance and diabetes due to a mutation in AKT2[END_REF]. We conclude from our data that macrophages inhibit Akt2 action thereby diminishing the insulin survival signal and rendering fat cells sensitive to apoptosis triggers.

It has to be mentioned that treatment of 3T3-L1 cells with conditioned media from J774 macrophages protected fat cells from apoptosis induction [START_REF] Molgat | Macrophage-induced preadipocyte survival depends on signaling through Akt, ERK1/2, and reactive oxygen species[END_REF]. This apparent discrepancy to our findings can be explained by differences in apoptosis sensitivity between murine 3T3-L1 cells and human fat cells. 3T3-L1 preadipocytes and adipocytes are sensitive to apoptosis induction by serum deprivation [START_REF] Magun | The effect of adipocyte differentiation on the capacity of 3T3-L1 cells to undergo apoptosis in response to growth factor deprivation[END_REF], whereas human SGBS preadipocytes and adipocytes are characterized by a general resistance to apoptosis stimuli including serum withdrawal [START_REF] Fischer-Posovszky | Inhibition of death-receptor mediated apoptosis in human adipocytes by the insulin-like growth factor I (IGF-I)/IGF-I receptor autocrine circuit[END_REF]. This resistance to apoptosis stimuli has also been shown for primary human preadipocytes and adipocytes [START_REF] Fischer-Posovszky | Inhibition of death-receptor mediated apoptosis in human adipocytes by the insulin-like growth factor I (IGF-I)/IGF-I receptor autocrine circuit[END_REF]. We suggest that 3T3-L1 cells benefit from macrophagesecreted factors such as PDGF [START_REF] Molgat | Preadipocyte apoptosis is prevented by macrophageconditioned medium in a PDGF-dependent manner[END_REF] which are known to rescue cells from apoptosis. In A c c e p t e d M a n u s c r i p t 15 addition, it has been shown that macrophage media stimulate Akt phosphorylation in 3T3-L1 cells [START_REF] Molgat | Macrophage-induced preadipocyte survival depends on signaling through Akt, ERK1/2, and reactive oxygen species[END_REF], further supporting the differences between murine and human cells.

Our results show that the number of macrophages correlates with the number of apoptotic adipocytes in visceral as well as in subcutaneous human adipose tissue samples. In our study we used immunohistochemistry to detect cleaved caspase-3, which is a specific marker for apoptosis induction in situ. We identified apoptotic adipocytes in the center of CD11c + crown-like structures. Apoptosis is associated with typical biochemical events such as externalization of phosphatidylserine and cleavage of caspases [START_REF] Kroemer | Classification of cell death: recommendations of the Nomenclature Committee on Cell Death 2009[END_REF]. These characteristics are not found in other forms of cell death such as necrosis [START_REF] Kroemer | Classification of cell death: recommendations of the Nomenclature Committee on Cell Death 2009[END_REF]. Cinti et al. detected signs of necrotic cell death with crown-like structures in murine and human adipocytes as assessed by rupture of the plasma membrane and dilatation of the endoplasmic reticulum (ER) via electron microscopy [START_REF] Cinti | Adipocyte death defines macrophage localization and function in adipose tissue of obese mice and humans[END_REF]. The relative absence of neutrophils usually involved in necrosis-induced inflammation led the authors to suggest that the adipocyte cell death might share features of both apoptosis and necrosis [START_REF] Cinti | Adipocyte death defines macrophage localization and function in adipose tissue of obese mice and humans[END_REF]. It is well conceivable that features of necrosis occur as late, secondary events in the process of adipocyte death after a primary induction of apoptosis.

Interestingly, a recent study showed that blocking the function of Bid, a key pro-apoptotic molecule, resulted in a significant decrease in fat cell apoptosis, which prevented macrophage recruitment into adipose tissue and protected against development of insulin resistance in mice [START_REF] Alkhouri | Adipocyte apoptosis, a link between obesity, insulin resistance, and hepatic steatosis[END_REF]. In this experimental setting the primary event leading to apoptosis induction is unknown. Potential triggers of apoptosis include stress responses such as endoplasmic reticulum (ER) stress [START_REF] Hotamisligil | Endoplasmic reticulum stress and the inflammatory basis of metabolic disease[END_REF][START_REF] Schenk | Insulin sensitivity: modulation by nutrients and inflammation[END_REF] and hypoxia [START_REF] Ye | Emerging role of adipose tissue hypoxia in obesity and insulin resistance[END_REF][START_REF] Regazzetti | Hypoxia decreases insulin signaling pathways in adipocytes[END_REF]. Both of these pathological conditions have been shown to be implicated in the development of metabolic disease [START_REF] Wood | Cellular hypoxia and adipose tissue dysfunction in obesity[END_REF][START_REF] Hotamisligil | Endoplasmic reticulum stress and the inflammatory basis of metabolic disease[END_REF].

A c c e p t e d M a n u s c r i p t 16

Conclusion

We elucidated how macrophages affect apoptosis sensitivity of human fat cells during adipose tissue inflammation in vitro. The close neighborhood of both cell types in human adipose tissue strongly suggests biological interaction in vivo. We propose a model where macrophages contribute to maintain low grade inflammation in human adipose tissue in a feed-forward cycle. Macrophages, which have already infiltrated human adipose tissue upon stimuli discussed above, affect two, presumably associated, characteristics of functional fat cellsinsulin sensitivity and apoptosis resistance. In particular, secreted cytokines block Akt2 in neighboring fat cells thereby inducing insulin resistance. Simultaneously, macrophages produce death ligands such as TNF and CD95L [START_REF] Xu | Chronic inflammation in fat plays a crucial role in the development of obesity-related insulin resistance[END_REF][START_REF] Weigert | The liaison between apoptotic cells and macrophages--the end programs the beginning[END_REF]. This mechanism enables macrophages to induce apoptosis of fat cells that became insulin resistant and therefore apoptosis sensitive. The increased number of apoptotic adipocytes in turn attracts more macrophages, thereby accelerating the inflammatory process within human adipose tissue.

In conclusion, the discovery that macrophages induce apoptosis of human fat cells during adipose tissue inflammation is novel and has important implications for future investigations in the field of obesityrelated metabolic disorders. After 24 h cells were stained with Annexin V followed by flow cytometric analysis. (D) After seven days in adipogenic medium, SGBS adipocytes were incubated with vehicle or 50% MacCM for 24, 48 and 72 h. The content of hypodiploid DNA was determined by flow cytometry. (E) After seven days in adipogenic medium, SGBS adipocytes were incubated with increasing concentrations of MacCM. After 48 h, the content of hypodiploid DNA was determined by flow cytometry. For (A), (B), (D) and (E) specific apoptosis was calculated as described in Methods (Section 2.6.). All data are expressed as mean ± SEM of three independent, triplicate experiments. * p < 0.05 vs. vehicle.

