domain: info sdv

text:
Crispin Zavala-Alvarado

Antony T Vincent

Odile Sismeiro

Rachel Legendre

Hugo Varet

Giovanni Bussotti

Céline Lorioux

Jean-Yves Coppée

Frédéric J Veyrier

Mathieu Picardeau

Nadia Benaroudj
email: nadia.benaroudj@pasteur.fr

The oxidative stress response and virulence of pathogenic Leptospira are controlled by the interplay of two peroxide stress regulators

Keywords: Microbiology, transcription regulation, reactive oxygen species (ROS), oxidative stress, virulence, leptospirosis, RNASeq, non-coding RNAs, Leptospira, Spirochetes, PerR, Fur

Pathogenic Leptospira are the causative agents of leptospirosis, the most widespread zoonotic infectious disease. Leptospirosis is a potentially severe and life-threatening emerging disease with highest burden in sub-tropical areas and impoverish populations. Mechanisms allowing pathogenic Leptospira to survive inside a host and induce acute leptospirosis are not fully understood. The ability to resist deadly oxidants produced by the host during infection is pivotal for Leptospira virulence. We have previously shown that genes encoding defenses against oxidants in L. interrogans are repressed by PerRA (encoded by LIMLP_10155), a peroxide stress regulator of the Fur family. In this study, we describe the identification and characterization of another putative PerR-like regulator (LIMLP_05620) in L. interrogans.

Protein sequence and phylogenetic analyses indicated that LIMLP_05620 displayed all the canonical PerR amino acid residues and is restricted to pathogenic Leptospira clades. We therefore named this PerR-like regulator PerRB. In L. interrogans, the PerRB regulon is distinct from that of PerRA. While a perRA mutant had a greater tolerance to peroxide, inactivating perRB led to a higher tolerance to superoxide, suggesting that these two regulators have a distinct function in the adaptation of L. interrogans to oxidative stress. The concomitant inactivation of perRA and perRB resulted in a higher tolerance to both peroxide and superoxide and, unlike the single mutants, to the loss of Leptospira virulence.

Interestingly, this correlated with major changes in gene and non-coding RNA expression, only observed in the double perRAperRB mutant. Notably, several virulence-associated genes (clpB, ligA/B, and lvrAB) were repressed. By obtaining the first double mutant in a pathogenic

Leptospira strain, our study has uncovered for the first time the interplay of two PerRs, not only in the adaptation of Leptospira to oxidative stress, but also in their virulence and pathogenicity, most likely through the transcriptional control of a complex regulatory network.

Author summary

Leptospirosis is a widespread infectious disease responsible for over one million of severe cases and 60 000 fatalities annually worldwide. This neglected and emerging disease has a worldwide distribution, but it mostly affects populations from developing countries in sub-tropical areas. The causative agents of leptospirosis are pathogenic bacterial Leptospira spp. There is a considerable deficit in our knowledge of these atypical bacteria, including their virulence mechanisms. In addition to the Leptospira PerRA regulator that represses defenses against peroxide, we have identified and characterized a second PerR regulator in pathogenic Leptospira species (PerRB) that participates in Leptospira tolerance to superoxide.

Phenotypic and transcriptomic analyses of single PerRA and PerRB mutants suggest that the two PerRs fulfill distinct functions in the adaptation to oxidative stress. However, concomitant inactivation of PerRA and PerRB resulted in a higher tolerance to both peroxide and superoxide, but to a loss virulence. The absence of the two PerR regulators resulted in global and major changes in the transcriptional profile, including a dramatic decrease of several virulence factor expression.

Our study has demonstrated that PerRA and PerRB cooperate to orchestrate a complex regulatory network involved in Leptospira virulence.

Introduction

Pathogenic Leptospira spp. are aerobic Gram-negative bacteria of the spirochetal phylum that are the causative agents of leptospirosis, the most widespread zoonosis [START_REF] Haake | Leptospirosis in humans[END_REF]. More than one million cases of leptospirosis are currently reported annually in the world, with 10%

of mortality [START_REF] Costa | Global Morbidity and Mortality of Leptospirosis: A Systematic Review[END_REF]. This disease is considered as a health threat among impoverished populations in developing countries under tropical areas [START_REF] Costa | Global Morbidity and Mortality of Leptospirosis: A Systematic Review[END_REF], but the number of reported cases of leptospirosis are also on the rise in developed countries under temperate climates [START_REF] Pijnacker | Marked increase in leptospirosis infections in humans and dogs in the Netherlands[END_REF]. Rodents are asymptomatic reservoir for leptospires as the bacteria colonize the proximal renal tubules of these animals. Leptospires are shed in the environment through the urine of infected animals and leptospirosis is transmitted to other animal species and humans mostly by exposure to contaminated soils and water. Leptospira penetrate an organism through abraded skins and mucous membranes, subsequently disseminate within the bloodstream and rapidly spread to multiple tissues and organs (including kidney, liver, lungs). Clinical manifestations range from a mild flu-like febrile state to more severe and fatal cases leading to hemorrhages and multiple organ failure. Because of the lack of efficient tools for genetic manipulation of Leptospira spp. and their fastidious growth in the laboratory conditions, our understanding of the mechanism of pathogenicity and virulence as well as the basic biology of these pathogens have been greatly hampered [START_REF] Ko | Leptospira: the dawn of the molecular genetics era for an emerging zoonotic pathogen[END_REF][START_REF] Picardeau | Virulence of the zoonotic agent of leptospirosis: still terra incognita?[END_REF].

During their life cycle, most bacteria will be exposed to reactive oxidative species (ROS), such as superoxide anion (  O 2 -) and hydrogen peroxide (H 2 O 2 ), that are produced endogenously or encountered in the environment [START_REF] Imlay | Where in the world do bacteria experience oxidative stress?[END_REF]. The superoxide is formed during the aerobic respiratory chain upon the reduction of the dioxide (O 2 ). Dismutation of superoxide will give rise to H 2 O 2 which, in turn, can react with ferrous iron to generate the highly reactive hydroxyl radicals (  OH) through the Fenton reaction. These ROS are also produced together with hypochlorous acid (HOCl), and nitric oxide anion (  NO) as powerful and efficient weapons by eukaryotic innate immune response upon infection by pathogenic bacteria [START_REF] Winterbourn | Redox Reactions and Microbial Killing in the Neutrophil Phagosome[END_REF]. ROS cause oxidative damage to cellular components (proteins, DNA and lipids) and this would result in bacterial death if bacteria had not developed scavenging enzymes to counteract the deadly effect of ROS, including catalase, peroxidases and superoxide dismutase or reductase (SOD, SOR). Indeed, in response to any increase in ROS concentration, as occurred during environmental oxidative stress or when infecting a host, bacteria trigger an adaptive response allowing the fine-tuning of scavenging enzyme synthesis rate that appropriately adapts the defenses against ROS to the oxidant assault. ROS production is increased upon infection by Leptospira [START_REF] Marangoni | Production of reactive oxygen species and expression of inducible nitric oxide synthase in rat isolated Kupffer cells stimulated by Leptospira interrogans and Borrelia burgdorferi[END_REF] and with the development of severe leptospirosis in patients and infected animals [START_REF] Araújo | Oxidative stress markers correlate with renal dysfunction and thrombocytopenia in severe leptospirosis[END_REF][START_REF] Erdogan | Serum sialic acid and oxidative stress parameters changes in cattle with leptospirosis[END_REF]. In fact, the ability to detoxify H 2 O 2 is essential for Leptospira virulence as inactivation of the catalase-encoding gene led to virulence attenuation in L. interrogans [START_REF] Eshghi | Leptospira interrogans Catalase Is Required for Resistance to H2O2 and for Virulence[END_REF].

The oxidative stress adaptive response in bacteria is controlled by specialized ROS sensing transcription factors. OxyR and PerR are the main peroxide-sensing regulators found in Gram-negative and Gram-positive bacteria, respectively [START_REF] Dubbs | Peroxide-sensing transcriptional regulators in bacteria[END_REF]. OxyR and PerR are functional equivalents but they do control expression of peroxide scavenging enzymes by different mechanisms. OxyR is a transcriptional regulator from the LysR family. Tetrameric OxyR binds promoters and activates the expression of genes coding for catalase and peroxidases through the oxidation of its H 2 O 2 sensing cysteine residue [START_REF] Choi | Structural Basis of the Redox Switch in the OxyR Transcription Factor[END_REF][START_REF] Zheng | Activation of the OxyR Transcription Factor by Reversible Disulfide Bond Formation[END_REF]. PerR belongs to the Fur (Ferric uptake regulator) transcriptional factor family. It is a metalloprotein that binds DNA in the presence of iron or manganese and represses the expression of catalase and peroxidases [START_REF] Herbig | Roles of metal ions and hydrogen peroxide in modulating the interaction of the Bacillus subtilis PerR peroxide regulon repressor with operator DNA[END_REF][START_REF] Jacquamet | Structural characterization of the active form of PerR: insights into the metal-induced activation of PerR and Fur proteins for DNA binding[END_REF][START_REF] Traoré | Crystal structure of the apo-PerR-Zn protein from Bacillus subtilis[END_REF]. In the presence of peroxide, PerR is oxidized on the histidine residues participating in iron coordination, PerR dissociates from DNA and peroxide-scavenging enzyme repression is alleviated [START_REF] Lee | The PerR transcription factor senses H2O2 by metal-catalysed histidine oxidation[END_REF][START_REF] Traoré | Structural and functional characterization of 2-oxo-histidine in oxidized PerR protein[END_REF]. In Gram-negative bacteria, SOD expression is controlled by the SoxRS system [START_REF] Nunoshiba | Two-stage control of an oxidative stress regulon: the Escherichia coli SoxR protein triggers redox-inducible expression of the soxS regulatory gene[END_REF][START_REF] Wu | Two-stage induction of the soxRS (superoxide response) regulon of Escherichia coli[END_REF] by a redox-sensing mechanism and oxidation of a Fe-S cluster [START_REF] Gu | The SoxRS response of Escherichia coli is directly activated by redox-cycling drugs rather than by superoxide[END_REF]. In Grampositive bacteria, SOD expression regulation probably involves the alternative Sigma Factor SigB among other mechanisms [START_REF] Price | Genome-wide analysis of the general stress response in Bacillus subtilis[END_REF]. Pathogenic Leptospira spp. are among the few examples of Gram-negative bacteria where the expression of peroxide-scavenging enzymes is regulated by a PerR (encoded by LIMLP_10155/LIC12034/LA1857) and not an OxyR [START_REF] Lo | Transcriptional response of Leptospira interrogans to iron limitation and characterization of a PerR homolog[END_REF]. Moreover, their genomes do not contain any ORF encoding a SOD and they do not exhibit canonical SOD activity [START_REF] Austin | Distribution of superoxide dismutase, catalase, and peroxidase activities among Treponema pallidum and other spirochetes[END_REF][START_REF] Fouts | What Makes a Bacterial Species Pathogenic?: Comparative Genomic Analysis of the Genus Leptospira[END_REF].

We have very recently characterized the transcriptional response to hydrogen peroxide in the pathogen L. interrogans and shown that these bacteria respond to sublethal H 2 O 2 concentration by increasing the expression of catalase and of two peroxidases (Alkylperoxiredoxin (AhpC) and Cytochrome C peroxidase (CCP)) [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF]. When Leptospira were exposed to deadly H 2 O 2 concentration, additional enzymes with a putative role as antioxidants and/or in repair of oxidized cysteines in proteins were up-regulated, including several thiol oxidoreductases (thioredoxin, glutaredoxin, DsbD, and Bcp-like proteins) [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF].

