F Lenoir

C Hauck

S Mongellaz

P Fabrega

M. D Nitschke

J Esposti

Schwartzentruber

Emmanuel Martin

Noé Palmic

Sylvia Sanquer

Christelle Lenoir

Fabian Hauck

Cédric Mongellaz

Sylvie Fabrega

Patrick Nitschké

Mauro Degli Esposti

Jeremy Schwartzentruber

Naomi Taylor

Jacek Majewski

Nada Jabado

Robert Wynn

Capucine Picard

Alain Fischer

Peter Arkwright

Sylvain Latour
email: sylvain.latour@inserm.fr

CTP synthase 1 deficiency in humans reveals its central role in lymphocyte proliferation

published or not. The documents may come   L'archive ouverte pluridisciplinaire

characterized by an impaired capacity of activated T and B cells to proliferate in response to antigen receptor-mediated activation. In contrast, proximal and distal TCR signaling events and responses were only weakly affected by the absence of CTPS1. Activated CTPS1-deficient cells exhibited decreased levels of CTP. Normal T-cell proliferation was restored in CTPS1-deficient cells by expressing wild-type CTPS1 or by addition of exogenous CTP or its nucleoside precursor, cytidine. CTPS1 expression was found to be low in resting T cells, but rapidly upregulated following TCR activation. These results highlight a key and specific role of CTPS1 in the immune system by its capacity to sustain the proliferation of activated lymphocytes during the immune response. CTPS1 may therefore represent a therapeutic target of immunosuppressive drugs that could specifically dampen lymphocyte activation.

We initially studied two unrelated families (family 1 and 2) originating from the northwest region of England, whose four children suffered from severe and recurrent Epstein-Barr virus (EBV) infection, in whom known primary immunodeficiencies have been excluded [START_REF] Wynn | Treatment of Epstein-Barr-virus-associated primary CNS B cell lymphoma with allogeneic T-cell immunotherapy and stem-cell transplantation[END_REF] (Fig. 1a and Table 1). Four additional patients (family 3 to 5) originating from the same geographical area were identified thereafter (Methods). All patients had early onset of severe chronic viral infections, mostly caused by herpes viruses, including EBV and Varicella Zooster Virus (VZV) and, also suffered from recurrent encapsulated bacterial infections, a spectrum of infections typical of a combined deficiency of adaptive immunity (CID) [START_REF] Notarangelo | Functional T cell immunodeficiencies (with T cells present)[END_REF] (Table 1 and data not shown). Overall, the clinical phenotype is severe with 3 patients have died. Six of 8 patients have undergone hematopoietic stem cell transplantation. Of note, none of the patients had extra-hematopoietic manifestations (Table 1).

Immunological investigations showed that most of patients had variable lymphopenia which was exacerbated during infection episodes with inversed CD4:CD8 T-cell ratio, while other blood cell counts were usually normal (Extended Data Table 1 and data not shown). Their immunoglobulin levels were normal or elevated with increased IgG but low IgG2 levels with low antibody titers to Streptococcus pneumoniae. Further analyses were performed in patient P1. 2 showing naive CD4 + T-cell lymphopenia, increased numbers of effector memory T cells, low numbers of memory CD27 + B cells, a complete absence of both invariant T cell populations (CD3 + Vα24 + Vβ11 + ) iNKT and (CD3 + CD161 high Vα7.2 + ) MAIT cells, as well as an impaired PHA-and antigen-induced proliferation of peripheral blood mononuclear cells (PBMCs) (Extended Data Table 2).

To identify the gene defect underlying the immunodeficiency in these patients, we performed whole exome sequencing (WES) in three patients (P1.1, P1.2 and P2.1). Intersection of the genetic variations found in the three patients pointed to an unique common homozygous G to C mutation in the CTPS1 gene encoding the CTP synthase 1 at position 41475832 in chromosome 1 with an assigned rsID (rs145092287) in the dbSNP database (Fig. 1b and Extended Data Fig. 1a,b). CTPS1 encodes a 67-kDa protein containing a CTP synthetase domain and a glutamine amide transfer domain promoting the formation of CTP from UTP and glutamine [START_REF] Kursula | Structure of the synthetase domain of human CTP synthetase, a target for anticancer therapy[END_REF] . The identified mutation affects a splice donor site at the junction of intron 17-18 and exon 18 (IVS18-1 G>C) leading to the expression of an abnormal transcript lacking exon 18 (Extended Data Fig. 1b,c). This splice mutation was found to be deleterious since CTPS1 protein expression could not be detected in lysates of EBV-transformed B cells and T-cell blasts from patients (Fig. 1c, Fig. 2c and Extended Data Fig. 2). In contrast, CTPS2 was expressed normally in patient cell lysates. In the five affected families, all patients were homozygous for the IVS18-1 G>C mutation and all parents and tested healthy siblings were heterozygous carriers (Fig. 1a,b and data not shown). Sequencing of a cohort of 752 healthy individuals from the northwest of England gave an estimated frequency of homozygosity of 1:560,000. This represents more than a 10fold increase compared to the frequency estimated from available exome databases. WES data and analysis of polymorphic microsatellite markers in all patients revealed a common region of homozygosity of 1.1 Mb surrounding the IVS18-1 G>C mutation (Supplementary information). All these data were indicative of a founder effect. These observations led us to conclude that the immunodeficiency resulting from the CTPS1 mutation in these patients could be primarly associated with a T-cell immunodeficiency.

We next examined CTPS1 expression in normal tissues. CTPS1 mRNA expression was comparable between the different tissues, except for T cells in which CTPS1 expression was strongly up regulated after cell activation in response to TCR-CD3 and CD28 co-stimulation (Fig. 1d). Interestingly, in lysates from T-cell blasts and T cells from PBMCs, CTPS1 protein was almost undetectable (Fig. 2a-d). In contrast, CTPS2 expression was readily detected. Activation of T cells by anti-CD3 antibody or phorbol 12-myristate 13-actate (PMA) and ionomycin stimulations induced CTPS1 protein expression while activation with IL-2 and/or IL-15 resulted in only weak effect. Under the same experimental conditions, CTPS2 expression was also induced but to a lesser extent. In TCR-CD3-stimulated T-cell blasts, CTPS1 protein expression was enhanced from 12 hours and persisted for up to 96 hours as a consequence of CTPS1 gene transcription activation (Fig. 1d, inset and Fig. 2b).

As expected, no expression of CTPS1 was detected in T-cell blasts from the CTPS1deficient patient (P1.2) contrasting with detection of CTPS1 mRNA and suggesting protein instability (Fig. 2c and Extended Data Fig. 1c). These data indicate that T-cell activation through the TCR results in a rapid and sustained CTPS1 protein expression. Of note, in B cells activated by anti-BCR and CpG, IL-4 and CD40L or PMA and ionomycin, CTPS1 was also found to be upregulated (Fig. 2d and Extended Data Fig. 3a,b).

To further characterize the consequences of the CTPS1 deficiency in T cells, we investigated proximal T-cell activation signals as well as late responses. CTPS1-deficient T cells exhibited normal early and late responses with the exception of ERK1/2 phosphorylation and CD25 and CD69 upregulation which were found to be decreased (Extended Data Fig. 4). Basal and activation-induced cell death was also slightly increased (Extended Data Fig. 4g). These data suggest that CTPS1 deficiency had limited consequences in signaling downstream of TCR-CD3. Because the pool of CTP is potentially a limiting factor for DNA synthesis [START_REF] Fairbanks | Importance of ribonucleotide availability to proliferating T-lymphocytes from healthy humans. Disproportionate expansion of pyrimidine pools and contrasting effects of de novo synthesis inhibitors[END_REF][START_REF] Traut | Physiological concentrations of purines and pyrimidines[END_REF] , we carefully analyzed proliferation of CTPS1-deficient T-cells. In response to activation by antigens, anti-CD3 antibody or co-stimulation by anti-CD3 and anti-CD28 antibodies, CTPS1-deficient cells from three patients (P1.1, P1.2 and P2.2) failed to sustain proliferative responses as measured by [START_REF] Higgins | Regulation of human cytidine triphosphate synthetase 1 by glycogen synthase kinase 3[END_REF] H-thymidine uptake and CFSE or violet cell tracer dye dilution (resulting in a weak index of cell proliferation) (Fig. 2e,f and Extended Data Table 1 and Extended Data Fig. 5,6). Uptakes of 3 H-Uridine and 3 H-Cytidine were also found to be impaired in activated CTPS1-deficient T cells. This suggests that both RNA and DNA synthesis were affected (Extended Data Fig. 6). Defective proliferation of CTPS1deficient T cells was associated with a lack of cell cycle progression since a majority of cells were arrested in the G1 phase (Fig. 2g). Proliferation of CTPS1-deficient B cells activated by anti-BCR and CpG was also found to be defective whereas that of IL-2-activated NK cells seemed to be less affected (Fig. 2h and Extended Data Fig. 5b). Down-regulation of CTPS1 expression in control T cells, by lentiviral transduction of two distinct shRNA together with a GFP reporter gene, led to a specific decrease in the CD3mediated proliferation of GFP-positive cells (Fig. 3a). No changes in proliferation were detected in non-targeted GFP-negative cells or in cells targeted with a scramble shRNA. The diminished proliferation resulting from the inhibition of CTPS1 expression led to a selective cell growth disadvantage with a decreased number of GFP targeted cells over time (middle panel). A similar decrease in proliferation rate was also observed in the Jurkat T-cell line in which CTPS1 expression was down-regulated (Extended Data Fig. 7).

Together, these results indicate that CTPS1 deficiency causes a defect in T-cell proliferation in response to TCR-CD3 activation. To formally prove the causal relationship between CTPS1 deficiency and defective T-cell proliferation, we carried out reconstitution experiments with wild-type CTPS1 or by direct addition of CTP or its cytidine precursor that acts on CTP levels via the salvage pathway. Expression of ectopic CTPS1 in CTPS1deficient T-cells fully restored proliferation upon CD3 stimulation (Fig. 3b) and enabled cells to expand selectively as shown by the accumulation of GFP-positive cells expressing CTPS1 (Fig. 3c, left panels). No such effect was detected in CTPS1-deficient cells transduced with an empty vector or in control cells transduced with the CTPS1-containing vector.

