Catherine Baugé
email: catherine.bauge@unicaen.fr

Elise Duval

David Ollitrault

Nicolas Girard

Sylvain Leclercq

Philippe Galéra

Karim Boumédiene

Ea4652 Microenvironnement

Type II TGFβ receptor modulates chondrocyte phenotype

Keywords: TGFbeta receptors, chondrocytes, dedifferentiation/differentiation, Sp1, osteoarthritis, aging

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INTRODUCTION

Articular cartilage consists of an abundant extracellular matrix, which is produced by chondrocytes. These highly differentiated cells are responsible for the maintenance of the cartilage as a functional entity. Aging cause changes in the properties of this tissue, notably by modifying the matrix composition or the growth factor responsiveness (Kempson,1982 ;[START_REF] Thonar | Maturation-related differences in the structure and composition of proteoglycans synthesized by chondrocytes from bovine articular cartilage[END_REF][START_REF] Dozin | Response of young, aged and osteoarthritic human articular chondrocytes to inflammatory cytokines: molecular and cellular aspects[END_REF][START_REF] Loeser | Reduction in the chondrocyte response to insulin-like growth factor 1 in aging and osteoarthritis: studies in a non-human primate model of naturally occurring disease[END_REF][START_REF] Rosen | Differential effects of aging on human chondrocyte responses to transforming growth factor beta: increased pyrophosphate production and decreased cell proliferation[END_REF][START_REF] Guerne | Growth factor responsiveness of human articular chondrocytes in aging and development[END_REF]. During embryonic development, cartilage formation is initiated with condensation of mesenchymal cells, followed by progression of chondrocyte differentiation toward proliferation, prehypertrophy and hypertrophy.

During osteoarthritis (OA), an articular disease strongly correlated with ageing (Buckwalter and Mankin, 1998, van der Kraan et van den berg, 2008), chondrocytes undergo morphogenic alterations accompanied by profound biochemical changes including the loss of production of cartilage-specific macromolecules. For instance, type II collagen expression decreased wheareas type I collagen synthesis increases. Some of the alterations of chondrocyte phenotype can be reproduced in vitro. Passages induce morphogenic alterations of cells, reduction of type II collagen, and induction of type I collagen synthesis [START_REF] Benya | Independent regulation of collagen types by chondrocytes during the loss of differentiated function in culture[END_REF][START_REF] Archer | Phenotypic modulation in sub-populations of human articular chondrocytes in vitro[END_REF][START_REF] Pei | Expression of collagen type I, II and III in loose body of osteoarthritis[END_REF][START_REF] Mcdougall | Surface adhesion-mediated regulation of chondrocyte-specific gene expression in the nontransformed RCJ 3.1C5.18 rat chondrocyte cell line[END_REF][START_REF] Homicz | Effects of serial expansion of septal chondrocytes on tissue-engineered neocartilage composition[END_REF]. However, this model is not able to mimic others modifications undergone by OA chondrocytes, and notably the induction of type X collagen, specific of hypertrophy stage. On the other hand, the chondrocyte-specific phenotype can be recovered when the cells are cultured in three-dimensional culture system, such as alginate matrices [START_REF] Reginato | Formation of nodular structures resembling mature articular cartilage in long-term primary cultures of human fetal epiphyseal chondrocytes on a hydrogel substrate[END_REF][START_REF] Bonaventure | Reexpression of cartilage-specific genes by dedifferentiated human articular chondrocytes cultured in alginate beads[END_REF].

Several studies suggest that modifications of chondrocyte phenotype during OA result from alteration of Transforming Growth Factor β (TGFβ) signaling. Thus, the expression of TGFβ and its signaling molecules is altered with age, and the subsequent lack of TGFβ responsiveness might be at the root of OA development (Blaney- [START_REF] Davidson | Reduced transforming growth factor-beta signaling in cartilage of old mice: role in impaired repair capacity[END_REF]. Besides, TGFβ signaling is important for chondrogenesis. It promotes the differentiation of embryonic chick limb cartilage [START_REF] Schonherr | Extracellular matrix and cytokines: a functional unit[END_REF], and upregulates a number of molecules associated with prechondrogenic condensation [START_REF] Serra | Expression of a truncated, kinase-defective TGF-beta type II receptor in mouse skeletal tissue promotes terminal chondrocyte differentiation and osteoarthritis[END_REF][START_REF] Chimal-Monroy | Expression of N-cadherin, N-CAM, fibronectin and tenascin is stimulated by TGF-beta1, beta2, beta3 and beta5 during the formation of precartilage condensations[END_REF][START_REF] Grimaud | Recent advances in TGF-beta effects on chondrocyte metabolism. Potential therapeutic roles of TGF-beta in cartilage disorders[END_REF]. In addition, primary chondrogenesis derived from mesenchymal stem cells (MSCs) needs TGFß signals [START_REF] Reddi | Bone and cartilage differentiation[END_REF][START_REF] Andriamanalijaona | Differentiation potential of human muscle-derived cells towards chondrogenic phenotype in alginate beads culture[END_REF].

Members of TGFß family transduce their signal via two serine/threonine receptors, type I (TßRI) and type II (TßRII) receptors [START_REF] Wrana | Mechanism of activation of the TGF-beta receptor[END_REF]. Upon ligand-binding, TßRI is phosphorylated by TßRII, resulting in activation of intracellular proteins, named Smads. Receptor-regulated Smads (R-Smads: Smad2 and Smad3) are phosphorylated by TßRI. Phosphorylated R-Smads form heterodimers with common-partner Smad (Co-Smads: Smad4), and translocate into the nucleus where they bind to specific DNA sequences for activation of TGFß target gene transcription [START_REF] Shi | Mechanisms of TGF-beta signaling from cell membrane to the nucleus[END_REF]. Inhibitory Smads (I-Smads: Smad7) inhibit phosphorylation of R-Smads by competing with their binding to activated TßRI [START_REF] Nakao | Identification of Smad7, a TGFbeta-inducible antagonist of TGF-beta signalling[END_REF]. Recently, it was shown that besides signaling via TßRI (Activin-like kinase 5, ALK5), TGF-beta can also signal via ALK1, thereby activating Smad1/5/8 instead of Smad2/3 (Blaney- [START_REF] Davidson | Increase in ALK1/ALK5 ratio as a cause for elevated MMP-13 expression in osteoarthritis in humans and mice[END_REF][START_REF] Finnson | ALK1 opposes ALK5/Smad3 signaling and expression of extracellular matrix components in human chondrocytes[END_REF].

We hypothesised that the maintenance of chondrocyte phenotype, which is altered with age, required a functional TGFβ signalling. We examined the influence of members of TGFß signalling on chondrocyte phenotype. We showed that TßRII is implicated in the loss of TGFß responsiveness observed during chondrocytes passages. TßRII ectopic expression permits to restore TGFß responsiveness as well as to increase the expression of some matrix components. We also found that TβRII expression level is intimately linked to differentiation status of chondrocytes.

Finally, we established a relationship between TGFβ-induced effects on matrix genes, and expression level of Specific protein 1 (Sp1) transcription factor, which expression is decreased in the course of dedifferentiation process [START_REF] Duval | Asporin expression is highly regulated in human chondrocytes[END_REF] and is require to ensure constitutive expression of TβRII in chondrocytes [START_REF] Baugé | NFkappaB mediates IL-1betainduced down-regulation of TbetaRII through the modulation of Sp3 expression[END_REF].

MATERIAL AND METHODS

Reagents.

Reagents were supplied by Invitrogen (Fisher Bioblock Scientific, Illkirch, France) unless otherwise noted.

Mithramycin (Sigma-Aldrich, St. Quentin Fallavier, France) and TGFß1 (R&D Systems, Lille, France) were resuspended in PBS or PBS-HCl, respectively. Alginate was provided by Sigma-Aldrich, and oligonucleotides were supplied by Eurogentec (Angers, France).

Chondrocyte culture

Human articular chondrocytes (HAC) were prepared from femoral head of post-mortem individuals with no known joint diseases or after femoral head fractures (ages between 45 and 80 years with median age of 68). The causes of donor's death were multiple and all of them signed agreement forms in the context of a body donation program according to local legislations. Similarly, donors from arthroscopy signed the agreement for this study according to local ethical comitee (CPP). Chondrocytes were released by enzymatic digestion as previously described [START_REF] Baugé | Interleukin-1beta impairment of transforming growth factor beta1 signaling by down-regulation of transforming growth factor beta receptor type II and up-regulation of Smad7 in human articular chondrocytes[END_REF]. Cells were seeded at 4x10 4 cells/cm 2 and cultured in Dulbecco's modified Eagle's medium (DMEM) (4.5 g⁄L D-glucose) supplemented with 10% heat-inactivated foetal calf serum (FCS), 100 IU/ml penicillin, 100 µg/ml streptomycin and 0.25 µg/ml fungizone, in a 5 % CO2 atmosphere. For dedifferentiation, HAC were passaged up to five times (P5) (about 7-8 population doubling) with trypsin-EDTA after reaching confluency.

For re-differentiation studies, chondrocytes, which had underwent three passages (about 4-5 population doubling), were encapsulated in alginate beads. Cells were suspended in alginate at the density of 2.5x10 6 cells/ml, then slowly dropped into a 100 mM CaCl2 solution. After instantaneous gelation, the beads were allowed to polymerize further for a period of 10 min in CaCl2 solution. After three washes in PBS, they were finally placed in DMEM supplemented with 10% foetal calf serum. After 3 weeks, the beads were rinsed with PBS and dissolved in a solution composed of 55 mM sodium citrate and 25 mM EDTA. The suspension was incubated at 37°C until the beads were completely dissolved (approximately 10 min.). Centrifugation of the suspension at 800 g for 10 min allows separating the cells from their alginate matrix.

Culture of human bone marrow-derived stem cells

Human bone marrow-derived stem cells were collected from the same patients as donors used for HAC isolation.

