Jaime Gómez-Laguna

Fernando Cardoso-Toset

Jazmín Meza-Torres

Javier Pizarro- Cerdá

Juan J Quereda
email: juan.quereda@uchceu.es

Virulence potential of Listeria monocytogenes strains recovered from pigs in Spain

Keywords: isolates, invasion, replication, virulence factor, tonsil, meat

Background: Listeria monocytogenes is a foodborne bacterial pathogen that causes listeriosis, an infectious disease of animals and humans, with pigs acting as asymptomatic reservoirs. In August 2019 a listeriosis outbreak associated with the consumption of pork meat caused 222 human cases in Spain. Determining the diversity as well as the virulence potential of strains from pig origin is important for public health concerns.

Methods:

The behavior of twenty-three L. monocytogenes strains recovered from pig tonsils, meat and skin was compared by studying: (1) Internalin A (InlA), Internalin B (InlB), Listeriolysin O (LLO), Actin assembly-inducing protein (ActA) and PrfA expression levels, and

(2) invasion and intracellular growth in eukaryotic cells.

Results: Marked differences were found in the expression of the selected virulence factors and the invasion and intracellular replication phenotypes of the L. monocytogenes strains. Strains obtained from meat samples and belonging to ST 1/2a did not have InlA anchored to the peptidoglycan. Some strains expressed higher levels of the studied virulence factors and invaded and replicated intracellularly more efficiently than an epidemic L. monocytogenes reference strain (F2365).

Conclusion:

This study demonstrates the presence of virulent L. monocytogenes strains in pigs, with valuable implications in veterinary medicine and food safety.

Introduction

Listeria monocytogenes is a Gram-positive facultative intracellular pathogen responsible for listeriosis, a food-borne zoonotic disease of animals and humans that causes gastroenteritis, meningoencephalitis and abortions, with a high mortality rate. L. monocytogenes has been isolated worldwide from humans, animals, environmental sources and food sources like meat.

Food-producing animals are significant reservoirs in disseminating this bacterium. Although all L. monocytogenes strains are currently treated equally for regulatory purposes, some strains are highly virulent and are more often associated with epidemics, while others are less virulent and are infrequently related to epidemics [START_REF] Smith | Listeria monocytogenes isolates from ready to eat plant produce are diverse and have virulence potential[END_REF][START_REF] Fravalo | Characterisation of InlA truncation in Listeria monocytogenes isolates from farm animals and human cases in the province of Quebec[END_REF][START_REF] Velge | Variability of Listeria monocytogenes virulence: a result of the evolution between saprophytism and virulence?[END_REF] . Particularly, in human listeriosis outbreaks, isolates belonging to the serotype (ST) 4b are overrepresented, with a lower number of cases associated with isolates from the ST 1/2a which is usually associated with food and food-related L. monocytogenes isolates [START_REF] Maury | Uncovering Listeria monocytogenes hypervirulence by harnessing its biodiversity[END_REF][START_REF] Cossart | Illuminating the landscape of host-pathogen interactions with the bacterium Listeria monocytogenes[END_REF] .

In 2018, 28 countries reported 2,549 confirmed listeriosis cases in the EU/EEA with more than 229 associated deaths [START_REF] Ecdc | The European Union One Health 2018 Zoonoses Report[END_REF] . In 2017-2018 a listeriosis outbreak related to ready-to-eat processed meat products in South-Africa affected 1,060 human patients, of who 216 died [START_REF]Republic-of-South-Africa[END_REF] . More recently, in August 2019 the Regional Health Authorities in Andalusia, Spain, reported a listeriosis outbreak associated with the consumption of pork meat which caused 222 confirmed cases with 3 deaths, 6 miscarriages and one travel-related case in France [START_REF] Who | [END_REF] . Therefore, from a public health point of view, it is essential to evaluate the virulence potential of strains that colonize foodproducing animals, since differences in virulence between L. monocytogenes strains may influence infection and clinical outcome [START_REF] Smith | Listeria monocytogenes isolates from ready to eat plant produce are diverse and have virulence potential[END_REF][START_REF] Jensen | Processing plant persistent strains of Listeria monocytogenes appear to have a lower virulence potential than clinical strains in selected virulence models[END_REF] . L. monocytogenes possess several virulence factors which, after ingestion of contaminated food, play a significant role in the cellular internalisation, vacuolar escape, and cell-to-cell spread. The major bacterial virulence factors for the successive steps of the cell infectious process are the surface proteins Internalin (Inl) A (InlA) and InlB for the internalisation process, the poreforming toxin Listeriolysin O (LLO) for the escape from the vacuole and the Actin assemblyinducing protein (ActA) for the actin-based intracellular motility. PrfA is the major transcriptional regulator of virulence genes (reviewed in [START_REF] Cossart | Illuminating the landscape of host-pathogen interactions with the bacterium Listeria monocytogenes[END_REF] ). Importantly, during the last decades, the majority of L. monocytogenes studies in bacterial pathogenesis compared three major laboratory strains: EGD, EGD-e, and 10403S, all of them belonging to the ST 1/2a [START_REF] Becavin | Comparison of widely used Listeria monocytogenes strains EGD, 10403S, and EGD-e highlights genomic variations underlying differences in pathogenicity[END_REF][START_REF] Quereda | Bacteriocin from epidemic Listeria strains alters the host intestinal microbiota to favor infection[END_REF][START_REF] Quereda | Role in virulence of phospholipases, listeriolysin O and listeriolysin S from epidemic Listeria monocytogenes using the chicken embryo infection model[END_REF][START_REF] Freitag | Regulation of the prfA transcriptional activator of Listeria monocytogenes: multiple promoter elements contribute to intracellular growth and cell-to-cell spread[END_REF][START_REF] Garner | Exposure to salt and organic acids increases the ability of Listeria monocytogenes to invade Caco-2 cells but decreases its ability to survive gastric stress[END_REF] . Major listeriosis epidemics have been preferentially associated with L. monocytogenes of ST 4b [START_REF] Maury | Uncovering Listeria monocytogenes hypervirulence by harnessing its biodiversity[END_REF][START_REF] Jeffers | Comparative genetic characterization of Listeria monocytogenes isolates from human and animal listeriosis cases[END_REF] . Moreover, concerns about the adequacy of laboratory-adapted reference strains for the study of field isolates pathogenesis have risen from 1) the plasticity of bacterial genomes which enables bacteria to rapidly adapt to in vitro conditions; and 2) the continuous sub-culturing of bacteria which may lead to the loss of important pathogenic characteristics [START_REF] Fux | Can laboratory reference strains mirror "realworld" pathogenesis?[END_REF] .

L. monocytogenes is a ubiquitous bacterium that has been previously isolated along the pork production chain either from environmental or animal samples [START_REF] Giovannacci | Listeria monocytogenes in pork slaughtering and cutting plants. Use of RAPD, PFGE and PCR-REA for tracing and molecular epidemiology[END_REF][START_REF] Meloni | Listeria monocytogenes in five Sardinian swine slaughterhouses: prevalence, serotype, and genotype characterization[END_REF] . The pork production sector has been hit hard by a series of L. monocytogenes-related food poisoning outbreaks, and many studies have determined the serotypes or PCR serogroups most commonly found in pig slaughterhouses [START_REF] Felix | Population Genetic Structure of Listeria monocytogenes Strains Isolated From the Pig and Pork Production Chain in France[END_REF][START_REF] Ortiz | The Connection between Persistent, Disinfectant-Resistant Listeria monocytogenes Strains from Two Geographically Separate Iberian Pork Processing Plants: Evidence from Comparative Genome Analysis[END_REF] . However, no studies are evaluating the potential of L. monocytogenes isolates from pig origin to express virulence factors, invade and grow intracellularly in human cells. We thus decided to study the variability in the expression of InlA, InlB, LLO, ActA and PrfA in twenty-three L. monocytogenes strains of different ST (1/2a and 4b) recovered from pig tonsils, meat and skin in Spain as well as their phenotype during human host cell infections.

Material and methods

Ethics approval statement

This experiment and all animal procedures were performed according to the guidelines of the European Union (Directive 2010/63/EU). As the data used in this study were part of the routine slaughterhouse activity no ethical committee approval was needed.