Proliferation and CD25 expression of CTPS1-deficient cells also recovered to a normal level by addition of CTP or cytidine (Fig. 3d and data not shown). In contrast, addition of a mix of UTP, GTP and ATP or uracil, guanine and adenosine did not result in increased proliferation of CTPS1-deficient cells. To determine the influence of the CTPS1 defect on the de novo pyrimidine synthesis pathway, we measured the incorporation of carbon from [START_REF] Ben-Sahra | Stimulation of de novo pyrimidine synthesis by growth signaling through mTOR and S6K1[END_REF] C-aspartate into nucleic acids of activated CTPS1-deficient T cells, which is a specific assay for de novo pyrimidine synthesis [START_REF] Ben-Sahra | Stimulation of de novo pyrimidine synthesis by growth signaling through mTOR and S6K1[END_REF] (Fig. 3e and Extended Data Fig. 6 b,c). Incorporation of 3 Hthymidine and 3 H-uridine was analyzed in parallel as control of the global RNA and DNA synthesis. TCR-CD3 activation-mediated incorporation of 14 C-aspartate, [START_REF] Higgins | Regulation of human cytidine triphosphate synthetase 1 by glycogen synthase kinase 3[END_REF] H-thymidine and 3 H-uridine was significantly decreased in CTPS1-deficient T cells. Addition of exogenous CTP or cytidine that bypassed the de novo synthesis pathway restored incorporation of 3 Hthymidine but not of 14 C-aspartate in CTPS1-deficient cells, thus demonstrating that the de novo CTP synthesis pathway is impaired in the absence of CTPS1. Deazauridine, an analogue of UTP and a known inhibitor of CTP synthetase activity [START_REF] Mcpartland | Cytidine 5′-triphosphate synthetase as a target for inhibition by the antitumor agent 3-deazauridine[END_REF] completely blocked Tcell proliferation of control cells in response to CD3 activation without affecting proximal TCR-CD3-mediated responses, similar to results observed in CTPS1-deficient cells (Fig. 3f and data not shown). As expected, inhibition of T-cell proliferation by deazauridine was fully reverted by addition of CTP and partially by UTP, but not by ATP or GTP. Analysis of nucleotides pools in activated CTPS1-deficient T-cell blasts and CTPS1-deficient B/EBV cell lines revealed decreased levels of CTP, as also observed in activated normal cells treated with deazauridine (Fig. 3g,h). Defective CTPS1 expression or addition of deazauridine also led to reduced pools of ATP, GTP and UTP in activated T cells (Extended Data Fig. 8) suggesting interconnection in the nucleotide pools [START_REF] Qiu | Mycophenolic acid-induced GTP depletion also affects ATP and pyrimidine synthesis in mitogen-stimulated primary human T-lymphocytes[END_REF] . In contrast, CTP as well as ATP, GTP and UTP were found to be normal or increased in resting CTPS1-deficient T cells as the salvage pathway is predominent in quiescent cells [START_REF] Van Den Berg | The roles of uridine-cytidine kinase and CTP synthetase in the synthesis of CTP in malignant human T-lymphocytic cells[END_REF] . Expression of wild-type CTPS1 in CTPS1-deficient B/EBV cell lines restored levels of CTP comparable to control cells and confered to cells a selective cell growth advantage (Fig. 3h,i).

This study reveals a critical role for CTPS1 in promoting the proliferation of T cells following their activation. However, proliferation of B cells was also found to be dependent of CTPS1. This may directly participate to the susceptibility to encapsulated bacterial infections seen in CTPS1-deficient patients and account for the low titers of S. pneumoniae antibodies as it is a T-independent B-cell response. The role of CTPS1 in B cells coud be different or/and less important than in T cells. Of note, CTPS1-deficient B cells preserve an intact capacity to expand upon transformation by EBV and patients had normal Ig levels and/or elevated IgG. Decreased expansion of NK cells and low numbers of iNKT and MAIT cells might also contribute to the CTPS1 immunodeficiency as these cells have been proposed to play a role in a broad range of immune responses including anti-EBV immunity [START_REF] Bourhis | MAIT cells, surveyors of a new class of antigen: development and functions[END_REF][START_REF] Vivier | Functions of natural killer cells[END_REF][START_REF] Chung | Innate immune control of EBV-infected B cells by invariant natural killer T cells[END_REF][START_REF] Brennan | Invariant natural killer T cells: an innate activation scheme linked to diverse effector functions[END_REF] . The finding that CTPS1-deficiency causes no other significant clinical consequences favors a redundancy with CTPS2 activity in other cell lineages and tissues. Interestingly, pyrimidine pools including CTP have been previously shown to be strongly expanded in PHA-stimulated T cells via de novo pathways including increased CTPS activity [START_REF] Fairbanks | Importance of ribonucleotide availability to proliferating T-lymphocytes from healthy humans. Disproportionate expansion of pyrimidine pools and contrasting effects of de novo synthesis inhibitors[END_REF][START_REF] Van Den Berg | Cytidine triphosphate (CTP) synthetase activity during cell cycle progression in normal and malignant T-lymphocytic cells[END_REF] . The induction of CTPS1 expression in activated T cells reported here thus appears as the major determinant of CTP pool increase. Indeed, proliferation of CTPS1deficient T cells was restored to normal level by addition of CTP. The exact mechanism(s) by which TCR signaling induces a rapid expression and activation of CTPS1 in T cells remains to be determined, although we showed that the ERK pathway is required as well as tyrosine phosphorylation signals (Extended data Fig. 3c). It is interesting to note that T cell differentiation does not appear to be severely impaired by CTPS1 deficiency, suggesting that CTP pools in thymocytes may originate from the nucleoside salvage pathway and/or the CTPS2 activity [START_REF] Fairbanks | Importance of ribonucleotide availability to proliferating T-lymphocytes from healthy humans. Disproportionate expansion of pyrimidine pools and contrasting effects of de novo synthesis inhibitors[END_REF][START_REF] Toy | Requirement for deoxycytidine kinase in T and B lymphocyte development[END_REF][START_REF] Marijnen | Studies on the incorporation of precursors into purine and pyrimidine nucleotides via 'de novo' and 'salvage' pathways in normal lymphocytes and lymphoblastic cellline cells[END_REF][START_REF] Austin | Nucleoside salvage pathway kinases regulate hematopoiesis by linking nucleotide metabolism with replication stress[END_REF] . Notably though, CTPS1 activity is critical for the intense cell division induced by antigenic stimulation as exemplified by massive proliferation and expansion of CD8 + T cells during viral infections [START_REF] Murali-Krishna | Counting antigen-specific CD8 T cells: a reevaluation of bystander activation during viral infection[END_REF][START_REF] Hislop | Cellular responses to viral infection in humans: lessons from Epstein-Barr virus[END_REF] .

In the absence of CTPS1, we showed that de novo pyrimidine synthesis pathway is impaired but not totally abrogated. This residual activity is likely dependent of CTPS2. Recently, the de novo pyrimidine synthesis pathway was shown to be dependent on post-transcriptional regulation by mTORC1 and S6 protein (S6K) kinases that activate the first enzymatic steps required for pyrimidine synthesis [START_REF] Ben-Sahra | Stimulation of de novo pyrimidine synthesis by growth signaling through mTOR and S6K1[END_REF][START_REF] Huang | De novo synthesis of pyrimidine nucleotides; emerging interfaces with signal transduction pathways[END_REF][START_REF] Robitaille | Quantitative phosphoproteomics reveal mTORC1 activates de novo pyrimidine synthesis[END_REF] . Thus, distinct regulatory mechanisms control de novo pyrimidine synthesis. Based on the present study, CTPS1-mediated tuning of CTP synthesis in lymphocytes appears to be a key element in enabling adaptive immune responses. If CTPS1-specific inhibitors can be designed, they would potentially be highly specific immunosuppressive drugs able to inhibit auto-or allogenic-specific T and B cell responses without additional toxicity given the lymphocyte specificity of the CTPS1deficiency phenotype. In conclusion, our results provide the first in vivo evidence of a role of the de novo pyrimidine synthesis pathway as a critical step for proliferation of T and B lymphocytes when activated by antigens.

METHODS

Cohorts of patients

Beside the four initially identified patients, four additional patients were identified by screening 10 patients (9 families) originating from the northwest of England with severe chronic viral infections, mostly caused by herpes viruses, including EBV and VZV. Furthermore, 24 patients (24 families) originating from different geographical areas with the same phenotype were also tested for all exons of CTPS1 in order to identify other mutations and none was found to be a carrier of CTPS1 mutations.

Exome sequencing and analysis

Exome capture was performed according to the manufacturer's protocol using the Illumina TruSeq exome enrichment kit and sequencing of 100 bp paired end reads on an Illumina HiSeq. Approximately 10 Gb of sequence were obtained for each subject such that 90% of the coding bases of the exome defined by the consensus coding sequence (CCDS) project were covered by at least 10 reads. Adaptor sequences and quality trimmed reads were removed using the Fastx toolkit (http://hannonlab.cshl.edu/fastx_toolkit/) and a custom script was then used to ensure that only read pairs with both mates present were subsequently used. Reads were aligned to hg19 with BWA31, and duplicate reads were marked using Picard (http://picard.sourceforge.net/) and excluded from downstream analyses. Single nucleotide variants (SNVs) and short insertions and deletions (indels) were determined using samtools (http://samtools.sourceforge.net/) pileup and varFilter32 with the base alignment quality (BAQ) adjustment disabled, and they were then quality filtered to require at least 20% of reads supporting the variant call. Variants were annotated using both ANNOVAR33 and custom scripts to identify whether they affected protein coding sequences, and whether they had previously been seen in the dbSNP, the 1000 Genomes data set (1092 genomes), or in approximately 2073 exomes previously sequenced at our center. A variant detected in a patient was considered to be a candidate mutation if it had not been reported or had a frequency below 0.001% in the three databases indicated above. At the time the homozygous G>C mutation in CTPS1 (at position chr1:41475832) was identified by WES in the three patients (P1.1, P1.2 and P2.1), it was not described as a dbSNP or assigned to a rsID. Since, this mutation has been identified in the NHLBI GO Exome Sequencing Project (http://evs.gs.washington.edu/EVS/) with the assigned rsID: rs145092287. In the NHLBI GO Exome Sequencing Project, the rs145092287 is present three times in an heterozygous status among 4300 genomes from an European-American population and not found in the 2203 genomes of an African-American population. The rs145092287 was not found in an homozygous or heterozygous status in other available genome databases (NCBI, 1000 Genomes project and the 3519 genomes of our center). Homozygosity regions around the rs145092287 were determined in the exomes by looking at the homozygous variations. Between the positions chr1:40737516 (rs6677717) and chr1:42008077 (rs 63729761) a succession of 97 homozygous variations (without heterozygous variations) was found to be shared by the three patients.