The harvested cells were washed twice with α-MEM, and isolated by adherence over 24h to tissue culture plastic and were expanded in monolayer culture with α-MEM supplemented with glutamine, FGF (1 ng/ml), 10% FCS and antibiotics. After amplification by successive passages (until P5), the absence of hematopoietic markers (such as CD45) was checked by RT-PCR (data not shown). Thereafter, bone marrow-derived stem cells were directed toward chondrogenesis by culture in 3-D scaffold (alginate bead).

For chondrogenic differentiation, isolated human bone marrow derived stem cells were encapsulated in alginate beads, according to the same protocol than HAC, and cultured with a chondrogenic medium consisting of DMEM containing 100 nM dexamethasone, 50 µg/ml ascorbic acid-2-phosphate, 40 µg/ml proline, 1mM sodium pyruvate and Insulin Transferin Selenium supplement (ITS, Sigma), for 21 days and the media were changed twice a week.

Plasmids and transfection experiments

p3TPLux, pCMV5-TßRII, pEVR2-Sp1 expression vectors were described earlier [START_REF] Baugé | NFkappaB mediates IL-1betainduced down-regulation of TbetaRII through the modulation of Sp3 expression[END_REF]. Chondrocytes were transiently transfected by the calcium phosphate precipitation method as previously described [START_REF] Baugé | Interleukin-1beta impairment of transforming growth factor beta1 signaling by down-regulation of transforming growth factor beta receptor type II and up-regulation of Smad7 in human articular chondrocytes[END_REF]. After overnight transfection, the culture medium was replaced by DMEM containing 10% FCS, and maintained in culture for 24 h.

Luciferase assay

Luciferase assay was performed with Luciferase Assay kit from Promega (Charbonnières, France), as previously described [START_REF] Baugé | Interleukin-1beta impairment of transforming growth factor beta1 signaling by down-regulation of transforming growth factor beta receptor type II and up-regulation of Smad7 in human articular chondrocytes[END_REF].

Gene silencing experiments

1.5 µg of double strand siRNA targeting Sp1, provided from Santa-Cruz Biotechnology (through Tebu, Le Perray en Yvelines, France), were nucleofected in HAC, according to the manufacturer's protocol with human chondrocyte Nucleofector solution (Amaxa, Köln, Germany). Negative control siRNA known to not lead to specific degradation of any cellular message was also used. After 6 h of transfection, the culture medium was replaced with DMEM containing 10% FCS and the chondrocytes cultures were incubated for 24 h.

RNA extraction and real-time RT-PCR.

Total RNA from primary HAC cultures were extracted using Trizol. Following extraction, 1 µg of DNase-I treated RNA was reverse transcribed into cDNA in the presence of random hexamers (Applied Biosystems, Courtaboeuf, France) and Moloney murine leukaemia virus reverse transcriptase. The reaction was carried out at 37°C for 1h followed by a further 10-min step at 95°C. Amplification of the generated cDNA was performed by real-time PCR in Applied Biosystems SDS7000 apparatus with appropriate primers designed with Primer Express software. The relative mRNA level was calculated with the 2 -∆∆CT method.

Protein extraction and Western-blot.

Cells were rinsed, scrapped in RIPA lysis buffer supplemented with phosphatase and protease inhibitors. The extracts (30 µg protein) were subjected to fractionation in a 8 or 10% SDS-PAGE, transferred to PVDF membranes (Amersham Biosciences, Orsay, France), and reacted with rabbit aggrecan, TβRII or Sp1 polyclonal antibodies. Subsequently, membranes were incubated with anti-rabbit secondary peroxidase-conjugated antibody. The signals were revealed with SuperSignal West Pico Chemiluminescent Substrate (Pierce Perbio Science, Brébières, France) and exposed to X-ray film. The membranes were also reacted with anti β-actin to verify equal loading.

Synthesis and transfection of decoy oligonucleotides.

Sp1 decoy oligonucleotides sequences are: Forward 5'-ATTCGATCGGGGCGGGGCGAGC -3' and Reverse 5'-CGACGCCCCGCCCCGATCGAAT -3' (consensus sequences are underlined). This decoy oligonucleotide was previously shown to bind Sp1 transcriptional factor [START_REF] Baugé | NFkappaB mediates IL-1betainduced down-regulation of TbetaRII through the modulation of Sp3 expression[END_REF]. These oligonucleotides were annealed overnight and transfected into cells using the Oligofectamine reagent according to the manufacturer's instructions.

Nuclear extracts and EMSA

Nuclear extracts were prepared as described previously [START_REF] Dignam | Accurate transcription initiation by RNA polymerase II in a soluble extract from isolated mammalian nuclei[END_REF]. For EMSA, 5 fentomoles of (γ-32 P)ATP-labelled probes (Sp1 binding sequence) and 4 µg of nuclear extracts were incubated at room temperature for 15 min in binding buffer (20 mM Tris-HCl pH 7.5, 1 mM DTT, 100 mM NaCl, 5% bovine serum albumin, 0.1% Nonidet P-40, 10% glycerol, 25 µM ZnCl2) and 2 µg of poly(dI-dC).(dI-dC). Samples were resolved on native polyacrylamide gel and exposed for autoradiography.

Statistical analysis:

All experiments were repeated at least with 3 different donors, and similar results were obtained. Representative

RESULTS

Chondrocyte passages reduce the TGFß responsiveness and decrease TßRII expression.

In order to ascertain that subculturing chondrocytes led to the loss of cartilaginous phenotype in our hands, we monitored the decrease of type II collagen and aggrecan mRNA expression after successive passages. Their expressions were strongly decreased as soon as the first passage (about 2 cell doubling). The maximal reduction was obtained at P2 (approximatively 3 doubling) and was maintained at least until P5. At opposite, collagen type I and III mRNA levels were higher at P2 (suppl figure 1A). Therefore, we limited the following experiments to 2 or 3 passages (3-5 cell doubling).

To investigate chondrocyte responsiveness to TGFß during dedifferentiation process, we focused on the expression of Plasminogen Activator Inhibitor-1 (PAI-1), which is rapidly induced by TGFβ in chondrocytes [START_REF] Baugé | Interleukin-1beta impairment of transforming growth factor beta1 signaling by down-regulation of transforming growth factor beta receptor type II and up-regulation of Smad7 in human articular chondrocytes[END_REF]. Human articular chondrocytes which underwent three passages were incubated for 4 h with TGFβ1 (2 ng/ml).

Figure 1A shows that passages induced a significant decrease of PAI-1 induction by TGFß.

TGFß initiates its cellular action by binding to TßRII, which in turn activates TßRI. Thus, we investigated the effect of passages on mRNA expression of both receptors. Whereas TßRI transcripts were increased in comparison to primary chondrocytes (P0) (fig. 1B), TβRII is dramatically decreased following dedifferentiation (fig. 1C). This reduction of TßRII expression was also observed at protein level (fig 1D).

Passages increases the expression of the R-Smad, Smad3, and decreases that of the I-Smads, Smad7.

Since signalling by TGFß is mediated by Smads, we explored the potential alteration of Smads expression during chondrocyte dedifferentiation. Our result (fig 2) showed that serial passages of chondrocytes in serum supplemented medium did not modulate significantly Smad2 and 4 mRNA levels. Moreover, they increased Smad3 mRNA levels whereas they decreased the expression of the I-Smad, Smad7. These data suggest that the loss of TGFß responsiveness observed in dedifferentiated chondrocytes is independent of Smads expression level, since an upregulation of Smad3 and down-regulation of Smad7 is usually associated with activation of TGFß signalling.

Ectopic expression of TßRII restores TGFß responsiveness in passaged chondrocytes and increases expression of matrix genes.

To confirm the crucial role of TßRII expression alteration in the reduction of TGFß sensitiveness of passaged chondrocytes, we examined the effect of forced TßRII expression on the induction of PAI-1 by TGFß (fig 3A). HAC were transfected with a TβRII expression vector (pCMV5-TβRII) or its corresponding insertless vector (pCMV5) for 24h and further incubated with TGFβ. Dedifferentiation clearly causes reduced TGFβ-induced PAI-1 expression, which was restored by TβRII ectopic expression. In addition, transfection of TβRII expression vector in passaged chondrocytes induces an increased mRNA level the expression of aggrecan and collagen type II (x2-3) (fig 3B -C).

TßRII expression increases during chondrocyte differentiation

Since TßRII levels decreased upon dedifferentiation, we hypothesized that its expression could also be modulated during chondrocyte differentiation process. Therefore, we pursued by examination of TßRII expression in redifferentiating chondrocytes. For that, the TßRII mRNA level was evaluated prior dedifferentiation (P0), after dedifferentiation (P3) and after redifferentiation in alginate. TßRII mRNA level was increased after a 3 week-culture in alginate (fig 4A), time which permit to restore (at least partially) a chondrocyte phenotype in this system [START_REF] Duval | Asporin expression is highly regulated in human chondrocytes[END_REF].

Similar experiments were performed using human bone marrow-derived mesenchymal stem cells, which underwent chondrogenesis by culture in alginate microbeads (Duval E/Boumediene K, unpublished data; suppl data 2B). This system is well-known to induce chondrogenic differentiation from mesenchymal stem cells from different origins [START_REF] Andriamanalijaona | Differentiation potential of human muscle-derived cells towards chondrogenic phenotype in alginate beads culture[END_REF]. As shown in figure 4B, TßRII mRNA levels strongly increased with the culture time, and subsequently increased during differentiation of hBM derived-MSC into chondrocyte-like phenotype.

Sp1 is important for TβRII expression and the maintenance of chondrocyte phenotype

Given that Sp1 expression decreases during chondrocyte dedifferentiation and is restored by alginate culture [START_REF] Duval | Asporin expression is highly regulated in human chondrocytes[END_REF], we proposed that Sp1 expression and activity could increase during differentiation of stem cells. Sp1 mRNA level was raised during human bone marrow derived stem cells differentiation into chondrocytes (Fig 5A).

