Sébastien Viudes

Rémy Zamar

Vincent Burlat

Fabrice Roux

Christophe Dunand
email: christophe.dunand@univ-tlse3.fr

Genome wide association study of Arabidopsis seed mucilage layers at a regional scale

Keywords: Arabidopsis thaliana, Environmental factor, GWAS, Mucilage, Natural accessions

The myxospermous species Arabidopsis thaliana extrudes a polysaccharidic mucilage from the seed coat epidermis during imbibition. The whole seed mucilage can be divided into a seed-adherent layer and a fully soluble layer, both layers presenting natural genetic variations. The adherent mucilage is variable in size and composition, while the soluble mucilage is variable in composition and physical properties. Studies reporting both the genetic architecture and the putative selective agents acting on this natural genetic variation are scarce. In this study, we set up a Genome Wide Association study (GWAS) based on 424 natural accessions collected from 166 natural populations of A. thaliana located southwest of France and previously characterized for a very important number of abiotic and biotic factors.

We identified an extensive genetic variation for both mucilage layers. The adherent mucilage was mainly related to precipitation and temperature whereas the non-adherent mucilage was unrelated to any environmental factors. By combining a hierarchical Bayesian model with a local score approach, we identified 55 and 28 candidate genes, corresponding to 26 and 10 QTLs for the adherent and nonadherent mucilages, respectively. Putative or characterized function and expression data available in the literature were used to filter the candidate genes. Only one gene among our set of candidate genes was already described as a seed mucilage actor, leaving a large set of new candidates putatively implicated inseed mucilage synthesis or release. The present study lay out foundation to understand the influence of regional ecological factors acting on seed mucilage in A. thaliana.

Introduction

Seed mucilage is a polysaccharidic hydrogel extruded around seeds upon imbibition by the outermost epidermal cell layer of the seed coat in myxospermous species [START_REF] Phan | New insights into the composition and structure of seed mucilage[END_REF]. Seed mucilage can help plant development from germination until seedling establishment in biotic and abiotic stress conditions [START_REF] Viudes | Seed mucilage evolution: Diverse molecular mechanisms generate versatile ecological functions for particular environments[END_REF]. Depending on plant species, the interaction between seed mucilage and biotic stresses can be direct, e.g. escaping ant predation by sticking to soil [START_REF] Pan | Mucilage binding to ground protects seeds of many plants from harvester ants: A functional investigation[END_REF] or facilitating passage through the digestive tract of birds [START_REF] Kreitschitz | The role of mucilage envelope in the endozoochory of selected plant taxa[END_REF], or indirect, e.g. mobilizing soil bacteria that limit the development of plant pathogen fungi, which lead to reduced disease symptoms and a better plant development [START_REF] Meschke | Streptomyces lividans inhibits the proliferation of the fungus Verticillium dahliae on seeds and roots of Arabidopsis thaliana[END_REF]. Concerning abiotic stresses, seed mucilage is known to influence or to rely on water around the seed by regulating the incoming flow, and by changing soil water retention capacity [START_REF] Deng | The effect of natural seed coatings of Capsella bursa-pastoris L. Medik. (shepherd's purse) on soil-water retention, stability and hydraulic conductivity[END_REF][START_REF] Saez-Aguayo | Local evolution of seed flotation in Arabidopsis[END_REF][START_REF] Teixeira | Myxospermous seed-mucilage quantity correlates with environmental gradients indicative of water-deficit stress: Plantago species as a model[END_REF]. The ecological function of seed mucilage was historicaly studied in several plant species. In contrast, the genetics and development of seed mucilage was nearly exclusively studied in the model plant Arabidopsis thaliana.

In A. thaliana, around one hundred genes are participating to mucilage secretory cell formation and mucilage biosynthesis and release [START_REF] Viudes | Myxospermy evolution in brassicaceae: A highly complex and diverse trait with Arabidopsis as an uncommon model[END_REF]. A large majority of these genes have been characterized through classical forward and reverse genetic studies. First of these were major regulators or main actors of mucilage establishment that lead to drastic phenotypes when mutated [START_REF] Western | Isolation and characterization of mutants defective in seed coat mucilage secretory cell development in Arabidopsis[END_REF].

Progressively, the newly discovered genes had more subtle roles for myxospermy, displaying more discrete phenotypes such as specific polysaccharide proportions or decorations when mutated.