Genomic DNA sequencing

Genomic DNA from peripheral blood cells, EBV-B cell lines and/or fibroblasts of patients, their parents, and other family members was isolated according to standard methods. Genomic DNAs of 752 healthy control subjects born in the North West of England were obtained from the UK 1958 Birth Cohort (http://www2.le.ac.uk/projects/birthcohort/ 1958BC-About). The estimated frequency of the CTPS1 mutation in the populations was calculated according to the Hardy-Weinberg law. Oligonucleotide primers flanking the 3′ region of intron 17-18 and exon 18 of CTPS1 were used to amplify genomic DNA: Forward 5′-AGAGTTGGTGGTAGGGTGTGTGAC-3′ and reverse 5′-CTTGCAATCGCAGTGTGTTATCAC-3′. PCR products were amplified using high fidelity Platinum Taq DNA Polymerase (Invitrogen) according to the manufacturer's recommendations, purified with the QIAquick gel extraction kit (Qiagen) and sequenced using the ABI PRISM BigDye Terminator Cycle Sequencing Ready Reaction Kit (PerkinElmer) according to the manufacturer's recommendations. All collected sequences were analysed using 4peaks software (Version 1.7.2; A. Griekspoor and T. Groothuis, http:// nucleobytes.com/index.php/4peaks).

Analysis of microsatellite markers

Microsatellite markers were genotyped using UniSTS sequences and mapping information available from the NCBI (http://www.ncbi.nlm.nih.gov). Genomic DNA from patients was used as templates to amplify by PCR with specific fluorescent labelled oligonucleotides, the polymorphic repeats corresponding to the microsatellite markers. PCR products were evaluated using an ABI 3100 DNA Fragment Analyzer (Applied Biosystems), and data were analyzed using Genescan and Genotyper software (Applied Biosystems).

Gene expression analysis

Total RNA was isolated from EBV-B cell and activated T cell blasts of P1.2, P2.1 patients and control donors using the RNeasy Mini kit (QIAGEN). The samples were depleted of genomic DNA and reverse transcription was performed using Superscript II First Strand Synthesis System (Invitrogen). cDNAs were used as a template to perform PCR amplifications of exon 15 to exon 19 of CTPS1 or exon 4 of ACTIN with the following primers using standard protocols: CTPS1 forward primer: 5′-GAGAGGCACCGCCACCGATTTG-3′, CTPS1 reverse primer: 5′-GCCAGTACACGTGATGGGACATGC-3′, ACTIN forward primer: 5′-CTCCTTAATGTCACGCACGAT-3′; ACTIN reverse primer: 5′-CTCCTTAATGTCACGCACGAT-3′. PCR to amplify full lengh CTPS1 cDNAs from control and patients cells were also performed using the following primers: forward primer: 5′-AGCTCTGTCGCTGACGGGAGGAT-3′ (exon1); reverse primer: 5′-GCCAGTACACGTGATGGGACATGC-3′ (exon 19). PCR products were verified by sequencing revealing the expression of an abnormal CTPS1 transcript lacking exon 18 in patients cells. Multiple tissue cDNA panels from Ozyme (Human MTC panel I, II and Human Immune Sytem MTC panel) were analyzed for CTPS1 and CTPS2 gene expression by RT-qPCR. Gene expression assays were performed with Assays-on-Demand probe and primer combinations (CTPS1, Hs01041858; CTPS2, Hs00219845; GAPDH, Hs027558991) from Applied Biosystem labeled with 6-carboxy-fluorescein (FAM) dye, and universal reaction mixture. Real time quantitative PCRs for GAPDH, CTPS1 and CTPS2 were performed in triplicate using a LightCycler VIIA7 System (Roche). Expression levels were determined by relative quantification using the comparative threshold cycle method 2 ΔΔCt in which ΔΔCt is determined as followed: (Ct target gene -Ct reference gene ) target tissue -(Ct target gene -Ct reference gene ) calibrator tissue. The results shown in arbitrary units (A.U.) have been normalized to GAPDH gene expression and are presented as the relative change in gene expression normalized against the calibrator sample corresponding to leukocyte tissue.

Cell culture and stimulation

Peripheral blood mononuclear cells (PBMCs) collected from patients and healthy donors were isolated by Ficoll-Paque density gradient (Lymphoprep, Proteogenix) from blood samples using standard procedures. Expansion of T-cell blasts were obtained by incubating PBMCs for 72 h with phytohaemagglutinin (PHA) (2.5 μg ml -1 , Sigma-Aldrich) in RPMI 1640 GlutaMax medium (Invitrogen) supplemented with 5% human AB serum (Etablissement Français du Sang), penicillin (100 U ml -1 ) and streptomycin (100 μg ml -1 ). After 3 days, dead cells were removed by Ficoll-Paque density gradient and blasts were maintained in culture with IL-2 (100 UI ml -1 ). For proliferation and cell cycle analyses, blasts were washed and cultured without IL-2 for 72 hours to synchronize the cells. Blasts or PBMCs were then cultured during 4 to 6 days in complete medium alone or in the presence of 0.1, 1 or 10 μg ml -1 coated anti-CD3 antibody (clone OKT3, eBiosciences), anti-CD3/ CD28-coated beads (Invitrogen), PHA (2.5 μg ml -1 , 10 -5 M ionomycin (Sigma-Aldrich) plus 10 -7 M Phorbol 12-myristate 13-acetate (PMA, Sigma-Aldrich), Candidin (5 μg ml -1 , Bio-Rad), tetanus toxoid (1/8000 dilution, Statens Serum Institute) or tuberculosis antigen (50 μg ml -1 , Statens Serum Institute). Proliferation and cell cycle were analysed at the indicated time points. 40 μM of 3-Deazauridine (DAZ, Sigma-Aldrich) was added for 12 hours before the stimulation. In complementation experiments, blasts were incubated with 100 μM of CTP, UTP, GTP or ATP (New England Biolabs) separately or in combination, or with 200 μM of Cytidine, Uridine, Guanosine or Adenosine (Sigma-Aldrich) separately or in combination 12 hours before the stimulation. For dosage of nucleotides, blasts were deprived of IL-2 for 72 hours before stimulation or not with anti-CD3/CD28-coated beads for 48 h and cell lysates were prepared. Jurkat cells [START_REF] Gülow | HIV-1 trans-activator of transcription substitutes for oxidative signaling in activation-induced T cell death[END_REF] , 293-T cells and B EBV cell lines from patients were cultured in RPMI 1640 GlutaMax medium supplemented with 10 % heat-inactivated fetal calf serum (GIBCO), penicillin (100 U ml -1 ) and streptomycin (100 μg ml -1 ). Cells were free of mycoplasma contamination.

Proliferation and cell cycle assays

Nucleic acids and de novo pyrimidine synthesis assays

PBMCs were stimulated in the presence of 1 μg ml -1 coated anti-CD3 antibody (clone OKT3, eBiosciences) or 2.5 μg ml -1 PHA (Sigma-Aldrich) for 3 days or in the presence of candidin (5 μg ml -1 , Bio-Rad), tetanus toxoid (1/8000 dilution, Statens Serum Institute) or PPD (tuberculin) (50 μg ml -1 , Statens Serum Institute) for 6 days. 0.074 MBq ml -1 of [ 3 H]thymidine, [ 3 H]-cytidine, [ 3 H]-uridine or [ 3 H]-leucine or 0.185 MBq ml -1 U-[ 14 C]-aspartate were added during the last 18 h of stimulation. For [ 3 H]-cytidine, this corresponds to 0.133 μM, which not allows to restore normal proliferation of CTPS1-deficient cells that required 50μM. Cells were harvested with a Filter Mate harvester (PerkinElmer) on filters for labelled cell assays (Unifilter™ plates, PerkinElmer) that retain nucleic acids and filters were then washed. Radioactivity (cpm) on filters (corresponding to radiolabelled compounds incorporated in nucleic acids) was measured by liquid scintillation counting with TopCount NXT beta counter (PerkinElmer).

Apoptosis assay

Apoptosis was determined by evaluating phosphatidylserine (PS) exposure in the outer leaflet of the cytoplasmic membrane with PE-conjugated Annexin-V in combination with 7-AAD Viaprobe (Apoptosis Detection Kit I, BD) 12 hours after stimulation with coated-OKT3 (0.1, 1 or 10 μg ml -1 ). Apoptosis was based on the percentage of Annexin V + /7AAD -cells to exclude necrotic cells. Cells were analyzed by flow cytometry.

Cytokine production and degranulation

For intracellular staining of cytokines, cells were stimulated overnight with PMA and ionomycin or anti CD3/CD28 beads in the presence of brefeldin A (GolgiPlug, BD). Cells were then fixed and permeabilized using the BD cytofix/cytoperm plus kit (BD Pharmingen) according to the manufacturer's instructions. Cells were labelled with PE-anti-IL-2 (rat IgG2a, MHQ1-17H12), PE/Cy7-anti-TNF-α (mouse IgG1; MAb11), APC-anti-IFN-γ (mouse IgG1, 4S.B3) and isotype-matched mAbs purchased from Biolegend and analysed by flow cytometry. Degranulation was determined by analysis of the expression of CD107/ LAMP, a marker of the exocytosis of lytic granules. Blasts were stimulated for 3 hours in the presence of 0.3, 3 or 30 μg ml -1 coated-OKT3 and simultaneously labelled with eFluor660anti-CD107a (mouse IgG1; eBioH4A3), eFluor660-anti-CD107b (mouse IgG1; eBioABL-93) or isotype matched mAbs purchased from eBiosciences. Thereafter, cells were harvested, washed and stained with FITC-anti-CD3 and PE-anti-CD8 mAbs and analyzed by flow cytometry.

Flow cytometry

Cell staining and the flow-cytometry-based phenotypic analyses of PBMCs and cells were performed according to standard flow cytometry methods. The following mAbs were conjugated to fluorescein isothiocyanate (FITC), phycoerythrin (PE), phycoerythrin-Cyanine7 (PE/Cy7), phycoerythrin-Cyanine5, phycoerythrin-Cyanine5.5, allophycocyanin (APC), allophycocyanin-Vio7, View blue or View green: anti-CD25 (M-A251), anti-CD27 (M-T271), anti-CD31 (WM59), anti-CD45RA (HI100), anti-CD45RO (UCHL1), anti-CD197/CCR7 (3D12), anti-TCRαβ (T10B9), anti-TCRγδ(B1), anti-CD95 (DX2), anti-CD19 (HIB19), anti-CD21 (B-ly4), anti-IgM (G20-127), anti-IgD (IA6-2), anti-CD56 (B159) and anti-CD16 (3G8), all purchased from BD Biosciences and anti-CD3 (BW264/56), anti-CD4 (VIT4), anti-CD8 (BW135/80) and anti-CD69 (FN50) from Miltenyi Biotec. iNKT cells were detected by staining with anti-Vα24 (C15) and anti-Vβ11 (C21) (Beckman Coulter) and MAIT cells by staining with anti-Vα7.2 (3C10) and anti-CD161 (HP-3G10 (Biolegend). All data were collected on a FACS-Canto II cytometer (BD Biosciences) and analysed using FlowJo Version 9.3.2 software (TreeStar).

