Jing Zhang

Megan Teh

Jamie Kim

Megan M Eva

Romain Cayrol

Rachel Meade

Anastasia Nijnik

Xavier Montagutelli

Danielle Malo
email: danielle.malo@mcgill.ca

Jean Jaubert
email: jean.jaubert@pasteur.fr

A Loss-of-Function Mutation in the Integrin Alpha L ( Itgal ) Gene Contributes to Susceptibility to Salmonella enterica Serovar Typhimurium Infection in Collaborative Cross Strain CC042

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

Introduction

Salmonella enterica is a relatively common Gram-negative bacteria that is generally transmitted via the consumption of contaminated food or water [START_REF] Kang | Enterobacteria and host resistance to infection[END_REF]. Infection with Salmonella can lead to a variety of pathologies with worldwide health and economic costs. Human restricted Salmonella strains S. Typhi and S. Paratyphi result in typhoid fever causing an estimated 190, 000 deaths per year and is typically observed in nations lacking adequate sanitation and clean drinking water programs (2,[START_REF] Radhakrishnan | Introductory Article on Global Burden and Epidemiology of Typhoid Fever[END_REF]. Symptoms of typhoid include fever, abdominal pain and general malaise [START_REF] Larock | Salmonellae interactions with host processes[END_REF]. In contrast, non-typhoidal strains such as S. Typhimurium lead to 93.8 million cases of gastroenteritis annually [START_REF] Majowicz | The Global Burden of Nontyphoidal Salmonella Gastroenteritis[END_REF]. Symptoms of gastroenteritis involve diarrhea, vomiting and nausea [START_REF] Kang | Enterobacteria and host resistance to infection[END_REF]. In immunocompromised patients, non-typhoidal strains can also result in systemic and invasive infections involving bacteremia and sepsis [START_REF] Dhanoa | Non-typhoidal Salmonella bacteraemia: epidemiology, clinical characteristics and its' association with severe immunosuppression[END_REF].

Study of Salmonella in mouse models is typically conducted with S. Typhimurium as it is known to induce systemic infections in mice similar to the bacteremia observed in immunocompromised patients [START_REF] Kang | Enterobacteria and host resistance to infection[END_REF]. After systemic infection with S. Typhimurium, the bacteria are rapidly cleared from the bloodstream (within 2h), followed by localization of approximately 10% of the inoculum within macrophages and polymorphonuclear cells of visceral organs such as the spleen and liver where it can replicate efficiently. In order to resolve the resulting systemic infection, the host must activate a robust innate and adaptive immune response [START_REF] Kang | Enterobacteria and host resistance to infection[END_REF][START_REF] Dougan | Immunity to salmonellosis[END_REF].

Many factors are known to be involved in the clinical outcomes and the ability of the host to clear Salmonella infection in both humans and mouse models. Factors include the bacterial strain, the dosage of infection, and the host immune status, microbiome and genetic makeup [START_REF] Kang | Enterobacteria and host resistance to infection[END_REF][START_REF] Dhanoa | Non-typhoidal Salmonella bacteraemia: epidemiology, clinical characteristics and its' association with severe immunosuppression[END_REF][START_REF] Gilchrist | Genetic susceptibility to invasive Salmonella disease[END_REF][START_REF] Vogt | Gut microbiota-mediated protection against diarrheal infections[END_REF]. Host genetics is increasingly being recognized as a crucial element involved in host susceptibility to infection. While many genes such as toll like receptor 4 (TLR4), interleukin 12 (IL12) and signal transducer and activator of transcription 4 (STAT4) have been implicated in the in vivo response to Salmonella, the complete cohort of genes involved has yet to be determined [START_REF] Gilchrist | Genetic susceptibility to invasive Salmonella disease[END_REF][START_REF] Eva | Altered IFN-gamma-mediated immunity and transcriptional expression patterns in N-Ethyl-N-nitrosourea-induced STAT4 mutants confer susceptibility to acute typhoid-like disease[END_REF][START_REF] Mastroeni | Interleukin-12 is required for control of the growth of attenuated aromatic-compound-dependent salmonellae in BALB/c mice: role of gamma interferon and macrophage activation[END_REF][START_REF] Qureshi | Endotoxin-tolerant mice have mutations in Toll-like receptor 4 (Tlr4)[END_REF]. Nonetheless, establishing the genetic factors that influence disease is essential for the elucidation of host immune response pathways and will enable the identification of novel targets for therapeutic drug development. Recent genome-wide association studies have revealed the contribution of host genetic determinants of enteric fever and invasive non-typhoidal Salmonella infection in human populations [START_REF] Dunstan | Variation at HLA-DRB1 is associated with resistance to enteric fever[END_REF][START_REF] Alvarez | Human genetic variation in VAC14 regulates Salmonella invasion and typhoid fever through modulation of cholesterol[END_REF][START_REF] Gilchrist | Risk of nontyphoidal Salmonella bacteraemia in African children is modified by STAT4[END_REF].

One approach used for the detection of novel genes involved in complex traits such as Salmonella susceptibility utilizes a murine genetic reference population known as the Collaborative Cross (CC) [START_REF] Morgan | Informatics resources for the Collaborative Cross and related mouse populations[END_REF]. While traditional models tend to use highly homogenous mouse populations, the CC has been designed to model the range of genetic variation of human population [START_REF] Roberts | The polymorphism architecture of mouse genetic resources elucidated using genome-wide resequencing data: implications for QTL discovery and systems genetics[END_REF]. The CC is a panel of recombinant inbred mice derived from eight founder strains including five laboratory strains and three wild-derived inbred strains [START_REF] Churchill | The Collaborative Cross, a community resource for the genetic analysis of complex traits[END_REF] resulting in highly variable phenotypes. The genomes of the CC strains feature relatively well dispersed recombination sites and balanced allele origins from all eight founder strains [START_REF] Aylor | Genetic analysis of complex traits in the emerging collaborative cross[END_REF] allowing for the genetic dissection of complex trait [START_REF] Saul | High-Diversity Mouse Populations for Complex Traits[END_REF]. Moreover, the CC serves as a platform to develop improved models of infectious disease and to map loci associated with variations in susceptibility to pathogens [START_REF] Noll | The Collaborative Cross: A Systems Genetics Resource for Studying Host-Pathogen Interactions[END_REF].

We previously utilized the CC to demonstrate that host genetic factors contribute to significant variation in Salmonella susceptibility [START_REF] Zhang | Identification of new loci involved in the host susceptibility to Salmonella Typhimurium in collaborative cross mice[END_REF]. Following challenge of 35 CC strains with S. Typhimurium we showed that the bacterial burdens of the spleen and liver were significantly different between strains [START_REF] Zhang | Identification of new loci involved in the host susceptibility to Salmonella Typhimurium in collaborative cross mice[END_REF]. One strain in particular known as CC042/GeniUnc (CC042) was shown to be extremely susceptible to S. Typhimurium infection with greater than 1000-fold higher colony forming units (CFU) in the spleen and liver as compared to the highly susceptible C57BL/6J reference strain [START_REF] Zhang | Identification of new loci involved in the host susceptibility to Salmonella Typhimurium in collaborative cross mice[END_REF]. It has been shown that a missense mutation in the Solute Carrier Family 11 member 1 gene (Slc11a1) contributes to the high susceptibility of C57BL/6J strain [START_REF] Vidal | The Ity/Lsh/Bcg locus: natural resistance to infection with intracellular parasites is abrogated by disruption of the Nramp1 gene[END_REF] and CC042 strain inherited this missense mutation from C57BL/6J founder strain [START_REF] Zhang | Identification of new loci involved in the host susceptibility to Salmonella Typhimurium in collaborative cross mice[END_REF]. While the Slc11a1 mutation partially accounts for the high susceptibility of CC042 mice, other host genetic variants are required to explain the extreme CC042 phenotype and have yet to be identified. The current work reports the characterization of the CC042 immunophenotype, the mapping of two loci associated with the susceptibility phenotype and the identification of a causal variant. CC042 mice were found to have a primary immunodeficiency with alterations in spleen, thymus and bone marrow hematopoietic cell populations. Quantitative trait locus mapping showed that the Salmonella susceptibility phenotype was controlled by at least two regions on chromosome 7 (Stsl6 and Stsl7). Within the Stsl7 locus, a de novo 15 bp deletion mutation in the intron 1 splice acceptor site of the integrin alpha L chain gene (Itgal) resulting in complete protein abrogation was shown to increase susceptibility to Salmonella infection. This study provides a foundation for investigating the role of Itgal in the response to Salmonella and illustrates how the CC can serve to identify mechanisms of infectious and immunological traits.

Results

Clinico-pathological characterization of CC042

To characterize the observed Salmonella susceptibility in CC042 mice, clinicopathological parameters were evaluated in naïve CC042 mice where C57BL/6J mice were used as reference as they have a well-characterized immune phenotype prior and during Salmonella infection. While C57BL/6J mice are typically considered susceptible to infection, they were previously shown to be more resistant to Salmonella Typhimurium infection relative to CC042 mice [START_REF] Zhang | Identification of new loci involved in the host susceptibility to Salmonella Typhimurium in collaborative cross mice[END_REF].

CC042 did not present any other overt visible phenotypes prior to infection with the exception of a characteristic white head spot, most likely inherited from the WSB/EiJ CC founder parent [START_REF] Zhang | HTreeQA: Using Semi-Perfect Phylogeny Trees in Quantitative Trait Loci Study on Genotype Data[END_REF]. Age and sex matched CC042 mice had comparable body weights to C57BL/6J mice (Fig. 1a). However, CC042 mice displayed significantly smaller spleens and thymi, in terms of mass, in comparison to the C57BL/6J (Fig. 1b and1c). Examination of haematological parameters in naïve CC042 mice indicated lower total white blood cell and lymphocyte counts compared to C57BL/6J (Table 1). CC042 mice presented a small but significant increase in the number of red blood cells, hemoglobin and mean corpuscular hemoglobin concentration (MCHC) (Table 1). In addition, the splenic microarchitecture of CC042 mice was assessed using H&E staining and was shown to present moderate extramedullary hematopoiesis with prominent presence of megakaryocytes in comparison to C57BL/6J (Fig. 2a). Histopathological examination of CC042 liver, and kidney was normal (data not shown). The extramedullary erythropoiesis may explain the increased numbers of circulating RBCs and may be indicative of bone marrow failure.

CC042 mice display a primary immunodeficiency

To identify if the reduced splenic size (Fig. 3a) observed in CC042 mice may be due to alterations in the cellular immune compartment, flow cytometry for lymphoid and myeloid cells was performed on the splenocytes of CC042 and C57BL/6J naïve mice (gating scheme shown in Supplementary Fig. S1a). The mean total splenocyte counts in CC042 mice was significantly reduced to 31.2  5.1 x 10 6 cells compared to 75.1  3.0 x 10 6 for the C57BL/6J spleens (Fig. 3b).

CC042 mice showed significant alterations in the splenic lymphoid compartment. Despite carrying similar numbers of splenic CD4 + T cells compared to C57BL/6J mice, CC042 mice displayed a reduction in the number of activated CD4 + T cells, as measured by CD69 expression (Fig. 3c and3d). In addition, numbers of splenic CD8 + T cells and activated CD69 expressing CD8 + T cells were also significantly reduced in CC042 mice compared to C57BL/6J (Fig. 3e and3f). Further CD4 + and CD8 + T cells effector function was assessed by measuring intracellular staining of IFNγ and TNFα after activation with anti-CD3 and anti-CD28. We observed diminished IFNγ and TNFα production from CC042 T cells both in naïve mice and after infection (Fig. 3g-3j). Reduced numbers of neutrophils, monocytes, macrophages and B cells were also observed in CC042 prior to infection (Fig. 3k-3n). Overall, CC042 mice present an abnormal immunophenotype characterized by reduced spleen size and generalized reduction of spleen cells affecting lymphoid and myeloid compartments. This reduction in splenic immune cells together with the observed leukopenia indicates a potential defect in leukocyte development.

