Olaya Rendueles

Christophe Beloin

Patricia Latour-Lambert

Jean-Marc Ghigo
email: jmghigo@pasteur.fr

A new biofilm-associated colicin with increased efficiency against biofilm bacteria

Keywords: Bacterial competition, Biofilm, Colicin, Escherichia coli, SOS response Subject category: Microbe-microbe and microbe-host interactions

Formation of bacterial biofilm communities leads to profound physiological modifications and increased physical and metabolic exchanges between bacteria. It was previously shown that bioactive molecules produced within the biofilm environment contribute to bacterial interactions. Here we describe new pore-forming colicin R, specifically produced in biofilms formed by the natural isolate Escherichia coli ROAR029 but that cannot be detected under planktonic culture conditions. We demonstrate that an increased SOS stress response within mature biofilms induces SOS-dependent colicin R expression. We provide evidence that colicin R displays increased activity against E. coli strains that have a reduced lipopolysaccharide length, such as the pathogenic enteroaggregative E. coli LF82 clinical isolate, therefore pointing to LPS size as an important determinant for resistance to colicins. We show that colicin R toxicity toward E. coli LF82 is increased under biofilm conditions compared to planktonic susceptibility, and that release of colicin R confers a strong competitive advantage in mixed biofilms by rapidly outcompeting sensitive neighboring bacteria.

This work therefore identifies the first biofilm-associated colicin that preferentially targets biofilm bacteria. Furthermore, it indicates that the study of antagonistic molecules produced in biofilm and multispecies contexts could reveal unsuspected, ecologically relevant bacterial interactions influencing population dynamics in natural environments.

INTRODUCTION

Biofilms are widespread mixed-species bacterial communities in which high cell density, reduced diffusion and heterogeneous structure favor physical and metabolic contact between bacteria. Biofilms are therefore ideal environments for development of antagonistic and synergistic interactions [START_REF] Hibbing | Bacterial competition: surviving and thriving in the microbial jungle[END_REF][START_REF] James | Interspecies bacterial interactions in biofilms[END_REF][START_REF] Rendueles | Multi-species biofilms: how to avoid unfriendly neighbors[END_REF]. It was hypothesized early on that the particular conditions prevailing within biofilms could lead to the emergence of biofilm-specific competition mechanisms [START_REF] Ghigo | Are there biofilm-specific physiological pathways beyond a reasonable doubt?[END_REF][START_REF] Valle | The amino acid valine is secreted in continuous-flow bacterial biofilms[END_REF]. Consistently, biofilm communities were shown to release molecules that affect bacterial access to space and resources. Examples of such compounds range from metabolites and amino acids [START_REF] Kolodkin-Gal | D-amino acids trigger biofilm disassembly[END_REF][START_REF] Kolodkin-Gal | A selfproduced trigger for biofilm disassembly that targets exopolysaccharide[END_REF][START_REF] Valle | The amino acid valine is secreted in continuous-flow bacterial biofilms[END_REF] to anti-adhesion molecules [START_REF] Bendaoud | Broad Spectrum Biofilm Inhibition by Kingella kingae Exopolysaccharide[END_REF][START_REF] Rendueles | Screening of Escherichia coli Species Biodiversity Reveals New Biofilm-Associated Antiadhesion Polysaccharides[END_REF][START_REF] Rendueles | Antibiofilm polysaccharides[END_REF] and bacteriocins [START_REF] Kreth | Transcriptional analysis of mutacin I (mutA) gene expression in planktonic and biofilm cells of Streptococcus mutans using fluorescent protein and glucuronidase reporters[END_REF][START_REF] Yan | Biofilm-specific cross-species induction of antimicrobial compounds in bacilli[END_REF].

Production of antagonistic molecules is a widespread trait across the microbial world [START_REF] Cascales | Colicin biology[END_REF][START_REF] Chater | Streptomyces inside-out: a new perspective on the bacteria that provide us with antibiotics[END_REF][START_REF] Duquesne | Structural and functional diversity of microcins, gene-encoded antibacterial peptides from enterobacteria[END_REF][START_REF] Hibbing | Bacterial competition: surviving and thriving in the microbial jungle[END_REF][START_REF] Lee | Lantibiotics, class I bacteriocins from the genus Bacillus[END_REF][START_REF] Wenzel | Myxobacteria--'microbial factories' for the production of bioactive secondary metabolites[END_REF]. In the specific context of biofilm development, bacteriocin production has been researched at length in oral biofilms [START_REF] Bucci | The evolution of bacteriocin production in bacterial biofilms[END_REF][START_REF] Kreth | Transcriptional analysis of mutacin I (mutA) gene expression in planktonic and biofilm cells of Streptococcus mutans using fluorescent protein and glucuronidase reporters[END_REF][START_REF] Merritt | The mutacins of Streptococcus mutans: regulation and ecology[END_REF] in which Gram-positive bacteria are predominant, however, little is known about the occurrence and efficacy of bacteriocins produced within Gram-negative biofilms.

It is estimated that at least 50% of all E. coli strains synthesize colicins, plasmidencoded bacteriocins secreted by Escherichia coli and closely related enterobacteria, and microcins (chromosomally encoded low-molecular-weight bacteriocins). Secretion of colicins is considered to provide an ecological advantage to colicin-producing bacteria compared to their colicin-sensitive neighbors [START_REF] Cascales | Colicin biology[END_REF][START_REF] Gillor | Colicins and microcins: the next generation antimicrobials[END_REF].

Colicinogenic bacteria are resistant to colicin biocidal action via a constitutively expressed immunity gene typically coded downstream the colicin gene. Colicin release into the extracellular medium is mediated by cell lysis, presumably encoded by the kil or lysis gene carried in the plasmid. For lethal activity, colicin interactions with target cells are mediated upon binding to receptors at the surface and subsequent colicin internalization. Colicins have two distinct mechanisms of action, they can either, form pores in the inner membrane (activity encoded in the C terminal of the colicin) or display an enzymatic activity that will degrade DNA or RNA, both of which will result in cell death (for review on colicin biology [START_REF] Cascales | Colicin biology[END_REF][START_REF] Riley | Bacteriocins: evolution, ecology, and application[END_REF]). Colicin expression is tightly regulated by the SOS response [START_REF] Gillor | The role of SOS boxes in enteric bacteriocin regulation[END_REF]. In neutral environments, the dimerized repressor LexA inactivates the SOS regulon by binding to a DNA sequence called the LexA box.

Upon DNA damage, RecA activates LexA self-cleavage allowing gene expression of more than 80 genes under RecA/LexA control [START_REF] Courcelle | Comparative gene expression profiles following UV exposure in wild-type and SOS-deficient Escherichia coli[END_REF][START_REF] De Henestrosa | Identification of additional genes belonging to the LexA regulon in Escherichia coli[END_REF].

Here we describe a new colicin detectable only within biofilms formed by natural E. coli ROAR029 and that was identified while screening for molecules inhibiting E. coli adhesion and growth [START_REF] Rendueles | Screening of Escherichia coli Species Biodiversity Reveals New Biofilm-Associated Antiadhesion Polysaccharides[END_REF]. We demonstrate that ROAR029 colicin (hereafter referred to as colicin R) is a pore-forming, plasmid-encoded colicin the production of which is not detected under planktonic conditions. We show that the increased SOS stress response induced in mature biofilms leads to induction of colicin R expression and to local concentration in biofilms. We demonstrate that lipopolysaccharides (LPS) integrity is a determinant for resistance, since colicin R is mainly active toward bacteria producing short LPS. Finally, we show that release of colicin R confers a strong competitive advantage upon mixed biofilms and that colicin R toxicity toward the enteroaggregative E. coli LF82 clinical isolate is increased under biofilm conditions compared to planktonic susceptibility. This study therefore identifies the first biofilmassociated colicin targeting biofilm bacteria.

MATERIALS AND METHODS

Bacterial strains, plasmids, and liquid growth conditions. E. coli isolates used in this study originated from published collections [START_REF] Escobar-Paramo | Identification of forces shaping the commensal Escherichia coli genetic structure by comparing animal and human isolates[END_REF][START_REF] Skurnik | Effect of human vicinity on antimicrobial resistance and integrons in animal faecal Escherichia coli[END_REF], and compiled in [START_REF] Rendueles | Screening of Escherichia coli Species Biodiversity Reveals New Biofilm-Associated Antiadhesion Polysaccharides[END_REF]. Bacterial strains and plasmids used are listed in Supplementary Tables S1 andS2. Bacteria were cultured in 0.4% glucose M63B1 minimal medium (M63B1 Glc ) or in LB lysogeny broth medium at 37°C. Antibiotics were used as follows: ampicillin, 100 µg mL -1 ; apramycin, 30 µg mL -1 ; chloramphenicol, 25 µg mL -1 ; spectinomycin, 50 µg mL -1 ; zeocin, 50 µg mL -1 ; apramycin, 30 µg mL -1 ; and kanamycin, 50 µg mL -1 .