Bacterial strains and eukaryotic cell lines

The L. monocytogenes strains from pigs used in this study were selected from a previous study (21) . The origin of the bacterial isolates and their ST are listed in Table 1. The L. monocytogenes strain F2365 (ST 4b) responsible for the 1985 California listeriosis outbreak associated to contaminated cheese and characterised in recent studies [START_REF] Quereda | Bacteriocin from epidemic Listeria strains alters the host intestinal microbiota to favor infection[END_REF][START_REF] Quereda | Role in virulence of phospholipases, listeriolysin O and listeriolysin S from epidemic Listeria monocytogenes using the chicken embryo infection model[END_REF][START_REF] Quereda | Reassessing the role of internalin B in Listeria monocytogenes virulence using the epidemic strain F2365[END_REF][START_REF] Quereda | Listeriolysin S Is a Streptolysin S-Like Virulence Factor That Targets Exclusively Prokaryotic Cells In Vivo[END_REF] was used as a reference strain (24) . The L. monocytogenes strains were grown overnight in brain heart infusion (BHI). The epithelial cell line JEG-3 (ATCC HTB-36), commonly used in L. monocytogenes studies [START_REF] Quereda | Role in virulence of phospholipases, listeriolysin O and listeriolysin S from epidemic Listeria monocytogenes using the chicken embryo infection model[END_REF][START_REF] Portillo | Association of ActA to peptidoglycan revealed by cell wall proteomics of intracellular Listeria monocytogenes[END_REF] was propagated in Dulbecco's modified Eagle medium (DMEM) supplemented with 10 % foetal bovine serum (FBS).

Western blot analysis

Bacteria grown overnight in BHI to stationary phase in shaking conditions at 37 °C were pelleted and re-dissolved in 1 ml of PBS and complete protease inhibitor and lysed by sonication. For cell wall protein analysis, subcellular fractions containing protoplasts and peptidoglycan associated proteins were obtained by mutanolysin treatment as previously described [START_REF] Portillo | Association of ActA to peptidoglycan revealed by cell wall proteomics of intracellular Listeria monocytogenes[END_REF][START_REF] Mariscotti | Contribution of sortase A to the regulation of Listeria monocytogenes LPXTG surface proteins[END_REF] . Extracts were adjusted to load the same number of bacteria of all strains. Load control was performed using the BioRad stain-free imaging technology (proteins are made fluorescent directly in the gel with a short photoactivation, allowing the immediate visualisation of proteins). A volume of protoplasts (5μl) and the cell wall fraction (2μl) was analysed by SDS-PAGE followed by Western blot using standard protocols for membrane blocking and antibody incubation [START_REF] Impens | N-terminomics identifies Prli42 as a membrane miniprotein conserved in Firmicutes and critical for stressosome activation in Listeria monocytogenes[END_REF] . The following primary antibodies were used: mouse monoclonal anti-InlA (dilution 1:1000) [START_REF] Mengaud | Antibodies to the leucine-rich repeat region of internalin block entry of Listeria monocytogenes into cells expressing E-cadherin[END_REF] , rabbit polyclonal anti-InlB R25 (dilution 1:10000) [START_REF] Impens | N-terminomics identifies Prli42 as a membrane miniprotein conserved in Firmicutes and critical for stressosome activation in Listeria monocytogenes[END_REF] , rabbit polyclonal anti-LLO R176 (dilution 1:20000) [START_REF] Samba-Louaka | Listeria monocytogenes dampens the DNA damage response[END_REF] , rabbit polyclonal anti-ActA R32 (dilution 1:1000) [START_REF] Steffen | Listeria monocytogenes ActA protein interacts with phosphatidylinositol 4,5-bisphosphate in vitro[END_REF] , and rabbit polyclonal antibody raised against PrfA R78 (dilution 1:1000) (Covalab UK Ltd, United Kingdom). HRPconjugated antibodies (AbCys S.A., France) were used as secondary antibodies (dilution 1:8000). Proteins were revealed using Pierce ECL 2 Western Blotting Substrate (Thermo Fisher, USA).

Spectrophotometric growth analysis

The kinetics of L. monocytogenes strains grown in BHI was recorded using an automated 96well plate reader (Sunrise Tecan, Switzerland) to measure OD 600 .

Infection of eukaryotic cell lines and determination of invasion and proliferation rates

Cell infections were performed by using gentamicin protection assays as previously described (23) . Briefly, JEG-3 cell suspensions were seeded in 96-well tissue culture plates and grown for 24 h in an antibiotic-free medium. On the day of infection, cells were then incubated with the bacteria at a multiplicity of infection of 5 for 1 h at 37°C. Following this incubation, the cells were washed, and extracellular bacteria were neutralized by adding complete medium containing 40 μg/ml of gentamicin. At 1 h 30 min or 5 h post-infection, cells were washed with PBS and finally lysed and the number of viable intracellular L. monocytogenes was calculated. In vitro invasion rates were calculated as the number of viable intracellular bacteria counted at 1 h 30 min post-infection. Intracellular replication was determined as the number of viable intracellular bacteria counted at 5 h post-infection (hpi). The index of proliferation was calculated by dividing the number of viable intracellular bacteria at 5 h by the number of viable intracellular bacteria obtained at 1 h 30 min. These experiments employed 6 technical replicates per bacterial strain and were repeated three times with independent clones of each of the strains.

Statistical analysis

The values were evaluated for approximate normality of distribution by the D'Agostino & Pearson omnibus normality test (GraphPad Prism v7.0, USA). Since data regarding growth kinetics, invasion and proliferation rates of L. monocytogenes strains followed a normal distribution, comparison between groups was assessed by ANOVA test followed by Fisher's LSD test (GraphPad Prism v8.0, USA). Differences with a P<0.05 were considered to be statistically significant.

Results

InlA, InlB, LLO, ActA and PrfA expression levels in L. monocytogenes strains

Twenty-three L. monocytogenes strains isolated from pigs of ST 1/2a and 4b were selected for characterisation of the expression of major virulence factors based on their origin, sample source and serotype [START_REF] Morales-Partera | Prevalence and diversity of Salmonella spp., Campylobacter spp., and Listeria monocytogenes in two free-range pig slaughterhouses[END_REF] . The L. monocytogenes strain F2365 (ST 4b) associated with a Mexican-style cheese listeriosis outbreak (142 human cases, 48 deaths) was included as an epidemic reference strain for comparison [START_REF] Quereda | Bacteriocin from epidemic Listeria strains alters the host intestinal microbiota to favor infection[END_REF][START_REF] Quereda | Role in virulence of phospholipases, listeriolysin O and listeriolysin S from epidemic Listeria monocytogenes using the chicken embryo infection model[END_REF][START_REF] Quereda | Reassessing the role of internalin B in Listeria monocytogenes virulence using the epidemic strain F2365[END_REF][START_REF] Quereda | Listeriolysin S Is a Streptolysin S-Like Virulence Factor That Targets Exclusively Prokaryotic Cells In Vivo[END_REF][START_REF] Linnan | Epidemic listeriosis associated with Mexican-style cheese[END_REF] . Marked differences were found at the protein level for InlA, InlB, LLO and ActA among the strains (Figure 1 and Supplementary Figure 1). Conversely, PrfA levels were quite homogenous across L. monocytogenes strains, suggesting that the differences in the expression level of the analysed virulence factors are not due to differential amounts of this transcriptional regulator (Figure 1 and Supplementary Figure 1). No pattern in terms of InlA, InlB, LLO, ActA or PrfA expression profiles could be found to differentiate L. monocytogenes 1/2a and 4b strains. Interestingly, three strains (G6C6, G8C13, G10C14) obtained from meat samples and belonging to ST 1/2a did not have InlA anchored to the peptidoglycan (Figure 1 and Supplementary Figure 1). However, this finding was not observed in other ST 1/2a strains. As previously reported, InlB was not detected in L. monocytogenes F2365 due to a nonsense mutation in inlB [START_REF] Quereda | Role in virulence of phospholipases, listeriolysin O and listeriolysin S from epidemic Listeria monocytogenes using the chicken embryo infection model[END_REF][START_REF] Nightingale | Listeria monocytogenes F2365 carries several authentic mutations potentially leading to truncated gene products, including inlB, and demonstrates atypical phenotypic characteristics[END_REF] (Figure 1 and Supplementary Figure 1). According to our findings, no clear correlation could be found between the protein levels of the virulence factors analysed: for example, the strains G3T6 and G3T7 displayed high levels of LLO and InlA whilst the strains G8T7 and G10T1 presented high levels of LLO but low levels of InlA. Another example is given by the strain G8T4 which displays low levels of LLO and ActA and the strain G3T7 which showed high levels of LLO but low levels of ActA. The distinctive InlA, InlB, LLO, ActA and PrfA profiles of each L. monocytogenes strain are summarised in Table 1.