To define whether DNA binding activity of Sp1 is also increased in these conditions, EMSAs were performed using the consensus DNA binding site as a probe (fig 5B). One delayed protein-DNA complex was strongly enhanced by hBM-MSC culture in alginate beads after 72h or 96h. The binding activity of the complex to the probe was decreased in the presence of Sp1 antibody (and not in the presence of irrelevant antibodies, namely against cKrox or Sp3), indicating that this complex contains Sp1 factor.

Then, we investigated the effect of inhibition of Sp1 on TßRII expression. First, we blocked the binding activity of Sp1 family members using decoy oligonucleotides or mithramycin [START_REF] Jia | Molecular basis of the synergistic antiangiogenic activity of bevacizumab and mithramycin A[END_REF] Mithramycin also reduced TßRII protein expression. Interestingly, aggrecan expression was also decreased by mithramycin (figure 6D).

Finally, in figure 7, we showed that Sp1 ectopic expression is able to restore TßRII expression in dedifferentiated chondrocytes and counteracts inhibition of aggrecan expression induced by passages.

DISCUSSION

Modifications of chondrocyte phenotype occur with aging and during osteoarthritis. They can be mimicked, in part, in vitro by subculturing monolayer cultures of chondrocytes [START_REF] Homicz | Effects of serial expansion of septal chondrocytes on tissue-engineered neocartilage composition[END_REF][START_REF] Pei | Expression of collagen type I, II and III in loose body of osteoarthritis[END_REF][START_REF] Mcdougall | Surface adhesion-mediated regulation of chondrocyte-specific gene expression in the nontransformed RCJ 3.1C5.18 rat chondrocyte cell line[END_REF][START_REF] Duval | Asporin expression is highly regulated in human chondrocytes[END_REF]. This culture model provides a way to understand some of the molecular mechanisms responsible for chondrocyte senescence and dedifferentiation, and to elucidate the transition from young/healthy to old/osteoarthritic cartilage. Conversely, studies of mechanism of re-differentiation of chondrocyte or differentiation of mesenchymal cells into chondrocytes can permit to bring new insights in order to develop the therapeutic use of mesenchymal stem cells (MSC) in cartilage disorders. In this study, we showed that chondrocyte dedifferentiation is accompanied by TßRII expression alteration, whereas redifferentiation of passaged chondrocytes or differentiation of MSC into chondrocytes are accompanied by TßRII production. Moreover, we demonstrated that forced TßRII expression is able to restore totally or partially the expression of matrix genes, suggesting a key-role of this receptor and TGFß signal in the maintenance of chondrocyte phenotype.

Our in vitro experiments show the key-role of TßRII expression in chondrocyte phenotype maintenance. This result completes in vivo data which showed TßRII-deficient mice have a reduced proliferation of chondrocytes and an accelerated early hypertrophic differentiation relative to control [START_REF] Seo | Deletion of Tgfbr2 in Prx1-cre expressing mesenchyme results in defects in development of the long bones and joints[END_REF]. In addition, conditional transgenic mice in which a dominant negative form of TßRII is expressed, develop a hypoplastic cartilage [START_REF] Hiramatsu | Expression of dominant negative TGF-β receptors inhibits cartilage formation in conditional transgenic mice[END_REF], confirming that TßRII is required for normal cartilage development, and subsequently chondrocyte phenotype.

Chondrocyte dedifferentiation, which occurs with aging, is likely associated to alteration of cell responsiveness to TGFß by reduction of TßRII expression. Of interest, we have recently demonstrated that interleukin-1ß, a proinflammatory cytokine which is increased in OA process, rendered cells insensitive to TGFß by reducing TßRII expression [START_REF] Baugé | Interleukin-1beta impairment of transforming growth factor beta1 signaling by down-regulation of transforming growth factor beta receptor type II and up-regulation of Smad7 in human articular chondrocytes[END_REF]. Together, these results corroborate in vivo data, which indicated that TßRII expression is dramatically reduced in old mice (Blaney [START_REF] Davidson | Reduced transforming growth factor-beta signaling in cartilage of old mice: role in impaired repair capacity[END_REF] and during OA [START_REF] Boumediene | Decrease of cartilage transforming growth factor-beta receptor II expression in the rabbit experimental osteoarthritis--potential role in cartilage breakdown[END_REF][START_REF] Verdier | Immunohistochemical analysis of transforming growth factor beta isoforms and their receptors in human cartilage from normal and osteoarthritic femoral heads[END_REF], and confirmed the relationship between TßRII expression and the reduction of cell responsiveness to TGFß observed in OA and old cartilage. Concerning TßRI expression, it increases with passages, suggesting that this receptor is not involved in the reduction of cell responsiveness to TGFß in passaged chondrocytes. This result was unexpected since TßRI expression is reduced during experimental OA induced in mice by destabilization of the medial meniscus model (Blaney- [START_REF] Davidson | Increase in ALK1/ALK5 ratio as a cause for elevated MMP-13 expression in osteoarthritis in humans and mice[END_REF].

In addition, we examined the expression levels of TGFß signalling intracellular mediators, namely Smads. We showed that Smad3 expression is increased during chondrocytes dedifferentiation whereas that of Smad2 is not modified. Similar effects were obtained in vivo with aging (Blaney- [START_REF] Davidson | Reduced transforming growth factor-beta signaling in cartilage of old mice: role in impaired repair capacity[END_REF]. Smad3 acts as one important TGFß signalling pathway members to develop and/or maintain the phenotype of chondrocytes. Smad3, but not Smad2, stimulates primary chondrogenesis [START_REF] Furumatsu | Smad3 induces chondrogenesis through the activation of SOX9 via CREB-binding protein/p300 recruitment[END_REF]. In addition, Smad3 deficiency accelerates chondrocytes maturation and leads to osteoarthritis [START_REF] Yang | TGF-beta/Smad3 signals repress chondrocyte hypertrophic differentiation and are required for maintaining articular cartilage[END_REF][START_REF] Van De Laar | Mutations in SMAD3 cause a syndromic form of aortic aneurysms and dissections with early-onset osteoarthritis[END_REF]. Therefore, we speculate that the increased Smad3 expression observed during dedifferentiation and aging could be a compensatory mechanism to promote the chondrocyte phenotype. Moreover, these results obtained on Smad expression could be due to the loss of TGFß response since TGFß reduces Smad3 expression [START_REF] Mori | Modulation of endogenous Smad expression in normal skin fibroblasts by transforming growth factor-beta[END_REF][START_REF] Baugé | Modulation of transforming growth factor beta signalling pathway genes by transforming growth factor beta in human osteoarthritic chondrocytes: involvement of Sp1 in both early and late response cells to transforming growth factor beta[END_REF].

We also observed the TßRII expression during in vitro chondrogenesis. Three-dimensional scaffold was used to redifferentiate dedifferentiated chondrocytes or to differentiate MSC toward chondrogenic phenotype. This culture system was described earlier to permit the induction of chondrogenic markers [START_REF] Reginato | Formation of nodular structures resembling mature articular cartilage in long-term primary cultures of human fetal epiphyseal chondrocytes on a hydrogel substrate[END_REF][START_REF] Bonaventure | Reexpression of cartilage-specific genes by dedifferentiated human articular chondrocytes cultured in alginate beads[END_REF][START_REF] Andriamanalijaona | Differentiation potential of human muscle-derived cells towards chondrogenic phenotype in alginate beads culture[END_REF][START_REF] Lemare | Dedifferentiated chondrocytes cultured in alginate beads: restoration of the differentiated phenotype and of the metabolic responses to interleukin-1beta[END_REF][START_REF] Cai | Ectopic osteogenesis and chondrogenesis of bone marrow stromal stem cells in alginate system[END_REF][START_REF] Duval | Asporin expression is highly regulated in human chondrocytes[END_REF]. We showed that TßRII expression is induced during redifferentiation of subcultured chondrocytes and during chondrogenic process of stromal stem cells. In this regard, Mizuta and coll. found that expression of both TßRI and TßRII receptors was upregulated during periosteal chondrogenesis induced by exogenous TGFß1 treatment [START_REF] Mizuta | The spatiotemporal expression of TGF-beta1 and its receptors during periosteal chondrogenesis in vitro[END_REF]. Our data corroborate another in vitro study which proposes that TGFß signal transduction may be involved in the early stage of bone marrow-MSC chondrogenesis [START_REF] Huang | Temporal expression patterns and corresponding protein inductions of early responsive genes in rabbit bone marrow-derived mesenchymal stem cells under cyclic compressive loading[END_REF], and are consistent with recent studies, which demonstrated that TßRII down-regulation induced by retinoid acid treatment inhibits chondrogenesis of mouse embryonic palate mesenchymal cells [START_REF] Yu | All-trans retinoic acid inhibited chondrogenesis of mouse embryonic palate mesenchymal cells by down-regulation of TGF-beta/Smad signaling[END_REF] and that TGFß signalling is needed in the chondrogenic differentiation of MSC [START_REF] Longobardi | Effect of IGF-I in the chondrogenesis of bone marrow mesenchymal stem cells in the presence or absence of TGFbeta signaling[END_REF].

Finally, we investigated the role of Sp1 role in the regulation of TßRII expression and matrix production. This ubiquitous zinc-finger protein has earlier been proposed to regulate matrix production by adult articular chondrocytes maintained in primary and expansion culture [START_REF] Duval | Asporin expression is highly regulated in human chondrocytes[END_REF][START_REF] Dharmavaram | Detection and characterization of Sp1 binding activity in human chondrocytes and its alterations during chondrocyte dedifferentiation[END_REF][START_REF] Oh | Downregulation of transcription factor, Sp1, during cellular senescence[END_REF]. Furthermore, we have demonstrated loss of Sp1 gene expression as a function of chondrocyte passage number and that Sp1 protein synthesis is restored in part by re-differentiation in 3D-alginate culture [START_REF] Duval | Asporin expression is highly regulated in human chondrocytes[END_REF]. Here, we showed that the decrease of Sp1 induced by serial passaging of chondrocytes is responsible for TßRII reduction.