Fig. 5:

Macrophages induce apoptosis in human preadipocytes and adipocytes

THP-1 cells were differentiated into macrophages as described in Methods and added in increasing rates to cultures of (A, C) SGBS preadipocytes or (B, D) adipocytes. (A) After 24 h cells were stained with anti-CD221-PE and Annexin V-FITC. CD221 + -preadipocytes were gated and analyzed for Annexin V expression. (B) Adipocytes were gated according to their FSC/SSC profile and analyzed for Annexin V.

  ice-cold PBS, harvested with 100 mM NaOH and incorporation of [ 14 C]-2-deoxy-Dglucose or [ 14 C]-D-glucose was measured on a -counter.

  spontaneous apoptosis in vehicle (%)] / [100%spontaneous apoptosis in vehicle (%)]. Externalization of phosphatidylserine was measured by Annexin V staining (Roche, Grenzach, Germany) according to the manufacturer's instructions and analyzed by flow cytometry.

  to a protein array to study the phosphorylation state of insulin-stimulated down-stream kinases. Out of several kinases studied, phosphorylation of Akt kinases was most robustly inhibited by macrophagesecreted factors in both the basal and the insulin-stimulated state (Fig. A.1). Likewise, phosphorylation of Akt down-stream targets, e.g. glycogen synthase kinase 3 alpha/beta (GSK3-) and 70-kDa S6 kinase (p70S6K) was down-regulated. Other signaling pathways including c-Jun N-terminal kinases (JNK), p38 MAPK (p38) and p44/42 MAPK (ERK) were either not or only slightly affected (Fig. A.1).

  In the present study, we demonstrate that macrophages induce apoptosis in human fat cells by interfering with the anti-apoptotic insulin signaling pathway at the level of Akt. Studying human subcutaneous and visceral adipose tissue samples we found a co-localization of CD11c + -macrophages and apoptotic adipocytes indicating a causal relationship of macrophage infiltration and apoptosis of fat cells.

  Figure legends

Fig. 2 :

 2 Fig. 2: Macrophage-secreted factors inhibit adipogenic differentiation and induce insulin resistance in human adipocytes SGBS preadipocytes were incubated with conditioned media from THP-1 macrophages (MacCM) during adipogenic differentiation. After 10 days cells were analyzed. (A) Rate of adipogenic differentiation was determined as described in methods. (B) The expression of the fatty acid binding protein 4 (FABP4) was analyzed by qPCR. (C) SGBS adipocytes were pre-incubated with 50% MacCM or vehicle for 24 h. [ 14 C]-2-deoxy-D-glucose uptake was determined with different insulin concentrations (0 -1000 nM). (D) Adipocytes were pre-incubated with escalating doses of MacCM (0 -50%) for 24 h. [ 14 C]-2-deoxy-Dglucose uptake was determined with or without insulin (100 nM). (E) SGBS adipocytes were incubated with vehicle or 50% MacCM. After 24 h basal and insulin-stimulated de novo lipogenesis was assessed by incorporation of [ 14 C]-D-glucose into lipids in the presence of different insulin concentrations (0 -1000 nM). (F) After eight days in adipogenic medium SGBS adipocytes were incubated with escalating doses of MacCM (0 -50%). After 24 h basal and insulin-stimulated de novo lipogenesis was assessed by incorporation of [ 14 C]-D-glucose into lipids in the presence or absence of insulin (100 nM). All data are the mean ± SEM of three independent experiments. For (C) -(F) results are presented as fold change of insulin-stimulated to basal glucose uptake or glucose incorporation. * p < 0.05 vs. vehicle.

Fig. 3 :

 3 Fig. 3: Macrophage-secreted factors inhibit basal and insulin-stimulated Akt phosphorylation (A) After seven days in adipogenic medium SGBS adipocytes were pre-incubated with 50% MacCM or vehicle. After 24 h cells were treated with indicated insulin concentrations for 15 min before lysis. Cell lysate (100 μg protein/lane) was separated by 4-12% gradient SDS-page. Phosphorylation of Akt, ERK, and p38 (pAkt, pERK and pp38) was analyzed by Western blot analysis. One representative experiment is shown. Densitometric analysis for the basal phosphorylation (B) or insulin-stimulated phosphorylation (100nM, C) of Akt, ERK and p38 are presented as mean + SEM of three independent experiments. Phosphorylation of the vehicle control was set to one. * p < 0.05 vs. vehicle.

Fig. 4 :

 4 Fig. 4: Macrophage-secreted factors induce apoptosis in human preadipocytes and lipid-laden adipocytes (A) SGBS preadipocytes were incubated with vehicle or 50% MacCM. After 24, 48 and 72 h, the content of hypodiploid DNA was determined by flow cytometry. (B) SGBS preadipocytes were incubated with increasing concentrations of MacCM (0 -50%) for 48 h. The content of hypodiploid DNA was determined by flow cytometry. (C) SGBS preadipocytes were incubated with vehicle or 50% MacCM.

  and (B) data are presented as % of control and as mean + SEM of three independent experiments. * p < 0.05 vs. control. (C, D) After 72 h of co-culture the number of adherent SGBS preadipocytes (C) or adipocytes (D) was assessed as described in methods. Data are expressed as % of control and presented as mean + SEM of four independent experiments. * p < 0.05 vs. control. (E) SGBS adipocytes grown on chamber slides were co-cultured with THP-1 macrophages (ratio 1:1). After 24 h cells were stained with anti-CD11c-PE (macrophages, red), BODIPY (lipid droplets, green), and DAPI (nuclei, blue). Cells were analyzed with a fluorescence microscope. One representative experiment is shown. (F) Microphotographs of CD11c and active caspase-3 staining of serial sections from human fat tissue of one representative patient (black circles: positive cells for indicated staining, magnification: 10x (left panel), 40x (right panel)). For the relationship between CD11c and active Caspase-3 positive cells, macrophages and apoptotic cells in adipose sections of 54 patients were quantified by counting the number of positive cells in 12 random microscopic fields. Spearman`s correlation analysis was performed between percentage of macrophages and apoptotic fat cells for subcutaneous (G) and visceral adipose tissue (H). Data were natural-log transformed before correlation analysis. Positive cells for both staining were detected in subcutaneous fat tissue of 19 patients and in visceral fat tissue of 30 patients (details see Table A.1).