Purification of planktonic and biofilm extracts. Planktonic supernatants. Bacteria were cultured for 24 h in M63B1 Glc and adjusted to an optical density (OD 600nm ) of 3. Cultures were centrifuged for 10 min at 7,500 rpm at 4°C and supernatants were filtered through 0.2 µm filters. Biofilm extracts. Sixty-milliliter microfermentors containing a removable glass stand were configured as continuous-flow culture bioreactors with a 40-mL h -1 flow rate, which maximizes biofilm development and minimizes planktonic growth of the bacteria (see at http://www.pasteur.fr/recherche/unites/Ggb/biofilmfermenter.html for more detailed information) [START_REF] Ghigo | Natural conjugative plasmids induce bacterial biofilm development[END_REF][START_REF] Rendueles | Screening of Escherichia coli Species Biodiversity Reveals New Biofilm-Associated Antiadhesion Polysaccharides[END_REF]. Inoculation was performed by dipping the glass spatula for 5 min in a culture adjusted to an OD 600nm of 1 from overnight bacterial cultures. The spatula was then reintroduced into the microfermentor. After 72 h at 37°C, 45 mL of microfermentor medium (OD 600nm < 0.1) was carefully discarded, and biofilm biomass was rapidly resuspended in the remaining 15 mL medium by vortexing for 30 s.

The final biofilm biomass was measured, adjusted to an OD 600nm of 3, centrifuged and filtered through 0.22-µm filters (this biofilm conditioned media is hereafter referred to as biofilm extract). Alternatively, microtiter plates were inoculated with overnight diluted cultures (1:100) and grown for 24 to 48 h at 37°C under static conditions. E. coli ROAR029 microtiter plate biofilms were resuspended in their own supernatant and directly spotted onto agar plates adequately supplemented with antibiotics to inhibit further growth of ROAR029.

Biofilm-related assays. Biofilm inhibition assays. Cultures diluted overnight in M63B1 Glc were inoculated into 96-well polyvinyl chloride (PVC) plates and filter-sterilized biofilm extract or planktonic supernatant was added at a 1:1 ratio. Biofilms were grown for 24 h at 37°C before quantification. Biofilm dispersion assays. Static biofilms were grown in 96well PVC microtiter plates for 48 h at 37°C. Free cells and used media were removed by inversion of the microtiter plate, biofilm was washed using M63B1 and 100 µL of ROAR029 biofilm extract were added to each biofilm for 2 h at 37°C before quantification.

Biofilm quantification. Unbound cells were removed by inversion of the microtiter plate; biofilms were washed with water and adhering cells were then stained with 1% crystal violet for 20 min. Excess stain was removed by washing with deionized water 3 times. Quantification of attached biofilms was performed by addition of acetone-ethanol (20:80) and the dissolved crystal violet was measured at an OD of 595 nm.

Colicin susceptibility assays on planktonic and biofilm cells. i. Planktonic. Bacterial cultures were grown around 6 h or 16 h (exponential and stationary, respectively) in M63B1 Glc . Bacteria were then harvested by centrifugation and resuspended in the same volume of M63B1 Glc (control planktonic) or in a 1:1 mix of M63B1 Glc and ROAR029 biofilm extract to obtain a final cell quantity of 1.10 8 (1.10 9 CFU/ml). One hundred µL were then transferred to microtiter plate wells. After 2 h of incubation at 37°C, serial dilution plating (CFU counts) was used to assess colicin susceptibility of the treated planktonic population versus a planktonic overnight untreated control.

ii. Biofilm. Biofilms were formed for 48 h in microtiter plates as described above.

Unattached and planktonic bacteria were removed by inversion of the microtiter plates and gentle tapping. One hundred µL of either a 1:1 mix of M63B1 Glc and ROAR029 biofilm extract or M63B1 Glc (control) were added to each well in which the preformed biofilm (ca.

1.10 8 CFU, 1.10 9 CFU/mL) had developed. The treated biofilms were incubated for 2 h at 37°C, and then resuspended by vigorous pipetting. Cell survival was determined by CFU count. iii. Biofilm resuspended cells. Biofilms were formed for 48 h in microtiter plates as described above. Unattached and planktonic bacteria were removed, biofilms were washed by adding (without resuspending) and removing M63B1. Then, the biofilms were resuspended in 100 µL of either a 1:1 mix of M63B1 Glc and ROAR029 biofilm extract or M63B1 Glc (control). Cells were then incubated at 37°C for 2 h and bacterial survival was monitored as above.

Growth inhibition test.

The inhibitory effect of filter-sterilized supernatants and biofilm extracts was evaluated by spotting 8 µL on an overlay of targeted cells on appropriate media. Plates were incubated overnight at 37°C. Batch growth in microtiter plates was also monitored by diluting overnight cultures to a final OD 600 nm of 0.1 and mixed at a 1:1 ratio with cell-free biofilm extract and grown in microtiter plates for 24 h at 37°C. Generation of deletion mutants. Gene deletions in the different strains of E. coli were generated by the λ-red linear DNA gene replacement system [START_REF] Chaveroche | A rapid method for efficient gene replacement in the filamentous fungus Aspergillus nidulans[END_REF][START_REF] Derbise | A rapid and simple method for inactivating chromosomal genes in Yersinia[END_REF] or by P1vir transduction from the Keio collection to E. coli MG1655 when indicated (Supplementary Table S2) [START_REF] Baba | Construction of Escherichia coli K-12 in-frame, single-gene knockout mutants: the Keio collection[END_REF]. Primers used are listed in Supplementary Table S3. ROAR029 constructs were checked by PCR and sequencing. Construction of ROAR029_KmRExTET_LexA were lexA is under the control of with an anhydrotetracycline inducible promoter was performed as described in [START_REF] Da Re | Tight modulation of Escherichia coli bacterial biofilm formation through controlled expression of adhesion factors[END_REF].

Construction of an LPS reconstituted mutant in MG1655. Construction of O-antigenreconstituted E. coli K-12 strains was performed using O-antigen trans-complementation between two related E. coli K-12 backgrounds, EMG2 and WG1 [START_REF] Liu | Escherichia coli K12 regains its O antigen[END_REF].

Both backgrounds are O-antigen-deficient, either by IS insertion in wbbL for EMG2 (rfb-50) or by a deletion encompassing the end of rfbB for WG1 (rfb-51). The KmFRT selectable genetic cassette from the pKD4 plasmid [START_REF] Datsenko | One-step inactivation of chromosomal genes in Escherichia coli K-12 using PCR products[END_REF] was inserted between the gnd and intact wbbL genes of WG1 strain using long primers (wbbL.KmFRT.long.L-5 and wbbL.KmFRT.long.L-3, Supplementary Table S3) and a λ-red recombination. Primers hybridizing two-hundred base pairs before and after wbbL (wbbL200.ext-5 and wbbL200.ext-3 Supplementary Table S3) were then used to amplify the KmFRT-wbbL region that was used to replace the wbbL-deficient region of EMG2 derivatives MG1655 and TG1 strains. Restoration of functional wbbL gene and O-antigen production was verified by sequencing and LPS visualization on gel (Figure 5A).

Random mutagenesis. The ROAR029 genome. Transposon mutagenesis of ROAR029

was performed using random insertion of the EZ-Tn5™ <KAN-2> Transposome™ (Epicentre Biotechnologies). A total of 5,000 clones were analyzed. ROAR029 plasmid.

Plasmid pColR was mutagenized by EZ-Tn5™ <KAN-2> transposition. Pools of mutagenized plasmids were electroporated into E. coli TG1 electrocompetent cells. In EZ-Tn5™ <KAN-2> the kanamycin gene is not followed by a transcription terminator allowing possible transcriptional read-through.

Mixed biofilm assays. ROAR029 bearing pColR_Km, pColR-or pColR+ and MG1655F' cells were grown overnight in M63B1 Glc , washed twice in M63B1 to remove traces of colicin R (if any) and diluted to OD 600nm 0.1. Cells were mixed at different initial ratios of ROAR029 derivatives to MG1655F' (1:1, 5.10 7 CFU/mL : 5.10 7 CFU/mL ; 1:10, 5.10 6 CFU/mL : 5.10 7 CFU/mL ; 1:100, 5.10 5 CFU/mL: 5.10 7 CFU/mL and 1:10 000, 5.10 3 CFU/mL : 5.10 7 CFU/mL), and inoculated into 96-well PVC plates for 24 h at 37°C. We counted the proportion of each strain in the 24-h-grown biofilms after washing, resuspension and serial dilution plating in appropriate antibiotics.

SDS-polyacrylamide gel for LPS observation. Cell preparation.

Cells were harvested by centrifugation, resuspended in deionized water and normalized to a final OD 600nm of 1.

Samples were then heated for 10 min at 100°C and treated with proteinase K (2mg/mL) for 1 h at 37°C. LPS gel and staining. SDS-polyacrylamide gels were made according to [START_REF] Lugtenberg | Electrophoretic resolution of the "major outer membrane protein" of Escherichia coli K12 into four bands[END_REF]. Gels were run at 75 mA for 60 to 90 min. LPS were stained with silver nitrate as described previously [START_REF] Tsai | A sensitive silver stain for detecting lipopolysaccharides in polyacrylamide gels[END_REF].

Sequence of ROAR029 colicin plasmid, colicin R sequence and structure analysis. The sequence of the pColR plasmid was deposited in Genbank (accession number KF137578).

Sequence alignments.

Alignments were performed by MultiAlin software described in [START_REF] Corpet | Multiple sequence alignment with hierarchical clustering[END_REF]. Phylogenetic dendrogram. Colicin sequences were directly obtained from GenBank (http://www.ncbi.nlm.nih.gov/gene). Sequences were aligned following ClustalW standards and further examined in order to create phylogenetic dendrograms with the program MEGA 5 software developed by [START_REF] Tamura | MEGA5: Molecular Evolutionary Genetics Analysis using Maximum Likelihood, Evolutionary Distance, and Maximum Parsimony Methods[END_REF]. Plasmid map. pColR plasmid map was constructed using Seqgene software. Colicin tertiary structure. Colicin R structure was simulated using PHYRE software [START_REF] Bennett-Lovsey | Exploring the extremes of sequence/structure space with ensemble fold recognition in the program Phyre[END_REF].