Invasion and intracellular replication of L. monocytogenes strains in eukaryotic cells

All the L. monocytogenes strains from this study demonstrated similar growth characteristics in BHI (Figure 2A). The strain F2365 grew slower than other L. monocytogenes strains as previously reported (24) (Figure 2A).

To evaluate the ability of the L. monocytogenes strains to invade and replicate within eukaryotic cells, gentamicin assays in JEG-3 cells were performed. Eight different L. monocytogenes strains (4 strains of ST 1/2a and 4 strains of ST 4b) were selected based on their distinctive InlA, InlB, LLO and ActA profiles (Table 1). The strains G5T14 (ST 1/2a), G8T7 (ST 4b), G10T1 (ST4b) and G10T14 (ST 1/2a) were significantly more invasive than the F2365 strain (ST 4b) (p<0.05) (Figure 2B). Accordingly to the absence of InlA in the cell wall, the strains G6C6 (ST 1/2a) and G10C14 (ST 1/2a) were less invasive than the rest of strains and the F2365 strain (p<0.05) (Figure 2B). Regarding intracellular proliferation, the strains G3T6 (ST 4b), G3T7 (ST 4b), G5T14 (ST 1/2a), G6C6 (ST 1/2a) and G10C14 (ST 1/2a) replicated more efficiently than the epidemic F2365 strain (ST 4b) (p<0.05) (Figure 2C and2D). Although the strains G6C6 and G10C14 entered less efficiently in JEG-3 cells due to the absence of InlA in the cell wall, their index of proliferation within cells was similar to other strains demonstrating the normal activity of the virulence factors involved in the intracellular lifestyle (e.g. LLO and ActA) (Figure 2B-2D). No specific pattern in terms of invasion or intracellular replication profiles could be found to differentiate L. monocytogenes 1/2a and 4b strains; however, as mentioned above, L. monocytogenes strains isolated from meat samples were less invasive but proliferated equally (or more) than those coming from other sampling sources. Altogether, these results show the existence in pigs of a diverse population of L. monocytogenes strains in terms of eukaryotic cell invasion and intracellular replication abilities, revealing the presence of strains (e.g. G5T14) that invade and replicate intracellularly more efficiently than a L. monocytogenes strain that was able to cause an outbreak (F2365).

Discussion

L. monocytogenes strains present in pigs can be transferred to food processing environments, and thus contaminate food and it is, therefore, important to evaluate the virulence of these strains.

Importantly, not all L. monocytogenes strains are equally virulent, some strains are highly virulent and are more often associated with epidemics, while others are less virulent and are infrequently related to epidemics [START_REF] Smith | Listeria monocytogenes isolates from ready to eat plant produce are diverse and have virulence potential[END_REF][START_REF] Fravalo | Characterisation of InlA truncation in Listeria monocytogenes isolates from farm animals and human cases in the province of Quebec[END_REF][START_REF] Velge | Variability of Listeria monocytogenes virulence: a result of the evolution between saprophytism and virulence?[END_REF] . Previous studies have shown that Lineage I (or serotype 4b) occurs more frequently among clinical isolates than lineage II (or serotypes 1/2b, 1/2a and 1/2c), relative to the frequency of these categories in food [START_REF] Maury | Uncovering Listeria monocytogenes hypervirulence by harnessing its biodiversity[END_REF] . Besides intrinsic L. monocytogenes virulence other extrinsic factors play equally an important role in listeriosis outbreaks, such as individual susceptibility, food category, good hygiene practices and good manufacturing practices, and persistence into the environment or biofilm formation among others [START_REF] Farber | Health risk assessment of Listeria monocytogenes in Canada[END_REF] .

Understanding virulence aspects of L. monocytogenes in animals and food supply chains can help regulators and operators to design more effective microbial surveillance and prevention strategies [START_REF] Smith | Listeria monocytogenes isolates from ready to eat plant produce are diverse and have virulence potential[END_REF] . In this sense, more strict control measures, such as lower limit of bacterial load in food could be implemented when especially virulent L. monocytogenes strains are identified. In the present study, we addressed the issue of the general virulence potential of L. monocytogenes strains from pig samples in Spain.

Major foodborne listeriosis epidemics have been preferentially associated with L. monocytogenes of ST 4b [START_REF] Maury | Uncovering Listeria monocytogenes hypervirulence by harnessing its biodiversity[END_REF][START_REF] Jeffers | Comparative genetic characterization of Listeria monocytogenes isolates from human and animal listeriosis cases[END_REF] . However, during the last decades, the majority of L. monocytogenes studies in host-pathogen interactions have compared three major laboratory strains: EGD, EGD-e, and 10403S all of them belonging to the same ST (1/2a) [START_REF] Becavin | Comparison of widely used Listeria monocytogenes strains EGD, 10403S, and EGD-e highlights genomic variations underlying differences in pathogenicity[END_REF][START_REF] Quereda | Bacteriocin from epidemic Listeria strains alters the host intestinal microbiota to favor infection[END_REF][START_REF] Quereda | Role in virulence of phospholipases, listeriolysin O and listeriolysin S from epidemic Listeria monocytogenes using the chicken embryo infection model[END_REF][START_REF] Freitag | Regulation of the prfA transcriptional activator of Listeria monocytogenes: multiple promoter elements contribute to intracellular growth and cell-to-cell spread[END_REF][START_REF] Garner | Exposure to salt and organic acids increases the ability of Listeria monocytogenes to invade Caco-2 cells but decreases its ability to survive gastric stress[END_REF] . Besides, the occurrence of different strains of L. monocytogenes in reported cases of listeriosis together with the recovery of diverse isolates from the food industry environment [START_REF] Meloni | Listeria monocytogenes in five Sardinian swine slaughterhouses: prevalence, serotype, and genotype characterization[END_REF][START_REF] Morales-Partera | Prevalence and diversity of Salmonella spp., Campylobacter spp., and Listeria monocytogenes in two free-range pig slaughterhouses[END_REF][START_REF] Allam | Whole-Genome Sequences of Listeria monocytogenes Sequence Type 6 Isolates Associated with a Large Foodborne Outbreak in South Africa, 2017 to 2018[END_REF] highlights the necessity of evaluating the phenotypic diversity of L. monocytogenes isolates from food-producing animals. Here, using molecular and cell biology techniques, a high diversity was found regarding the expression of L. monocytogenes virulence factors in isolates from pigs in Spain. No specific pattern in terms of virulence factor expression, invasion or intracellular replication profiles could be observed in our study to distinguish L. monocytogenes 1/2a and 4b food isolates.

The strains G6C6, G8C13, G10C14 obtained from pig meat samples did not have InlA anchored to the peptidoglycan and consequently were less invasive than the rest of strains and the F2365 strain (Figure 1, Figure 2B and Supplementary Figure 1). Several mutations causing premature stop codons have been reported for the gene inlA [START_REF] Van Stelten | Development and implementation of a multiplex singlenucleotide polymorphism genotyping assay for detection of virulence-attenuating mutations in the Listeria monocytogenes virulence-associated gene inlA[END_REF][START_REF] Mariscotti | The Listeria monocytogenes LPXTG surface protein Lmo1413 is an invasin with capacity to bind mucin[END_REF] . These premature stop codons cause loss of anchorage of InlA to the peptidoglycan and invasion attenuation [START_REF] Nightingale | inlA premature stop codons are common among Listeria monocytogenes isolates from foods and yield virulence-attenuated strains that confer protection against fully virulent strains[END_REF] . Premature stop codons in the inlA gene are more frequent in food isolates (≈30%) than in human clinical isolates (<2%) (37- 39) . Accordingly, in our study, the strains without InlA anchored to the cell wall were those strains isolated from meat samples (G6C6, G8C13, G10C14) but not those strains from the tonsils or the skin.