CC042 mice have reduced thymic cellularity and altered T cell development

To assess if the reduction in CD8 + T cells and total activated T cells in the spleen of CC042 mice may potentially be due to defective T cell maturation, flow cytometry of the thymus was carried out (gating scheme shown in Supplementary Fig. S1b). In agreement with the previous observation that CC042 mice have reduced thymus size, CC042 mice displayed a two-fold reduction in total thymocyte counts (Fig. 4a). While the percentages of DP (double positive for CD4 and CD8), SP (single positive for either CD4 or CD8) CD4+ and CD8+ T cells were comparable between CC042 and C57BL/6J mice due to the reduction in thymocyte numbers in CC042 mice, a significant decrease in DN (double negative for CD4 and CD8), DP, SP CD4+ and CD8+ T cells (25) cell counts was observed in CC042 (Fig. 4b and4c). Further examination of the DN subset revealed significant alterations in CC042 mice (Fig. 4d and4e). While no significant difference in the proportion of DN1 cells was observed, a significant reduction in the proportion of DN2 stage thymocytes was present in CC042 thymi compared to C57BL/6J controls.

Unexpectedly, the diminished proportion of DN2 cells did not result in suppression of downstream subsets as a comparable fraction of DN3 thymocytes and a significantly increased proportion of DN4 cells were observed in CC042 mice compared to C57BL/6J controls. Given the reduction in total thymocytes, a significant decrease in DN1, DN2, DN3 and DN4 total cell counts was observed in CC042 mice compared to C57BL/6J controls. Overall, CC042 mice displayed a reduction in thymocyte numbers and altered progression of T cell precursors through maturation, specifically in the DN stages.

CC042 showed alterations in the hematopoietic progenitor populations

The observed reduction in peripheral blood leukocytes and cellularity of the spleen and thymus in CC042 mice suggested that the hematopoietic development of these populations may be impaired. To investigate haematopoiesis in CC042 mice, flow cytometry (Table 2) of bone marrow haematopoietic progenitor populations was conducted.

Bone marrow haematopoietic stem cells (HSCs) differentiate to multipotent progenitors (MPPs) sub-populations defined based on the expression of the CD150, CD34, CD48 and Flt3 cell surface markers (Table 2 and Fig. 5a) [START_REF] Wilson | Hematopoietic stem cells reversibly switch from dormancy to self-renewal during homeostasis and repair[END_REF]. CC042 bone marrow contained less cells (Fig. 5b) and a reduced number of LSK (Lin -Sca-1 + c-Kit + ) cells compared to C57BL/6J with equivalent numbers of haematopoietic stem cells (HSCs) (Fig. 5c and5d). However, examination of LSK subsets revealed significant alterations in the proportions of multipotent progenitors (MPP) subpopulations. The CC042 LSK compartment displayed significant reduction in downstream MPP3 and MPP4 cells in comparison to C57BL/6J controls (Fig. 5c and5d). The depletion of MPP3 and MPP4 populations in CC042 bone marrow suggests a maturation block in the progression from MPP1 to MPP3 and MPP4 cell stages.

MPP sub-populations give rise to common lymphoid progenitors (CLPs) and common myeloid progenitors (CMPs) (Fig. 5a and5e) [START_REF] Cabezas-Wallscheid | Identification of regulatory networks in HSCs and their immediate progeny via integrated proteome, transcriptome, and DNA methylome analysis[END_REF]. CMPs form both megakaryocyte-erythrocyte progenitors (MEPs) which produce RBCs and platelets and granulocyte-macrophage progenitors (GMPs) [START_REF] Cabezas-Wallscheid | Identification of regulatory networks in HSCs and their immediate progeny via integrated proteome, transcriptome, and DNA methylome analysis[END_REF]. CC042 mice showed a decreased number of CMP progenitor cells but an increase in downstream GMPs compared to C57BL/6J mice (Fig. 5f). No significant difference was observed in the number of CLPs between CC042 and C57BL/6J mice (Fig. 5g). The maturational blocks observed upstream of MPP3, MPP4 and CMP progenitors could partially explain the reduced hematopoietic cell populations present in peripheral organs.

CC042 mice mount a distinct response to infection

To characterize the impact of infection on spleen and liver, histopathological examination of CC042 spleens and livers was conducted three days after Salmonella infection. The livers of C57BL/6J harbored typical lesions for this strain characterized by multifocal necrotic lesions with aggregation of histiocytes and neutrophils typical of granulomas (Fig. 2b). The necrotic hepatocyte foci were found to be smaller and less abundant in CC042 mice (2-4 foci per 40X field of view) than in C57BL/6J mice (4-5 foci per field of view) (Fig. 2b). CC042 spleens were also observed to have reduced neutrophil infiltration during infection compared to C57BL/6J spleens (Fig. 2b). The increased megakaryocyte numbers and erythropoiesis observed in the spleen of uninfected CC042 mice (Fig. 2a) was also present in the spleen of CC042 infected mice (Fig. 2b).

During infection, CC042 mice displayed a severe splenic immunodeficiency with an important reduction in the number of activated CD4 + T cells, CD8 + T cells, activated CD8 + T cells, neutrophils, monocytes, macrophages and B cells compared to C57BL/6J (Fig. 3c-3j). The reduction in CD8 + T cell counts combined with the generalized defect in total T cell activation and myeloid cell numbers may account for the observed increase in Salmonella susceptibility.

A mutation within Itgal is responsible for susceptibility of CC042 mice to Salmonella infection

To identify the genes which, in addition to Slc11a1, control the extreme susceptibility to Salmonella Typhimurium of CC042 mice, we produced an F2 cross between CC042 and C57BL/6NCrl, a strain closely related with C57BL/6J, with the same Slc11a1 missense mutation. C57BL/6N was separated from C57BL/6J in 1951 (https://www.jax.org/strain/005304). Both strains have accumulated their own variants and are considered as substrains. The genome wide polymorphisms between the two C57BL/6 substrains is estimated to ~10,700 SNPs based on C57BL/6N sequencing data [START_REF] Simon | A comparative phenotypic and genomic analysis of C57BL/6J and C57BL/6N mouse strains[END_REF]. This allows to find polymorphic markers between the two substrains, and to perform the mapping of quantitative trait loci (QTL) even in regions where CC042 genome is of C57BL/6J origin. Infected F1 mice showed intermediate bacterial loads in the liver and spleen compared with the parental strains (Fig. 6a and6b), while bacterial loads in F2 mice spanned over the parental range. Ninety-four F2 mice with the highest or lowest liver bacterial loads were selected for QTL mapping. Since bacterial loads in liver and spleen were strongly correlated in the 94 selected individuals (Fig. 6c), QTL mapping was performed on liver bacterial loads.

QTL mapping identified only two significant QTLs (at 0.05 genome wide significance level), both on Chromosome 7 (Fig. 6d and6e), which were named Salmonella Typhimurium susceptibility locus-6 (Stsl6) and Stsl7. Details for each QTL are given in Table 3. Stsl6 (peak position at 46.23Mb) showed semi-dominant mode of inheritance, with heterozygotes having an intermediate bacterial load compared to the two types of homozygotes (Fig. 6f). The CC042 allele at Stsl7 (peak at 123.78Mb) was recessive and homozygotes had 10 times higher liver bacterial loads (Fig. 6g).

CC042 inherited Stsl6 and Stsl7 QTL regions from the WSB founder (see Supplementary Fig. S2). Other CC strains also inherited WSB alleles in either or both QTL regions but had much lower bacterial loads than CC042 (Supplementary Fig. S2). In particular, CC035 also carried WSB-derived alleles at Stsl6 and Stsl7. This result suggested that the susceptibility alleles present in CC042 at these two loci could result from de novo mutations which occurred during the development of the CC042 strain. These private variants have been previously identified by the sequencing of CC strains and are publicly available [START_REF] Srivastava | Genomes of the Mouse Collaborative Cross[END_REF]. Three private variants were identified in the Stsl6 confidence interval in genes Ush1c, Ccdc123, 4930435C17Rik and only one within Stsl7 interval in the integrin alpha L(Itgal) gene, also known as Cd11a. This variant is a 15 base pair deletion located at the 3' end of intron 1 and resulting in the disruption of the intron 1 splice acceptor site (Fig. 7a). We hypothesized that the intron 1 splice donor would attack the intron 2 splice acceptor resulting in skipping of exon 2 during splicing. Joining exon 1 to exon 3 would create a frame shift and a downstream premature stop codon, resulting in a truncated form of ITGAL protein.

To verify this prediction, reverse transcription and amplification of Itgal transcripts was conducted in C57BL/6J and CC042 cells using primers flanking exon 2. C57BL/6J cells were found to express a 338 bp transcript indicative of the inclusion of exon 2 (Fig. 7b). In contrast, CC042 cells produced a 238 bp transcript corresponding to the loss of the 100 bp long exon 2. To confirm the predicted loss of ITGAL protein, flow cytometry of splenic leukocytes labelled with a fluorescently conjugated anti-ITGAL antibody was conducted. As predicted, none of the CC042 leukocyte populations examined, including CD4 + and CD8 + T cells, expressed the ITGAL protein (Fig. 7c). In contrast, all C57BL/6J leukocytes examined expressed the ITGAL protein.

To confirm in vivo the role of the Itgal deletion in the extreme susceptibility of CC042 strain to S. Typhimurium, we performed a quantitative complementation test. Compound heterozygous mice carrying a KO Itgal allele and a CC042 deleted Itgal allele had significantly higher liver bacterial loads (~6.3 Log CFUs/g liver, Fig. 8a) compared with mice heterozygous for either of these two alleles (~5.1-5.3 Log CFUs/g liver). Therefore, there was no significant difference between Itgal + / Itgal + and Itgal + / Itgal KO mice in the B6 background (red line), while the difference between Itgal + / Itgal del and Itgal KO / Itgal del mice was highly significant (blue line, p = 2.1 10 -6 ). These results demonstrate that the Itgal deletion present in CC042 mice contributes to their increased susceptibility. Remarkably, the Itgal KO mutation had only a mild impact on liver bacterial load in a B6 inbred background with no significant difference between Itgal + / Itgal + and Itgal KO / Itgal KO mice, but unexpected slightly higher bacterial load in Itgal + / Itgal KO mice compared with Itgal + / Itgal + mice (Fig. 8b).

Discussion

The CC is a murine genetic reference population that was developed to reflect the genetic diversity and complexity of the human population [START_REF] Aylor | Genetic analysis of complex traits in the emerging collaborative cross[END_REF]. In a previous study, we utilized the CC to investigate the role of genetic factors on host susceptibility to Salmonella Typhimurium infection [START_REF] Zhang | Identification of new loci involved in the host susceptibility to Salmonella Typhimurium in collaborative cross mice[END_REF]. The CC042 strain was identified as being highly susceptible to S. Typhimurium infection [START_REF] Zhang | Identification of new loci involved in the host susceptibility to Salmonella Typhimurium in collaborative cross mice[END_REF]. In this study, we sought to characterize the immunophenotype of the CC042 strain and to identify the underlying genetic factors contributing to the extreme susceptibility phenotype. We demonstrated that CC042 mice display a generalized immunodeficiency with reduced cellularity of the spleen, thymus and bone marrow. CC042 mice also exhibited alterations in immune progenitor development in both the thymus and the bone marrow. Using genetic linkage mapping and in vivo complementation testing we identified a de novo 15 bp deletion in the intron 1 of the Itgal gene as causatively linked to the Salmonella susceptibility phenotype in CC042 mice. This mutation results in exon 2 skipping and premature stop codon.