However, we cannot exclude that Sp1 acts indirectly. Sp1 might permit the re-expression of TßRII in dedifferentiated chondrocytes through Sox9, which can be induced by Sp1 [START_REF] Piera-Velazquez | Regulation of the human SOX9 promoter by Sp1 and CREB[END_REF]. Sox9 was first identified as a transcription factor which specifies the chondrocyte lineage and is absolutely required for complete cartilage differentiation [START_REF] Bi | Sox9 is required for cartilage formation[END_REF]. In our study, Sp1 was not able to counteract the decrease of type II collagen (data not shown) whereas it increases aggrecan mRNA level in passaged chondrocytes. Since it was previously established that TßRII gene expression occurred earlier than that of type II collagen gene expression during differentiation of bone marrow-MSC under dynamic compressive loading [START_REF] Huang | Temporal expression patterns and corresponding protein inductions of early responsive genes in rabbit bone marrow-derived mesenchymal stem cells under cyclic compressive loading[END_REF], we hypothesize that longer time of Sp1 overexpression could restore the aggrecan or type II collagen expression in dedifferentiated chondrocytes. In order to validate this speculation, it will be necessary to reproduce this experiment using stable transfection system, which could permit to maintain a high level of Sp1 during a long period.

In conclusion, we demonstrated that TßRII expression level is essential to maintain and/or promote chondrocyte phenotype, and established a key-role of Sp1 in this process. These findings may help in understanding the behaviour of aged chondrocytes. In addition, it brings original insights in the process of MSC differentiation into chondrocytes and subsequently permits to envisage innovative strategies to optimize the therapeutic use of chondrocytes or MSC in cartilage disorders. A: Primary HAC (P0) or passaged twice (P2) were transfected with TβRII expression vector (pCMV5-TβRII) or with the insertless plasmid (5 µg). After 15 h, the medium was changed for DMEM+2% FCS for 24 h and cells were incubated with TGFβ1 addition (2 ng/ml) for 4 h. PAI-1 expression was analyzed by real-time RT-PCR. B-C: HAC were cultured and transfected as above. 24 h later, col2A1 and aggrecan mRNA expression was analyzed and normalized with GAPDH level (B). Aggrecan and TβRII protein levels were also analysed by Western-blot (C).

FIGURE LEGENDS

  experiments are shown in the figures. Data are presented as the mean + SD. Statistical significance was determined by Student's t-test. P values less than 0.05 were considered significant. ***= P <0.001;**= P < 0.01; *= P < 0.05 versus controls.

  and showed that the loss of Sp1 binding to DNA inhibited TßRII mRNA level expression (fig 6A-B). Furthermore, we established that the specific reduction of Sp1 expression by RNA silencing, checked by Western-blot, reduced TßRII expression (fig 6C).

Figure 1 :

 1 Figure 1: Chondrocyte passages reduces the TGFß responsiveness and decreases TßRII expression.A: Primary (P0 : about 1-2 population doubling) or passaged (P1 (~ 2-3 population doubling), P2 (~ 3-4 population doubling) and P3 (~ 4-5 population doubling)) HAC were cultured for 5-6 days before incubation with TGFβ1 (2 ng/ml) for 4 h. Total RNA was isolated and analyzed by real-time RT-PCR, using PAI-1 and GAPDH primers. B-C: Total RNA and proteins from freshly isolated chondrocytes or from chondrocytes dedifferentiated by repeated passages were extracted. TßRI and TßRII mRNA levels were defined by real-time RT-PCR. D: TßRII protein level was analyzed by Western-blot.

Figure 2 :

 2 Figure 2: Passages up-regulates the R-Smad, Smad3, and down-regulates the I-Smads, Smad7. A-D: HAC were cultured as described in figure 1 and Smad2, 3, 4 and 7 mRNA levels were analyzed by real-time RT-PCR. Results are shown as histograms of relative expression versus GAPDH signal.

Figure 3 :

 3 Figure 3: Ectopic expression of TßRII restores TGFß responsiveness in passsaged chondrocytes and increases expression of matrix genes.

Figure 4 :

 4 Figure 4: TßRII expression increases during chondrocyte differentiation A: Primary chondrocytes (P0) or HAC dedifferentiated by 3 passages (P3) were cultured in monolayer or transferred into alginate beads for 21 days. Thereafter, TßRII mRNA level was analyzed by real-time RT-PCR. B: Human bone marrow-derived stem cells were cultured in alginate beads for different incubation-period and TßRII mRNA levels were assayed. Results are shown as relative gene expression versus GAPDH signal.

Figure 5 :

 5 Figure 5: Sp1 expression increases during chondrocyte differentiation A: Human bone marrow-derived stem cells were cultured in alginate beads for different incubation-period and Sp1 mRNA levels were assayed. Results are shown as histograms of relative gene expression versus GAPDH signal.B: Electrophoretic mobility shift assay (EMSA) reactions were performed using probe corresponding to Sp1 consensus DNA binding site, incubated with 4 μg nuclear extract from hBM-MSC cultured in alginate for 0h, 72h or 96h. EMSA reactions were also performed after incubation of nuclear extracts (from hBM cultured 72h in alginate) with Sp1, Sp3 and cKrox antibodies.

Figure 6 :

 6 Figure 6: Sp1 regulates TβRII expression.

  Figure 1:

ACKNOWLEDGMENTS

We would like to thank Dr. J. Massagué (Memorial Sloan-Kettering Cancer Center, New-York, USA) and Dr Suske (Institut fur Molekularbiologie and tumorforschung, Marburg, Germany) for providing pCMV5-TßRII and pEVR2-Sp1, respectively. This work was supported by a grant from European Community in the context of Genostem integrated project (6 th framework program -Contract grant number: 503161) and by Lions Club of Normandy. E Duval and D. Ollitrault were recipient of fellowship from "Conseil Régional de Basse-Normandie" and from "Ministère de l'Enseignement Supérieur et de la Recherche", respectively. CB conceived, carried out experiments, analyzed data and wrote paper. ED participated to data collection and analysis. DO, NG and SL participated to data collection. PG participated to data interpretation. KB conceived experiments, analyzed data and was involved in writing the paper. All authors had final approval of the submitted and published versions.

DISCLOSURE

The authors confirm that there are no conflicts of interest.

A: Primary HAC, transfected with a Sp1 consensus binding sequence (Sp1 decoy), were incubated in DMEM+2% FCS for 24 h. Thereafter, TβRII mRNA versus GAPDH level was determined by RT-PCR. B: Subconfluent cultures of chondrocytes were treated for 48 h in the presence or absence of mithramycin (100 and 150 nM). TβRII expression was analyzed at mRNA level by real-time RT-PCR. C: HAC were nucleofected with siRNA oligonucleotides targeted Sp1 or with control siRNA. Thereafter, the medium was replaced with DMEM+10% FCS for 24 h. Sp1 and TßRII protein levels were defined by Western-Blot. D: Chondrocytes were treated for 48 h in the presence or absence of mithramycin (100 and 150 nM). TβRII and aggrecan expression was analyzed at protein level by Western-Blot.

Figure 7: Sp1 regulates chondrocyte phenotype

A-B: HAC passaged twice (P2) were transfected with Sp1 expression vector (pEVR2-Sp1) or with the insertless plasmid (5 µg). After 15 h, the medium was changed for DMEM+2% FCS for 24 h. Total RNA was isolated and TßRII and aggrecan mRNA levels were analyzed and normalised with GAPDH level, in comparison to HAC with no passage (P0).

Suppl data 1:

Primary (P0) or passaged (until P5 ~ 6-7 population doubling) HAC were cultured in monolayer for 5-6 days.

Thereafter, COL1A1, COL2A1, COL3A1 and aggrecan mRNA levels were analyzed by real-time RT-PCR.

Suppl data 2:

A: P3 (~ 4-5 population doubling) chondrocytes were cultured in monolayer or transferred into alginate beads for 21 days. Thereafter, COL1A1, COL2A1, COL3A1 mRNA levels were analyzed by real-time RT-PCR. B: Human bone marrow-derived stem cells were cultured in alginate beads for 7 days and COL2A1 and aggrecan mRNA levels were assayed. Results are shown as histograms as relative gene expression versus GAPDH signal.


Pejman Peykani

Emran Mohammadi

Fatemeh Sadat

Seyed Esmaeili

Measuring Performance, Estimating Most Productive Scale Size, and Benchmarking of Hospitals Using DEA Approach: A Case Study in Iran

Keywords: Hospital Performance Evaluation, Data Envelopment Analysis (DEA), Health Care, Most Productive Scale Size (MPSS), Returns to Scale (RTS)

Background and Objectives:

The goal of current study is to evaluate the performance of hospitals and their departments. This manuscript aimed at estimation of the most productive scale size (MPSS), returns to scale (RTS), and benchmarking for inefficient hospitals and their departments.

Methods:

The radial and non-radial data envelopment analysis (DEA) approaches under variable returns to scale (VRS) assumption are applied for performance assessment of hospitals. Also, the MPSS model in DEA is employed to identify hospital with optimal scale size. Furthermore, the benchmarking for inefficient decision making units (DMUs) is introduced using the slack based measure (SBM) model.

Results:

In this manuscript, the DEA approaches are implemented at macro and micro levels in health care. At macro level, the performance of 15 Iranian hospitals is assessed and at micro level, the performance of 15 departments of one hospital is evaluated. It should be noted that the number of staff, the number of beds, location & infrastructures, and equipment & facilities were considered as the input variables and number of patients and number of surgeries were selected as output variables. According to the results, six hospitals at macro level and seven hospital departments at micro level were efficient. As a result, these hospitals and departments can be considered as a benchmark for other DMUs. Notably, only four hospitals at macro level and four hospital departments at micro level have the most productive scale size.