Immunoblotting and analysis CTPS1 protein expression

Cells (5. 10 6 cells/ml) were stimulated by anti-CD3 antibody (1 μg ml -1 ) cross-linking with a rabbit-anti-mouse IgG (2μg ml -1 ) or anti CD3/CD28 beads for the indicated time periods. Cells were then lysed in 1% NP40, 50 mM Tris pH 8, 150 mM NaCl, 2 0 mM EDTA, 1 mM Na3VO4, 1 mM NaF and complete protease inhibitor cocktail (Roche), as previously described [START_REF] Latour | Regulation of SLAM-mediated signal transduction by SAP, the X-linked lymphoproliferative gene product[END_REF] . Protein concentrations were quantitated by BCA assay (BIO-RAD). 80 μg of proteins were separated by SDS-PAGE and transferred on PVDF membranes (Millipore). Membranes were blocked with milk or BSA before incubation with antibodies. The following mAbs and rabbit polyclonal antibodies were used for immunoblotting: anti-PLC-γ1 (#2822S), anti-phosphorylated PLC-γ1 (#2821S), anti-phosphorylated ERK1/2 (clone E10, #9106S), anti-ERK1/2 (#4695S) anti-phosphorylated IκB-clone 5A5, #9246S), antiphosphorylated PKCtheta (#9377S), NFκB (clone D14E12) anti-phosphorylated AKT (Serine 473, clone 587F11) and anti-phosphorylated tyrosine (4G10) purchased from Cell Signaling Technology and rabbit polyclonal antibodies anti-ACTIN (#A2066) and anti-CTPS1 raised against the residues 341 to 355 (#SAB111071) or 416-430 (#SAB111072) and anti-CTPS2 (#HPA017437) purchased from Sigma Aldrich. Anti-CTPS1 rabbit polyclonal antibodies (K-21) from Santa Cruz were also tested. Membranes were then washed and incubated with anti-mouse or anti-rabbit HRP-conjugated secondary antibodies from Cell Signaling and GE Healthcare, respectively. Pierce ECL western blotting substrate was used for detection. For inhibition assays of the signaling pathways after TCR-CD3 activation, cells have been stimulated with anti-CD3/CD28 beads for 48 h in the presence of 100 nM of the MAPK/ERK inhibitor PD0325901, 10μM of the Src family protein tyrosine kinase inhibitor PP1, 10μM of the Src family protein tyrosine kinase inhibitor PP2, 10μM of the selective Ca ++ chelator 1,2-bis-(2-aminophenoxy) ethane-N,N,N′,N′-tetraacetic acid tetra(acetoxymethyl) ester (BAPTA/AM), 10μM of the IκBα phosphorylation inhibitor Bay 11-7085 or 10μM of PI3Kdelta inhibitor IC87114. All were from Sigma-Aldrich, excepted IC87114 from Calbiochem. The concentrations used were typical and previously reported. After 48 h incubation with the different inhibitors, cell viability was verified and was more than 90% in each condition. The activity and the selectivity of the inhibitors was verified in parallel by immunoblotting for phospho-tyrosine (for PP1 and PP2), IκBα phosphorylation (for Bay Bay 11-7085), ERK phosphorylation (for PD0325901) and AKT phosphorylation (for IC87114) (data not shown).

Calcium flux analysis

Ca 2+ responses were assessed by flow cytometry, as previously described [START_REF] Picard | Hypomorphic mutation of ZAP70 in human results in a late onset immunodeficiency and no autoimmunity[END_REF] . Briefly, cells were loaded with 5 μM Indo-1 AM (Molecular Probes), washed, incubated with anti-CD4-APC and anti-CD8-PE mAbs, stimulated by anti-CD3 antibody (0.125 μg ml -1 ) crosslinking with F(ab') 2 rabbit-anti-mouse IgG (10 μg ml -1 ) and then incubated with ionomycin (1 μM). Cells were analyzed with a FACS ARIA flow cytometer (BD Biosciences). Ca 2+ flux data were obtained using kinetic analyses of FlowJo software package (TreeStar). Intracellular Ca 2+ levels correspond to the normalized ratio of Ca 2+ -bound to Ca 2+ -free Indo-1 fluorescence and are plotted as a function of time.

Plasmid constructs, cell transfections and infections

A full length cDNA encoding wild-type CTPS1 and a full length cDNA encoding the mutant CTPS1delta18 were obtained by RT-PCR from control blasts and blasts from patient 1.2 respectively using the forward 5′-CGGGATCCCACCATGAAGTATATTCTGGTT-3′ and reverse 5′-CCGCTCGAGTCAGTCATGATTTAT TGA-3′ (for wild-type) and 5′-CCGCTCGAGTTAAAGAAAGTCTCCAAGC-3′ (for CTPS1delta18) specific primers. The cDNAs were verified by sequencing and inserted into a bicistronic lentiviral expression vector encoding the green fluorescent protein (GFP) as a reporter (pLenti7.3/V5-TOPO, Invitrogen). Viral particles for infection were obtained by coexpression of the lentiviral vector containing CTPS1 with third-generation lentiviral plasmids containing Gag-Pol, Rev and the G protein of the vesicular stomatitis virus (VSVG) into HEK 293T using calcium phosphate. Viral supernatants were collected every 12 h on 2 consecutive days, starting 48 h after transfection, and viral particles were concentrated by ultracentrifugation at 49,000 g for 1.5 h at 12°C. Cells were infected with viral particles at a minimal titer of 10 7 TU ml -1 and 48 h after infection, cells were deprived of IL-2 during 72 h for proliferation assays. To assess the selective advantage of GFP expression during long-term expansion, blasts were re-stimulated with antiCD3/CD28 beads (Invitrogen) every 48 h during 8 days. For CTPS1 gene knockdown, blasts or Jurkat cells were infected at day 3 of PHA stimulation with the pLKO.1 lentiviral vector containing a CTPS1-specific shRNA (Openbiosystems, n °TRCN0000045349 and n°TRCN0000045350) or a scrambled shRNA in which the puromycin resistance gene was replaced by the GFP gene. Proliferation was analyzed in Ligue contre le cancer (France). We are also grateful to the UK 1958 Birth Cohort (http://www2.le.ac.uk/projects/ birthcohort/1958BC-About) for providing DNA from 752 individuals born in the North West of England. Access to these resources was enabled via the 58READIE Project funded by the Wellcome Trust and Medical Research Council (grant numbers WT095219MA and G1001799). A full list of the financial, institutional and personal contributions to the development of the 1958 Birth Cohort Biomedical resource is available at http://www2.le.ac.uk/ projects/birthcohort.

  Cell proliferation was monitored by labelling cells with the cell trace violet dye (CellTrace violet proliferation kit, Invitrogen) or CFSE (5μM, Invitrogen) prior to stimulation, according to the manufacturer's instructions. After 4 or 6 days of culture, cells were harvested and CellTrace violet or CSFE fluorescence dilution was assessed by flow cytometry. The division index of proliferation was calculated using Flowjo software (Tree Star) and corresponds to the average number of cell divisions per cell including the undivided peak. T-cell responses within total PBMCs were also measured by [ 3 H]-thymidine incorporation after 6 days of stimulation. 0.074 MBq ml -1 of [ 3 H]-thymidine was added during the last 18 h of stimulation. Cell proliferation was determined by cpm of [ 3 H] thymidine incorporated in cells that were counted with TopCount NXT beta counter (PerkinElmer). Cell cycle analysis was determined by measuring the incorporation of the nucleoside analogue 5-ethynyl-2-deoxyuridine (EdU) into newly synthesized DNA, according to the manufacturer's instructions (Click-iT EdU, Invitrogen) after 48 h of anti-CD3 stimulation. EdU incorporation in cells was measured following conjugation of EdU to azide-modified Alexa Fluor 647 dye. Cells were analyzed by flow cytometry with a FACS-Canto II flow cytometry system (BD Biosciences).

Figure 1 .

 1 Figure 1. Identification of CTPS1 deficiency in patients with a combined immunodeficiency a, Pedigrees of the families in which an homozygous IVS18-1 G>C mutation in CTPS1 were identified. When known, the genotype of each individual is indicated. Black boxes represent affected individuals and diagonal bars indicate deceased individuals. Each patient (P) is identified by a number. b, Diagram of the CTPS1 intron-exon organization and protein domains with the serine phosphorylation sites (S) indicated and the coding exons in grey. DNA electropherograms showing the region containing the mutation in CTPS1 in family 1. The homozygous IVS18-1 G>C mutation is indicated by an arrow. c, Immunoblots for CTPS1 and CTPS2 expression in non stimulated EBV B-cell lines from healthy controls and CTPS1-mutated individuals (P1.1, P1.2 and P2.1). ACTIN serves as loading control. d, CTPS1 mRNA expression in normal tissues monitored by RT-qPCR in arbitrary units (A.U.). The inset shows the kinetic of CTPS1 mRNA expression following anti-CD3/CD28 beads stimulation.

Figure 2 .

 2 Figure 2. Induction of CTPS1 expression during T-cell activation and defective proliferation of activated CTPS1-deficient T-cells a-d, Immunoblots for CTPS1 and CTPS2 expression (a) in control T-cells (from an healthy donor) stimulated with various stimuli or (b) stimulated with anti-CD3 for different periods of time, (c) in control (Ctr.) or CTPS1-deficient cells from patient P1.2 stimulated with anti-CD3 for different periods of time and (d) in normal PBMCs sorted B-and T-cells stimulated with indicated stimuli. ACTIN serves as loading control. e, Representative dot plots showing cell divisions by dilution of the violet dye and expression of CD25 of control (Ctr.) or

Figure 3 .

 3 Figure 3. CTPS1 is required for proliferation of T-cells in response to TCR-CD3 activation a, Proliferation of T-cells in which CTPS1 expression was silenced with vectors containing shRNA for CTPS1 (Sh CTPS1#1 or Sh CTPS1#2) or containing a scramble shRNA (Sh scramble) with GFP gene reporter. Representative dot plots of GFP + cells corresponding to transduced cells (left upper panels). Representative histograms of violet dye dilution showing the cell divisions after stimulation (left lower panels). Curves showing the percentage of GFP + transduced cells in long-term expansions after repeated stimulation (middle panel). Immunoblots for CTPS1 and CTPS2 expression in transduced cells (right

Nature. Author manuscript; available in PMC 2019 April 26.