The Itgal mutation identified in CC042 mice was demonstrated to result in complete loss of ITGAL protein expression on all leukocytes examined. ITGAL is an integrin α-chain protein which in conjunction with the ITGB2 β-chain (ITGB2 can dimerize with any of ITGAL, ITGAM, ITGAD or ITGAX) forms the lymphocyte function associated antigen 1 (LFA-1) integrin complex [START_REF] Schittenhelm | beta2 Integrins As Regulators of Dendritic Cell, Monocyte, and Macrophage Function[END_REF]. The LFA-1 complex is expressed on all leukocytes and plays a critical role in various immunological functions [START_REF] Schittenhelm | beta2 Integrins As Regulators of Dendritic Cell, Monocyte, and Macrophage Function[END_REF] Circulating leukocytes utilize LFA-1 to bind to its cognate ligand ICAM-1 expressed on endothelial cells [START_REF] Evans | Integrins in immunity[END_REF][START_REF] Hogg | The insider's guide to leukocyte integrin signalling and function[END_REF]. Binding allows leukocytes to anchor in the bloodstream prior to extravasation into lymph nodes or inflamed tissues [START_REF] Evans | Integrins in immunity[END_REF][START_REF] Hogg | The insider's guide to leukocyte integrin signalling and function[END_REF]. LFA-1 also enables stabilization of the immunological synapse during T cell activation, thus permitting prolonged T cell receptor (TCR):major histocompatibility complex (MHC) interactions [START_REF] Evans | Integrins in immunity[END_REF][START_REF] Hogg | The insider's guide to leukocyte integrin signalling and function[END_REF].

Given that LFA-1 plays a crucial role in leukocyte migration and T cell activation, it is unsurprising that absence of ITGAL function may result in ineffective immune responses to Salmonella infection in CC042 mice.

CC042 mice displayed significant defects in splenic, thymic and bone marrow immune cell populations. A study by Bose et al. similarly reported that Itgal -/-mice had reduced splenic and thymic cellularity but did not observe a reduction in bone marrow cell counts [START_REF] Bose | CD11a is essential for normal development of hematopoietic intermediates[END_REF]. Moreover, Itgal -/-mice or mice treated with anti-LFA-1 antibodies are consistently reported to have reductions in splenic T cells [START_REF] Bose | CD11a is essential for normal development of hematopoietic intermediates[END_REF][START_REF] Bose | CD11a Regulates Effector CD8 T Cell Differentiation and Central Memory Development in Response to Infection with Listeria monocytogenes[END_REF][START_REF] Revilla | Treatment with anti-LFA-1 alpha monoclonal antibody selectively interferes with the maturation of CD4-8+ thymocytes[END_REF]. Multiple reports specifically highlight defects in the CD8 + T cell subset in Itgal -/-mice in terms of numbers, activation, proliferation and recruitment to infected tissues [START_REF] Bose | CD11a Regulates Effector CD8 T Cell Differentiation and Central Memory Development in Response to Infection with Listeria monocytogenes[END_REF][START_REF] Revilla | Treatment with anti-LFA-1 alpha monoclonal antibody selectively interferes with the maturation of CD4-8+ thymocytes[END_REF][START_REF] Rutigliano | Treatment with anti-LFA-1 delays the CD8+ cytotoxic-T-lymphocyte response and viral clearance in mice with primary respiratory syncytial virus infection[END_REF]. CC042 mice were similarly observed to have a more significant impairment in CD8 + T cells in comparison to CD4 + T cell populations. Defects in T cell activity may be due to the inability of T cells to sustain long-lived interactions with APCs. Itgal -/-T cells are noted to traffic at a quicker rate through lymph nodes while T cells expressing low levels of LFA-1 have been shown to only transiently form contacts with APCs resulting in diminished expression of effector proteins including IFN-γ [START_REF] Capece | A novel intracellular pool of LFA-1 is critical for asymmetric CD8(+) T cell activation and differentiation[END_REF][START_REF] Reichardt | A role for LFA-1 in delaying T-lymphocyte egress from lymph nodes[END_REF]. This is consistent with our observation of low expression of IFN-γ by CD4 + and CD8 + T cells after TCR stimulation in CC042 mice. Notably, both CC042 and Itgal -/-mice have previously been shown to display suppression of IFN-γ in lung tissue upon M. tuberculosis infection [START_REF] Ghosh | The LFA-1 adhesion molecule is required for protective immunity during pulmonary Mycobacterium tuberculosis infection[END_REF][START_REF] Smith | Tuberculosis Susceptibility and Vaccine Protection Are Independently Controlled by Host Genotype[END_REF]. Production of IFN-γ by T cells is crucial for the clearance of intracellular bacterial infections such as Salmonella as IFN-γ activates bacterial killing within infected macrophages [START_REF] Ingram | Interferon-gamma in Salmonella pathogenesis: New tricks for an old dog[END_REF][START_REF] Ramirez-Alejo | Innate defects of the IL-12/IFN-gamma axis in susceptibility to infections by mycobacteria and salmonella[END_REF]. Notably, IFN-γ production has been shown to be inducible by ISG15 through an ITGAL dependent signalling mechanism [START_REF] Swaim | Extracellular ISG15 Signals Cytokine Secretion through the LFA-1 Integrin Receptor[END_REF]. Thus, it is most likely that the immunosuppressed phenotype observed in CC042 mice in response to Salmonella may be due to reduced T cell activation and suppression of IFN-γ production via the ITGAL dependent pathway.

Alterations in thymocyte maturation were also observed in CC042 mice potentially explaining the corresponding reduction in peripheral T cell counts. ITGAL has been implicated in thymopoiesis as Itgal -/-mice exhibit a general reduction in all thymocytes while use of anti-ITGAL antibodies has been shown to result in impairment of double positive CD4 + /CD8 + and single positive CD8 + T cell development [START_REF] Bose | CD11a is essential for normal development of hematopoietic intermediates[END_REF][START_REF] Fine | The role of LFA-1/ICAM-1 interactions during murine T lymphocyte development[END_REF]. Thymocyte defects may be due to alterations further upstream as bone marrow hematopoiesis was also defective in CC042 mice. Hematopoietic progenitor populations have also been observed to be altered in Itgal -/-mice [START_REF] Bose | CD11a is essential for normal development of hematopoietic intermediates[END_REF]. In competitive reconstitution assays in irradiated mice, Itgal -/-derived bone marrow progenitors were unable to compete against WT cells presenting a role for ITGAL in hematopoietic cell generation [START_REF] Bose | CD11a is essential for normal development of hematopoietic intermediates[END_REF]. It should be noted that the mutation in Kitl also inherited from WSB/EiJ may be contributing to the defects in hematopoietic progenitor generation.

Several animal studies have delineated a significant role for ITGAL in the response to infection. Itgal -/-mice have been reported to be highly susceptible to bacterial infection by both intracellular M. tuberculosis and extracellular Streptococcus pneumoniae as characterized by reduced survival, increased bacterial burdens and defects in leukocyte recruitment to infected tissues [START_REF] Ghosh | The LFA-1 adhesion molecule is required for protective immunity during pulmonary Mycobacterium tuberculosis infection[END_REF][START_REF] Prince | The Differential Roles of LFA-1 and Mac-1 in Host Defense Against Systemic Infection with Streptococcus pneumoniae[END_REF]. During M. tuberculosis infection, the absence of the ITGAL protein was shown to result in impaired containment of the bacteria demonstrated by diffuse lung granuloma formation [START_REF] Ghosh | The LFA-1 adhesion molecule is required for protective immunity during pulmonary Mycobacterium tuberculosis infection[END_REF]. Interestingly, upon M. tuberculosis infection, CC042 mice displayed significantly increased bacterial burdens of the lung and spleen, increased necrotic lung granuloma formation and more rapid disease progression compared to C57BL/6J mice [START_REF] Smith | Tuberculosis Susceptibility and Vaccine Protection Are Independently Controlled by Host Genotype[END_REF]. This phenotype is likely partly explained by the Itgal deletion we have reported here. Notably, in a recent study (https://www.biorxiv.org/content/10.1101/785725v1?rss=1), Smith et al. reported their work on variants controlling susceptibility to M. tuberculosis in the Collaborative Cross and that Itgal gene was involved in CC042 extreme susceptibility to M. tuberculosis. Studies utilizing anti-LFA-1 antibodies to neutralize ITGAL protein activity have shown that ITGAL function is necessary for timely clearance of respiratory syncytial virus (RSV) and for control of parasitemia during Trypanosoma cruzi infection [START_REF] Rutigliano | Treatment with anti-LFA-1 delays the CD8+ cytotoxic-T-lymphocyte response and viral clearance in mice with primary respiratory syncytial virus infection[END_REF][START_REF] Ferreira | LFA-1 Mediates Cytotoxicity and Tissue Migration of Specific CD8(+) T Cells after Heterologous Prime-Boost Vaccination against Trypanosoma cruzi Infection[END_REF]. In contrast, two separate studies have reported that Itgal -/- mice display increased resistance to Listeria monocytogenes infection [START_REF] Bose | CD11a Regulates Effector CD8 T Cell Differentiation and Central Memory Development in Response to Infection with Listeria monocytogenes[END_REF][START_REF] Miyamoto | Neutrophilia in LFA-1-deficient mice confers resistance to listeriosis: possible contribution of granulocyte-colony-stimulating factor and IL-17[END_REF]. Counterintuitively, resistance to L. monocytogenes occurs despite Itgal -/-mice displaying a significant reduction in CD8 + T cell numbers and lytic activity during L. monocytogenes infection [START_REF] Bose | CD11a Regulates Effector CD8 T Cell Differentiation and Central Memory Development in Response to Infection with Listeria monocytogenes[END_REF]. The discrepancy in response to L. monocytogenes as compared to other pathogens may be due to differential methods of leukocyte recruitment in response to various inflammatory stimuli. One study found that neutrophil recruitment during S. Typhimurium and L. monocytogenes infection operates in ITGB2 dependent and independent manners respectively (48).

To our surprise, mice homozygous for the Itgal KO mutation on the B6 inbred background did not show increased liver bacterial load while increased susceptibility was observed in compound heterozygous mice on a B6/CC042 F1 background. The mild increase of bacterial load (0.28 log CFUs) in Itgal + / Itgal KO mice compared with Itgal + / Itgal + mice has no simple explanation and would require confirmation experiments and further investigations. The major differences between the B6 and B6xCC042 backgrounds suggest that the impact of ITGAL loss of function may depend on the genotype at other loci. There are in fact many examples were the phenotype induced by the inactivation of a gene is influenced by the genetic background [START_REF] Montagutelli | Effect of the genetic background on the phenotype of mouse mutations[END_REF][START_REF] Nadeau | Modifier genes and protective alleles in humans and mice[END_REF].

One of the modifier loci could be Stsl6 which we identified in the F2 cross. While the comparison of the founder haplotypes in CC042 and other CC strains suggested that, like for Stsl7, the causative variant was likely a de novo mutation proper to CC042, we failed to identify a candidate private variant for Stsl6 from the published data. Moreover, due to the linkage between these two loci, the F2 did not allow to investigate genetic interactions between the two loci. However, loci other than Stsl6 are likely genetic modifiers of the Itgal loss of function mutation and could explain the difference of phenotype between B6 and (B6xCC042)F1 genetic backgrounds. Their identification could use a two-generation cross where all mice are homozygous for a deficient Itgal allele. The strong effect of the Itgal mutation observed in the (Itgal-KO x CC042)F1s suggests that the CC042 alleles at these modifiers act in a dominant fashion. Therefore, evaluating the susceptibility to S. Typhimurium of progeny from a (Itgal-KO x CC042)F1 x Itgal-KO backcross could identify genomic regions involved in the variable impact of the Itgal mutation. The identification of the causal genes would require combining positional information from genetic mapping with other approaches including immunophenotyping of the Itgal-KO and (Itgal-KO x CC042)F1 parents and gene expression analysis in relevant tissues or cell types. One of the main advantages of the CC is the ability to model complex traits in genetically diverse populations [START_REF] Saul | High-Diversity Mouse Populations for Complex Traits[END_REF], in particular for studying host-pathogen interactions [START_REF] Noll | The Collaborative Cross: A Systems Genetics Resource for Studying Host-Pathogen Interactions[END_REF]. The hope is that genetic factors uncovered through CC mouse studies may be applicable to studies of human disease [START_REF] Threadgill | The collaborative cross: a recombinant inbred mouse population for the systems genetic era[END_REF].