Fig. 6 :

 6 Fig. 6: Insulin resistance sensitizes human fat cells to apoptosis by macrophage-secreted factors (A) Decreased phosphorylation of Akt by the pharmacological inhibitor of the PI3-kinase (LY294002) was controlled by western blot analysis. SGBS adipocytes were incubated for 1h with or without 20 µM LY294002. One representative out of three independent experiments performed is shown. (B) After seven days in adipogenic medium SGBS adipocytes were incubated with 20 µM LY294002. LY was either supplied in vehicle or 10% MacCM. After 48 h content of hypodiploid DNA was determined by flow cytometry. * p < 0.05 vs. LY or MacCM. (C) Lentiviral mediated knockdown of two different Akt2 shRNA constructs in SGBS preadipocytes was controlled by western blot analysis. One representative experiment out of three performed is shown. (D) Nonsense, Akt2#1 and Akt2#2 cells were incubated with 50% MacCM or vehicle. After 48 h, the content of hypodiploid DNA was determined by flow cytometry. * p < 0.05 vs. nonsense. For (B) and (D), specific apoptosis was calculated as described in Methods (Section 2.6.) and data of three independent, triplicate experiments are expressed as mean + SEM.



  Figure 1

  Figure 5

Acknowledgements

We thank A. Killian for excellent technical assistance and Dr. Patrick Gonzales for help with immunofluorescence. This study was supported by the German Research Association (WA1096/3-2) and the Centre of Excellence Baden-Wuerttemberg "Metabolic Disorders" to MW. PFP is funded by a Margarete von Wrangell scholarship financed by the Baden-Wuerttemberg Ministry of Science, Research and Arts, the European Social Fund, and Ulm University.

Disclosure Statement

AD and KA are employed by Hoffman-La Roche. Remaining authors declare no conflict of interest.


P Collet
email: collet@cpht.polytechnique.fr

Servet Martínez
email: smartine@dim.uchile.cl

Sylvie Méléard
email: sylvie.meleard@polytechnique.edu

Jaime San Martín

Stochastic models for a chemostat and long time behavior

We introduce two stochastic chemostat models consisting in a coupled population-nutrient process reflecting the interaction between the nutrient and the bacterias in the chemostat with finite volume. The nutrient concentration evolves continuously but depending on the population size, while the population size is a birth and death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long time behavior of the bacterial population conditioned to the non-extinction. We prove the global existence of the process and its almost sure extinction. The existence of quasi-stationary distributions is obtained based on a general fixed point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.

Introduction

Since some decades and the first work of J. Monod [START_REF] Monod | La technique de culture continue[END_REF] and Novik and Szilar [START_REF] Novick | Experiments with the Chemostat on Spontaneous Mutations of Bacteria[END_REF], [START_REF] Novick | Description of the Chemostat[END_REF], see also [START_REF] Smith | The Theory of the Chemostat[END_REF], biologists have developed procedures which allow to maintain a bacterial population at a stationary finite size while at the same time, the bacteria have a constant individual growth rate. The procedure is based on a chemostat: bacteria live in a growth container of constant volume in which liquid is injected continuously. The entering liquid contains a fixed concentration of nutrient but no bacteria (fresh liquid). In the container, the nutrient is consumed by the bacteria. We assume that the chemostat is well stirred, so that the distribution of bacteria and nutrient are spatially uniform. Since the container has a finite volume and fresh liquid enters permanently, an equal amount of liquid pours out containing both unconsumed nutrients and bacteria. These chemostats are extremely useful in the study of bacterial population dynamics, in particular in the study of the selection of the fastest growing species or the fixation of advantageous mutations (see [START_REF] Masci | Continuous selection of the fastest growing species in the chemostat[END_REF], [START_REF] Masci | Competitive exclusion in the generalized Droop model with N species[END_REF], [START_REF] Dykhuizen | Selection in chemostats[END_REF] or [START_REF] Leenheer | Competition in the chemostat: Some remarks[END_REF]). In the literature, their study is mainly based on deterministic models where both nutrient and bacteria population dynamics are described by a coupled deterministic continuous process. Deterministic approximations for the bacterial population's size are justified by a large population assumption.

In this work, we develop stochastic chemostat models based on a previous work of Crump and O' Young [START_REF] Crump | Some stochastic features of bacterial constant growth apparatus[END_REF] taking into account that the bacteria population may not be large enough so that a deterministic approximation can be justified. We introduce a coupled population-nutrient process reflecting the interaction between the nutrient and the bacterias in the chemostat. The nutrient concentration evolves continuously but depending on the population size, while the population size is a birth and death process with coefficients depending on time through the nutrient concentration. Moreover, the time derivative of the nutrient concentration jumps simultaneously with the population size. We thus take into account the random fluctuations of this population size due to the individual births and deaths. The bacteria need nutrient to reproduce. We will consider two cases. In the first one, the bacterial population dies instantaneously if the nutrient is missing. In the second case bacteria can survive without nutrient by undergoing some kind of hibernation and may wake up once nutrient reappears. The nutrient is shared by the bacteria. This creates an indirect competition between bacteria and leads to a regulation of their population size. In our models, the fluctuations due to the random births and deaths of individuals and the size regulation make the population go almost surely to extinction. Therefore, the long time behavior of the population's size is obvious and the interesting questions concern firstly the rate of extinction and secondly the long time behavior conditioned to the non-extinction which is captured by the notion of quasi-stationary distribution.

To our knowledge, the models introduced in this paper are the first stochastic chemostat models where interaction between bacteria is taken into account leading to extinction. The study of quasi-stationarity gives nevertheless a description of a quasi-stability which can happen in a faster time scale than extinction. This work concerns monotype individuals but could be generalized to a multi-type population.

In Section 2, we describe the two population-nutrient models described above and prove in Section 3 their global existence. We also show the extinction of the population when time increases. The existence of quasi-stationary distributions is obtained in Section 4. Our main theorem is based on a general argument proved in [START_REF] Collet | Quasi-stationary distributions for structured birth and death processes with mutations[END_REF]. In Section 5, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.

The Stochastic Population-Nutrient Process

We consider a stochastic discrete population process describing the dynamics of a bacteria population for which individuals develop and reproduce depending on the quantity of nutrient y in the solution. The dynamics of the nutrient is related to the consumption of the individuals. We assume that the concentration of nutrient in the injected solution (without bacteria) is a constant equal to y * . The chemostat has a finite volume equal to one. The liquid enters in the chemostat free of bacteria and pours out after being well stirred in the container. The pouring out liquid contains bacteria. The dilution coefficient of nutrient in the fresh liquid per unit of time is D. Since the liquid is well stirred, around N(t)D bacterias will be washed out in the pouring out, when N(t) is the size of the bacteria population at time t. Thus D is also the rate at which an individual will disappear due to the evacuation of liquid. We consider coupled processes in which the nutrient concentration evolves continuously while the bacteria population size evolves as a time-continuous birth and death process with coefficients depending on the nutrient concentration. We assume that the nutrient is partly consumed during the reproduction of bacteria.