Several genes of the LexA regulon (recA, recN, dinP) and other genes with putative role in DNA repair (mutS, radC) had a higher expression as well as genes encoding for canonical chaperones (dnaK/dnaJ/grpE, groES/EL, and hsp15/20) [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF]. Only genes coding for the catalase and peroxidases were under the control of PerR and our study has revealed a complex regulatory network independent of PerR involving other transcriptional regulators, sigma factors, two component systems and non-coding RNAs [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF]. During the course of this study, we noticed that an ORF encoding a Fur-like regulator (LIMLP_05620/LIC11158/LA2887) was up-regulated when Leptospira were exposed to a deadly concentration of H 2 O 2 . Here, we report the functional characterization of this pathogenic Leptospira-specific Fur-like regulator in the adaptation of Leptospira to oxidative stress. Phenotypic analyses of a mutant inactivated in LIMLP_05620 indicates a role of this Fur-like regulator in Leptospira tolerance to superoxide. This, together with the presence of the canonical amino acid residues of the PerR DNA binding helix and H 2 O 2 sensitivity, strongly suggest that LIMLP_05620 encodes a second PerR-like regulator in pathogenic Leptospira species. By obtaining a double mutant where LIMLP_05620 and perR (LIMLP_10155) are concomitantly inactivated, we have also investigated the interplay between these two PerR-like regulators in the adaptation to oxidative stress and virulence of L. interrogans. We have demonstrated a cooperation of LIMLP_05620 with PerR in controlling several Leptospira virulence-associated genes, including ligA, ligB, lvrAB, and clpB, perhaps illustrating the importance of functional redundancy in pathogenic Leptospira virulence and pathogenicity.

Results

Identification of an ORF that encodes a novel putative PerR regulator in pathogenic

Leptospira

Identification of differentially-expressed genes upon perRB inactivation.

To understand the role of PerRB in L. interrogans tolerance to ROS, we compared the global transcriptional profiles of the perRB mutant and WT strains. Differential gene expression analysis revealed changes in the transcription of 123 genes, with 59 and 64 downand up-regulated, respectively (see S2 Table for a complete set of data). However, perRB inactivation did not lead to dramatic changes in gene expression as the majority of Log 2 FC (108 out of 123) ranged between -1 and 1 (S2 Table ). These findings indicate that the absence of an active PerRB did not lead to substantial significant changes in genes expression when Leptospira are cultivated in the laboratory conditions (in EMJH medium at 30°C) and during the exponential phase.

Nevertheless, when examining the nature of the highest differentially-expressed genes in the perRB mutant, some tendencies could be observed. Many of the differentially-expressed

ORFs were annotated as protein with unknown function and did not have homologs in the saprophyte L. biflexa strain (S2 1. Differentially-expressed ORFs upon perRB inactivation Selected up-and down-regulated genes in the LIMLP_05620 (perRB) mutant with an adjusted p-value cutoff of 0.05. a Gene numbering is according to Satou et al [START_REF] Satou | Complete Genome Sequences of Low-Passage Virulent and High-Passage Avirulent Variants of Pathogenic Leptospira interrogans Serovar Manilae Strain UP-MMC-NIID, Originally Isolated from a Patient with Severe Leptospirosis, Determined Using PacBio Single-Molecule Real-Time Technology[END_REF]. Corresponding genes of L. interrogans serovar Lai strain 56601 and serovar Copenhageni Fiocruz strain L1-130 are indicated in parenthesis. b Closest ortholog in the saprophytes L. biflexa serovar Patoc strain Patoc1. The absence of syntheny is indicated in italic. Genes that are down-regulated upon perRA inactivation as determined previously [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF] are indicated in bold. Down (*) and up (**) -regulated genes upon exposure to lethal H 2 O 2 dose as determined previously [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF].

Among the highest up-regulated genes, two ORFs (LIMLP_04075 and LIMLP_18760) encoded lipoproteins with a putative adhesin function. These proteins contain DUF1566 domain repeats which is also share by Lsa25, a Leptospiral surface adhesin that binds extracellular matrix (ECM) [START_REF] Domingos | Features of two proteins of Leptospira interrogans with potential role in hostpathogen interactions[END_REF].

Several genes involved in the metabolism of c-di GMP were differentially-expressed upon perRB inactivation. C-di GMP is a secondary messenger in bacteria that regulates a variety of processes such as biofilm formation, motility, stress adaptation, and virulence. C-di GMP synthesis is catalyzed by diguanylate cyclases (DGCs) whereas its hydrolysis is catalyzed by phosphodiesterases (PDEs). DGCs and PDEs are numerous in pathogenic Leptospira, suggesting that c-di GMP fulfills an important role in sensing environmental signals when

Leptospira infect and colonize a host. C-di GMP has been recently shown to regulate biofilm formation, motility and protection against environmental stress in pathogenic Leptospira [START_REF] Thibeaux | The zoonotic pathogen Leptospira interrogans mitigates environmental stress through cyclicdi-GMP-controlled biofilm production[END_REF].

Four DGCs (LIMLP_05450, LIMLP_05455, LIMLP_05460, LIMLP_05485) were upregulated upon perRB inactivation (Table 1). These DGC-encoding ORFs are located in a gene cluster (LIMLP_05485-05450) that contains 7 ORFs coding for DGCs. LIMLP_05450, LIMLP_05455, LIMLP_05460, and LIMLP_05485 display the typical diguanylate cyclase GGDEF and sensory PAS domains. A DGC activity was demonstrated in vitro for LIMLP_05450, LIMLP_05455, LIMLP_05460 [START_REF] Xiao | Identification and Characterization of c-di-GMP Metabolic Enzymes of Leptospira interrogans and c-di-GMP Fluctuations After Thermal Shift and Infection[END_REF]. Three PDE-encoding ORFs (LIMLP_05845, LIMLP_9580, and LIMLP_18375) were also up-regulated in the perRB mutant.

LIMLP_13165 was among the most down-regulated ORFs when perRB was inactivated. It encodes a secreted protein with sphingomyelinase C and hemolytic activities [START_REF] Narayanavari | Role of sph2 Gene Regulation in Hemolytic and Sphingomyelinase Activities Produced by Leptospira interrogans[END_REF]. Another significantly down-regulated ORF encoded a protein with an acyl CoA acetyl tranferase domain annotated as a putative hemolysin (LIMLP_15470). This ORF is up-regulated when L. interrogans is cultivated in DMC implemented in rats [START_REF] Caimano | A Model System for Studying the Transcriptomic and Physiological Changes Associated with Mammalian Host-Adaptation by Leptospira interrogans Serovar Copenhageni[END_REF].

An ORF encoding an AraC transcriptional regulator (LIMLP_19320), and two ORFs of unknown function (LIMLP_18755 and 19325) were among the most up-regulated. The orthologs of LIMLP_19320 and LIMLP_19325 in L. interrogans serovar Lai belongs to a genomic island (Lai GI B, LA1747-1851) that can excise from the chromosome and form an independent replicative plasmid [START_REF] Bourhy | A Genomic Island of the Pathogen Leptospira interrogans Serovar Lai Can Excise from Its Chromosome[END_REF][START_REF] He | Genetic diversity among major endemic strains of Leptospira interrogans in China[END_REF]. Among the down-regulated genes, several ORFs encode factors related to oxidative stress.

LIMLP_04255 is part of a gene cluster (LIMLP_04240-04285) which code for a putative TonB-dependent transport system repressed by PerRA. We have previously shown that some genes of this cluster (LIMLP_04245, LIMLP_04270 and LIMLP_04280) are involved in L. interrogans tolerance to superoxide [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF]. LIMLP_11400 encodes the DNA repair protein RadC and LIMLP_14715 is a homolog of the E. coli Dps, a protein that sequesters iron and protects DNA from oxidative damage. LIMLP_06190 encodes a putative disulfide oxidoreductase with the N-terminal ScdA domain (DUF1858). In S. aureus, ScdA is a di-iron protein involved in repair of oxidatively damaged iron-sulfur cluster proteins [START_REF] Overton | Widespread Distribution in Pathogenic Bacteria of Di-Iron Proteins That Repair Oxidative and Nitrosative Damage to Iron-Sulfur Centers[END_REF].

LIMLP_14595 encodes a putative transmembrane lipoprotein with a cytochrome-like domain that shows homology with the CcoP subunit of the cytochrome C oxidase and could function in the respiratory chain or be an enzyme cofactor.

Only 7 out of the 123 differentially-expressed genes in the perRB mutant were also differentially-expressed upon perRA inactivation with a similar inclination (S3 Fig) [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF]. LIMLP_02010 and LIMLP_04325 were up-regulated whereas LIMLP_04255, LIMLP_11810, LIMLP_14225, LIMLP_15470 and LIMLP_18235 were down-regulated in the two mutants.

Notably, 82 out of the 123 differentially-expressed ORFs in the perRB mutant were also differentially-expressed upon exposure of L. interrogans to H 2 O 2 (S3 Fig) [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF]. Thus, 66% of the PerRB regulon is also regulated by the presence of H 2 O 2 . Interestingly, the majority of ORFs down-regulated in the perRB mutant, including the RadC and the Dps-encoding ORFs, were up-regulated in the presence of H 2 O 2 (with Log 2 FCs of 3.46 and 1.10, respectively) (S3 Fig and Table 1). On the contrary, many up-regulated ORFs in the perRB mutant had a lower expression in the presence of H 2 O 2 . For instance, the ORFs that code for Cas5 (LIMLP_02880), Cas3 (LIMLP_02885), and two DGCs (LIMLP_05450 and LIMLP_05455) were down-regulated upon exposure to H 2 O 2 with Log 2 FCs lower than -1.21 (S3 Fig and Table 1).

Concomitant inactivation of perRA and perRB leads to a higher resistance to ROS but to a lower virulence

In order to investigate whether PerRA and PerRB cooperate in regulating the adaptive response to ROS, we inactivated perRA by allelic exchange in the perRB mutant (S4 Fig).

This allowed obtaining a double perRAperRB mutant in a L. interrogans strain, the first double mutant constructed in a pathogenic Leptospira.

The double perRAperRB mutant had a growth rate comparable to that of the single perRA and perRB mutants and WT strains when L. interrogans were cultivated in EMJH medium (Fig Therefore, the concomitant inactivation of perRA and perRB resulted in a loss of virulence in L. interrogans and expressing only perRB in the double mutant restored Leptospira virulence.

Altogether, these results demonstrate that despite a higher resistance to ROS, the double perRAperRB mutant exhibited a dramatically reduced virulence. and LigB (LIMLP_15415) proteins. These surface-exposed proteins are involved in adhesion to host cells through ECM binding [START_REF] Choy | Physiological osmotic induction of Leptospira interrogans adhesion: LigA and LigB bind extracellular matrix proteins and fibrinogen[END_REF] and participate in the immune evasion through binding to the host complement Factor H and C4b binding protein [START_REF] Castiblanco-Valencia | Leptospiral Immunoglobulin-like Proteins Interact With Human Complement Regulators Factor H, FHL-1, FHR-1, and C4BP[END_REF]. Simultaneous downregulation of ligA and ligB expression led to attenuation of Leptospira virulence [START_REF] Pappas | Control of Gene Expression in Leptospira spp. by Transcription Activator-Like Effectors Demonstrates a Potential Role for LigA and LigB in Leptospira interrogans Virulence[END_REF]. LigA and ligB were down-regulated in the perRAperRB mutant (Log2FC of -3 and -2. ).

Concomitant inactivation of

LvrA (LIMLP_08490) and lvrB (LIMLP_08485) encode a hybrid histidine kinase and a hybrid response regulator, respectively. Inactivation of the lvrAB operon led to virulence attenuation in L. interrogans [START_REF] Adhikarla | Lvr, a Signaling System That Controls Global Gene Regulation and Virulence in Pathogenic Leptospira[END_REF]. LvrA and lvrB had both a decreased expression in the perRAperRB mutant (Log2FC of -2.

3) (Fig 8D and S5 Table).

Additional ORFs encoding chaperones (the small heat shock proteins Hsp15 and Hsp20) or enzymes involved in protein folding (the disulfide isomerase DsbD and the peptidyl-prolyl cis-trans isomerase SlyD) and degradation (HtpX) were down-regulated in the perRAperRB mutant. The involvement of these factors in Leptospira virulence has not been demonstrated but small Hsps participate in M. tuberculosis growth in macrophages [START_REF] Yuan | The 16-kDa α-crystallin (Acr) protein of <em>Mycobacterium tuberculosis</em> is required for growth in macrophages[END_REF]. DsbD and these two small Hsps were up-regulated in L. interrogans upon exposure to H 2 O 2 [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF]. All these factors might protect Leptospira proteostasis under adverse conditions as encountered during infection inside a host. The down-regulation of these virulence-associated genes together with the differential expression of several other genes was confirmed by RT-qPCR (S5 Fig).

Interestingly, gene expression of the virulence-associated genes (lvrA, lvrB, ligA, ligB, clpB, hsp15, and hsp20) was increased to the WT level (or even to a higher level) in the double perRAperRB mutant strain complemented in trans only with perRB (Fig 9)

.

Taken together, these findings indicate that the loss of virulence resulting from the concomitant inactivation of perRA and perRB correlated with a global deregulation of a complex gene network, including genes associated with virulence.

Identification of differentially-expressed non-coding RNAs in the perRB and perRAperRB mutants.

Intergenic regions were also analyzed to identify differentially expressed predicted non-coding RNAs (ncRNAs). As observed for coding sequences, inactivation of perRB led to the deregulation of only a few putative ncRNAs and most of the changes in expression were below two folds (see S6 Table for a complete set of data). Nonetheless, a few numbers of ncRNAs were significantly down-regulated with a Log 2 FC below -1 (S7 Table ). Some of the differentially-expressed ncRNAs (LepncRNA36, LepncRNA87, LepncRNA89, LepncRNA109, LepncRNA139) were located in the proximate vicinity of differentiallyexpressed ORFs in the perRB mutant. Three ncRNAs (LepncRNA35, LepncRNA89 and LepncRNA109) were also differentially expressed upon perRA inactivation (S7 Table) [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF].

55 putative ncRNAs were differentially-expressed (with a Log 2 FC cutoff of 1) in the perRAperRB mutant and several of them were adjacent or overlapped differentially-expressed ORFs (S8 Table ). Only a few of these differentially-expressed ncRNAs had an altered expression in the single perRA and perRB mutant (S8 Table) [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF].

Among the most highly differentially-expressed ncRNAs was LepncRNA38 that was located downstream ccp, a highly up-regulated ORF in the perRAperRB mutant (Fig 10 andS8 Table ). LepncRNA38 and ccp were also up-regulated in the perRA mutant [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF]. The ncRNA LepncRNA49, which was down-regulated in the perRAperRB mutant, overlapped with exbB (LIMLP_04255), an ORF that was also down-regulated in the double perRAperRB mutant as well as in the single perRA and perRB mutants ( Altogether, these findings indicate that the absence of both PerRA and perRB triggers major changes in the transcriptional activity of many ncRNAs in L. interrogans, that could consequently alter the expression of many ORFs.

Discussion

Virulence mechanisms are poorly characterized in pathogenic Leptospira. These bacteria possess a high number of genes encoding proteins of unknown function (40% of the genomes) and many of them are pathogen-specific. Pathogenic Leptospira spp. lack many classical virulence factors, such as a type III to type VI secretion systems, and it is unclear which factors are important for its pathogenesis. It is therefore generally agreed that these pathogens possess unique virulence factors. Nonetheless, studying heme oxygenase and catalase mutants have shown that, in vivo, iron acquisition and defense against peroxide stress are important virulence-associated mechanisms in L. interrogans [START_REF] Eshghi | Leptospira interrogans Catalase Is Required for Resistance to H2O2 and for Virulence[END_REF][START_REF] Murray | Leptospira interrogans requires heme oxygenase for disease pathogenesis[END_REF]. Catalase is repressed by PerRA [START_REF] Lo | Transcriptional response of Leptospira interrogans to iron limitation and characterization of a PerR homolog[END_REF][START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF] and genes encoding factors involved in iron uptake are very likely controlled by regulators of the Fur-family.

In addition to PerRA, pathogenic Leptospira contain three other ORFs annotated as Furs. In the present study, we have characterized the L. interrogans Fur-like regulator encoded by LIMLP_05620 and showed that it exhibits characteristic features of a PerR regulator. We consequently named this ORF perRB. Sequence alignment and phylogenetic analyses revealed that PerRB is the closest relative to the already characterized PerRA, and perhaps more importantly, they both do exhibit the canonical amino acid residues that are the hallmark of a PerR. The H 2 O 2 sensing histidine and aspartate residues are conserved in Leptospira

PerRA and PerRB and, interestingly, both genes are H 2 O 2 -responsive, albeit with different apparent sensitivity. This is consistent with a mechanism whereby PerRA and PerRB would repress their own transcription and dissociate from their promoter upon oxidation by H 2 O 2 , leading to alleviation of repression. Moreover, the higher survival of the perRB mutant in the presence of superoxide suggests a derepression of genes encoding defenses against ROS and therefore the participation of PerRB in controlling the adaptation to oxidative stress. Neither perRA nor perRB expression was up-regulated in iron-limiting condition [START_REF] Lo | Transcriptional response of Leptospira interrogans to iron limitation and characterization of a PerR homolog[END_REF]. Although the putative lipoprotein LIMLP_18755 was significantly up-regulated in the perRB mutant and under iron-limiting condition, there was no strong overlap between PerRB regulon and the transcriptional response to iron-limiting condition [START_REF] Lo | Transcriptional response of Leptospira interrogans to iron limitation and characterization of a PerR homolog[END_REF]. Altogether, these findings could argue in favor of LIMLP_05620 encoding a PerR-like regulator rather than a Fur. However, because iron homeostasis and oxidative stress are intertwined, a certain functional relationship has been observed between PerR and Fur. In several bacteria where PerR and Fur coexist, including B. subtilis and C. jejuni, the PerR regulon overlaps with that of Fur [START_REF] Hayashi | The H2O2 Stress-Responsive Regulator PerR Positively Regulates srfA Expression in Bacillus subtilis[END_REF][START_REF] Palyada | Characterization of the oxidative stress stimulon and PerR regulon of Campylobacter jejuni[END_REF]. In addition, fur and several Fur-regulated genes are also differentially expressed in the presence of H 2 O 2 [START_REF] Mostertz | Transcriptome and proteome analysis of Bacillus subtilis gene expression in response to superoxide and peroxide stress[END_REF][START_REF] Zhou | Hydrogen peroxide-induced oxidative stress responses in Desulfovibrio vulgaris Hildenborough[END_REF]. In fact, PerR represses fur, whose expression is up-regulated in the presence of H 2 O 2 as a consequence of dissociation of PerR from the fur promotor [START_REF] Fuangthong | Regulation of the Bacillus subtilis fur and perR Genes by PerR: Not All Members of the PerR Regulon Are Peroxide Inducible[END_REF][START_REF] Horsburgh | In Staphylococcus aureus, Fur Is an Interactive Regulator with PerR, Contributes to Virulence, and Is Necessary for Oxidative Stress Resistance through Positive Regulation of Catalase and Iron Homeostasis[END_REF]. Metalcatalyzed oxidation of the H 2 O 2 sensing residues will be fundamental in establishing that the Per-like regulator encoded by LIMLP_05620 is a bona fide PerR.

To the best of our knowledge, this is the first report that has identified the coexistence of two PerR regulators in a pathogenic Gram-negative bacterium. The coexistence of three PerR-like regulators has been reported only in Gram-positive bacteria such as B. licheniformis and M.

smegmatis. It was shown that the three B. licheniformis PerRs sense hydrogen peroxide by histidine oxidation, although with different sensitivity [START_REF] Kim | Bacillus licheniformis Contains Two More PerR-Like Proteins in Addition to PerR, Fur, and Zur Orthologues[END_REF] . In M. smegmatis, three Fur-like paralogs displayed the canonical PerR Asp residue involved in H 2 O 2 sensitivity, exhibited H 2 O 2 sensing by metal-catalyzed histidine oxidation and a higher H 2 O 2 resistance when their genes were inactivated [START_REF] Lee | Roles of three FurA paralogs in the regulation of genes pertaining to peroxide defense in Mycobacterium smegmatis mc2155[END_REF].

One important question was to understand the mechanism that have led to the coexistence of PerRA and PerRB exclusively in highly virulent species (P1 clade). Virulent mammalianadapted strains in the Leptospira genus might have originated from a free-living ancestor inhabiting soils. The phylogenetic analysis presented here indicates that the coexistence of PerRA and PerRB is not due to gene duplication. Indeed, PerRA was already present in the leptospirales ancestor whereas PerRB was probably acquired by pathogenic species by horizontal transfer from a soil/aquatic bacterium of another phylum. In this scenario, PerRA would had been lost by the P2 clade intermediate species but maintained together with PerRB by the P1 clade species to establish full virulence.

We had previously shown that when perRA was inactivated, L. interrogans acquired a higher resistance to H 2 O 2 explained by the derepression of katE, ahpC and ccp [START_REF] Lo | Transcriptional response of Leptospira interrogans to iron limitation and characterization of a PerR homolog[END_REF][START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF]. Here, we have demonstrated that inactivating perRB resulted in a higher survival of L. interrogans in the presence of the superoxide but it did not affect the survival of L. interrogans in the presence of H 2 O 2 . Therefore, even though perRB is up-regulated upon exposure to H 2 O 2 as perRA, the consequence of perRB inactivation is different than that of perRA, suggesting that The low number of significantly differentially-expressed genes in the perRB mutant when L.

interrogans are cultivated in vitro led us to propose that PerRB exerts its function during oxidative stress or upon host-related conditions. Consistent with this hypothesis is the upregulation of perRB in the presence of lethal H 2 O 2 dose. It is worth noting that there is, to some extent, an overlap between the PerRB regulon and the differentially-expressed genes upon exposure to lethal H 2 O 2 dose [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF]. Moreover, the exclusive presence of PerRB in the pathogenic Leptospira clades strongly suggests that PerRB function is more related to regulating adaptation to infection-related conditions rather than to environmental survival.

Very interestingly, when the perRB mutant was cultivated in the host conditions using the DMC implants, a higher number of genes were differentially-expressed with greater fold changes than when the perRB mutant was cultivated at 30°C in the laboratory conditions [START_REF] Grassmann | The Fur-like regulators PerRA and PerRB control a complex signaling network required for mammalian host-adpatation and virulence of Leptospira interrogans[END_REF].

Notably, in the host condition, the absence of perRB also led to the deregulation of several genes involved in signaling and regulation, which is consistent with a role of PerRB in regulating adaptation to the host environment [START_REF] Grassmann | The Fur-like regulators PerRA and PerRB control a complex signaling network required for mammalian host-adpatation and virulence of Leptospira interrogans[END_REF].

One feature of Leptospira genus is the genetic and functional redundancy where multiple genes commonly encode for a similar function. The development of genetic tools has made random and targeted mutagenesis possible, albeit with a low efficiency. Due to this limitation, only a few Leptospira virulence factors have been identified and have fulfilled Koch's molecular postulates. The present study is the first to report the concomitant inactivation of two genes and complementation of a double mutant in a pathogenic Leptospira. Obtaining a double perRAperRB mutant gave us the unique opportunity to investigate the functional relationship between two PerR-like regulators in a pathogenic bacterium.

In many pathogens which contain only one PerR paralog, such as N. gonorrhoeae, S.

pyogenes, and S. aureus, PerR was shown to be involved in virulence [START_REF] Horsburgh | In Staphylococcus aureus, Fur Is an Interactive Regulator with PerR, Contributes to Virulence, and Is Necessary for Oxidative Stress Resistance through Positive Regulation of Catalase and Iron Homeostasis[END_REF][START_REF] Brenot | The PerR regulon in peroxide resistance and virulence of Streptococcus pyogenes[END_REF][START_REF] Rea | Disruption of putative regulatory loci in Listeria monocytogenes demonstrates a significant role for Fur and PerR in virulence[END_REF][START_REF] Ricci | The Regulator PerR Is Involved in Oxidative Stress Response and Iron Homeostasis and Is Necessary for Full Virulence of <em>Streptococcus pyogenes<[END_REF][START_REF] Wu | PerR controls Mn-dependent resistance to oxidative stress in Neisseria gonorrhoeae[END_REF]. The single L. interrogans perRA and perRB mutants still retain full virulence in the model for acute leptospirosis (this study and [START_REF] Lo | Transcriptional response of Leptospira interrogans to iron limitation and characterization of a PerR homolog[END_REF]). Interestingly, virulence attenuation was only observed upon the concomitant inactivation of perRA and perRB, suggesting an interplay in controlling (directly or indirectly) L. interrogans virulence-associated genes. The loss of virulence correlated with a large differential gene and ncRNA expression compared not only with the WT but also with the single mutant strains. In other words, the double perRAperRB displayed differential gene expression that were not observed in the single perRA and perRB mutants.

This could indicate that a subset of genes and ncRNAs is controlled by both PerRA and PerRB. The absence of one regulator could be compensated by the other and most of the genes and ncRNAs that can be regulated by the two regulators would not be differentially expressed in the single mutants. The few genes (LIMLP_02010, LIMLP_04255, LIMLP_04235, LIMLP_11810, LIMLP_14225, LILP_15470, and LIMLP_18235) and ncRNAs that display differential expression in the single mutants in our transcriptomic studies (this study and [START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF]) indicate a certain functional redundancy of the two regulators even if the phenotypic analyses of the mutants suggest distinct functions. The change in expression of a few regulators when PerRA and PerRB are both absent could lead to major changes in expression of many ORFs or ncRNAs in cascade. One can also speculate that among the large differentially-expressed genes only observed in the double perRAperRB mutant some deregulation might be due to a compensatory effect to maintain a productive fitness.

Despite a higher ability to resist ROS, the double perRAperRB mutant has lost its virulence; it could not trigger acute leptospirosis-associated morbidity. This could be obviously explained by a significant lower expression of several virulence-associated factors in the double perRAperRB mutant, such as LigA, LigB, LvrA, LvrB, and ClpB. In addition, other dramatically down-regulated genes encode factors such as small Hsps (Hsp15 and Hsp20) for which a role in bacterial virulence is demonstrated in other bacteria including M. tuberculosis [START_REF] Stewart | The stress-responsive chaperone α-crystallin 2 is required for pathogenesis of Mycobacterium tuberculosis[END_REF]. Moreover, several differentially-expressed ORFs of unknown function could also be responsible for the loss of virulence of the double perRAperRB mutant. It is noteworthy that the double perRAperRB mutant was also unable to colonize mice [START_REF] Grassmann | The Fur-like regulators PerRA and PerRB control a complex signaling network required for mammalian host-adpatation and virulence of Leptospira interrogans[END_REF] and, consistent with what is observed with in vitro-cultivated perRAperRB mutant, this correlated with dramatic gene dysregulation including down-regulation of ligA and ligB [START_REF] Grassmann | The Fur-like regulators PerRA and PerRB control a complex signaling network required for mammalian host-adpatation and virulence of Leptospira interrogans[END_REF].

In summary, this study has allowed to identify a second PerR-like regulator in pathogenic L.

interrogans strains that cooperates with PerRA to control the adaptation to oxidative stress and virulence. By concomitantly inactivating perRA and perRB and establishing the molecular Koch' postulates, we have unveiled a complex regulatory network that reveals, for the first time, a functional relationship between PerR regulators and Leptospira virulence, most likely through the regulation of virulence-and pathogenicity-associated factors.

Materials and Methods

Bacterial strains and growth condition

L. interrogans serovar Manilae strain L495, perRA (M776), perRB (M1474), and the double perRAperRB mutant strains (see S9 Table for a complete description of the strains used in this study) were grown aerobically at 30°C in Ellinghausen-McCullough-Johnson-Harris medium (EMJH) [START_REF] Ellinghausen | Nutrition of Leptospira pomona and growth of 13 other serotypes: a serum-free medium employing oleic albumin complex[END_REF] with shaking at 100 rpm. Leptospira growth was followed by measuring the absorbance at 420 nm. 2163 and 1 E. coli strains were cultivated at 37°C in Luria-Bertani medium with shaking at 37°C in the presence of 0.3 mM thymidine or diaminopimelic acid (Sigma-Aldrich), respectively. When needed, spectinomycin, kanamycin and gentamycin were added at the respective concentration of 50 µg/ml, 30 µg/ml, and 8 µg/ml.

Concomitant inactivation of perRA (LIMLP_10155) and perRB (LIMLP_05620).

PerRA gene (LIMLP_10155/LIC12034/LA1857) was inactivated in the perRB (LIMLP_05620/LIC11158/LA2887) mutant strain (M1474, perRB::Km R ) by introduction of a spectinomycin resistance cassette (S4 Fig) . For this, a spectinomycin resistance cassette flanked by 0.8 kb sequences homologous to the sequences flanking perRA was created by gene synthesis (GeneArt, Life Technologies) and cloned into a kanamycin-resistant Escherichia coli vector unable to replicate in Leptospira. The obtained suicide plasmid (pKperRA) (S10 Table ) was introduced in the perRB mutant strain by electroporation as previously described [START_REF] Picardeau | First evidence for gene replacement in Leptospira spp. Inactivation of L. biflexa flaB results in non-motile mutants deficient in endoflagella[END_REF] using a Gene Pulser Xcell (Biorad). Individual spectinomycinresistant colonies were selected on EMJH plates containing 50 µg/ml spectinomycin and screened by PCR (using the P1 and P2 primer set, see S11 Table ) for proper replacement of the perRA coding sequence by the spectinomycin resistance cassette. PerRA inactivation in the double perRAperRB mutant was verified by western blot using an anti-PerRA serum (S4

Fig)

.

Complementation of the single perRB and double perRAperRB mutants

The perRB mutant (perRB::Km R ) complementation was performed by expressing the perRB ORF in the pMaORI replicative vector [START_REF] Pappas | A Replicative Plasmid Vector Allows Efficient Complementation of Pathogenic Leptospira Strains. Parales RE, éditeur[END_REF]. The perRB (LIMLP_05620) ORF together with its native promoter region (200 bp upstream region) were amplified from genomic DNA of L.

interrogans serovar Manilae strain L495 (using the ComPerR2_5Not and ComPerR2_3Xba

primer set, S11 Table ) and cloned between the Not1 and Xba1 restriction sites in the pMaORI vector. The absence of mutation in the perRB ORF in the obtained plasmid (pNB139) was checked by DNA sequencing and the pNB139 plasmid was introduced in the perRB mutant (M1474) by conjugation using the E. coli β2163 conjugating strain as previously described [START_REF] Picardeau | Conjugative transfer between Escherichia coli and Leptospira spp. as a new genetic tool[END_REF]. Leptospira conjugants were selected on EMJH plates containing 50 µg/ml spectinomycin and resistant colonies were then inoculated into liquid EMJH medium supplemented with spectinomycin for further analysis. The restoration of PerRB expression in the complemented perRB mutant was verified by RT-qPCR.

The double perRAperRB mutant (perRA, perRB::Km R ) complementation required the construction of a gentamycin resistant pMaORI complementation vector. A gentamycin resistant cassette [START_REF] Poggi | Antibiotic resistance markers for genetic manipulations of Leptospira spp[END_REF] was cloned into the pMaORI between the Not1 and Xba1 restriction sites and the integrity of the gentamycin resistance cassette in the pMaORIgenta plasmid (S10 Leptospira mutant strain by conjugation using the E. coli β2163 conjugating strain as previously described [START_REF] Picardeau | Conjugative transfer between Escherichia coli and Leptospira spp. as a new genetic tool[END_REF]. Leptospira conjugants were selected on EMJH plates containing 8 µg/ml gentamycin and resistant colonies were then inoculated into liquid EMJH medium supplemented with gentamycin and spectinomycin for further analysis. The restoration of PerRB expression in the trans-complemented double perRAperRB mutant was verified by RT-qPCR (S4 Fig).

Phylogenetic analyses

The sequences homologous to the LIMLP_10155 (PerRA), LIMLP_05620 (PerRB),

RNA purification

Virulent L. interrogans serovar Manilae strain L495 and perRB (M1474) mutant strains with less than three in vitro passages were used in this study. Four independent biological replicates of exponentially grown L. interrogans WT, perRB (M1474) and double perRAperRB mutant strains were harvested and resuspended in 1 ml TRIzol (ThermoFisher Scientific) and stored at -80°C. Nucleic Acids were extracted with chloroform and precipitated with isopropanol as described earlier [START_REF] Zavala-Alvarado | The Single-Step Method of RNA Purification Applied to Leptospira[END_REF]. Bioinformatics analyses were performed using the RNA-seq pipeline from Sequana [START_REF] Cokelaer | Sequana': a Set of Snakemake NGS pipelines[END_REF].

Reads were cleaned of adapter sequences and low-quality sequences using cutadapt version 1.11 [START_REF] Martin | Cutadapt removes adapter sequences from high-throughput sequencing reads[END_REF]. Only sequences at least 25 nt in length were considered for further analysis. Bowtie version 1.2.2 [START_REF] Langmead | Ultrafast and memory-efficient alignment of short DNA sequences to the human genome[END_REF], with default parameters, was used for alignment on the reference genome (L. interrogans serovar Manilae strain UP-MMC-NIID LP, from MicroScope Platform).

Genes were counted using featureCounts version 1.4.6-p3 [START_REF] Liao | featureCounts: an efficient general purpose program for assigning sequence reads to genomic features[END_REF] from Subreads package (parameters: -t gene -g locus_tag -s 1).

Count data were analyzed using R version 3.5.1 [START_REF]R, a language and environment for statistcial computing[END_REF] and the Bioconductor package DESeq2 version 1.20.0 [START_REF] Love | Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2[END_REF]. The normalization and dispersion estimation were performed with DESeq2 using the default parameters and statistical tests for differential expression were performed applying the independent filtering algorithm. Differential expressions were expressed as logarithm to base 2 of fold change (Log 2 FC). A generalized linear model including the replicate effect as blocking factor was set in order to test for the differential expression between Leptospira samples. Raw p-values were adjusted for multiple testing according to the Benjamini and Hochberg (BH) procedure [START_REF] Benjamini | Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing[END_REF] and genes with an adjusted pvalue lower than 0.05 and a Log 2 FC higher than 1 or lower than -1 were considered differentially expressed. Heat maps and Volcano plots were generated using the Galaxy platform (https://usegalaxy.eu/).

Quantitative RT-PCR experiments

cDNA synthesis was performed with the cDNA synthesis kit (Biorad) according to the manufacturer's recommendation. Quantitative PCR was conducted in triplicate with the SsoFast EvaGreen Supermix (Biorad) as previously described. LIMLP_06735 was used as a reference gene.

Non-coding RNA identification

Sequencing data from the L. interrogans WT, perRB (M1474) and double perRAperRB mutant strains were processed with Trimmomatic [START_REF] Bolger | Trimmomatic: a flexible trimmer for Illumina sequence data[END_REF] to remove low-quality bases and adapter contaminations. BWA mem (version 0.7.12) was used to discard the reads matching Leptospira rRNA, tRNA or polyA sequences and to assign the resulting reads to Leptospira replicons. Then Rockhopper [START_REF] Mcclure | Computational analysis of bacterial RNA-Seq data[END_REF] was used to re-align reads corresponding to separate replicons and to assemble transcripts models. The output was filtered to retain all transcripts longer than 50 nucleotides not overlapping within 10 nucleotides with NCBI annotated genes on the same orientation, and showing a minimum Rockhopper raw count value of 50 in at least two isolates. This high-quality set of new sRNA was subjected to differential expression analysis with Rockhopper, adopting a Benjamini-Hochberg adjusted P-value threshold of 0.01. For each non-coding RNAs, putative function was identified by BLAST using the Rfam database [START_REF] Kalvari | Rfam 13.0: shifting to a genome-centric resource for non-coding RNA families[END_REF].

Infection experiments

WT and mutant L. interrogans strains were cultivated in EMJH medium until the exponential phase and counted under a dark-field microscope using a Petroff-Hauser cell. 10 4 or 10 6 bacteria (in 0.5 ml) were injected intraperitoneally in groups of 4-8 male 4 weeks-old Syrian Golden hamsters (RjHan:AURA, Janvier Labs). Animals were monitored daily and sacrificed by carbon dioxide inhalation when endpoint criteria were met (sign of distress, morbidity). (http://www.sbg.bio.ic.ac.uk/~phyre2/html/page.cgi?id=index; [START_REF] Kelley | The Phyre2 web portal for protein modeling, prediction and analysis[END_REF]). Secondary structures are numbered as in (A). The WT containing the empty pMaORIgenta vector (black circles), the perRAperRB mutant containing the empty pMaORIgenta vector (pink circles) and the complemented perRAperRB mutant (blue circles) strains were cultivated in EMJH medium until the exponential phase.

  species. Regulators of the Fur family are homodimeric metalloproteins with a two-domain organization composed of an amino-terminal DNA binding domain and a carboxy-terminal dimerization domain (Fig 1A). The DNA binding domain contains a winged helix-turn-helix (HTH) DNA binding motif (H2-H4, in Fig1A) where DNA binding is mediated by the H4 helix. The dimerization domain consists of an / domain. The regulatory iron that controls DNA binding and dissociation is coordinated by histidine, aspartate, and glutamate residues located in a loop at the hinge of the amino and carboxy-terminal domains. Most of Fur-like regulators also coordinate a structural metal (zinc) through 2 cysteinate motifs (CxxC, where x designates any AA). This structural metal allows correct folding and dimeric assembly of the regulator.Due to a great conservation in protein folding, metal coordination and similarity in the metalinduced conformational switch controlling DNA binding, it is difficult to distinguish members of the Fur family on the sole basis of their primary sequence. However, in B. subtilis, a single amino acid residue in the H4 helix of the amino-terminal DNA binding domain (N61 and R61 for B. subtilis PerR and Fur, respectively) allows PerR and Fur to discriminate between their respective DNA sequence targets (PerR and Fur boxes, respectively)[START_REF] Caux-Thang | Single asparagine to arginine mutation allows PerR to switch from PerR box to fur box[END_REF] (Fig 1B

  6A). However, entry in exponential phase was delayed if the culture medium was inoculated with stationary phase-adapted perRAperRB mutant (S5 Fig).We had already shown that a perRA mutant had a higher ability to grow and survive in the presence of deadly concentration of H 2 O 2 but a slower growth in the presence of the superoxide-generating paraquat ([START_REF] Kebouchi | Structure and function of the Leptospira interrogans peroxide stress regulator (PerR), an atypical PerR devoid of a structural metal-binding site[END_REF], and see in S2 Fig andFigs 6B and 6C). On the contrary, inactivating perRB leads to a higher resistance to paraquat (Fig5and Fig 6C). The concomitant inactivation of perRA and perRB led to a greater growth in the presence of both H 2 O 2 (Fig 6B) and paraquat (Fig 6C). To complement the double perRAperRB mutant strain, the PerRB-encoding ORF LIMLP_05620 was expressed in trans using the replicative pMaORIgenta vector (S4 Fig). The growth of the trans-complemented perRAperRB mutant resumed to an impaired growth, as that of the WT and single perRA mutant strains, in the presence of paraquat (Fig 6D). Therefore, the double perRAperRB mutant exhibited cumulative phenotypes of the respective single perRA and perRB mutants when L. interrogans are cultivated in the presence of ROS. We then tested whether perRA and perRB inactivation had any influence on L. interrogans virulence in the animal model of acute leptospirosis. All hamsters infected intraperitoneally with 10 4 bacteria of the perRA or perRB single mutant strains exhibited morbidity sign after 7-8 days (Fig 6E), similarly to the WT strain. In contrast, all animals infected intraperitoneally with 10 4 bacteria of the double perRAperRB mutant strain did not show any sign of morbidity three weeks post-infection (Fig 6E), even when the animals were infected with 10 6 bacteria of the double perRAperRB mutant strain (Fig 6F). Virulence was restored in the double perRAperRB mutant strain complemented in trans only with perRB (Fig 6F).

  Fig 8B). All these up-regulated factors (except DoxX) were also up-regulated upon exposure

  44, respectively) (Fig 8D and S5 Table

Fig 10 )

 10 . The down-regulated LepncRNA105 and LepncRNA130 ncRNAs were located downstream the hsp20-15 operon and gst, respectively, three ORFs whose expression is decreased is the perRAperRB mutant (Fig 10 and S8 Table). It is worth noting that LepncRNA38, LepncRNA105 and LepncRNA130 are up-regulated by H 2 O 2 as were ccp, hsp20-15 and gst ([27]; Fig 10).

  PerRA and PerRB have a distinct and non-redundant function in Leptospira adaptation to oxidative stress. The distinct repartition of PerRA and PerRB in the Leptospira genus and differences in their respective regulon support the hypothesis of a non-redundant function in the adaptation to oxidative stress. The PerRA and PerRB regulons determined when L. interrogans are cultivated inside a host using DMC implemented in rats confirmed the limited overlap between PerRA and PerRB regulons[START_REF] Grassmann | The Fur-like regulators PerRA and PerRB control a complex signaling network required for mammalian host-adpatation and virulence of Leptospira interrogans[END_REF].Phenotypic studies suggest that PerRB represses (directly or indirectly) genes encoding defenses against superoxide. Highly pathogenic Leptospira species (from the P1 clade) do not encode any SOD or SOR that could be responsible for detoxification of superoxide whereas intermediated species (P2 clade) and saprophyte non-pathogenic Leptospira do have such enzymes. Understanding how pathogenic Leptospira detoxify superoxide encountered during infection is a very important question. Overall, the differentially-expressed genes upon perRB inactivation are mostly Leptospira-specific and poorly characterized. Examining the PerRB regulon determined when Leptospira are cultivated in laboratory conditions did not allow to draw a conclusive hypothesis on the identity of the factors that could participate in superoxide detoxification in L. interrogans. The highest differentially-expressed ORFs were mainly involved in regulation and cell signaling (transcription and sigma factors, adenylate/diguanylate cyclase, TCSs) and could be involved in regulating the adaptation to various challenging stress encountered in the environment or within a host. Further studies will be required to determine whether superoxide detoxification in L. interrogans is mediated by enzymatic detoxification or metal-dependent scavenging mechanisms and to clarify the exact role of PerRB in controlling those pathways.

  LIMLP_18590 and LIMLP_04825 proteins were searched with BLASTP version 2.10.0 among the other Leptospira species (Fig 3 and S1 Table) or among the protein sequences of 11,070 representative genomes (Fig 2), as previously described [68]. In that case, only the sequences with an e-value less than 1e-10 and a percentage of similarity greater than 60% were retained. Sequences with percent identity equal to 100% were clustered by CD-HIT version 4.8.1 and only one sequence was retained. The resulting 1671 sequences were subsequently aligned by MAFFT version 7.471. A phylogenetic tree was finally built with IQ-TREE version 2.1.1 under the best-fit model LG + R10. A second phylogenetic tree was made with a subset of sequences to improve the resolution of the separation between PerRA and PerRB. The same procedure was followed, except that the best-fit model used for phylogenetic reconstruction is LG + R5. Both trees were visualized with FigTree version 1.4.4 (https://github.com/rambaut/figtree).

  Contaminating genomic DNA was removed by DNAse treatment using the RNAse-free Turbo DNA-free turbo kit (ThermoFisher Scientific) as described by the manufacturer. The integrity of RNAs (RIN > 8.0) was verified by the Agilent Bioanalyzer RNA NanoChips (Agilent technologies, Wilmington, DE). RNA Sequencing rRNA were depleted from 0.5 μg of total RNA using the Ribo-Zero rRNA Removal Kit (Bacteria) from Illumina. Sequencing libraries were constructed using the TruSeq Stranded mRNA Sample preparation kit (20020595) following the manufacturer's instructions (Illumina). The directional libraries were controlled on Bioanalyzer DNA1000 Chips (Agilent Technologies) and concentrations measured with the Qubit dsDNA HS Assay Kit (ThermoFisher). Sequences of 65 bases were generated on the Illumina Hiseq 2500 sequencer.

S4Fig.

  Characterization and complementation of the double perRAperRB mutant strain. (A) Schematic representation of the double perRAperRB mutant construction. PerRA (LIMLP_10155) was inactivated by allelic exchange in the transposon perRB mutant. The kanamycin (Km) and spectinomycin (Spc) resistance cassettes inactivating perRB and perRA, respectively, are indicated. (B) Production of PerRA in the WT, in the single perRA and perRB mutants and in the double perRAperRB mutant strains. L. interrogans strains were cultivated in EMJH medium at 30°C until the logarithmic phase and lyzed by sonication. 10 g of total lyzates were resolved on a 15% SDS-PAGE and transferred on nitrocellulose membrane. PerRA was detected by immunoblot using a 1/2000 antibody dilution as described previously [30]. (C) PerRB expression in the WT, in the double perRAperRB mutant and in the trans-complemented perRAperRB mutant with the perRB ORF. RNAs were extracted from exponentially-grown L. interrogans strains and perRB expression was assessed by RT-qPCR in triplicate using flaB gene (LIMLP_09410) as reference gene. PerRB expression in the perRAperRB and trans-complemented mutant strains were normalized against that in the WT strain. (D) Growth of stationary phase-adapted WT, perRAperRB and transcomplemented perRAperRB mutant strains. L. interrogans WT (black circles), perRAperRB mutant (pink squares) and perRAperRB mutant trans-complemented with the perRB ORF (blue triangles) were cultivated in EMJH medium at 30°C until late stationary phase (7 days after the entry in the stationary phase) and used to inoculate EMJH medium. Bacteria were then cultivated at 30°C and growth was assessed by absorbance at 420 nm. Data are means and standard errors of three independent biological experiments. S5 Fig. RT-qPCR experiments in the double perRAperRB mutant.

Fig 1 .

 1 Fig 1. Analysis of the four Fur-like regulators of L. interrogans.

Fig 2 .

 2 Fig 2. (A) Phylogenetic tree with a cartoon representation showing the distribution of the

Fig 3 .

 3 Fig 3. Distribution of the four Fur-like regulators of L. interrogans in the genus

Fig 4 .

 4 Fig 4. Increased PerRA and perRB expression upon exposure to hydrogen peroxide.

Fig 5 .Fig 6 .Fig 9 .

 569 Fig 5. Effect of perRB inactivation on Leptospira growth in the presence of superoxide-

  Cells were harvested and RNAs were purified, and the expression of lvrA, lvrB, ligA, ligB, clpB, hsp20 and hsp15 was measured by RT-qPCR. Gene expression was normalized by that in the WT strains. Data are the means and standard errors of three independent biological replicates. P-values obtained by a Two-way Anova test indicates the statistical significance in comparison with WT samples.

Fig 10 .

 10 Fig 10. Non-coding RNAs expression in the double perRAperRB mutant.

Phylogenetic analysis of PerRA and PerRB in Leptospira species.

  . interrogans genome encodes 4 ORFs that share homology with regulators of the Fur family. Among those Leptospira Fur-like regulators, LIMLP_10155 is the only Leptospira Fur-like regulator that is functionally and structurally characterized. It encodes a PerR that represses genes encoding defenses against peroxide, including catalase and other peroxidases[START_REF] Lo | Transcriptional response of Leptospira interrogans to iron limitation and characterization of a PerR homolog[END_REF][START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF][START_REF] Kebouchi | Structure and function of the Leptospira interrogans peroxide stress regulator (PerR), an atypical PerR devoid of a structural metal-binding site[END_REF]. Our previous structural characterization of the PerR from L. interrogans indicates that this member of the Fur family lacks a structural metal binding site despite a correct folding and functional dimeric assembly[START_REF] Kebouchi | Structure and function of the Leptospira interrogans peroxide stress regulator (PerR), an atypical PerR devoid of a structural metal-binding site[END_REF].

	of sequence identity. LIMLP_05620 shares about 27% identity with the well-characterized B. The sequences of LIMLP_10155 and LIMLP_05620 ORFs from the strain L.
	subtilis PerR. The putative H4 helix in LIMLP_05620 (Leu63-Ser75) is relatively well interrogans serovar Manilae were searched and compared in all available genomes from
	conserved with that of B. subtilis (Val56-Ser69) (Fig1B-C) and LIMLP_05620 displays a the Leptospira genus (S1 Table). As seen in Fig 3, LIMLP_10155 is present in the
	typical regulatory metal coordination site (His44-Asp93-His99-His101-Asp112). As the saprophytes S1 and S2 clades and in the P1 clade (highly virulent strains). This ORF is absent
	LIMLP_10155-encoded PerR, LIMLP_05620 lacks the cysteinate motif involved in structural from most of the P2 clade (intermediate strains). However, there are two exceptions in the P2
	metal coordination [30] (Fig 1B). On the contrary, both LIMLP_04825 and LIMLP_18590 clade species since homologues of LIMLP_10155, which might have been acquired by a
	have one or two cysteinate motifs for structural metal coordination (C 113 xxC 116 and C 86 xxC 89 -recent horizontal gene transfer, are present in L. dzoumognesis and L. wolffii. Additionally,
	C 122 xxC 125 , respectively). Therefore, LIMLP_05620 encodes a putative PerR-like regulator this ORF is also present in other bacteria from the order Leptospirales such as Turneriella
	closely related to the PerR-encoding LIMLP_10155 whereas LIMLP_04825 and parva and Leptonema illini (Fig 2B). This suggests that LIMLP_10155 (PerRA) was present
	LIMLP_18590 could encode other type of Fur-like regulators (Fur, Zur, or Nur). We therefore in Leptospirales ancestor before Leptospira divergence and lost in the P2 clade. On the other
	annotated the LIMLP_10155 and LIMLP_05620 ORFs as perRA and perRB, respectively. side, LIMLP_05620 ORF is only present in P1 and P2 clades and absent in all species from
	S1 and S2 clades (Fig 3). LIMLP_05620 is also not found in other bacteria from the
	LIMLP_10155 Leptospirales order (Fig 2B and S1 Fig). This restricted distribution suggests that the ancestor (LIC12034/LA1857 encoding a PerR) and LIMLP_05620
	(LIC11158/LA2887), exhibit the canonical asparagine (N60 and N68, respectively) and To get a better understanding of the evolutionary relationship of the four Fur-like of pathogenic strains (P1 and P2 clades) has likely acquired LIMLP_05620 after their
	aspartate (D103 and D112, respectively) residues of a typical PerR. The third ORF encoding a regulators in pathogenic Leptospira, we undertook phylogenetic analyses by searching for divergence with other Leptospira. Overall, both PerR-encoding LIMLP_10155 and
	putative Fur-like regulator, LIMLP_04825 (LIC11006/LA3094), possesses the two Fur homologous sequences of the LIMLP_10155, LIMLP_05620, LIMLP_18590 and LIMLP_05620 ORFs only coexist in P1 clade that comprises the highly virulent Leptospira
	typical residues, R76 and E121, respectively. The fourth ORF encoding a putative Fur-like LIMLP_04825 proteins among the representative genomes present in GenBank. This revealed species. Altogether, these findings indicate that pathogenic Leptospira strains encode a
	regulator, LIMLP_18590 (LIC20147/LB183) possesses the typical Fur arginine residue in its a large phylogenetic distribution with several branches (Fig 2A). The sequences homologous second putative PerR-like regulator that is absent in saprophytes.
	putative H4 DNA binding helix (R51) but a typical PerR aspartate residue in the carboxy-to the LIMLP_04825 and LIMLP_18590 proteins form two distinct groups (red and orange,
	terminal domain (D94). Of note, LIMLP_18590 has a glutamate residue at the position 96. respectively) separated by a common ancestor. To get better definition of phylogenetic
	). In addition, D104 residue in the carboxy-terminal domain is Fold prediction suggests that the three Fur regulators encoded by LIMLP_05620, relationships of PerR-like homologues, we performed analysis with only a subset of sequence
	pivotal in the PerR sensitivity to H 2 O 2 . The corresponding residue in Fur is a glutamate and LIMLP_04825 and LIMLP_18590 adopt the two-domain organization typical of the Fur (Fig 2B). This phylogenetic analysis shows two separated groups composed of the sequences
	mutating this residue in aspartate leads to H 2 O 2 sensitivity [29] (Fig 1B). Therefore, N61 and family depicted in the crystal structure of LIMLP_10155 (Fig 1C). of LIMLP_10155 (PerRA) and LIMLP_05620 (PerRB) (see S1Fig for a more complete and
	The closest relative of the PerR-encoding LIMLP_10155 is LIMLP_05620 with about 26% of detailed tree).
	sequence identity, and LIMLP_04825 and LIMLP_18590 are closest relatives that share 20%

D104 both allow to differentiate a PerR from a Fur in B. subtilis and the presence of these canonical amino acid residues in PerR sequences is well-conserved in other bacterial species

[START_REF] Caux-Thang | Single asparagine to arginine mutation allows PerR to switch from PerR box to fur box[END_REF][START_REF] Parent | Single glutamate to aspartate mutation makes ferric uptake regulator (Fur) as sensitive to H2O2 as peroxide resistance regulator (PerR)[END_REF]

.

L

Sequence alignment of these ORFs with the Fur and PerR from B. subtilis shows a good conservation of the two-domain organization and residues involved in the regulatory metal coordination (Fig 1B). Interestingly, two of the 4 Fur-like ORFs of L. interrogans,

Role of PerRB in L. interrogans tolerance to ROS.

	We have previously shown that inactivating perRA led to the derepression of katE, ahpC and
	ccp and to a higher tolerance to H 2 O 2 [27,30] (see S2 Fig). The perRA mutant exhibited a
	reduced ability to grow in the presence of the superoxide-generating compound paraquat [30].
	A mutant with a transposon inserted into the PerRB-encoding LIMLP_05620 ORF was
	available in our random transposon mutant library and was used to investigate L. interrogans
	tolerance of ROS in the absence of PerRB. When cultivated in EMJH medium, the
	perRB mutant did not reach the same density than the WT strain at stationary phase (Fig 5A
	and S2 Fig). Inactivating perRB did not have any effect on the ability of L. interrogans to
	tolerate deadly concentration of H 2 O 2 (S2 Fig); however, it increases the capability of
	Leptospira to grow in the presence of paraquat (Fig 5B). Complementing in trans the perRB
	mutant restored the WT strain phenotype in the absence of paraquat (Fig 5A) and it decreased
	the growth rate of the cells to a somehow lower level than that of the WT in the presence of
	paraquat (Fig 5B). This indicates that PerRB is involved in Leptospira tolerance to
	superoxide, very likely by repressing (directly or indirectly) defenses against this ROS.
	As demonstrated previously [27], when L. interrogans are exposed to a subtlethal dose
	of H 2 O 2 (10 µM for 30 min) perRA expression is increased by a 7-fold whereas that of perRB
	is unchanged (Fig 4). In the presence of a higher dose of H 2 O 2 (1 mM for 1h), expression of
	both perRA and perRB was increased significantly by a 6-fold (Fig 4). This suggests that
	PerRB, like PerRA, responds to deadly H 2 O 2 dose.

Therefore, PerRA and PerRB have distinct function in pathogenic Leptospira survival in the presence of oxidants.

Table and Table 1

 and1 ).

	Table
	ORF ID a	Gene	L. biflexa b	Function	Log2FC Adjusted
						p-value
	Down-regulated genes
	LIMLP_02490* (LIC12988/LA0587)		LEPBI_I0886	Lipase putative extracellular lipoprotein	-1.608	6.11e-04
	LIMLP_02845 (LIC12920)			Hypothetical	-1.084	3.46e-02
	LIMLP_03640** (LIC12763/LA0865)			Hypothetical	-1.102	1.06e-02
	LIMLP_03790* (LIC12736/LA0905)			Hypothetical	-1.244	1.42e-05
	LIMLP_04255 (LIC10892/LA3244)	exbB	LEPBI_I0149 Biopolymer transport protein ExbB/TolQ	-0.947	1.07e-02
	LIMLP_06190** (LIC11265/LA2751)		LEPBI_I3113 Disulfide oxidoreductase	-0.723	6.96e-03
	LIMLP_11400** (LIC12297/LA1456)	radC		DNA repair protein RadC	-0.619	1.17e-02
	LIMLP_13165** (LIC12631/LA1029)	sph2		Sphingomyelinase C	-1.152	8.10e-03
	LIMLP_14595* (LIC10628/LA3571)		LEPBI_I2694 Cytochrome oxidase CcoP subunit	-0.583	3.40e-02
	LIMLP_14715** (LIC10606/LA3598)	dps	LEPBI_I2540	DNA-binding stress protein Dps	-0.896	1.61e-03
	LIMLP_15470** (LIC10454/LA3793)		LEPBI_I0671 Hemolysin (N-acyltransferase domain)	-1.317	3.40e-03
	LIMLP_15890 (LIC10377/LA0430)			Hypothetical putative lipoprotein	-1.353	3.90e-10
	LIMLP_16695 (LEPIC3326/LA4096)			Hypothetical	-1.124	8.14e-04
	LIMLP_18235** (LIC20078/LB099)			Hypothetical	-1.098	1.69e-02
	Up-regulated genes
	LIMLP_02880* (LIC12912/LA0688)	cas5		CRISPR-associated protein Cas5	0.881	5.84e-03
	LIMLP_02885* (LIC12911-12910/LA0689-0690) cas3		CRISPR-associated protein Cas3	0.834	4.51e-03
	LIMLP_04075** (LIC12680/LA0974)			Adhesin/FimH-like protein/DuF1566 domain	1.110	4.29e-03
	LIMLP_05450* (LIC11125/LA2933)			Diguanylate cyclase	0.983	2.12e-04
	LIMLP_05455* (LIC11126/LA2932)			Diguanylate cyclase	0.757	1.81e-02
	LIMLP_05460* (LIC11127/LA2930)			Diguanylate cyclase	0.922	1.11e-03
	LIMLP_05480 (LA2928)			Hypothetical	0.968	2.67e-02
	LIMLP_05485** (LIC11131/LA2926)			Diguanylate cyclase	0.679	8.14e-04
	LIMLP_05845 (LIC11203/LA2827)			Diguanylate phosphodiesterase	0.608	3.85e-02
	LIMLP_09580 (LIC11921/LA1980)		LEPBI_I1269 Diguanylate phosphodiesterase	0.547	1.61e-02
	LIMLP_17875 (LIC20015/LB017)	hemN	LEPBI_I1166	Coproporphyrinogen III oxidase HemN	0.727	9.53e-03
	LIMLP_18375** (LIC20106/LB133)			Diguanylate phosphodiesterase	0.593	1.99e-02
	LIMLP_18755 (LIC20176/LB225)			Hypothetical putative lipoprotein	1.453	3.77e-04
	LIMLP_18760 (LIC20177/LB226)			Adhesin/FimH-like protein/ DUF1566 domain-containing protein 1.095	3.39e-02
	LIMLP_19320* (LA1770)			AraC family transcriptional regulator	1.279	4.51e-03
	LIMLP_19325* (LA1771)			Hypothetical	1.262	4.05e-02

Table ) .

 ) Strikingly, several down-regulated ORFs in the double perRAperRB mutant such as clpB, ligA, ligB, and the operon lvrAB have been associated with Leptospira virulence. As in many bacteria, leptospiral ClpB ATPase is involved in disaggregating protein aggregates arising upon stress-induced protein denaturation[START_REF] Krajewska | Characterization of the molecular chaperone ClpB from the pathogenic spirochaete Leptospira interrogans[END_REF]. ClpB expression is increased upon exposure to H 2 O 2 and it is required for Leptospira virulence[START_REF] Zavala-Alvarado | The transcriptional response of pathogenic Leptospira to peroxide reveals new defenses against infection-related oxidative stress[END_REF][START_REF] Lourdault | Inactivation of clpB in the pathogen Leptospira interrogans reduces virulence and resistance to stress conditions[END_REF]. The ClpB-encoding ORF (LIMLP_10060) is dramatically down-regulated in the perRAperRB mutant (Log 2 FC of -2.99)

(Fig 8D and S4 Table).

Another virulence factors in Leptospira are the immunoglobulin-like LigA (LIMLP_15405)

Table )

 )

	was checked by DNA sequencing. Then, the perRB ORF (LIMLP_05620) together
	with its native promoter region (248 bp upstream region) were amplified (using the F-ApaI-
	PerR2genta and R-ApaI-PerR2genta primer set, S11 Table) from genomic DNA of L.
	interrogans serovar Manilae strain L495 and cloned into the PCR-blunt II TOPO vector (Zero
	Blunt TOPO PCR cloning kit, Invitrogen). The perRB locus was subsequently subcloned into
	the pMaORIgenta plasmid at the ApaI restriction site. The obtained plasmid (pCZ3, S10

Table )

 ) or the empty pMaORIgenta plasmids were introduced in perRAperRB double

Strains used in this study S10 Table. Plasmids used in this study S11 Table. Primers used in this study

RNAs were extracted from exponentially-grown L. interrogans strains WT or double S9 Table.

Acknowledgement

The authors would like to thank Melissa Caimano and André Grassmann for fruitful discussions.

Funding Crispin Zavala-Alvarado was part of the Pasteur -Paris University (PPU) International PhD Program. This project has received funding from the European Union's Horizon 2020 research and innovation programme (https://ec.europa.eu/programmes/horizon2020/en) under the Marie Sklodowska-Curie grant agreement No 665807 (https://ec.europa.eu/research/mariecurieactions) and from Fondation Etchebès-Fondation France (S-CM 16008) (https://www.fondationdefrance.org/fr/fondation/fondation-etchebes).