Indications that ITGAL may be important for responses to Salmonella infection in humans is supported by the finding that ITGAL is upregulated upon infection with S. Typhi in human volunteers [START_REF] Mcarthur | Activation of Salmonella Typhi-specific regulatory T cells in typhoid disease in a wild-type S. Typhi challenge model[END_REF]. Moreover, ITGAL has been implicated in a range of inflammatory diseases in humans. A recent genome wide association study identified an ulcerative colitis risk allele at the Itgal locus resulting in upregulation of ITGAL protein expression [START_REF] De Lange | Genome-wide association study implicates immune activation of multiple integrin genes in inflammatory bowel disease[END_REF]. ITGAL has also shown to be upregulated in systemic lupus erythematosus likely due to reported DNA hypomethylation at the ITGAL promoter [START_REF] Lu | Demethylation of ITGAL (CD11a) regulatory sequences in systemic lupus erythematosus[END_REF][START_REF] Balada | Clinical and serological findings associated with the expression of ITGAL, PRF1, and CD70 in systemic lupus erythematosus[END_REF]. Meanwhile, gene pathway analysis has uncovered a multiple sclerosis susceptibility gene network involving both Itgal and its cognate receptor gene Icam1 [START_REF] Damotte | A gene pathway analysis highlights the role of cellular adhesion molecules in multiple sclerosis susceptibility[END_REF]. Furthermore, deficiency of ITGAL's dimerization partner, ITGB2, results in leukocyte adhesion deficiency I (LAD-I) which is classically characterized by recurrent bacterial infections [START_REF] Hanna | Leukocyte adhesion deficiencies[END_REF]. These studies suggest that overexpression of ITGAL may result in excessive activation of the immune system resulting in inflammatory disease while deficiency may result in immunosuppression.

Characterization of the CC042 mouse strain has provided a foundation for future use of the CC in infection susceptibility studies. We have used this model to identify and characterize the novel Salmonella susceptible CC042 strain and to identify Itgal as a gene critical in the Salmonella response pathway. CC042/GeniUnc (CC042) mice were originally generated through the CC at Geniad [START_REF] Morahan | Establishment of "The Gene Mine": a resource for rapid identification of complex trait genes[END_REF] in Australia and then maintained at the Institut Pasteur (Paris) and McGill University (Montréal) under specific-pathogen-free conditions. C57BL/6J (B6) mice were purchased from the Jackson Laboratory (stock #000664). C57BL/6NCrl (B6N) mice used to generate (B6NxCC042)F2s were purchased from Charles River. B6.129S7-Itgal tm1Bll /J (Itgal KO) mice used to perform complementation tests were purchased from the Jackson Laboratory (stock #005257). F1s were obtained by crossing CC042 males with B6N females and were intercrossed to produce F2s.

Material and methods

Ethics statement and animals

Animal

Salmonella Typhimurium infection

The infectious dose was generated via culture of frozen S. Typhimurium strain SL1344 stocks in trypticase soy broth at 37°C until an optical density of 0.1-0.2 at 600 nm was reached.

To determine the specific bacterial concentration, the bacterial suspension was diluted in saline and plated on trypticase soy agar (TSA). Using the resulting concentration, an inoculum of 1000 CFU in 200 μL was administered to mice via intravenous injection into the caudal vein. All healthy mice aged between 7-12 weeks were included. Investigators were blinded to genotypes during the monitoring of Salmonella-infected mice. The dose was verified via serial dilution of the inoculum and bacterial culture on TSA. Following infection, mice were monitored and assessed using body condition scoring. Mice were humanely euthanized at different time points post Salmonella infection for sample collection. To determine bacterial loads, the spleen and liver were aseptically removed, weighed and added to 0.9% saline. The organs were homogenized using a Polytron homogenizer (Kinematica, Bohemia, NY), serially diluted in PBS and cultured on TSA.

Hematology

For complete blood count and white blood cell differential, blood samples were collected in EDTA tubes from mice aged 8-10 weeks. Analyses were performed at the Comparative Medicine Animal Resources Centre, McGill University.

Histology

Tissues were collected and placed in 10% formalin at room temperature for 24 hours to enable fixation. The organs were then transferred to 70% ethanol solution at 4°C followed by organ processing, embedding and sectioning at the Goodman Cancer Research Center histology facility, McGill University. The resulting tissue sections were stained with hematoxylin and eosin prior to microscopic examination.

Flow cytometry analyses

For flow cytometry analyses, whole spleens and thymi were aseptically collected, weighed and transferred to 3 mL of phosphate buffered saline (PBS). Tissues were mechanically dissociated using the backend of 3 ml syringes into a 70 μm strainer (Fisherbrand). The resulting cell suspensions were passed through a second 70 μm strainer and transferred to a 15 mL conical tube. The cells were spun at 1400 revolutions per minute (RPM) for 5 minutes at 4°C. The supernatant was discarded and the pellets were resuspended in 10 mL (spleen) or 5 mL (thymus) of ammonium-chloride-potassium (ACK) lysis buffer at room temperature. The suspension was spun at 1400 RPM for 5 minutes at 4°C and the supernatant was discarded. The cell pellets were resuspended in 10 mL PBS and passed through a third cell strainer to generate single cell suspensions. For bone marrow cells preparations, femurs were aseptically collected. The bones were cleaned of flesh using a scalpel blade. The epiphyses of the femurs were cut and a 25 G needle and syringe was used to pass 2 mL of PBS through the bone to displace the bone marrow.

The bone marrow containing media was transferred to a 50 mL tube and 2 mL of red blood cell lysis buffer (Sigma Aldrich) was added. After 1 minute of gentle mixing, 15 mL of PBS was added and the tubes were centrifuged for 7 minutes at 1400 RPM and the supernatant discarded.

The resulting cell pellet was resuspended in 2 mL of PBS.

The following monoclonal antibodies were used for flow cytometric analysis of spleen, thymus and bone marrow samples. Invitrogen: CD4-PE-Cy7 (GK1.5), CD11b-APC (M1/70), Ly6C-PE (HK1.4). BioLegend: B220/CD45R-PerCP-Cy5.5 (RA3-GB2), CD8α-APC-Cy7 (53-6.7), CD11a-AlexaFluor488 (M17/4), CD49b-pacific blue (DX5), c-Kit-PE-Cy7 (2B8), F4/80-PE-Cy7 (BM8), Flt3-PE (A2F10), IL7R-PE (A7R34), Ly6G-APC-Cy7 (1A8), Sca-1-APC-Cy7 (D7), TER119-PerCP-Cy5.5 (TER-119). eBioscience: B220/CD45R-APC (RA3-6B2), B220/CD45R-PerCP-Cy5.5 (RA3-GB2), CD11b-PerCP-Cy5.5 (M1/70), CD11c-eFluor450 (N418), CD16/CD32-FITC (93), CD25-AlexaFluor488 (eBio3C7), CD34-eFluor450 (RAM34), CD44-PE (IM7), CD48-HM48.1 (FITC), CD69-FITC (H1-2F3), CD150-alexafluor647 (mShad150), MHCII-FITC (AF6-120.1), Sca-1-APC (D7), TCRβ-PE-Cy5 (H57.597).

Live cells in cell suspensions prepared from spleen, thymus and bone marrow were counted using a hematocytometer, plated in 96 well plates and washed twice with PBS. The cells were stained with 1:400 Zombi Aqua Fixable viability dye (BioLegend) for 15 minutes at room temperature in the dark followed by washing with PBS. Cells were incubated with antibody for 20 minutes at 4°C. The cells were fixed using Cytofix/Cytoperm (BD) according to the manufacturer's directions, washed in PBS and resuspended in PBS containing 2% fetal bovine serum (FBS). Cell acquisition was completed using the FACS canto II (BD) or LSR Fortessa cell analyzer (BD) flow cytometers (McGill Flow Cytometry Core Facility) using FACs Diva software (BD). Compensation was completed using OneComp eBeads (Invitrogen) and data was analyzed using FlowJo software version 10.0.8r1.

For cytokine intracellular staining, splenocytes were stimulated with pre-coated anti-CD3 (5μg/ml, eBioscience) and soluble anti-CD28 (2μg/ml, eBioscience) in the presence of Protein Transport Inhibitor Cocktail (eBioscience) for 6 hours at 37°C and 5% CO2 in complete RPMI (10% FBS, 1mM sodium pyruvate, 1 Non-essential amino acids, 1% P/S, and β-mercaptoethanol).

Cells were incubated with the following antibodies: α-CD4 PE (GK1.5), α-CD8α BV421 (53-6.7, BioLegend), and α-TCRβ FITC (H57-597). Cells were then fixed and permeabilized as per manufacturer's protocol (Cytofix/Cytoperm, BD) and stained intracellularly with the following antibodies: α-TNFα PerCP-Cy5.5 (MPG-XT22, BioLegend) and α-IFNγ APC (XMG1.2) for 30 min at 4°C. Cells were acquired on an eight-color FACSCanto II using FACS Diva software (BD).

The data was analyzed using FlowJo version 10.0.8r1software. Doublets were removed by SSC-H versus SSC-W gating.

Bone marrow derived macrophage (BMDM) cell culture, RNA extraction, reverse transcription, PCR and gel electrophoresis

Bone marrow was extracted from mouse femurs as described above (flow cytometry analyses). The 2 mL cell suspension that was recovered was split into two petri dishes and 7 mL of complete RPMI (RPMI, 10% FBS, 1% pen/strep) and 4 mL of macrophage colony stimulating factor (M-CSF) was added to each dish. The cells were incubated at 37°C at 5% CO2. At day 3 of culture, the supernatant was removed and replaced with 6 mL RPMI. The dishes were scraped and 10 mL of RPMI and 8 mL of M-CSF was added. The suspension was split into 2 petri dishes. At day 6, the supernatant was removed and the cells were rinsed once in 5 mL PBS before being collected in 5 mL of PBS. The cells were centrifuged for 7 minutes at 1400 RPM. The supernatant was discarded and the pellets were resuspended in TRIzol (Invitrogen, Cat #15596-018) for RNA extraction. Reverse transcription of 2.5 μg of RNA was conducted using the ABM 5X All-In-One RT MasterMix (Cat #G486) and amplification of the resulting cDNA was completed using a standard PCR reaction using EasyTaq DNA polymerase (Transgen Biotech, Cat #AP111-01) (forward primer: 5' CCATGCAAGAGAAGCCACCAT 3', reverse primer: 5' AGTGATAGAGGCCTCCCGTGT 3'). RNA was visualized using gel electrophoresis on 2% agarose gels.