We will denote by (Y (t), t ≥ 0) the concentration of nutrient and by (N(t) : t ≥ 0) the population size process. The stochastic process Z = (Z(t) := (N(t), Y (t)) : t ≥ 0) describes both the population size and the nutrient concentration in the chemostat.

Let us now define the parameters of the model.

If y is the quantity of nutrient, then the birth and death parameters driving the dynamics of the population are as follows.

• The birth rate per individual is b(y), where the function b : R + → R + is assumed to be an increasing continuous function and such that b(0) = 0 and b(y) > 0 for y > 0. We assume that b is bounded with an upper-bound b ∞ .

A usual example of a function b is, for some constant

K > 0, b(y) = b ∞ y K + y .
An extra hypothesis that we will add for some results is that b is differentiable and db dy (0) > 0. • The background death rate per individual is d(y), so it is supposed to be a function of the concentration of nutrient. The function d : R + → R + ∪ {∞} is assumed to be continuous non-increasing, strictly positive, and d(0) is the unique value that can be infinite.

• The dilution makes each individual disappear at rate D independently of the birth and death events.

• The per individual rate of consumption of nutrient for reproduction is b(y) R , where R denotes the biomass yield. Furthermore, individuals consume nutrient during their life and the quantity of nutrient consumed per individual will be denoted by η ≥ 0. • We will consider two cases. In the first case, individuals need nutrient to survive. Then we will assume that their death is instantaneous as soon as nutrient is missing, therefore d(0) = ∞. In the second case, bacteria enter in some kind of hibernation if nutrient is missing. That means that d 0 (0) can be finite. In both cases, we will set b(y) = b(y) R + η 1 y>0 .

Let us now describe the process. In both cases the nutrient concentration process Y = (Y (t) : t ≥ 0) evolves according to

dY (t) dt = D(y * -Y (t)) -b(Y (t)) N(t) . (1)
The process Z has the following infinitesimal generator: for (n, y)

∈ N × R + , Lf (n, y) = b(y) n f (n + 1, y) + (D + d(y)) n f (n -1, y) -(b(y) + D + d(y)) n f (n, y) + D(y * -y) -n b(y) ∂ y f (n, y) . (2)
We refer to [START_REF] Campillo | Stochastic models of the chemostat[END_REF] for the numerical study of similar models.

Below, we show that the hypotheses we gave on the coefficients leading the process, guarantee that Z is well-defined and takes values in N × [0, y * ].

Proposition 2.1. The process Z is well defined and takes values in

N × R + for all positive time t ∈ R + . Moreover, N + × [0, y * ] is an invariant set for the process Z, so Y (0) ∈ [0, y * ] implies Y (t) ∈ [0, y * ] for all t ∈ R + .
Proof. The process N = (N(t) : t ≥ 0) has a pathwise representation driven by a Point Poisson measure N (dθ, ds) defined on R + × R + :

N(t) = N 0 + R + ×(0,t]
1 θ≤b(Ys) N (dθ, ds)-

R + ×(0,t]
1 b(Ys)≤θ≤b(Ys)+D+d(Ys) N (dθ, ds);

Y (t) = Y (0) + t 0 D(y * -Y (s)) -b(Y (s)) N s ds. (3)
It is obvious that the process N is stochastically upper-bounded by a birth process with individual birth rate b ∞ . This latter does not explode, so does N.

Let us now study the nutrient concentration Y = (Y (t) : t ≥ 0). Note that the assumptions on the parameters and Equation ( 1) ensure that Y (t) ≥ 0. Indeed, the derivative of Y (t) at y = 0 cannot be negative.

Let us show that

N × [0, y * ] is invariant. Take Y (0) ∈ [0, y * ]. A standard comparison theorem yields Y (t) ≤ v(t) where v ′ (t) = D(y * -v(t)) ; v(0) = Y (0). But in that case, v(t) = y * -(y * -Y (0))e -Dt ≤

Study of Extinction

We are now interested in studying the extinction of the population or the complete consumption of the nutrient or other specific states of the populationnutrient process and the associated hitting times. The hitting time of the boundaries will be denoted in such a way that the reference to process N will be avoided, the contrary for Y where we will explicit it. More precisely,

T 0 = T {0}×R + = inf{t ≥ 0 : N(t) = 0}, T Y =0 = T N×{0} = inf{t ≥ 0 : Y (t) = 0}. ( 4
) We will also denote by T ≤m = T {0,..,m}×R + = inf{t ≥ 0 : N(t) ≤ m}. Analogously for T <m .

Note that the set {0}×R + is an absorbing set, that is N(t) = 0 for all t ≥ T 0 .

After T 0 the nutrient Y (t) is absorbed linearly at y * , in fact:

Y (t) = y * for t ≥ T 0 + (y * -Y (T 0 ))/D y * and Y (t) = Y (T 0 ) + D(t -T 0 ) for t ∈ [T 0 , T 0 + (y * -Y (T 0 ))/D].
Let us see what happens after T Y =0 . In the case η ≥ Dy * the nutrient Y (t) remains at 0 up to the extinction of the population, so Y (t) = 0 when T Y =0 < t < T 0 . The evolution of the nutrient after T 0 was already described. Now assume n η > Dy * for some n ≥ 1, and denote n 0 the minimal of these values, so (n 0 -1) η ≤ Dy * < n 0 η. In this case Y (t) = 0 when T Y =0 < t < T <n 0 . In the case where d 0 (0) = ∞, we get

T Y =0 ≤ T 0 =⇒ T Y =0 = T 0 because all individuals die instantaneously.
Let us firstly study the stationary nutrient concentration states at fixed population size. (5)

Then, (i) If η = 0, then for any n ∈ N, Equation (5) has a unique simple root y n , which belongs to [0, y * ]. In addition, the sequence (y n : n ∈ N * ) of the roots decreases to 0.

(ii) If η > 0 then Equation (5) has no root for n > Dy * η and admits a simple root y n for n ≤ Dy * η .

Proof. Let us fix n. Obviously for n = 0, Equation ( 5) has the trivial root y 0 = y * , so we restrict n to be in N * . By assumption, the function G n is strictly decreasing so for each n ∈ N * there exists at most one root. Note that for all such n we have G n (y * ) < 0 and so there is no root to (5) in the set [y * , ∞).

