Alex Mercier

Florence Carpentier

Clémentine Duplaix

Annie Auger

Jean-Marc Pradier

Muriel Viaud

Pierre P Gladieux

Anne-Sophie Walker
email: anne-sophie.walker@inra.fr

The polyphagous plant pathogenic fungus Botrytis cinerea encompasses host-specialized and generalist populations

Keywords: Host specialization, grey mould, population structure, molecular epidemiology, crosspathogenicity, gene flow

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Introduction

Fungi are ubiquitous in ecosystems, but most fungal species are not globally distributed and display some kind of population structure [START_REF] Taylor | Eukaryotic microbes, species recognition and the geographic limits of species: examples from the kingdom Fungi[END_REF]. In fungal plant pathogens, investigating population structure in relation to host range is particularly relevant, as it is essential for disease management to recognize the different populations of a pathogen, which can have different life history traits, such as resistance to fungicide or phenology [START_REF] Milgroom | Population biology of plant pathogens: the synthesis of plant disease epidemiology and population genetics[END_REF]. Beyond the practical interest in understanding the structure of pathogen populations, studying barriers to gene flow between populations within species may reveal key features of adaptive divergence before they become confounded by other factors. The host plant is often the main variable explaining population subdivision in fungal plant pathogens, because specialization to hosts contributes to reduce gene flow between populations associated with different hosts. Host specialization acts against gene flow due to the reduced viability of migrants and hybrids, and the role of specialization as a barrier to gene flow is reinforced in species that reproduce within their hosts, which is the case for many plant pathogens [START_REF] Giraud | Linking the emergence of fungal plant diseases with ecological speciation[END_REF][START_REF] Giraud | Importance of the life cycle in sympatric host race formation and speciation of pathogens[END_REF]Gladieux et al., 2010). Hence, polyphagous fungal pathogens can form collections of specialist populations, each associated with a limited number of hosts (i.e. polyspecialists, such as the rice blast fungus Pyricularia oryzae; [START_REF] Gladieux | Gene flow between divergent cereal-and grass-specific lineages of the rice blast fungus Magnaporthe oryzae[END_REF][START_REF] Valent | Pyricularia graminis-tritici is not the correct species name for the wheat blast fungus: response to Ceresini et al. (MPP 20:2)[END_REF]. Depending on time-based, geographic and genetic factors [START_REF] Nosil | Ecological explanations for (incomplete) speciation[END_REF], speciation can then unfold relatively rapidly between pathogen populations [START_REF] Giraud | Linking the emergence of fungal plant diseases with ecological speciation[END_REF]. The phytopathological literature provides many examples of adaptive radiations [START_REF] Schluter | The ecology of adaptive radiation[END_REF] in pathogens, with many sibling pathogen species associated with distinct hosts [START_REF] Giraud | Importance of the life cycle in sympatric host race formation and speciation of pathogens[END_REF][START_REF] Tellier | Theory and examples of reciprocal influence between hosts and pathogens, from short-term to long term interactions: coevolution, cospeciation and pathogen speciation following host shifts[END_REF]. Alternatively, in some pathogens, infection of a novel host is not associated with a specialization process, and merely results in the expansion of the host range of a generalist species. The domestication of plant pathogens, i.e. the effective and durable control over pathogen reproduction by humans, remains a major goal in agricultural sciences and knowledge of the intrinsic and extrinsic factors underlying the emergence of new populations specialized onto new hosts -or expansions in host range in generalist species-should help reaching this goal.

Botrytis cinerea (Ascomycota) is a textbook example of a highly polyphagous fungal plant pathogen, causing grey mold on more than 1400 known hosts, in 586 plant genera and 152 botanical families, including high-value crops such as grapevine and tomato (polyphagy index = 54 ; Elad et al., 2016; Supplementary Information 1). B. cinerea is a necrotrophic pathogen that asexually produces conidia that form a sporulating grey layer on the surface of infected tissues. Conidia are mainly dispersed during spring and summer by wind and human activity. During winter, the fungus develops as a saprobe on plant debris, where it can proceed to undergo sexual reproduction and form melaninpigmented sclerotia [START_REF] Carisse | Epidemiology and aerobiology of Botrytis spp[END_REF]. The analysis of population structure of B. cinerea populations infecting wild (e.g. [START_REF] Giraud | Two sibling species of the Botrytis cinerea complex, transposa and vacuma, are found in sympatry on numerous host plants[END_REF][START_REF] Rajaguru | Genetic differentiation between hosts and locations in populations of latent Botrytis cinerea in southern England[END_REF] or cultivated (e.g. [START_REF] Karchani-Balma | Geography, plants and growing systems shape the genetic structure of Tunisian Botrytis cinerea populations[END_REF][START_REF] Leyronas | Assessing host specialization of Botrytis cinerea on lettuce and tomato by genotypic and phenotypic characterization[END_REF] host plants revealed differentiation between sympatric populations from different hosts. However, no correlation was found between population structure and crosspathogenicity, suggesting that the observed differentiation between populations was not related to host specialization, or that it was instead linked to components of specialization that have not been measured. For example, Leyronas et al. (2015) showed lack of genetic differentiation between B. cinerea populations found on tomato (Solanum lycopersicum) and lettuce (Lactuca sativa). Patterns of pathogenicity were also consistent with generalism, as strains showed significant differences in quantitative pathogenicity on tomato hosts, but not on lettuce. Recently, we have found strong genetic differentiation between isolates collected in the greenhouse on tomato on the one hand and outdoor on bramble (Rubus fruticosus) and grapevine (Vitis vinifera) on the other hand, suggesting possible specialization of the pathogen to these hosts [START_REF] Walker | Population structure and temporal maintenance of the multihost fungal pathogen Botrytis cinerea: causes and implications for disease management[END_REF]. Here, we test the hypothesis of host specialization in B. cinerea, by further investigating the genetic structure of B. cinerea populations sampled in four French regions on sympatric hosts, including previously tested hosts (e.g. grapevine, tomato and bramble ;[START_REF] Walker | Population structure and temporal maintenance of the multihost fungal pathogen Botrytis cinerea: causes and implications for disease management[END_REF] and new hosts (e.g. strawberry -Fragaria x ananassa-and hydrangea -Hydrangea macrophylla), and by measuring quantitative pathogenicity on various hosts using cross-pathogenicity testings. More specifically, we addressed the following questions: (i) What is the population structure of B. cinerea on grapevine, tomato, bramble, strawberry and hydrangea in France? (ii) Are there differences in qualitative or quantitative pathogenicity between identified populations? (iii) Are patterns of population structure and pathogenicity consistent with the specialization of B. cinerea to some hosts? P-values in bold are significant at the 5% confidence level. d.f., degrees of freedom.

Results

Population subdivision

We characterized 683 isolates from 5 hosts in 15 sites representing 4 areas of France, using 8 microsatellite markers. To examine population structure, we first characterized the partitioning of genetic variation among different factors (host plant and area) using hierarchical analyses of molecular variance (AMOVA) (Error! Reference source not found.). With samples grouped by host, variation among hosts was found highly significant (P<0.001; F ST =0.18) and accounted for 18.4% of the molecular variance. Variation due to geography was also found significant but accounted for only 5.5% of molecular variance (P<0.001; F ST =0.06). With samples grouped by host and nested within areas, variation within populations accounted for most of the molecular variance (77,7%; P<0.001; F ST =0.07). Variation among hosts, and among hosts within areas were significant (P<0.001) and accounted for 16% and 6% of the molecular variance, respectively. Variation among areas was not significant (P=0.772), but variation among areas within hosts was significant (P<0.001) and accounted for 23.1% of the molecular variation, respectively. These results suggest that hosts (tomato, grapevine, strawberry, bramble and/or hydrangea) and areas (Alsace, Champagne, Loire Valley and Provence) explain a significant proportion of the genetic variance in the dataset, with a stronger effect of the host. We investigated patterns of population subdivision using (i) the Bayesian clustering method implemented in STRUCTURE, assuming a model with admixture and correlated allele frequencies, (ii) the non-parametric multivariate clustering method implemented in the discriminant analysis of principal components (DAPC), and (iii) the maximum likelihood clustering method implemented in SNAPCLUST (Figure 1). The three methods converged towards the same clustering solutions at all K value. Models with K=5 captured the most salient features of the structure of the dataset, with two clusters mostly associated with grapevine and tomato, respectively, and three clusters found on all hosts but tomato. Increasing K above 5 did not reveal new clusters but mostly introduced heterogeneity in clustering patterns (see also Supplementary Information 4 & 5). We calculated the mean pairwise F st between clusters for each K value (Supplementary Information 3) and at K=5 the F st is close to its maximum and displays very close values across all three clustering methods. The ΔK Evanno method (Supplementary Information 6) favors K=4, while goodness of fit statistics (Supplementary Information 7) favors K=5. All subsequent analyses assumed K=5 and isolates were assigned to clusters using the membership proportions estimated with STRUCTURE.

Using membership proportions greater than 0.7 to assign multilocus genotypes, clusters tended to be associated with one main host (Figure 2A). Clusters C1 and C2 were almost exclusively associated with tomato and grapevine, respectively (96% of isolates from C1 were collected on tomato; 84% of isolates from C2 collected on grapevine). The three other clusters were less specific, with the majority of C3 isolates collected on grapevine (53%), and the majority of C4 and C5 isolates collected on bramble (50 and 60%, respectively). Isolates from strawberry were mostly distributed in C3 and C4, whereas isolates collected from hydrangea were mostly assigned to C4 and C5. Pairwise Jost's D between clusters inferred with STRUCTURE were all significantly different from zero and ranged between 0.320 and 0.869 at K=5 ( Tomato, grapevine, grapevine, bramble and bramble were found as the main hosts for C1, C2, C3, C4 and C5, respectively (Error! Reference source not found.). Isolates were assigned into clusters when their membership coefficient was greater than 0.7 for a given cluster. Nonassigned isolates weren't used in this analysis. *indicates that the pairwise D is significant at the 5% confidence levels.

Patterns of genetic diversity and reproductive mode

We investigated how genetic variability varied across the clusters defined at K=5, by estimating the mean number of alleles per locus Ar, genic diversity He, and Shannon's diversity index H (Figure 2B). The tomato-associated cluster C1 was the least variable (Ar=2.12; He=0.26), while the bramble-associated cluster C4 was the most variable (Ar =11.45; He=0.77). Intermediate levels of variability were observed for the grapevine-associated clusters C2 and C3 (Ar=5.31 and 6.57; He=0.51 and 0.69) and the other bramble-associated cluster C5 (Ar=4.01; He=0.44). Multilocus genotypes were evenly represented in clusters C1, C2, C3 and C5 (1.18<H<1.5), but not in C4 (H=0.37).

To investigate the reproductive mode of B. cinerea, we computed the proportion of genotypes repeated multiple times (clonal fraction 1-G/N) and we estimated multilocus linkage disequilibrium (ݎҧ D ; Figure 2A). The highest clonal fraction was found in the tomato-associated cluster C1 (CF=0.75), which also displayed a relatively high level of linkage disequilibrium among markers (ݎҧ D = 0.18). Clonal fraction was low and in the same order of magnitude (range: 0.11-0.26) across other clusters. ݎҧ D values were also low for these clusters (range: 0.008-0.20), although significantly different from zero.

Cross-pathogenicity tests

On leaves. We assessed differences in aggressiveness between clusters using cross-pathogenicity tests on strawberry, grapevine, bramble, tomato and hydrangea leaves, with bean used as a naive host (no isolates were sampled on this host). Mean necrosis diameters are reported in Figure 3 and differences between pairs of hosts of origin are presented in Each boxplot represents a group of 7-33 isolates collected on strawberry (pink), hydrangea (blue), bramble (purple), tomato (red), grapevine (green) and a reference isolate used as a control (grey). These isolate subgroups were tested on leaves from strawberry, hydrangea, bean, bramble, tomato and grapevine (test hosts). Lesions were scored between 2 and 9 days after inoculation, depending on the test host.

. , *, ** and *** indicates tests of difference of mean necrosis diameter between two groups of isolates with P-value lower than the 10%, 5%, 1% and 1‰, respectively.

On strawberry leaves, the aggressiveness of isolates collected on hydrangea was significantly reduced (range: -0.75 --4.47mm) compared to isolates collected on grapevine, for three out of the four scoring dates. Aggressiveness of other groups of isolates on strawberry were similar to each other. Isolates from strawberry did not exhibit greater growth of necrosis on this host compared to other hosts.

On hydrangea leaves, no pairwise differences between isolates collected from different hosts were observed, including for isolates collected on this host.

On bramble leaves, isolates collected from bramble did not exhibit the best performance, at any scoring time. Isolates from tomato were more aggressive on bramble than both grapevine isolates (range of mean diameter necrosis: 0.36-12.7 mm; significant at all scoring times) and bramble isolates (range of mean diameter necrosis: 0.2-11.80; significant for 3 out of 6 scoring times). On tomato leaves, isolates collected from this host exhibited the highest mean aggressiveness, for the three latest scoring dates. Aggressiveness of isolates collected on bramble and grapevine was significantly reduced on tomato (range of mean diameter necrosis: -0.25 --13.20 mm for bramble isolates, -0.33 --13.60 mm for grapevine isolates; statistically significant at all dates but the earliest), compared to isolates collected on tomato. Isolates from bramble and grapevine hosts performed similarly on tomato leaves.

On grapevine leaves, aggressiveness of isolates was highly variable, as indicated by the high proportion of boxplot outliers. No significant difference in aggressiveness was observed among isolates from different hosts on grapevine, but the biggest mean necrosis diameters were measured for isolates originating from grapevine at the two latest scoring times. In summary, the results of cross-pathogenicity tests support host specialization of B. cinerea to tomato, and to a lower extent, to grapevine; the differences in aggressiveness between B. cinerea populations associated with tomato and grapevine were also observed on bramble leaves. On berries. We assessed differences in aggressiveness between clusters using pathogenicity tests on tomato berries after 16 days of inoculation and grapevine berries, respectively after 16 and 12 days. Necrosis percentages for each cluster are reported in Error! Reference source not found. and differences between pairs of hosts of origin are presented in Table 4. Our statistical model detected a significant effect of the host of origin. On tomato berries, isolates collected from this host exhibited the highest mean aggressiveness (46.6% of mean necrosis). Aggressiveness of isolates collected on bramble (28% of mean necrosis) was significantly reduced on tomato compared to isolates collected on tomato. No significant difference was detected with isolates collected on grapevine (38.1% of mean necrosis). Each boxplot represents a group of 16-34 isolates collected on bramble (purple), tomato (red), grapevine (green), a reference isolate used as a control (grey) and a not inoculated control (NIC, black). These isolate subgroups were tested on tomato and grapevine berries (test hosts) and percentage of necrosis were measured 16 days and 12 days, respectively, after inoculation.

. , *, ** and *** indicates tests of difference necrosis percentage between two groups of isolates with P-value lower than 10%, 5%, 1% and 1‰, respectively. . ,*, ** and *** indicates that the P-value is significant at the 10%, 5%, 1% and 1‰ confidence levels, respectively. -indicates that this interaction was not tested.

On grapevine berries, percentages of necrosis were lower than on tomato berries. Isolates collected from this host exhibited higher mean aggressiveness (23.7% of mean necrosis) than isolates collected on tomato (12.2%, P-value=0.064) but lower mean aggressiveness than isolates collected on bramble (33.3%, P-value=0.052).

Discussion

Host specialization in B. cinerea populations infecting grapevine and tomato

Several previous studies investigating the population structure of B. cinerea proposed that host plant should be considered as a potent factor structuring populations of the pathogen, based on observed associations between patterns of population subdivision and the host of origin of isolates (see review and examples in [START_REF] Walker | Diversity within and between species of Botrytis[END_REF]. In a previous [START_REF] Walker | Population structure and temporal maintenance of the multihost fungal pathogen Botrytis cinerea: causes and implications for disease management[END_REF] and the present study, we confirmed that most of the genetic variation segregating in B. cinerea populations is explained by hosts, and that the effect of geography is weaker. Two of the clusters identified were associated with tomato and grapevine, respectively, while other clusters were observed on all hosts but tomato, although in different frequencies. Crosspathogenicity experiments were consistent with analyses of population subdivision, with patterns of quantitative pathogenicity consistent with specialization of tomato isolates to tomato host, and specialization of grapevine isolates to grapevine hosts. This pattern of quantitative pathogenicity was observed after inoculation on leaves, but also in berry tests, which represents the type of symptoms more commonly found in the field. A pattern of pathogenicity partially consistent with specialization was previously observed in indoor samples of B. cinerea collected on lettuce and tomato but differences in pathogenicity were not associated with population subdivision [START_REF] Leyronas | Assessing host specialization of Botrytis cinerea on lettuce and tomato by genotypic and phenotypic characterization[END_REF]. To the best of our knowledge, this is the first report of host-associated ecological divergence in B. cinerea, with patterns of population structure mirroring patterns of quantitative pathogenicity. Indeed, the association between population structure and pathogenicity features suggests that host specialization contributes to reduce gene flow between populations associated with different hosts, due to the reduced viability of migrants and hybrids. The association between adaptation to hosts and barriers to gene flow would be favored by limited migration of gametes (especially non-mobile sclerotia developing on plant debris) between phases of selection by the host and sexual reproduction on (or inside) the host's tissue.

Comparison of pairwise Jost's D between clusters suggest that the tomato host induces greater population differentiation, i.e. greater reduction of gene flow between populations, compared to other hosts. This may rely on functional variations into the diverse and sophisticated biochemical arsenal of B. cinerea.

Indeed, this pathogen shows a large repertoire of secreted proteins, secondary metabolites and small interfering RNA (siRNA) that act as effectors in the interactions with its hosts [START_REF] Amselem | Genomic analysis of the necrotrophic fungal pathogens Sclerotinia sclerotiorum and Botrytis cinerea[END_REF][START_REF] Mbengue | Emerging trends in molecular interactions between plants and the broad host range fungal pathogens Botrytis cinerea and Sclerotinia sclerotiorum[END_REF][START_REF] Reino | Virulence-toxin production relationship in isolates of the plant pathogenic fungus Botrytis cinerea[END_REF][START_REF] Siewers | Functional analysis of the cytochrome P450 monooxygenase gene bcbot1 of Botrytis cinerea indicates that botrydial is a strainspecific virulence factor[END_REF][START_REF] Valero-Jiménez | Comparative genomics of plant pathogenic Botrytis species with distinct host specificity[END_REF][START_REF] Weiberg | Fungal small RNAs suppress plant immunity by hijacking host RNA interference pathways[END_REF]. The possible role of the different fungal effectors in host specialization remain to be investigated, as well as the possible role of the host phytoalexins. Indeed, contrasted pathogenic interactions between fungal pathogens (e.g. Cladosporium fulvum and Fusarium oxysporum f. sp. lycopersici) and tomato were explained by variation of tomatine content produced by the host and the production of tomatinase, an enzyme produced by the pathogen and able to detoxify the saponin toxin produced by tomato [START_REF] Melton | Heterologous expression of Septoria lycopersici tomatinase in Cladosporium fulvum: effects on compatible and incompatible interactions with tomato seedlings[END_REF][START_REF] Sarhan | Tomatine and phenol production associated with control of fusarial wilt of tomato by the NO3-nitrogen, lime, and fungicide integrated systems[END_REF]. Then, many molecular mechanisms could be involved in the incomplete host specialization of the grey mold fungus. Actually a recent genome-wide association mapping of B. cinerea showed that its virulence against a range of domesticated and wild tomatoes is highly polygenic and involves a diversity of mechanisms [START_REF] Breeze | 97 shades of gray: genetic interactions of the gray mold, Botrytis cinerea, with wild and domesticated tomato[END_REF][START_REF] Soltis | Interactions of tomato and Botrytis cinerea genetic diversity: parsing the contributions of host differentiation, domestication, and pathogen variation[END_REF]. Moreover, the previous observation that conidia of B. cinerea strains isolated from tomato have a higher germination rate on tomato cutin than conidia of strains isolated from grapevine [START_REF] Cotoras | Differences in the initial events of infection of Botrytis cinerea strains isolated from tomato and grape[END_REF] suggests that some of the molecular determinants underlying host specialization must be involved from the very early stage of the plant-pathogen interaction. Altogether, this pleads for elucidating the molecular determinants of host specialization in B. cinerea and to determine their specificity towards host plants, such as tomato but also within the Solanaceae family. From an evolutionary perspective, disentangling the molecular evolution of these determinants would contribute to unveil speciation in the Botrytis genus, in the light of the domestication of solanaceous plants. The biological material characterized in this study, as well as the high quality genomic resources available for B. cinerea should allow populations genomic analysis investigating the direction and density of selection exerted by the host, tomato in particular.

Lack of host specialization in B. cinerea populations infecting bramble, strawberry and hydrangea

In this study we extended previous analyses to new hosts, with the aim to better characterize the pattern of host specialization in B. cinerea. We found no evidence for host specialization in isolates of B. cinerea collected on hydrangea in outdoor nurseries, neither from population subdivision nor from crosspathogenicity tests. However, our capacity to detect ecological divergence in pathogens from hydrangea may have been limited by the small number of isolates collected on this host, associated with a relatively high proportion of clones. Further prospections may help determining the pathogenicity profile of hydrangea-associated isolates, in particular in environmental conditions favoring Botrytis population isolation in hydrangea cropping, such as in cold climatic chambers used to keep plants during vernalization.

The lack of population cluster specifically associated with strawberry was surprising, since Leroch et al. ( 2013) identified using multiple-gene sequencing, a novel clade of B. cinerea called group S, associated with strawberry in Germany. Using the detection marker designed by Leroch et al. ( 2013), we did detect a non-negligible proportion of group S isolates in our dataset (data not shown). However, group S isolates were not assigned into a specific cluster, suggesting that the pattern of population subdivision previously observed, and which led to the description of group S, may result from specific selection pressures such as resistance to fungicides, rather than from host specialization. Indeed, crops from our dataset, with the exception of bramble, were regularly sprayed with chemical fungicides, which might have favored the maintenance of group S strains in populations. The lack of specific interactions between strawberry and strawberrycollected isolates in the cross-pathogenicity test also supports this hypothesis.

The case of bramble-associated isolates is more puzzling, given that populations sampled from this host were differentiated from other populations in a previous study [START_REF] Walker | Population structure and temporal maintenance of the multihost fungal pathogen Botrytis cinerea: causes and implications for disease management[END_REF] and partially in this one, but tests did not reveal higher aggressiveness of isolates collected from bramble, compared to isolates from other hosts. Surprisingly, the difference in aggressiveness between tomatoand grapevine-collected strains was observed when inoculating on bramble, whereas it belongs to a botanical family, the Rosaceae, distant from Solanaceae and Vitaceae. However, plant susceptibility to Botrytis cinerea shows very little association to the evolutionary distances between the plant species [START_REF] Caseys | A generalist pathogen view of plant evolution[END_REF]. Henceforth, theses aggressiveness variations could be linked to species-specific interactions. We may assume that the decrease in gene flow is not sufficient to induce host specialization on bramble, compared to tomato and grapevine. Considering host specialization as a dynamic process, adaptation to bramble might be only in the early stages, consistent with the recent domestication of blackberry and the restricted cultivation areas. In comparison, grapevine was domesticated more than 2000 years ago and tomato more than 400 years ago and both represent large cultivation areas worldwide.

Concluding remarks

Our study shows that B. cinerea is a model of polyphagous fungal plant populations, whose populations are often comprised of generalist individuals, but can also be specialized to particular hosts (e.g. on tomato). While revealing key aspects of the process of adaptive divergence, understanding host specialization in B. cinerea may also help forecasting disease emergence and implementing sound management strategies. Indeed, host specialization suggests synchrony between host and pathogen development, determining time-shifted epidemics onsets in sympatric hosts within a landscape. Some cultivated hosts may also "filter" some specialized populations, within the collection of generalist and specialist individuals constituting B. cinerea. Limited gene flow between populations specialized on different crops may then be accentuated by optimizing spatial crop organization within an agricultural area. Limiting the density of putative high-risk host plants (with low host specialization) in the vicinity of high value crops may help decreasing disease propagation, as well as delaying fungicide resistance evolution.

Experimental procedures

Sample collection -Population genetics

Botrytis cinerea samples were collected in four areas of France (Alsace, Champagne, Loire Valley and Provence), with one to five collection sites per area (Error! Reference source not found.). Collection sites were 1 to 173 km apart within areas, and 180 to 703 km apart between areas. Populations from Champagne and Provence were chosen as a subset of a previously published dataset [START_REF] Walker | Population structure and temporal maintenance of the multihost fungal pathogen Botrytis cinerea: causes and implications for disease management[END_REF], keeping the sample size similar to the number of newly characterized strains (n=331) to avoid biases related to unbalanced sample size. Populations from Alsace and the Loire Valley (n=350) were collected specifically for the present study and include hosts that have not been previously characterized. Samples were collected in June in years 2002, 2005, 2006, 2007, 2014 and 2015. Spring is favorable to the sampling of B. cinerea populations from various hosts, as spring conditions facilitate primary infections by ascospores and wintersurviving macroconidia. Primary infections can occur on flowers and young ovaries by penetrating through scars left by floral pieces. Samples were collected from five different hosts: (i) strawberry (Fragaria x ananassa; diseased berries), (ii) grapevine (Vitis vinifera; asymptomatic flower caps), (iii) wild blackberry (Rubus fruticosus; asymptomatic berries and floral pieces) from bushes surrounding vineyards and fields, (iv) hydrangea (Hydrangea macrophylla; symptomatic and asymptomatic flower buds and stems) from outside nurseries and (v) tomato (Solanum lycopersicum ; symptomatic fruits or petioles; Error! Reference source not found.). Isolates from asymptomatic hosts (only strawberry and tomato hosts systematically exhibited symptomatic berries at the collection time) were collected after incubating dead flowers and/or young fruit with supposed B. cinerea infection in a moist chamber at room temperature until sporulation. Single spores were isolated for each sample and cultured on malt-yeast-agar (MYA; 20 g.L -1 malt extract, 5 g.L -1 yeast extract, 15 g.L -1 agar) at 23°C, under continuous light until sporulation. Six hundred and eighty one single-spored isolates were produced and kept as conidial suspensions in glycerol 20% at -80°C until use.

Microsatellite genotyping

DNA was extracted after 7 days of cultivation on MYA (malt extract 20 g.L -1 ; yeast extract 5 g.L -1 ; agar 15 g.L -1 ) medium at 21 °C in the dark. DNA was extracted using the "DNeasy® 96 Plant Kit" (QIAGEN®) following manufacturer protocol. Strains were genotyped at eight microsatellite markers (Bc1, Bc2, Bc3, Bc4, Bc5, Bc6, Bc7 and Bc10;Fournier et al., 2002) in multiplex polymerase chain reactions by Eurofins Scientific (Nantes, France). Markers with overlapping allelic size ranges bore distinct fluorochromes, as detailed in Supplementary Information 2. Allele size was scored automatically after binning analysis. Allele scoring for Bc4 enabled to remove from the dataset isolates of Botrytis pseudocinerea, the sister species living in sympatry with B. cinerea, based on their private allele at this locus (Fournier et al., 2002).

Population subdivision

Hierarchical AMOVA in Arlequin v3.5 [START_REF] Excoffier | Arlequin suite ver 3.5: a new series of programs to perform population genetics analyses under Linux and Windows[END_REF]) was used to estimate population differentiation within and between samples grouped according to their area or host plant of origin. Three different methods were used to investigate population subdivision. The model-based Bayesian clustering implemented in the STRUCTURE V2.3.4 software was used first [START_REF] Pritchard | Inference of population structure using multilocus genotype data[END_REF]. STRUCTURE was run with admixture and correlated allele frequencies options, with 500,000 Markov Chain Monte Carlo iterations, following 100,000 burn-in steps, and results were processed with STRUCTURE HARVESTER [START_REF] Earl | STRUCTURE HARVESTER: a website and program for visualizing STRUCTURE output and implementing the Evanno method[END_REF]. Second, the Discriminant Analysis of Principal Components DAPC was used as a multivariate analysis of population subdivision implemented in the ADEGENET 2.1.1 [START_REF] Jombart | adegenet 1.3-1: new tools for the analysis of genome-wide SNP data[END_REF] package in R 3.4.3. Third, the population structure was inferred using the maximum likelihood method SNAPCLUST implemented in the ADEGENET 2.1.1 package in R 3.4.3. Genetic differentiation between the clusters identified with STRUCTURE was measured using Weir & Cockerham's F st in GENEPOP 4.7.0 [START_REF] Rousset | genepop'007: a complete reimplementation of the genepop software for Windows and Linux[END_REF] to choose an appropriate number of clusters (K) and Jost's D in GENALEX 6.5 [START_REF] Peakall | GenAlEx 6.5: genetic analysis in Excel. Population genetic software for teaching and research-an update[END_REF].

Genetic diversity and mode of reproduction

All calculations were performed on clusters identified with STRUCTURE, considering only genotypes with a membership coefficient greater than or equal to 0.7 in one cluster. POPPR 2.8.2 [START_REF] Kamvar | Poppr: an R package for genetic analysis of populations with clonal, partially clonal, and/or sexual reproduction[END_REF][START_REF] Kamvar | Novel R tools for analysis of genome-wide population genetic data with emphasis on clonality[END_REF] was used to calculate the number of unique genotypes G and the clonal fraction 1 -G/N with N the number of individuals in the population, and the ݎҧ D estimator of multilocus linkage disequilibrium. Allelic richness Ar, the mean number of alleles per locus, was estimated using a rarefaction method to standardize sample size in ADZE v1.0 [START_REF] Szpiech | ADZE: a rarefaction approach for counting alleles private to combinations of populations[END_REF]. Gene diversity (He, called expected heterozygosity, in diploids) was estimated using GENETIX (Belkhir et al., 1996(Belkhir et al., -2004). Shannon's diversity index, which captures the richness and evenness of allelic profiles, was computed using GENALEX 6.5 [START_REF] Peakall | GenAlEx 6.5: genetic analysis in Excel. Population genetic software for teaching and research-an update[END_REF]. b Isolates collected on bramble, grapevine and hydrangea were retrieved from asymptomatic flower parts or young fruits.

Isolates from tomato and strawberry were isolated from symptomatic fruits.

Sample collection -Pathogenicity tests

Cross-pathogenicity tests were carried out using single-spored isolates previously collected from sympatric grapevine, bramble and tomato hosts [START_REF] Walker | Population structure and temporal maintenance of the multihost fungal pathogen Botrytis cinerea: causes and implications for disease management[END_REF], and single-spored isolates newly collected from strawberry, hydrangea and grapevine hosts (this study). Seventy-three isolates (7 to 33 isolates per host plant) were used in total (Supplementary Information 8). Only isolates with membership coefficient greater than or equal to 0.7 in the cluster they represent were used, and the laboratory reference strain B05.10 [START_REF] Buttner | Variations in ploidy among isolates of Botrytis cinerea: implications for genetic and molecular analyses[END_REF], whose genome is sequenced, was used as a control. Strains were cultivated and kept as described in previous sampling procedure.

Cross-pathogenicity tests

Isolates were cultivated on MYA at 23°C in the dark to prevent sporulation. Inoculations were performed using 1mm-diameter young mycelial plugs collected on the margin of the colonies, deposited on detached leaves (the mycelium facing the plant tissue), between leaf veins, and without wounding the leaf. cultivar Caruso; 2 week-old plantlets cultivated at 20°C with 8h dark) was used as a naive test host, differing from the previous collection hosts. Two sets of tests were conducted (Table 1): isolates from Champagne were inoculated onto grapevine, bramble, tomato and bean; isolates from Alsace and Pays de Loire were compared together on grapevine, strawberry and hydrangea.

Inoculations were performed between May and July, when grapevine, strawberry and hydrangea have young leaves. Leaves were detached with sterilized scissors on the day of inoculation, and they were chosen to be as similar as possible in terms of size, age, and stalk position. Each plant species was challenged with all isolates within a week. Experiments were repeated twice on bean, bramble, strawberry and hydrangea, and four repeats were carried out on tomato and grapevine. For each test, ten plugs per isolate were randomly distributed across leaves; the reference isolate B05.10 was also deposited as a control on each leaf. The number of plugs (from different isolates) on each leaf differed across hosts, as they differ in leaf surface: 5 per tomato leaf, 3 per bean leaflet, 3 per grapevine and strawberry leaf, and 4 per hydrangea leaf. Inoculated leaves were incubated in 30 x 30 x 2 cm transparent plastic boxes, over paper moisturized with 40 ml sterile water and these moist chambers were incubated in climate chambers at 23°c with 8h of daily obscurity. Two diameters of each lesion surrounding plug inoculation were measured daily, during between 2 to 9 days depending on the plant species and the speed of necrosis maturation. Lesions were scored until they coalesced with neighboring lesions. For each test and each lesion, plug, leaf and box numbers were recorded in order to include this information in statistical tests. Altogether, the dataset on leaves was comprised of 25,863 measurements of lesion diameter.

For berry tests, isolates were cultivated on MYA at 23°C under continuous light to promote sporulation. Only isolates collected from tomato, grapevine and bramble were used in berry tests (a subset of isolates described in Supplementary Information 8). Ten-day old cultures were used to produce spore suspensions in sterile water at 5.10 5 spores/ml, the day of the test. 10 µl droplets of spore suspension were deposited on the unwounded apical side of grapevine (white cultivar Mélissa) and tomato (unknown cultivar of cherry tomato) ripe berries. Berries were previously cleaned twice for 5 min in sterile water with 10% Tween80, rinsed two times for 5 min in pure water and dried before use. Ten inoculated berries per isolate were stabilized using individual gaskets in a 24 x 18 x 10 cm transparent plastic box over paper moisturized with 20 ml sterile water. A set of ten non-inoculated berries was added in each test to detect natural contamination of the berries. These moist chambers were incubated as for leaf tests. The test was repeated two times for each isolate. The percentage of necrotic area was scored for each berry after 16 and 12 days for tomato and grapevine, respectively. Altogether, the dataset on berries was comprised of 1600 measurements of necrosis surface.

Statistical analysis

Hosts of origin were tested for their effect on leaf necrosis. For each tested host and for each day after inoculation, a linear mixed model was used, with diameter of necrosis measured on leaf as the response variable, the host of origin as fixed effect and the isolate, leaf and test box as random effects (using the R packages lmer and lmerTest; [START_REF] Bates | Fitting linear mixed-effects models using lme4[END_REF][START_REF] Kuznetsova | lmerTest package: tests in linear mixed effects models[END_REF]. The mean of diameter of necrosis was compared between hosts of origin of isolates using a post-hoc Tukey test (using the R package emmeans; [START_REF] Lenth | Gray mold populations in German strawberry fields are resistant to multiple fungicides and dominated by a novel clade closely related to Botrytis cinerea[END_REF].

The effect of hosts of origin was further tested on percentage of necrosis on fruit. A generalized linear mixed model with percentage of necrosis as the response variable, the host of origin and the test as fixed effects and isolate as a random effect was used for each tested host and for each day after inoculation. The necrosis percentage was compared between hosts of origin of isolates using a post-hoc Tukey test.

Figure 1 .

 1 Figure 1. Barplots of the STRUCTURE, DAPC and SNAPCLUST analysis showing genetic subdivision of the 681 B. cinerea isolates collected between 2002 and 2015 into a varying number of genetic clusters (K=2 to K=10).

  Individuals are sorted out by their area of origin (black vertical bars) and then by their sampling host (grey vertical bars). The areas names are indicated at the top of the chart and the host of origin at the bottom in abbreviated form (Vv: V. vinifera; Rf: R. fruticosus; Fxa: F. x ananassa; Sl: S. lycopersicum and Hm; H. macrophylla). Each cluster is colored according to the most represented host of its members (red for the tomato-associated cluster, shades of green for grapevineassociated clusters, shades of purple for bramble-associated clusters and shades of pink for strawberry-associated clusters).

Figure 2 .

 2 Figure 2. Patterns of diversity and reproduction mode measured in the five clusters delimited by the STRUCTURE partitioning analysis at K=5.(A) Proportions of sampling hosts represented in inferred clusters (left), clonal fraction (middle) and ݎҧ D (estimation of multilocus linkage disequilibrium ; right). Isolates from tomato Sl are colored in red, grapevine Vv in green, bramble Rf in purple, hydrangea Hm in blue and strawberry Fxa in pink. (B) Diversity is estimated via allele richness A r (mean number of alleles per locus; left), the expected heterozygosity He (mean expected heterozygosity over the eight loci; middle) and the Shannon's diversity index H (mean over the eight loci; right).

Figure 3 .

 3 Figure 3. Evolution of mean necrosis diameter over time, for B. cinerea isolates representative of their host of origin and inoculated on a range of test hosts.

Figure 5 .

 5 Figure 5. Percentage of necrosis for B. cinerea isolates representative of their host of origin and inoculated on grapevine and tomato hosts.

Figure 4 .

 4 Figure 4. Map of B. cinerea populations collected from four French areas on various host plants between 2002 and 2015. Each color represents an area with Alsace in red, Champagne in blue, Loire Valley in green and Provence in yellow.

Table 3

 3

				). Highest values were observed
	for pairs including tomato-associated cluster C1
	(range 0.663-0.869) and bramble-associated cluster
	C5 (range 0.658-0.869). The smallest differentiation
	was observed between grapevine-associated clusters C2 and C3 (D=0.320).
	Cluster C1	C2	C3	C4
	C2	0,754*
	C3	0,633* 0,320*
	C4	0,782* 0,623* 0,579*
	C5	0,869* 0,666* 0,701* 0,658*
	Table 2. Pairwise Jost's D between clusters inferred
	with STRUCTURE at K=5.

Table 3 .

 3 Results of cross-pathogenicity leaf tests observed over post-inoculation time for B. cinerea isolates representative of their host of collection.Isolates collected on strawberry, hydrangea, grapevine, bramble and tomato (hosts of origins, in columns) and a reference isolate used as a control were inoculated on strawberry, hydrangea, bean, bramble, tomato and/or grapevine leaves (test hosts, in row). Necrosis were scored between 2 and 16 days after inoculation, depending on the test host. . Our statistical model detected significant effects of the host of origin on the mean necrosis diameters, for several time points after inoculation. Mean necrosis diameters increased over time for all interactions and clusters, and especially for the reference isolate B05.10, indicating that tests were properly carried out. For a given scoring time, aggressiveness of the reference isolate differed among test hosts, bean and tomato showing faster growth of necrosis compared to other hosts. No significant differences (P-values > 5%) in aggressiveness were observed among isolates from different clusters inoculated onto the naive host bean, at any scoring time.

Table 3 .

 3 Results of cross-pathogenicity leaf tests observed over post-inoculation time for B. cinerea isolates representative of their host of collection. Isolates collected on strawberry, hydrangea, grapevine, bramble and tomato (hosts of origins, in columns) and a reference isolate used as a control were inoculated on strawberry, hydrangea, bean, bramble, tomato and/or grapevine leaves (test hosts, in row). Necrosis were scored between 2 and 16 days after inoculation, depending on the test host. a P-value indicating if lesion size according to host of origin was significantly different, on a given test host. b Significance of the P-value, indicating if test conditions (test, leaf and box repeat) significantly interacted with the lesion size on a given test host. c Values indicate the mean difference of necrosis diameter (in mm) between isolates collected on a host (2 nd row) compared to isolates collected on a second one (3 rd row), on given test host. Either positive or negative, these values indicate on which test host a given group of isolates had better growth.

								Control vs.				Strawberry vs.			Hydrangea vs.		Bramble vs.	Tomato
	Test host	Days after inoculation	Effect of the host of collection a	Test effect	b	Strawberry	Hydrangea	Bramble	Tomato	Grapevine	Hydrangea	Bramble	Tomato	Grapevine	Bramble	Tomato	Grapevine	Tomato	Grapevine	Grapevine
							Leaf tests : Pairwise difference in mean necrosis diameter for individuals from various hosts of collection (mm)	c
			0.005**	***		0.49	1.00	-	-	0.25	0.51	-	-	-0.25	-	-	-0.75**	-	-	-
	Strawberry		0.057. 0.3			0.27 -0.07	1.32 1.48	--	--	-0.27 -1.34	1.05 1.55	--	--	-0.53 -1.27	--	--	-1.58* -2.82	--	--	--
			0.035*			-1.20	2.33	-	-	-2.14	3.53	-	-	-0.94	-	-	-4.47*	-	-	-
			0.46	.		0.43	0.36	-	-	0.44	-0.07	-	-	0.017	-	-	0.08	-	-	-
	Hydrangea		0.31 0.17	** ***		1.74 2.82	0.61 -0.22	--	--	1.81 2.77	-1.13 -3.04	--	--	0.08 -0.05	--	--	1.20 2.99	--	--	--
			0.17	***		4.09	-0.61	-	-	3.81	-4.70	-	-	-0.28	-	-	4.42	-	-	-
			0.3	***		-	-	2.41	1.65	1.46	-	-	-	-	-	-	-	-0.75	-0.95	-0.19
	Bean		0.3 0.28	*** ***		--	--	5.57 8.30	3.89 4.86	3.81 5.93	--	--	--	--	--	--	--	-1.67 -3.44	-1.76 -2.36	-0.08 1.08
			0.099.			-	-	5.93	-1,00	7.54	-	-	-	-	-	-	-	-6.93	1.60	8.54 .
			0.049*			-	-	0.27	0.05	0.41	-	-	-	-	-	-	-	-0.2	0.14	0.36*
			0.029*	***		-	-	0.76	0.11	0.79	-	-	-	-	-	-	-	-0.65	0.03	0.68*
	Bramble		0.0031** 0.0084**	*** ***		--	--	2.58 3.84	0.29 0.57	2.39 3.78	--	--	--	--	--	--	--	-2.29* -3.28*	-0.20 -0.06	2.10** 3.21*
			0.005**			-	-	9.04	-2.78	9.96	-	-	-	-	-	-	-	-11.80*	0.92	12.70**
			0.044*	***		-	-	8.1	0.49	9.15	-	-	-	-	-	-	-	-7.60	1.05	8.65*
			0.091.	***		-	-	0.33	0.08	0.41	-	-	-	-	-	-	-	-0.25	0.08	0.33•
	Tomato		1.1e-06*** 0.00024***	*** ***		--	--	5.54* 9.37	1.24 1.10	5.40* 9.67	--	--	--	--	--	--	--	-4.31*** -8.27**	-0.15 0.31	4.16*** 8.57***
			0.0006***	***		-	-	15.70	2.44	16,00	-	-	-	-	-	-	-	-13.20**	0.34	13.60***
			0.4	***		0.20	0.08	0.42	0.21	0.35	-0.12	0.22	0.02	0.15	0.336	0.132	0.27	-0.20	-0.07	0.13
			0.26	***		1.33	1.11	1.37	0.65	0.67	-0.22	0.04	-0.68	-0.66	0.26	-0.456	-0.43	-0.72	-0.69	0.02
	Grapevine		0.12	***		2.36	2.25	3.15	1.45	1.14	-0.11	0.79	-0.90	-1.22	0.895	-0.798	-1.11	-1.69	-2.01	-0.31
			0.037*	***		3.82	4.09	4.03	1.93	0.73	0.27	0.22	-1.89	-3.09	-0.0541	-2.16	-3.36	-2.11	-3.31	-1.20
			0.15	***		-	-	14.2	8.74	4.76	-	-	-	-	-	-	-	-5.50	-9.47	-3.98
									Berry tests : Pairwise log odds ratio from various hosts of collection
	Tomato fruit	16	0.02401 *	-		-	-	0.52	-0.77	-0.24	-	-	-	-	-	-	-	-1.30*	-0.77	0.52
	Grapevine	12	0.000663 ***	***		-	-	-0.57	0.74	0.097	-	-	-	-	-	-	-	1.31***	0.66•	-0.64•

Table 4 .

 4 Populations of B. cinerea collected in four French areas, on four host plants between 2002 and 2015.

	Area a	Location	Host plant b	Cultivar	No isolates
	Alsace	Balbronn	Fragaria x ananassa	Clery	40
			Rubus fruticosus	Wild	27
			Vitis vinifera	Riesling	36
		Sigolsheim	Fragaria x ananassa	Candiss	31
			Vitis vinifera	Pinot auxerrois 51
	Champagne Courceroy	Solanum lycopersicum	Moneymaker	21
		Courteron	Rubus fruticosus	Wild	15
			Vitis vinifera	Pinot meunier 30
		Foissy-sur-Vanne	Rubus fruticosus	Wild	19
			Solanum lycopersicum	Moneymaker	21
		Hautvilliers	Vitis vinifera	Pinot noir	31
		Vandières	Rubus fruticosus	Wild	9
			Vitis vinifera	Pinot noir	31
	Loire Valley Angers	Hydrangea macrophylla Unknown	33
		Beaulieu-sur-Layon Vitis vinifera	Chenin	39
		Cheverny	Fragaria x ananassa	Darselect	29
			Rubus fruticosus	Wild	10
			Vitis vinifera	Pinot noir	2
		Cour-Cheverny	Fragaria x ananassa	Gariguette	27
			Rubus fruticosus	Wild	10
			Vitis vinifera	Pinot noir	15
	Provence	Lançon-Provence	Rubus fruticosus	Wild	5
			Solanum lycopersicum	Grenache	7
			Vitis vinifera	Alison	43
		La Farlède	Vitis vinifera	Rolle	21
		Pierrelatte	Solanum lycopersicum	Hipop	6
		Sarrians	Rubus fruticosus	Wild	20
			Solanum lycopersicum	Emotion/Alison 18
			Vitis vinifera	Grenache	34
	Total				681

a Isolates were sampled at multiple dates between 2002 and 2015 in Alsace (June 2015), Champagne and Provence (September 2005 to June 2007) and Loire Valley (June 2002 to June 2015).

Acknowledgements

AM was supported by a grant from the Doctoral School « Sciences du Végétal », Université Paris-Saclay. We thank Marguerite Cuel, for preliminary data analysis of the cross-pathogenicity tests. We thank Claire Amiraux (Planète Légumes), Marie-Laure Panon (Comité Champagne), Jean-Marie Guichardon and Michel Badier (Chambre d'agriculture du Loire et Cher) and Elisabeth Fournier (UMR BGPI, INRA) for help with the sampling of B. cinerea isolates. We are also grateful to Thomas Perron (IRHS, INRA, Angers), the Vegepolys Pôle de Compétitivité, the Region Pays de la Loire (Physi'Ho project) and the local horticultors for providing the isolates from hydrangea.

B. anthophila Synthesis after (Elad et al., 2016). Polyphagy index calculated as follows: PI = (A x B) 1/2 , with A, number of genera of host plants, and B, taxonomic diversity of the hosts. Primer pairs and fluorochromes initially described in Fournier et al. (2002).

5 µl of DNA (5 ng/µl) and 1.25 µl of each primer (2 µM) was used per multiplex PCR reaction, with the Taq polymerase Type-it (Qiagen), according to the manufacturer's recommendations. The PCR program was:

-Initial denaturation: 94°C for 5 mn -Elongation (35 cycles): 94°C 30s, 55°C 30s, 72°C 2mn -Final elongation: 72°C 2mn Amplicons were observed with the GeneScan and Genotyper programs (Life Technologies) after capillary electrophoresis achieved on an ABI3130xl sequencer, using the Liz500 size standard (Life Technologies).

Supplementary Information 3. F st and number of unassigned individuals with increasing K values.

Differentiation between clusters (F st ) and number of 'unassigned' genotypes ('assigned' genotypes with >90% membership proportion or membership probability in a single group) as a function of the number of clusters K with Each plot represent the proportion of individuals sampled from different hosts colored accordingly (Solanum in red, Vitis in green, Rubus in purple, Hydrangea in blue and Fragaria in pink) at K values from 3 to 5. The upper bar corresponds to the STRUCTURE assignation, the middle one to DAPC and lower one to SNAPCLUST. Supplementary Information 6. Evanno's ΔK statistic.

The ΔK statistic [START_REF] Evanno | Detecting the number of clusters of individuals using the software STRUCTURE: a simulation study[END_REF] is calculated on STRUCTURE results along K values from 2 to 10. Supplementary Information 7. Goodness of fit statistics.

Each line corresponds to one of the goodness of fit statistic available with SNAPCLUST [START_REF] Jombart | adegenet: a R package for the multivariate analysis of genetic markers[END_REF]. The solid black line corresponds the AIC, the dotted black line to the AICc, the solid dark gray line to the BIC and the solid light gray line to the KIC. June 2014 Riesling 14-V-007 14-V-019 14-V-022 14-V-028 14-V-029 14-V-063 14-V-220


Rodrigo Osuna-Orozco

Nathanael Machicoane

Peter D Huck

Alberto Aliseda

FEEDBACK CONTROL OF THE SPRAY LIQUID DISTRIBUTION OF ELECTROSTATICALLY ASSISTED COAXIAL ATOMIZATION

Keywords: real-time feedback control, two-fluid coaxial atomization, electrostatic atomization, multiphase flow, reduced order modeling, principal component analysis

Electrostatic actuation is used for real-time multiphysics feedback control of two-fluid coaxial atomization. This actuation is added to the modulation of the axial and angular momentum of the turbulent coaxial gas stream, for a total of three actuators to control atomization. We characterize the spray real-time response through optical scattering measurements of radial liquid distribution.

We apply principal component analysis on the scattering radial profiles and correlate the first three principal components to the three control inputs. The control algorithm continuously adjusts the three inputs to minimize the difference between a goal radial profile representing the desired spray state and the profile observed in real time. This real-time multiphysics (gas momentum plus electrostatic repulsion) control on the liquid distribution in a two-fluid coaxial spray is a novel contribution to the archival literature on this technology.

INTRODUCTION

Liquid atomization is ubiquitous in both the natural world and technological applications (e.g. fuel injection, thin film coating, irrigation, manufacturing). Yet, in spite of its widespread use, the development of real-time feedback control strategies for spray shaping remains a challenge, with many open questions and opportunities for development in this area [START_REF] Arai | The Possibility of Active Attitude Control for Fuel Spray[END_REF]. In particular, controlling the liquid volume fraction distribution in time and space is still an elusive objective.

This paper extends the authors' earlier work (Osuna-Orozco et al., 2019) on real-time feedback control of coaxial atomization by adding an electrostatic actuation as well as expanding the range of parameters over which the spray is controlled.

Both coaxial [START_REF] Aliseda | Atomization of viscous and non-newtonian liquids by a coaxial, high-speed gas jet. Experiments and droplet size modeling[END_REF][START_REF] Eggers | Physics of liquid jets[END_REF][START_REF] Lasheras | Break-up and atomization of a round water jet by a high-speed annular air jet[END_REF][START_REF] Marmottant | On spray formation[END_REF] and electrostatic [START_REF] Cloupeau | Electrohydrodynamic spraying functioning modes: a critical review[END_REF][START_REF] Gañán-Calvo | Review on the physics of electrospray: From electrokinetics to the operating conditions of single and coaxial Taylor cone-jets, and AC electrospray[END_REF][START_REF] Grace | A review of liquid atomization by electrical means[END_REF][START_REF] Kien Nguyen | Control and improvement of jet stability by monitoring liquid meniscus in electrospray and electrohydrodynamic jet[END_REF][START_REF] Manna | Primary atomization of electrified water sprays[END_REF][START_REF] Rosell-Llompart | Electrosprays in the cone-jet mode: From Taylor cone formation to spray development[END_REF][START_REF] Verdoold | A generic electrospray classification[END_REF] atomizers have been the subject of intensive study in the last few decades. More recently, there has been increasing efforts to develop hybrid atomizers that include gas flow forcing as well as an electrostatic component [START_REF] Fei | Graphene Oxide Involved Air-Controlled Electrospray for Uniform, Fast, Instantly Dry, and Binder-Free Electrode Fabrication[END_REF][START_REF] Kourmatzis | Combined aerodynamic and electrostatic atomization of dielectric liquid jets[END_REF][START_REF] Li | Temporal instability of coflowing liquid-gas jets under an electric field[END_REF][START_REF] Patel | An Volume x[END_REF][START_REF] Patel | Plausibility of variable coverage high range spraying: Experimental studies of an externally air-assisted electrostatic nozzle[END_REF][START_REF] Seong | A hybrid aerodynamic and electrostatic atomization system for enhanced uniformity of thin film[END_REF]. These studies have revealed a wide range of atomization regimes that can be attainable with a rather simple atomizer geometry. Moreover, there has been an intense effort in developing control strategies for a variety of sprays, most notably, for fuel injection in combustion systems [START_REF] Billoud | Adaptive Active Control of Combustion Instabilities[END_REF][START_REF] Coker | Active instability control effectiveness in a liquid fueled combustor[END_REF][START_REF] Conrad | Feasibility of "intermittent" active control of combustion instabilities in liquid fueled combustors using a "smart" fuel injector[END_REF][START_REF] Jones | Closed-loop active control of combustion instabilities using subharmonic secondary fuel injection[END_REF][START_REF] Mcmanus | A review of active control of combustion instabilities[END_REF][START_REF] Muruganandam | Active control of lean blowout for turbine engine combustors[END_REF][START_REF] Murugappan | Optimal control of a swirl-stabilized spray combustor using system identification approach[END_REF].

Here, we present results for an electrostatically assisted two-fluid coaxial atomizer, where three control inputs shape the spray liquid distribution in real-time feedback. The sensing of the spray state is done through optical scattering, measuring a radial profile of liquid volume fraction at 18.5 gas diameters downstream of the nozzle. Applied voltage on the far field is actuated on, in addition to the actuation on the swirl and no-swirl gas flow rates.

Atomization and Sprays

EXPERIMENTAL SETUP

Atomizer and flow loop

The two-fluid coaxial atomizer used in this work has been characterized extensively [START_REF] Machicoane | Synchrotron radiography characterization of the liquid core dynamics in a canonical two-fluid coaxial atomizer[END_REF]Osuna-Orozco et al., 2019). For the electrostatic actuation, the nozzle is placed between two parallel metallic plates and a voltage, variable between 0 and 25 kV, is equally applied to both of them, while the nozzle remains grounded This electrohydrodynamic multiphase flow can be characterized in terms of three dimensionless groups: the gas-to-liquid momentum ratio M = ρ g U 2 g /ρ l U 2 l , the swirl ratio SR = Q ns /Q sw , and the ratio of electric to surface tension energy, Γ =

σ 2 0 / 0 γ/r0
, where the subscripts g and l denote the gas and liquid phases respectively, ρ i are the mass densities. The mean velocities, U i , are calculated from the volume flow rates Q i , the subscripts ns and sw denote the no-swirl and swirling components of the gas flow rate respectively, and γ is the surface tension between the liquid and gas phases. For the electrostatic variables, σ 0 is the surface charge density at the liquid-gas interface, 0 is the permittivity of free space and r 0 the liquid needle radius.

Our working fluids were ambient air and water with a small concentration of sodium chloride dissolved, with a conductivity of 2000 µS/cm. For all the results presented, the liquid flow rate was kept at Q l = 99(±2.5)ml/min, corresponding to a liquid Reynolds number Re l = 1200.

The total gas flow rates explored in this study were in the range 150 -860 SLP M , resulting in velocities between 34 and 200 m/s and momentum ratios between 5 and 175. For a given total gas flow rate, the swirl ratio was varied in the range 0 to 1. Charge densities up to 3 C/m 3 were observed.

Light scattering measurements

The control sensing consists of light scattering measurements along a radial profile through the spray. In contrast with the attenuation measurements used in our previous work, light scattering measurements allow for a more robust detection of spray features at lower volume fractions, which enables detection of lower mass loading reliably, at higher gas momentum ratio. A collimated light sheet is shined through the spray at an angle of 30 degrees with the horizontal plane, intersecting the spray between 18 and 19 d g downstream from the nozzle (the gas inner diameter, dg = 1 cm and the plates are 35d g apart).

The light refracted by the spray droplets at 120 o forward scattering is collected by a linear CCD array camera (Thorlabs model LC100), forming a radial transverse profile , perpendicular to the spray axis and normal to the plane of the metallic plates. A Zeiss 100 mm macro lens was used for visualization, yielding a resolution of 77 µm, with a field of view of 16 cm wide. images, with the scattered intensity at the edges approaching zero.

The optical scattering real-time sensing is able to capture the different features of the spray over the range of control inputs studied. We explored momentum ratios from 5 to 175, swirl ratios from 0 to 1 and applied voltages of 0 to 25 kV (corresponding to Γ = 0 -1000). Appropriate filtering and sliding-time averaging reveal consistent profiles for varying actuation in spite of the unsteadiness of the instantaneous liquid volume fraction signal, intrinsic to turbulent flows. Figure 3 shows a small set of representative profiles that illustrate some of the features of the spray at different values of the control inputs. As the momentum ratio increases, scattering increases because the droplet number-density increases, especially near the centerline of the spray. Increasing the swirl ratio broadens the spray as droplets are transported away from the spray centerline, but may result in a double bump feature in the scattered transverse profiles.

Finally, we observe a broadening of profiles as we increase the applied voltage, resulting both from the electrostatic repulsion among charged spray droplets and by the electric field established between the grounded nozzle and charged plates.

Principal Component Analysis of light scattering profiles

Principal Component Analysis was performed onto a large data set collected over a wide range of control inputs, in order to arrive at a reduced-order representation of the data. The dataset included measurements for 140 different spray conditions with around 250 profiles for each condition. The parameter space sampled was comprised of combinations of 7 momentum ratios between 5 and 170, 5 swirl ratios between 0 and 1 and 4 voltages between 0 and 25 kV. We used a standard implementation of PCA using MATLAB's Singular Value Decomposition builtin functions and standard algorithms as the ones detailed by Kutz. [START_REF] Kutz | Data-Driven Modeling & Scientific Computation : Methods for Complex Systems & Big Data[END_REF]. Most of the features of the collected profiles could be adequately and sparsely represented by the main three principal components which contained about 84% of the energy. As shown in Figure 4 accurate reconstructions were achieved with the first three principal components.

More importantly, the distance between the two profiles as measured by the integral over the profile of the normalized root-mean-squared difference and by the sum of the normalized rootmean-squared difference of their three principal components closely match each other (Figure 4b)). This property guarantees that two profiles that are close in the principal component parameter space will indeed be very similar in physical space, such that convergence of the principal component coefficients ensures that the desired profile goal will be achieved.

The resulting reduced-order representations were correlated to the control inputs using a least squares linear fit, in a 3 by 3 matrix, in order to obtain a linear map between the observed profiles and the control variables.

REAL-TIME FEEDBACK CONTROL IMPLEMENTATION

Control Performance

The goal of the spray control is to drive the experimental spray towards a desired state, charac- real-time error. This results in an closed-loop iterative approach (with a frequency of iteration being approximately 100 Hz) with control input adjustments that decrease as the desired profile is approached (Osuna-Orozco et al., 2019). The algorithmic gains are set as a compromise between convergence speed and stability/robustness to process noise.

As seen in figure 5 (and in the movie included in the supplementary material), our control algorithm leads to convergence upon a wide variety of specified goals.

CONCLUSIONS

We demonstrate multi-physics control, using an external electric field to both charge the spray droplets and attract the charged droplets radially outwards as they flow downstream of the nozzle. This extends previous work on real-time feedback control of the spray liquid distribution in two significant ways: 1) adopting a more robust sensing technique that captures features of the spray more accurately, even in less dense sprays, over a broader range of momentum ratios and swirl ratios; 2) and by adding an electrostatic actuation. To the best of the authors' knowledge, this is the first instance of real-time feedback control on the liquid distribution of a two-fluid coaxial spray using multiphysics actuation. Future work will explore more robust control algorithms, additional multi-physics actuations and will establish more closely the link between the scattering measurements and the physical variables that characterize the spray (e.g. volume fraction, area fraction, mass flux).

  FIG. 1: (a) Plan view of the nozzle along the gas inlets plane showing the liquid channel in the middle and the eight inlets for gas. (b) Side view of the atomizer including the parallel plates (not to scale).

  figure 1. The liquid nozzle internal and external diameters are respectively 2 and 3 mm, while the gas nozzle internal diameter is 10 mm.

Figure 2

 2 FIG. 2: Geometry of light scattering for spray visualization, showing the collimated plane of light that crosses the axis of the spray at an angle of 60 degrees.

FIG. 3 :

 3 FIG. 3: Radial profiles of light scattering (representing liquid volume fraction) with (red) and without (blue) electrostatic actuation for swirl ratios of zero (solid lines) and swirl ratio of one (dashed lines) Left and right are respectively at a momentum ratio of 25 and 50.

  FIG. 4: (a) First Three Principal Components normalized by their maxima (Left). Measured profiles (solid lines) and principal component reconstructions (dashed lines) (Right). (b) Normalized root mean square error time series between the goal and the real-time measurement computed from the full data (solid) and the PCA decomposition (dashed).

  FIG. 5: (a) Goal profiles used to test the control algorithm as shown in the supplementary video. (b) Example of control algorithm convergence, progressing from profile 6 to 1. (c) Normalized root mean square errors between real-time profiles and goals (desired light profiles in the spray). The legend indicates the starting profile as numbered in the supplementary video and the goal profile.

  Table 1 below presents the mean convergence error, ε , and the standard deviation of the convergence error , ε , for the control examples in the supplementary video and in figure5. The standard deviation of the normalized convergence error is a fraction of the convergence error. The signal shows greater variance at higher values of the swirl ratio, which is also apparent in higher variance of the

	convergence error for such cases.
	Atomization and Sprays

TABLE 1 :

 1 Closed loop control performance metrics.

	M	SR Voltage (kV)	ε	ε	100 • ε ε
	59 0.21	0.0	0.10 0.01	8.94
	87 0.40	20.7	0.05 0.01	26.44
	48 0.70	8.9	0.10 0.03	25.89
	49 0.24	0.0	0.09 0.01	9.22
	27 0.38	6.2	0.21 0.03	16.29
	22 0.56	8.8	0.05 0.01	28.35

Volume x, Issue x, 2019

ACKNOWLEDGMENTS

This work was sponsored by the Office of Naval Research (ONR) as part of the Multidisciplinary University Research Initiatives (MURI) Program, under grant number N00014-16-1-2617. The views and conclusions contained herein are those of the authors only and should not be interpreted as representing those of ONR, the U.S. Navy or the U.S. Government.


Julien Fontchastagner
email: julien.fontchastagner@ensem.inpl-nancy.fr

Frédéric Messine
email: frederic.messine@n7.fr

Interval Branch and Bound Algorithms dealing with Black-Box Constraints

Keywords: Interval Arithmetic, Branch and Bound, Black-box constraint 1

Interval Branch and Bound Algorithms have shown their intrinsic interest to solve exactly some difficult mixed non-convex and non-linear programs. A code named IBBA was developed to solve some design problems for electromechanical actuators. The idea of this paper is to extend the exact global optimization algorithm by inserting the direct use of an automatic numerical tool (based on a finite element method). This new methodology permit to solve more rationally design problems. Some numerical examples validate the interest of this new approach.

Introduction

IBBA is a code that we developed in order to solve some particular design problems of electromechanical actuators (such as electrical machines), [START_REF] Messine | A Deterministic Global Optimization Algorithm for Design Problems, chapter in Essays and Surveys in Global Optimization[END_REF]. Because IBBA is based on interval arithmetic, it needs all the expressions of the equations of the optimization problem. That was exploited by using analytical models of actuators developed by researchers in electrical engineering since the 80th. Associating analytical models and IBBA code yields to an efficient and rational approach to solve these design problems which can be formulated as non-homogeneous mixed constrained global optimization ones:      min (x,y)∈(X,Y )⊆IR n ×IN m f (x, y) u.c. g i (x, y) ≤ 0, ∀i ∈ P = {1, ..., p} h j (x, y) = 0, ∀j ∈ Q = {1, ..., q}

where f is a real function, IR and IN are respectively the real and the positive integer sets.

Normally the discrete variables can also be boolean or categorical ones [START_REF] Messine | A Deterministic Global Optimization Algorithm for Design Problems, chapter in Essays and Surveys in Global Optimization[END_REF]. However, it is sufficiently general for this paper to use only these classical mixed formulation.

A large part of these kind of design problems were solved efficiently by using IBBA code with analytical models [START_REF] Fitan | The Electromagnetic Actuator Design Problem : A General and Rational Approach[END_REF]. Nevertheless, before making the prototype of the optimized actuator it is preferable to validate the optimal solution by using a finite element code (because for developing analytical models some assumptions have to be done and it is less the case for a numerical model). Thus, some differences generally occur between some important characteristical values yielding the designer to correct some parameters of the solution. In [START_REF] Fontchastagner | Design of Electrical Rotating Machines by Associating Deterministic Global Optimization Algorithm With Combinatorial Analytical and Numerical Models[END_REF], we developed an extension of IBBA, named IBBA+NUMT to solve a problem of type (1) but where one constraint has to be satisfied via a finite element code NUMT. This was extended recently when more than one constraint have to be taken into account.

In this paper, we will present in a more general case this extension of IBBA, that we name IBBA BBC for IBBA with Black-Box Constraints (constraints which do not have an explicit form but which has to be computed via an algorithm).

Formulations of the problem

The problem that we are interested with is the following:

             min (x,y)∈(X,Y )⊆IR n ×IN m f (x, y) u.c. g i (x, y) ≤ 0, ∀i ∈ I ⊆ P h j (x, y) = 0, ∀j ∈ J ⊆ Q g A i (x, y) ≤ 0, ∀i ∈ P \ I h A j (x, y) = 0, ∀j ∈ Q \ J (2)
where A on the constraints indicates that they are computed using an algorithm.

Unfortunately, this problem is actually impossible to be solved by a code based on IBBA; because for the black-box constraints, it is not possible to compute bounds. Hence, we introduce a new more constrained program but related to Program (2):

                     min (x,y)∈(X,Y )⊆IR n ×IN m f (x, y) u.c. g i (x, y) ≤ 0, ∀i ∈ I ⊆ P h j (x, y) = 0, ∀j ∈ J ⊆ Q g A i (x, y) ≤ 0, ∀i ∈ P \ I h A j (x, y) = 0, ∀j ∈ Q \ J g i (x, y) ≤ s i , ∀i ∈ P \ I r j ≤ h j (x, y) ≤ r j , ∀j ∈ Q \ J (3)
where s i , r j and r j associated with the analytical equations define a zone where the global solution has to be searched. Of course, analytical constraints g i , ∀i ∈ P \ I and h j , ∀j ∈ Q \ J as to be related to the numerical ones, respectively g A i , ∀i ∈ P \ I and h A j , ∀j ∈ Q \ J.

IBBA BBC Algorithm

The following corresponds to the iterations of IBBA BBC Algorithm:

1. Set (X, Y ) := the initial hypercube. In this algorithm, one has: -f which represents the current best known solution of program (3); -L is the list which contain all the boxes which can possibly have the global solution at each stage of the iterations; -lb represents a technique using interval arithmetic which allows to compute lower bounds of a function (explicitly known) over a box; -the constraint propagation code which is used is detailed in [START_REF] Messine | Deterministic Global Optimization using Interval Constraint Propagation Techniques[END_REF] and it is based on the calculus trees of the constraints (it is only for constraints which are explicitly known). Remark that there a few differences between IBBA and IBBA BBC; the differences are underlined in the above algorithm. Now, we want to prove that this algorithm IBBA BBC find the global solution of Program (3). In order to prove that point, we introduce the following relaxed program:

             min (x,y)∈(X,Y )⊆IR n ×IN m f (x, y) u.c. g i (x, y) ≤ 0, ∀i ∈ I ⊆ P h j (x, y) = 0, ∀j ∈ J ⊆ Q g i (x, y) ≤ s i , ∀i ∈ P \ I r j ≤ h j (x, y) ≤ r j , ∀j ∈ Q \ J (4)
where the black-box constraints are eliminated from Program (3). In fact, the main principle of IBBA BCC is to solve the relaxed program (4) and just to consider the Program (3) for improving the current solutions f. Such a Branch and Bound algorithm proceeds by exclusions of boxes when it is proven that the global solution cannot be inside it. Thus, suppose that a box Z which contains the global minimum of Program ( 3) is discarded by IBBA BBC. Z has only 3 distinct possibilities to be eliminated: (i) Z = ∅ after the propagation step. This implies that at least one constraint of the relaxed problem ( 4) is unsatisfiable, see [START_REF] Messine | Deterministic Global Optimization using Interval Constraint Propagation Techniques[END_REF]. Hence, no point in Z can satisfy all the constraints of problem ( 4) and therefore of problem (3). Thus, this possibility cannot occur; (ii) using interval arithmetic, an analytical constraint is proved to be unsatisfiable for all points in Z. It is also an impossible case such as in (i); (iii) f ≥ z := lb(f, Z). This implies that no point in Z can have a value lower than f, therefore Z does not contain a global minimum. Therefore, we show that these 3 possibilities cannot occur and then that such a box Z cannot exist. Hence, IBBA BBC provides a global minimum of Program (3).

Numerical experiments

To illustrate our methodology, we consider the design of an electromechanical machine with permanent magnets. See [START_REF] Fitan | The Electromagnetic Actuator Design Problem : A General and Rational Approach[END_REF] for details about its mixed non-convex analytical formulation.

For the computations of the electromagnetic torque of the machine, we use NUMT which is an efficient finite element numerical algorithm that we developed, [START_REF] Fontchastagner | Design of Electrical Rotating Machines by Associating Deterministic Global Optimization Algorithm With Combinatorial Analytical and Numerical Models[END_REF]. Thus, here we have just one black-box constraint.

In Table 1, parameters of the machine are: -D its bore diameter, -L its length, -l a the thickness of the magnets, -E the winding thickness, -C the thickness of yoke, -β the polar arc factor,-k d the filling factor, -p the number of poles pairs, -m the number of slots per pole and phase, -b r the rotoric configuration, -b f the kind of electromotive force waveform, -σ m the type of permanent magnet, -σ mt the type of magnetic conductor. On this example, we minimize the mass (see [START_REF] Fontchastagner | Design of Electrical Rotating Machines by Associating Deterministic Global Optimization Algorithm With Combinatorial Analytical and Numerical Models[END_REF] for complementary results), the main constraint is about the torque of the machine which must be equal to 10 N•m (with a tolerance about ±0.2 N•m). In the column named IBBA, we solve program [START_REF] Fitan | The Electromagnetic Actuator Design Problem : A General and Rational Approach[END_REF] with only the analytical model and just using IBBA. Remark that the numerical value for the torque (computed using NUMT) is equal to 9.12 which is quite far from imposed value 10. In column IBBA BBC, we solve Program (3) with a zone of 10% around 10 for the torque and using NUMT. Then, we obtain 9.83 N•m for the numerical torque; notice that the analytical value is 10.21. We emphasize in bold all the parameters which have changed between these two resolutions (nine parameters have changed including the integer ones).

In Table 2, we represent the solutions found when the zone is progressively increased: 2%, 5% and so on. We note that a zone about 5% is sufficient to solve efficiently this design problem.

Conclusion

We show in this paper that it is possible to extend IBBA code in order to take into account global optimization problems with black-box constraints. We prove that our new algorithm provides the global minimum of such a problem. Our new method is validated by solving a problem of design of an electromechanical machine with permanent magnets.

2 .

 2 Set f := +∞ and set L := (+∞, (X, Y )). 3. Extract from L the lowest lower bound. 4. Bisect the considered box chosen by its midpoint: V1, V2. 5. For j:=1 to 2 do (a) Vj :=Propagation of the analytical constraints on Vj,[START_REF] Messine | Deterministic Global Optimization using Interval Constraint Propagation Techniques[END_REF].(b) if Vj 6 = ∅ then i. Compute vj := lb(f, Vj).ii. Compute all the lower and upper bounds of all the analytical constraints on Vj.iii. if f ≥ vj and no analytical constraint is unsatisfied then insert (vj , Vj) in L. set m the midpoint of Vj if m satisfies all the analytical constraints and then if the black-box constraints are also satisfied then f := min( f, f (m)). if f is changed then remove from L all (z, Z) where z > f and set ỹ := m.6. If fmin(z,Z)∈L z < ǫ (where z = lb(f, Z)) then STOP. Else GoTo Step 4.

Table 1 .

 1 Results for the minimization of the Mass.

	Parameter	Bounds	Unit	IBBA	IBBA BBC
	D	[0.01, 0.3]	m	0.1430	0.1320
	L	[0.01, 0.3]	m	0.0491	0.0519
	la	[0.003, 0.01]	m	0.0034	0.0047
	E	[0.005, 0.03]	m	0.0075	0.0082
	C	[0.003, 0.02]	m	0.0040	0.0039
	β	[0.7, 0.9]		0.73	0.79
	k d	[0.4, 0.6]		0.5020	0.5024
	p	[[3, 10]]		8	9
	m	{1, 2}		1	1
	br	{0, 1}		0	1
	b f	{0, 1}		1	1
	σm	{1, 2}		2	2
	σmt	{1, 2}		2	2
	Mass	kg	2.92	3.35
	Analytical Torque	N•m	9.81	10.21
	Numerical Torque	N•m	9.12	9.83
	CPU -Time	min	0'51	34'44
	Iterations		152,126	216,623
	Iterations of NUMT		-	2,898

Table 2 .

 2 Results when the zone varies.

	zone: pc = Mass	Γem NUMT	Time	Its Its of NUMT
	IBBA	2.92	9.81	9.12	0'51 152,126	-
	2%	3.44 10.18	9.92	7'59 223,769	585
	5%	3.35 10.21	9.83	17'08 213,094	1,404
	10%	=	=	=	34'44 216,623	2,898
	20%	=	=	=	79'29 223,118	7,162
	30%	=	=	= 159'25 228,324	14,751
	40%	=	=	= 281'09 231,513	25,004