F2 cross and QTL mapping

One hundred and ninety-six offspring from an F2 cross of CC042 and C57BL/6NCrl mice were infected with Salmonella Typhimurium and euthanized at day 4 post-infection. Spleen and liver bacterial loads were determined as described above. Ninety-four animals with extreme phenotypes (half with high and half with low bacterial loads) were selected for genotyping. SNP genotyping was performed by Neogen Inc. (Lincoln, NE), using the Mouse Universal Genotyping Array (MUGA) containing 7.8K SNPs [START_REF] Morgan | The Mouse Universal Genotyping Array: From Substrains to Subspecies[END_REF]. All statistical tests (Pearson's R correlation coefficient, QTL mapping) were performed using R statistical software. QTL mapping on liver bacterial load was performed by using the J/qtl interface 1.3.5 for R/qtl software version running under R 3.2.2 [START_REF] Broman | R/qtl: QTL mapping in experimental crosses[END_REF].

Significance thresholds of LOD scores were estimated by 10,000 permutations of experimental data.

CC042 private variants in the QTL regions

Most CC strains were recently sequenced [START_REF] Srivastava | Genomes of the Mouse Collaborative Cross[END_REF] and data are publicly available, including the list of private variants specific to each CC strain. We retrieved the CC042 private variants localized within the Stsl6 and Stsl7 QTLs. Three CC042 private variants were present within Stsl6 (Ush1c, Ccdc123, 4930435C17Rik) and only one within Stsl7 (Itgal).

Itgal genotyping

Amplification of the region containing the CC042 Itgal deletion was conducted using a standard polymerase chain reaction (PCR) (forward primer: 5' TGCTTGGGTGTAGGCAGCCTCA 3', and reverse primer: 5' CTTCAATCTGCAAGACCTGGTA 3'). DNA amplicons were digested using FauI with CutSmart buffer (New England BioLabs, R0651S) for 4 hours at 55°C. The reaction was stopped by incubating the samples at 80°C for 15 minutes. The digested DNA was run on 1.5% agarose gel in tris-borate-EDTA (TBE) buffer.

Quantitative complementation testing

CC042 mice were crossed with C57BL/6J and Itgal KO mice to produce Itgal B6 /Itgal CC042 and Itgal KO /Itgal CC042 genotypes on a B6/CC042 genetic background. C57BL/6J were crossed with Itgal KO mice to produce Itgal B6 /Itgal KO genotype on a B6 genetic background. Mice were infected with S. Typhimurium and bacterial loads were determined at day 4 post-infection as described above.

Tables Table 1 footnote

Hematology data for C57BL/6J and CC042 naïve mice aged 14-16 weeks. Values indicate the mean ±SEM. populations were defined as follows: CD4 + T cells (TCRb + CD4 + ), CD8 + T cells (TCRb + CD8α + ), neutrophils (CD11b + Ly6G + Ly6C lo ), monocytes (CD11b + Ly6G -Ly6C hi ), macrophages (CD11b + Ly6G -Ly6C + F4/80 + ), and B cells (B220 + MHCII + ). Analysis was conducted using Benjamini, Krieger and Yekutieli correction for multiple testing (Two-Way ANOVA) where significance is indicated as follows, *p < 0.05, **p < 0.01, ***p < 0.001, and ****p < 0.0001. See Table 3 for details on each QTL. The proximal Stsl6 QTL acted semi-dominantly on liver (F) and on spleen (I) bacterial loads, while the CC042-inherited allele at the distal Stsl7 QTL had a recessive mode of action both for liver (G) and for spleen (J). For both QTLs, the CC042inherited allele was associated with increased bacterial load. B corresponds to the B6 allele while C corresponds to the CC042 allele.

  experiments performed at McGill University were conducted in accordance to guidelines provided by the Canadian Council on Animal Care (CCAC). Guidelines include the Guide to Care and use of experimental animals, vol 1, 2nd edition, choosing an appropriate endpoint in experiments using animals for research, teaching and testing, euthanasia of animals used in science, husbandry of animals in science, laboratory animal facilitiescharacteristics, design and development, and training of personnel working with animals in science. The animaluse protocol was approved by the McGill University Facility Animal Care Committee (protocol no. 5797). Animal experiments performed at the Institut Pasteur were conducted in compliance with French and European regulations and were approved by the Institut Pasteur Ethics Committee (project #2014-0050) and authorized by the French Ministry of Research (decision #8563).

FiguresFigure 1 |

 1 Figures

Figure 2 |

 2 Figure 2 | Pathologic changes in the spleen and liver of infected C57BL/6J and CC042 mice.

Figure 3 |

 3 Figure 3 | CC042 mice have significantly reduced total splenocyte numbers. Flow cytometry

Figure 4 |

 4 Figure 4 | CC042 mice show significantly decreased total thymocyte numbers. Total

Figure 5 |

 5 Figure 5 | CC042 mice have altered bone marrow resident hematopoietic progenitor

Figure 6 |

 6 Figure 6 | Susceptibility of CC042 mice to S. Typhimurium is controlled by two linked loci

Figure 7 |

 7 Figure 7 | Itgal mutation in CC042 mice resulted in exon 2 skipping and absence of protein

Figure 8 |

 8 Figure 8 | Quantitative complementation test confirmed the role of the CC042 Itgal loss-of-

Acknowledgements

We thank Line Larivière and Hyejin Park for technical assistance and Patricia D'Arcy (Mc Gill), Isabelle Lanctin and Tommy Penel (DTPS/C2RA-Central Animal Facility, Institut Pasteur) for mouse breeding and screening. D.M. is a recipient of a Canadian Institutes of Health Research (CIHR) project grant (MOP133700) and a Natural Sciences and Engineering Research Council (NSERC) discovery grant (RGPIN-2017-04717). A.N. is a Canada Research Chair Tier II in Hematopoiesis and Lymphocyte Differentiation and a recipient of CIHR project grant PJT-153016 and NSERC discovery grant RGPIN-2016-05657". Funding had been provided to M.T. by the NSERC -Undergraduate Student Research Award and the Fonds de Recherche du Québec Nature

Data availability

All relevant data that support the findings of this study are within the paper and its supporting information files.

Competing interests

The authors declare no competing financial interests.


B Zohidov

M Servières

N Normand

H Andrieu

J Langlois

Rainfall in Urban and Natural Systems 10th International Workshop on Precipitation in Urban Areas UR15-85 Tomographic reconstruction of rainfall maps using attenuation data from cellular networks: the first results based on the Mojette Transform

The real time monitoring of rainfall over urban areas is a critical issue for hydrological applications such as flood warning and water resource systems. Despite the fact that traditional techniques such as networks of rain gauges and weather radars are used to measure rainfall many cities worldwide are not well equipped with these devices. However, they are generally equipped with mobile telecommunication networks. Mobile networks use atmospheric Hyper-Frequency (HF) links whose transmitted signal power is attenuated by rainfall. Measuring that signal attenuation along each link could allow the measurement of path-averaged rainfall. As HF links are concentrated in cities, these networks could constitute a self-suf

We propose a simulation approach in order to evaluate the feasibility of reconstruction of rainfall maps by means of tomographic processing applied to attenuation measurements from HF links. A discrete tomographic algorithm based on the Mojette Transform (MT) is applied to reconstruct 2D rainfall map. Before applying the algorithm, specific adjustment procedures, which take into account the geometry of the network topology and non-uniform distribution of links frequency and lengths, are performed. The study domain is the city of Nantes (France) where the density of HF links is greatest. A series of rainfall fields recorded at high spatial (250m x 250m) and temporal (5 minute) resolutions are used to simulate the attenuation data and are considered as reference rainfall fields. These maps are obtained from C-band weather radar located about 10 km north of the center of Nantes (France) and are considered as reference rainfall fields.

We carry out more than 100 reconstruction tests in light rain, shower and storm events to compare the reconstructed rainfall fields to reference ones. Initially obtained results are a subject to further works: (i) improvement on the adjustment procedures of the tomographic reconstruction algorithm, (ii) Implementation of the method in the presence of the error sources.

Introduction

The real time monitoring of rainfall is an important issue for urban hydrological applications such as flood warning and water resource management systems. Even though, rain gauges and weather radars are widely used for rainfall measurement, most urban areas worldwide lack such devices. Instead, they are equipped with cellular communication networks. These networks make use of atmospheric Hyper-Frequency (HF) links whose transmitted signal is attenuated by rainfall. The signal attenuation measured along the multiple HF links can be used to retrieve rainfall fields. The retrieval principle is based on the conversion of 1D attenuation vector into 2D rainfall map using well known k-R empirical relation [START_REF] Olsen | The aR b relation in the calculation of rain attenuation[END_REF]. However, such procedure cannot be directly performed since it is not straightforward and requires a specific inversion technique. The use of tomographic reconstruction algorithms, in this context, is an original approach proposed by [START_REF] Giuli | Tomographic reconstruction of rainfall fields through microwave attenuation measurements[END_REF] about two decades ago. In the same line of this study, taking advantages of a large number of HF links [START_REF] Cuccoli | Retrieving rainfall fields through tomographic processing applied to radio base network signals[END_REF] and [START_REF] Zinevich | Estimation of rainfall fields using commercial microwave communication networks of variable density[END_REF] tomographic reconstruction algorithms which process signal attenuation measured along the HF links to retrieve rainfall fields. However, the challenges in the tomographic approach have not been completely explored yet. Therefore, following the same direction as previously cited studies, we evaluate the feasibility of rainfall retrieval using a discrete tomographic technique, which uses the Mojette Filtered Back projection (Guédon & Normand, 2005), applied to attenuation data from HF links in a simulation framework. Section 2 presents how the signal attenuation caused by rain is simulated and the tomographic algorithm used to reconstruct rainfall maps based on the simulated attenuation data. A brief description of the study area and data are given in section 3. The results are detailed in section 4, and conclusion and further works are presented in section 5.

Methods

To formulate the problem, the power law model is adopted to relate a signal attenuation to a rain intensity as follows [START_REF] Olsen | The aR b relation in the calculation of rain attenuation[END_REF]:

A = L aR (1)
Where, A -total signal attenuation, dB; L -link length, km; R -rain intensity along the link, mm/hour; a and b -power law coefficient which is a function of frequency, polarization, temperature and drop size distribution. In Figure 1b, the network area is represented by square grids in 2D space [START_REF] Zinevich | Estimation of rainfall fields using commercial microwave communication networks of variable density[END_REF]. A rain rate value R in the th pixel is assumed to be constant. Each HF link connects two locations at a distance L . We assume that the microwave signal is a straight line travelling through the network area and intersected at a length l in the th pixel:

A = a l R + , i = 1…m (2)
Where, n -number of pixels crossed by the links; R -the rain rate in jth pixel, [mm/hour]; l -the length of the crossed part of the ith link in jth pixel, [km]; a and b -the power law coefficients at the th link frequency and -measurement error [START_REF] Atlas | Path-and area-integrated rainfall measurement by microwave attenuation in the 1-3 cm band[END_REF]. As a starting point, we assume that the power law relation is linear and attenuation along the HF link is not influenced by the measurement error. The exponents of power law relations at all frequencies are assumed to be homogenous and linear, that b is equal to 1. Then, the eq.( 2) will take the following form for m number of links:

… = x … + … (3)
As a starting point, a total attenuation along the HF link was simulated using eq.( 3) in the absence of . The procedure is applied to 256 real HF links operating at 18, 23 and 38 GHz. The power law coefficients a, b are obtained from (ITU-R, 2005).