Assume η = 0. For all n we have G n (0) > 0 then there exists a unique root which is denoted y n , so it satisfies G n (y n ) = 0. On the other hand, we have

G n+1 (y n ) = D(y * -y n ) -(n + 1) b(y n ) = -b(y n ) . Then, G n+1 (0) > 0, G n+1 (y n ) < 0. We deduce 0 < y n+1 < y n < y * . Let y ∞ = lim n→∞ y n . By continuity D(y * -y ∞ ) -n b(y ∞ ) = lim n→∞ D(y * -y n ) -n b(y n ) = 0.
Then, necessarily b(y ∞ ) = 0 and so y ∞ = 0, and b(y

n ) ∼ Dy * R n as n → ∞.
Assume η > 0. Then for all n > Dy * /η we have G n (0) < 0 and so (5) has no solution. Hence the same argument as before gives the existence of a finite set of roots (y n : 1 ≤ n ≤ ⌊Dy * /η⌋) decreasing with n, where ⌊Dy * /η⌋ the biggest integer that is smaller or equal to Dy * /η.

Note that when n 0 = Dy * /η then y n 0 = 0 and if Dy * < η, there is no root.

From Lemma 3.1, we know that the set

S = {y ∈ R + : ∃n ∈ N * , D(y * -y) -b(y)n = 0}. ( 6
)
is a countable set included in [0, y * ). If η = 0, it is infinite and accumulates at 0 and if η > 0, it is finite.

In the sequel, when we refer to y n , we will assume implicitly that it exists, namely we are in the case η = 0 or η > 0 but n ∈ {1, ..⌊Dy * /η⌋}.

Corollary 3.2. (i) The set N × [0, y 1 ] is invariant for the process Z T 0 = (Z(t) : t ≤ T 0 ) up to extinction, that is if Z(0) = (N(0), Y (0)) ∈ N × [0, y 1 ] then Z(t) = (N(t), Y (t)) ∈ N × [0, y 1 ] for all t ≤ T 0 .
(ii) The set N × [0, y n ] is invariant for the process Z T<n = (Z(t) : t ≤ T <n ).

Proof. Let us show the first part. We have

Z(t) = (N(t), Y (t)) ∈ N * × [0, y 1 ]
for all t < T 0 because dY (t)/dt ≤ 0 when y ≥ y 1 and n ≥ 1, and so if the trajectory arrives to y 1 the variable Y (t) immediately decreases. For n = 0 it is evident, because we stop the process at T 0 , and so

Y (T 0 ) = Y (T - 0 ) but Y (T - 0 ) ≤ y 1 since N(T - 0 ) = 1. Part (ii)
is shown in a similar way as (i).

Let us state a useful lemma. Lemma 3.3. For any n 0 ∈ N * , there exists t 0 > 0 such that inf y∈[0,y * ], 1≤n≤n 0 P (n,y) T 0 < t 0 > 0 .

Proof. This follows at once from the fact that the population process is stochastically dominated by a birth and death process with birth rate b(y * ) and death rate D + d(y * ).

Theorem 3.4. We have extinction of the population almost surely, namely for any y ∈ [0, y * ] and any integer n,

P (n,y) T 0 < ∞ = 1 .
Proof. Remark that an obvious comparison theorem as used in the previous proof cannot be applied. Indeed the birth and death rates b(y * ) and D+d(y * ) could correspond to a supercritical case. The effect of the chemostat through the nutrient is a regulation of the population. We will exhibit an integer n 0 such that the population process will spend an infinite amount of time below n 0 + 1.

Let ỹ > 0 be such that b(ỹ) < D + d(ỹ). Note that by monotonicity, for all y ∈ [0, ỹ] we have b(y) < D + d(y). Let us define n 0 as an integer such that y n 0 < ỹ if some exists or equal to [Dy * / b(0)] + 1 otherwise. Let τ be the random time defined by

sup t≥τ N(t) ≤ n 0 .
Then, it follows from Lemma 3.3 and the Markov property that if P (n,y) τ < ∞ > 0, then

P (n,y) T 0 < ∞ τ < ∞ = 1 .
Assume now P (n,y) τ = ∞ = 1. Let τ ′ be the random time defined by inf t≥τ ′ N(t) > n 0 . Assume P (n,y) τ ′ < ∞ > 0. Let y 0 (t) be the solution of the differential equation

dy 0 dt = D(y * -y 0 (t)) -(n 0 + 1) b(y 0 (t)) ,
with initial condition y 0 (0) = y * . Let t 1 > 0 be the finite solution of

y 0 (t 1 ) = ỹ .
The time t 1 is finite from the choice of n 0 . It is easy to verify that for any integer valued measurable function n(t) ≥ n 0 + 1, the solution y(t) of the equation

dy dt = D(y * -y(t)) -n(t) b(y(t)) ,
with initial condition y(0) ≤ y * , satisfies y(t) ≤ ỹ for any t ≥ t 1 . On the set τ ′ < ∞ , the process (N(t), t ≥ τ ′ + t 1 ) is dominated by a linear birth and death process with birth rate b(ỹ) and death rate D+d 0 (ỹ) (from the monotonicity of the functions). This birth and death chain attains n 0 almost surely in finite time since b(ỹ) < D + d 0 (ỹ), see [START_REF] Van Doorn | Quasi-stationary distributions and convergence to quasistationarity of birth-death processes[END_REF]. Hence on τ ′ < ∞ , the process (N(t), t ≥ τ ′ + t 1 ) should also attains n 0 in finite time. This contradicts our assumption P (n,y) τ ′ < ∞ > 0. It remains to consider the case τ = τ ′ = ∞ almost surely. In this case there exist two infinite sequences of random times

T 1 < S 1 < T 2 < S 2 < • • • such that N(t) ≤ n 0 for t ∈ [T j , S j ) , N(t) > n 0 for t ∈ [S j , T j+1 ) .
Since we visit the set {N ≤ n 0 } infinitely many times, and at each visit we have a uniformly positive probability of extinction, it follows by the Markov property and the Borel Cantelli Lemma that

P (n,y) T 0 < ∞ τ = τ ′ = ∞ = 1 .
One of our main objectives of this work is to study the processes up to the moment the population is extinct Z T 0 = (Z(t) : t ≤ T 0 ), or before the moment of extinction Z T - 0 = (Z(t) : t < T 0 ). All the statements related to quasi-stationary distributions depend on Z T - 0 .