Statistical methods. Statistical analyses were performed using unpaired t-test with Prism v5.0 (GraphPad Software).

RESULTS

Commensal E. coli ROAR029 produces a biofilm-associated bacteriotoxic molecule

We had previously screened a collection of 122 E. coli natural isolates from different phylogenetic groups, hosts and origins for their ability to produce biofilm-specific molecules inhibiting biofilm formation of other competing strains [START_REF] Rendueles | Screening of Escherichia coli Species Biodiversity Reveals New Biofilm-Associated Antiadhesion Polysaccharides[END_REF].

That study revealed that biofilm extracts from commensal E. coli ROAR029 isolated from an impala antelope (Aepyceros melampus) in Gabon (phylogenetic group D) inhibited biofilm formation and planktonic growth of the biofilm-forming MG1655F'tet∆traD strain (referred to as MG1655F') [START_REF] Rendueles | Screening of Escherichia coli Species Biodiversity Reveals New Biofilm-Associated Antiadhesion Polysaccharides[END_REF][START_REF] Valle | Broad-spectrum biofilm inhibition by a secreted bacterial polysaccharide[END_REF] (Figure 1A). While filter-sterilized planktonic supernatant of E. coli ROAR029 did not show biocidal activity, a filter-sterilized extract from ROAR029 biofilms (hereafter referred to as biofilm extract) produced a growth inhibition halo when spotted on an MG1655F' lawn (Figure 1B). Production of the bacteriotoxic compound was also detected in microtiter plates and colony biofilm models [START_REF] Anderl | Role of antibiotic penetration limitation in Klebsiella pneumoniae biofilm resistance to ampicillin and ciprofloxacin[END_REF] (Figure 1C and data not shown). This biofilmassociated bacteriotoxic effect could not be attributed to the previously reported presence of valine in E. coli biofilms, since ROAR029 biofilm extract retained activity on the MG1655 ilvG+ strain in which the corrected ilvG frameshift confers resistance to valine (Figure 1D) [START_REF] Valle | Broad-spectrum biofilm inhibition by a secreted bacterial polysaccharide[END_REF]. Finally, determination of colony-forming units of a MG1655F' planktonic culture upon exposure to ROAR029 biofilm extract in a 1:1 ratio for 2 h showed no surviving bacteria, indicating bacteriotoxic rather than bacteriostatic activity of the ROAR029 biofilm extract (data not shown).

The ROAR029 biofilm-associated bacteriotoxic molecule is a new pore-forming colicin

To identify the genetic determinants of ROAR029 bacteriotoxic activity, we performed random transposon mutagenesis and searched for ROAR029 mutants unable to produce active biofilm extract. We identified one transposon mutant inserted in pcnB, a gene coding for a poly-A polymerase involved in mRNA poly-A adenylation and plasmid copy number regulation [START_REF] Liu | Genetics and sequence analysis of the pcnB locus, an Escherichia coli gene involved in plasmid copy number control[END_REF]. Introduction of a pcnB deletion into ROAR029 abolished the bacteriotoxic activity of microtiter-plate-or microfermentorgrown biofilm extracts (Figure 2A). Since ROAR029 ΔpcnB did not show significant growth or biofilm defects (Supplementary Figure S1A), we hypothesized that the genetic determinants of the ROAR029 bacteriotoxin could be carried by a plasmid; indeed, we detected the presence of a ca. 5.8 kb plasmid in ROAR029 (Figure 2B). We subjected the ROAR029 plasmid to random in vitro mutagenesis and we electroporated pools of mutagenized plasmids into E. coli K-12 strain TG1. We found that 67% of kanamycinresistant transformants produced active bacteriotoxic compound in microtiter plate biofilm extracts, demonstrating that this plasmid carries genes involved in synthesis of ROAR029 bacteriotoxin. The remaining 33% kanamycin-resistant TG1 transformants did not produce active bacteriotoxic compounds. Sequence analysis showed that inactive mutagenized plasmids carried transposon insertions in DNA regions homologous to pore-forming colicins U and Y encoding genes, including genes potentially coding for a 69.3 kDa colicin, as well as lysis and immunity proteins (Table 1 and Figure 2CD, Supplementary Figure S2, Supplementary Figure S3, Supplementary Figure S4) [START_REF] Riley | The newly characterized colicin Y provides evidence of positive selection in pore-former colicin diversification[END_REF][START_REF] Smajs | Colicin U, a novel colicin produced by Shigella boydii[END_REF]. We further tested whether the immunity encoded by ROAR029 plasmid conferred resistance to other colicins and indeed, it diminished the killing of colicin U and completely blocked growth inhibition of colicin Y (Supplementary Figure S4A), but had no effect against unrelated colicin E2. Sequence and phylogenetic analyses further showed that while immunity genes of pore-forming colicins U, Y and ROAR029 cluster together, key aminoacids for interaction between the immunity protein and its colicin partner [START_REF] Smajs | Recognition of pore-forming colicin Y by its cognate immunity protein[END_REF] appeared more similar between ROAR029 and colicin Y than with colicin U (Supplementary Figure S4BC).

We then replaced the original marker-less ROAR029 colicin-encoding plasmid (pColR), hereafter referred to as colicin R for ROAR029, with an inactive kanamycinresistant plasmid by transformation (pColR-). Plasmid maintenance upon antibiotic selection led to exclusion of the native plasmid. Biofilm extracts of ROAR029 strains with inactive kanamycin-resistant plasmid did not display bacteriotoxic activity (Figure 2A).

Additionally, one mutagenized plasmid seemed to confer an increased killing effect, referred to as pColR+ (Supplementary Figure S5).

To further confirm the colicinogenic nature of the antagonistic molecule, we checked that TG1 transformed with pColR were resistant to biofilm extracts produced by pColR and pColR+ strains by virtue of the immunity protein carried in the plasmid (Supplementary Figure S6).

Taken together, our results demonstrated that ROAR029 biofilm extracts bacteriotoxic activity was associated with production of colicin R encoded by the 5.8 kb plasmid pColR.

Colicin R is a group A Tol-dependent colicin

Colicins are classified into groups A and B based on use of either Tol or Ton translocation systems, respectively [START_REF] Cascales | Colicin biology[END_REF]. Although E. coli MG1655 Ton system mutants ∆tonB, ∆exbB and ∆exbD were sensitive to colicin R, a ∆tolA mutant displayed full resistance to colicin R as it did for colicin U (Figure 3). Consistently, we identified a conserved TolA box consensus motif (DGTGW), as well as several repeated TolA motifs involved in colicin internalization in the colicin R gene sequence (data not shown). Other Tol mutants (∆tolQ, ∆tolR, ∆tolC and ∆tolB) showed various levels of tolerance to colicin R (Supplementary Figure S7). Residual activity of colicin R in other Tol mutants might be explained by partial phenotype complementation of Ton proteins. Indeed, the E. coli mutant strain carrying double ton and tol deletions displayed full resistance to colicin R (data not shown). Finally, we showed that, like for colicin U, an ompA mutation in E. coli MG1655 led to colicin R resistance (Figure 3 and Supplementary Figure S7). Interestingly, closely related colicins U and Y both used OmpF as a translocator (data not shown). However, colicin R activity remained unaffected in an ompF mutant (Supplementary Figure S7) as well as in an ompC mutant (data not shown), hence, we cannot exclude the possibility that OmpA acts as a colicin R receptor, translocator or both. Accordingly, E. coli TG1 transformed with pColR inhibited E. coli MG1655 but not OmpA and TolA mutants (data not shown).

Biofilm specificity of colicin R might be due to SOS regulatory activity

Colicins have been shown to be tightly regulated by the SOS system [START_REF] Gillor | The role of SOS boxes in enteric bacteriocin regulation[END_REF]. Consistently, we identified two overlapping LexA boxes in the promoter region located upstream of the ROAR029 colicin gene (Figure 4A). LexA is a repressor of SOS-regulated genes cleaved by RecA upon induction of the SOS response. We observed that overexpression of LexA correlated with decreased levels of colicin in E. coli ROAR029 biofilm extracts (Figure 4B), and addition of mitomycin C to either biofilm extracts or inactive E. coli ROAR029 planktonic cultures led to derepression of colicin R production (Figure 4C and data not shown, respectively).

We had previously shown that the SOS system is upregulated within E. coli biofilms [START_REF] Beloin | Global impact of mature biofilm lifestyle on Escherichia coli K-12 gene expression[END_REF][START_REF] Bernier | Starvation, together with the SOS response, mediates high biofilm-specific tolerance to the fluoroquinolone ofloxacin[END_REF]. Thus, colicin R biofilm-specific production could be directly linked to an increased SOS response. To investigate this, we used three different plasmids derived from wild type pColR: (i) pColR-, unable to produce colicin R due to transposon-based interruption of the colicin gene, (ii) pColR+ in which colicin gene transcription is constitutive due to a probable combination of read-through from the promoter of the kanamycin gene from the EZ-Tn5TM <KAN-2> transposon and dirsruption of LexA binding to LexA box 1 caused by transposon insertion (Figure 4A) and (iii) pColR_Km, active kanamycin-resistant pColR (Supplementary Figure S5). These plasmids were transformed into E. coli TG1 and TG1lexAind3, with the latter unable to induce an SOS response due to a non-cleavable version of the LexA protein. As expected, biofilm extract from strains carrying pColR-did not show growth-inhibiting activity, whereas biofilm extracts from strains bearing pColR+ displayed strong SOS-independent growth-inhibiting activity (Figure 4D). In contrast, the growth-inhibiting activity of strains carrying the pColR_Km plasmid was totally dependent on a functional LexA protein (Figure 4D). Taken together, these results suggest that increased release of colicin R is a consequence of induction of the SOS response within biofilms.