Although internalisation has been previously associated to be mediated by InlA, InlB, LLO and ActA [START_REF] Cossart | Illuminating the landscape of host-pathogen interactions with the bacterium Listeria monocytogenes[END_REF][START_REF] Portillo | Association of ActA to peptidoglycan revealed by cell wall proteomics of intracellular Listeria monocytogenes[END_REF][START_REF] Quereda | Role of Listeria monocytogenes Exotoxins in Virulence[END_REF][START_REF] Vadia | The pore-forming toxin listeriolysin O mediates a novel entry pathway of L. monocytogenes into human hepatocytes[END_REF] , here we have demonstrated that the strain G3T6 possessing an InlA high InlB high LLO high ActA med PrfA high profile invaded less efficiently than the G5T14 (InlA med InlB high LLO low ActA low PrfA low ), G8T7 (InlA low InlB high LLO high ActA med PrfA med ), G10T1 (InlA low InlB high LLO high ActA high PrfA med ) and G10T14 (InlA low InlB high LLO med ActA high PrfA med ) strains. These results suggest that invasion could be also mediated by the activity of additional virulence factors (i.e. the invasin LPXTG surface protein Lmo1413) [START_REF] Mariscotti | The Listeria monocytogenes LPXTG surface protein Lmo1413 is an invasin with capacity to bind mucin[END_REF] and the interaction of all of these factors simultaneously. Recent studies regarding bacterial population genomics have harnessed the biodiversity of L. monocytogenes and uncovered multiple new putative virulence factors [START_REF] Maury | Uncovering Listeria monocytogenes hypervirulence by harnessing its biodiversity[END_REF][START_REF] Kuenne | Reassessment of the Listeria monocytogenes pan-genome reveals dynamic integration hotspots and mobile genetic elements as major components of the accessory genome[END_REF] . These discoveries will be helpful for further pathogenesis, diagnostic, phylogenetic and functional research studies.

Regarding intracellular proliferation, we showed that the intracellular CFU numbers of the epidemic strain F2365 at 5 hpi were lower than for other pig strains which suggest that one or more of the 20 mutations resulting in premature stop codons of this F2365 strain could play a fundamental role in its intracellular lifestyle [START_REF] Nightingale | Listeria monocytogenes F2365 carries several authentic mutations potentially leading to truncated gene products, including inlB, and demonstrates atypical phenotypic characteristics[END_REF] . Interestingly, the strain G8T7 which expresses high levels of LLO and moderate levels of ActA proliferated less intracellularly than the strain G5T14 which showed low protein levels of LLO and ActA. These results point out that, as previously described for the invasion process, more virulence factors besides the classical ones (i.e. LLO and ActA) could play an important role during intracellular proliferation. Even though the meat pool strains G6C6 and G10C14 invaded eukaryotic cells less efficiently due to the lack of InlA in the cell wall, they proliferated intracellularly similar to other strains used in the present study or even more than the epidemic strain F2365. This is important because: 1) a L. monocytogenes clinical isolate expressing a truncated form of InlA was responsible for a case of septicemia [START_REF] Jonquieres | The inlA gene of Listeria monocytogenes LO28 harbors a nonsense mutation resulting in release of internalin[END_REF] ; and 2) the strain F2365 responsible for the 1985 California listeriosis outbreak also lacked one of the main virulence factors involved in entry (due to a nonsense mutation in inlB), but caused one of the largest epidemics of listeriosis (142 human cases, 48 deaths) [START_REF] Quereda | Reassessing the role of internalin B in Listeria monocytogenes virulence using the epidemic strain F2365[END_REF][START_REF] Linnan | Epidemic listeriosis associated with Mexican-style cheese[END_REF] .

Other studies have also compared the ability of L. monocytogenes strains from different origins (human isolate, cow faeces, ground pork and cheese; and frozen whole eggs, cheeses, egg, ice cream, milk and frozen yolk, respectively) to invade cells, revealing widely different levels of invasion among the tested strains [START_REF] Zilelidou | Highly Invasive Listeria monocytogenes Strains Have Growth and Invasion Advantages in Strain Competition[END_REF] . However, other studies that analysed L. monocytogenes strains from sausages, fish processing plants and a human clinical case [START_REF] Jensen | Processing plant persistent strains of Listeria monocytogenes appear to have a lower virulence potential than clinical strains in selected virulence models[END_REF] did not find differences among strains' ability to proliferate intracellularly, since all the tested L. monocytogenes strains presented identical intracellular growth rates in Caco-2 cells. It is now apparent that the restricted use of the most frequently used laboratory L. monocytogenes strains has masked the existence of this high diversity of isolates.

The virulence potential of L. monocytogenes strains recovered from pigs and their related food products is unknown. As far as we know, this is the first report documenting the virulence characteristics of L. monocytogenes recovered from pigs in Spain. Our analysis of twenty-three L. monocytogenes strains recovered from pig tonsils, meat and skin establishes the existence of a diverse population of L. monocytogenes in terms of virulence factors expression, eukaryotic cell invasion, and intracellular replication abilities. No specific pattern in terms of virulence factor expression, invasion or intracellular replication profiles could be found to differentiate L. monocytogenes 1/2a and 4b strains or from different sources, beyond the lower invasion for strains isolated from meat samples. Moreover, these results show the existence of some strains from pigs that invade and replicate intracellularly more efficiently than an epidemic L. load the same number of bacteria of all isolates. Load control was performed using the BioRad stain-free imaging technology (proteins are made fluorescent directly in the gel with a short photoactivation, allowing the immediate visualisation of proteins). The InlA, InlB and ActA panels in this figure are the same as in figure 1 but highly exposed.

Figure 1 .Figure 2 .

 12 Figure 1. InlA, InlB, LLO, ActA and PrfA protein levels detected by Western blot. Bacteria

Table 1 .

 1 L. monocytogenes isolates origin (21), serotype and distinctive InlA, InlB, LLO, ActA, and PrfA profiles. InlB high LLO med ActA high PrfA med #10 G10C14 1/2a Meat pool InlA neg InlB high LLO med ActA low PrfA med

	Farm Lab code Serotype	Source	Virulence factor Profile
	#3	G3T6	4b	Tonsils	InlA high InlB high LLO high ActA med PrfA high
	#3	G3T7	4b	Tonsils	InlA high InlB low LLO high ActA low PrfA med
	#3	G3T8	4b	Tonsils	InlA low InlB low LLO high ActA low PrfA med
	#4	G4T2	1/2a	Tonsils	InlA high InlB high LLO med ActA low PrfA low
	#4	G4T5	1/2a	Tonsils	InlA high InlB low LLO low ActA low PrfA low
	#4	G4T6	1/2a	Tonsils	InlA high InlB med LLO low ActA med PrfA med
	#5	G5T14	1/2a	Tonsils	InlA med InlB high LLO low ActA low PrfA low
	#6	G6C6	1/2a	Meat pool	InlA neg InlB high LLO high ActA med PrfA high
	#7	G7T1	4b	Tonsils	InlA med InlB low LLO low ActA low PrfA low
	#7	G7T6	4b	Tonsils	InlA high InlB low LLO low ActA low PrfA low
	#8	G8T4	4b	Tonsils	InlA low InlB med LLO low ActA med PrfA med
	#8	G8T7	4b	Tonsils	InlA low InlB high LLO high ActA med PrfA med
	#8	G8T14	4b	Tonsils	InlA high InlB low LLO high ActA med PrfA med
	#8	G8E6	4b	Skin abrasive sponge InlA low InlB low LLO high ActA low PrfA high
	#8	G8C13	1/2a	Meat pool	InlA neg InlB high LLO med ActA med PrfA med
	#9	G9T6	4b	Tonsils	InlA med InlB low LLO med ActA low PrfA low
	#9	G9T10	4b	Tonsils	InlA med InlB low LLO med ActA med PrfA med
	#9	G9T12	4b	Tonsils	InlA high InlB low LLO low ActA med PrfA high
	#10 G10T1	4b	Tonsils	InlA low InlB high LLO high ActA high PrfA med
	#10 G10T4	4b	Tonsils	InlA low InlB med LLO med ActA med PrfA med
	#10 G10T5	4b	Tonsils	InlA low InlB low LLO high ActA med PrfA high
	#10 G10T14 1/2a	Tonsils	InlA low

Funding and competing interest statement

This study was supported by the Agency for Innovation and Development of Andalucía (IDEA; project references 351504/550859), by the University of Córdoba Research Program (Spain), by the Andalusian FEDER Operational Funding Program, by Generalitat Valenciana (Project reference GV/2018/A/183) and by the Spanish Ministry of Science and Innovation (Project reference PID2019-110764RA-I00). Gómez-Laguna J. is supported by a "Ramón y Cajal" contract of the Spanish Ministry of Economy and Competitiveness (RYC-2014-16735). Quereda J.J. is supported by a "Ramón y Cajal" contract of the Spanish Ministry of Science, Innovation and Universities (RYC-2018-024985-I). The funding bodies had no role in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript.