Existence of Quasi-Stationary Distributions

A quasi-stationary distribution ν (with respect to the absorbing time T 0 ) is a probability measure defined on N * × R + that verifies

P ν (Z(t) ∈ D | T 0 > t) = ν(D) , ∀D ∈ B(N * × R + ) . (7)
It is known that starting from a quasi-stationary distribution, the time of absorption is exponential, that is P ν (T > t) = e -λt where λ = λ(ν) > 0. Let M b = {f : N * × R + → R bounded and measurable}. Equation ( 7) can be written as,

∃ λ > 0 : E ν (f (Z(t)), T 0 > t) = e -λt ∞ 0 f (y)ν(dy) ∀t > 0, f ∈ M b . (8)
Denote by

κ n = ν({n} × R + ) = P ν (N(0) = n) ,
and the probability measure conditioned to have n individuals by

ν n (B) = ν({n} × B | {n} × R + ) = P ν (Y (0) ∈ B | N(0) = n) ∀B ∈ B(R + ) . Then ν(D) = n∈N * κ n ν n (D∩{n}×R + ), ν n (R + ) = 1 ∀n ∈ N * and n∈N * κ n = 1.
In order that the probability measure ν is a quasi-stationary distribution, it must satisfy the infinitesimal condition deduced from [START_REF] Leenheer | Competition in the chemostat: Some remarks[END_REF], which is given by,

∃ λ > 0 : n∈N * κ n ∞ 0 dν n (y) Lf (n, y) -λf (n, y) = 0 ∀f ∈ M b .
In terms of the adjoint operator L † of L defined in (2) the equation to be a quasi-stationary distribution is,

∃ λ > 0 : L † ν = -λ ν . ( 9
)
Theorem 4.1. Assume there exists 0 ≤ σ < 1 such that lim sup

yց0 y σ d(y) < ∞.
Then there exists a quasi-stationary distribution. Moreover there exists at least a quasi-stationary distribution such that Y is supported in [0, y * ].

Proof. It suffices to show the existence of a quasi-stationary distribution. In fact, the last part of the statement follows when the existence proof is applied to the process Z taking values in the invariant set N × [0, y * ].

The idea for showing the existence is to use the abstract Theorem 4.2 proved in [START_REF] Collet | Quasi-stationary distributions for structured birth and death processes with mutations[END_REF]. We assume y ∈ [0, y * ]. We define a function

ϕ 1 (y, n) = 1 n≥1 .
Then a simple computation (it is the same computation as for a birth and death process since ϕ 1 does not depend on y) leads to

Lϕ 1 (y, n) = 0 if n > 1 , -D -d(y) if n = 1 .
Therefore if d 0 is bounded above by a constant d * 0 , then

e -t(D+d * 0 ) ϕ 1 ≤ e tL ϕ 1 ≤ ϕ 1 .
If d 0 is not bounded, we will prove the lower bound inf

n 0 ≥1, y 0 ∈]0,y * ] P n 0 ,y 0 (N(1) ≥ 1) ≥ Q .
for a constant Q > 0. By the Markov property it suffices to prove that there exists Q > 0 such that inf

y 0 ∈[0,y * ] P 1,y 0 (N(t) = 1, ∀ 0 ≤ t ≤ 1) ≥ Q .
Since N = 1 on the whole time interval [0, 1], y(t) satisfies the differential equation

dy dt = D(y * -y) -b(y) , (10)
with initial condition y 0 . Since there neither birth, nor death on the time interval [0, 1] we get

P 1,y 0 (N(t) = 1, ∀ 0 ≤ t ≤ 1) = e -1 0 d(y(t))+b(y(t) dt .
Since b(y) is bounded uniformly in y, the above quantity does not vanish if

1 0 d(y(t)) dt < ∞ .
It is easy to show that there is a constant c > 0 such that for any y * ≥ y 0 ≥ 0, we have y(t) ≥ ct for any t ∈ [0, 1]. Since d(y) ≤ O(1)y -σ for y small, we get

1 0 d(y(t)) dt ≤ O(1) 1 0 y(t) -σ dt < ∞ .

It now follows immediately that

Q ϕ 1 ≤ e L ϕ 1 ≤ ϕ 1 .
The second function is

ϕ 2 (y, n) = 1 n≥1 e a(y)n ,
with a(y) = αy + a 0 , α > 0 and a 0 > 0. A simple computation for n > 1 (it is the same computation as for a birth and death process since ϕ 2 does not depend on y) leads to,

Lϕ 2 (y, n) = n e a(y)n Ξ(n, y) ,
where Ξ(n, y) = b(y) e a(y) -1-αn R + (D+d(y)) e -a(y) -1 +Dα(y * -y)-αη1 y>0 .

Let us show that there exists A > 0 and N 0 such that for all n > N 0 , sup

y∈[0,y * ] Ξ(n, y) ≤ -A. Define ζ(y) = (D + d(y
)) e -a(y) -1 +Dα(y *y) -αη1 y>0 . We choose a 0 and α such that D (e -a 0 -1) +Dαy * < 0. Then ζ(0) < 0. It follows that ∀y ∈ [0, y * ], ζ(y) < -A for some A > 0. Consider N0 such that e a(y * ) -1 -α N0 R < 0. We still have for n > N0 that e a(y * ) -1 -αn R < 0. Then for n > N0 , we get Ξ(n, y) ≤ -A. Therefore for any C > 0 there exists N(C) such that for any n > N(C),

Lϕ 2 (y, n) ≤ -C ϕ 2 (y, n) .
Therefore, for any C > 0 there exists Γ(C) > 0 (finite) such that

Lϕ 2 ≤ -C ϕ 2 + Γ(C) ϕ 1 .
(the estimate for n = 1 is by direct computation taking Γ(C) adequately large enough). Hence,

e tL ϕ 2 ≤ e -tC ϕ 2 + Γ(C) C ϕ 1 .
In the case where Proof. We recall that a quasi-limiting distribution ν is a probability measure on N * × R + such that for some initial point (n 0 , y 0 )

∈ N * × R + ν(A) = lim t→∞ P (n 0 ,y 0 ) (Z(t) ∈ A | T 0 > t) , ∀A ∈ B(N * × R + ).
If y 0 ∈ [0, y * ], the assertion follows from Proposition 2.1. Let us now assume that y 0 > y * . We introduce the function t → v(t) defined by

dv dt (t) = D(y * -v) -b(y * ) ; v(0) = y 0 .
Let τ 0 (y 0 ) be defined by v(τ 0 (y 0 )) = y * . For t ≤ T 0 ∧ τ 0 (y 0 ), we have

Y (t) ≤ v(t). It follows from Proposition 2.1 that P (n 0 ,y 0 ) (Y (t) > y * |T 0 > t) = 0 , ∀t ≥ τ 0 (y 0 )
, which concludes the proof.

Theorem 5.2. For all n ∈ N * and any quasi-stationary distribution, the probability measure ν n is absolutely continuous with respect to the Lebesgue measure, with C ∞ -density on the set R + \ {0, y n }.

The proof of this theorem is obtained from the following lemmas. Recall that the set S has been defined in [START_REF] Dykhuizen | Selection in chemostats[END_REF].

Lemma 5.3. For all n ∈ N * , the measure ν n satisfies

dν n = c n 0 δ 0 + j∈N * c n j δ y j + u n (y) dy (11)
with c n j ≥ 0 for j ∈ N and u n is the density of the absolutely continuous part of ν n (so it is a non-negative integrable function) and it is a C ∞ function outside S ∪ {0}.

Proof. The measures ν n satisfy in the sense of distributions the differential equations

∂ y D(y * -y) -n b(y) ν n = b(y)n ν n-1 +(D+d(y))n ν n+1 -(b(y)+D+d(y))n ν n +λν n . (12)
Since the right hand side is a measure, we conclude by a recursive argument that the measures ν n have a C ∞ density on (S ∪ {0}) c . This also shows that these measures have no singular part, and the Lebesgue decomposition theorem gives relation [START_REF] Novick | Description of the Chemostat[END_REF]. Proof. Assume that there is an interval I not intersecting S ∪ {0} such that for some integer n ∈ N * , ν n (I) = 0 vanishes on I. From [START_REF] Smith | The Theory of the Chemostat[END_REF] we deduce that ν n+1 and ν n-1 also vanish at I. Therefore, we conclude recursively that ν j (I) = 0 for all j ∈ N * .

Lemma 5.5. The probability measure ν n is absolutely continuous on R + \ {0, y n } and its density is bounded on any compact set contained in R + \ {0, y n }.

Proof. Let us show that if j = n, ν n cannot have a Dirac mass in y j . We will do it by contradiction, so assume it does. Let f be a C ∞ function with compact support containing y j and such that its support does not contain any other point of S ∪ {0} except y j . By using formula [START_REF] Novick | Description of the Chemostat[END_REF]

It is not difficult to construct a sequence of functions (f q : q ∈ N * ) contained in C ∞ with support in a fixed small enough neighborhood of y j and such that,

f ′ q (y j ) = 1; lim q→∞ sup y∈R + |f q (y)| = 0; sup y∈R + |f ′ q (y)| ≤ 1 ∀q; lim q→∞ |f ′ q (y)| = 0 ∀y = y j .
This leads to a contradiction, when we take f = f q in (13) and make q tend to infinity. Now, using again equation ( 12), we deduce easily the boundedness of the density of ν n outside a neighborhood of {y n , 0}.

Lemma 5.6. The probability measure ν n cannot have a Dirac mass in y n .

Proof. Assume it does. Then in a neighborhood of y n , by using (11) since c n j = 0 for j = 0, n and by writing c n = c n n , we can write

dν n = c n 0 δ 0 + c n δ yn + u n (y) dy .
Let f be a function C ∞ with compact support containing y n but that does not contain 0. We have

-f ′ (
+c n f (y n ) λ -(b(y n ) + D + d(y n )) n .
We now construct a sequence (f q : q ∈ N * ) of C ∞ functions with support in a fixed small enough neighborhood of y n such that for some constant C ′ ,

f q (y n ) = 1 ∀q ; sup y∈R + |f q (y)| = 1 ; lim q→∞ |f q (y)| = 0 ∀y = y n and sup y∈R + |y -y n ||f ′ q (y)| ≤ C ′ ∀q ; lim q→∞ (y -y n )f ′ q (y) = 0 ∀y = y n .
Such a sequence can be easily constructed.

Recall that ν n-1 and ν n+1 are absolutely continuous with C ∞ density near y n . We conclude that if

c n = 0 λ = (b(y n ) + D + d(y n )) n .
In the case d is constant, it is known that the eigenvalue λ satisfies λ = κ n (D + d) < D + d, and so it is strictly less than (b(y n ) + D + d(y n )) n and we obtain a contradiction.

If d is not constant, the proof of the contradiction is more intricate.

Coming back to Equation ( 12) for u n in a neighborhood of y n but outside that point we get

∂ y G n (y) u n (y) = f n (y) G n (y) u n (y) + h n (y) with (see Lemma 3.1) G n (y) = D(y * -y) -n b(y) = β n (y -y n ) + O (y -y n ) 2 ,
with β n < 0, and

f n (y) = n b(y n ) -b(y) + d(y n ) -d(y) G n (y) = O(1) ,
Proof. In the proof we will assume y 0 < y ′ (the case y 0 ∈ (y ′ , y 1 ) is shown similarly). Let β > 0 be smaller than min(y 0 , y ′y 0 )/2 and let t > 0 be fixed. Then, there is ǫ = ǫ(n 0 , β) such that the following event has a strictly positive probability:

-On the interval time [0, ǫ] there are exactly n 0 -1 deaths and there is no other jump of N, and so N(t) decreases from n 0 to N

(ǫ) = 1; -Y (ǫ) belongs to (Y (0) -β, Y (
dY (t) dt = D(y * -Y (t)) -b(Y (t)) is bounded below by D(y * -y ′ ) -b(y ′ ). Take t = (y ′ -Y (0) + β)/(D(y * - y ′ ) -b(y ′ ))
, let us see that the number t(y ′ ) = ǫ + t′ makes the job. In fact, we have ensured that in a time smaller or equal to t(y ′ ) we have attained {m} × [y ′γ, y ′ + γ]. For any time bigger than t(y ′ ) it suffices to modify slightly the above argument and allow a sequence of jumps up to the moment that Y (t) has negative derivate and in this way we can postpone the time of attaining the set {m} × [y ′γ, y ′ + γ] from t(y ′ ) to a prescribed time t > t(y ′ ).

Proposition 5.10. For all n ∈ N * , the probability measure ν n has a support dense in (0, y 1 ).

Proof. Denote by n ⊗ ν n the probability measure defined on N × R + by n ⊗ ν n ({m} × B) = δ n (m)ν n (B) for all B ∈ B(R + ). From Lemma 5.9 and the quasi-stationary distribution property (8) we have Then, the result follows.

6 Bound on the asymptotic survival rate Proof. In Section 3 in [START_REF] Collet | Quasi-stationary distributions for structured birth and death processes with mutations[END_REF], it was shown that this extinction rate is an eigenvalue of the dual problem associated with the probability measure

L † ν = -λν.
For simplicity, we will prove inequality (14) in case of n = 1. The general case proved in a similar way is left to the reader. We introduce the notation

  y * then it remains in [0, y * ] forever proving the invariance.For the initial condition N(0) ∈ N * and Y (0) ∈ R + \ [0, y * ] we have that at time t = (Y (0)y * )/ b(y * ) the process has already attained the invariant set [0, y * ] × R + or became extinct.