Evidence of natural diversity for seed mucilage was first shown with the characterisation of an A. thaliana natural genotype, originated from Tajikistan, which is not able to extrude mucilage (Macquet et al., 2007). Among 22 A. thaliana populations originating from central Asia and Scandinavia, seven were reported to be fully defective for mucilage release [START_REF] Saez-Aguayo | Local evolution of seed flotation in Arabidopsis[END_REF]. Total seed mucilage can have great polysaccharide composition changes among natural populations, regarding mannose, galactose, galacturonic acid and rhamnose monosaccharide contents [START_REF] Voiniciuc | Extensive natural variation in Arabidopsis seed mucilage structure[END_REF]. The A. thaliana mucilage has two distinct layers, the inner layer is highly adherent to the seed thanks to cellulosic microfibrils, and the outer layer is fully soluble [START_REF] Tsai | Identification and characterization of Arabidopsis seed coat mucilage proteins[END_REF]. The volume of adherent mucilage was shown to be also variable among natural populations probably linked to the observed cellulose crystalline structure changes [START_REF] Voiniciuc | Extensive natural variation in Arabidopsis seed mucilage structure[END_REF]. The non-adherent mucilage exhibits also variable compositions and physical properties [START_REF] Poulain | Composition and physicochemical properties of outer mucilage from seeds of Arabidopsis natural accessions[END_REF]. Interestingly, the intra-species loss of myxospermy can occur several times due to independent genetic mutations [START_REF] Saez-Aguayo | Local evolution of seed flotation in Arabidopsis[END_REF]. Two independent Genome-Wide Association Studies (GWAS), based on A. thaliana adherent and non-adherent seed mucilage phenotyping, highlighted 21 and 11 significant SNPs, respectively, supporting the polygenic architecture of myxospermy [START_REF] Fabrissin | Natural variation reveals a key role for rhamnogalacturonan I in seed outer mucilage and underlying genes[END_REF][START_REF] Voiniciuc | Extensive natural variation in Arabidopsis seed mucilage structure[END_REF].

Due to the lack of availability of accurate environmental data for many A. thaliana natural populations, natural genetic variation in mucilage characteristics has so far never been associated with environmental data through a statistical approach.

Here, we aimed at using GWAS to further investigate the genetics underlying natural genetic variation of A. thaliana myxospermy. Our hypothesis was that this approach without a priori will highlight new candidate genes that were not yet linked to seed mucilage. The quantitative and independent phenotyping of both mucilage layers should increase the efficiency of GWAS and also provide the opportunity to study if a correlation exists between the two layers of mucilage. In addition, the availability of climatic conditions and soil composition for the studied populations created a unique opportunity to address the potential ecological role of myxospermy in A. thaliana.

Material and methods

Plant material

Our study was based on 168 natural populations of A. thaliana identified in spring 2014 and located in the Midi-Pyrénées region (south-west of France) [START_REF] Bartoli | In situ relationships between microbiota and potential pathobiota in Arabidopsis thaliana[END_REF][START_REF] Frachon | A genomic map of climate adaptation in Arabidopsis thaliana at a micro-geographic scale[END_REF]. The average distance between these populations is ~100 km. These populations have been characterized for six noncorrelated variables retrieved from the ClimateEU database [START_REF] Frachon | A genomic map of climate adaptation in Arabidopsis thaliana at a micro-geographic scale[END_REF], 13 chemical and 1 physical soil properties with soil samples collected in situ in late autumn 2015 and late winter 2015 [START_REF] Frachon | Adaptation to plant communities across the genome of Arabidopsis thaliana[END_REF], leaf and root bacterial microbiota and pathobiota in late autumn 2015 and late winter 2015 [START_REF] Bartoli | In situ relationships between microbiota and potential pathobiota in Arabidopsis thaliana[END_REF], and plant communities in late spring 2015 [START_REF] Frachon | Adaptation to plant communities across the genome of Arabidopsis thaliana[END_REF].

In addition, based on a Pool-Seq approach with an average of 15.3 plants per population, the 168 populations have been whole-genome sequenced with the Illumina technology, leading to the identification of 1,638,649 SNPs [START_REF] Frachon | A genomic map of climate adaptation in Arabidopsis thaliana at a micro-geographic scale[END_REF]. For the purpose of this study, we used 166 out of the 168 initial populations, each being composed of one to three accessions, thereby bringing the total of phenotyped seed lots to 424. Differences in the maternal effects among the 424 seed lots were reduced by growing one plant of each accession for one generation with the following steps: (i) for each accession, several seeds were sown in one 7 x 7 x 6 cm plastic pot (Soparco®, Condé-sur-Huisne, France) filled with damp standard culture soil (PROVEEN MOTTE 20,Soprimex®,France) on November 1 st 2016, (ii) seeds were stratified at 4°C for four days in order to break primary dormancy and promote germination, (iii) pots were transferred to greenhouse conditions (22°C, 16 h photoperiod) on November 4 th 2016, (iv) seedlings were thinned to one on November 25 th 2016, (v) pots were placed on a field station at the INRAE campus of Castanet-Tolosan (France) on December 5 th 2021 in order to promote flowering through natural vernalization, (vi) at the onset of flowering, plants were transferred in a greenhouse mimicking the outdoor conditions (no additional light and heating) but protecting plants from rainfall, and (vii) plants were separated from each other by aratubes in order to avoid cross-pollination among accessions. All seed lots were harvested during a 3-week period from late April to early May 2017 and were then stored at 4°C until the set-up of the phenotyping experiments.