Tomographic reconstruction algorithm

The objective is to reconstruct rainfall map values , , … , using attenuation data , , … , see eq.( 3). The tomographic reconstruction algorithm used to solve the problem is based on the principles of the Mojette Transform (MT) developed by Guédon et al. The MT is a discrete version of Radon Transform. It uses simple linear operations such as addition, subtraction to reconstruct any image from a number of projections (p i , q i ). A projection value (called a "bin" as in tomography) on a projection (p, q) is the sum of pixels value centered on the line as shown in Figure 1:

Where, ( , ) -a pixel value in ( , ) indices of MxN image; ( , ) -a set of projection direction defined by Farey-Haros series [START_REF] Servières | The Mojette transform: Discrete angles for tomography[END_REF]; b -a projection value; Let us consider an example of a 3x3 image in which pixels represent rainfall rate … , see Figure 1a. The transform domain for the image consists of two projections, namely , = (1,0), , = (1,1). The way of taking projection from the sample image is shown for those two projection directions. Along each projection, bin values are computed by summing up the pixel values in the image. Unlike other tomographic algorithms, the MT has an exceptional property that a number of bins ( ) in each projection depend on the chosen angle ( = tan ):

( ) = ( 1)| | ( 1) + 1 (5)
The Mojette Filtered Back Projection algorithm is applied to reconstruct the rainfall map. However, the algorithm is not directly applicable due to (i) the arbitrary geometry of the network topology, (ii) non-uniform distribution of links frequency and lengths. Therefore, we do specific adjustment procedures before applying the algorithm. In total, the reconstruction process consists of 4 steps including the adjustments: Step 1. Grids selection: The objective is to select valid grids for Mojette projection (acquisition) in the network area. Only zones totally crossed by the HF link are chosen to be reconstructed. We divide the reconstructed zone into sub-grid so that each zone can be reconstructed independently with respect to the attenuation measured through it. In Figure 1b, this is depicted as black rectangles with different area sizes over the study region. In this preliminary study these regions are selected visually.

Step 2. Projections direction matching: A set of projections , with discrete Mojette directions are found for the MT acquisition in each grid. We choose discrete (p, q) projections directions fitting the best HF link directions and minimizing the number of bins. These possible directions depend on the reconstructed region size. However, only very few bins onto each projection are filled;

Step 3. Interpolation: An angular interpolation technique proposed by (Servières, et al., 2006) is applied to fill empty bins in each projection based on nearby values. The interpolation is performed in all selected grids of the network area; Step 4. Back Projection: Mojette Filtered Back Projection algorithm (Guédon & Normand, 2005) is used to reconstruct the attenuation map from the projection values. These 4 steps are performed in all selected grids of the network. Reconstructed images are assembled, and then normalized with the maximum attenuation value. In the end, the reconstructed map represents a specific attenuation field that could be converted to rainfall map.

Evaluation of the system performance

The reconstructed rainfall map is compared with reference rainfall field using root mean square error (RMSE), which represents the standard deviation of the residuals:

= ( ) (6)
Where, -reconstructed rainfall rate; -reference rainfall rate; -a length of a rain vector;

Data

The study domain is the city of Nantes (France), where the density of HF links is greatest (Figure 1b. We have, at our disposal, hundreds of weather radar images recorded at high spatial (250m x 250m) and temporal (5 minute) resolutions in 7 different periods (Table 1). This C band weather radar of Treillières is located about 10 km north of the center of Nantes. The radar reflectivity images were converted to rain fields with a power law Z-R relation and considered as reference rainfall maps [START_REF] Emmanuel | Temporal and spatial variability of rainfall at the urban hydrological scale[END_REF].

Results

To evaluate the monitoring system performance, we carry out a series of rainfall retrieval tests for various rain events such as light rain, shower, organized and unorganized storm. Figure 2 shows a comparison between reconstructed and reference rainfall map by weather radar in a storm event in a linear case with the absence of measurement noise. It is worth noting that the MT algorithm was applied to an inhomogeneous network topology for the first time. One can see that the reconstruction by the MT is generally consistent with reference rainfall. The poor performance of the algorithm is due to the following reasons: (i) the accuracy of the MT algorithm strongly depends on a number of projections which are not sufficient along the HF links. In other words, this is called under-determined problem; (ii) the grids choice was manually selected in this study. However, the algorithm is fast enough due to the fact that it uses very simple addition and subtraction operation to reconstruct a rain field.

Conclusions

The objective of this paper was to demonstrate the feasibility of rainfall mapping by means of tomographic reconstruction applied to simulated signal attenuation from HF links. Initially obtained results lead to further works in the reconstruction algorithm development part: improvement on a grid selection in step 1, modifying the way of taking projection in step 2, test of other interpolation methods in step 3. Further, the definition of a priori knowledge about the state of the problem needs to be addressed since the problem to be solved is highly under-determined in tomography context. A sensitivity analysis to the application conditions, associated error sources in both measurement and reconstruction, spatial and temporal rainfall field structure.

Fig 1 :

 1 Fig 1: a) An example of Mojette projection at two angles: ( , ) = (1,0), ( , ) = (1,1); b) Microwave links at 18, 23 and 38 GHz in Nantes city. Black rectangles in different sizes illustrate the selected grids for projection and reconstruction of attenuation data.

Fig 2 :

 2 Fig 2: Comparison between reference and reconstructed rainfall maps at 2x2 km2 resolution in storm.

  also applied adaptive

	Rainfall in Urban and Natural Systems
	10th International Workshop on Precipitation in Urban Areas
	UR15-85

Table 1 .

 1 Characteristics of the selected rainfall events.

	Rain type	Duration	Number of fields
	Light rain	9h 25 min	113
	Unorganized storm	2h 50 min	34
	Organized storm	2h 50 min	34
	Shower	2h 10 min	26


PhD. Serge Mordon
email: serge.mordon@inserm.fr.

PhD. Anne Sophie Vignion-Dewalle

Low-irradiance red light compared to conventional red light in photodynamic therapy of actinic keratosis: a way to reduce pain during treatment

Conventional photodynamic therapy (C-PDT), which requires an irradiation with 37 J/cm 2 of red light after 3 hours of Methyl aminolevulinate incubation is an effective protocol, approved in Europe, for thin and non-pigmented AK on the face and scalp [START_REF] Pariser | Topical methyl-aminolevulinate photodynamic therapy using red light-emitting diode light for treatment of multiple actinic keratoses: A randomized, double-blind, placebo-controlled study[END_REF][START_REF] Szeimies | Topical methyl aminolevulinate photodynamic therapy using red light-emitting diode light for multiple actinic keratoses: a randomized study[END_REF]. However C-PDT is associated with high pain scores [START_REF] Morton | Intraindividual, right-left comparison of topical methyl aminolaevulinate-photodynamic therapy and cryotherapy in subjects with actinic keratoses: a multicentre, randomized controlled study[END_REF][START_REF] Rubel | Daylight photodynamic therapy with methyl aminolevulinate cream as a convenient, similarly effective, nearly painless alternative to conventional photodynamic therapy in actinic keratosis treatment: a randomized controlled trial[END_REF][START_REF] Wiegell | Continuous activation of PpIX by daylight is as effective as and less painful than conventional photodynamic therapy for actinic keratoses; a randomized, controlled, single-blinded study[END_REF].

This letter aims to review and discuss clinical studies (Table 1) dealingwith PDT involving irradiation with low-irradiance (lower than 20 mW/cm²) red light (Li-PDT).

In 2012, Ibbotson et al evaluated the Ambulight® device emitting 7 mW/cm 2 red light [START_REF] Ibbotson | Ambulatory photodynamic therapy using low irradiance inorganic light-emitting diodes for the treatment of non-melanoma skin cancer: an open study[END_REF]. Twenty three patientsreceived both C-PDTand Ambulight which was programmed to remain switched off for threehours and then to switch on automatically and remain on for a further threehours. The mean pain score for was 1 (minimum: 0; maximum: 7) for PDT using the Ambulight® device and 5 (minimum: 1.5; maximum: 9) for C-PDT.At one year, lesion clearance rate was 84 %.

Gholam et al have alsoevaluated Li-PDT using the BF-RhodoLED® lamp system with an irradiance of 15.4 mW/cm²for40 minutes in order to achieve a lightdose of 37 J/cm 2 .Thirty-one patientswere enrolled [START_REF] Gholam | Low irradiance compared with conventional photodynamic therapy in the treatment of actinic keratoses[END_REF].Li-PDT resulted in significant less pain scores than C-PDT(2.8 for Li-PDT vs 7.6 for C-PDT) while maintaining a similar efficacy four weeks after treatment: percentage reduction of 69.9%for Li-PDT vs 74.3%for C-PDT.

Vicentini et al performed a study evaluating PDT using a light-emitting fabric device delivering fractionated red light at a low irradiance of 12 mW/cm²(Li-PDT) [START_REF] Vicentini | Photodynamic therapy for actinic keratosis of the forehead and scalp: a randomized controlled clinical study evaluating the non-inferiority of a new protocol applying irradiation with a light-emitting, fabric-based device (the Flexitheralight protocol) compared to the conventional protocol using the Aktilite CL 128 lamp[END_REF]. After 30 minutes of MAL incubation, the Li-PDT devicewas switched on and illumination was performed for 2h30 giving a lightdose of 37J/cm².With a mean pain score of 0.4, Li-PDT was found to be almost pain free compared to C-PDT (5.0). Efficacy was 84 % for % Li-PDT and 76.8% for C-PDT at6 month-follow up.

The 3 studiesfound a significant and clinically relevant reduction inpain for Li-PDT compared to C-PDTatthe samelight dose. The data reported by these three studies were pooled in order to evaluate the relationship between irradiance (mW/cm²) and pain (Figure 2). A linear relationship was observed with a coefficient of determination (R-squared) of 0.726.

It is usually admitted that pain is correlated with PpIX production. Regarding daylight-PDT, it is presumed that continuous production and photoactivation of small amounts of PpIX instead of fast photo activation of large amounts decreases pain [START_REF] Wiegell | Continuous activation of PpIX by daylight is as effective as and less painful than conventional photodynamic therapy for actinic keratoses; a randomized, controlled, single-blinded study[END_REF]. This explains that pain score reported in Study #3 where pain (0.4) is lower than those reported in Studies #1 (7 mW/cm² ; pain: 1.0) and #2 (15 mW/cm² ; pain: 2.8) . In Study #3, illumination was performed after 30minutes of incubation whereas the two others studies uses an incubation time of 3 hours. Martin found that a 15minutes incubation resulted in significantly less pain than a 75minutes incubation time [START_REF] Martin | In-office Painless Aminolevulinic Acid Photodynamic Therapy: A Proof of Concept Study and Clinical Experience in More Than 100 Patients[END_REF].Comparatively, Daylight PDT is associated to pain scores, typically around 2.0 [START_REF] Wiegell | Daylight photodynamic therapy for actinic keratosis: an international consensus: International Society for Photodynamic Therapy in Dermatology[END_REF].

In conclusion, since pain is always reported as the main drawback of C-PDT, irradiation with low irradiance red light should be considered by dermatologists. The controlled light delivery of Li-PDT is an advantage over daylight-PDT, which has also been explored as a less painful means to deliver PDT.

However, the limitation of continuous low irradiance is prolonged treatment times.