Acknowledgements

The authors thank the patients, their families and the healthy donors for cooperation. We thank S. Rigaud, S. Gérart, C. Synaeve and R. Rodriguez for help with experiments and P. Revy for discussion. This work was supported by grants from INSERM, ANR (ANR-08-MIEN-012-01, ANR-2010-MIDI-005-02 and ANR-10-IAHU-01), Fondation ARC (France), the European Research Council (ERC-2009-AdG_20090506 n°FP7-249816) and the Rare Diseases Fondation (France). S. L. is a senior scientist of CNRS (France). E. M. is supported by ANR (France) and

Nature. Author manuscript; available in PMC 2019 April 26.

Europe PMC Funders Author Manuscripts

Europe PMC Funders Author Manuscripts GFP + and GFP -blasts after 4 days of stimulation with anti-CD3/CD28 beads as previously described. For survey of loss of GFP expression in long-term expansions, blasts were repeatedly stimulated with anti-CD3/CD28 beads every 48 h during 8 days. Jurkat cells were maintained in culture after infection during 26 days. The proportions percentages of GFP+ cells were determined by flow cytometry.

Quantification of intracellular nucleotides

Intracellular pools of nucleotides were quantified based on previously described methods [START_REF] Luo | Simultaneous determination of multiple intracellular metabolites in glycolysis, pentose phosphate pathway and tricarboxylic acid cycle by liquid chromatography-mass spectrometry[END_REF][START_REF] Scavennec | Purine and pyrimidine ribonucleoside monophosphate patterns of peripheral blood and bone marrow cells in human acute leukemias[END_REF] . Briefly, 5 millions of cells were washed in 0.1 M phosphate buffer (pH=7.4) and lysed in 60 μL HClO 4 1M, containing 2 μM 8-bromo-AMP (8-BrAMP) as an internal standard. After 12,000 g centrifugation for 5 min at 4°C, supernatants were transferred to a 384-wells plate and kept at 4°C in auto sampler before injection. Aliquots of 5 μL were injected onto a separation column (ACQUITY UPLC BEH300 C18, 1.7 μm, 2.1×100 mm reversed-phase column, Waters) with a flow rate of 0.5 mL/min and analyzed with a tandem mass spectrometry system consisting of an Acquity Ultra Performance Liquid Chromatography (UPLC Waters) interfaced with a xevo-TQ-S tandem quadrupole mass spectrometer (Waters). Mobile phase A was 0.1% formic acid in water and mobile phase B, 0.1% formic acid in acetonitrile. A programmed mobile phase-gradient was used during a 7min run: 0 min, 1% B; 5 min, 10% B; 5.1 min, 100% B; 6 min, 100% B; 6.1 min, 1% B; 7 min, 1% B. The content of the 4 nucleotides ATP, GTP, UTP and CTP was quantified in the electrospray negative ion mode with multiple reaction monitoring (MRM). Transitions of m/z 505.9>408 and 505.9>272.9 were used for quantification and confirmation of ATP, respectively, and those of 521.9>158.9 and 521.9>177 for GTP, 482.8>158.9 and 482.8>79 for UTP, and 481.8>158.9 and 481.8>384 for CTP. Concentrations were determined by using calibration curves of the 4 nucleotides. The linearity, exactitude and variability were determined for the technical validation of this assay. The linearity gave a correlation coefficient of the linear regression curves greater than 0.99 for the 4 nucleotides. The minimum and maximum recovery of spiked samples with the 4 nucleotides at a concentration of 90 mg/L and 250 mg/L ranged from 72% to 123%. The maximum intraand inter-assay variability was of 22% and 23%, respectively.

Statistical analysis

P values were calculated with the Students t-test using PRISM software (GraphPad Software), with a two-tailed distribution. The variance was similar between the groups that have been statistically compared.

Supplementary Material

Refer to Web version on PubMed Central for supplementary material. Nature. Author manuscript; available in PMC 2019 April 26.

Table 1

Clinical features of patients


Florence Dupin De Saint-Cyr

Francis Faux

Integrating evolutionary prejudices in belief function theory *

This paper deals with belief change in the framework of Dempster-Shafer theory in the context where an agent has a prejudice, i.e., a priori knowledge about a situation. This situation is modeled as a sequence (p, m) where p reflects the prejudices of an agent and m is a mass function that represents the agent's uncertain beliefs. In contrast with the Latent Belief Structure introduced by Smets where a mass is decomposed into a pair of separable mass functions called respectively the confidence and diffidence, m can be any mass function (i.e., not necessarily separable) and p is not a mass. The aim of our study is to propose a framework in which the evolution of prejudices and beliefs are described through the arrival of new beliefs. Several cases of prejudice are described: the strong persistent prejudice (which never evolves and forbids beliefs to change), the prejudice that is slightly decreasing each time a belief contradicts it, etc.

Introduction

When dealing with information pervaded with uncertainty, several frameworks can be used: probabilities, possibilities, ... with their variants. The most general framework in which uncertainty can be expressed is belief-function theory. It is well suited to epistemic analysis in situations where there is little information to assess a probability, or where information is nonspecific, ambiguous or contradictory. This theory makes it possible to express that there is evidence in favor of a set of events A without specifying the precise degree of certainty of each element of A, whereas in a probabilistic setting, the probability of each event of A should be known.

Belief function theory also known as Dempster-Shafer theory was first introduced by Arthur P. Dempster in the context of statistical inference, then developed by Glenn Shafer into a formal framework for representing and reasoning with uncertain information [START_REF] Shafer | A mathematical theory of evidence[END_REF]. G. Shafer viewed belief functions as the result of the conjunctive combination of pieces of evidence such as (more or less unreliable) testimonies from different sources, in order to form a representation of beliefs about certain aspects of the world.

This theory has been well studied and developped in order to reason with several sources of information. However one can be interested in combining positive evidence and personal a priori convictions (coming from moral values, tastes, and past experiences), that are called "prejudice". A prejudice can be defined as an a priori favorable or unfavorable 1 "opinion adopted without examination, imposed by an environment, an education" (Montaigne, Essais, II, 12, ed. P. Villey and V.-L. Saulnier, p.506). Unlike a belief, a prejudice can be disproved on the basis of facts. Prejudices can be more or less strong, the strongest they are the more difficult it is to disprove them and the more they will influence the reasoning. In his famous book The Nature of prejudice the American psychologist Gordon Allport [START_REF] Willard Allport | The nature of prejudice. Addison-wesley Reading[END_REF] asserted that "prejudice is essentially a by product of the necessary mental shortcuts the human brain uses to process the vast amount of information it takes in".

In the framework of Dempster-Shafer theory, there were attempts to encode the retraction of information (which turns out to be wrong): the operation of removal (or retraction) was proposed by many authors [START_REF] Matthew | Non-monotonic reasoning using Dempster's rule[END_REF][START_REF] Kramosil | Probabilistic Analysis of Belief Functions[END_REF][START_REF] Prakash | Conditional independence in valuation-based systems[END_REF][START_REF] Smets | The canonical decomposition of a weighted belief[END_REF] in order to decrease the degree of belief by retracting some piece of evidence. In particular, the model called "Latent Belief Structure" introduced by [START_REF] Smets | The canonical decomposition of a weighted belief[END_REF] then studied in [START_REF] Pichon | On latent belief structures[END_REF] considers a pair of belief functions, one representing the confidence part, and the other the diffidence part playing the role of a moderator that can annihilate, via retraction, some information supplied by the former.

Recently, this model was reinterpreted by [START_REF] Dubois | Prejudice in uncertain information merging: Pushing the fusion paradigm of evidence theory further[END_REF] in terms of prejudice of the receiver, and retraction was considered as a special kind of belief change. Its role is to weaken the support of some focal sets of a belief function, possibly stemming from the fusion of the incoming information. The authors suggest that prejudices are due to some prior knowledge that is more entrenched than incoming new pieces of uncertain evidence. They detail how such prior information can affect a belief function. They encode a prejudice by a negative mass function which should be combined with a positive mass function (representing a testimony). However the result of this combination must be a positive mass (because a negative mass is not interpretable in this framework). Moreover this approach is only able to combine some particular forms of prejudices and beliefs where the prejudices are against some part of a focal set of a simple mass function. In that case a revision is performed on the beliefs. The revision operation is a classical topic broadly studied in knowledge representation literature [START_REF] Katsuno | Propositional knowledge base revision and minimal change[END_REF][START_REF] Carlos E Alchourrón | On the logic of theory change: Partial meet contraction and revision functions[END_REF], and also in the particular context of belief functions by [START_REF] Dubois | Conditioning in Dempster-Shafer theory: prediction vs. revision[END_REF].

Example 1 Let us consider the beliefs of a doctor about 3 diseases that can be encountered: pyelonephritis noted 1, urinary infection noted 2 and lumbago noted 3. Assume that the doctor has observed a stomach ache. In that case there is a prejudice against urinary infection (2) and lumbago (3) encoded by the orange rectangle. Suppose now that the doctor learns from health test results that the patient may have a disease among pyelonephritis (1) and lumbago (3) (blue oval). Then we may wonder how the prejudice can be taken into account for representing the final belief state of the doctor. The aim of this paper is to study how the beliefs and the prejudice may evolve, according to their strength and to their incompatibility.

Prejudice of -2/3 against {2, 3} and evidence of 1/3 for {1, 3}

Prejudice of -2/3 against {1, 3} and evidence of 1/3 for {1, 3}

In this example, we consider that the prejudice has a strength of -2/3 (on a scale from 0 to -∞) with a confidence in the new piece of information of 1/3. It means that the prejudice is stronger than the beliefs hence the doctor should transfer its beliefs towards pyelonephritis (1), the prejudice against urinary infection (2) and lumbago (3) can then either decrease or remain depending on its nature (stubborn prejudice will not be questioned by any new piece of information while open minded prejudice may decrease).

Another more classic instance of this example would assume that the prejudice is against (1) and (3), and the piece of evidence is about the same set (1) and [START_REF] Denoeux | Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence[END_REF]. When the prejudice is -2/3 and evidence is 1/3 then the prejudice being stronger it remains against (1) and (3) (maybe attenuated) but the evidence is canceled, while when the prejudice is weaker (-2/3) than the evidence (say 0.5), the prejudice is canceled but the evidence is only integrated with a strength of 1/6.