The authors have no conflict of interest to declare.

monocytogenes strain (F2365). These results have valuable implications in terms of veterinary medicine and food safety, since they highlight the importance of determining L. monocytogenes virulence to accordingly implement rigorous control measures and point out the necessity of identifying all the virulence factors involved in L. monocytogenes pathogenesis.

Supplementary material

Supplementary Figure 1: InlA, InlB and ActA protein levels detected by Western blot. Bacteria were grown in BHI to stationary phase in shaking conditions at 37 °C. Extracts were adjusted to


Olivier Rahavandrainy
email: olivier.rahavandrainy@univ-brest.fr

All bi-unitary perfect polynomials over F 2 with at most four irreducible factors

We give, in this paper, all bi-unitary perfect polynomials over the prime field F 2 , with at most four irreducible factors.

We consider the following polynomials over F 2 :

Theorem 1.2. Let A ∈ F 2 [x] be b.u.p. such that ω(A) = 4. Then A, A ∈ {C j : 8 ≤ j ≤ 13} ∪ {D 1 , D 2 }.

Introduction

Let S ∈ F 2 [x] be a nonzero polynomial. We say that S is odd if gcd(S, x(x + 1)) = 1, S is even if it is not odd. A Mersenne (prime) is a polynomial (irreducible) of the form 1 + x a (x + 1) b , with gcd(a, b) = 1. A divisor D of S is called unitary if gcd(D, S/D) = 1. We denote by gcd u (S, T ) the greatest common unitary divisor of S and T . A divisor D of S is called bi-unitary if gcd u (D, S/D) = 1. We denote by σ(S) (resp. σ * (S), σ * * (S)) the sum of all divisors (resp. unitary divisors, bi-unitary divisors) of S. The functions σ, σ * and σ * * are all multiplicative. We say that a polynomial S is perfect (resp. unitary perfect, bi-unitary perfect) if σ(S) = S (resp. σ * (S) = S, σ * * (S) = S). Finally, we say that S is indecomposable bi-unitary perfect (i.b.u.p.) if it is bi-unitary perfect but it is not a product of two coprime nonconstant biunitary perfect polynomials. As usual, ω(S) designates the number of distinct irreducible factors of S. Several studies are done about perfect and unitary perfect. In particular, we gave ( [START_REF] Gallardo | There is no odd perfect polynomial over F 2 with four prime factors[END_REF], [4], [START_REF] Gallardo | All unitary perfect polynomials over F 2 with at most four distinct irreducible factors[END_REF]) the list of all (unitary) perfect polynomials A over F 2 (even or not), with ω(A) ≤ 4.

In this paper, we are interested in bi-unitary perfect polynomials (b.u.p. polynomials) A with ω(A) ≤ 4. If A ∈ F 2 [x] is nonconstant b.u.p., then x(x + 1) divides A so that ω(A) ≥ 2 (see Lemma 2.5). Moreover, the only b.u.p. polynomials over F 2 with exactly two prime factors are x 2 (x + 1) 2 and x 2 n -1 (x + 1) 2 n -1 , for any nonnegative integer n ([1], Theorem 5). We prove (Theorems 1.1 and 1.2) that the only b.u.p. polynomials A ∈ F 2 , with ω(A) ∈ {3, 4}, are those given in [START_REF] Beard | Bi-Unitary Perfect polynomials over GF (q), Annali di Mat[END_REF], plus four other ones. Note that all odd irreducible divisors of the C j 's are Mersenne primes (there is a misprint for C 6 , in [START_REF] Beard | Bi-Unitary Perfect polynomials over GF (q), Annali di Mat[END_REF]).

In the rest of the paper, for S ∈ F 2 [x], we denote by S the polynomial obtained from S with x replaced by x + 1: S(x) = S(x + 1). As usual, N (resp. N * ) denotes the set of nonnegative integers (resp. of positive integers). For S, T ∈ F 2 [x] and n ∈ N * , we write:

S n ∥T if S n |T but S n+1 T .
Finally, let M denotes the set of all Mersenne primes.

Preliminaries

We need the following results. Some of them are obvious or (well) known, so we omit their proofs.

Lemma 2.1. Let T be an irreducible polynomial over

F 2 and k, l ∈ N * . Then, gcd u (T k , T l ) = 1 (resp. T k ) if k ̸ = l (resp. k = l). In particular, gcd u (T k , T 2n-k ) = 1 for k ̸ = n, gcd u (T k , T 2n+1-k ) = 1 for any 0 ≤ k ≤ 2n + 1. Lemma 2.2. Let T ∈ F 2 [x] be irreducible. Then i) σ * * (T 2n ) = (1 + T )σ(T n )σ(T n-1 ), σ * * (T 2n+1 ) = σ(T 2n+1 ). ii) For any c ∈ N, T does not divide σ * * (T c ). Proof. i): σ * * (T 2n ) = 1 + T + • • • + T n-1 + T n+1 + • • • + T 2n = (1 + T n+1 )σ(T n-1 ) = (1 + T )σ(T n )σ(T n-1 ), σ * * (T 2n+1 ) = 1 + T + • • • + T 2n+1 . ii) follows from i). Corollary 2.3. Let T ∈ F 2 [x] be irreducible. Then i) If a ∈ {4r, 4r + 2}, where 2r -1 or 2r + 1 is of the form 2 α u -1, u odd, then σ * * (T a ) = (1 + T ) 2 α • σ(T 2r ) • (σ(T u-1 )) 2 α , gcd(σ(T 2r ), σ(T u-1 )) = 1. ii) If a = 2 α u-1 is odd, with u odd, then σ * * (T a ) = (1+T ) 2 α -1 •(σ(T u-1 )) 2 α .
Corollary 2.4. i) The polynomial σ * * (x a ) splits over F 2 if and only if a = 2 or a = 2 α -1, for some α ∈ N * . ii) Let T ∈ F 2 [x] be odd and irreducible. Then σ * * (T c ) splits over F 2 if and only if (T is Mersenne, c = 2 or c = 2 γ -1 for some γ ∈ N * ). Lemma 2.5. If A is a nonconstant b.u.p. polynomial over F 2 , then x(x +1) divides A so that ω(A) ≥ 2.

Lemma 2.6. If

A = A 1 A 2 is b.u.p. over F 2 and if gcd(A 1 , A 2 ) = 1, then A 1 is b.u.p. if
n, m ∈ N. i) If σ(P 2n ) = Q m , then m ∈ {0, 1}. ii) If σ(P 2n ) = Q m T , with m > 1 and T ∈ F 2 [x] is nonconstant, then deg(P ) > deg(Q). iii) If P is a Mersenne prime and if P = P * , then P ∈ {M 1 , M 4 }. iv) If σ(x 2n ) = P Q and P = σ((x + 1) 2m ), then 2n = 8, 2m = 2, P = M 1 and Q = P (x 3 ) = 1 + x 3 + x 6 . v) If any irreducible factor of σ(x 2n ) is a Mersenne prime, then 2n ≤ 6. vi) If σ(x 2n ) is a Mersenne prime, then 2n ∈ {2, 4}. vii) If σ(x n ) = σ((x + 1) n ), then n = 2 h -2, for some h ∈ N * .
Lemma 2.9. [see [START_REF] Gallardo | Characterization of Sporadic perfect polynomials over F 2[END_REF], Lemma 2.6] Let m ∈ N * and T be a Mersenne prime. Then, σ(x 2m ), σ((x + 1) 2m ) and σ(M 2m ) are all odd and squarefree.

The following equalities (obtained from Corollary 2.3) are useful.