Conclusions:

The current study presents a functional pattern to managers at macro and micro levels in health care systems to better planning for capacity development and resource saving.

Introduction

Hospital is one of the most important parts in the health care network and the major share of health care from gross domestic product (GDP) is spent in hospitals. In other words, hospitals as one of the most important and largest providers of health care services, play a crucial role in health economics. Therefore, measuring the performance and productivity of hospitals is mandatory.

Data envelopment analysis (DEA) is one of the applicable tools that can be applied to evaluate the performance of hospitals. [START_REF] Aldalbahi | Methodological approach and theoretical concept for measuring hospital technical efficiency[END_REF][START_REF] Kohl | The use of data envelopment analysis (DEA) in healthcare with a focus on hospitals[END_REF][START_REF] Peykani | Fuzzy Data Envelopment Analysis: An Adjustable Approach[END_REF] DEA is one of the most effective and powerful methods widely used in many real-world applications. [START_REF] Peykani | Interval network data envelopment analysis model for classification of investment companies in the presence of uncertain data[END_REF][START_REF] Peykani | Window network data envelopment analysis: an application to investment companies[END_REF][START_REF] Peykani | Stock evaluation under mixed uncertainties using robust DEA model[END_REF][START_REF] Peykani | Utilizing robust data envelopment analysis model for measuring efficiency of stock, a case study: Tehran Stock Exchange[END_REF][START_REF] Peykani | A novel fuzzy data envelopment analysis based on robust possibilistic programming: possibility, necessity and credibility-based approaches[END_REF][START_REF] Peykani | Fuzzy data envelopment analysis approach for ranking of stocks with an application to Tehran stock exchange[END_REF] The initial idea of DEA approach was raised by Farrell [START_REF] Farrell | The measurement of productive efficiency[END_REF] and then developed by Charnes et al. [START_REF] Charnes | Measuring the efficiency of decision making units[END_REF] and Banker et al. [START_REF] Banker | Some models for estimating technical and scale inefficiencies in data envelopment analysis[END_REF] under constant returns to scale (CRS) and variable returns to scale (VRS) assumptions, respectively. [START_REF] Emrouznejad | A survey and analysis of the first 40 years of scholarly literature in DEA: 1978-2016[END_REF] It should be noted that so far, DEA has been applied in many studies to assess the performance of Iranian hospitals. [START_REF] Bahadori | The evaluation of hospital performance in Iran: a systematic review article[END_REF][START_REF] Mahdiyan | Hospitals' efficiency in Iran: A systematic review and meta-analysis[END_REF][START_REF] Rezaei | Iranian hospital efficiency: a systematic review and meta-analysis[END_REF] Table 1 summarizes the studies that applied DEA approach for performance assessment of hospitals and health care centers in Iran:

Data Envelopment Analysis (DEA)

In this section, the background of DEA models based on radial and non-radial approaches are introduced. Assume that there are n homogenous DMUs that each D M U j ( j = 1, ..., n ) uses m inputs

x ij (i = 1,..., m ) to produce s outputs y rj ( r = 1,..., s ) and k is the index of DMU under evaluation.

Radial Approach

In radial approach, DEA calculates the maximum proportional reduction in all inputs under input orientation (increment in all outputs under output orientation) to increase the efficiency of DMU under evaluation. It should be noted that the BCC model is a popular radial DEA model under variable returns to scale. [START_REF] Banker | Some models for estimating technical and scale inefficiencies in data envelopment analysis[END_REF] The input and output-oriented BCC models are presented as Models ( 1) and ( 2), respectively:

j =1,...,n j=1 j  0, j=1 j =1 (1) j =1,...,n j=1 j  0, j =1 j =1 (2)
Note that in output-oriented BCC model presented in Model (2), the efficiency of DMU under investigation is calculated by 1 * .

Non-Radial Approach

In non-radial approach in contrast with radial approach, DEA model reduces inputs under input orientation (increases outputs under output orientation) non-proportionally. In other words, non-radial DEA model consider the input excesses and the output shortfalls using slack variables. [START_REF] Tone | A slacks-based measure of efficiency in data envelopment analysis[END_REF] The input and output-oriented SBM models are introduced as Models ( 3) and ( 4), respectively: i =1,...,m j =1,...,n

j=1 s i  0, j=1 j =1
(3)

m i=1 x ik s i r =1,...,s j =1,...,n j=1 t r  0, j=1 j =1 (4) s r=1 y rk t r
It should be noted that the s and the t , respectively, represent the amount of i th input surplus i r and r th output shortages. Also, the efficiency of DMU under evaluation based on output-oriented SBM model that presented in Model (4), is calculated by 1 * .

Estimating Most Productive Scale Size (MPSS)

The type of returns to scale for each DMUs under VRS production possibility set (PPS) is a very important issue. [START_REF] Banker | Estimation of returns to scale using data envelopment analysis[END_REF] Because, the type of RTS presents the direction of rescaling needed for improving the efficiency.

If the RTS of DMU is an increasing returns to scale (IRS), the expansion of the DMU under investigation will be suggested. And if the RTS of DMU is a decreasing returns to scale (DRS), contraction of DMU will be suggested. Finally, if the RTS of DMU is CRS, it is believed that the DMU under investigation operates as the most productive scale size (MPSS). The concept of MPSS has been introduced by Banker [START_REF] Banker | Estimating most productive scale size using data envelopment analysis[END_REF] for the first time. Cooper et al. [START_REF] Cooper | Extensions and new developments in DEA[END_REF] presented Model (5) for estimating MPSS:

It should be stated that the DMU under evaluation is the MPSS, if and only if in the fractional

objective function  * = 1 . 65  ,   0 j  0, j = 1, ..., n j=1 j =1 j =1 (5)
 Max  = 

Real Case Studies and Numerical Results

In this section, the implementation of DEA approach using real-world data at macro level (hospitals) and micro level (departments of hospital) from Iranian health care system are presented.

Based on literature review [START_REF] Azreena | A systematic review of hospital inputs and outputs in measuring technical efficiency using data envelopment analysis[END_REF] , and expert opinions, inputs and outputs of DEA approach are selected. The input variables include the number of staff (doctors, nurses, and ancillary personnels), the number of beds, location & infrastructures, and equipment & facilities. Also, the output variables include number of patients and number of surgeries.

The Macro Level: Hospital

As previously discussed, the goal at macro level is to assess the performance of hospitals. To reach this goal, 15 Iranian hospitals are selected and data from these hospitals are gathered.

Summary of real-world data for inputs and outputs variables of these hospitals are presented in Table 2. Then, after collecting data, the radial and non-radial models under input and output orientation are run and MPSS and RTS for each hospital are estimated. The results of performance appraisal of hospitals are illustrated in Table 3. As seen in Table 3, 4 hospitals including H02, H10, H11, and H15 are the MPSS. In addition to these hospitals, H08 and H13 are also efficient. Finally, by applying input and output-oriented SBM models, input-excess, output-shortfall, and reference-set for 9 inefficient hospitals are proposed in Table 4: It is worth mentioning that the type of input and output variables is integer and the values that presented in Table 4, should be approximated to the nearest integer value.

The Micro Level: Hospital Departments

In this subsection, the performance of hospital departments as a micro level is evaluated. For Then, data from these departments were gathered and the summary of real-world data are introduced in Table 5: Now, by employing the DEA models that were proposed in Models (1) to (4), the efficiency score for each department are calculated. Additionally, type of RTS for each department as well as MPSS are estimated. The results of performance assessment of hospitals are shown in Table 6: As seen in Table 7, the results introduce a functional pattern for managers in health care systems to better planning for capacity development and resource saving. Since in most cases, it is easier for hospital managers to reduce inputs such as the number of staff than to increase outputs such as the number of patients. Consequently, the input-oriented model is recommended.

Managerial Insights, Conclusions, and Future Research Directions

Performance appraisal of hospitals is one of the major concerns of managers. The reason for this is the identification of hospitals with a desirable performance as a benchmark for inefficient hospitals. Undoubtedly, DEA is one of the most powerful approaches that can be applied to performance assessment, ranking and benchmarking in health care. In this research, the radial and non-radial DEA models including BCC and SBM are implemented for a real case study in Iran to assess the performance of hospitals and their departments. Also, the most productive scale size and returns to scale are estimated for hospitals and their departments. Finally, input-excess, outputshortfall, and reference-set for each of inefficient DMUs are proposed. For the future studies, hybrid and novel multiple criteria decision making (MCDM) approaches can be employed for performance appraisal of hospitals. [START_REF] Torkzad | Evaluating and prioritizing hospital service quality[END_REF][START_REF] Sedady | A novel MCDM model for prioritizing the renewable power plants' construction[END_REF][START_REF] Beheshtinia | A hybrid MCDM approach for performance evaluation in the banking industry[END_REF] Also, in cases where the number of hospitals is considerable, clustering or grouping approach can be applied to categorize hospitals in different groups so that in each group, there are at least one benchmark hospital and other related hospitals.

Moreover, uncertain DEA models such as fuzzy DEA, stochastic DEA, and robust DEA can be applied for performance measurement of hospitals in the presence of data uncertainty and ambiguity. [START_REF] Cheraghi | Integrated planning for blood platelet production: a robust optimization approach[END_REF][START_REF] Samani | A multilateral perspective towards blood network design in an uncertain environment: Methodology and implementation[END_REF][START_REF] Samani | A hybrid algorithm for a two-echelon locationrouting problem with simultaneous pickup and delivery under fuzzy demand[END_REF][START_REF] Hosseini-Motlagh | Robust and stable flexible blood supply chain network design under motivational initiatives[END_REF][START_REF] Hosseini-Motlagh | Blood supply chain management: robust optimization, disruption risk, and blood group compatibility (a real-life case)[END_REF]

  this purpose, 15 departments of hospitals including intensive care unit (ICU), intensive care unitopen heart (ICU-OH), neonatal intensive care unit (NICU), coronary care unit (CCU), post coronary care unit (Post CCU), very important person (VIP), kidney transplant, head and neck surgery, neurology and physical medicine, internal, orthopedics and urology, general surgery, heart, pregnancy and maternity, and pediatric, are considered.