In classical belief function theory, two mass functions m and m ′ that are considered as two sources of evidence, are combined by using Dempster rule m ′′ = m ∩ ⃝m ′ . Dempster's rule aims at gathering the two sources, in a conjunctive way, but this raises several issues when prejudices are taken into account:

• what if we combine two sources that have both a prejudice and a mass: (p, m) and (p ′ , m ′ ):

this kind of combination is considered as out of the scope of the paper because we reason only from the point of view of one agent who receives an incoming information. In this paper, the incoming information is delivered under the form of a mass function, with no prejudice explicitly present (however it could be done with Latent belief structure). Note that, in this context, if we have no prejudice at start, i.e, p = 0, then the combination (p, m) with m ′ (i.e., the combination with (0, m ′ )) should yield (0, m ∩ ⃝m ′ ).

• what are the links between reasoning with prejudices and belief revision ?

In this paper we propose to extend the previous work of [START_REF] Dubois | Prejudice in uncertain information merging: Pushing the fusion paradigm of evidence theory further[END_REF] with a general formalism that considers any prejudice and beliefs using a pair (p, m) containing an encoding of the prejudice p and an encoding of the beliefs by a mass m. We are interested in the evolution of this pair when new information arrives, this is why the study is related to Dempster's revision (recalled in Section 2.3). Some necessary background on belief functions is introduced in Section 2. After defining specific rules and constraints governing the evolution of beliefs (Section 3), we conclude with a comparison with the literature and some perspectives.

Background

Basics about belief functions

Let us consider a finite set Ω = {ω 1 , . . . , ω N }, called the frame of discernment, whose elements represent descriptions of possible situations, states of the world, one of which corresponding to the truth. In Dempster-Shafer theory [START_REF] Shafer | A mathematical theory of evidence[END_REF], the uncertainty concerning an agent's state of belief on the real situation is represented by a mass function defined as a mapping m : 2 Ω -→ [0, 1] such that m(∅) = 0 and verifying A⊆Ω m(A) = 1. m(A) expresses the proportion of evidence that the current state is in A. Each subset A ⊆ Ω such as m(A) > 0 is called a focal set of m.

An elementary testimony T with weight (1 -α) in favor of a non-contradictory and nonuniversal proposal A ∈ 2 Ω \ {Ω, ∅} is represented by the simple mass function m : 2 Ω -→ [0, 1] denoted by m = A α in [START_REF] Smets | The canonical decomposition of a weighted belief[END_REF][START_REF] Denoeux | Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence[END_REF] such that

m = A α denotes m(A) = 1 -α m(Ω) = α
Here, α evaluates the lack of confidence in the testimony T also called mistrust.

In presence of multiple sources of information or multiple uncertain testimonies, the result of the conjunctive combination of two mass functions m 1 and m 2 , noted m 1 ∩ ⃝2 , is defined as follows:

m 1 ∩ ⃝ 2 (A) = A 1 ∩A 2 =A (m 1 (A 1 ).m 2 (A 2 ))
In particular, m 1 ∩ ⃝2 (∅) represents the conflict between the mass functions. We will denote ⊕ the normalized conjunctive combination rule called Dempster's rule (which assigns a zero mass to the empty set and divides all the masses of the focal elements by 1 -m 1 ∩ ⃝2 (∅)). A belief function Bel(A) is a non-additive set function which represents the total quantity of pieces of evidence supporting the proposition A ⊆ Ω and is defined by

Bel(A) = ∅̸ =E⊆A m(E)
The plausibility P l(A) is the dual set-function of Bel(A) where P l(A) = 1 -Bel(A), i.e., : P l(A) = E∩A̸ =∅ m(E). A mass function m can be equivalently represented by its associated commonality function defined for all A ⊆ Ω by Q(A) = B⊇A m(B) The commonality function Q(A) represents the total quantity of incomplete evidence that makes all elements of A possible. [START_REF] Shafer | A mathematical theory of evidence[END_REF] calls a separable support function, a belief function m = ⊕ k i=1 A d i i resulting from Dempster rule combination of simple mass functions

A d i i , with A i ̸ = Ω, 0 < d i < 1, i = 1, . . . , k.
Each single mass represents an independent testimony.

Defiance, retraction and latent structures

In 1995, [START_REF] Smets | The canonical decomposition of a weighted belief[END_REF] extended the range of defiance functions δ initially defined on [0, 1] to the interval ]0, +∞) and has defined the notion of retraction for δ > 1. The retraction of a simple mass function B y , y < 1 supporting B from a simple support function

A x , x < 1, denoted by ̸ ∩
⃝ is defined such that :

A x ̸ ∩ ⃝B y = A x ∩ ⃝B 1/y
and yields the diffidence function

δ = δ A /δ B with δ A (E) = x if E = A, 1/x for E = Ω and 1 otherwise and δ B (E) = y if E = B, 1/y for E = Ω and 1 otherwise. However A x ̸ ∩ ⃝B y is a belief function if only if A = B and x/y < 1. Indeed the result A x ̸ ∩
⃝B y is NOT a belief function in general as the mass function induced by δ may fail to be positive. Retraction also fails if the set to retract is not focal. It is not possible neither when the focal set A to be retracted intersects some other focal set B without being included in it (i.e., A ∩ B ̸ = A). In other words retraction is only possible on a set (A ∩ B) that is an intersection of two focal sets (A and B).

Hence, a necessary and sufficient condition required for retraction is that the set of focal sets of m should be closed under intersection. Note that it is a necessary but not sufficient condition of separability. Indeed retracting a focal set E J from a separable mass function m affects and may delete all focal sets E I ⊂ E J as well, namely all combinations between the merging of information E J from sources indexed in J, with information from other sources.

Based on the canonical decomposition of belief functions and the retraction operation, the concept of latent belief structure (LBS) [START_REF] Smets | The canonical decomposition of a weighted belief[END_REF][START_REF] Pichon | On latent belief structures[END_REF] was defined as a pair of separable non dogmatic2 masses m c and m d called respectively the confidence and diffidence components such that m =

m c ̸ ∩ ⃝m d with m c = ∩ ⃝ A∈C A w(A) and m d = ∩ ⃝ A∈D A 1 w(A) . The disjoint subsets C and D come from a partition of 2 Ω such that C = {A : A ⊂ Ω, w(A) ∈ (0, 1]} and D = {A : A ⊂ Ω, w(A) ∈ (1, ∞)}.
The diffidence component may be interpreted as a prejudice against the subset D. However, only a few particular cases of prejudice can be modeled by LBS because the constraints linked to the retraction operation detailed above are very restrictive. When dealing with statistical data, [START_REF] Dubois | Conditioning in Dempster-Shafer theory: prediction vs. revision[END_REF] differentiates revision and prediction with respect to the new piece of information. The initial information corresponds to a belief function encoded by a mass function (the assignment of non-negative weights m(E) to subsets E of Ω). This mass m is modified by taking the new observation saying that the states in C are observed. When this new piece of information is totally certain then it is a revision and the Dempster conditioning is used for handling it. [START_REF] Dubois | Conditioning in Dempster-Shafer theory: prediction vs. revision[END_REF] defines this operation as the revision m(.||C) of a mass function m by a totally certain new piece of information C, as follows:

Conditioning in Dempster and revision

m(B||C) = E:B=C∩E̸ =∅ m(E)
In other words m(B||C) = P l(E ∩C). Moreover [START_REF] Dubois | Conditioning in Dempster-Shafer theory: prediction vs. revision[END_REF] defines P l(B||C) = P l(B ∩C)/P l(C) and Bel(B||C) = 1 -P l(B||C).

Example 2 (Ellsberg's paradox) We consider an urn with three kinds of balls: white, black and red. We know that 1/3 are reds, the universe is all the possible outputs obtained after the event to draw a ball from the urn: Ω = {ω 1 , ω 2 , ω 3 } where ω 1 (resp. ω 2 , ω 3 ) represents the fact that the ball is white ( resp. black and red). The mass representing the initial information is named m in Table 1. We don't know the proportions of balls of each kind let us call α the proportion of white among the white and black balls. We learn that the ball that is extracted is not black, C = {ω 1 , ω 3 }. It yields the results presented in Table 1 column 2. Note that in this example, in both revision and prediction cases P l = Bl, which translate the fact that focal elements after revision/prediction are singletons.

In the case called "revision" by [START_REF] Dubois | Conditioning in Dempster-Shafer theory: prediction vs. revision[END_REF], Dempster conditioning transfers the full mass of each focal set E to E ∩ C ̸ = ∅ (followed by a renormalisation). This means that the new information C modifies the initial mass function in such a way that P l(C) = 0: situations where C is false are considered as impossible. In the "prediction" case, only a proportion of the mass of E is transferred to E ∩ C, but after normalization the rest is distributed over the new focal elements (the ones in E ∩ C).

Revising by a new mass function m I :

[12] introduced a "revision operator" • s.t. given two mass functions m and m I over Ω defined by:

for any

E ̸ = ∅, m • m I (E) = A∩B=E σ(A, B)m I (B) (1)
where σ(A, B), called specialization matrix, is s.t. σ(A, B) = 0 when A ∩ B = ∅ and otherwise:

σ(A, B) =    m(A) P l(B)
for P l(B) > 0 0 for P l(B) = 0 and In other words, it flows down a portion of m I (B) to A ∩ B, making the revision result a "specialization" 3 . An example of specialization matrix is depicted on the right of Table 1, it is the one of the mass m representing the Ellsberg's paradox (Example 2).

A ̸ = B 1 for P l(B) = 0 and A = B E m P l m ||C m I m • m I m I ′ m • m I ′ ∅ 0 0 0 0 0 0 0 {1} 0 2 3 2 3 0 2 3 0.5 0.5+ 0.8 3 {2} 0 2 3 0 0 0 0.1 0.1 {3} 1 3 1 3 1 3 0 1 3 0 0.4 3 {1, 2} 2 3 2 3 0 0 0 0 0 {1, 3} 0 1 0 1 0 0.4 0 {2, 3} 0 1 0 0 0 0 0 Ω 0 1 0 0 0 0 0 σ(A, B) {1} {2} {3} {1, 2} {1, 3} {2, 3} Ω {1} 0 0 0 0 0 0 0 {2} 0 0 0 0 0 0 0 {3} 0 0 1 0 1 3 1 3 1 3 {1, 2} 1 1 0 1 2 3 2 3 2 3 {1, 3} 0 0 0 0 0 0 0 {2, 3} 0 0 0 0 0 0 0 Ω 0 0 0 0 0 0 0
As we can see in Table 1, the two revision operators are equivalent when the new piece of information can be represented by a simple mass function. However if the new piece of information is more complex, only • can be applied, hence • is a refinement of the revision operator based on Dempster conditioning.