Compared to daylight-PDT, Li-PDT consistently delivers the same light dose and can be conducted in all weather conditions, in all geographic locations, all year round. [START_REF] Vicentini | Photodynamic therapy for actinic keratosis of the forehead and scalp: a randomized controlled clinical study evaluating the non-inferiority of a new protocol applying irradiation with a light-emitting, fabric-based device (the Flexitheralight protocol) compared to the conventional protocol using the Aktilite CL 128 lamp[END_REF] Figure 2: Pain scores (0-10) vs Irradiance (mW/cm²). Data were obtained from refs: [START_REF] Vicentini | Photodynamic therapy for actinic keratosis of the forehead and scalp: a randomized controlled clinical study evaluating the non-inferiority of a new protocol applying irradiation with a light-emitting, fabric-based device (the Flexitheralight protocol) compared to the conventional protocol using the Aktilite CL 128 lamp[END_REF]) [START_REF] Ibbotson | Ambulatory photodynamic therapy using low irradiance inorganic light-emitting diodes for the treatment of non-melanoma skin cancer: an open study[END_REF]) [START_REF] Gholam | Low irradiance compared with conventional photodynamic therapy in the treatment of actinic keratoses[END_REF]

  Figures: 2

Table Author

 Author

		Comparative study	Number of	Duration	Light dose	Pain	Efficacy
			patients	illumination			(% reduction)
	Gholam(Gholam						At 1 month FU
	et al., 2018)	RhodoLED 15.4 mW/cm²)	31	40 minutes	37 J/cm²	2.8 ± 1.5	67.9 ± 11.8
		vs	Actinic	vs	vs	vs	vs
		Aktilite (66.2 mW/cm²)	Keratosis	9min and 19 s	37 J/cm²	7.6 ± 1.5	69.9 ± 2.1
			Grade I -III
	Vicentini						At 6 month FU
	(Vicentini et al.,	Flexitheralight 12 mW/cm²	25	50 minutes	37 J/cm²	0.4 ± 0.6	84.0%
	2018)	vs	Actinic	vs	vs	vs.	vs
		Aktilite (65 mW/cm²)	Keratosis	9min and 50 s	37 J/cm²	5.0 ± 2.6	76.8 %
			Grade I -II
	Ibbotson(Ibbots						At 1 year FU
	on & Ferguson,	Ambulight (7 mW/cm²)	23	180 minutes	75 J/cm²	1.0 (0-7)	84 %
	2012)	Vs	small	vs	vs	vs	vs
		Aktilite® (80-90 mW/cm²)	lesions of	20 minutes	75 J/cm²	5.0 (1.5-9)	?
			Bowen's
			disease
			and
			superficial
			BCC

Table 1 :

 1 Data reported for the clinical studies included in the review: from refs:[START_REF] Vicentini | Photodynamic therapy for actinic keratosis of the forehead and scalp: a randomized controlled clinical study evaluating the non-inferiority of a new protocol applying irradiation with a light-emitting, fabric-based device (the Flexitheralight protocol) compared to the conventional protocol using the Aktilite CL 128 lamp[END_REF][START_REF] Ibbotson | Ambulatory photodynamic therapy using low irradiance inorganic light-emitting diodes for the treatment of non-melanoma skin cancer: an open study[END_REF][START_REF] Gholam | Low irradiance compared with conventional photodynamic therapy in the treatment of actinic keratoses[END_REF]. Grade I Actinic Keratosis are slightly palpable, grade II are moderately thick, and grade III AK are very thick and/or obvious

Figures

Figure 1: Light-emitting fabric device (from ref


Vincent Savaux
email: vincent.savaux@b-com.com

Luc Le Magoarou
email: luc.lemagoarou@b-com.com

On the Computation of Integrals of Bivariate Gaussian Distribution

Keywords: Bivariate Gaussian distribution, Digital communications, Signal processing, PSK

This paper deals with the computation of integrals of centred bivariate Gaussian densities over any domain defined as an angular sector of R 2 . Based on an accessible geometrical approach of the problem, we suggest to transform the double integral into a single one, leading to a tractable closed-form expression only involving trigonometric functions. This solution can also be seen as the angular cumulative distribution of bivariate centered Gaussian variables (X, Y ). We aim to provide a didactic approach of our results, and we validate them by comparing with those of the literature.

I. INTRODUCTION

The problems of estimating unknown parameters from noisy (disturbed) Gaussian observations and characterizing the performance of estimators are ubiquitous in a vast majority of quantitative fields. They have been thoroughly studied in an unified way by detection and estimation theory [START_REF] Lehmann | Testing statistical hypotheses[END_REF]- [START_REF] Van Trees | Detection, estimation, and modulation theory, part I: detection, estimation, and linear modulation theory[END_REF], and are frequently met in signal processing and digital communications [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF], [START_REF]Fundamentals of Statistical Signal Processing: Detection Theory[END_REF] as well. The problems involving Gaussian observations usually lead to the computation of multiple integrals of multivariate Gaussian distributions. However, these integrals generally do not admit closed-form solutions, and are costly to approximate.

In this paper, we focus on integrals of dimension two, i.e. we consider integrals of bivariate Gaussian densities over any angular sector of R 2 . This problem has been widely tackled in literature [START_REF] Rose | Mathematical Statistics with Mathematica[END_REF]- [START_REF] Dharmawansa | Envelope and Phase Distribution of Two Correlated Gaussian Variables[END_REF], but leads to simple specific solutions or intractable general solutions. Indeed, authors in [START_REF] Rose | Mathematical Statistics with Mathematica[END_REF], [START_REF] Stuart | Kendall's Advanced Theory of Statistics[END_REF] propose a closed-form expression of the integral in the case where two correlated Gaussian variables have the same unitary variance, and the domain of integration is one quadrant of R 2 (e.g. (R + , R + )). In [START_REF] Dharmawansa | Envelope and Phase Distribution of Two Correlated Gaussian Variables[END_REF], the distributions of the envelope and the phase of two correlated Gaussian variables are derived in a very general case of non-zero mean and non-unitary variance. However, these distributions have no closed-form nor tractable expressions, and the resulting probabilities even require the integration of these ones.

In the present paper, we tackle the problem of the integral of a bivariate Gaussian centred distribution over any angular sector of R 2 delimited by two linear functions, and in the case where two correlated Gaussian variables have different variances. To do so, we use a geometrical approach to rewrite the double integral as a single one, leading to an accessible and tractable closed-form solution. To the best of our knowledge, this approach is original. Furthermore, this leads us to define an angular cumulative distribution function (CDF) of zeromean bivariate Gaussian variables. Our geometrical approach of the problem is didactically illustrated, and our results are compared with those of the literature [START_REF] Rose | Mathematical Statistics with Mathematica[END_REF]- [START_REF] Dharmawansa | Envelope and Phase Distribution of Two Correlated Gaussian Variables[END_REF].

The rest of the paper is organized as follows: Section II presents the general problem to be solved and gives the existing special solutions. In Section III, we derive the generalized solution, and Section V concludes this paper.

Notations: Vectors and matrices are respectively written in lower and upper case in boldface: x, and X. The scalars are written in normal font: x or X. Moreover, (.) T and |.| stand for the transpose and determinant operators, respectively, and E{.} for the mathematical expectation.

II. BACKGROUND AND COMMON RESULTS

Let X and Y be two zero-mean Gaussian random variables with variances σ 2

x = E{X 2 } and σ 2 y = E{Y 2 }, respectively. Moreover, the two variables X and Y are correlated, such that E{XY } = ρσ x σ y , where ρ ∈ [-1, 1] is the so-called correlation coefficient. The joint probability density function of (X, Y ), denoted by g X,Y , can be expressed as

g X,Y (x) = 1 2π |Σ| exp(- 1 2 x T Σ -1 x), (1)
where x = (x, y) ∈ R 2 , and Σ is the covariance matrix defined as

Σ = E{(X, Y ) T (X, Y )} = σ 2 x ρσ x σ y ρσ x σ y σ 2 y , (2)
such that |Σ| = σ 2 x σ 2 y (1 -ρ 2 ). Computing any probability involving both X and Y (e.g. P(X ≶ 0 ∩ Y ≶ 0) or P(X ≶ 0 ∪ Y ≶ 0)) requires the integral of (1) over subsets of R 2 . In the following, we consider the integral of g X,Y (x) over any angular sector in R 2 , and we define the integral

M = Ω g X,Y (x)dxdy, ( 3
)
where Ω is the angular sector which can be defined by

Ω = {f 1 (x) ≶ y ≶ f 2 (x), x ∈ R}, (4)
and f 1 (x) and f 2 (x) are linear functions such as illustrated in Fig. 1. Note that such angular divisions of R 2 are usually considered in digital communications, for instance to define phase shift keying (PSK) modulations.

It must be reminded that a straightforward closed form of M is obtained when Σ is a scaled identity matrix. Moreover, a closed-form solution of the double integral (3) has been suggested in [START_REF] Rose | Mathematical Statistics with Mathematica[END_REF], [START_REF] Stuart | Kendall's Advanced Theory of Statistics[END_REF] for any ρ value, but only in the case where

σ 2 x = σ 2 y = 1 and Ω = (R + , R + ). In that case, we obtain M = 1 4 + arcsin(ρ) 2π
. We hereby generalize the result to any case where σ 2

x and σ 2 y are non-unitary variances, as well as any domain of integration defined as Ω. Moreover, this leads us to define an angular cumulative distribution function of bivariate Gaussian random variables.

III. A CLOSED-FORM SOLUTION TO (3)

In this section, we derive a general solution of (3) in Theorem 1. To this end, let us first consider the change of basis (x, y) → (ξ, ν) where (ξ, ν) correspond to semi-axes of the ellipse defined by the equation

(x, y) T Σ -1 m,n (x, y) = 1, (5)
such as illustrated in Fig. 2. This ellipse is a level set of the function g X,Y . Without loss of generality, we assume that ξ and ν correspond to the semi-major and semi-minor axes, respectively. Moreover, θ 1 and θ 2 are arbitrarily defined as the angles between the semi-major axis ξ and the lines f 1 (x) and f 2 (x), respectively. After the aforementioned change of basis, M in (3) can be rewritten as

M = Ω ξ,ν 1 2πσ ξ σ ν exp - 1 2 ξ σ ξ 2 + ν σ ν 2 dξdν, (6)
where Ω ξ,ν is the new domain of integration of ξ and ν, which corresponds to the rotated angular sector Ω in the new basis. The standard deviations σ ξ and σ ν are the length of

ξ ν θ I σ ν A(θ 1 ,θ 2 ) σ ξ θ 2 f 1 (x) f 2 (x) y x φ Fig. 2.
Sector area of an ellipse in the general case.

the semi-major or semi-minor axes of the ellipse [START_REF]Fundamentals of Statistical Signal Processing: Detection Theory[END_REF]. Using the general ellipse equation and straightforward geometrical developments, they are expressed as

σ ξ = σ 2 x cos 2 (ϕ) + σ 2 y sin 2 (ϕ) + ρσ x σ y sin(2ϕ) 1 2 σ ν = σ 2 x sin 2 (ϕ) + σ 2 y cos 2 (ϕ) -ρσ x σ y sin(2ϕ) 1 2 , ( 7
)
where ϕ is the angle between the axes x and ξ, and is defined as

ϕ =        1 2 arctan 2ρσxσy σ 2 x -σ 2 y , if σ 2 x > σ 2 y π 2 + 1 2 arctan 2ρσxσy σ 2 x -σ 2 y , if σ 2 x > σ 2 y π 4 , if σ 2 x = σ 2 y . (8)
Theorem 1. Let p ∈ [0, 1] be defined as the ratio of the semiminor and semi-major axes of the ellipse in (5), i.e. p = σν σ ξ . Then, for any θ 1 , θ 2 ∈ [-π, π], M can be expressed as

M = Ω g X,Y (x)dxdy = Q(θ 1 , θ 2 ) π , (9)
where Q(θ 1 , θ 2 ) is the function such that the area A(θ 1 , θ 2 ) of the elliptic sector bounded by (θ 1 , θ 2 ) is defined by

A(θ 1 , θ 2 ) = σ ν σ ξ Q(θ 1 , θ 2 ).
For simplicity purpose, we denote

θ M = max(|θ 1 |, |θ 2 |) and θ m = min(|θ 1 |, |θ 2 |). Then, if θ 1 , θ 2 ∈ [-π 2 , π 2
], we have

Q(θ 1 , θ 2 ) = u(|θ M |) -u(|θ m |), if sign(θ 1 ) = sign(θ 2 ) u(|θ M |) + u(|θ m |), if sign(θ 1 ) = sign(θ 2 ) , ( 10
) where u(θ) = 1 2 arctan( 1 p tan(θ)). If θ 1 ∈ [-π, -π 2 ] ∪ [ π 2 , π] and θ 2 ∈ [-π 2 , π 2
] (θ 1 and θ 2 are exchangeable), then we get

Q(θ 1 , θ 2 ) =              π 2 -u(π -|θ M |) -u(|θ m |), if sign(θ 1 ) = sign(θ 2 ) π 2 -u(π -|θ M |) + u(|θ m |), if sign(θ 1 ) = sign(θ 2 ) . ( 11
) ξ ξ(z) z dz 0 C A(ξ(z),θ 1 ,θ 2 )
Fig. 3. The integral M can be computed by integrating A(ξ(z), θ 1 , θ 2 ) over the axis z.