Colicin R is only active toward E. coli strains with short LPS

To investigate the activity spectrum of colicin R, we tested a panel of pathogenic and commensal E. coli as well as other Gram-negative strains for their sensitivity to ROAR029 biofilm extract (Supplementary Table S1). We observed that all tested strains were resistant to colicin R except for E. coli K-12 EMG2 and B derivatives, and adherentinvasive E. coli LF82. E. coli K-12 EMG2 derivatives are characterized by an insH-7 insertion element in wbbL resulting in rough LPS [START_REF] Liu | Escherichia coli K12 regains its O antigen[END_REF]. Reintroduction of a wild type wbbL allele in E. coli K-12 MG1655 (MG1655 wbbL+) and E. coli TG1 (TG1 wbbL+) restored O-antigen production (Figure 5A) and led to total resistance to colicin R in bacterial lawns, batch cultures and microtiter plate biofilms (Figure 5B and Supplementary Figure S8). Furthermore, E. coli LF82 naturally lacks O-antigen due to a deletion of wbbLKIJH_glf_rfbX_rfbC genes [START_REF] Miquel | Complete genome sequence of Crohn's disease-associated adherent-invasive E. coli strain LF82[END_REF] (Supplementary Figure S9), which probably explains its colicin R sensitivity. We also tested whether colicins closely related to colicin R, i.e. colicins U and Y, or another colicin, colicin E2, had bacteriotoxic effects against E. coli K-12 MG1655 and E. coli MG1655 wbbL+; we observed that these three colicins were also inactive against the O-antigen LPS-restored MG1655 wbbL + strain (data not shown). Colicins U, Y and E2 as well as colicin R produced in TG1_pColR were also active against LF82 strain (data not shown).

We further investigated the influence of LPS length on colicin R activity using E. coli 536, a uropathogenic strain resistant to colicin R and its ∆rfaH, ∆manB, ∆waaG and ∆waaC mutants displaying various LPS sizes [START_REF] Beloin | The transcriptional antiterminator RfaH represses biofilm formation in Escherichia coli[END_REF] (Figure 5C). Addition of ROAR029 biofilm extract to E. coli 536 mutants revealed that resistance increased with LPS size (Figure 5DE) and that restoration of whole LPS in a complemented rfaH mutant (rfaHc) led to total resistance to colicin R (Figure 5DE). Interestingly, uropathogenic E. coli CFT073 and 536 mutants unable to synthesize other extracellular polymeric surface structures like colanic acid and capsules remained fully resistant to colicin R (Figure 5DE and data not shown). Taken together, these results showed that a reduction in LPS size correlated with increased sensitivity to colicin R.

ROAR029 colicin excludes other E. coli from mixed biofilms

To determine whether colicin R production confers a competitive advantage upon ROAR029 within structured communities, we compared the ability of ROAR029 carrying pColR_Km, pColR-or pColR+ plasmids to grow in mixed biofilms in competition with the biofilm-forming colicin-R-sensitive E. coli MG1655F' strain [START_REF] Ghigo | Natural conjugative plasmids induce bacterial biofilm development[END_REF]. It was verified that ROAR029 strains expressing differing levels of colicin R did not display significantly different growth rates in liquid-and biofilm-forming abilities (Supplementary Figure S10). ROAR029 derivatives and MG1655F' cultures were adjusted to different initial ratios and grown as mixed biofilms for 24 h. Although ROAR029 pColR-was naturally able to outcompete MG1655F' due to a higher growth rate of ROAR029 derivatives (see Supplementary Figure S10), bacterial survival showed that ROAR029 pColR_Km and ROAR029 pColR+, colicin-producing strains, more efficiently outcompeted MG1655F' (Figure 6). In order to further confirm the prominent role of colicin R in the outcompetition within biofilm, transformed TG1 with pColR plasmids was competed against a streptomycin-resistant TG1 at different initial ratios. Accordingly, TG1 pColR outcompeted TG1. This effect was increased when TG1 carried pColR+ plasmid, and abolished when pColR-was used (Supplementary Figure S11).

We further characterized temporal dynamics of ROAR029/MG1655F' interactions by inoculating the two strains at a 1:1 ratio, frequencies at which bacteriocin production provides the greatest advantage [START_REF] Inglis | Spite and virulence in the bacterium Pseudomonas aeruginosa[END_REF]. To avoid potential accumulation of colicin R in the medium, we used continuous flow microfermentors and observed a rapid reduction in MG1655F' and complete out-competition after 8 h (Supplementary Figure S12). This strong outcompetition was not due to higher ability of ROAR029 pColR_Km to initially adhere to glass spatula since, on the contrary, the initial adhesion of MG1655F', probably by virtue of the production of the strong F pilus adhesion factor [START_REF] Ghigo | Natural conjugative plasmids induce bacterial biofilm development[END_REF], was ∼20 fold higher than the one of ROAR029 pColR_Km (adhesion of ROAR029_pColR_Km and MG1655F' are respectively 0.10 +/-0.037 % and 2.22 +/-0,76%, p= 0.00838, unpaired t-test). These results demonstrated that colicin R production and higher growth rate provide a strong competitive advantage to ROAR029 in mixed biofilms. This suggests that production of colicin R could play a role in population dynamics in natural situations.

Colicin R is more active against biofilms than planktonic cultures

The specific release of colicin R within E. coli ROAR029 biofilms led us to hypothesize that its production could correlate with increased sensitivity to colicin R in E. coli biofilms.

To test this, we compared sensitivity to colicin R of planktonic and biofilm bacteria for two different biofilm-forming pathogens with smooth (full-length) and rough (lacking Oantigen) LPS, E. coli 55989 and E. coli LF82, respectively (Supplementary Figure S9A).

Whereas E. coli 55989 in planktonic and biofilm conditions remained unaffected by colicin R, LF82 biofilms displayed increased sensitivity compared to LF82 planktonic cultures (Figure 7). This increased sensitivity was not due to further reduction of LPS length during biofilm growth, since LF82 planktonic and biofilm LPS profiles were identical (Supplementary Figure S9A). In addition, we tested whether resistance to colicin R could be enhanced by the characteristic 3D structure and architecture displayed by biofilm cells by assessing the sensitivity of biofilm bacteria resuspended in liquid medium, maintaining the physiological specificities of biofilm bacteria, but not their characteristic structured environment. Resuspended LF82 biofilm bacteria were more sensitive to colicin R planktonic bacteria and marginally more sensitive than biofilms (Figure 7), indicating that colicin R preferentially targets biofilm cells of susceptible bacteria.

DISCUSSION

In the present study, we characterized an unknown antagonistic antimicrobial molecule released within biofilms formed by E. coli natural isolate ROAR029. We demonstrated that this biofilm-associated molecule corresponds to a new 5.8 kb group I plasmid-encoded colicin (colicin R), exhibiting group A colicin characteristics [START_REF] Cascales | Colicin biology[END_REF]. Phylogenetic analysis of colicin-associated genes of ROAR029 showed that they cluster with two other pore-forming colicins, U and Y, isolated from Shigella boydii, and a natural E. coli isolate from the Amazonian region, respectively [START_REF] Riley | The newly characterized colicin Y provides evidence of positive selection in pore-former colicin diversification[END_REF][START_REF] Smajs | Colicin U, a novel colicin produced by Shigella boydii[END_REF]. Additionally, colicin R immunity protein confers full cross-immunity to colicin Y and partial immunity to colicin U, but no immunity to colicin E2.

Biofilm-specificity of colicin R production

Here we demonstrate that induction of the SOS response, previously shown to be upregulated in E. coli biofilms [START_REF] Beloin | Global impact of mature biofilm lifestyle on Escherichia coli K-12 gene expression[END_REF][START_REF] Bernier | Starvation, together with the SOS response, mediates high biofilm-specific tolerance to the fluoroquinolone ofloxacin[END_REF], is directly correlated with increased production of colicin R in biofilms. Other reports have highlighted the fact that colicins are also upregulated in late stationary phase independently of the SOS system and are triggered by nutrient depletion in the medium [START_REF] Eraso | Increased production of colicin E1 in stationary phase[END_REF][START_REF] Kuhar | Transcription regulation of the colicin K cka gene reveals induction of colicin synthesis by differential responses to environmental signals[END_REF], a common situation in the deep layers of biofilms [START_REF] Boles | Endogenous oxidative stress produces diversity and adaptability in biofilm communities[END_REF][START_REF] Stewart | Physiological heterogeneity in biofilms[END_REF]. Recently, Cornforth and Forster proposed that bacteriocin production is correlated with the stress response as a potential reaction to the presence of ecological competitors and probably plays an important role in ecological interference within structured bacterial populations [START_REF] Cornforth | Competition sensing: the social side of bacterial stress responses[END_REF].