Table 1 :

 1 The Application of DEA in Iranian Hospitals and Health Care Centers: A Literature Review

	DEA Approach	Application

  [START_REF] Hajialiafzali | Efficiency measurement for hospitals owned by the Iranian social security organisation[END_REF]

  [START_REF] Hatam | The efficiency of general public hospitals in Fars Province, Southern Iran[END_REF]

  [START_REF] Jandaghi | Efficiency evaluation of Qom public and private hospitals using data envelopment analysis[END_REF]

  [START_REF] Ketabi | Efficiency measurement of cardiac care units of Isfahan hospitals in Iran[END_REF]

  [START_REF] Shahhoseini | Efficiency measurement in developing countries: application of data envelopment analysis for Iranian hospitals[END_REF]

  [START_REF] Khani | Measuring the relative efficiency of Ilam hospitals using data envelopment analysis[END_REF]

  [START_REF] Marnani | Performance assessment indicators: how DEA and Pabon Lasso describe Iranian hospitals' performance[END_REF]

  [START_REF] Sheikhzadeh | Public and private hospital services reform using data envelopment analysis to measure technical, scale, allocative, and cost efficiencies[END_REF]

  [START_REF] Yusefzadeh | The efficiency and budgeting of public hospitals: Case study of Iran[END_REF]

  [START_REF] Kalantary | A robust data envelopment analysis model for ranking: A case of hospitals of Tehran. Data Envelopment Analysis and Performance Measurement[END_REF]

  [START_REF] Lotfi | Various indicators for the assessment of hospitals' performance status: differences and similarities[END_REF]

  [START_REF] Mehrtak | Pabon Lasso and Data Envelopment Analysis: a complementary approach to hospital performance measurement[END_REF]

  [START_REF] Torabipour | Hospitals productivity measurement using data envelopment analysis technique[END_REF]

  [START_REF] Haji-Sami | Robust DEA under discrete uncertain data: An application for Iranian hospital emergency[END_REF]

  [START_REF] Jahangoshai Rezaee | Do geographical locations affect in hospitals performance? A multi-group data envelopment analysis[END_REF]

  [START_REF] Kheirollahi | Chance constrained input relaxation to congestion in stochastic DEA. An application to Iranian hospitals[END_REF]

  [START_REF] Rezapour | Technical efficiency and resources allocation in university hospitals in Tehran, 2009-2012[END_REF]

  [START_REF] Kakeman | Technical efficiency of hospitals in Tehran, Iran[END_REF]

  [START_REF] Kalhor | Factors affecting the technical efficiency of general hospitals in Iran: data envelopment analysis[END_REF]

  [START_REF] Nabilou | The productivity and its barriers in public hospitals: case study of Iran[END_REF]

  [START_REF] Rabbani | A bootstrap interval robust data envelopment analysis for estimate efficiency and ranking hospitals[END_REF]

  [START_REF] Rezaei | Do Hospitals Affiliated With the Kurdistan University of Medical Sciences Perform Efficiently? Non-parametric Data Envelopment Analysis[END_REF]

  [START_REF] Abadi | The use of resource allocation approach for hospitals based on the initial efficiency by using data envelopment analysis[END_REF]

  [START_REF] Ameryoun | Factor selection for service quality evaluation: a hospital case study[END_REF]

  [START_REF] Farzianpour | Determining the technical efficiency of hospitals in tabriz city using data envelopment analysis for 2013-2014[END_REF]

  [START_REF] Kakemam | Association between managers' competency and technical efficiency in general hospitals in Iran[END_REF]

  [START_REF] Kheirollahi | An input relaxation model for evaluating congestion in fuzzy DEA[END_REF]

  [START_REF] Mirmozaffari | Ranking of heart hospitals using cross-efficiency and two-stage DEA[END_REF]

  [START_REF] Mirmozaffari | Window analysis using two-stage DEA in heart hospitals[END_REF]

  [START_REF] Raei | Patterns of productivity changes in hospitals by using Malmquist-DEA Index: a panel data analysis (2011-2016)[END_REF]

  [START_REF] Shafaghat | Efficiency determination of hospitals of Shiraz university of medical sciences using simple and super efficiency DEA Models[END_REF]

  [START_REF] Alinezhad | Malmquist productivity index using two-stage DEA model in heart hospitals[END_REF]

  [START_REF] Bahrami | Data envelopment analysis for estimating efficiency of intensive care units: a case study in Iran[END_REF]

  [START_REF] Hadipour | How efficient rural healthcare centres work in Iran?[END_REF]

  [START_REF] Ketabi | Efficiency measurement of emergency departments in Isfahan, Iran[END_REF]

  [START_REF] Kiani | Efficiency evaluation of university hospitals in Bushehr province before and after the implementation of the health system development plan[END_REF]

  [START_REF] Pirani | Hospital efficiency measurement before and after health sector evolution plan in southwest of Iran: a DEA-Panel Data Study[END_REF]

Table 2 :

 2 Data Set for 15 Hospitals in Iran

	Hospital			Inputs		Outputs
	Code	Number of	Number of	Score of Location	Score of Equipment	Number of	Number of
		Staff	Beds	& Infrastructures	& Facilities	Patients	Surgeries
	H01	356	120	72	56	58734	1867
	H02	878	452	94	92	420434	19321
	H03	333	141	66	60	76489	3218
	H04	399	285	68	75	146468	3080
	H05	362	122	62	58	142352	555
	H06	186	76	45	52	48854	1800
	H07	315	125	59	47	78278	4223
	H08	175	94	35	53	97932	1155
	H09	223	89	66	47	108205	1752
	H10	76	20	35	24	60628	85
	H11	100	24	45	34	78954	432
	H12	212	78	46	39	96451	2185
	H13	95	30	33	35	46196	630
	H14	320	175	58	78	136582	5837
	H15	107	73	43	37	85139	2062

Table 3 :

 3 The Results of Hospitals Evaluation

	Hospital	MPSS	RTS	Radial	Non-Radial

Table 4 :

 4 The Benchmarking for Hospitals

				Input-Oriented			Output-Oriented
	Hospital		Input-Excess		Reference-Set	Output-Shortfall	Reference-Set
	Code
		s 1	s 2	s 3	s 4		t 1	t 2
	H01	205.70	59.98	31.53	25.70	H02, H10	96813.64	2801.79	H02, H11
	H02	0.00	0.00	0.00	0.00	H02	0.00	0.00	H02
	H03	126.38	50.64	21.39	24.92	H02, H10	95813.50	2377.58	H02, H11
	H04	131.66	161.94	18.92	34.78	H02, H10	65656.91	5518.48	H02, H15
	H05	103.84	3.88	13.60	18.55	H02, H10	14791.35	4202.05	H02, H11
	H06	83.11	10.02	3.06	16.72	H10, H15	62145.78	820.57	H02, H10, H11
	H07	66.48	12.07	11.31	8.37	H02, H10	80159.69	654.07	H02, H10, H11
	H08	0.00	0.00	0.00	0.00	H08	0.00	0.00	H08
	H09	56.36	28.34	16.80	8.03	H02, H11	22609.28	1548.66	H02, H11
	H10	0.00	0.00	0.00	0.00	H10	0.00	0.00	H10
	H11	0.00	0.00	0.00	0.00	H11	0.00	0.00	H11
	H12	48.45	10.84	4.56	7.58	H02, H10	17367.40	537.61	H02, H10, H11
	H13	0.00	0.00	0.00	0.00	H13	0.00	0.00	H13
	H14	44.36	19.10	3.84	28.97	H02, H15	41000.70	905.60	H02, H11, H15
	H15	0.00	0.00	0.00	0.00	H15	0.00	0.00	H15

Table 5 :

 5 Data Set for 15 Departments of a Hospital in Iran

				Inputs		Output	s
	Department
	Code	Number of	Number of	Score of Location	Score of Equipment	Number of	Number of
		Staff	Beds	& Infrastructures	& Facilities	Patients	Surgeries
	D01	31	26	82	63	2128	207
	D02	21	11	67	48	656	138
	D03	19	18	58	52	1024	12
	D04	22	26	67	69	2714	472
	D05	11	12	48	58	771	161
	D06	14	29	63	39	1208	725
	D07	15	27	64	48	725	518
	D08	25	50	88	93	4313	3255
	D09	30	38	70	59	1955	46
	D10	25	34	64	73	1771	92
	D11	54	78	88	96	5486	2921
	D12	30	62	84	47	4669	1967
	D13	37	60	93	94	4439	1794
	D14	15	16	34	29	2887	92
	D15	18	18	39	59	1116	115

Table 6 :

 6 The Results of Evaluating the Hospital Departments

	Department	MPSS	RTS	Radial	Non-Radial

Table 7 :

 7

Table 7 :