Finally, if θ 1 , θ 2 ∈ [-π, -π 2 ] ∪ [ π 2 , π], we obtain Q(θ 1 , θ 2 ) =              u(π -|θ m |) -u(π -|θ M |), if sign(θ 1 ) = sign(θ 2 ) π -u(π -|θ M |) -u(π -|θ m |), if sign(θ 1 ) = sign(θ 2 )
.

Proof. It must be emphasized that, although the expression of M seems to be more tractable in [START_REF] Rose | Mathematical Statistics with Mathematica[END_REF] than in (3), it can still not be computed in closed form, due to the complicated domains of integration. However, it can be noticed that the value M of the double integral corresponds to the volume under the curve

g(ξ, ν) = 1 2πσ ξ σ ν exp - 1 2 ( ξ σ ξ ) 2 + ( ν σ ν ) 2 ,
in the angular sector Ω (or equivalently Ω ξ,ν ). This volume can also be assessed by summing (integrating) the areas A of the elliptical sectors corresponding to the angular sector Ω (see Figs. 2 and3). For any angles θ 1 , θ 2 and any length of the semi-major axis ξ, given that p = ν ξ , the sector area A(ξ, θ 1 , θ 2 ) of the ellipse is given by

A(ξ, θ 1 , θ 2 ) = ξνQ(θ 1 , θ 2 ) = ξ 2 pQ(θ 1 , θ 2 ). ( 13
)
The expression of A is proved in Appendix A. It follows from (13) that the sector area of the ellipse only depends of the angular sector defined by (θ 1 , θ 2 ), and the length ξ. By setting z = g(ξ, ν = 0) and ξ ∈ R + , then for any z ∈ [0, C = 1 2πσν σ ξ ] the following bijection holds:

ξ(z) =g -1 (z) = 2σ 2 ξ ln( C z ) 1 2 , ( 14
)
allowing us to write ξ ∈ [0, +∞[ in function of z. Such as aforementioned, M in ( 6) can then be rewritten as the integral of the sector areas A(ξ(z), θ 1 , θ 2 ) of the ellipse over z ∈ [0, C = 1 2πσν σ ξ ] (see Fig. 3), which yields

M = C 0 A(ξ(z), θ 1 , θ 2 )dz. ( 15
)
Hence, considering the substitution X = ln( C z ) and p = σν σ ξ , we obtain

M = C 0 2pQ(θ 1 , θ 2 )σ 2 ξ ln( C z )dz = +∞ 0 2pQ(θ 1 , θ 2 )σ 2 ξ Ce -X dX = pQ(θ 1 , θ 2 )σ 2 ξ πσ ν σ ξ = Q(θ 1 , θ 2 ) π , (16)
which concludes the proof.

It is worth mentioning that the suggested closed-form solution only involves the computation of the well-known trigonometric functions tan and arctan, and avoids the computation of the integral of the distributions in [START_REF] Dharmawansa | Envelope and Phase Distribution of Two Correlated Gaussian Variables[END_REF], or the double integral as in [START_REF] Van Trees | Detection, estimation, and modulation theory, part I: detection, estimation, and linear modulation theory[END_REF]. In fact, the latter should usually be approximated by truncating discrete sums or by using a Monte-Carlo method, both methods being computationally expensive. In the following, we provide general results related to the analytic solution of the integral M, and we show that it reduces to known results given in [START_REF] Rose | Mathematical Statistics with Mathematica[END_REF], [START_REF] Stuart | Kendall's Advanced Theory of Statistics[END_REF] in specific cases. Furthermore, an example of application is given as well.

IV. RESULTS RELATED TO M

A. Angular Cumulative Distribution Function M

A first important result is that M in (9) can be interpreted as the angular cumulative distribution (CDF) of bivariate Gaussian variables (X, Y ). Thus, Fig. 4 illustrates M in function of (θ 1 , θ 2 ) ∈ [-π, π] 2 , for p = 1 3 . It can be observed that the CDF M is symmetric with respect to the line θ 2 = θ 1 , for which M = 0, and that M = 1 for θ 2 = -θ 1 = ±π. Note that this representation of the angular CDF M depends on the values of ρ, σ x , and σ y only through the p value, since Fig. 4 holds in the basis (ξ, ν). Different behaviors of M would be obtained in the basis (x, y), such as discussed afterward.

Alternatively to the two-dimensional representation of M, Fig. 5 shows the CDF M versus θ 1 ∈ [-π, π] for a fixed θ 2 = -π. Note that in that case, the expression of M in (9)-(12) simplifies to: Three behaviors of M, taking p ∈ { 1 3 , 1 10 , 10 11 } are shown in Fig. 5. It can be observed that M tends to the linear function M(θ 1 ) = 1 2 + θ1 2π when p approaches 1. This highlights the fact that p = 1 corresponds to an uncorrelated couple of Gaussian variables (X, Y ) with σ 2

M(θ 1 ) =        u(π+θ1) π , if θ 1 ∈ [-π, -π 2 ] 1 2 + u(θ1) π , if θ 1 ∈ [-π 2 , π 2 ] 1 -u(π-θ1) π , if θ 1 ∈ [ π 2 , π] . (17)
x = σ 2 y (i.e. the ellipse (5) reduces to a circle), therefore having an uniformly distributed phase. Otherwise when p approaches 0, M approaches a step function, highlighting that the ellipse becomes a line.

B. Particular Cases

In this section we show that the general expression of M in (9)-(12) simplifies to known results given in [START_REF] Rose | Mathematical Statistics with Mathematica[END_REF], [START_REF] Stuart | Kendall's Advanced Theory of Statistics[END_REF] when σ 2

x = σ 2 y = 1 and when Ω is one of the four quadrants of R 2 . If we assume that ρ ≥ 0 without loss of generality, we have ϕ = π 4 and then p = 1-ρ 1+ρ . Consider the quadrant (R + , R + ), then θ 1 = -π 4 , and θ 2 = π 4 , hence M in (9)-(10) becomes

M = 1 π arctan 1 + ρ 1 -ρ . (18)
Then, by using some relationship between trigonometric functions (e.g. the half-angle formula), we obtain the equivalent expressions:

M = 1 π arctan 1 + ρ 1 -ρ 2 = 1 2 - 1 π arctan 1 -ρ 2 1 + ρ = 1 2 - 1 2π arccos(ρ) = 1 4 + 1 2π arcsin(ρ). (19)
Similarly, it can be straightforwardly shown that if ρ ≥ 0 and Ω = (R + , R -) (or equivalently ρ ≤ 0 and Ω = (R + , R + ))

we have M = 1 2π arccos(ρ). These results also match those obtained by the integration of the PDF f θ

(θ) = √ 1-ρ 2 2π(1-ρ sin(2θ))
given in [START_REF] Dharmawansa | Envelope and Phase Distribution of Two Correlated Gaussian Variables[END_REF] (see [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF] in [START_REF] Dharmawansa | Envelope and Phase Distribution of Two Correlated Gaussian Variables[END_REF] considering zero-mean variables):

π 2 0 f θ (θ)dθ = 1 4 + 1 2π arcsin(ρ). (20)
Moreover, it must be noted that these expressions only depend on ρ, and we can show that this behavior holds for any variance of the variables X and Y as well. In fact, we directly have

P = P(X ≶ 0 ∩ Y ≶ 0) = P( 1 σ x X ≶ 0 ∩ 1 σ y Y ≶ 0), (21)
where 1 σx X and 1 σy Y have unitary variance. In the following, we show that the suggested general expression is mandatory to calculate more elaborate probabilities than (21).

C. Application to More Elaborate Probabilities

Let us consider zero-mean Gaussian variables X and Y with variances σ 2

x and σ 2 y , respectively, and correlation coefficient ρ. For any (α, β) ∈ R 2 + , the suggested analysis allows us to calculate probabilities such as

P = P(X ≥ 0 ∩ Y ≥ 0 ∩ αY ≥ βX), ( 22
)
where P is not independent of the variances of X and Y anymore due to the additional event αY ≥ βX. To reduce the number of degrees of freedom of the problem, let us set r = σx σy , and γ = β α . Then, the orientation φ and the ratio p of the ellipse depends on both ρ and r, and the domain of integration Ω (partially) depends on γ, such as illustrated in Fig. 6. Thus, in the example given in (22), we have θ 2 = π

Fig. 1 .

 1 Fig. 1. Example of domain of integration Ω.

Fig. 4 .

 4 Fig. 4. M interpreted as the angular CDF of bivariate Gaussian variables (X, Y ), for p = 1/3.

Fig. 5 .

 5 Fig. 5. Angular CDF M versus θ 1 ∈ [-π, π], θ 2 = -π. Comparison for different p values.

and θ 1 = arctan(γ).

To better illustrate the dependence of P in (22) on the variances of X and Y , we consider the following numerical application: ρ = 0.5, γ = 1, and r ∈ { 1 4 , 1, 4}, and we imposed σ x σ y = 4. Fig. 7 shows the level sets of the three considered bivariate Gaussian distributions. The domain of integration Ω is above the line y = x. The corresponding values of probability P have been reported on Fig. 7, and it can be shown that the lower r, the larger ϕ and therefore the larger P, since the sector areas A(ξ, θ 1 , θ 2 ) of the ellipses in Ω are greater.

Such as mentioned in Section II and stated in [START_REF] Dharmawansa | Envelope and Phase Distribution of Two Correlated Gaussian Variables[END_REF], such results can be applied in the domains of digital communications and signal processing. For instance, it allows for an accurate characterization of the error rate of PSK-modulated signals with inphase/quadrature (I//Q) correlated noise. The main difference of the suggested analysis with results given in [START_REF] Dharmawansa | Envelope and Phase Distribution of Two Correlated Gaussian Variables[END_REF] is that we carry out our developments in the basis (ξ, ν) instead of (x, y), leading to tractable closed-form expressions of M, where the parameters σ x , σ y , and φ inherent to the basis (x, y) are included in the geometrical parameter p in the basis (ξ, ν).

To complete this study, non-zero means of X and Y could be included in a similar geometrical approach of the problem. Furthermore, this analysis could be extended in dimensions larger than two (involving ellipsoides), in order to tackle the problem of the integrals of multivariate Gaussian distributions.

V. CONCLUSION

In this paper we suggested a geometrical approach to solve the problem of integrating centered bivariate Gaussian densities over any domain defined as an angular sector of R 2 . This led to a tractable solution composed of trigonometric functions, which can be seen a the angular CDF of a couple of centered Gaussian variables (X, Y ). The developments were illustrated to provide a didactic approach of our results. Furthermore, we compared them to those of the literature in order to validate our solution. Future works will investigate the possible extension of the suggested geometrical approach to non-centred correlated Gaussian variables, as well as problems of dimension larger than two.

APPENDIX

A. Proof of the expression of A(ξ, θ)

We consider an ellipse with semi-major and semi-minor axes ξ and ν, respectively. The area A of the elliptical sector between the semi-major axis and the radius r with angle dθ is equal to A = r 2 2 dθ. The parametric equation of the ellipse is expressed as x = ξ cos(t) y = ν sin(t).

Thus, any radius with length r and angle φ (corresponding to parameter t = t φ ) with respect to the semi-major axis admits the equations: