Marie Marchal

Romain Briandet

David Halter

Sandrine Koechler

Michael S Dubow

Marie-Claire Lett

Philippe N Bertin
email: philippe.bertin@unistra.fr

Subinhibitory Arsenite Concentrations Lead to Population Dispersal in Thiomonas sp

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Introduction

To cope with environmental stresses, microorganisms have evolved multiple survival strategies, including the synthesis of detoxification enzymes and the acquisition of resistance genes by horizontal gene transfer. These defence mechanisms may also rely on multicellular behaviours, such as those observed inside biofilms, where a part of the bacterial population can die to promote the survival of the remaining cells [START_REF] Nadell | The sociobiology of biofilms[END_REF]. Biofilms, which are surfaceattached microbial communities embedded in an extracellular matrix, exhibit significantly higher survival capacities in extreme conditions compared to their planktonic counterparts [START_REF] Davey | Microbial biofilms: from ecology to molecular genetics[END_REF]. They constitute the most common lifestyle of microbial communities, providing them with several advantages. For example, the extracellular matrix allows the development of a structured community with subsequent physiology and genetic diversity [START_REF] Lewis | Multidrug tolerance of biofilms and persister cells[END_REF]. To date however, little is known regarding the behaviour of environmental bacterial biofilms despite their roles in numerous processes, either deleterious, e.g. biocorrosion and remobilisation of toxic elements, or beneficial, e.g. bioremediation [START_REF] Flemming | The biofilm matrix[END_REF][START_REF] Singh | Biofilms: implications in bioremediation[END_REF]. Due to the significance of these processes, their understanding and the prediction of their evolution are of prime importance and require a better knowledge of the mechanisms governing biofilm formation and persistence.

Members of the Thiomonas genus are ubiquitous in extreme environments contaminated by arsenic [START_REF] Battaglia-Brunet | Oxidation of arsenite by Thiomonas strains and characterization of Thiomonas arsenivorans sp[END_REF][START_REF] Bruneel | Mediation of arsenic oxidation by Thiomonas sp. in acid-mine drainage (Carnoule `s, France)[END_REF], one of the most toxic inorganic elements currently known [START_REF] Hughes | Arsenic toxicity and potential mechanisms of action[END_REF]. The acid mine drainage (AMD) waters of the former mine of Carnoule `s, Gard (France), contain high arsenic levels (up to 4.66 mM) mainly in the form of arsenite As(III). Interestingly, this site is subjected to a natural process of remediation in which Thiomonas bacterial strains play a key role by oxidizing As(III) into As(V) which co-precipitates with iron and sulfur [START_REF] Bruneel | Mediation of arsenic oxidation by Thiomonas sp. in acid-mine drainage (Carnoule `s, France)[END_REF][START_REF] Bertin | Metabolic diversity between main microorganisms inside an arsenic-rich ecosystem revealed by meta-and proteo-genomics[END_REF]. These Betaproteobacteria, defined as facultative chemolithoautotrophs, which grow optimally in mixotrophic media containing reduced inorganic sulfur compounds (RISCs) and organic supplements, are also of particular interest for their use in bioremediation [START_REF] Michel | Biofilms of As(III)-oxidising bacteria: formation and activity studies for bioremediation process development[END_REF]. Indeed, they have developed several resistance and adaptive mechanisms to cope with arsenic stress, mainly by the means of detoxification enzymes. In addition, these strains have been shown to possess the ability to form biofilms [START_REF] Arse `ne-Ploetze | Structure, function, and evolution of the Thiomonas spp. genome[END_REF][START_REF] Bryan | Carbon and arsenic metabolism in Thiomonas strains: differences revealed diverse adaptation processes[END_REF]. However, little is known regarding the factors influencing the development of these structures and how these organizations may play a role in the persistence of strains in such extreme environments.

With this aim in view, we explored the effects of subinhibitory As(III) concentrations on both biofilm formation and development in Thiomonas sp. CB2. This strain, previously isolated from the Carnoule `s AMD waters, is not capable to grow in the absence of any inorganic energy source, i.e. reduced inorganic sulfur compounds (RISCs) or As(III), and its genome carries an operon aox coding for the arsenite oxidase enzyme. Interestingly, Thiomonas sp. CB2 has been shown to form flocs when cultivated under dynamic conditions and to form important biofilms under static growth conditions [START_REF] Arse `ne-Ploetze | Structure, function, and evolution of the Thiomonas spp. genome[END_REF]. In the present study, using multimodal confocal laser scanning microscopy, we show that the structural dynamics of the biofilm is markedly changed when bacteria are exposed to As(III). Importantly, our results highlight how this process may allow the strain to survive and also develop in such toxic ecosystems.

Materials and Methods

Bacterial strain and culture conditions

Thiomonas sp. CB2 was previously isolated from the acid mine drainage waters of Carnoule `s, Gard (France) [START_REF] Arse `ne-Ploetze | Structure, function, and evolution of the Thiomonas spp. genome[END_REF]. The strain was cultivated on m126 (modified 126 medium) liquid or solid medium as previously described [START_REF] Bryan | Carbon and arsenic metabolism in Thiomonas strains: differences revealed diverse adaptation processes[END_REF]. Arsenic was added at the desired concentration from sterile stocks of 667 mM of the metalloid ion in deionized water, from either NaAsO 2 [As(III)] or Na 2 HA-sO 4 N7H 2 O [As(V)] salts (Prolabo, Fontenay-sous-bois, France). Planktonic growth was assessed in liquid medium in the absence and in the presence of either 1.33 or 2.67 mM As(III) as follows: an exponential culture was diluted to an OD 600 of 0.001 on a Spectronic Genesis spectrophotometer and growth was then assessed by measuring daily OD 600 of the cultures. The minimum inhibitory concentration (MIC) of As(III) for this strain has been previously determined to be 10 mM as follows: cultures were performed in triplicate on gelified medium supplemented with 10, 5, 2.25, 1.25 or 0 mM As(III) and were incubated at 30uC for up to 10 days. MIC was defined as the minimal As(III) concentration tested inhibiting growth [START_REF] Arse `ne-Ploetze | Structure, function, and evolution of the Thiomonas spp. genome[END_REF][START_REF] Bryan | Carbon and arsenic metabolism in Thiomonas strains: differences revealed diverse adaptation processes[END_REF].

Biofilm cultures were performed in microscopic quality 35-mm polystyrene dishes (ibiTreat low 35-mm m-Dish, Integrated BioDiagnostics, Martinsried, Germany) as follows: 1 mL of m126 medium, supplemented or not with As(III), was inoculated with 10 mL of an exponentially growing culture. Incubation was performed at 30uC under stagnant culture conditions. The medium was renewed every day to avoid growth limitation due to nutrient depletion.

Confocal Laser Scanning Microscopy

Biofilms were grown in microscopic quality 35-mm polystyrene dishes as described above. Before staining, samples were gently rinsed with 1 mL of 150 mM NaCl to remove loosely adherent planktonic cells. Cell viability was assessed by probing the cells for 15 min with 50 mL of a stock solution of SYTO9 (6 mM), a permeable nucleic acid fluorescent probe dye, and SYTOX Red (5 nM), a cell impermeable probe (Molecular Probes, Invitrogen Corporation, Cergy Pontoise, France) [START_REF] Roth | Bacterial viability and antibiotic susceptibility testing with SYTOX green nucleic acid stain[END_REF]. This combination was chosen for the spectral separation of these dyes, as previously demonstrated for equivalent stains [START_REF] Biggerstaff | New methodology for viability testing in environmental samples[END_REF]. Polysaccharides within the exopolymeric matrix of the biofilm were stained with 50 mL of Alexa Fluor 633 conjugated to Concanavalin A (ConA) or Wheat Germ Agglutinin (1 mg mL 21 ) (Molecular Probes, Invitrogen Corporation).

Z-stacks of horizontal plane images of biofilm were acquired with a step of 1 mm using multimodal confocal scanning laser microscopy (Leica TCS SP2 AOBS, Leica Microsystems, Nanterre, France) with a 663 (1.4 N.A.) oil immersion lens. Images were recorded at an excitation wavelength of 488 nm and emission wavelengths from 500 to 530 nm for SYTO9 and at an excitation wavelength of 633 nm and emission wavelengths from 657 to 757 nm for SYTOX Red and Alexa Fluor 633 conjugated to Concanavalin A or Wheat Germ Agglutinin. The surface coverage was quantified using NIH ImageJ analysis software (http://rsbweb.nih.gov/ij/). A manual threshold was applied to binarize images. Simulated 3D fluorescence projections were generated using IMARIS 7.0.0 software (Bitplane, Zu ¨rich, Switzerland). Dead (SYTOX Red), total cells (SYTO9) and exopolysaccharide (ConA) biovolumes were quantified from images series using PHLIP, a freely available Matlab-based image analysis toolbox (www.phlip.org). The biovolume represents the overall volume of a biofilm (mm 3 ) and may be used to estimate its total biomass. It has been defined as the number of foreground pixels in an image stack multiplied by the voxel volume, which is the product of the squared pixel size and the scanning step size [START_REF] Kuehn | Automated confocal laser scanning microscopy and semiautomated image processing for analysis of biofilms[END_REF]. To limit the temporal variability between biofilm samples, all replicates for confocal microscopy experiments were performed during a single week. In each condition, six stacks were randomly acquired on two independent samples from six microscopic fields of the central 35-mm dish zone.

Nuclease treatment of biofilms

To test the role of extracellular nucleic acids on the biofilm structure, BenzonaseH Nuclease (Sigma-Aldrich, St Louis, MO, USA) was added to 72 h-old biofilms to a final concentration of 125 U mL 21 (a control without BenzonaseH Nuclease was also included). After 1 h of incubation at 37uC, biofilm supernatant was removed to eliminate non-attached cells. Biofilm integrity was determined, after staining of the attached cells with SYTO9 as described above, at a magnification of 10006under oil immersion using a Leica DM 4000 B epifluorescence microscope equipped with a Leica DFC300 FX digital camera (Leica Microsystems). Images were recorded at an excitation wavelength of 488 nm and emission wavelengths from 500 to 550 nm.

As(III) resistance assay

As(III) resistance of 72 h-old biofilms grown in the absence or in the presence of 1.33 mM As(III) was assessed by plating biofilm cells (washed in 150 mM NaCl) on solid m126 medium containing 10.7 mM As(III). The minimal inhibitory concentration for this strain has been previously determined to be 10 mM [START_REF] Arse `ne-Ploetze | Structure, function, and evolution of the Thiomonas spp. genome[END_REF]. Total biofilm cell concentrations were obtained by plating serial dilutions of the biofilm population on solid m126 medium containing no As(III). Colony forming units (CFUs) were counted after 2 weeks of incubation at 30uC. As(III) resistance of the biofilm cell population was expressed as the percent of cells able to grow in the presence of 10.7 mM As(III).

Identification of potential exopolysaccharide synthesis coding genes

Genes possibly involved in exopolysaccharide synthesis were searched for in Thiomonas sp. CB2 genome using available genomic data, i.e. the genome sequence of Thiomonas sp. 3As (http://www. cns.fr/agc/microscope/mage/index.php?) and comparative genomic hybridization (CGH) data between Thiomonas sp. CB2 and Thiomonas sp. 3As (available in the ArrayExpress database with the accession number E-MEXP-2260) [START_REF] Arse `ne-Ploetze | Structure, function, and evolution of the Thiomonas spp. genome[END_REF].

RNA extraction

Biofilms were grown as described above, in the absence or in the presence of 1.33 mM As(III). After 72 h of incubation, biofilm cells were manually recovered by scratching the dishes. Cells were then harvested by centrifugation and stored at 280uC to prevent RNA degradation. Pellets were suspended in 400 mL of suspension solution (25 mM Tris-HCl pH 7.6-10 mM EDTA and 20% glucose (vol/vol)) and transferred into microtubes containing 500 mL of acidic phenol (pH 4.5) and 0.4 g of 0.1 mm-diameter glass beads (VWR, Fontenay-sous-Bois, France). Cells were mechanically broken with a ''FP120 FastPrep Cell Disruptor'' (Bio 101, Q-BIOgene, Illkirch, France) (two 30 sec cycles of homogenization at maximum speed with 1 min interval on ice). Total RNA were then extracted with TRIzol Reagent (Invitrogen Corporation) according to the manufacturer's instructions. RNA aliquots were purified with the RNeasy Plus mini kit (Qiagen, Courtaboeuf, France) to ensure the elimination of genomic DNA. RNA integrity was checked by automated electrophoresis with the Experion system (Bio-Rad, Marnes-la-Coquette, France). Total RNA concentration was determined spectrophotometrically with a Nanodrop spectrophotometer (NanoDrop Technologies, Wilmington, DE, USA).

Quantitative real time PCR

Total RNA (2 mg) for each condition was reverse transcribed using MaximaH First Strand cDNA Synthesis Kit for RT-qPCR (Fermentas, Saint-Re ´my-le `s-Chevreuse, France). The cDNA synthesis reaction was completed in 30 min at 50uC. Real time quantitative PCR was carried out with a MyiQ single-color Real-time PCR detection system (Bio-Rad). The reaction mixture contained 12.5 mL of iQ TM SYBRHGreen Supermix (Bio-Rad), 5 mL of cDNA (diluted 1/3) and 300 nM of each primer in a total volume of 25 mL. Thermocycling conditions were as follow: 5 min at 95uC and 40 cycles of 15 sec at 95uC, 15 sec at 62uC and 1 min at 72uC. For each quantitative PCR run, non-template controls were performed to identify false positives. Negative controls without reverse transcriptase were also performed for each cDNA synthesis reaction and verified in real time PCR to determine the presence of contaminating genomic DNA. Absence of primer dimers was checked. Two biological replicates (independent cultures) and two technical quantitative PCR replicates were performed for each experiment. The data were analyzed with the Relative Expression Software Tool [START_REF] Pfaffl | Relative expression software tool (REST) for group-wise comparison and statistical analysis of relative expression results in real-time PCR[END_REF]. Statistical significance was defined at p-value,0.05.

House-keeping genes were chosen on the basis of previous transcriptomic experiments on arsenic stress [START_REF] Cleiss-Arnold | Temporal transcriptomic response during arsenic stress in Herminiimonas arsenicoxydans[END_REF]. Genes with a stable expression between cultures performed with and without As(III) were identified and homologous genes were then searched for in Thiomonas sp. CB2 genome using Thiomonas sp. 3As genome and the previously performed CGH experiments [START_REF] Arse `ne-Ploetze | Structure, function, and evolution of the Thiomonas spp. genome[END_REF]. Primers were designed using Thiomonas sp. 3As genome to generate amplification products smaller than 200 bp in size. The pairs of primers used are listed in Table S1. Three housekeeping genes, i.e. THI_2580 coding for a putative S-adenosyl-L-methionine-dependent methyltransferase, THI_0556 coding for a DnaA initiator-associating protein and THI_1854 coding for a putative transcriptional regulator, were used as standards to obtain a normalized ratio for tested genes in As(III) induced samples compared to non induced. The PCR efficiency of the genes of interest and internal control genes were optimized to be similar enough by adjusting the primer concentrations to 300 nM each (data not shown).

Statistical analysis

The ANOVA test was used to compare strain growth parameters and to evaluate the disparity between biofilm replicates. The Student's two-tailed t-test was used to compare the surface coverage rates and also the ConA/SYTO9 and SYTOX Red/SYTO9 biovolume ratios. These ratios were calculated by averaging the data of the six microscopic fields analyzed per biofilm culture condition. To compare the ratios between each culture conditions, Student's two-tailed t-tests were performed using an n of 6, assuming unequal variances equality between the different conditions. A Fisher test was performed to assess the variance equality of the different parameters between each condition. Significance of all three statistical tests was set at pvalue,0.05 (P,0.05).

Results

Increased extracellular matrix synthesis under arsenite exposure

Thiomonas sp. CB2 has the capacity to form flocs when cultivated under planktonic conditions. This propensity was observed to be enhanced in the presence of 1.33 mM As(III) but not in the presence of 1.33 mM As(V) (data not shown). To gain insight into the possible effects of As(III) on biofilm formation and development, the strain was cultivated under stagnant conditions in polystyrene dishes in the absence or in the presence of subinhibitory arsenite concentrations, i.e. 1.33 and 2.67 mM As(III). In the range of concentrations tested here, arsenite did not significantly affect Thiomonas sp. CB2 planktonic growth (ANOVA test, P.0.05, Figure S1). To avoid growth limitation due to nutrient depletion, the growth medium was renewed every day. Both biofilm formation and development were investigated using multimodal confocal scanning and a cell viability staining.

The bacterial adhesion began after about 24 hours of culture and, after 48 hours of growth, cell adhesion was not significantly influenced by the presence of arsenite (t-test, P.0.05). The percentage of bacterial surface coverage was of 11.565.2 in the absence of As(III) and of 15.764.7 and 12.964.2 (mean 6 SD) in the presence of 1.33 and 2.67 mM of As(III). Microcolonies were present in both the presence and absence of metalloid, the most developed multicellular structures being filled with living cells, i.e. labelled with SYTO9 but not with SYTOX Red (Figure S2).

To get further structural information about these bacterial cell aggregates, two fluorescently labelled lectin, i.e Concanavalin A (ConA) binding selectively a-mannopyranosyl and a-glucopyranosyl residues and Wheat Germ Agglutinin (WGA) binding Nacetyl-D-glucosamine and N-acetylneuraminic acid, were used to stain the polysaccharides present within the exopolymeric matrix of the biofilm. Interestingly, after 72 hours incubation, these microcolonies were surrounded by a higher amount of extracellular polysaccharides in the presence of arsenite. These exopolysaccharides were revealed by ConA but not by WGA, which did not mark any extracellular component (data not shown). The exopolysaccharide volume revealed by ConA compared to the total cell volume revealed by SYTO9 staining was 2.1 and 6.2-fold greater in the presence of 1.33 and 2.67 mM As(III), respectively, than without As(III) (Figure 1). These data suggest that the exposure to the metalloid increases the extracellular polymer biosynthesis in Thiomonas sp. CB2. Genes possibly involved in the extracellular polymer synthesis were searched for using the genome of a closely-relative strain, Thiomonas sp. 3As, and previous CGH array data (see Material and Methods section). Four genes were identified, i.e. rfbABCD, which are organized in an operonlike manner in Thiomonas sp. 3As genome (THI_2966 to THI_2969) and code for dTDP-rhamnose biosynthesis, a potential exopolysaccharide precursor. The expression of rfbB and rfbD, THI_2969 and THI_2968, respectively, in the Thiomonas sp. 3As genome, was measured by quantitative RT-PCR in biofilm cells exposed or not to 1.33 mM As(III). These two genes were expressed in both conditions, although no significant variation in their expression could be detected between the two conditions tested (data not shown).

Cell death, As(III)-exposed biofilm architecture and the role of extracellular nucleic acids in the biofilm structural integrity

When we examined the internal organization of the microcolonies within 72 h-old biofilms, we observed that they were filled with dead cells only in the presence of arsenite. The proportion of dead cells increased with arsenite concentration. The dead cell volume, compared to the total cell volume, was 8.4 and 83.2-fold greater in the presence of 1.33 and 2.67 mM As(III), respectively, than without As(III) (Figure 2). The biofilm physiological heterogeneity increased significantly with As(III) concentration (F-tests, P,0.05), variance of the SYTOX Red/SYTO9 biovolume ratios being 0.03, 0.33 and 4.05 in the absence of As(III) and in the presence of 1.33 and 2.67 mM As(III), respectively.

This physiological heterogeneity was reinforced by cell lysis events which occurred inside arsenite exposed microcolonies after cell death events, generating hollow voids (Figure 3 and Figure S3).

Concomitantly to cell lysis, the three-dimensional biofilm structures underwent complex rearrangements. This phenomenon began after 72 hours of culture (Figure 3) and increased with time. Indeed, after 7 days of culture, we observed tall three dimensional biofilm structures only in the presence of 1.33 and 2.67 mM As(III) (Figure 4). These structures, up to 90 mm high, were covered by large amounts of exopolysaccharides.

Cell lysis events are known to result in the liberation of nucleic acids. In order to determine the role of extracellular nucleic acids in Thiomonas sp. CB2 biofilm architecture, 72 h-old biofilms were treated with a nuclease. This treatment led to the disintegration of the biofilms grown in the presence of 1.33 mM As(III), removing all non-surface attached cells. However, nuclease treatment had no effects on non-As(III)-exposed biofilms (Figure S4 and data not shown). This result demonstrates an essential role of extracellular nucleic acids in Thiomonas sp. CB2 three-dimensional biofilm structure.

Promotion of biofilm dispersal under arsenite exposure

Exploring the three-dimensional As(III)-exposed biofilm structures revealed the existence of subpopulations of highly motile cells inside the hollow biofilm voids generated by cell lysis events (Figure 5 and Movie S1). These remobilized bacteria were present in at least 72 h-old biofilms only in the presence of As(III) and promoted biofilm dispersal as they were observed to evacuate from the hollow voids of 7 days-old 2.67 mM As(III)-exposed biofilms (Movie S2).

In order to determine whether this bacterial As(III)-exposed biofilm population had acquired specific capacities regarding Figure 1. As(III)-induced exopolysaccharide biosynthesis. 72 h-old Thiomonas sp. CB2 biofilms were stained using SYTO9 (cells, green) and a labeled lectin, ConA (exopolysaccharides, red). A. Three-dimensional confocal reconstruction of representative microcolonies using IMARIS software in the absence and in the presence of 2.67 mM As(III). B. Biovolume ConA/SYTO9 as a function of As(III) concentration. Cells (SYTO9) and exopolysaccharide (ConA) volumes were quantified with light emission intensity using PHLIP software (www.phlip.org). For each experiment, six random microscope fields were analyzed from two replicates. An ANOVA test was performed to evaluate the disparity between replicates, indicating P.0.05 for each parameter. The two replicates per condition could also be considered as not significantly different. The presence of arsenite significantly influenced the biovolume ConA/SYTO9 (t-test, p = 0.047 between biofilms non-exposed to arsenite and biofilms exposed to 2.67 mM As(III)). Error bars represent standard deviations. Scale bars: 10 mm. doi:10.1371/journal.pone.0023181.g001 arsenic stress, we measured As(III) resistance of the cellular population originating from 72 h-old biofilm grown either in the absence of As(III) or in the presence of 1.33 mM As(III). Biofilm cells were plated on 10.7 mM As(III) containing solid medium, i.e. 0.7 mM more than the minimal inhibitory concentration initially determined for the strain. The formed colonies displayed the same convex and undulate morphology as those formed by the wild-type strain without As(III). The percentage of cells able to grow in the presence of 10.7 mM As(III) was of 0.260.05 and 19.967.28 (mean 6 SD) for the population originating from biofilm grown in the absence and in the presence of As(III), respectively. The biofilm structural dynamics under As(III) exposure led also to development of cells exhibiting an higher As(III) resistance than that observed for the initial strain.

Discussion

Bacteria of the Thiomonas genus are ubiquitously found in arsenic contaminated sites, including AMD waters. We have recently shown that they are particularly well suited to survive in such extreme environments, combining several defence mechanisms, including arsenite metabolism and DNA repair. We have hypothesized that these strains, in particular Thiomonas sp. CB2, form biofilms in response to As(III) stress [START_REF] Arse `ne-Ploetze | Structure, function, and evolution of the Thiomonas spp. genome[END_REF]. Indeed, during planktonic growth the CB2 strain formed flocs, i.e. bacterial aggregates, and this propensity was increased in the presence of As(III) but not in the presence of As(V). In the present study, the Thiomonas sp. CB2 biofilm development kinetics was compared in the absence and in the presence of As(III) at subinhibitory concentrations under static culture conditions. The overall biofilm development appeared to be strongly modified in response to As(III).

Under such conditions, the biofilm integrity may be reinforced by the high quantities of exopolysaccharide synthesised in the presence of As(III), these compounds being known to play an essential role in biofilm cohesion [START_REF] Flemming | The biofilm matrix[END_REF]. Even if the molecular determinants regulating this increased exopolysaccharide synthesis have not been precisely identified due to the current lack of mutagenesis tools for Thiomonas spp., four genes involved in dTDPrhamnose biosynthesis, an exopolysaccharide precursor, may play a role in this process. Indeed, these genes are also present in closely-related strains, i.e. Thiomonas sp. 3As [START_REF] Arse `ne-Ploetze | Structure, function, and evolution of the Thiomonas spp. genome[END_REF] and Acidithiobacillus ferrooxidans [START_REF] Barreto | Identification of a gene cluster for the formation of extracellular polysaccharide precursors in the chemolithoautotroph Acidithiobacillus ferrooxidans[END_REF]. However, these genes were not found to be significantly over-expressed in Thiomonas sp. CB2 biofilm cells under As(III) exposure. This suggests that these genes are not the main genetic determinants involved in the exopolysaccharide synthesis induction highlighted by the ConA staining in the presence of As(III).

During biofilm formation, the presence of As(III) led to localized cell death and lysis events inside microcolonies. Such events have been previously described in other strains and several hypotheses have been proposed, including the coexistence of subpopulations of cells differing in their physiological state [START_REF] Bayles | The biological role of death and lysis in biofilm development[END_REF][START_REF] Pamp | Tolerance to the antimicrobial peptide colistin in Pseudomonas aeruginosa biofilms is linked to metabolically active cells, and depends on the pmr and mexAB-oprM genes[END_REF]. In Thiomonas sp. CB2, a similar mechanism may explain the cell death events we observed inside the microcolonies of the arsenite-exposed biofilms. Indeed, arsenic-tolerance requires active cell physiology, i.e. the extensive synthesis of detoxification enzymes, efflux pumps, and DNA repair systems [START_REF] Stolz | Arsenic and selenium in microbial metabolism[END_REF]. The cells located in the centre of the microcolonies, known to exhibit low metabolic activity [START_REF] Stewart | Physiological heterogeneity in biofilms[END_REF], may also be more susceptible to this stress.

Localized cell death and lysis events are thought to contribute to the increased antibacterial compound tolerance in biofilm cells, in part because the lysis of a cell subpopulation provides nutrients and cellular polymers to protect the remaining cells [START_REF] Bayles | The biological role of death and lysis in biofilm development[END_REF][START_REF] Lopez | Cannibalism enhances biofilm development in Bacillus subtilis[END_REF]. More importantly, these events appear to be of prime importance in the subsequent developmental processes we observed in the presence of arsenite in Thiomonas sp. CB2 biofilm. Indeed, extracellular nucleic acids were shown to be essential components of the biofilm matrix, playing a key structural role in this strain. Their liberation during cell lysis may also reinforce the structures and allow the subsequent biofilm developmental steps which led to the formation of the tall biofilm structures we observed only in the presence of As(III).

The three-dimensional development of the biofilm under As(III) exposure enhanced the biofilm heterogeneity and may also favour the development of distinct subpopulations of cells [START_REF] Stewart | Physiological heterogeneity in biofilms[END_REF]. This hypothesis is supported by the presence of subpopulations of remobilized cells inside the hollow voids formed during the cell lysis events. These highly motile cells promoted the seeding dispersal of the strain by evacuating from the biofilm hollow voids. Such an active biofilm dispersal mechanism allows the strain to colonize new environments [START_REF] Mai-Prochnow | Ecological advantages of autolysis during the development and dispersal of Pseudoalteromonas tunicata biofilms[END_REF]. Moreover, about 20% of the bacterial As(III)-exposed biofilm population exhibited an As(III) resistance higher than that of the wild-type strain. Even if this increased resistance is probably mainly linked to the induction of enzymatic defence mechanisms [START_REF] Stolz | Arsenic and selenium in microbial metabolism[END_REF], a part of this subpopulation of cells may also represent genetic variants, as described in other biofilms, e.g. those formed by Pseudomonas aeruginosa and Streptococcus pneumoniae [START_REF] Boles | Endogenous oxidative stress produces diversity and adaptability in biofilm communities[END_REF][START_REF] Allegrucci | Formation of Streptococcus pneumoniae non-phasevariable colony variants is due to increased mutation frequency present under biofilm growth conditions[END_REF]. A high frequency of genomic rearrangement events inside Thiomonas spp. biofilms may also explain the high genomic diversity of the strains belonging the Thiomonas genus, which has been previously shown by comparative genomic analysis [START_REF] Arse `ne-Ploetze | Structure, function, and evolution of the Thiomonas spp. genome[END_REF]. This hypothesis is currently under investigation.

In conclusion, we showed that the overall Thiomonas sp. CB2 biofilm development was markedly modified in response to subinhibitory arsenite concentrations leading to an increased As(III) resistance and, more importantly, to a rapid dispersal of part of the community. Our results suggest also that, in Thiomonas sp. CB2, the formation of a biofilm plays several roles in the arsenite resistance process. It is tempting to speculate that the survival and persistence of the strain in its extreme environment relies on a rapid biofilm development followed by the dispersal of a more resistant population. Thiomonas spp. may be persistent in AMD waters and thus assure a long-term role in the natural remediation process occurring in this site. More importantly, these processes may confer an important selective advantage by allowing the strain to survive in harsh conditions and also to colonize more favourable environments. These biofilm-associated processes presumably constituting an adaptive strategy may be widely conserved among bacteria in response to adverse conditions, including in pathogenic biofilms exposed to antimicrobials compounds.

Figure 2 .Figure 3 .

 23 Figure 2. Cell death in microcolonies of 72 h-old Thiomonas sp. CB2 biofilms under As(III) exposure. A. Cell death occurs in localized regions in microcolonies. Confocal micrograph of a 1.33 mM As(III)-exposed biofilm microcolony visualised using viability stains. B. Biovolume SYTOX Red/SYTO9 as a function of As(III) concentration. Dead (SYTOX Red) and total cell (SYTO9) volumes were quantified using light emission intensity with PHLIP software. For each experiment, six random microscope fields from two replicates were analyzed. An ANOVA test was performed to evaluate the disparity between replicates, indicating P.0.05 for each parameter. The two replicates per condition could also be considered as not significantly different. The presence of arsenite presence influenced significantly the biovolume SYTOX Red/SYTO9 (t-test, p = 0.032 between biofilms nonexposed to arsenite and biofilms exposed to 1.33 mM As(III)). Error bars represent standard deviations. Scale bar: 10 mm. doi:10.1371/journal.pone.0023181.g002

Figure 4 .

 4 Figure 4. Three dimensional confocal reconstruction of 7 day-old biofilms. Thiomonas sp. CB2 biofilms were cultivated in the absence (A) or in the presence of 1.33 mM (B) and 2.67 mM As(III) (C). Biofilms were stained using SYTO9 (cells, green) and ConA (exopolysaccharide, red). Arsenic exposure induced altered biofilm architecture, leading to the formation of sponge-(B) and coral-like structures (C). Scale bars: 20 mm. doi:10.1371/journal.pone.0023181.g004

Figure 5 .

 5 Figure 5. Motile bacteria swimming inside hollow voids. A. 2Dconfocal representation of hollow voids of a 72 h-old 2.67 mM As(III)exposed biofilm structure containing swimming bacteria. Cells were stained with SYTO9. Scale bar: 20 mm. B. 2D-time progression, in red, from point 1 to point 2 of the cell circled on panel (A). The representation is a projection of 16 confocal images taken at 1.63 second intervals. doi:10.1371/journal.pone.0023181.g005

PLoS ONE | www.plosone.org

August 2011 | Volume 6 | Issue 8 | e23181

Acknowledgments

The authors would like to thank Dr. Odile Bruneel for providing the Thiomonas sp. CB2 strain and Julien Deschamps for processing the confocal data. Confocal laser scanning microscopy was performed at the Microscopy and Imaging Platform (MIMA2, INRA, Massy, France).

MM was supported by a studentship from the ANR COBIAS project (PRECODD 2007}, and by Agence Nationale de la Recherche (http://cobias.brgm.fr/). DH was supported by a grant from the French Ministry of Education. This work was performed as part of the program of the « Groupement de Recherche -Me ´tabolisme de l'Arsenic chez les Microorganismes (GDR2909-CNRS)» (http://gdr2909.alsace.cnrs.fr/). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Supporting Information

Thiomonas sp. CB2 planktonic growth in the absence (grey) or in the presence of either 1.33 (white) or 2.67 mM As(III) (blue). Due to the strain propensity to form flocs during planktonic growth, including in the absence of As(III), growth parameters were assessed by measuring every 24 h during 4 days the optical density at 600 nm of the cultures. Cultures were performed in liquid medium in independent triplicates. The presence of As(III) did not significantly influence strain growth (t-tests, P.0.05). Error bars represent standard deviations. (PPT)


Renaud Travadon

Ivan Sache

Christian Cyril Dutech

Anna Stachowiak

Bruno Marquer

Lydia Bousset

Cyril Dutech

Absence of isolation by distance patterns at the regional scale in the fungal plant pathogen Leptosphaeria maculans

Keywords: Gene flow, Indirect measures, Minisatellites, Spatial genetic structure, Spore dispersal Highlights

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

Introduction

Dispersal has an important influence on patterns of local adaptation of plant pathogens to their host (Kaltz & Shykoff 1998), especially in agricultural ecosystems where migration of virulent pathogen genotypes is favoured compared to immobile, cultivated resistant host genotypes. These unbalanced migration rates between host plants and pathogens eventually lead to a reduced efficacy of plant resistance (Gandon et al. 1996). Durable disease control, therefore, requires the deployment of resistant varieties in a spatial arrangement that limits disease transmission [START_REF] Aubertot | Diggle Improved resistance management for durable disease control: a case study of phoma stem canker of oilseed rape (Brassica napus)[END_REF]. The spatial scale of relevance for such an arrangement should match the scale of gene flow between pathogen populations. For wind-dispersed, plantpathogenic fungi, spore dispersal usually occurs both at long distances over hundreds of kilometres (lowfrequency, atmospheric dispersal) and at short distances within a few hundred metres (high-frequency, local dispersal) [START_REF] Aylor | Irwin Aerial dispersal of pests and pathogens: implications for integrated pest management[END_REF]. Locally, the number of deposited spores decreases with increasing distance from the inoculum source (Gregory 1973). Spore trapping experiments, however, do not always yield consistent estimates at increasing distances from the inoculum source, thereby underestimating rare events of long-distance dispersal (McCartney et al. 2006).

Comment citer ce document : Travadon, R., Sache, I., Dutech, C. C., Stachowiak, A., Marquer, B., Bousset, L. (2011). Absence of isolation by distance patterns at the regional scale in the fungal plant pathogen Leptosphaeria maculans. Fungal Biology, 115 (7), 649-659. , DOI : 10.1016/j.funbio.2011.03.009

This file is the post-print of : Travadon et al. Fungal Biology (2011) 115: 649-659 Doi: 10.1016/j.funbio.2011.03.009 2 When direct measures fail, the scale of spore dispersal can be inferred from indirect genetic measures. If local dispersal is predominant, the resulting pattern of neutral genetic differentiation between populations is expected to follow the 'isolation by distance' (IBD) model (Wright, 1943[START_REF] Slatkin | Slatkin Gene flow and the geographical structure of natural populations[END_REF][START_REF] Rousset | Rousset Genetic differentiation and estimation of gene flow from F-statistics under isolation by distance[END_REF]. Assuming that neutral genetic structure is primarily due to genetic drift and is counterbalanced by gene dispersal (in a two-dimensional environment), and that it reflects the dispersal capacities of the studied species, a linear correlation is expected between the logarithm of the geographical distance and estimated between pairs of populations (Fst being the index of genetic differentiation among populations; Wright 1943). The detection of an IBD pattern requires a precise framework, assuming mutationdrift equilibrium, and considering spatial distances ranging between σ and 20σ, where σ 2 is the second moment of parent-offspring distance dispersal [START_REF] Rousset | Rousset Genetic differentiation and estimation of gene flow from F-statistics under isolation by distance[END_REF]. Estimations through this method have been shown to be robust to the mutational process of microsatellite loci (i.e. allele size homoplasy), and to spatial and temporal heterogeneities of demographic parameters (i.e. variation in dispersal and density) (Leblois et al. 2003(Leblois et al. , 2004)). Furthermore, analysis conducted at a local scale allows reducing the influence of selection pressure on genetic differentiation [START_REF] Rousset | Rousset Inferences from spatial population genetics[END_REF]. For several tropical tree species, indirect gene dispersal estimates were consistent with direct estimates of seed and pollen dispersal (Hardy et al. 2006).

Among wind-dispersed fungi, IBD patterns have been tested at inter-continental scales, at which the spatial genetic structure could either reflect past, stochastic, and rare long-distance colonisation events (Linde et al. 2002;Rivas et al. 2004;[START_REF] Zaffarano | Global hierarchical gene diversity analysis suggests the Fertile Crescent is not the center of origin of the barley scald pathogen Rhynchosporium secalis[END_REF], or the presence of physical barriers to gene flow (mountains, seas, and deserts) (Hayden et al. 2007). At broad spatial scales, the effects of selection, mutation, and demographic variation on genetic differentiation are not negligible; thus, translating measures of genetic differentiation such as Fst and related indices into a quantitative evaluation of spore dispersal when transformed as a direct measure of the product of effective population size and number of successful migrants per generation, could often lead to misinterpretation (see for example [START_REF] Whitlock | Indirect measures of gene flow and migration: F-ST not equal 1/(4Nm+1)[END_REF]. One way to test for the relevance of the IBD model to plant-pathogenic fungi when σ is not known is to compare spatial scales. For instance, in populations of the poplar rust fungus Melampsora larici-populina, Barrès et al. (2008) detected IBD patterns within Europe, but population genetic structures from Iceland and Canada were shown to result from rare long-distance dispersal events.

Leptosphaeria maculans (Desm.) Ces. & de Not. (anamorph Phoma lingam Tode ex Fr.), is a heterothallic, haploid fungus causing Phoma stem canker, a disease responsible of severe yield losses of Brassica, especially oilseed rape (Brassicanapus) world-wide (Fitt et al. 2006). In French oilseed rape crops, the life-cycle of the fungus matches the host presence, which is from Sep. to Jun. A single, sexual reproduction cycle per year occurs during summer on infected stubble (Fitt et al. 2006). While the resulting ascospores were once thought to travel on distances up to 10 km (McGee 1977), direct measures recently suggested that most ascospores were indeed concentrated within a few hundred metres from the previous year's crop [START_REF] Salam | Spatial spread of blackleg in canolaa regional scale simulation model[END_REF], Marcroft et al., 2004). From previous indirect measures, contradictory conclusions were drawn on the dispersal abilities of L. maculans. At the inter-continental scale, Amplified Fragment Length Polymorphism (AFLP) analysis differentiated Australian, North American, and European L. maculans populations (Purwantara et al. 2000). In Australia, AFLP and Restriction Fragment Length Polymorphism markers did not reveal a genetic east-west differentiation among populations (Barrins et al. 2004), a differentiation which was subsequently revealed by microsatellite and minisatellite markers (Hayden et al. 2007), and explained as a result of the separation of these two regions by an arid desert. In Canada, Random Amplified Polymorphic DNA markers differentiated two populations sampled from 20 km distant fields (Mahuku et al. 1997). In contrast, a study conducted in France, including four minisatellite markers used in the present study, showed low genetic differentiation among L. maculans populations (Gout et al. 2006). Findings of Gout et al. (2006) are based on only four populations with a scatter distribution across France. This sampling was not designed to infer spore dispersal parameters through indirect genetic measures, but rather allowed the examination of levels and distribution of within-population genetic diversity. The low measures of genetic differentiation reported by Gout et al. (2006), who estimated that >99.5 % of genetic diversity measured at four minisatellite markers was present within two square metres field plots, do not allow the assessment of the minor variation in genetic differentiation within a local spatial scale, which could possibly reflect the evolution of L. maculans populations under an IBD model.

The aim of the present study was to test the relevance of the IBD model to a windborne plantpathogenic fungus. Indeed, it remains to be proven that IBD patterns can be detected in agricultural ecosystems characterized by pathogen populations with a high multiplication rate, a large population size alternating with recurrent yearly bottlenecks, or with a recent range expansion. Based on a priori knowledge of the organism's dispersal abilities, the genetic structure of L. maculans was investigated at a spatial scale where gene flow was expected to occur. Direct measures of ascospore dispersal distances suggest that L. maculans populations separated by geographic distances ranging from several hundred metres to 10 km evolve under IBD [START_REF] Salam | Spatial spread of blackleg in canolaa regional scale simulation model[END_REF], Marcroft et al., 2004). We developed four new minisatellite markers and genotyped 693 isolates at eight loci. We sampled 29 field populations in two regions of France, covering distances ranging from a few hundred metres to one hundred kilometres in each region. Our objective was to test for the presence of IBD pattern in the genetic structure of L. maculans populations. Also, we tested whether the detection of IBD pattern was not erased by a recent range expansion of the pathogen. To this aim, we compared pathogen genetic structure in Region Brittany, where the cultivation of B. napusand thus the presence of its L. maculans pathogenis less than 25-y-old, with that of Region Centre, where oilseed rape has been intensively cultivated for more than 60 y.

Materials and methods

Sample collection and DNA extraction

In This file is the post-print of : Travadon et al. Fungal Biology (2011) 115: 649-659 Doi: 10.1016/j.funbio.2011.03.009 4 Minisatellite identification A systematic search approach described by Eckert et al. (2005) was undertaken in order to identify minisatellite loci in Leptosphaeria maculans. BAC-end sequence data, from the v23.1.3 genomic DNA HindIII BAC library [START_REF] Attard | Analysis of molecular markers genetically linked to the Leptosphaeria maculans avirulence gene AvrLm1 in field populations indicates a highly conserved event leading to virulence on Rlm1 genotypes[END_REF], were screened for regions with tandem repeat using TANDEM REPEAT FINDER (Benson 1999). Primers were designed using Oligo6 (MedProbe, Norway). Genomic DNA of 56 isolates was extracted, comprising 50 L. maculans 'brassicae', one L. maculans 'lepidii' (IBCN84), as well as one Leptosphaeria biglobosa 'brassicae' (IBCN93), 'thlaspii' (IBCN65),'canadensis' (IBCN82), 'australensis' (IBCN91), and 'erysimii' (IBCN83). These isolates were obtained from existing collections maintained at INRA Versailles, and previously described (Gall et al. 1994;Balesdent et al., 2001, Mendes-Pereira et al., 2003). In order to analyse the genetic control of minisatellite segregation, progeny from in vitro cross a.2 × H5 (Balesdent et al. 2001) was used. TANDEM REPEAT FINDER highlighted 203 out of 2550 BAC-end sequences analysed as sequences with putative tandem repeats, for which primer pairs specific to flanking regions of tandem repeats were designed. These primer pairs were first tested on genomic DNA from a set of 7 L. maculans 'brassicae' isolates originating from diverse geographical locations. This first screen resulted in 23 primer pairs generating polymorphic products (2-4 variants of alleles). Then, these 23 primer pairs were tested on 47 L. maculans isolates of diverse geographical origin. Moreover, 12 of them displayed polymorphism between a.2 and H5 parental isolates. Meiotic segregation of minisatellite markers confirmed that these 12 new markers were single-locus sequences. Among these 12 markers, four were chosen according both to their polymorphism, and to their specificity to L. maculans.

Minisatellite amplification

Multilocus genotypes were characterized with minisatellite loci MinLm2, MinLm4, MinLm5, MinLm6 (Eckert et al. 2005), and MinLm8, MinLm9, MinLm632, MinLm1377 for which primers were designed (Table 1). Forward primers were fluorescently-labelled. PCR reactions were performed in a 20-μl final volume containing 10-30 ng of genomic DNA, 1× PCR buffer, 200 μM of each dNTPs, 1.5 mM MgCl2, each primer at 0.5 μM, 0.25 U GoTaq ® DNA Polymerase (Promega, WI, USA). Amplifications were carried out in a G-STORM GS4 thermocycler (GRI Ltd., Braintree, UK) using one cycle of 94 °C for 4 min, followed by 30 cycles of 94 °C for 30 s, annealing temperature for 30 s (Table 1), 72 °C for 1 min, and 5 min of final extension at 72 °C. Allele assignments were performed on an ABI PRISM ® 3130xl sequencer using manufacturer's instructions with the GENEMAPPER 3.7 software (Applied Biosystems). Absence of isolation by distance patterns at the regional scale in the fungal plant pathogen Leptosphaeria maculans. Fungal Biology, 115 (7), 649-659. , DOI : 10.1016/j.funbio.2011.03.009

Table 1. Minisatellite markers and PCR primers used to characterise molecular diversity in populations of

This file is the post-print of : Travadon et al. Fungal Biology (2011) 115: 649-659 Doi: 10.1016/j.funbio.2011.03.009 5 Data analysis Multilocus haplotypes were constructed for each isolate. All isolates collected within the same field were considered as one population. We determined the number of identical multilocus haplotypes within and across populations, using GENCLONE 2.0 [START_REF] Belkhir | GENCLONE: a computer program to analyse genotypic data, test for clonality and describe spatial clonal organization[END_REF]. The same program was used to estimate the probability that identical multilocus haplotypes were the result of a distinct sexual reproductive event (Psex), rather than clonal spread, based on the number of loci and allelic frequencies. For each locus within each population, mean number of alleles (Nall), Nei's gene diversity corrected for sample size (h) (Nei 1987) and allele frequencies were estimated using SPAGEDI 1.2 (Hardy & Vekemans 2002). At a given locus, Nei's gene diversity (h) is defined as the probability that two sampled alleles are different. Mean gene diversity (H) was calculated as the mean of h across loci within each population. In each population, haplotypic richness (R) was estimated as R = (G -1)/ (N -1), where G is the number of unique haplotypes and N is the number of isolates (Dorken & Eckert 2001).

For each pair of loci within each population, gametic linkage disequilibria were computed using GENEPOP 4.0 (Raymond & Rousset 1995). The index of multilocus linkage disequilibrium rd [START_REF] Agapow | Burt Indices of multilocus linkage disequilibrium[END_REF]) was computed for each population. The significance of rd, which is expected to equal zero if there is no linkage disequilibrium, was tested by comparing the observed variance with the distribution of the variance expected under the null hypothesis of random mating, as determined from 1000 randomizations of the haplotype data implemented in MULTILOCUS 1.3 [START_REF] Agapow | Burt Indices of multilocus linkage disequilibrium[END_REF].

For each locus and across all populations, Nei's estimator of genetic differentiation (GST) was calculated with FSTAT 2.9.3 (Goudet 1995). GST indicates the proportion of the total genetic variation attributable to population differentiation. Values close to zero indicate little differentiation while values close to unity indicate nearly complete differentiation. The null hypothesis of no genetic differentiation between field populations was also tested using FSTAT to estimate θ [START_REF] Weir | Cockerham Estimating F-statistics for the analysis of population structure[END_REF], an unbiased estimator of the parameter Fst. Significance levels were determined after Bonferroni corrections based on the adjusted P-value following 8000 permutations. Genetic differentiation was estimated with a hierarchical analysis of molecular variance (AMOVA) using ARLEQUIN 3.1 (Excoffier et al. 2005). AMOVA was used to determine the proportion of genetic variation partitioned among the two regions, among fields within a region, or among individuals within a field. The number of permutations to test for significance was set at 5000.

IBD between populations was tested using the method described by [START_REF] Rousset | Rousset Genetic differentiation and estimation of gene flow from F-statistics under isolation by distance[END_REF], under which a correlation is expected between the logarithm of the geographical distance and the genetic distance θ/ (1-θ), assuming gene flow in a two-dimensional environment. Data analysis involves the regression of FST / (1-FST) estimates for each pair of subpopulations. Pair-wise geographical distances among field populations were calculated without correcting for the Earth's curvature because of the limited spatial distance among fields. Significance was tested using a Mantel test (1000 permutations) implemented in GENEPOP.

A test to detect evidence of recent demographic fluctuations was performed with BOTTLENECK (Piry et al. 1999). For each population and for each locus, we estimated gene diversity as expected under the assumption of mutation-drift equilibrium (expected gene diversity), from the number of alleles and sample size, and compared it to gene diversity calculated directly from the allele frequencies (Hobs, observed gene diversity). In a population that has undergone a bottleneck, gene diversity is higher than expected at equilibrium, due to a high-frequency of rare alleles (Cornuet & Luikart 1996). Gene diversity was estimated under three mutation models: the infinite allele model (IAM), the stepwise mutation model (SMM), and the two-phase model (TPM) (Rienzo et al. 1994). In the IAM, each mutation creates a novel allele at constant rate. In the SMM, each mutation creates a novel allele by adding or deleting a single repeated unit of the minisatellite. TPM is an offshoot of SMM developed to account for a proportion of larger mutation events (addition or deletion of several units). We tested these three mutation models because patterns of mutation for minisatellites are not perfectly known and can vary from one locus to another. The proportion of alleles attributed to SMM under TPM was 70 %, with a variance of 30 (default parameters). Estimations were based on 1000 replications. The Wilcoxon sign-rank test (Luikart et al. 1997) was performed to determine if the allele frequency distribution deviates significantly from that expected under mutation-drift equilibrium. This file is the post-print of : Travadon et al. Fungal Biology (2011) 115: 649-659 Doi: 10.1016/j.funbio.2011.03.009 6

Results

Genetic diversity within populations

Across all 29 populations, the number of alleles at each locus ranged from 3 (MinLm4) to 21 (MinLm632), and only one locus (MinLm4) presented one allele occurring at a frequency greater than 95 %. Average gene diversity across populations ranged from 0.05 (MinLm4) to 0.87 (MinLm2) (Table 2). Mean number of alleles per locus (Nall) was 5.88 (±0.45, standard deviation), ranging from 4.88 to 7 in the Ce11 (Region Centre) and Br14 (Region Brittany) field populations, respectively. Mean gene diversity H across all loci was high (0.66 ± 0.02) and homogeneous (Table 3). Among all 693 isolates analysed, 687 multilocus haplotypes were obtained. Isolates sharing identical multilocus haplotypes were recovered from the same population (Br06, Ce08, and Br16) and likely resulted from clonal spread, as the probability that these repeated multilocus haplotypes originated from distinct sexual reproduction events was low (Psex < 0.005). In contrast, isolates sharing identical multilocus haplotypes were identified from different populations in the same region (Br16-Br06, Br04-Br13) or from different populations in different regions (Ce10-Br15) and likely resulted from distinct sexual reproduction events (Psex > 0.096). Numbers of haplotypes equalled numbers of isolates in 26 field populations (Table 3), and haplotypic richness (R) was high in all field populations (0.96-1). Absence of isolation by distance patterns at the regional scale in the fungal plant pathogen Leptosphaeria maculans. Fungal Biology, 115 (7), 649-659. , DOI : 10.1016/j.funbio.2011.03.009

Table 2. Frequency of the most common (frequency greater than 0.05) minisatellite alleles, gene diversity (h), number of alleles at the eight minisatellite loci, and Nei's estimator of genetic differentiation (GST) per locus across the 29 field populations of

This file is the post-print of : Travadon et al. Fungal Biology (2011) 115: 649-659 Doi: 10.1016/j.funbio.2011.03.009 7

Table 3. Field population code, number of analysed isolates (N), number of observed haplotypes (G), haplotypic richness (R), mean number of alleles over the eight loci (Nall), mean gene diversity over the eight loci (H) and index of multilocus linkage disequilibrium (rd).

a Field population from region Brittany (Br01-Br17) and from region Centre (Ce01-Ce12), geographically localized in Fig 1.

Linkage disequilibrium

Seventeen populations showed random association of all alleles at all loci. Linkage disequilibrium among alleles was not significant in 801 of 812 pair-wise comparisons (98.6 %). The index of multilocus linkage disequilibrium rd was low (|rd| < 0.041) and not significant, and the null hypothesis of random mating was not rejected for all field populations (Table 3). When the complete data set was analysed as a single population, none of the pair-wise comparisons showed evidence of significant linkage disequilibrium.

Genetic diversity among populations

Hierarchical AMOVA revealed that ca. 97 % of the genetic variability was distributed within populations, compared to only 2.56 % among populations within a region, and 0.06 % among regions. None of these three factors contributed significantly to total molecular variance (P > 0.05). Similarly, none of the pair-wise comparisons revealed significant population differentiation among populations (θ = -0.02 to 0.05; P > 0.05), with an average Fst value of 0.003 (±0.004). Fst maximal value (0.05) was reached for the comparison between field populations Ce08 and Br10, two field populations from the two distinct regions sampled, separated by one of the farthest distance in the overall sampling (318 km; Fig 1). Nonetheless, there was no significant correlation between logarithmic geographic distance and genetic distance FST / (1-FST), neither among all populations, nor within regions. Among all populations, the maximum distance between two populations was 364.2 km; the linear regression between logarithmic geographic distance and pair-wise genetic distance was y = 1.13 × 10 -4 x + 0.0015 (r 2 = 1.7 × 10 -4 ; P = 0.35) (Fig 2A). Among populations from Region Brittany, the maximum distance between two populations was 99.6 km; the linear regression between logarithmic

Fig 2. Genetic differentiation among Leptosphaeria maculans populations. Multilocus estimates (θ) of genetic differentiation

Fst expressed as FST / (1-FST) are plotted against logarithm of geographic distance (km) for (A) each pair of L. maculans field populations (maximum distance between two populations is 364.2 km; the regression is y = 1.13 × 10 -4 x + 0.0015; r 2 = 1.7 × 10 -4 ; P = 0.35), (B) each pair of L. maculans field populations from region Brittany (maximum distance between two populations is 99.6 km; the regression is y = -1.5 × 10 -3 x + 0.018; r 2 = 0.016; P = 0.88), and (C) each pair of L. maculans field populations from region Centre (maximum distance between two populations is 122.2 km; the regression is y = 3.4 × 10 -4 x + 2.53 × 10 -5 ; r 2 = 1.5 × 10 -3 ; P = 0.38). Fst was estimated according to [START_REF] Weir | Cockerham Estimating F-statistics for the analysis of population structure[END_REF].

Comment citer ce document : Travadon, R., Sache, I., Dutech, C. C., Stachowiak, A., Marquer, B., Bousset, L. ( 2011). Absence of isolation by distance patterns at the regional scale in the fungal plant pathogen Leptosphaeria maculans. Fungal Biology, 115 (7), 649-659. , DOI : 10.1016/j.funbio.2011.03.009

This file is the post-print of : Travadon et al. Fungal Biology (2011) 115: 649-659 Doi: 10.1016/j.funbio.2011.03.009 9

Effective population size fluctuations Comparisons between observed and expected gene diversity revealed significant gene diversity excess for 20 of 29 populations, under TPM (Wilcoxon sign-rank test, P < 0.05; Table 4) and IAM (P < 0.05). Conversely, under SMM, none of the populations displayed significant gene diversity excess (P > 0.05). For the three models, no differences were found between regions. Table 4. Probability for an excess in observed gene diversity (Hobs) compared to that expected under mutationdrift equilibrium, in each field population, tested with Wilcoxon sign-rank test, under three models: the IAM, the TPM, with 70 % of alleles attributed to SMM and the SMM. P-values after 1000 simulation replicates are presented.

Discussion

In characterizing the genetic structure of Leptosphaeria maculans populations in France, we intended to demonstrate that, in the case of this wind-dispersed phytopathogenic fungus, the scale of spore dispersal could be obtained from the theoretical migration model IBD. Despite the fact that our sampling scheme was based on a priori knowledge of spore dispersal, however, we did not detect a significant IBD pattern among populations. We will discuss how deviations from model assumptions occurring in populations, including departure from genetic equilibrium or gene flow higher than previously assumed, are likely to explain the inapplicability of the IBD model at the spatial scale studied.

The detection of IBD pattern strongly depends on the spatial scale studied, as the studied area should match the spatial scale at which gene dispersal occurs [START_REF] Rousset | Rousset Genetic differentiation and estimation of gene flow from F-statistics under isolation by distance[END_REF]) and sampling at larger distances leads to an overestimation of demographic parameters (Leblois et al. 2003). Our sampling scheme was designed assuming that migration was due to ascospore dispersal over a few hundred metres, distances estimated from Absence of isolation by distance patterns at the regional scale in the fungal plant pathogen Leptosphaeria maculans. Fungal Biology, 115 (7), 649-659. , DOI : 10.1016/j.funbio.2011.03.009

This file is the post-print of : Travadon et al. Fungal Biology (2011) 115: 649-659 Doi: 10.1016/j.funbio.2011.03.009 10 spore [START_REF] Salam | Spatial spread of blackleg in canolaa regional scale simulation model[END_REF]disease (Marcroft et al. 2004) gradients. As both methods are known to be inaccurate at large distances, over several kilometres (McCartney et al. 2006), it is possible that our sampling scale was too small, and that we underestimated ascospore dispersal. Our findings suggest that ascospore dispersal occurs at distances greater than several hundred metres. Undetected by direct methods, widespread spore dispersal at continental scale by atmospheric turbulence could occur for this fungus. Such long-distance dispersal events are known to be the main cause of the world-wide spread of plant diseases (Brown & Hovmøller 2002). But assuming that these events have a low-frequency compared to local dispersal, based on reduced fungal spore viability under atmospheric conditions (Rotem et al. 1985), an IBD pattern should be detectable at the studied regional spatial scales.

Alternatively, it is possible that human transport of infected seeds is responsible for long-distance dispersal of L. maculans (Hall et al. 1996). Long-distance dispersal via human transport and, thus, high migration rates that reduce genetic differentiation between populations [START_REF] Slatkin | Slatkin Gene flow and the geographical structure of natural populations[END_REF], were suggested to explain the world-wide spread of other plant pathogens, such as the banana black leaf streak fungus Mycosphaerella fijiensis (Rivas et al. 2004), the wheat leaf blotch fungus Mycosphaerella graminicola (Linde et al. 2002), and the poplar rust fungus Melampsora larici-populina (Barrès et al. 2008). Nonetheless, long distance human-mediated transport of infected plant material would have to be regular and intense to erase population differentiation, which is locally driven by genetic drift in the absence of selection. Further investigations of infected seed material are necessary to elucidate its effect on the spread of L. maculans.

The absence of demographic equilibrium in the studied populations may have prevented the detection of IBD patterns, though empirical data showed that demographic estimates through the method used were robust to population density changes that occur within just a few generations (Leblois et al. 2004). In Region Brittany, oilseed rape acreage increased from 0 ha in 1983 to ca. 50 000 ha in 2007 [START_REF] Agreste | Tableaux de l'agriculture bretonne Direction régionale de l'agriculture et de la forêt de Bretagne[END_REF]), yet a matching increase in effective population sizes was not detected in this region. As the test implemented in the BOTTLENECK software only allows the detection of recent size fluctuations, pathogen spread in Brittany, which possibly dates back from about 25 generations ago assuming one sexual cycle per season, may be too ancient for detection. In Region Centre, where oilseed rape has been intensively grown for more than 60 y, we might expect that coevolving L. maculans populations have reached demographic equilibrium. Nonetheless, annual, recurrent bottlenecks could hinder the detection of IBD patterns. For annual crop pathogens, this decrease in effective population size is expected when host plant material available for pathogen development dramatically decreases between growing seasons (Burdon 1993). However, the high levels of within-population genetic diversity revealed in this study, concordant with previous studies (Mahuku et al., 1997, Barrins et al., 2004, Gout et al., 2006, Hayden et al., 2007), do not support recent or recurrent bottlenecks. Furthermore, with SMM, considered the most conservative mutation model in the test conducted (i.e. the null hypothesis of no population size fluctuation is rejected only when a strong bottleneck signal is detected; Cornuet & Luikart 1996), none of the populations showed significant recent changes in size. Multiple sampling dates and years are necessary to characterize the genetic diversity of L. maculans populations and to further examine their local and temporal demography.

The absence of both bottleneck signatures and significant spatial genetic structure among French populations of L. maculans suggest that these populations do not undergo strong loss of genetic diversity through genetic drift during the saprophytic stage on oilseed rape stubble. Instead, high proportion of genetic diversity distributed within-population (>97 %) suggests that the examined populations have large effective population sizes. Large effective population sizes reduce genetic differentiation among populations (Criscione et al. 2005) and hence limit the chance to detect IBD patterns. The substantial quantities of stubble that is not buried, but is instead left on the soil surface (Lydia Bousset, personal observation), can explain the maintenance of large effective population size in L. maculans.

The absence of linkage disequilibrium and the uniform distribution of mating types in all populations (data not shown) confirm the significance of sexual reproduction in the life-cycle of L. maculans, and are consistent with previous research identifying ascospores, and not conidia (i.e. spores resulting from asexual multiplication), as the main source of inoculum to initiate epidemics (Fitt et al. 2006). Nonetheless, three repeated haplotypes were detected in the same field and were unlikely to originate from distinct sexual reproduction events. These findings provide indirect evidence for the spread of isolates through splashdispersal of conidia in field conditions, hence supporting the involvement of conidia in the epidemiology of phoma stem canker on oilseed rape [START_REF] Travadon | Sache Splash dispersal of Leptosphaeria maculans pycnidiospores and the spread of blackleg on oilseed rape[END_REF]. In contrast, similar associations of alleles in repeated haplotypes detected in distant fields were likely the result of random mating. Overall, genetic diversity indices estimated in all field populations are in agreement with the hypothesis that, in France, populations of L. maculans operate as one panmictic population.

As discussed in this study, the absence of IBD patterns in a wind-dispersed, plant-pathogenic fungus may not be only due to dispersal of spores or infected plants across large geographic distances. Instead, a focus on the inter-generation evolution of local genetic diversity may reveal the contribution of large effective population sizes and the absence of demographic equilibrium as factors that mask IBD patterns. Buffering the effects of genetic drift, the effective population size has a large influence on overall levels of genetic diversity (Criscione et al. 2005). For plant-pathogenic fungi not experiencing strong effects of genetic drift during their life-cycle, either because of constant host availability (i.e. perennial hosts), or because of their ability to survive saprophytically between host growing seasons, or due to their ability to infect multiple hosts, and thus to survive independently of one single host persistence, high levels of within-population genetic diversity and low levels of among-population differentiation are expected. In these cases the IBD model does not seem appropriate to infer dispersal distances.

Large effective population sizes, high dispersal rates and random mating confer a high evolutionary potential to L. maculans (McDonald & Linde 2002), and such characteristics are consistent with its rapid adaptation to major resistance genes [START_REF] Sprague | Howlett Major gene resistance in Brassica napus (oilseed rape) is overcome by changes in virulence of populations of Leptosphaeria maculans in France and Australia[END_REF]. Since effective population sizes affect local adaptive potential to evolve in response to host resistance, it is necessary to reduce effective population size in L. maculans populations. Accordingly, management practices between growing seasons, especially burial of the oilseed rape stubble from which the pathogen fruits, is recommended [START_REF] Aubertot | Diggle Improved resistance management for durable disease control: a case study of phoma stem canker of oilseed rape (Brassica napus)[END_REF]. Disease control strategies aiming at either the prevention of virulent spore dispersal from resistant crops to new resistant crops through spatial cultivar deployment, or at the reduction of sexual reproduction via stubble management, are essential to preserve the efficacy of the few available specific resistance genes against L. maculans in B. napus.

  autumn 2004 and 2005, 693 isolates were collected from 29 commercial oilseed rape fields in two regions [western France (Region Brittany, n = 17) and central France (Region Centre, n = 12) (Fig 1)]. In 2007, 66 405 ha of oilseed rape were grown in the Department Cher (Region Centre), compared to only 12 950 ha in the Department Ille-et-Vilaine (Region Brittany) (Prolea 2008). Pair-wise distances between field populations ranged from 0.44 km to 364 km (from 0.44 km to 99.7 km and from 1.32 km to 114.5 km in Region Brittany and in Region Centre, respectively). In each field, one infected leaf displaying typical phoma leaf spots was collected from each of 23-25 plants. Monospore mycelial cultures were increased on V8 juice agar at 20 °C for 21 d (Chèvre et al. 2008), then scraped with a sterile scalpel. DNA was extracted from lyophilized mycelia following the standard protocol supplied in the Nucleospin ® Plant II kit (Macherey-Nagel, Düren, Germany).

Fig 1 .

 1 Fig 1. Map of France showing the location of the 29 French field populations sampled (○) in Regions Brittany and Centre. The main towns in the sampling areas are marked with a × on the map.

  Leptosphaeria maculans sampled in two regions of France. a Eckert et al. 2005. b GenBank database accession number. Comment citer ce document : Travadon, R., Sache, I., Dutech, C. C., Stachowiak, A., Marquer, B., Bousset, L. (2011).

  Comment citer ce document : Travadon, R., Sache, I., Dutech, C. C., Stachowiak, A., Marquer, B., Bousset, L. (2011). Absence of isolation by distance patterns at the regional scale in the fungal plant pathogen Leptosphaeria maculans. Fungal Biology, 115 (7), 649-659. , DOI : 10.1016/j.funbio.2011.03.009

  Leptosphaeria maculans. a Alleles are scored as the number of repeat units. b Nei's estimator of genetic differentiation. Comment citer ce document : Travadon, R., Sache, I., Dutech, C. C., Stachowiak, A., Marquer, B., Bousset, L. (2011).

  Comment citer ce document : Travadon, R., Sache, I., Dutech, C. C., Stachowiak, A., Marquer, B., Bousset, L. (2011).

Acknowledgements

This work was supported by « ANR-Agence Nationale de la Recherche -The French National Research Agency » program « Agriculture et Développement Durable », grant « ANR-05-PADD-05, CEDRE », and by a grant from INRA (Divisions Santé des Plantes et Environnement and Écologie des Forêts, Prairies et milieux Aquatiques: Projet Épidémiologie et dynamique des populations). A. Stachowiak was funded by Marie-Curie FellowshipHPMT-CT-2001-00395 'FUNGENE'. We thank K. Baumgartner (USDA-ARS, Davis, CA) for review of this manuscript.

Version postprint

Comment citer ce document : Travadon, R., Sache, I., Dutech, C. C., Stachowiak, A., Marquer, B., Bousset, L. (2011). Absence of isolation by distance patterns at the regional scale in the fungal plant pathogen Leptosphaeria maculans. Fungal Biology, 115 (7), 649-659. , DOI : 10.1016/j.funbio.2011.03.009

This file is the post-print of : Travadon et al. Fungal Biology (2011) 115: 649-659 Doi: 10.1016/j.funbio.2011.03.009 12 Balesdent et al., 2001 M.H. Balesdent, A. Attard, D. Ansan-Melayah, R. Delourme, M. Renard, T. Rouxel Genetic control and host range of avirulence toward Brassica napus cultivars Quinta and Jet Neuf in Leptosphaeria maculans Phytopathology, 91 (2001), pp. 70-76 Barrès et al., 2008 B. Barrès, F. Halkett, C. Dutech, A. Andrieux, J. Pinon, P. Frey Absence of isolation by distance patterns at the regional scale in the fungal plant pathogen Leptosphaeria maculans. Fungal Biology, 115 (7), 649-659. , DOI : 10.1016/j.funbio.2011.03.009

This file is the post-print of : Travadon et al. Fungal Biology (2011) 115: 649-659 Doi: 10.1016/j.funbio.2011.03 Absence of isolation by distance patterns at the regional scale in the fungal plant pathogen Leptosphaeria maculans. Fungal Biology, 115 (7), 649-659. , DOI : 10.1016/j.funbio.2011.03.009

This file is the post-print of : Travadon et al. Fungal Biology (2011) 115: 649-659 Doi: 10.1016/j.funbio.2011.03.009


Tomas Lagos Jenschke

Georgios Z Papadopoulos

Remous-Aris Koutsiamanis

Nicolas Montavont

Strat égies d'anc être commun pour les r éseaux RPL multi-chemins

Keywords: Multi-path, RPL, PRE, Determinism, LeapFrog Colaboration, LLN

Le protocole de routage IPv6 pour les réseaux à faible puissance et fort taux de pertes (RPL) est conc ¸u pour les réseaux Internet des objets (IoT) afin de générer des itinéraires entre les appareils avec un traitement minimal. Ce protocole crée une topologie de réseau DODAG (Destination Oriented Directed Acyclic Graph) grâce à l'utilisation de paquets de contrôle DODAG Information Object (DIO). Le DODAG achemine les paquets de données en amont vers le périphérique de destination. Afin d'obtenir un réseau fiable, nous implémentons la réplication et l'élimination des paquets (PRE) pour effectuer une transmission de données à chemins multiples via plusieurs périphériques parents. Cependant, il n'existe aucun moyen standard de sélectionner un chemin alternatif. Ce document présente trois types de sélection de parent alternatif (AP) suivant un modèle triangulaire. Nous nous concentrons sur l'analyse de ses performances en termes de retard et de compromis entre trafic réseau et fiabilité.