 7 The Benchmarking for a Hospital Departments

	Code	 *	Yes	No	DRS	CRS	IRS	*	1 *	*	1 *
	D01	2.01		•			•	0.59	0.65	0.52	0.27
	D02	2.38		•			•	1.00	1.00	1.00	1.00
	D03	3.17		•			•	0.78	0.34	0.71	0.04
	D04	1.42		•			•	0.74	0.82	0.66	0.57
	D05	2.22		•			•	1.00	1.00	1.00	1.00
	D06	1.93		•			•	1.00	1.00	1.00	1.00
	D07	3.23		•			•	0.89	0.60	0.85	0.46
	D08	1.00	•			•		1.00	1.00	1.00	1.00
	D09	2.95		•			•	0.50	0.51	0.47	0.05
	D10	2.71		•			•	0.57	0.49	0.50	0.09
	D11	1.00	•			•		1.00	1.00	1.00	1.00
	D12	1.00	•			•		1.00	1.00	1.00	1.00
	D13	1.32		•		•		0.88	0.93	0.76	0.71
	D14	1.00	•			•		1.00	1.00	1.00	1.00
	D15	2.63		•			•	0.89	0.39	0.78	0.38
				Input-Oriented				Output-Oriented
	Department Code		Input-Excess		Reference-Set	Output-Shortfall		Reference-Set
		s 1	s 2	s 3	s 4			t 1	t 2
	D01	15.64	8.76	46.04	31.67	D08, D14	98.71	1088.33	D02, D08, D14
	D02	0.00	0.00	0.00	0.00		D02	0.00	0.00		D02
	D03	4.00	2.00	24.00	23.00		D14	717.35	565.97	D02, D05, D08
	D04	5.80	5.92	26.51	32.31	D08, D14	0.00	700.10	D02, D08, D14
	D05	0.00	0.00	0.00	0.00		D05	0.00	0.00		D05
	D06	0.00	0.00	0.00	0.00		D06	0.00	0.00		D06
	D07	0.00	7.93	18.63	1.57	D05, D08, D14	1624.15	38.28	D05, D08, D14
	D08	0.00	0.00	0.00	0.00		D08	0.00	0.00		D08
	D09	15.00	22.00	36.00	30.00		D14	1830.20	1692.61	D08, D12, D14
	D10	10.00	18.00	30.00	44.00		D14	1735.80	1705.13	D05, D08, D14
	D11	0.00	0.00	0.00	0.00		D11	0.00	0.00		D11
	D12	0.00	0.00	0.00	0.00		D12	0.00	0.00		D12
	D13	9.02	4.05	14.87	47.26	D08, D12, D14	292.71	1339.98	D08, D11, D12
	D14	0.00	0.00	0.00	0.00		D14	0.00	0.00		D14
	D15	2.93	1.75	4.61	29.53	D08, D14	1680.33	202.58	D05, D08, D14

Competing Interests

The authors declare no competing interests.


Fausto Galetto
email: fausto.galetto@gmail.com

Independent Researcher

Syracuse_Collatz Conjecture: Comparison of two Markov approaches towards the proof

Keywords: Quality Methods, Numerical Methods, Hailstone Conjecture, SPQR

We analyse two probabilistic methods for the proof of the conjecture and we provide a comparison of the proofs, using the Reliability Integral Theory and the SPQR Principle.

Introduction

A problem posed by L. Collatz in 1937 (also called the 3x+1 mapping), states that the system of the two difference equations, involving natural numbers, [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF] given the initial condition y 0 (any integer positive number) arrives after some (n is a number not known in advance) "continued" iterations to the value y n =1. Numerical experiments confirmed the validity of the conjecture for extraordinarily large values of the starting integer y 0 : it always reached 1 for all numbers up to 5.48 10 18 . (Oliveira e Silva 2008) The system (1) can be reduced to a non-linear difference equation, as the following one [START_REF] Galetto | Proof of the Syracuse_Collatz Conjecture[END_REF] The numbers y k+1 of the sequence provided by the previous (Collatz) equations are sometimes named hailstone numbers. It is considered a very difficult problem to be solved, in spite of its very simple definition; they say that Erdős commented that "mathematics is not yet ready for such problems". In this document we compare two approaches, the 1 st based on the paper Quantifying the degree of average contraction of Collatz orbits. [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF], and the 2 nd based on the document Proof of the Syracuse_Collatz Conjecture. [START_REF] Galetto | Proof of the Syracuse_Collatz Conjecture[END_REF]; both of them use a Markov approach to make the proof "plausible". We will use excerpts of them to make clear the difference. Before that we provide a little of matrix algebra. Let u(k) an infinite dimensional row vector, with all entries u i (k)=0, but one entry u y (k)=1: it is a unit vector of vector space. The vector u(k) refers to the k-th iteration of a mapping T: the result of the mapping T to the vector u(k) is denoted u(k+1)=u(k)T. The vector u(k+1) is unit vector with all entries u j (k+1)=0, but one entry u y* (k+1)=1, where we have the subindexes y*y. The subindexes are according to (1): if u y (k)=1, then y=y k and the index y* of entry u y* (k+1)=1 of the vector u(k+1) has index y*=y k /2 IF y k is even, and y*=3y k +1 IF y k is odd. The mapping T is provided by an infinite-dimensional matrix P=[a ij ], named transition matrix (with infinite rows and columns); rows and columns are indexed by the natural numbers 1, 2, 3, 4, ..., n, n+1, ...; every a ij entry is 0, except , ........ [START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF] where the indexes i and j are given by ( 1). Accordingly we have u(k+1)=u(k)P (4)

In the figure 1 we show the transition matrix; the 3 by 3 matrix with rows and columns indexed by the numbers 1, 2, 4, is highlighted due to its importance:  when the system is in the state 1, the next transition is to state 4: 1  4  when the system is in the state 2, the next transition is to state 1: 2  1  when the system is in the state 4, the next transition is to state 2: 4  2 All this means that when the system enters one of those 3 states [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF][START_REF] Galetto | Proof of the Syracuse_Collatz Conjecture[END_REF][START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF] it never leaves out of them, the system (or the process) circulates in the set [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF][START_REF] Galetto | Proof of the Syracuse_Collatz Conjecture[END_REF][START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF] forever. It is a "periodic process". state 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ……………………………… 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 4 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ……………………………… 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ……………………………… 6 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 10 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 12 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 18 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… 20 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ……………………………… … 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ……………………………… Figure 1. The transition matrix P [only a part is shown]

The matrix P can be partitioned into 4 submatrices, written simply as where P11 and P22 are square matrices, given more explicitly by (notice that P11 refers to the states 1, 2, 4)

It is important to notice that P 3 , the 3 rd power of the matrix P, is such that the submatrix [START_REF] Galetto | The SPQR («Semper Paratus ad Qualitatem et Rationem[END_REF] is the identity matrix; when the system reaches the set 1, 2, 4 of the states it remains there forever. It follows that = .

The ideas in Quantifying the degree of average contraction of Collatz orbits. arXiv:1612.07820.v1

The paper is very interesting. We use various excerpts from the paper. The authors say:

In this paper, we provide a novel argument to support the validity of the Collatz conjecture. To anticipate our findings, we shall demonstrate that the third iterate T°3 of the Collatz map admits three fixed points, 1, 2 and 4. These latter elements define the, supposedly unique, attracting cycle conjectured by Collatz. The third iterate map is naturally defined on the mod8 congruence classes of positive integers. We thus quantify the factor of relative compression or expansion, as follows the application of T°3, on each of the eight congruence classes obtained under the mod8 operation. In the second part of the paper, we show that orbits are on average bound to asymptotically shrink in size so heading towards the deputed equilibrium. We will further enhance the resolution of the measure by reducing to an arbitrary extent the degree of imposed coarse graining, i.e. working on the congruence classes mod8 m for any chosen m. Working in this generalised setting, we will prove that the average Collatz dynamics is contracting, for m large as sought, namely shrinking the congruence classes arbitrarily close to the singletons corresponding to each integer.

Markov processes based on congruence classes invariant under application of T have been previously considered in […] and revisited by Lagarias in his comprehensive survey on the (3x + 1) problem […]

. It is however the combined usage of (i) the third iterate map T°3, (ii) the representation of numbers in mod8 , (iii) the idea of employing a Markov chain constructed from T°3 via a suitably defined measure, that allows us to draw a rigorous bound for the contraction factor, which is not just heuristically guessed.

To guide the reader through the text we shall hereafter provide a schematic outline of the main steps involved in the analysis, by making explicit reference to specific key results.

 We will begin by defining the third iterate of the Collatz map hereby named S = T°3.  We will determine the action of S on integers expressed in mod8. We will obtain class-dependent, B(i; 8) (i = 0; : : : ; 7), expansion/contraction factors that exemplify the action of S, see Eqs. [START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF]. Working in this setting we will also show that 1; 2; 4 are the only fixed points of the deterministic map S. The subsequent analysis is targeted to showing that the trajectories of S are bound to converge on average to one of the above fixed points.  To this end we first introduce a finite states Markov chain which runs on the eight congruence classes B(i; 8) (i = 0; : : : ; 7). The transition probabilities are given by equation ( 8) and have been obtained using the Sinvariant measure  inv on the classes B(j; 8m), m 1 and j = 0; : : : ; 8 m -1.  The measure  inv is defined by equation ( 12) (or equivalently equation ( 14)). The invariance of the measure under S is proved in Theorem 5.  Since the transition probabilities are computed from the S-invariant measure, it is possible to draw conclusion on the iterates of S (namely its restriction in mod8) by iterating forward the Markov process. This observation follows from a straightforward application of the Chapman-Kolmogorov equation, as discussed in Proposition 9. The explicit form of the stochastic matrix Q* that characterises the introduced Markov chain is given in Proposition 10. The stochastic chain does not account for the specificity of 1; 2; 4, the equilibria of S.

It will hence allow us to elaborate on the out-of-equilibrium dynamics of S, prior (possible) convergence to the asymptotic Collatz equilibrium.  The stationary distribution of the Markov chain is computed and given by formula (29). Recall that by

iterating forward the Markov chain one can inspect the equilibrium dynamics of S, in its mod8 representation, see Proposition 9.  By using the expansion/contraction factors associated to each of the classes B(i; 8) (i = 0; : : : ; 7) one can show that the deterministic trajectories are on average contracting. This is substantiated by formula (30).  The analysis is generalised by working on the congruence classes mod8 m , for any m. By operating in this setting, we will proof that the average Collatz dynamics is contracting, for arbitrarily large m, i.e. shrinking the size of the congruence classes as sought. Remarkably, the estimated upper bound for the contraction factor is shown to be independent on m. Excerpt 1. From the paper "Quantifying … of Collatz orbits. arXiv:1612.07820.v1"

We give the information connecting excerpt 1 with the Introduction:

The mentioned mapping T is represented (in the Introduction) by the Matrix P. The mentioned third iterate of the Collatz map hereby named S = T°3 is represented (in the Introduction) by the Matrix P 3 . The infinite matrix P 3 is reduced to a matrix 8 by 8 Q*, using a congruence mod8 on the integer numbers. 8 equivalence classes are obtained B(i, 8) and indicated as B(0, 8), B(1, 8), B(2, 8), B(3, 8), B(4, 8), B(5, 8), B(6, 8), B(7, 8). The matrix Q* is stochastic as given in the excerpt 2 [rows and columns are indexed by the classes B(i, 8)] Excerpt 2. From the paper "Quantifying … of Collatz orbits. arXiv:1612.07820.v1"

The authors show that there is a steady state probability vector, named P stat , of being in the equivalence classes B(i, 8) P stat =(1/6; 1/12; 1/6; 1/12; 1/6; 1/12; 1/6; 1/12) (29) Excerpt 3. From the paper "Quantifying … of Collatz orbits. arXiv:1612.07820.v1" From P stat it is clear that the process does arrive into the set B(1, 8), B(2, 8), B(4, 8) BUT it does not stay there "forever" after its arrival. The process does not end in the set 1, 2, 4, that is, process does not end in the three fixed points, 1, 2 and 4. (as the authors say)! Therefore the authors are forced to prove that that the deterministic trajectories are on average contracting.

Eventually they write in the Conclusion

In this paper we have provided an analytical argument to support the validity of the so called Collatz conjecture, a long standing problem in mathematics which dates back to 1937. The analysis builds on three main pillars. In short, we (i) introduced the (forward) third iterate of the Collatz map (so to reduce the analysis of the period 3 cycle to a search for a fixed point) and considered the equivalence classes of integer numbers modulo 8; (ii) defined a Markov chain (based on a suitable non trivial measure) which runs on a set of finite states and whose transition probabilities reflect the deterministic map; (iii) showed that orbits are on average contracting, as follows strict bound that combines the visiting frequencies, as derived in the framework of the aforementioned stochastic picture, and the contraction/expansion factors associated to each transition among classes. Notice that the conclusion reached holds for any level of imposed coarse graining, i.e. by computing the visiting frequencies on the partition in mod8m classes, with m large as wished. Despite the measure introduced cannot be extended to weight individual singletons, we can proof that the Collatz dynamics is contracting on uniform partitions of the natural numbers in classes. These partitions can be refined to approximate singletons with suited accuracy, without eventually converging to them. Excerpt 4. From the paper "Quantifying … of Collatz orbits. arXiv:1612.07820.v1"

The ideas in Proof of the Syracuse_Collatz Conjecture. 2019. <hal-02048821> and Academia.Edu

Some of them have been given in the Introduction, in order to make comparable the ideas of T. Carletti, D. Fanelli [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF] with the ones of Galetto, F. [START_REF] Galetto | Proof of the Syracuse_Collatz Conjecture[END_REF]. Please see them there: the transition stochastic matrix P of figure 1, partitioned into 4 submatrices, = 0 where P11 and P22 are square matrices (noticing that P11 refers to the set 1, 2, 4 of the states), given more explicitly by the following matrix in the introduction.

The process is a "periodic" with period 3: when the system enters one of the 3 states [1, 2, 4] it never leaves the set 1, 2, 4, the system (or the process) circulates in the set 1, 2, 4 forever. P 3 , the 3 rd power of the matrix P, is such that the submatrix [see formula [START_REF] Galetto | The SPQR («Semper Paratus ad Qualitatem et Rationem[END_REF]] is the identity matrix; when the system reaches the set 1, 2, 4 of the states it remains there forever because the rectangular submatrix in the upper right corner has only 0 entries. The process is bound to enter the set S 1 =1, 2, 4 because the rectangular submatrix in the lower left corner has only one 1 entry [the other entries are all 0]. The "periodic process" circulating in the set S 1 =1, 2, 4 is ruled by the submatrix P 11 . The infinite set S 2 =3, 5, 6, 7, …. comprises all the other states. The transitions are given in figure 2. By using the theory given in the books [START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF][START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF] related to Reliability Integral Theory [RIT] one can find two vectors z 1 and z 2 defined, as follows,  z 1 is the vector of the (steady state) probabilities of entering into one of the states 1, 2, 4 [S 1 ], when there is a transition S 2 =3, 5, 6, ..., n, n+1, ....  S 1 =1, 2, 4.  z 2 is the vector of the (steady state) probabilities of entering into one of the states 3, 5, 6, ..., n, n+1, .... [S 2 ], when there is a transition from S 1 =1, 2, 4  S 2 =3, 5, 6, ..., n, n+1, ..... z 1 is by definition a three-dimensional row vector [0, 0, 1] related to the set S 1 ; see the figure 2. The system enters into the set S 1 only through the state 4.

Comparison of the findings by the ideas …

We rearrange the states [8 equivalence classes are obtained B(i, 8)] of the finite process described by the previous stochastic matrix Q* (given in the excerpt 2) as follows By using the theory given in the books [START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF][START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF] related to Reliability Integral Theory [RIT] one can find two vectors z 1 and z 2 defined, as follows,  z 1 is the vector of the (steady state) probabilities of entering into one of the UP, when there is a transition Down  UP.  z 2 is the vector of the (steady state) probabilities of entering into one of the Down states, when there is a transition from UP  Down. z 1 is a three-dimensional row vector related to the UP set (see the figure 3): z 1 0.27273 0.36364 0.36364 so that the MUT (Mean Up Time) is MUT=1.818182 z 2 is a five-dimensional row vector related to the Down set (see the figure 3 The two authors (T. Carletti, D. Fanelli [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF]) then consider a number 8 m of equivalent classes B(i, 8 m ), generated by the congruence mod8 m . By increasing m, one generates more and more classes [process states], always with the Collatz cycle 1; 2; 4. The two authors prove that there is a stochastic matrix Q(m) [a 8 m by 8 m matrix] which provides a stationary row vector P stat (m)=[a, b, a, b, …, a, b] solution of the equation P stat (m)=P stat (m)Q(m), where a=0.1667/8 m-1 and b=0.0833/8 m- Again we see that to "prove" the Collatz Conjecture one should prove that A SS1,2,4 =1 and MUT 1,2,4 =, from the above ideas ... In F. Galetto opinion, the Collatz Conjecture is not made plausible by T. Carletti and D. Fanelli [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF], because as m the probability that the process is in the Collatz cycle 1; 2; 4 tends to 0 and not to 1… Perhaps F. Galetto did not understand… Then the following remarkable conclusion … is in doubt …

The remarkable conclusion is therefore that the third iterate of the Collatz map is always contracting, when seen on the equivalence classes B(i; 8 m ), for m large as sought, and that the estimated bound for the contraction factor is independent on the classes index m. In other words, we can make the number of classes as large as wished (and consequently reduce their size so to approach the singletons with arbitrary accuracy), while still detecting a contracting deterministic dynamics, with a constant (independent on m.) bound for the rate of contraction. As previously remarked when the limit for m that goes to infinity is performed, the measure of the classes, and hence the singletons, converges to zero. Despite the fact the contracting factors stays constant for any, arbitrarily large m, it seems that we cannot rule out the existence of zero measure orbits that violate this constraint. Excerpt 8. From the paper "Quantifying … of Collatz orbits. arXiv:1612.07820.v1" Now let's go the F. Galetto method [START_REF] Galetto | Proof of the Syracuse_Collatz Conjecture[END_REF], as sketched in the section 3. See the figure 2, as well. The reader is asked to see the transition stochastic matrix P of figure 1, partitioned into 4 submatrices, = 0 where P11 refers to the set 1, 2, 4 of the states (named Collatz cycle by T. Carletti, D. Fanelli [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF]). The process is a "periodic" with period 3: when the system enters one of the 3 states of set 1, 2, 4 it never leaves the set, the system (or the process) circulates in the set 1, 2, 4 forever. P 3 , the 3 rd power of the matrix P, is such that the submatrix [see formula [START_REF] Galetto | The SPQR («Semper Paratus ad Qualitatem et Rationem[END_REF]] is the identity matrix; when the system reaches the set 1, 2, 4 of the states it remains there forever because the rectangular submatrix in the upper right corner has only 0 entries. The process is bound to enter the set S 1 =1, 2, 4 because the rectangular submatrix in the lower left corner has only one 1 entry [the other entries are all 0]. The "periodic process" circulating in the set S 1 =1, 2, 4 is ruled by the submatrix P 11 . The infinite set S 2 =3, 5, 6, 7, …. comprises all the other states. The transitions are given in figure 2. After entering S 1 the probability of being in the states 1, 2, 4 is given by a vector =[1/3, 1/3, 1/3] solution of the relationship =P 11 , which means that the mean recurrence time in each state is 3. z 2 is by definition an infinite dimensional row null vector [0, 0, 0, ...., 0, ...] related to the set S 2 ; see the figure 2. The system never enters into the set S 2 from S 1 . This means that  IF a person choose a number y 0  S 2 =3, 5, 6, ..., n, n+1, ....  and applies the rules then the Markov system enters the state y 0  S 2 for the first time,  it makes all the needed transitions within the set S 2 (figure 2)  until it goes into the set S 1 =1, 2, 4 (figure 2)  and it remains there, with probability vector : periodic process

Conclusion

Having applied the SPQR («Semper Paratus ad Qualitatem et Rationem») Principle, the author thinks that his probabilistic method is able to show the (probabilistic) proof of the Syracuse_Collatz Conjecture. On the contrary, the Collatz Conjecture (in spite of an interesting paper) is not made plausible by T. Carletti and D. Fanelli [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF], because as m the probability that the process is in the Collatz cycle 1; 2; 4 tends to 0 and not to 1… Therefore the mean recurrence time in each state is infinite. As a matter of fact to "prove" the Collatz Conjecture one should prove that A SS1,2,4 =1 and MUT 1,2,4 = ...