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Ethics Statement

The protocol for animal experimentation was reviewed by the Institut Pasteur (Paris, France), the competent authority, for compliance with the French and European regulations on Animal Welfare and with Public Health Service recommendations. This project has been reviewed and approved (CETEA #2016-0019) by the Institut Pasteur ethic committee for animal experimentation, agreed by the French Ministry of Agriculture.

S2 Table. Complete set of ORF expression in Leptospira interrogans WT and M1474

perRB mutant. containing the pMaORIgenta vector expressing LIMLP_05620 (blue triangles) in the presence of 2 µM paraquat. Growth was assessed by measure of absorbance at 420 nm and the data are means and standard errors of three independent biological replicates. (E) Virulence was assessed by infecting hamsters (n=4) by peritoneal route with 10 4 of WT (black circles), single perRA or perRB mutants (cyan and green triangles, respectively), or the double perRAperRB mutant (pink squares) as described in Material and Methods section. (F) Complementation of the double perRAperRB mutant with perRB was performed by infecting hamsters (n=8) by peritoneal route with 10 6 of WT containing the empty pMaORIgenta vector (black circles), of the double perRAperRB mutant containing the empty pMaORIgenta vector (pink squares) or the double perRAperRB mutant containing the pMaORIgenta vector expressing LIMLP_05620 (blue triangles). Red and blue dots indicate significantly up-and down-regulated genes, respectively, with a Log 2 FC cutoff of 1 (dashed vertical lines) and p-value<0.05. Selected genes are labelled. Expression of selected genes of the TonB-dependent transport cluster (A), involved in

S3 Table. Complete set of ORF expression in


Pierre Gaillard
email: pierre.gaillard@u-bourgogne.fr

Multi-parameters rational solutions to the mKdV equation

Keywords: mKdV equation, wronskians, rational solutions PACS numbers : 33Q55, 37K10, 47.10A-, 47.35.Fg, 47.54.Bd

come

Introduction

We consider the modified Korteweg-de Vries (mKdV) equation

u t -6u 2 u x + u xxx = 0, (1)
with u t = ∂ t u, u x = ∂ x u and u xxx = ∂ 3 x u. The mKdV equation has many applications in various fields as in the study of waves propagating in plasma [START_REF] Khater | Bäcklund transformations and exact solutions for Alfven solitons in a relativistic electronpositron plasma[END_REF], the dynamics of traffic flow [START_REF] Li | Analysis of stability and density waves of traffic flow model in an ITS environment[END_REF] and fluid mechanics [START_REF] Helal | Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics[END_REF]. In particular, it is used in nonlinear optics as for example to model supercontinuum generation in optical fibres [START_REF] Leblond | Models for supercontinuum generation beyond the slowly-varying-envelope approximation[END_REF] or to describe pulses consisting of a few optical cycles [START_REF] Leblond | Few-optical-cycle solitons: Modified Kortewegde Vries sine-Gordon equation versus other non-slowly-varying-envelopeapproximation models[END_REF]. Various methods have been used to contruct solutions to the mKdV equation. Hirota [START_REF] Hirota | Exact envelope-soliton solutions of a nonlinear wave equation[END_REF] constructed the exact soliton for the mKdV equation in 1972. In the same year, Tanaka [START_REF] Tanaka | Modified Korteweg. de Vries Equation and Scattering Theory[END_REF] was the first to solve the mKdV equation by using the inverse scattering technique. Wadati using this same method succeeded to obtain 1 the exact N-soliton solution for the mKdV [START_REF] Wadati | The Exact Solution of the Modified Korteweg-de Vries Equation[END_REF]. In 1976, rational solutions to the mKdV equation [START_REF] Ono | Algebraic Soliton of the Modified Korteweg-de Vries Equation[END_REF] were derived by Ono by using Bäcklund transformation. A limiting procedure based on bilinear results gave other type of solutions [START_REF] Ablowitz | Solitons and rational solutions of nonlinear evolution equations[END_REF] in 1978. In 1983, rational solutions to the mKdV equation were determined by recurrence relations in [START_REF] Kametaka | On Rational Similarity Solutions o[ KdV and m.KdV Equations[END_REF] by Tanaka. Solitary wave solutions and cnoidal wave solutions were given in 1994 in [START_REF] Lou | Solitary wave solutions and cnoidal wave solutions to the combined KdV and mKdV equation[END_REF]. Some classes of periodic solutions of mKdV have been given in [START_REF] Kevrekidis | On some classes of mKdV periodic solutions[END_REF] in 2004. More recently, in 2012, solutions to the mKdV equation has been constructed via bilinear Bäcklund transformation in [START_REF] Sun | Rational Solutions with non-zero asymptotics of the modified Korteweg-de Vries Equation[END_REF] and rational solutions in terms of Wronskians were obtained. In 2016, periodic solutions and rational solution of first and second order were presented in [START_REF] Chowdury | Periodic and rational solutions of modified Korteweg-de Vries equation[END_REF] by using a Darboux transformation were constructed.

Here, Darboux transformation is used to construct different type of solutions. Representations of solutions in terms of wronskians of order N depending on 2N real parameters are given, using trigonometric or hyperbolic functions. Rational solutions are obtained in performing a passage to the limit when one of these parameters goes to 0. So rational solutions as a quotient of polynomials in x and t, depending on 2N parameters are constructed. We give explicit solutions in the simplest cases N = 1, 2, 3 and some particular rational solutions for N = 1 until 6.

2 N-order solutions to the mKdV equation in terms of wronskians 2.1 N-order solutions in terms of wronskians of hyperbolic sine functions

We consider the mKdV equation

u t -6u 2 u x + u xxx = 0.
We recall that the wronskian of order N of the functions f 1 , . . . , f N is the determinant denoted W (f 1 , . . . , f N ), defined by det

(∂ i-1 x f j ) 1≤i≤N, 1≤j≤N , ∂ i x
being the partial derivative of order i with respect to x and ∂ 0 x f j being the function f j . We consider a j , b j arbitrary real numbers 1 ≤ j ≤ N . We have the following result :

Theorem 2.1 Let f j be the functions defined by

f j (x, t) = sinh 1 2 a j x - 1 2 a 3 j t + b j , for 1 ≤ i ≤ N, (2)
then the function u defined by

u(x, t) = ∂ x ln W (∂ x (f 1 ), . . . , ∂ x (f N )) W (f 1 , . . . , f N ) (3)
is a solution to the mKdV equation (1) depending on 2N real parameters a j , b j , 1 ≤ j ≤ N .

Some examples of solutions to the mKdV equation with sine hyperbolic generating functions

In the following we only give the solutions of order 1, 2 and 3 in the case of generating hyperbolic sinus functions.

with

n(x, t) = - 1 2 (a1 2 -a2 2 )[a2 sinh(-a1x + a1 3 t -2b1) -a1 sinh(-a2x + a2 3 t -2b2)])
and, d(x, t) = 2 (-sinh(-1/2 a1x+1/2 a1 3 t-b1) cosh(-1/2 a2x+1/2 a2 3 t-b2)a2+sinh(-1/2 a2x+ 1/2 a2 3 t-b2) cosh(-1/2 a1x+1/2 a1 3 t-b1)a1)(cosh(-1/2 a1x+1/2 a1 3 t-b1) sinh(-1/2 a2x+ 1/2 a2 3 t -b2)a2 -cosh(-1/2 a2x + 1/2 a2 3 t -b2) sinh(-1/2 a1x + 1/2 a1 3 t -b1)a1)

is a solution to the mKdV equation (1) with a 1 , a 2 , b 1 , b 2 arbitrarily real parameters.

Solution of order 3

In this case of order 3, we only present solution with

a 1 = 1, a 2 = 2, a 3 = 3, b 1 = 0, b 2 = 0, b 3 = 0 to shorten the paper. Proposition 2.3 The function u defined by u(x, t) = n(x, t) d(x, t) , (5)
with

n(x, t) = -150 (cosh(-1/2 x + 1/2 t)) 2 -384 (cosh(-x + 4 t -2)) 2 -54 (cosh(-3/2 x + 3 27 2 t-3)) 2 +300 sinh(-1/2 x+1/2 t) cosh(-3/2 x+ 27 2 t-3) cosh(-1/2 x+1/2 t) sinh(-3/2 x+ 27 2 t-3)-312 cosh(-3/2 x+ 27 2 t-3) sinh(-x+4 t-2) sinh(-3/2 x+ 27 2 t-3) cosh(-x+ 4 t -2) -180 (cosh(-1/2 x + 1/2 t)) 2 (cosh(-3/2 x + 27 2 t -3)) 2 + 480 (cosh(-x + 4 t - 2)) 2 (cosh(-1/2 x + 1/2 t)) 2 + 288 (cosh(-x + 4 t -2)) 2 (cosh(-3/2 x + 27 2 t -3)) 2 - 600 sinh(-1/2 x + 1/2 t) cosh(-x + 4 t -2) cosh(-1/2 x + 1/2 t) sinh(-x + 4 t -2)
and,

d(x, t) = 2 (16 sinh(-1/2 x+1/2 t) cosh(-x+4 t-2) sinh(-3/2 x+ 27 2 t-3)-9 sinh(-1/2 x+ 1/2 t) cosh(-3/2 x + 27 2 t -3) sinh(-x + 4 t -2) -5 cosh(-1/2 x + 1/2 t) sinh(-x + 4 t - 2) sinh(-3/2 x + 27 2 t -3))(-16 cosh(-1/2 x + 1/2 t) sinh(-x + 4 t -2) cosh(-3/2 x + 27 2 t-3)+9 cosh(-1/2 x+1/2 t) sinh(-3/2 x+ 27 2 t-3) cosh(-x+4 t-2)+5 sinh(-1/2 x+ 1/2 t) cosh(-x + 4 t -2) cosh(-3/2 x + 27 2 t -3))
is a solution to the mKdV equation (1).

Other types of solutions

We obtain similar results with other types of generating functions whose proofs are identical.

Solutions with hyperbolic cosine generating functions

Theorem 2.2 Let h j , h be the following functions

h j (x, t) = cosh 1 2 a j x - 1 2 a 3 j t + b j , for 1 ≤ i ≤ N, (6)
then the function u defined by

u(x, t) = ∂ x ln W (∂ x (h 1 ), . . . , ∂ x (h N )) W (h 1 , . . . , h N ) (7)
is a solution to the mKdV equation ( 1) with a j , b j 1 ≤ j ≤ N arbitrarily real parameters.

Solutions with trigonometric generating functions

Theorem 2.3 Let g j be the following functions

g j (x, t) = cos 1 2 a j x + 1 2 a 3 j t + b j , for 1 ≤ i ≤ N, (8)
then the function u defined by

u(x, t) = ∂ x ln W (∂ x (g 1 ), . . . , ∂ x (g N )) W (g 1 , . . . , g N ) (9)
is a solution to the mKdV equation ( 1) with a j , b j 1 ≤ j ≤ N arbitrarily real parameters.

Theorem 2.4 Let k j be the following functions

k j (x, t) = sin 1 2 a j x + 1 2 a 3 j t + b j , for 1 ≤ i ≤ N, (10)
then the function u defined by

u(x, t) = ∂ x ln W (∂ x (k 1 ), . . . , ∂ x (k N ) W (k 1 , . . . , k N ) ( 11
)
is a solution to the mKdV equation ( 1) with a j , b j 1 ≤ j ≤ N arbitrarily real parameters.

3 Rational solutions to the mKdV equation

In the following, we replace all parameters a j and b j , 1 ≤ j ≤ N by âj = N k=1 a k (je) 2k-1 and bj = N k=1 b k (je) 2k-1 with e an arbitrary real parameter. We realize this change to obtain rational solutions to the mKdV equation ( 1); for this, we perform a limit when the parameter e tends to 0. We get the following result : Theorem 3.1 Let ψ j be the functions

ψ j (x, t, e) = sinh 1 2 N k=1 a k (je) 2k-1 x - 1 2 N k=1 a k (je) 2k-1 3 t + N k=1 b k (je) 2k-1 , for 1 ≤ j ≤ N , then the function u defined by u(x, t) = lim e→0 ∂ x ln W (∂ x (ψ 1 ), . . . , ∂ x (ψ N )) W (ψ 1 , . . . , ψ N ) ( 12
)
is a rational solution to the mKdV equation [START_REF] Leblond | Models for supercontinuum generation beyond the slowly-varying-envelope approximation[END_REF].