Introduction

The use of Low Power and Lossy Network (LLN) technologies has impacted existing production modes within industry, influencing the automation of process chains. However, the quality of service that is required by Industry 4.0 conflicts with the nature of wireless technologies. This is because the industry is especially concerned about the reliability and determinism of a network, while wireless technologies are generally oriented to best effort. To adapt to these requirements, several standards and tools have been established. An existing standard that adapts wireless technologies to the industry is IPv6 Routing Protocol for Low-Power and Lossy Networks (RPL) [START_REF] Winter | RPL: IPv6 Routing Protocol for Low-Power and Lossy Networks[END_REF]. This routing protocol creates a hierarchic network called a Destination Oriented Directed Acyclic Graph (DODAG), where all the upstream traffic goes to the DODAG root. In this paper, we propose the use of multi-path routing in a RPL network to provide more opportunities for a data packet to reach its destination using the least number of re-transmissions. We evaluate three types of Alternative Parent (AP) selection algorithms taking as reference the Braided pattern [START_REF] Minet | Increasing Reliability of a TSCH Network for the Industry 4.0[END_REF] and the LeapFrog Collaboration (LFC) [START_REF] Papadopoulos | Leapfrog Collaboration: Toward Determinism and Predictability in Industrial-IoT applications[END_REF], [START_REF] Koutsiamanis | Meet the PAREO Functions: Towards Reliable and Available Wireless Networks[END_REF], [START_REF] Koutsiamanis | From Best-Effort to Deterministic Packet Delivery for Wireless Industrial IoT Networks[END_REF] algorithm.

Background

IPv6 Routing Protocol for Low-Power and Lossy Networks (RPL)

RPL is a distance vector protocol that generates link routes between LLN devices. This protocol comprises a hierarchical topology in which all traffic flows to the Root node. RPL uses three types of control packets: the DODAG Informational Solicitation (DIS), the Destination Advertisement Object (DAO), and the DODAG Information Object (DIO). The last one of the three (the DIO) is the most relevant for our work, since it carries the information required to incorporate a node to its DODAG. We add new information to these packets to enable the operation of braided multi-path routing.

Packet Replication and Elimination (PRE)

Packet Replication and Elimination (PRE) aims to improve the reliability of a network by transmitting multiple copies of the same packet through multiple routes, i.e., the Replication function. To avoid redundant information, PRE only forwards the first packet that was received. The rest of the packet copies that arrive later are dropped, i.e., the Elimination function. Figure 1: AP selection: Matching PPs (Fig. 1a), LFC (Fig. 1b), intersecting parent sets (Fig. 1c).

Problem Statement

Problems considered in LLN devices include their bounded power and their susceptibility to packet loss in transmissions. Given the multi-path policy, transmission reliability is improved but at the cost of more energy consumption. Therefore, a careful selection of multiple parents is required. In this section, we analyze the LFC algorithm, comparing it in scenarios that were missing or that were not taken into consideration in the original exposition of the algorithm [START_REF] Papadopoulos | Leapfrog Collaboration: Toward Determinism and Predictability in Industrial-IoT applications[END_REF].

Topology

In [START_REF] Papadopoulos | Leapfrog Collaboration: Toward Determinism and Predictability in Industrial-IoT applications[END_REF], the authors work with a topology with two parents per node. This environment limits the use of multi-path in RPL because only one node will be available to be an AP.

With LFC (also called Medium Common Ancestor (CA), see section 4), a node selects an AP if one element of its Parent Set (PS) is the parent of its Preferred Parent (PP) (i.e., its Preferred GrandParent (PGP)), as illustrated in Fig. 1b. If the node has more than two potential parents, then it can happen that the AP is not a node close to the PP, but farther away. As result, the AP may lead to the alternative path diverging far away from the preferred (i.e., main) path.

Flooding

The goal of LFC is to improve the reliability of a network through transferring copies of the same packet through its PP and an AP. If it is not possible to control the expansion of routes due to the use of multi-path, the number of copies will generate flooding in the network. It should be taken in consideration that LFC also uses overhearing and re-transmission, thus increasing the network traffic overhead of the approach.

Contribution

Pa re nt se t si ze 1 2 3 4 5 6 7 8 9 10 P S M C se t si ze In this paper, we define three types of algorithms to obtain an AP based on the braided pattern [START_REF] Jenschke | Alternative parent selection for multi-path rpl networks[END_REF]. In a nutshell, the braided pattern selects a node as an AP, if it has a Common Ancestor (CA) with the PP. As illustrated in Fig. 1, three algorithms are defined by their level of restrictions to select this AP. The nodes report a subset of their parent set in the PS extension [START_REF] Koutsiamanis | Common Ancestor Objective Function and Parent Set DAG Metric Container Extension[END_REF] of the Node State and Attribute (NSA) object in the Metric Container (MC) of DIO messages (referred to as the PS MC parent set here on). This PS MC extension contains a fixed number of addresses M, while the number of parents in the parent set is N, with 1 ≤ M ≤ N. The result of reporting just a subset of all the PS, instead of the whole set, is that AP selection becomes probabilistic.

The common general equations used to derive the probability P(∃CA) of having a CA and the probability P(∃AP) of having an AP in the next steps are:

P( AP) = (1-P(∃CA)) N-1 ⇒ P(∃AP) = 1-P( AP) = 1-(1-P(∃CA)) N-1
(1) Given a node u that intends to select an and a candidate AP v, the strict CA algorithm is characterized by selecting its AP if this one has the same PP as the actual PP of the node, i.e., PP(PP(u)) = PP(v). Fig. 1a illustrates the selection of an AP by using strict CA algorithm, where E will be selected as an AP since A is PP of E and D, thus the probability of finding an AP is represented by Equation 2.

P(∃CA) = 1 N (2)
where P(∃CA) is the probability of having at least one CA and P(∃AP) is the probability of having at least one AP (the complement of probability P( AP) of not being able to find any APs). Given these, the probability of finding an AP is calculated in Equation 3 and represented in Fig. 2a.

P(∃AP) = 1- N -1 N N-1 (3)
The medium CA algorithms selects a node as AP if this one has within its neighbor set the preferred grandparent of the actual node. This algorithm is represented in Fig. 1b, where E will be selected as an AP because it has A within its neighbors set. It is worth mentioning that A does not have to be selected as a parent, but that it only exists within the neighbors set. The probability of having an AP is given by Equation 4.

P(∃CA) = M N (4)
Using Equation 1, we obtain Equation 5where its values are represented in Fig. 2b.

P(∃AP) = 1- N -M N N-1 (5)
The soft CA aims to select an AP if this node has a neighbor that exists within the set of neighbors of the current PP. As shown in Fig. 1c, E will be selected as AP since its B is also a neighbor of D.

In Equation 6, the probability P(∃CA) of obtaining a CA depends on the probability of having a common node in the two PS MC parent sets via the PP and via the candidate AP.

P( CA) = N M • N-M M N M • N M = N-M M N M ⇒ P(∃CA) = 1-P( CA) = 1- N-M M N M (6)
Finally, using Equation 1, the probability of finding an AP is represented by Equation 7and is illustrated in Fig. 2c.

P(∃AP) = 1- N-M M N M N-1 (7)

Simulation setup

The simulations were executed using the COOJA simulator developed by as part of the Contiki OS. We take the topology of [START_REF] Jenschke | Alternative parent selection for multi-path rpl networks[END_REF], which consists of a matrix of 5x6 nodes, except for the source node and the root node located at the vertical ends of the topology. The routes are established vertically (from every node in layer l to every node in layer l -1) using RPL, while the sibling set corresponds to the nodes located horizontally (every node n in layer l has every other node at the same layer l as a sibling). Simulations were performed with several re-transmissions (RT) for single path (i.e., RT n where n is the number of re-transmissions and n+1 is the number of total transmissions), while for multi-path they were performed with RT1. To have a realistic set-up, 1000 data packets were transmitted with uniformly random link quality variation between 70% and 100%.

Performance results

In terms of PDR, the probability of having an AP directly affects performance, since a greater number of forwarding nodes leads to a larger number of packet copies, which in turn increases reliability. On the other hand, a greater number of re-transmissions also leads to an improvement in the PDR. This can be seen in Fig. 3a where RT4 and RT8 reach almost 100% and in the ascending results of each CA algorithm. In terms of delay, it is observed that increasing the number of re-transmissions leads to an increase of delay. This is due to the number of slotframes required for each re-transmission and, therefore, an increase in the delay, as it can be seen in Fig. 3b. Finally, as shown in Fig. 3c, the use of multi-path will increase the traffic of data packets depending on how strict the algorithm is, more precisely, between the strict CA and medium CA algorithms. This is because the probability of obtaining an AP in Strict CA is much lower than the probabilities that have Medium CA and Soft CA, which means a smaller number of nodes to use.

Conclusions

For the topology analyzed, the minimum probability of having an AP is 60%, increasing as the number of parents increases and as the number of addresses contained in the PS extension of the NSA metric in DIO messages increases. We identified that although the overall network power consumption is higher for our algorithms in comparison to the default single-path algorithms, the trade-off presented is useful for the intended industrial network performance applications. Therefore the trade-off must be balanced based on the requirements of a given system.

1

 1

Figure 2 :

 2 Figure 2: Probability of finding an AP through common ancestors vs. the PS size N and the PS MC set size M.

Figure 3 :

 3 Figure 3: Performance evaluation