  Let B(N × R + ) be the class of Borel sets of N × R + and B(R + ) the class of Borel sets of R + . For D ∈ B(N × R + ) we put by T D = inf{t ≥ 0 : Z(t) ∈ D} the hitting time of D by the process, with the usual convention ∞ = inf ∅.

Lemma 3 . 1 .

 31 Consider the equation G n (y) = 0 with G n (y) := D(y *y)n b(y) .

Proposition 5 . 1 .

 51 d is bounded above by d * < ∞, we now choose C > D + d * and t = 1 and apply Theorem 4.2 in [2]. In the case d unbounded, we choose C >log Q and apply Theorem 4.2 in [2]. 5 Properties of the Quasi-Stationary Distributions Any quasi-limiting distribution has support in N * × [0, y * ].

Lemma 5 . 4 .

 54 Let I be an open interval included in (S ∪ {0}) c . If there exists n ∈ N * such that ν n (I) = 0, then ν j (I) = 0 for all j ∈ N * .

  we get, -f ′ (y j ) D(y * -y j )n b(y)f ′ (y) D(y * -y)n b(y) ψ n (y)dy = f (y) b(y) n dν n-1 (y) + (D + d(y)) n dν n+1 (y) + f (y) λ -(b(y) + D + d(y)) n dν n (y) .

  y) D(y *y)n b(y) u n (y) dy = f (y) b(y) n dν n-1 (y) + (D + d(y)) n dν n+1 (y) + f (y) λ -(b(y) + D + d(y)) n u n (y) dy

P

  n⊗νn (N(0) = m, Y (0) ∈ (y ′ -γ, y ′ +γ) = e -λt P n⊗νn (N(t) = m, Y (t) ∈ (y ′ -γ, y ′ +γ) = e -λt P (n,y 0 ) (N(t) = m, Y (t) ∈ (y ′ -γ, y ′ +γ))dν n (y 0 ) > 0.

Theorem 6 . 1 .

 61 Let λ be the exponential extinction rate associated with a quasi-stationary distribution ν. Then λ < inf n (n(b(y n ) + D + d(y n )).(14)

G

  (y) = D(y *y) -b(y) H(y) = D + d(y) The quasi-stationary distribution equation for n = 1 is given by d dy G(y) u 1 (y)b(y) + H(y) u 1 (y) + 2H(y) u 2 (y) + λ u 1 (y) = 0 . Let a ∈]0, y 1 [. We have for y ∈]0, y 1 [ d dy G(y) u 1 (y) e y a (b(s)+H(s)-λ)/G 1 (s)ds = e y a (b(s)+H(s)-λ)/G(s)ds 2 H(y) u 2 (y) , and integrating between a and y we get G(y) u 1 (y) e y a (b(s)+H(s)-λ)/G(s)ds -G(a)u 1 (a) s)+H(s)-λ)/G(s)ds H(σ) u 2 (w) dw ≥ 0 . Therefore u 1 (y) ≥ G(a)u 1 (a) G(y) e -y a (b(s)+H(s)-λ)/G(s)ds . Let us study more carefully the quantity e -y a (b(s)+H(s)-λ)/G(s)ds . We recall that G only vanishes at y 1 and that G is decreasing since b is increasing. By a simple computation one gets b(s) + H(s)λ)/G(s) = b(y 1 ) + H(y 1 )λ (sy 1 ) G ′ (y 1 ) + O(1) , s) + H(s)λ)/G(s)ds = b(y 1 ) + H(y 1 )λ G ′ (y 1 ) log |yy 1 | + O(1) .

Finally, since G(y 1 ) 1 G 1 ,

 111 = 0, and G ′ (y 1 ) < 0, we get for y ≤ y (y)= (yy 1 ) G ′ (y 1 ) + O (yy 1 ) 2 = |yy 1 | |G ′ (y 1 )| + O(|yy 1 |) . Therefore u 1 (y) ≥ G(a)u 1 (a) G(y) |yy 1 | -(b(y 1 )+H(y 1 )-λ)/G ′ (y 1 ) e O(1) ≥ G(a)u 1 (a) |yy 1 | |G ′ (y 1 | |yy 1 | -(b(y 1 )+H(y 1 )-λ)/G ′ (y 1 ) e O(1) .As we have u 1 (a) > 0 by Theorem 5.7, the integrability of u 1 on [0,y 1 ] implies 1 + b(y 1 ) + H(y 1 )λ G ′ (y 1 ) <we finally get λ < b(y 1 ) + H(y 1 ) .

  0) + β); -On the interval of time [ǫ, ǫ + t] there is no jump of N (no birth and no death); -On the interval of time [ǫ + t, ǫ + t + β] there are exactly m -1 births and no other jump when m > 1, there is no jump if m = 1, or there is a unique death and no other jump when m = 0;

-|Y (ǫ + t) + β) -Y (ǫ + t)| < γ/2.

For t ∈ [ǫ, ǫ + t) we have N(t) = 1, then in this interval of time and before the process Y (t) has attained y ′ , the derivate

and recall that h n (y) is C ∞ near y n . The only solution which is integrable near y n is given by

y yn e -s yn fn(w)dw h n (s) ds .

If h n > 0 on a subset of positive measure of a small neighborhood of y n , we have u n < 0 which is a contradiction. Therefore, h n must vanish on both sides of y n . By the above result on the support, we conclude that u n vanishes in a neighborhood of y n as well as all the ν j with j = n (see Lemma 5.4). In particular, if we consider the equation for ν n+1 in this neighborhood (see ( 12)), we get

Theorem 5.7. On (0, y 1 ), the density of ν n satisfies u n > 0 except perhaps in y n .

The proof uses two lemmas.

Lemma 5.8. If ν n+1 or ν n-1 has a support dense in (0, y 1 ) then u n can be 0 only in y n .

Proof. The function G n has a simple zero in y n . Assume that for z ∈ (0, y 1 ), z = y 1 , u n (z) = 0 (and G n (z) = 0). Computation as in the proof of Lemma 5.6 gives

The conclusion follows.

Let us show that the process is irreducible up to extinction and before y 1 .

Lemma 5.9. Starting from any initial condition on N * × (0, y 1 ) the process Z has dense support on N × (0, y 1 ), that is, ∀m ∈ N, ∀y ′ ∈ (0, y 1 ), ∀γ > 0, ∃t(y ′ ) > 0 such that ∀t > t(y ′ ), ∀(y 0 , n 0 ) ∈ N * × (0, y 1 ) : P (n 0 ,y 0 ) (N(t) = m, Y (t) ∈ (y ′ -γ, y ′ +γ)) > 0.