In addition, several studies postulate that optimal efficiency of bacteriocins is achieved within structured environments, such as biofilms [START_REF] Chao | Structured habitats and the evolution of anticompetitor toxins in bacteria[END_REF][START_REF] Durrett | Allelopathy in Spatially Distributed Populations[END_REF][START_REF] Gardner | Bacteriocins, spite and virulence[END_REF]. In agreement with previous studies, we show that in mixed biofilm conditions, colicin R confers a competitive advantage against closely related directniche competitors consuming similar potentially limited resources.

In addition, enhanced colicin R production within biofilms could correspond to a specific stress-induced defense mechanism, since release of group A colicins is a lethal event for the cell [START_REF] Cascales | Colicin biology[END_REF]. Induction of the SOS system in biofilms upon stresses such as waste product accumulation, microaerobic areas or increased oxidative stress, could lead to colicin R release and subsequent cell death. Hence, biofilm-associated colicin production could have several benefits, including eliminating the most stressed bacteria from biofilms without negatively impacting growth or subsequent biofilm development of the overall population (Figure S10). Other suicidal mechanisms have been reported in structured populations, including phage release [START_REF] Carrolo | Prophage spontaneous activation promotes DNA release enhancing biofilm formation in Streptococcus pneumoniae[END_REF], nitric oxide production [START_REF] Barraud | Nitric oxide signaling in Pseudomonas aeruginosa biofilms mediates phosphodiesterase activity, decreased cyclic di-GMP levels, and enhanced dispersal[END_REF] and cannibalism toward B. subtilis cells [START_REF] Lopez | Cannibalism enhances biofilm development in Bacillus subtilis[END_REF].

Moreover, biofilm-specific colicin production could contribute to enabling kind discrimination mechanisms [START_REF] Strassmann | Kin discrimination and cooperation in microbes[END_REF]. Biofilms rely on cooperative acts such as matrix production and communication signals that can be exploited by other noncooperative individuals ("cheaters") who benefit from them, but do not share the associated cost. When uncontrolled, these cheaters can increase in frequency, undermine cooperation and even cause population crashes in situations where cooperation is important for survival [START_REF] Hamilton | The genetical evolution of social behaviour. I &II[END_REF]. In this context, biofilm-specific colicin production could ensure close relatedness between interacting bacteria and promote cooperation by excluding possible cheaters and reducing social exploitation (kin selection) [START_REF] Hamilton | The genetical evolution of social behaviour. I &II[END_REF][START_REF] Nadell | The sociobiology of biofilms[END_REF].

Colicin R targets LPS-deficient and preferentially biofilm cells

Like previously described colicins, ROAR029 colicin has a narrow spectrum of action [START_REF] Cascales | Colicin biology[END_REF]. Here we demonstrate that, in addition to receptor and translocator proteins on the surface of target bacteria, LPS may play an important role in determining sensitivity towards colicins. We show that E. coli strains with full-length LPS are resistant to colicin R, whereas strains without O-antigen display increased susceptibility. Interestingly, restoration of O-antigen led to resistance to two pore-forming colicins (U and Y) and to RNAse colicin E2 (data not shown), suggesting that intact Oantigen might be a general determinant of resistance to colicins. While the presence of Oantigen has been previously suspected to alter the effect of temperature or EDTA on colicin susceptibility of target cells by means of the change of receptor activity [START_REF] Gado | Effect of precultivation conditions on colicin susceptibility in Escherichia coli[END_REF] we postulate that colicin R resistance of full-length LPS strains could be due to charge or steric repulsion by O-antigen, therefore preventing colicin R interaction with its receptor at the cell surface. This phenomenon is consistent with observation that purified LPS fractions do not seem to interact directly with colicins [START_REF] Law | OmpF enhances the ability of BtuB to protect susceptible Escherichia coli cells from colicin E9 cytotoxicity[END_REF][START_REF] Weltzien | The nature of the colicin K receptor of Escherichia coli Cullen[END_REF].

In the case of colicin R this potential steric or charge repulsion seems specific to O-antigen, since deletion of other large surface molecules such as colanic acid and capsules did not enhance susceptibility to colicin R of uropathogenic E. coli strains 536 and CFT073.

Additionally, we show that a reduction in the LPS lipid A core size was correlated with E. coli increased sensitivity to colicin R, probably because of an increased accessibility of colicin R to its receptor. This is in contrast to increased resistance to colicin U displayed by E. coli mutants with a shorter lipid A LPS core [START_REF] Smajs | Colicin U, a novel colicin produced by Shigella boydii[END_REF] or cellular resistance of deep rough mutants to colicin N [START_REF] Sharma | Genome-wide screens: novel mechanisms in colicin import and cytotoxicity[END_REF] suggesting that lipid A core could play an important role in early interaction with certain colicins as shown for colicin E2 [START_REF] Osada | Lipopolysaccharide defective mutant of Escherichia coli with decreased sensitivity to Colicin E2[END_REF]. Hence, colicins U, E2 and R have in common to be inhibited by the presence of O-antigen, but colicins U and E2 need an intact lipid A outer core for binding to the outer membrane and subsequent insertion while colicin R efficacy is reduced by the increased size of the LPS lipid A core. Furthermore, we provide evidence that strains with naturally reduced LPS length display increased sensitivity to colicin R when grown as biofilms. In the clinical enteroaggregative E. coli LF82 strain, colicin R indeed killed biofilm cells more efficiently than planktonic cells. Despite this increased activity toward biofilm cells, colicin R was unable to dissolve biofilms (data not shown), suggesting that it can penetrate biofilms, but that the 3D structure of biofilms is conserved after the colicin R effect. However, biofilm 3D architecture and the presence of extracellular matrix slightly contributed to protecting E. coli LF82 biofilm from the effects of colicin R (Figure 7). Although the specific mechanism by which LF82 biofilm bacteria display a decreased survival ratio compared to planktonic cells remains to be elucidated, it does not seem to be linked to further LPS modifications. Indeed, LPS extracted from biofilms and planktonic E. coli LF82 appeared to be identical. Alternatively, the particular metabolic and physiological properties displayed by biofilm bacteria may account for their increased sensitivity. For instance, a reduced growth rate resulting in decreased outer membrane component turnover or increased stress in biofilm layers or niches could lead to enhanced sensitivity to colicin R.

Since biofilm bacteria typically display enhanced tolerance to antimicrobial agents [START_REF] Hoiby | Antibiotic resistance of bacterial biofilms[END_REF], this is, to our knowledge, the first example of biofilm bacteria being more sensitive to a bacteriotoxic compound than their planktonic counterparts.

Counterselection of LPS rough E. coli cells: ecological implications in species diversity

Production of antagonistic molecules has been proposed to play an important role in selection and evolution of strains [START_REF] Kirkup | Antibiotic-mediated antagonism leads to a bacterial game of rock-paper-scissors in vivo[END_REF][START_REF] Majeed | Competitive interactions in Escherichia coli populations: the role of bacteriocins[END_REF][START_REF] Riley | The ecological role of bacteriocins in bacterial competition[END_REF]. Indeed, colicin production promotes microbial diversity, both theoretically [START_REF] Czaran | Chemical warfare between microbes promotes biodiversity[END_REF] and empirically [START_REF] Walker | Transcriptional profiling of colicin-induced cell death of Escherichia coli MG1655 identifies potential mechanisms by which bacteriocins promote bacterial diversity[END_REF]. Here we demonstrated that colicin R can preferentially target strains with reduced LPS (absence of O-antigen and reduced lipid A core) probably by an increased ability in these strains to access its receptor. This suggests that, in mixed biofilm communities, selective pressure exerted by colicin R producers may favor bacteria with long LPS structures. In addition, it has been recently been shown that electrostatic interactions between LPS and other polysaccharides enhances and stabilizes biofilms [START_REF] Amini | Genetic dissection of an exogenously induced biofilm in laboratory and clinical isolates of E. coli[END_REF], and that LPS play a key role in mixed biofilm dynamics [START_REF] Bandara | Bacterial lipopolysaccharides variably modulate in vitro biofilm formation of Candida species[END_REF]. Taken together, colicin R could therefore directly impact population evolution. Interestingly, in P. aeruginosa and P. fluorescens biofilms, reduction of LPS O-antigen has been reported [START_REF] Ciornei | Biofilmforming Pseudomonas aeruginosa bacteria undergo lipopolysaccharide structural modifications and induce enhanced inflammatory cytokine response in human monocytes[END_REF][START_REF] Giwercman | Polyacrylamide gel electrophoresis analysis of lipopolysaccharide from Pseudomonas aeruginosa growing planktonically and as biofilm[END_REF][START_REF] Spiers | The Pseudomonas fluorescens SBW25 wrinkly spreader biofilm requires attachment factor, cellulose fibre and LPS interactions to maintain strength and integrity[END_REF], suggesting that such LPS modifications could also occur in E. coli biofilms, increasing relevance of producing a colicin against strains with short LPS.

In conclusion, our study contributes to the role played by colicins in representative multispecies contexts. It further establishes biofilms as a source of bioactive antagonistic molecules, as well as being a crucial site for ecological interactions that influence E. coli evolution and population dynamics in complex and relevant multispecies environments.

Figure 1 .

 1 Figure 1. Bacteriotoxic effect of ROAR029 extracts upon E. coli MG1655F'. A.

Figure 2 .

 2 Figure 2. ROAR029 bacteriotoxic activity is due to plasmid-encoded colicin regulated

Figure 3 .