Formalizing prejudices

In this section we propose to model prejudices against a piece of evidence. We propose to define what happens in the situation where the receiver already has some prejudices and some knowledge, the prejudice being characterized by a strength and a tenacity. We study how new incoming information can modify both the levels of prejudice and beliefs. We do not address the case of the integration of a new prejudice or the reinforcement of an existing one. More precisely the incoming information is only a testimony that can decrease some prejudice (or not affecting it at all) but cannot increase any prejudice or create a new one. The creation of prejudice is left for further studies.

In order to both encode prejudice and knowledge, we propose to consider the couple (p, m) where m is a mass function and p is a prejudice against some set A such that p(A) ≤ 0. This is done to be consistent with Smets retraction operation (recalled in Section 2.2).

Definition 1 (belief state) A belief state is a pair (p, m) where p, representing a prejudice against some piece(s) of evidence, is a prejudice function p : 2 Ω \ {Ω, ∅} → (-∞, 0] and m is a mass function. Intuitively ∀ ∅ ⊂ A ⊂ Ω, p(A) represents the threshold of evidence required to change one's mind about A ̸ = ∅:

• p(A) = 0 indicates the absence of prejudice against A • p(A) = -∞ means an unshakable prejudice against A.
p is extended to 2 Ω by setting p(Ω) = 1 -X⊂Ω p(X) and p(∅) = 0 (normalization).

In the following sections we are going to study the cases where there is only one focal set for the beliefs (called A), and only one prejudice that focuses on a set (called B). We describes all the situations where A and B intersects, namely A = B (case 1), A ⊂ B (case 2), B ⊂ A (case 3), and A \ B ̸ = ∅ and B \ A ̸ = ∅ (case 4). The prejudice is characterized by a threshold of evidence (under which the evidence is not affected) and a tenacity function that describes how this threshold evolves when new information contradicts it (this tenacity function is decreasing, because in this paper we restrict our study to the case where no piece of evidence can increase a prejudice).

Case 1 : Information and prejudice focused on the same subset A ̸ = ∅

and A ̸ = Ω

The simplest situation occurs when the prejudice and the information (testimony) are concerning the same set of pieces of evidence.

Definition 2 A simple belief state about A ̸ = ∅ has the form (p = A β , m = A α ) with α ∈ [0, 1], β ∈ [1, +∞)
where p is a prejudice against A, called simple prejudice, and m is a simple mass function on A. It can be simplified according to the following rules:

• αβ ≤ 1: the prejudice is canceled and the confidence in A decreases so the pair becomes (A 1 , A αβ ),

• αβ > 1 the prejudice decreases but the informative mass m(A) is deleted : the pair becomes (A αβ , A 1 ).

Note that, p = A β is a shortcut for p(A) = 1 -β and p(Ω) = β, hence due to β ≥ 1, p(A) is negative in accordance to Definition 1. In other words, a simple mass function m about A, m = A α in presence of a prejudice p against A, p = A β , can give three situations according to the incoming information A α :

1. The prejudice is deleted if the incoming information is sufficiently convincing. In other words, the threshold of persuasiveness required to change one's mind is overtaken, the prejudice was low compared to the strength of the evidence A α .

2. The prejudice remains but is possibly attenuated and the piece of evidence is rejected. So the attenuation of the prejudice is a function (called f now on) dependent of the strength β of the prejudice against A.

3. The prejudice is preserved whatever the incoming information (which is canceled).

Definition 3 (Evolving prejudice)

Given A ⊂ Ω with A ̸ = ∅, we denote by A β the prejudice of strength β which evolves according to f , where β ∈ [1, +∞] and f :

[0, 1] × [1, +∞] → [1, +∞] is a function s.t. f (α, β)
represents a new threshold of prejudice: f (α, β) replaces β when the prejudice is attenuated

• αβ ≤ 1 the prejudice is canceled and the level of evidence decreases so the pair becomes (A 1 , A αβ ),

• αβ > 1 the prejudice changes but m(A) is canceled: the pair becomes (A f (α,β) , A 1 ).

In this paper, due to the fact that we assume that prejudices can only decrease or stay still, f is a decreasing or constant function. Here are some examples of special cases for f : (1, αβ) decreases in function of the strength of the certainty on the incoming information, at most the prejudice is removed.

• f (α, β) = max(1, β -ε) decreases by ε ∈ [0, +∞[ after receiving each new evidence. • f (α, β) = max
• f (α, β) = β leads to the conservation of the prejudice in the case of a narrow-minded person yielding (A β , A 1 ),

If there is no prejudice against A, whatever the new piece of evidence that may arrive about A, it cannot create a new prejudice on A. So this amounts to having a simple mass function as shown in the following proposition: The two first columns of Table 2 shows a belief state (p b , m a ) where the initial belief is m a = {ω 1 , ω 3 } 0.5 (Kidney ache which is translated by an evidence for ω 1 or ω 3 , meaning the presence of pyelo-nephritis or urinary infection) and there is a prejudice against those two diseases because of a stomach ache p b = {ω 1 , ω 3 } 5/3 .

′ = A β • A α = A αβ E p b m a (p ′ b , m ′ a ) m c (p ′ b , m ′ c ) (p ′′ b , m ′′ c ) (p ′′′ b , m ′′′ c ) any f f = max(1, β -0.2) f = max(1, αβ) f (α, β) = β αβ = 5/6 ≤ 1 αβ = 10/9 > 1 E ∈ 2 Ω \ {{1, 3}, Ω} 0 0 (0, 0) 0 (0, 0) (0,0) (0,0) {1, 3} -
The third column shows the results of conjunctive combination of m a and p b as described in Section 2.1, i.e., the prejudice disappear (hence the belief state is a simple mass) and the mass on A decreases: p ′ b ({ω 1 , ω 3 }) = 0 and m ′ a ({ω 1 , ω 3 }) = 1 -αβ = 1/6. In the case where the belief is described by m c then the prejudice decreases (depending of f (α, β)) and the evidences are canceled. Column 6 and 7 respectively show the behavior of the prejudice for f = max(1, β -ε) (with ε = 0.2) and f = max (1, αβ). The last column presents the case of a narrow minded agent with f (α, β) = β.

Case 2: Evidence on A and prejudice against B with ∅ ⊂ A ⊂ B ⊂ Ω

In this case, we have A \ B = ∅ and B \ A ̸ = ∅. Note that there is a discontinuity: when A = B, there was an inconsistency between the information and the prejudice while in the current case A ⊂ B, the set B \ A is non-empty, hence the transfer is possible.

• αβ ≤ 1 the prejudice on A is deleted, the prejudice on B \A is maintained but the evidence on A decreases.

• αβ > 1 the mass on A is canceled and the prejudice against A is possibly decreased by f (α, β) while the prejudice on B \ A is maintained Definition 4 Given a belief state (p, m) such that p = B β and m = A α , (with ∅ ⊂ A ⊂ B ⊂ Ω) this state is simplified into (p ′ , m ′ ) such that:

• αβ ≤ 1 (the prejudice is weaker than the information):

(p ′ , m ′ ) = ((B \ A) β , A αβ )
• αβ > 1 (the prejudice is stronger than the information): information is canceled and the prejudice can be affected on the set A: (p ′ , A 1 ) with p ′ such that:

p ′ (A) = 1 -f (α, β) (as in the Case 1 Section 3.1)

-p ′ (B \ A) = 1 -β -p ′ (Ω) = 1 + f (α, β) + β 3.3 Case 3: Evidence on A and prejudice against B with ∅ ⊂ B ⊂ A ⊂ Ω In this case A \ B ̸ = ∅ and B \ A = ∅.
In the current case there is no contradiction between the target of the prejudice and the information, information can be transferred to A \ B, concerning the prejudice it can either decrease or remain the same depending on its strength.

• αβ ≤ 1: the prejudice on B is deleted, but the belief is transferred to A \ B. In this case, there is no evidence for B hence the mass on B remains 0 it cannot increase.

• αβ > 1: the mass on B is canceled and the prejudice about B is possibly decreased by f (α, β) while the prejudice on A \ B is maintained

Definition 5 Given a belief state (p, m) such that p = B β and m = A α with ∅ ⊂ B ⊂ A ⊂ Ω,
this state is simplified into (p ′ , m ′ ) such that:

• αβ ≤ 1 (weak prejudice): (p ′ , m ′ ) = (B 1 , (A \ B) α ) • αβ > 1 (strong prejudice): (p ′ , m ′ ) = (B f (α,β) , (A \ B) α )
Note that in the Definition 5, we could have considered that after removing the prejudice against B, the mass would remain on the entire set A, i.e., (p ′ , m ′ ) = (A 1 , A α ). However we think that this is a less cautious attitude, it would amount to forget completely the old prejudice. In this case, there is no precise information about B hence whatever the strength of the prejudice, the information is transferred from A to (A \ B).

Case 4: Evidence on

A ̸ = Ω and prejudice against B ̸ = Ω with (A \ B) ̸ = ∅ and (B \ A) ̸ = ∅
In all cases where A ̸ = B (cases 2, 3 and 4) there is no frontal contradiction between the beliefs and the prejudice, here A \ B ̸ = ∅ means that information can be transferred to A \ B, and B \ A ̸ = ∅ means that a prejudice can remain on B \ A.

• αβ ≤ 1 the prejudice on A \ B is deleted, but remains on B \ A, the belief is transferred to A \ B.

• αβ > 1 the mass on B is canceled and the prejudice about B ∩ A is possibly decreased by f (α, β) while the prejudice on

B \ A is maintained E p b m a (p ′ b , m ′ a ) m c (p ′ b , m ′ c ) (p ′′ b , m ′′ c ) (p ′′′ b , m ′′′ c ) any f f = max(1, β -0.2) f = max(1, αβ) f (α, β) = β αβ = 5/6 ≤ 1 αβ = 10/9 > 1 {∅, {1, 2}} 0
0 (0, 0) 0 (0, 0) (0,0) (0,0) {1} 0 0 (0,0.5) 0 (0,1/3) (0,1/3) (0,1/3) {2} 0 0 (-2/3,0) 0 (-2/3,0) (-2/3,0) (0,0) {3} 0 0 (0,0) 0 (-7/15,0) (-1/9,0) (0,0) {1, 3} 0 0.5 (0,0) 1/3 (0,0) (0,0) (0,0) {2, 3}

-2/3 0 (0,0) 0 (0,0) (0,0) (-2/3,0)

Ω
• A ∩ B = ∅ : (p ′ , m ′ ) = (p, m) : no change, • αβ ≤ 1 (weak prejudice): (p ′ , m ′ ) = ((B \ A) β , (A \ B) α ),
• αβ > 1 (strong prejudice): (p ′ , (A \ B) α ) with p ′ such that:

if f (α, β) = β then p ′ = p (strong and persistent prejudice) In the case where the belief is described by m c , the prejudice against {ω 3 } decreases (depending on f (α, β)) and is transferred to {ω 2 }. The evidence for {ω 3 } is canceled but transferred to {ω 1 }(m ′ c ({ω 1 }) = 0.5). The last column presents the case of a narrow minded agent with f (α, β) = β.