We have similar results with generating sine or trigonometric functions.

We can also give the expression of the rational solutions of the mKdV without the presence of a limit. We get the following result :

Theorem 3.2 Let ψ, ϕ j be the functions

ψ(x, t, e) = sinh 1 2 N k=1 a k e 2k-1 x - 1 2 N k=1 a k e 2k-1 3 t + N k=1 b k e 2k-1 , ϕ j (x, t) = ∂ 2j-1 ψ(x, t, 0) ∂ 2j-1 e , for 1 ≤ j ≤ N , then the function v defined by v(x, t) = ∂ x ln W (∂ x (ϕ 1 ), . . . , ∂ x (ϕ N ) W (ϕ 1 , . . . , ϕ N ) ( 13
)
is a rational solution to the mKdV equation ( 1) depending on 2N parameters a j , b j , 1 ≤ j ≤ N .

We give some examples of rational solutions in the following.

It must be pointed out that these resulting rational solutions are singular. These results are consequences of the previous result.

First order rational solutions

We have the following result at order N = 1 :

Proposition 3.1 The function v defined by v(x, t) = -a1 a1x + 2 b1 , ( 14
)
is a rational solution to the mKdV equation ( 1) with a 1 , b 1 , arbitrarily real parameters.

Second order rational solutions

with n(x, t) = -2 a1 5 x 3 -12 a1 4 b1x 2 -24 a1 3 b1 2 x + 2 a1(12 a2b1 + 6 a1 4 t -12 b2a1 -8 a1b1 3 ), and,

d(x, t) = a 1 5 x 4 + 8 a 1 4 b 1 x 3 + 24 a 1 3 b 1 2 x 2 + (24 a 1 2 b 1 3 + a 1 (24 a 2 b 1 + 12 a 1 4 t - 24 b 2 a 1 + 8 a 1 b 1 3 ))x + 2 b 1 (24 a 2 b 1 + 12 a 1 4 t -24 b 2 a 1 + 8 a 1 b 1 3
) is a rational solution to the mKdV equation ( 1

Rational solutions of order three

In this case, we only give the rational solution with some fixed parameters in reason of the length of the general solution. We choose a j = b j = j for 1 ≤ j ≤ 3.

We get the following rational solution given by : is a rational solution to the mKdV equation (1).

Proposition 3.3 The function v defined by v(x, t) = n(x, t) d(x, t) , (16)

Rational solutions of order four

With the same choices of parameters a j = b j = j for 1 ≤ j ≤ 4, we get the following rational solution given by : -462720 t+(-160-60 t)(180 t+960))x 10 + ((-160 -60 t)(2520 t + 3360) -1752000 t -696320 + (-360 t -240)(180 t + 960) + 14400 t 2 )x 9 + (-1013760 + 129600 t 2 -4440960 t + (-160 -60 t)(8064 + 15120 t) + (-192-720 t)(180 t+960)+(-360 t-240)(2520 t+3360))x 8 +(-302400 t 3 -7580160 t-1064960+(-192-720 t)(2520 t+3360)+(-360 t-240)(8064+15120 t)+(-64+720 t 2 -480 t)(180 t + 960) + (-160 -60 t)(13440 + 50400 t))x 7 + ((-64 + 720 t 2 -480 t)(2520 t + 3360) -4233600 t 3 -8248320 t -753664 + (-160 -60 t)(100800 t + 15360) + (-192 -720 t)(8064+15120 t)+(-360 t-240)(13440+50400 t))x 6 +(-25401600 t 3 -5114880 t-319488 + (-192 -720 t)(13440 + 50400 t) + (-360 t -240)(100800 t + 15360) + (-64 + 720 t 2 -480 t)(8064 + 15120 t) + (-160 -60 t)(120960 t + 11520))x 5 + ((-64 + 720 t 2 -480 t)(13440 + 50400 t) + (-192 -720 t)(100800 t + 15360) + (-160 -60 t)(302400 t 3 + 80640 t+5120)-61440-1382400 t-36288000 t 3 +(-360 t-240)(120960 t+11520))x 4 + ((-360 t -240)(302400 t 3 + 80640 t + 5120) + (-64 + 720 t 2 -480 t)(100800 t + 15360) + (-192-720 t)(120960 t+11520)+(-160-60 t)(1024+23040 t+604800 t 3 ))x 3 +((-64+ 720 t 2 -480 t)(120960 t+11520)+(-192-720 t)(302400 t 3 +80640 t+5120)+(-360 t-240)(1024+23040 t+604800 t 3 ))x 2 +((-64+720 t 2 -480 t)(302400 t 3 +80640 t+5120)+ (-192 -720 t)(1024 + 23040 t + 604800 t 3 ))x + (-64 + 720 t 2 -480 t)(1024 + 23040 t + 604800 t 3 ) is a rational solution to the mKdV equation [START_REF] Leblond | Models for supercontinuum generation beyond the slowly-varying-envelope approximation[END_REF].

Proposition 3.4 The function v defined by v(x, t) = n(x, t) d(x, t) , (17)

Rational solutions of orders five and six

We choose parameters to get the shorter expression of the solution. For this, we consider a j = j and b j = 0, for 1 ≤ j ≤ N , we get the following rational solution given by : Order 5 Proposition 3.5 The function v defined by

v(x, t) = n(x, t) d(x, t) , (18)
with n(x, t) = 5 x 24 +2280 tx 21 +352800 t 2 x 18 +4838400 t 3 x 15 +2794176000 t 4 x 12 +259096320000 t 5 x 9 + 5120962560000 t is a rational solution to the mKdV equation [START_REF] Leblond | Models for supercontinuum generation beyond the slowly-varying-envelope approximation[END_REF].

For solutions of order 5 and 6 depending respectively on 10 and 12 parameters, the structure of the polynomials of the numerators and denominators is the following. For order 5, the numerator contains 8370 terms and the denominator 10279; the degree of the numerator in x is 24, in t is 8; the degree of the denominator in x is 25, in t is 8. For order 6, the numerator contains 100409 terms and the denominator 119620; the degree of the numerator in x is 35, in t is 11; the degree of the denominator in x is 36, in t is 12. Unlike other equations like the nonlinear Schrödinger equation, we do not see a specific structure of polynomials defining these solutions.

Conclusion

Two types of representations of solutions to the mKdV equation have been given. First, we have constructed solutions as a quotient of a wronskian of order N by a wronskian of order N depending on 2N real parameters. Then rational solutions to the mKdV equation depending on 2N real parameters, performing a passage to the limit when one parameter goes to 0. We can mention some other recent works about this equation. In [START_REF] Chowdury | Periodic and rational solutions of mKdV equation[END_REF], first and second-order rational solutions are given as limiting cases of periodic solutions, the second one depending on one real parameter. In [START_REF] Wei | Periodic and rational solutions of the reduced Maxwell Bloch equations[END_REF] the reduced Maxwell-Bloch (RMB) equations are considered and N th-order rational solutions containing several free parameters are presented, in particular explicit expressions of these solutions from first to second order. Explicit periodic and rational solutions of first and second order are given in [START_REF] Wang | Conservation laws periodic and rational solutions for an extended mKdV equation[END_REF], and some typical nonlinear wave patterns are shown. The first four exact rational solutions of the set of rational solutions of the mKdV equation are presented in [START_REF] Ankiewicz | Rogue wave-type solutions of the mKdV equation and their relation to known NLSE rogue wave solutions[END_REF]. Multiple periodic solutions of the mKdV equation are given in [START_REF] Zhaquilao | Nonsingular complexiton and rational solutions for the mKdV equation and KdV equation[END_REF] and in particular first to third-order rational solutions.

All the solutions presented in this article are different from those proposed in previous references and are new. So we get an infinite hierarchy of multiparametric families of rational solutions to the mKdV equation as a quotient of a polynomials in x and t depending on 2N real parameters. It would be relevant to better understand the structure of polynomials defining the rational solutions of this equation.

Figure 1 .

 1 Figure 1. Solution of order 1 to (1), on the left a 1 = 1, b 1 = 1; on the right a 1 = 1, b 1 = 10 2 .

Proposition 3 . 2

 32 The function v defined by v(x, t) = n(x, t) d(x, t) ,

  ) dependant on 4 real parameters a 1 , a 2 , b 1 , b 2 .

Figure 2 .

 2 Figure 2. Solution of order 2 to (1), on the left a 1 = 1, a 2 = 2, b 1 = 10, b 2 = 2; in the center a 1 = 1, a 2 = 2, b 1 = 1, b 2 = 2; on the right a 1 = 1, a 2 = 2, b 1 = 10, b 2 = 0.

Figure 3 .

 3 Figure 3. Solution of order 3 to (1), on the left a 1 = 1, a 2 = 2, a 3 = 3, b 1 = 0, b 2 = 0, b 3 = 0; in the center a 1 = 1, a 2 = 2, a 3 = 3, b 1 = 10, b 2 = 2, b 3 = 3; on the right a 1 = 1, a 2 = 2, a 3 = 3, b 1 = 10, b 2 = 0, b 3 = 0.

Figure 4 .

 4 Figure 4. Solution of order 4 to (1), on the left a 1 = 1, a 2 = 2, a 3 = 3, a 4 = 4, b 1 = 0, b 2 = 0, b 3 = 0, b 4 = 10; in the center a 1 = 1, a 2 = 2, a 3 = 3, a 4 = 4, b 1 = 10, b 2 = 0, b 3 = 0, b 4 = 0; on the right a 1 = 1, a 2 = 2, a 3 = 3, a 4 = 4, b 1 = 0, b 2 = 0, b 3 = 0, b 4 = 4.

  12 + (-139776-74880 t)x 11 +(-358912+720 t 2

				10250240 +
	24192000 t 2 + 35481600 t)x 6 + (12300288 + 72576000 t 2	-29030400 t 3 + 60825600 t)x 5 +
	(-145152000 t 3 +145152000 t 2 +11182080+76032000 t)x 4 +(67584000 t-36288000 t 4 +
	193536000 t 2 + 7454720 -387072000 t 3 )x 3 + (-217728000 t 4 + 40550400 t + 3440640 +
	165888000 t 2	-580608000 t 3 )x 2 +(983040-435456000 t 4 +82944000 t 2	-464486400 t 3 +
	14745600 t)x+131072+2457600 t-217728000 t 5 +18432000 t 2	-154828800 t 3	-290304000 t 4
	and,

with n(x, t) = 4 x 15 + 120 x 14 + 1680 x 13 + (14560 + 600 t)x 12 + (14400 t + 87360)x 11 + (158400 t + 384384)x 10 + (36000 t 2 + 1056000 t + 1281280)x 9 + (3294720 + 648000 t 2 + 4752000 t)x 8 + (15206400 t + 5184000 t 2 + 6589440)x 7 + (-2419200 t 3 + d(x, t) = -x 16 -32 x 15 -480 x 14 + (-4480 -240 t)x 13 + (-29120 -6240 t)x

  6 x 6 -153628876800000 t 7 x 3 + 460886630400000 t 8 and, d(x, t) = -x 25 -600 tx 22 -100800 t 2 x 19 -6955200 t 3 x 16 -254016000 t 4 x 13 +39626496000 t 5 x 10 -365783040000 t 6 x 7 + 76814438400000 t 7 x 4 + 460886630400000 t 8 x is a rational solution to the mKdV equation (1). = -6 x 35 -6300 tx 32 -2419200 t 2 x 29 -381024000 t 3 x 26 -50295168000 t 4 x 23 + 192036096000 t 5 x 20 +1371137725440000 t 6 x 17 +129970029772800000 t 7 x 14 -116143430860800000 t 8 x 11 + 319394434867200000000 t 9 x 8 +2146330602307584000000 t 10 x 5 +128779836138455040000000 t 11 x 2 and, d(x, t) = x 36 + 1260 tx 33 + 544320 t 2 x 30 + 110073600 t 3 x 27 + 11430720000 t 4 x 24 + 329204736000 t 5 x 21 +119062379520000 t 6 x 18 +7189831434240000 t 7 x 15 +1393721170329600000 t 8 x 12 -59620294508544000000 t 9 x 9 -4292661204615168000000 t 10 x 6 +128779836138455040000000 t 12

	Order 6
	Proposition 3.6 The function v defined by
	v(x, t) =	n(x, t) d(x, t)	,	(19)
	with
	n(x, t)