 3 Figure 3. ROAR029 colicin is a group A colicin. A. Effect of addition of an 8 µL drop of

Figure 4 .

 4 Figure 4. ROAR029 biofilm-specific production of colicin R is due to an SOS

Figure 5 .

 5 Figure 5. ROAR029 colicin activity is dependent on LPS length. A. SDS-

Figure 6 .

 6 Figure 6. ROAR029 excludes MG1655F' from mixed biofilms. Percentage of ROAR029

Figure 7 .

 7 Figure 7. Colicin R has enhanced activity towards biofilms of LF82. Differential effect

ACKNOWLEDGEMENTS

We are grateful to Juraj Bosák and David Šmajs for providing colicin U-and colicin Y-producing strains. We thank Erick Denamur for the strain ROAR029. We thank Franziska Stressmann for critical reading of the manuscript. O. R. was supported by a fellowship from the Network of Excellence EuroPathogenomics; European Community Grant LSHB-CT-2005-512061. This work was supported by a grant from the Région Ilede-France (DIM Malinf) and from the French Government's Investissement d'Avenir program, Laboratoire d'Excellence "Integrative Biology of Emerging Infectious Diseases" (grant n°ANR-10-LABX-62-IBEID).

CONFLICT OF INTERESTS

The authors declared no conflict of interest

SUPPLEMENTARY INFORMATION

Supplementary information is available at ISMEJ's website


Bertrand Hassani
email: bertrand.hassani@gruposantander.com

Wei Yang
email: wei.yang@santander.co.uk

The Lila distribution and its applications in risk modelling

Keywords: probability distribution, parametric distribution, multimodal distribution, operational risk, market risk, pseudo bi-modal distribution

The opinions, ideas and approaches expressed or

Introduction

Since the Dodd-Frank Act was signed in 2010, there has been a wave of new regulations to reform the financial industry and to improve the stability of financial system (coming from the Bank of England, the European Banking Authority, the Basel Committee, the Federal Reserve, etc.). In order to meet formal expectations set by regulators, banks have elevate standards for governance and enterprise risk management. Internal risk reporting practices are required to be set up and risk data are aggregated according to supervisory recommendations. Standard models or advanced internal models approved by regulators are used to calculate the amount of capital which should be reserved, this type of capital is usually referred to as regulatory capital. Models relying on statistical distributions are also used to evaluate the economic capital and more importantly these models are used for management purposes, i.e. to make decisions. Indeed, any model for regulatory capital calculation requires a way of quantifying the level of risk. In traditional statistics, a commonly used risk measure is the standard deviation. Yet within the context of risk management and capital requirements, the most common measure of risk is a quantilebased risk measure Value-at-Risk (VaR). The VaR measures the maximum potential loss with a degree of confidence for a specified period. Given a confidence level α ∈ (0, 1), the V aR α is the relevant quantile of the loss distribution:

V aR α (X) = inf{x|P [X > x] ≤ 1 -α} = inf{x|F X (x) ≥ α}
where X is a risk factor admitting a loss distribution F X , for a systematic discussion on VaR see e.g. [START_REF]Value-at-Risk: Theory and Practice[END_REF]. Another choice of risk measure as an extension of the VaR is the Expected Shortfall (ES, also called Conditional VaR), which seeks to access what happens beyond its maximum loss threshold. Specifically, ES α is defined as the average of all losses which are greater than or equal to V aR α c.f. [START_REF] Rockafellar | Conditinal value-at-risk for general loss distributions[END_REF]:

ES α (X) = 1 1 -α 1 α V aR p dp.
While there remain a number of issues related to the selection of appropriate risk measures since every risk measure has drawbacks and limited applications, a prerequisite for any risk model is the modelling of the underlying risks from historical data. There is a wide range of distribution families available for risk modelling since many distributions, such as lognormal, Weibull, Gamma, and Generalised Hypobolic [START_REF] Barndorff-Nielsen | Exponentially decreasing distributions for the logarithm of particle size[END_REF] distributions, are successfully used to model real life random variables. However, irrelevant of risk types, most of these uni-modal parametric distributions fail to capture fat-tailed features of risk data, in the sense that upper quantitles are larger than those obtained from any fitted distributions. In order to model heavy-tailed risk data, extreme value distribution and heavy-tailed paretotype distributions with a shape parameter β have attracted a lot attention. The tails of the loss distribution functions are in the form:

P(X > x) = x -β h(x)
where β > 0 is the tail-index parameter and h is a slowly varying function; for detailed discussion see e.g. [START_REF] Embrechts | Modelling Extremal events for insurance and finance[END_REF]. Yet for example, the risk measure V aR, which is currently the financial industry practice, is sensitive to the value of β, especially when 0 < β < 1. In case 0 < β < 1, that is the distribution has a extremely heavy tail, the VaR may lead to ridiculously high capital charges, besides the fact that V aR no longer has the desired property of sub-additivity [START_REF] Neslehova | Infinite Mean Models and the LDA for Operational Risk[END_REF]. (The lack of sub-additivity of VaR contradicts the diversification effect.) Even in some cases, tails of risk data are too "fat" such that another "peak" is formed.

Remark 1.1. Though in this paper we are focusing on risk measures in the tail, by properly fitting the data, the modelling of the expected loss (represented by the mean or the median for example depending on how financial institutions perceive it) is also impacted, consequently the risk management decision might be significantly modified.

The characteristics of fat-tails and even multiple peaks of risk data make it impossible to appropriately fit and use most uni-modal parametric distributions. To tackle these issues, mixture distributions have been proposed and applied in the area of risk modelling. The importance of mixture distributions is remarked in a number of books on mixtures for example [START_REF] Lindsay | Mixture models: theory geometry and applications[END_REF][START_REF] Frühwirth-Schnatter | Finite Mixture and markov Switching Models[END_REF] and a diversity of publications. Mao [START_REF] Mao | Estimating population sizes for capture-recapture sampling with binomial mixtures[END_REF] identified a variety of sources of difficulties in statistical inference using mixture.

The application of mixture distributions has also seen some drawbacks in the area of risk modelling. The mixture of Gaussian distributions to create so-called Gaussian Mixture (GM) distributions are considered e.g. in [START_REF] Cohen | Estimation in Mixtures of Two Normal Distributions[END_REF][START_REF] Li | Mixture Density Estimation[END_REF]. GM distributions can accommodate heavy tails, asymmetry, and/or multimodality. Given enough components, the mixture of normal distributions can achieve great flexibility. For example, the mixture of two Normal distributions requires 5 parameters, 2 for each normal distribution and 1 weight parameter. However, the weight parameter in particular might be quite complicated to obtain. Besides, the support of the resultant distribution is infinite which makes it complicated to be applied to credit and operational risk, although this mixed distribution may be appropriate to model market risk. [START_REF] Guégan | A modified Panjer algorithm for operational risk capital calculations[END_REF] applied the mixture of a lognormal and a Generalized Pareto Distribution (GPD) for modelling operational risk and focused on the estimation of the threshold. Although the resulting distribution fits better operational risk data, the proposed method involves the estimation of 5 parameters. Particularly the estimation procedure for the truncated lognormal distribution, the estimation of the location parameter and the shape parameter of the GPD make it a complicated approach to undertake in practice.

Guégan and Hassani

One may consider to combine distributions of any kind. By this method, the number of parameters needed increases. Usually this method requires at least 5 parameters. The resulting distribution is very unstable as a slight change in the data set such as adding a new data point may require the modification of the parameters and/or new combination of distributions (a feature that multiple approved OpRisk models permit).

In this paper, we introduce a new class of probability distributions that features a parametric distribution with two peaks of different heights. The introduction is called the lowered-inside-leveraged-aside (Lila) distribution. The Lila distribution is inspired by the unusual phenomenon observed from the risk data. For example, in risk data sets, small losses have very high frequency and only a few extremely large losses appear. However, the extreme large losses are not negligible since the sum of the few large losses accounts for a large proportion of the total losses. Sometimes, the empirical distributions of loss data often exhibit two peaks which are of different heights and far apart from each other.

Informally, to construct the distribution with the desired features, the idea is to distort a cumulative distribution function of a random variable, by "re-distributing" the probabilities of events. We lower the probabilities of rare events in the body part and leverage those in the tail events to form a "tail peak". Mathematically speaking, a S-shaped function is used to assign weights to events and then it is composed with the distribution under consideration. Formally we call this new class of distributions lowered-insideleveraged-aside (Lila) distributions. Due to the feature of two peaks with different heights, we refer to this new class of Lila distributions as bi-modal parametric distributions. With abuse of terminology, the term modal is used for local maxima of the Lila density function at which the density's derivative vanishes and the second order derivative is negative.

The rest of the paper is organised as follows: in Section 2, we formally construct the Lila distributions and discuss some basic properties. The method is illustrated through an example in Section 3, with detailed analysis on the roles played by the parameters used. In Section 4, we fit two sets of real data to the Lila distribution as an application. Finally Section 5 concludes the paper.

Construction and properties of Lila family of distributions 2.1 Construction of the Lila distribution

Let X be a non-negative random variable with a cummulative distribution function F X (x) = P (X ≤ x) where lim x→0 F (x) = 0 and lim x→∞ F (x) = 1. Let F -1 X denote the inverse function3 of F X and f X the density function of F X .

Consider a distortion function G : [0, 1] → [0, 1] with G(0) = 0 and G(1) = 1 which is continuous and increasing with a differentiable and convex derivative function g : [0, 1] → [0, k], for a k > 0. This type of functions G are often referred to as sigmoid functions or S-shape functions. In the paper, we study the Lila distributions with the specific polynomial G in Eq. (3.1) and Eq. (3.3).

Definition 2.1. Define a function L : [0, ∞) → [0, 1] by L(y) := (G • F X )(y) = G(F X (y)), ∀y ∈ [0, ∞) (2.1)
where the notation • denotes the composition of two functions. Denote the density function of

L by l : [0, ∞) → [0, ∞) l(x) := (G • F X ) (x) = (g • F X )(x)f X (x). (2.2)
The distribution function L defined in (2.1) with the density function l in (2.2) is called the lowered-inside-leveraged-aside (Lila) distribution. Remark 2.2. Effectively, one can understand that the construction of Lila distributions is to "re-distribute" the density of the underlying random variable according to the "weight" (g • F X )(x) for each x.

We would like to stress that, in order to generate general Lila distributions, the distortion function G must satisfy the following conditions:

1. 0 < G(p) < 1, G(0) = 0 and G(1) = 1.
This condition ensures that for each x, (G • F X )(x) defines a valid probability.

2. G(p) is a differentiable and increasing function, i.e. g(p) ≥ 0. This condition ensures that the risk is re-distributed (or shifted) to the tail part and the distorted probability (G • F X )(x) defines another distribution.

3. The derivative function g(p) is non-negative, differentiable and convex. This condition is critical and ensures the resultant distribution is smooth and has two peaks.

The above method to construct a new distribution function is related to the method of transformation of random variables, that is using an invertible transformation function y(x) to transform a random variable X into another one Y . Then the density function of the new random variable Y can be implied from the transformation function. In contrast, our method directly distorts the distribution of the random variable X. One can find the transformation function y(x), which transforms a random variable X with the distribution f X (x) into a new random variable Y with the Lila distribution, by solving the following equation for y(x)

g(F X (y))f X (y) = f X (x(y)) dx dy . (2.3)
Another related method to construct new distributions is through the composition of distributions functions. For example, in [START_REF] Jones | Families of distributions arising from distributions of order statistics[END_REF], Jones chose Beta distributions for g with parameters a, b and constructed a family of distributions by specifying F X to be normal, Fréchet, Gumbel, exponential Weibull, exponential, Burr and power distributions. Yet the Beta distribution g in [START_REF] Jones | Families of distributions arising from distributions of order statistics[END_REF] is either convex or concave depending on the parameters a, b.

In [START_REF] Al-Hussaini | Composition of cumulative distribution functions[END_REF], AL-Hussaini generated distribution functions by composing a cumulative distribution function F X with several cumulative distribution functions G including Weibull, exponential, Rayleigh, Burr type XII, Lomax, compound Rayleigh, Gompertz and compound Gompertz CDFs. However, AL-Hussaini focused on the study of the so-called survival function in the form 1 -G[-ln F X (y)].

Our method is also different from constructing distortion risk measures which is defined as distorted expectation of any "non-negative loss4 " random variables

ρ g (X) = ∞ 0 g(1 -F X (x))dx
where the distortion function g is a non-decreasing function with g(0) = 0 and g(1) = 1 and g : [0, 1] → [0, 1], see e.g. [START_REF] Wang | A class of distortion operators for pricing financial and insurance risks[END_REF][START_REF] Wang | Premium calculation by transforming the layer premium density[END_REF][START_REF] Wang | Insurance pricing and increased limits ratemaking by proportional hazards transforms[END_REF][START_REF] Sereda | Distortion Risk Measures in Portfolio Optimization[END_REF][START_REF] Hardy | Distortion risk measure: coherence and stochastic dominance[END_REF][START_REF] Dennegerg | Non-additive measure and integral[END_REF][START_REF] Dennegerg | Premium calculation: why standard deviation should be replace by absolute deviation[END_REF][START_REF] Harlimann | Distortion risk measures and economic capital[END_REF]. Distortion risk measures were originally applied to a wide variety of insurance problems then later to some asset allocation problems related to the investment risks [START_REF] Hamada | Dynamic portfolio allocation, the dual theory of choice and probability distortion function[END_REF]. The focus is to search a distortion function g so that the resultant distortion risk measure ρ g satisfies certain properties such as coherence. 5 In order to generate a coherent risk measure for capital calculation, a concave distortion function g is often used to adjust the true probability measure to give more weight to higher risk events. The focus of this method is to construct distortion risk measures ρ g rather than the distorted distribution itself.

Cobb et al. in [START_REF] Koppstein | Estimation and moment recursion relations for multimodal distributions of the exponential family[END_REF] introduced multi-modal exponential distributions in the form

l(x) = ξ(b) exp - x 0 g(s) v(s) ds (2.4)
where the shape polynomial g(x) has the form g(x

) = b 0 + b 1 x + b 2 x 2 + • • • + b k
x k and v(x) can be polynomials of degree at most 2, and ξ is a normalisation function. The maximum number of possible modes of l in (2.4) is determined by the degree k > 0 of the shape polynomial g. In the special case when g

(x) = b 0 + b 1 x + b 2 x 2 + b 3 x 3 and v(x) = 1, λ = -b 2 3b 3 is a loca- tion parameter, σ = b -0.25 3 is a scale parameter, α = σg(λ) is a measure of skewness and β = -σ 2 (b 1 + b 2 λ) is a measure of kurtosis. If ( α 2 ) 2 -( β 3 ) 3 < 0, then l(x) in (2.

Some properties of the Lila distribution

In this subsection, some properties of Lila distributions are analysed, including quantile and the first moment.

Recall G is a S-shape distortion function with its derivative function g; F X is the cumulative distribution function of a positive random variable X. Let the newly constructed function L be the cumulative distribution function of a random variable Y , with a density function l.

Quantile: For a given number q ∈ [0, 1], the q-quantile y q of the random variable Y is such that q = L(y q ), that is

y q = L -1 (q) = (G • F ) -1 (q) = (G -1 • F -1 )(q).
Mean: By the construction of l in (2.2), we have

E[Y ] = ∞ 0 l(y)ydy = ∞ 0 (g • F X )(x)f X (x)xdx = E[g(F X (X))X] (2.5)
where the equality in (2.5) can be seen as the transformation of the random variable X. To find the transformation function y(x) so that Y = y(X), one needs to solve the equation (2.3) for y.

From (2.5), the following result can be obtained:

Proposition 2.1. (i) If the mean of the random variable X exists, i.e. E[X] < ∞, then the mean of Lila distributions is bounded and the following inequality holds:

min p∈[0,1] g(p)E[X] ≤ E[Y ] ≤ max p∈[0,1] g(p)E[X].
(ii) Conversely, if the mean of the random variable X does not exist, then the mean of the Lila distribution does not exist.

Proof. (i) To obtain the upper bound of the mean of Lila distribution:

E[Y ] = ∞ 0 l(y)ydy = ∞ 0 x (g • F )(x)f (x)dx ≤ max x∈[0,∞) (g • F )(x) ∞ 0 xf (x)dx = max p∈[0,1] g(p)E[X].
The similar calculation leads to the lower bound:

E[Y ] = ∞ 0 l(y)ydy = ∞ 0 x (g • F )(x)f (x)dx ≥ min x∈[0,∞) (g • F )(x) ∞ 0 xf (x)dx = min p∈[0,1] g(p)E[X].
(ii) The statement holds true due to the way of the construction of Lila distribution. Define k = arg x min g(F X (x)). If the mean of the random variable of X does not exist, it means that the tail of

X on [k, ∞) is "fat" such that ∞ k f X (x)dx = ∞ for k > k.
Then after the distortion of f X (x), see Eq. (2.2), the tail of the new random variable Y is even "fatter" and

∞ k l(x)dx = ∞ k g(F X (x))f X (x)dx = ∞
due to the boundedness of g(F X ). This means the mean of Y is infinite.

The above proposition concludes that Lila distributions do not change the infinite mean property of the underlying distributions. In case the mean of the underlying distribution exists, the mean value of the Lila distribution has bounded mean. The bounds of the mean depend on the distortion function G, more precisely on its derivative function g.

Examples of the Lila distribution

In this section we demonstrate the construction of the Lila distributions and analyse its basic features with specific underlying distributions F and distortion functions G.

The main example

Let F X be the lognormal cumulative distribution function with the location parameter µ and the scale parameter σ. The distortion function G is defined as

G(p) =    b -b 1 -p a 2n+1 , p ≤ a b + (1 -b) p-a 1-a 2n+1
, p > a In each graph of Figure 1, the parameter a is fixed. Increasing the value of the parameter b moves the function G from the yellow line at the bottom to the black line at the top, namely increasing the height of function G. In each graph of Figure 2, the parameter b is fixed. Increasing the value of the parameter a moves the function G from the yellow line at the left to the black line at the right, meanwhile maintaining the position of the turning point. In Figure 3, increasing the parameter n widens the "flat region".

More insights on the roles of the parameters a, b and n can be obtained by looking at the derivative function g of G:

g(p) =    (2n+1)b a 2n+1 (p -a) 2n , p ≤ a 2n+1(1-b) (1-a) 2n+1 (p -a) 2n , p > a (3.2)
The graphs of the derivative function g are illustrated in Figure 4, Figure 5 and Figure 6. In each graph of Figure 4, the parameter a is fixed and in each graph of Figure 5, the parameter b is fixed, where n = 1. One can see that when a is fixed, increasing b the values of the function g on the left and increases the values of the function g on the right; when b is fixed, increasing a shifting the saddle point or turning point of the function g to the right. In Figure 6, one can see that for fixed a and b, increasing n widens the flat region.

Recall that the function g will be used to re-attribute the weight to the original density f and to generate Lila distributions defined in (2.2). The parameter a decides the location where a minimum weight is assigned to; the parameter b controls the scale of weights on each side of the minimum value; the parameter n controls the width of the flat region. For these reasons, the parameter a is referred to as the turning location parameter and the parameter b is referred to as the tail weight parameter and the parameter n is referred to as the separate parameter. The example of the probability density function l has a distinctive feature, see in Figure 7: it has two local maximum points, which we refer to as two peaks, the body peak on the left and the tail peak on the right; the terminology pseudo bi-modal comes from this feature: two peaks with different heights. In each of these five graphs in Figure 9, it is possible to see that the parameter a adjusts the location of and the gaps between the two peaks. As a increases, the range of rare events between two peaks is enlarged and its location shifts to the right; further the height of the tail peak keeps the same for the same s as a change. As a consequence of the preservation of the total mass, the height of the body peak decreases as a increases. Meanwhile, across the five graphs from left top to bottom right, the parameter b increases from 0 to 1. We see that the height of tails decrease as b increases.

Next

In Figure 10, as n increases, the body and tail parts get further apart.

In summary, a Lila distribution constructed with the distortion function G in (3.1) has two peaks of different heights which are call a body peak and a tail peak. The parameters n and a control the range and the location of rare events between two peaks: as a increase the range of rare events between two peaks is enlarged and its location shifts to the right; the parameter b controls the height of tail peaks: as b increases the tail height decreases.

More examples

One can also consider F X to be either a Normal distribution function (exponential family) or Weibull distribution (sub-exponential family as the shape parameter is inferior to 1). Figure 11 and 12 exhibit an identical feature (two peaks) as the example above. Comparing the Gaussian distorted distribution and the Weibull distorted distribution, it is interesting to note that some of the figures look really similar, however the Weibull and the Gaussian distributions are not defined on the same support. Indeed, the Weibull distribution does not tale negative values, as a consequence applied to credit and operational risk problems, where the losses are all defined on ]0, ∞) (recall the definition of non-negative loss), the use of the Gaussian distribution even distorted would not be appropriate, and the risk measures would mechanically be impacted.

Next, we apply the method introduced in Subsection 2.1 with the function G

G(p) =    bp + (1 -b)a 1 -(1 -p a ) 2n+1 , p ≤ a bp + (1 -b) a + (1 -a)( p-a 1-a ) 2n+1 , p > a (3.3)
to construct a Lila distribution with a "shoulder" feature, which is illustrated in Figure 13. In the case when n = 1, the G in (3.3) the Double-Cubic seat with linear blend 7 . Due to a structure similar to the one in (3.1), we will not discuss G in (3.3) any further, and we will only briefly discuss the resulting Lila distributions. In Figure 13 (a), the parameter a is set to be 0.5. In the case a is fixed, the crossing two points of Lila with lognormal are fixed, say x * 1 and x * 2 with x * 1 < x * 2 . In the range (x * 1 , x * 2 ), the density of Lila is lower than lognormal and outside of the range (x * 1 , x * 2 ), the density of Lila is higher than the lognormal. As a decreases from 1 to 0, the gap between the Lila and the lognormal in the region (x * 1 , x * 2 ) increases and a "valley" is formed.

In Figure 13 (b), the parameter b is set to be 0.5. In the case b is fixed, changing a to adjust the shoulder's height and position. As a decreases from 1 to 0, Lila starts to have a shoulder on the right. At certain critical value of a, say a * ∈ (0, 1], the two humps have the same height. (The value of a * depends on the value of b). When a decreases further from a * to 0, a shoulder on the left start to appear, the main body shifts to the right.

In Figure 13 (c), the parameters a = 0.5 and b = 0.5. As n increases, the "shoulder" gets lower and two "shoulders" are formed.

It is worth stressing, in order to generate the desired distribution, that we require the derivative function g of G to be continuous, positive and convex (see in subsection 2.1). Here is an example of the function G which does not satisfy this condition:

G(p) =    b a a 2 -(p -a) 2 , p ≤ a 1 -1-b 1-a (1 -a) 2 -(p -a) 2 , p > a. (3.4)
This function is often referred to as a Double-Elliptic Seat. Although this example of G produces two "peaks", yet due to the indifferentiability of g(p) at the point p = a, the resulting density has some undesired phenomenon of unsmoothness (sudden changes) at certain points. 4 An application of the Lila distribution

In this section, as an application of the newly introduced Lila distribution, we will fit it to two sets of data. In order to find the best fit parameters of a Lila distribution to the sample data, we apply the quantile distance minimisation method, that is to find the parameters which give the minimum value of the square root of the sum of quantile differences between the empirical distribution and a Lila distribution8 .

We generate two sets of data: one set of data distinctively has a body peak and a tail peak, see the histogram in Figure 15, and another set of data with a shoulder feature (see the histogram in Figure 16). To fit a Lila distribution, for the data described in Figure 15,the distortion function G in (3.1) with n = 1 is used; for the data described in Figure 16, the distortion function G in (3.3) with n = 1 is used. In both graphs, the green line represents empirical distributions and the red line the fitted Lila distributions.

Conclusion

In this paper, we focus on the presentation of the Lila distribution by combining an S-shape distortion function with a lognormal distribution. The newly constructed Lila family of distributions allows capturing multiple peaks or several modes empirical risk data usually have. Yet the Lila family of distributions is neither limited to this kind of distortion function nor to this kind of underlying distributions. Indeed we compared the results with the application of the distortion function to Weibull and Gaussian distributions. The Lila family of distributions could naturally extended to other distribution and polynomial of order superior to 3 as soon as the resulting composition is a distribution.

Remark 2 . 1 .

 21 By construction, the function L defined in (2.1) satisfies the criterion of being a cumulative distribution function. That is, the function L is a non-decreasing continuous function with lim x→0 L(x) = 0 and lim x→∞ L(x) = 1.

  [START_REF] Koppstein | Estimation and moment recursion relations for multimodal distributions of the exponential family[END_REF]) is a bimodal exponential density function, and α indicates the relative heights of two modes and β indicates the relative separation of the two modes. In order to generate a bimodal distribution, 4 parameters (i.e. b 0 , b 1 , b 2 , b 3 ) are needed. It is worth noting that our bimodal Lila distributions defined in Definition 2.1 are sub-exponential, meaning we capture fatter tails than these exponential distributions introduced in[START_REF] Koppstein | Estimation and moment recursion relations for multimodal distributions of the exponential family[END_REF], in other words, sub-exponential distribution tails decrease more slowly than any exponential tail).

(3. 1 )

 1 with parameters a ∈ (0, 1), b ∈ [0, 1] and integer n ≥ 1. The function G defined in (3.1) is usually referred to as Double-Odd-Polynomial Seat. In the case when n = 1, (3.1) is referred to as the Double-Cubic Seat 6 . The S-shape feature of the function G defined in (3.1) are illustrated in Figure 1, Figure 2 and Figure 3. Specifically, set n = 1, Figure 1 illustrates the change in the function G by increasing the value of the parameter b from 0 to 1 when the value of the parameter a is fixed. For each graph in Figure 1, we fix the value of the parameter a then we take 7 values of the parameter b. Namely, in each figure, one sees 7 coloured lines corresponding to the 7 values of b.Similarly, set n = 1, Figure2illustrates the change in function G by increasing the value of the parameter a when the value of the parameter b is fixed. Figure3illustrates G with varying n when a = 0.5 and b = 0.5.

Figure 2 : 1 11Figure 3 : 5 12

 2135 Figure 2: Illustration of G for the fixed parameter b when n = 1

Figure 5 : 1 14Figure 6 :Figure 7 :

 5167 Figure 5: Illustration of g for the fixed parameter b when n = 1

Figure 8 : 1 16

 81 Figure 8: Effect of b varying given a is fixed and n = 1

Figure 9 :Figure 10 :

 910 Figure 9: Effect of a varying given b is fixed and n = 1

Figure 11 :Figure 12 :

 1112 Figure 11: Distorted Normal distribution with G in (3.1)

Figure 13 :

 13 Figure 13: Lila distributions with G in (3.2)

Figure 14 : 1 21

 141 Figure 14: Distortion of Lognormal PDF by G in Example 3.1

Figure 16 :

 16 Figure 15: data with 2 local peaks

Documents de travail du Centre d'Economie de la Sorbonne -2016.68

Documents de travail du Centre d'Economie de la Sorbonne -2016.68

Sometimes F -1 X is refered to as quantile function of X. 5 Documents de travail du Centre d'Economie de la Sorbonne -2016.68

As we are working with losses defined on ]0, ∞), we removed the sign for clarity. As a consequence, the losses are regarded as positive values.

A risk measure satisfying the four axioms of translation invariance, sub-additivity, positive homogeneity, and monotonicity is called coherent, c.f [2]. 7 Documents de travail du Centre d'Economie de la Sorbonne -2016.68

http : //www.f long.com/texts/code/shaperspoly/

http : //www.f long.com/texts/code/shaperspoly/ 20 Documents de travail du Centre d'Economie de la Sorbonne -2016.68

The maximum likelihood estimation approach has also been tested and is viable in most cases.22Documents de travail du Centre d'Economie de la Sorbonne -2016.68