                                                 σ * * (T 2 ) = (1 + T ) 2 , if T is irreducible For a, b ≥ 3, σ * * (x a ) = (1 + x) 2 α • σ(x 2r ) • (σ(x u-1 )) 2 α , with gcd(σ(x 2r ), σ(x u-1 )) = 1, if a = 4r, 2r -1 = 2 α u -1, (resp. a = 4r + 2, 2r + 1 = 2 α u -1), u odd σ * * ((x + 1) b ) = x 2 β • σ((x + 1) 2s ) • (σ((x + 1) v-1 )) 2 β , if b = 4s, 2s -1 = 2 β v -1, (resp. b = 4s + 2, 2s + 1 = 2 β v -1), v odd σ * * (x a ) = (1 + x) 2 α -1 • (σ(x u-1 )) 2 α , if a = 2 α u -1 is odd, with u odd σ * * ((x + 1) b ) = x 2 β -1 • (σ((x + 1) v-1 )) 2 β , if b = 2 β v -1 is odd, with v odd r, α, β ≥ 1.
(1) Moreover, we shall also (prove and) consider the following relations:

c ∈ {2, 2 γ -1 : γ ≥ 1}, σ * * (P c ) = (1 + P ) c (in Section 3). (2)
In Section 4.1:

c, d ∈ {2, 2 γ -1 : γ ≥ 1}, σ * * (P c ) = (1 + P ) c , σ * * (Q d ) = (1 + Q) d (3)
and in Section 4.2:

                       σ * * (P c ) = (1 + P ) 2 γ • σ(P 2t ) • (σ(P w-1
)) 2 γ , with gcd(σ(P 2t ), σ(P w-1 )) = 1, if c ∈ {4t, 4t + 2}, where 2t -1 or 2t + 1 is of the form 2 γ w -1, w odd

σ * * (P c ) = (1 + P ) 2 γ -1 • (σ(P w-1 )) 2 γ , if c = 2 γ w -1 is odd, with w odd d ∈ {2, 2 γ -1 : γ ≥ 1}, σ * * (Q d ) = (1 + Q) d = x u 2 d (x + 1) v 2 d P w 2 d r, α, β, u 2 , v 2 , w 2 ≥ 1, ε 1 = min(1, u -1), ε 2 = min(1, v -1), ε 1 , ε 2 ∈ {0, 1}. ( 4
)
3 Proof of Theorem 1.1

We set A = x a (x + 1) b P c , with a, b, c ∈ N * and P odd irreducible. We suppose that A is b.u.p.:

σ * * (x a ) • σ * * ((x + 1) b ) • σ * * (P c ) = σ * * (A) = A = x a (x + 1) b P c .
We show that P is a Mersenne prime. By direct (Maple) computations, we get our result from Lemma 3.4. Proof.

Put P = 1 + x u 1 (x + 1) v 1 . If a = 2, then b ≥ 3, σ * * (x a ) = (1 + x) 2 , x 2 ∥A = σ * * (A)
. By comparing a with the exponent of x in σ * * (A), we get

a = 2 β +u 1 c > 2 if b is even, a = 2 β -1+u 1 c if b is odd, with b = 2 β v -1. So, b is odd, β = u 1 = c = 1.
We also have: P = σ((x + 1) v-1 ) and c = 2 β ≥ 2, which is impossible.

b = 2 β v-1, where v is odd. Since σ((x+1) v-1 ) divides σ * * (A) = A, v ∈ {1, 3, 5} and 2 β -1 ≤ a ≤ 10. iii): Write a = 2 α u -1 and b = 2 β v -1, where u, v are odd. As above, u, v ∈ {1, 3, 5}. σ * * (x a (x + 1) b ) does not split, so u ≥ 3 or v ≥ 3. Moreover, α = 1 (resp. β = 1) if u ≥ 3 (resp. v ≥ 3). We also get: 2 β -1 ≤ a, 2 α -1 ≤ b. If α = 1 = β, then a, b ≤ 9. If α = 1 and v = 1, then b = 2 β -1 ≤ a ≤ 9 so that b ≤ 7. If u = 1 and β = 1, then a = 2 α -1 ≤ 7 and b ≤ 9.

Proof of Theorem 1.2

In this section, we set A = x a (x + 1) b P c Q d , with a, b, c, d ∈ N * , P, Q odd irreducible, and deg(P ) ≤ deg(Q). We suppose that A is b.u.p.:

σ * * (x a ) • σ * * ((x + 1) b ) • σ * * (P c ) • σ * * (Q d ) = σ * * (A) = A = x a (x + 1) b P c Q d .
We prove that P ∈ M (Lemma 4.1). Moreover, Q ∈ M or it is of the form 1 + x u 2 (x + 1)

): From ii), σ * * (Q d ) = (1 + Q) d so that (1 + Q) d divides A.
We may put: From the proof of iii), x du 2 and (x+1)

1 + Q = x u 2 (x + 1) v 2 P w 2 , for some u 2 , v 2 , w 2 ∈ N, u 2 , v 2 ≥ 1.
Q lie in U = {M 1 , M 2 , M 3 , M 4 , M 5 }. Proof. First, if m ≥ 1 and if σ(x 2m ) divides σ * * (A), then 2m ≤ 6 and σ(x 2m ) ∈ {M 1 , M 4 , M 2 M 3 }. If P, Q ̸ ∈ U, then neither P nor Q divides σ * * (x a )σ * * ((x + 1) b ). So, P | σ * * (Q d ), P = σ(Q 2m ) with m ≥ 1. It is impossible since deg(P ) ≤ deg(Q). If P ∈ U but Q ̸ ∈ U, then Q does not divide σ * * (x a )σ * * ((x + 1) b
). Hence, it must divide σ(P 2m ), for some m ≥ 1. Thus, Q = σ(P 2m ). We get the contradiction:

x u 2 (x + 1) v 2 = 1 + Q = 1 + σ(P 2m
) is divisible by P . If c is odd, then put c = 2 γ u -1, with u odd and γ ≥ 1. We also get a contradiction if u ≥ 3, since σ(P u-1 ) divides σ * * (A). The proof is similar for d.

Lemma 4.5. The exponents a, b, c and d satisfy:

a ∈ {4, 6, 8, 10, 12, 14}, c, d ∈ {1, 2, 3, 7}, if a is even b ∈ {2 β v -1 : β ∈ {1, 2, 3}, v ∈ {1, 3, 5, 7}}, if a is even and b odd a, b ∈ {1, 3 , 5, 7, 9, 11,

Case where Q ̸ ∈ M

We prove Proposition 4.6.

Proposition 4.6. If A is b.u.p., where

P ∈ M but Q ̸ ∈ M, then A, A ∈ {D 1 , D 2 }.

Useful facts

As in Lemma 3.1, one has: a ≥ 3 or b ≥ 3. Lemma 4.1 allows to write: iii): The polynomial U = 1 + x(x + 1) 2 ν -1 must be irreducible, so ν ∈ {1, 2} by Lemma 4.8. Thus, U ∈ {M 1 , M 4 }.

P = 1 + x u 1 (x + 1) v 1 and Q = 1 + x u 2 (x + 1) v 2 P w 2 , with u i , v j ,
If P = U = M 1 , then Q = 1 + x + x 4 = 1 + x(x + 1)P is irreducible. If P = M 1 and U = M 4 , then Q = 1 + x 3 (x + 1) 3 P is irreducible. If P = M 4 and U = M 1 , then Q = 1+x(x+1) 3 P = (x 6 +x 5 +x 4 +x 2 +1)M 1 is reducible. If P = U = M 4 , then Q = 1 + x 3 (x + 1) 9 P = (x 12 + x 9 + x 8 + x 7 + x 6 + x 4 + x 2 + x + 1)(1 + x + x 4 ) is reducible. Lemma 4.11. If P Q = σ(x 2n ), then (2n = 8, P = M 1 , Q = 1 + x 3 + x 6 ) or (2n = 24, P = M 4 , Q = 1 + x 5 (x 5 + 1) 3 ). Moreover, Q, Q ̸ ∈ {σ(x 2g ), σ(P 2g ) : g ≥ 1} and P Q ̸ ∈ {σ(x 2g ), σ((x + 1) 2g ) : g ≥ 1}.
Proof. Since P Q = σ(x 2n ), we get P = P * or P = Q * . But, here, deg(P ) < deg(Q). So, P = P * and Q = Q * . Since P is a Mersenne prime and P = P * , one has P = M 1 or P = M 4 . If P = M 1 , then by Lemma 2.8-iv), Q = 1 + x 3 (x + 1)P = 1 + x 3 + x 6 . If P = M 4 , then direct computations give

Q = 1 + x 5 (x + 1) 2 ν -1 P 2 ν -1 . Since Q is irreducible, we get from Lemma 4.10-i), ν = 2 and Q = 1 + x 5 (x 5 + 1) 3 . Thus, Q ̸ ∈ {σ(x 6 ), σ((x + 1) 6 )} (resp. Q ̸ ∈ {σ(x 20 ), σ((x + 1) 20 )} if P = M 1 (resp. if P = M 4 ). We also remark that deg(Q) deg(P ) ∈ {3, 5}. So, Q, Q ̸ ∈ {σ(P 2g ) : g ≥ 1}.
Lemma 4.12.