-else * p ′ (A ∩ B) = 1 -f (α,

Properties

In this section, we establish two propositions that are concerning the belief part of the cognitive state of the agent: it appears that when beliefs are contradicted by prejudice but not radically, i.e. there are sets of beliefs that can remain uncontradicted, then a transfer can be made towards the more specialized set of uncontradicted beliefs. However, when there is a radical opposition between beliefs and prejudices, i.e. when prejudices are against a set of worlds that contains the set of worlds we believe in, then a revision should be performed (Proposition 4).

We start by showing that our definition of belief state evolution under prejudices is conserving the belief masses. In this section, we first recall the approaches that deal with information deletion, namely retraction and updating. Indeed, belief function theory is made to add new pieces of evidence through Dempster combination rule, but the issue of deleting or modifying the agent's belief states when some evidence is invalidated or modified has deserved some attention.

In probability and possibility theory, the retraction operation consists of a division followed by a normalization. Since 1984, Ginsberg [START_REF] Matthew | Non-monotonic reasoning using Dempster's rule[END_REF] proposed a special case of retraction applied to belief functions in the simple case of a frame of discernment with only two elements. In the valuation-based system framework, Shenoy [START_REF] Prakash | Conditional independence in valuation-based systems[END_REF] defined removal as point-wise division followed by normalization (if normalization is possible). Kramosil [START_REF] Kramosil | Measure-theoretic approach to the inversion problem for belief functions[END_REF] generalized the notion of belief functions with basic signed measure assignment (BSMA) and proposed an operation inverse to Dempster's rule. He introduced the notion of q-invertibility that may be seen as generalizing non-dogmaticism. Pichon [START_REF] Pichon | Belief functions: canonical decompositions and combination rules[END_REF] pursued Kramosil's seminal work by defining the so-called conjunctive signed weight function. But the absence of a semantic, the lack of intuitive interpretation of such generalized belief functions and the fact that only the conjunctive rule is used to combine BSMAs (normalization cannot be applied) are obstacles to the potential use of this approach. Smets [START_REF] Smets | The canonical decomposition of a weighted belief[END_REF] generalized the concept of simple support function, allowing the diffidence values to range on the positive reals and introduced the retraction operation defined by the division of commonality functions. Smets defined then the concept of latent belief structure for non dogmatic mass functions. This concept is studied in more details by Pichon and Denoeux [START_REF] Pichon | On latent belief structures[END_REF]. Lukaszewski [START_REF] Lukaszewski | Updating the evidence in the Dempster-Shafer theory[END_REF] proposes an algorithm for what he calls updating which consists in removing or changing some pieces of evidence without carrying out all the combinations again except for the ones that have been deleted or modified. Dubois, Faux et Prade [START_REF] Dubois | Prejudice in uncertain information merging: Pushing the fusion paradigm of evidence theory further[END_REF] consider retraction as a special symmetric belief change operation that avoids the explicit use of negative mass functions. Table 4 presents an abstract example which deal with the three different approaches (negation viewed as a conjunctive combination with the complementary, revision by the complementary and retraction) that can be used to remove a piece of evidence on a set B and the same example dealt with a strong persistent prejudice on B. It is important to note that retraction of a focal set B differs (fifth column) from conjunctive combination with the complementary of this focal set B (third column). Indeed retraction allows us to focus and reduce or delete the mass on B (it is possible to find values for x, y, z and u such that m ̸ ∩ ⃝B u (B) = 0 with y, u ̸ = 0) while negation never allows us to cancel beliefs on B (since yu ̸ = 0 as soon as y, u ̸ = 0). In other words, integrating a piece of evidence on the complementary of B (B u ) is different from canceling an evidence for B (by integrating B 1/u ). Revision gives priority to the new piece of information, hence revising by B amounts to transfer pieces of evidence from A, B and A ∩ B to A \ B (since A \ B ⊂ B). Our approach takes a complementary point of view relatively to revision and retraction since it allows us to make evolve the beliefs either by transfer or by attenuation. The important difference is the introduction of a new dimension for prejudices allowing us to distinguish them from negative evidence and to handle their evolution independently. To sum up this paper presents a preliminary study about the integration of evidence in a belief state where the agent has some prejudices. We propose a bipolar model considering prejudices and uncertain beliefs. Prejudices have a strength and a tenacity. We study how new incoming information can modify both the levels of prejudice and belief. This model is compatible with revision and retraction operations.

Negation

Note that in a more general case, it may be impossible to represent masses and prejudices with a simple belief state (i.e., a simple prejudice and a simple mass). In that case, i.e., with complex masses and prejudices, in order to know if the prejudice is stronger than the beliefs, we could either consider a veto approach that takes into account only the focal sets with highest evidence that are concerned by the prejudice or a cumulative approach that would consider the sum of the beliefs on focal sets that are concerned by it. It could be interesting to study whether it is possible to recover a purely Dempster-Shafer framework from a general belief state made of complex masses and prejudices.

2. 3 . 1

 31 Revising by a sure observation C

Proposition 1 □Proposition 2

 12 The pair (A 1 , A α ) is equivalent to the simple mass function m = A α Proof : Here β = 1, hence αβ ≤ 1, according to definition 3, the pair becomes (A 1 , A α ). When αβ ≤ 1, the pair (A β , A α ) is equivalent to a revision of the simple mass function m = A β by the simple mass function A α yielding a simple mass m ′ such that m

Table 2 :□Example 3

 23 Kidney ache (a), stomach ache (b) and back ache (c) (p b is a prejudice against pyelonephritis and lumbago) Proof : Due to Equation 1, A β • A α (Ω) = σ(Ω, Ω)α and σ(Ω, Ω) = β. Let us consider a universe Ω = {ω 1 , ω 2 , ω 3 } where three diseases can be encountered: pyelonephritis noted ω 1 , urinary infection noted ω 2 and lumbago noted ω 3 . Three symptoms can be observed Kidney ache (a), stomach ache (b) and back ache (c). When stomach ache is observed there is a prejudice against pyelonephritis and lumbago encoded by p b .

  β) : the prejudice decreases as in the Case 1 and 2 * p ′ (B \ A) = 1 -β * p ′ (Ω) = 1 + f (α, β) + β Example 3 (continued): Let us now consider a prejudice against urinary infection and lumbago encoded by p b = {ω 2 , ω 3 } 5/3 . The fourth columns shows the results of the conjunctive combination of m a and p b as described in Section 3.4, i.e., the prejudice disappears on ω 3 but remains on ω 2 (p ′ b ({ω 2 }) = -2/3). The mass m a ({ω 1 , ω 3 } = 0.5 is transferred to m ′ a ({ω 1 }).

Proposition 3 □ 5

 35 Given a belief state (p, m) such that p = B β and m = A α , in the cases(3 and 4) where A \ B ̸ = ∅ there is conservation of masses without addition of further contradiction :m ′ (A) + m ′ (A \ B) = m(A)Proof : For the cases 3 and 4: in both cases αβ ≤ 1 and αβ > 1, m ′ is (A \ B) α (due to Definitions 5 and 6). Hence m′ (A) = 0 and m ′ (A \ B) = 1 -α.□ When the prejudice is weak, i.e., αβ ≤ 1, we recover Dubois-Denøeux revision: Proposition 4 Given ∅ ⊂ A ⊂ Ω and ∅ ⊂ B ⊂ Ω and a belief state (p, m) such that p = B β and m = A α , according to the respective position of the two sets A and B, when αβ ≤ 1 then • A \ B = ∅ (cases 1 and 2): m ′ = B β • A α (revision) • A \ B ̸ = ∅ (cases 3 and 4): m ′ = (A \ B) α (transfer) Proof : Due to Equation (1), p • m(Ω) = σ p (Ω, Ω)m(Ω) where σ p (Ω, Ω) = p(Ω) P lp(Ω) hence σ p (Ω, Ω) = β 1 = β. Moreover, m(Ω) = α this p • m(Ω) = αβ. Now, except from Ω, m(E) ̸ = ∅ only for E = A, hence p•m(A) = 1-αβ thus p•m = A αβ , it corresponds exactly to Definitions 2 and 4. The second item corresponds directly to Definitions 5 and 6. Discussion and Related work

1 - 1 -

 11 u (1 -x -y -z)(1 -u) x -y -z u (1 -x -y -z)u (1 -x -y -z)u (1 -x -y -z)/u (β, (1 -x))

Table 1 :

 1 Computing the revision m ||C by C = {ω 1 , ω 3 } of the mass functions m in the Ellsberg paradox example, where ω 1 means white, ω 2 is black and the revision m • m

I (resp. m • m I ′ ) of a given mass m by the simple mass m I (resp. with the more complex mass m I ′ ). The table on the right gives the specialization matrix for m. In this table ω i is abbreviated i for any i ∈ {1, 2, 3}.

Table 3 :

 3 Kidney ache (a), stomach ache (b) and back ache (c) (p b is a prejudice against pyelonephritis and urinary infection) Definition 6 Given a belief state (p, m) such that p = B β and m = A α with A, B ∈ 2 Ω \ {Ω} and (A \ B) ̸ = ∅ and (B \ A) ̸ = ∅, this state is simplified into (p ′ , m ′ ) such that:

Table 4 :

 4 Four different views of "negative information": negation (i.e., combination with B), revision, retraction and strong persistent prejudice

"(good or bad) opinion that one forms in advance" (Lanoue, Discours pol. et milit., 436 in Littré,1587)

A mass is dogmatic when m(Ω) = 0.

Specialization was introduced in[START_REF] Dubois | On the unicity of dempster rule of combination[END_REF], m specializes m ′ iff there exists a square matrix Σ with general term σ(A, B) being a proportion (i.e., verifying A σ(A, B) = 1, for any B. σ(A, B) > 0 implies A ⊆ B for any A, B) such that m(A) = B σ(A, B)m ′ (B) for all A. In[START_REF] Ma | Revision rules in the theory of evidence[END_REF], the definition of specialization matrix is taken in a broader sense: only imposing that σ(A, B) > 0 implies A ∩ B ̸ = ∅ for any A, B.