If Q = σ(x 2n ) with n ≥ 1, then for some ν ≥ 1, Q = 1 + x(x + 1) 2 ν -1 M 1 2 ν or Q = 1 + x(x + 1) 2 ν -1 M 4 2 ν . Moreover, Q, Q ̸ ∈ {σ(P 2g ) : g ≥ 1} and P Q ̸ ∈ {σ(x 2g ), σ((x + 1) 2g ) : g ≥ 1}.
Proof. By direct computations, one has, for some ν ≥ 1: 2n = 2 ν t, t ∈ {3, 5}, P = σ(x t-1 ) and Q

= 1 + x(x + 1) 2 ν -1 P 2 ν . Hence, P 2 ν ∥1 + Q. If P Q is of the form σ(x 2g ), then P ∥1 + Q or P 3 ∥1 + Q (Lemma 4.11), which is impossible. Since Q = σ(x 2m ), Lemma 4.14-i) implies that Q ̸ ∈ {σ(P 2m ), σ(P 2m )}. Lemma 4.13. If Q = σ(P 2n ), then 2n ≤ 4, P = M 1 , so that Q ∈ {1+x(x+ 1)M 1 , 1 + x 3 (x + 1) 3 M 1 }. Moreover, Q, P Q ̸ ∈ {σ(x 2g ), σ((x + 1) 2g ) : g ≥ 1}.
Proof. By direct computations, one has: 2n

= 2 ν , Q = 1 + P (1 + P ) 2 ν -1 , for some ν ≥ 1. Since Q is irreducible, we get ν ∈ {1, 2} and P = M 1 . Again, by direct computations, Q, P Q ̸ ∈ {σ(x 2g ), σ((x + 1) 2g ) : g ≥ 1}.

Maple Computations

The function σ * * is defined as Sigm2star, for the Maple code.

Case where ω(A) = 3

We have proved that P ∈ {M 1 , M 4 , M 5 }. By means of Lemma 3.4. We obtain C 1 , . . . , C 7 .

Lemma 3 . 1 .Corollary 3 . 2 .Lemma 3 . 3 .

 313233 The polynomial σ * * (x a (x + 1) b ) does not split, so that (a ≥ 3 or b ≥ 3) and(a ̸ = 2 n -1 or b ̸ = 2 m -1 for any n, m ≥ 1). Proof. If σ * * (x a (x + 1) b ) splits, then σ * * (x a (x + 1) b ) = x b (x + 1) a . Thus, a = b and σ * * (P c ) = P c . It contradicts Lemma 2.2-ii). If a, b ≤ 2 or (a = 2 n -1, b = 2 m -1 for some n, m ≥ 1), then σ * * (x a) and σ * * ((x + 1) b ) split. The polynomial P is a Mersenne prime, P ∈ {M 1 , M 4 , M 5 }. Moreover, c = 2 or c = 2 γ -1, for some γ ≥ 1 and c ≤ min(a, b).Proof. By Lemma 3.1, there exists m ≥ 1 such that σ(x 2m ) or σ((x + 1) 2m ) divides σ * * (A) = A. Moreover, P does not divide σ * * (P c ). We conclude that P ∈ {σ(x 2m ), σ((x + 1) 2m )}. Thus, 2m ≤ 4 by Lemma 2.8-vi). By Corollary 2.4, σ * * (P c ) must split. So, c takes the expected value. Furthermore, x c and (x + 1) c both divide σ * * (A) = A, because they divide (1 + P ) c = σ * * (P c ). So, c ≤ min(a, b). If a (resp. b) is even, then a ≥ 4 (resp. b ≥ 4).

Lemma 3 . 4 .

 34 i) If a is even, then a ∈ {4, 6, 8, 10} and c ∈ {1, 2, 3, 7}. ii) If a is even and b odd, then b ∈ {2 β v -1 : v ∈ {1, 3, 5}, β ∈ {1, 2, 3}}.iii) If a and b are both odd, then a, b ∈ {1, 3, 5, 7, 9} and c ∈ {1, 2, 3, 7}.Proof. i): Since a ≥ 4 (Lemma 3.3), put a = 4r or a = 4r + 2, with r ≥ 1. Then, σ(x 2r ) divides σ * * (A). So, 2r ≤ 4 and c ≤ a ≤ 10. ii): Write

  iv): a, b ≥ 3 because 1+x divide σ * * (x a ), x divides σ * * ((x+1) b ) and x(x+1) divides both σ * * (P c ) and σ * * (Q d ).

Lemma 4 . 4 .

 44 i) For T ∈ {P, Q} and m ≥ 1, σ(T 2m ) does not divide σ * * (A).ii) The exponents c and d lie in {2, 2 γ -1 : γ ≥ 1}.Proof. i): For example, if T = P and if σ(T 2m ) | σ * * (A) = A, then we must have: σ(T 2m ) = Q, which is impossible (see the proof of Lemma 4.3). ii): If c is even and c ̸ = 2, then put c = 4r or c = 4r + 2, with r ≥ 1. σ(P 2r ) divides σ * * (A), which contradicts i).

  13}, c, d ∈ {1, 2, 3, 7}, if a and b are both odd. Proof. We refer to Relations in (1) and in (3). -If a is even, then a ≥ 4, a = 4r or a = 4r + 2 and σ(x 2r ) divides σ * * (A). So, 2r ≤ 6 and c, d ≤ a ≤ 14. -If a is even and b odd, then 2 β -1 ≤ a ≤ 14 and v ≤ 7. -If a and b are both odd, then u ≥ 3 or v ≥ 3, u, v ≤ 7. As in the proof of Lemma 3.4, if u, v ≥ 3, then α = 1 = β, then a, b ≤ 13. If u ≥ 3 and v = 1, then b = 2 β -1 ≤ a ≤ 13 so that b ≤ 7. If u = 1 and v ≥ 3, then β = 1, then a = 2 α -1 ≤ 7 and b ≤ 13.

w 2 ≥

 2 1. We obtain Corollaries 4.20, 4.25 and 4.27. Only, the last of them gives b.u.p. polynomials, namely D 1 , D 2 , D 1 and D 2 (see Section 5).

Lemma 4 .

 4 14. i) For any m, n ∈ N * , σ(P 2m ) ̸ = σ(x 2n ), σ((x + 1) 2n ). ii) If σ(x 2n ) = σ((x + 1) 2n ), then σ(x 2n ) ̸ ∈ {Q, P Q}. Lemma 4.26. The integer a + b is odd, a, b ≤ 11, c ≤ 8 and d ≤ 3.Proof. We refer to Relations in[START_REF] Beard | Bi-Unitary Perfect polynomials over GF (q), Annali di Mat[END_REF] and in (4). Lemma 4.7 is also useful.If c is even, then 2m = 2t ≥ 2, σ(P 2t ) = Q. So, w = 1, d = 1. If c is odd, then Q = σ(P w-1 ), w ∈ {3, 5}, d = 2 γ . -Ifa and b are even, then a, b ≥ 4 (by Lemma 4.1-iv)), P = σ(x 2r ) = σ((x + 1) 2s ). Hence, u = v = 1, 2r = 2s = 2, a, b ≤ 6 and c = 2 + d (by considering the exponents of P ). We get a contradiction on the value of c. -If a and b are odd, then σ(x u-1 ), σ((x + 1) v-1 ) ∈ {1, P }, so that u, v ≤ 3. Moreover, if c is even, then σ(P 2t ) = Q, w = 1, d = 1 and c ∈ {1, 1 + 2 α , 1 + 2 β , 1+2 α +2 β }. It contradicts the parity of c. If c is odd, then Q = σ(P w-1 ), w ∈ {3, 5}, d = 2 γ , so that d = 2 and c ∈ {2, 2 + 2 α , 2 + 2 β , 2 + 2 α + 2 β }.We also get a contradiction on the value of c.-If a is even and b odd, then a ≥ 4 (Lemma 4.1), σ(x 2r ) = P = M 1 , u = 1, 2r = 2, a ≤ 6. Moreover, σ((x + 1) v-1 ) ∈ {1, P }, so v ≤ 3. We get β ≤ 2, b ≤ 11, d ≤ 3 and c ≤ 8 because 2 β -1 ≤ a ≤ 6, d ≤ a ≤ 6 and c ∈ {1 + d, 1 + 2 β + d}. The proof is similar if a is odd and b even.

Corollary 4 .

 4 27. If A is b.u.p., with Q of the form σ(P 2m ), then P = M 1 , Q ∈ {1 + x(x + 1)P, 1 + x 3 (x + 1) 3 P }, a + b is odd, a, b ≤ 11, c ≤ 8, d ≤ 3.

>

  Sigm2star1:=proc(S,a) if a=0 then 1;else if a mod 2 = 0 then n:=a/2:sig1:=sum(S^l,l=0..n):sig2:=sum(S^l,l=0..n-1): Factor((1+S)*sig1*sig2) mod 2: else Factor(sum(S^l,l=0..a)) mod 2:fi:fi:end: > Sigm2star:=proc(S) P:=1:L:=Factors(S) mod 2:k:=nops(L[2]): for j to k do S1:=L[2][j][1]:h1:=L[2][j][2]: P:=P*Sigm2star1(S1,h1):od:P:end: We search all S = x a (x+1) b P c or S = x a (x+1) b P c Q d such that σ * * (S) = S.

Lemma 2.8. Let

  and only if A 2 is b.u.p. P, Q ∈ F 2 [x] be such that P is irreducible and let

	Lemma 2.7. If A is b.u.p. over F 2 , then the polynomial A is also b.u.p.
	over F 2 .
	Lemma 2.8 below gives some useful results from Canaday's paper ([2],
	Lemmas 4, 5, 6, Theorem 8 and Corollary on page 728).

  v 2 P w 2 , where u 2 , v 2 , w 2 ≥ 1.

	Lemma 4.1. i) The polynomial P is a Mersenne prime.
	ii) The integer d equals 2 or it is of the form d = 2

δ -1, with δ ∈ N * . iii) The polynomial Q is of the form 1 + x u 2 (x + 1) v 2 P w 2 , where w 2 ∈ {0, 1}. iv) One has: a, b ≥ 3 and d ≤ min(a, b). v) If σ * * (P c ) does not split, then Q is its unique odd divisor.

Proof. i): We remark that 1+P divides σ * * (P c ). If 1+ P does not split over F 2 , then Q is an odd irreducible divisor of 1+P and we get the contradiction:

deg(Q) < deg(P ) ≤ deg(Q). ii): If d is even and if d ≥ 4,

then d is of the form 4r or 4r + 2. Thus, the odd polynomial σ(Q 2r ) divides σ * * (A) = A, so we must have P = σ(Q 2r ), which contradicts the fact: deg(P ) ≤ deg(Q). If d = 2 δ w -1 is odd (with w odd) and if w ≥ 3, then P = σ(Q w-1 ) and deg(P ) > deg(Q), which is impossible. iii

5.3 Case where ω(A) = 4 with P ∈ M, Q ̸ ∈ M We apply Corollaries 4.20, 4.25 and 4.27. 1) If Q or P Q is of the form σ(x 2m ), then we obtain no b.u.p. polynomials. 2) If Q is of the form σ(P 2m ), then we get D 1 , D 2 , D 1 and D 2 .

For any g ≥ 1, P Q is not of the form σ(P 2g ), because P does not divide σ(P 2g ). We shall see that it suffices to consider three cases (replace A by A, if necessary): P Q = σ(x 2m ), Q = σ(x 2m ), Q = σ(P 2m ), for some m ≥ 1. Lemma 4.7. i) Let n ≥ 1 be such that σ(x 2n ) (resp. σ((x + 1) 2n ), σ(P 2n )) divides σ * * (A), then σ(x 2n ) ∈ {P, Q, P Q} (resp. σ((x + 1) 2n ) ∈ {P, Q, P Q}, σ(P 2n ) = Q). ii) For any n ≥ 1, σ(Q 2n ) does not divide σ * * (A).

Proof. Recall that we suppose: σ * * (A) = A. i): σ(x 2n ), σ((x + 1) 2n ) and σ(P 2n ) are all odd and squarefree (Lemma 2.9). Hence, they belong to {P, Q, P Q} whenever they divide σ * * (A), with σ(P 2n ) ̸ ∈ {P, P Q}.

Proof. Since σ(P 2n ) is odd and square-free, Q must divide it. So Q = σ(P 2n ). Put: 2n = 2 γ h, with h odd. We get:

Thus, w 2 = 1 and (1 + P ) 2 γ -1 (1

). Thus, P (P +1)

), then by Lemma 4.11, one has: (2n = 8, P = M 1 and

Without loss of generality, by Lemmas 4.11, 4.12 and 4.13, it suffices to consider the following three cases:

In each case, we distinguish: (a, b both even), (a even, b odd), (a, b both odd). We shall compare a, b, c or d with all possible values of the exponents of x, x + 1, of P or of Q, in σ * * (A).

According to Corollary 2.3 and Lemma 4.1, we get Lemma 4.15 from Relations in [START_REF] Beard | Bi-Unitary Perfect polynomials over GF (q), Annali di Mat[END_REF] and in (4).

Lemma 4.15.

i) The polynomial P does not divide σ * * (P c ), but it may divide σ * * (Q d ).

ii) One has:

Case where

We get, from Lemma 4.11, Q, Q ̸ ∈ {σ(x 2g ), σ(P 2g ) : g ≥ 1}, (2m = 8, P = M 1 and Q = 1 + x 3 + x 6 = 1 + x 3 (x + 1)P ) or (2m = 24, P = M 4 and Q = 1 + x 5 (x 5 + 1) 3 = 1 + x 5 (x + 1) 3 P 3 ). We refer to Relations in [START_REF] Beard | Bi-Unitary Perfect polynomials over GF (q), Annali di Mat[END_REF] and in (4). Proof. Lemma 4.1-iv) implies that a, b ≥ 4. Moreover, P Q ∈ {σ(x 2r ), σ(x u-1 )}.

), then σ(x 2r ) = P (by Lemma 4.7), which is impossible since gcd(σ(x 2r ), σ(x u-1 )) = 1. Proof. As above, a even implies that a = 4r = 16 and P = M 1 . One has:

4.2.3

Case where Q = σ(x 2m ), for some m ≥ 1 One has (Lemma 4.12):

We consider Relations in [START_REF] Beard | Bi-Unitary Perfect polynomials over GF (q), Annali di Mat[END_REF] and in (4). Proof. If σ * * (P c ) does not split, then Q is the unique odd irreducible divisor of σ * * (P c ). It contradicts the fact that Q is not of the form σ(P 2g ). So, σ * * (P c ) splits and Proof. If a and b are both odd, then Q = σ(x u-1 ), Q ̸ = σ((x + 1) v-1 ) (by Lemma 4.14-ii)) and σ((x +1) v-1 ) ∈ {1, P }. Thus, v ∈ {1, 3, 5}, Proof. One has: Q ∈ {σ(x 2r ), σ(x u-1 )}.

-If Q = σ(x 2r ), then Q ̸ = σ((x + 1) 2s ) (by Lemma 4.14-ii)), Q does not divide σ(x u-1 ) since gcd(σ(x 2r ), σ(x u-1 )) = 1. So, Q∥σ * * (A). Therefore,

Here, Q = 1 + x(x + 1) 2 ν -1 P 2 ν , with P ∈ {M 1 , M 4 } and ν ≤ 2. By Lemma 4.10-ii), one has: (P = M 1 , ν = 2 and 2r = 12) or (P = M 4 , ν = 1 and 2r = 10). So, 20 ≤ a ≤ 26.

Lemma 4.24. The case where a is even and b odd does not happen.

Proof. If a is even and b odd, then Q ∈ {σ(x 2r ), σ(x u-1 )}.

-

4.2.4

Case where Q = σ(P 2m ), for some m ≥ 1 Lemma 4.13 implies that Q, P Q ̸ ∈ {σ(x 2g ), σ((x + 1) 2g ) : g ≥ 1}. P = M 1 and (Q = σ(P 2 ) = 1 + x(x + 1)P or Q = σ(P 4 ) = 1 + x 3 (x + 1) 3 P ). Thus,