Phenotyping experiments

Given the high number of accessions, the experiments were performed sequentially with several experimental batches distributed along a few months and randomly selected. The number of accessions in each batch was dependent on the phenotypic traits. In order to consider micro-environmental variations among the experimental batches, the same Columbia (Col-0) seed lot was used as an internal control that was systematically included in all experimental batches in order to normalize the results.

Adherent mucilage and seed area phenotyping

A total of 44 experimental batches, including Col-0 as a control and 11 different accessions, were set up during over 6 months. Because the adherent mucilage is strongly attached to the seed, we used a previously described protocol that allows comparing adherent mucilage area among accessions in the most standardized manner [START_REF] Francoz | Pectin demethylesterification generates platforms that anchor peroxidases to remodel plant cell wall domains[END_REF]. Briefly, to remove the non-adherent mucilage, 50 to 100 seeds per accession were vigorously shaken at 250 rpm in 1.6 ml of 0.01 M Tris-HCl pH 7.5 contained in a 2 ml tube. The remaining adherent layer of mucilage was then stained with a 0.02 % (w/v) ruthenium red solution in Tris-HCl. The seeds were rinsed in 0.01 M Tris-HCl pH 7.5, transferred in 24-well microplates containing 0.01 M Tris-HCl pH 7.5 and imaged using an Epson perfection V100 photo scanner at 6400 dpi resolution. The measurements of seed area including or not the adherent mucilage was realized with imageJ using a previously described macro [START_REF] Francoz | Pectin demethylesterification generates platforms that anchor peroxidases to remodel plant cell wall domains[END_REF]. The adherent mucilage area was calculated for each seed by subtracting the seed area alone from the seed area with the adherent mucilage. For each experimental batch, raw data for adherent mucilage area and seed area of each accession were standardized with the corresponding means of adherent mucilage and seed areas of Col-0.

Non-adherent mucilage area phenotyping

A total of 98 batches of experiments including Col-0 as a control and three or six different accessions, were set up over 6 months. The non-adherent mucilage is totally soluble in water and quickly released upon seed imbibition. To estimate its area, we used the MuSeeQ protocol [START_REF] Miart | MuSeeQ, a novel supervised image analysis tool for the simultaneous phenotyping of the soluble mucilage and seed morphometric parameters[END_REF]. Briefly, 5 to 10 dry seeds by accession were individually and synchronously disposed on perfectly flat 0.6 % (w/v) agarose media containing 0.00004 % (w/v) of toluidine blue O polychromatic stain (Sigma-Aldrich, Saint-Quentin-Fallavier, France). Upon the contact with the aqueous media, the non-adherent mucilage is released, spreading around the seed on the flat surface, and is simultaneously stained in pink contrasting with the blue colour of the media. Top-view pictures were taken 24 h after seed deposition with a camera (Canon DS 126271). The non-adherent mucilage area was measured for each seed using imageJ with a previously described macro [START_REF] Miart | MuSeeQ, a novel supervised image analysis tool for the simultaneous phenotyping of the soluble mucilage and seed morphometric parameters[END_REF]. For each experimental batch, raw data for non-adherent mucilage area of each accession were standardized with the corresponding mean of Col-0 non-adherent mucilage area.

Statistical analyses

Natural genetic variation

To estimate the extent of natural genetic variation for each mucilage trait, we run the following mixed models using the PROC MIXED procedure in SAS v9.4 (SAS Institute Inc., Cary, North Carolina, USA):

Model (1) Yijk = μ + populationi + accessionj(populationi) + seed areak + Ɛijk Where 'Y' is the mucilage area of seeds, 'μ' is the overall mean of mucilage area, 'population' accounts for genetic differences among the 166 populations, 'accession(population)' accounts for the mean genetic difference among accessions within population, 'seed area' is a covariate to control for the developmental effect of seed size on adherent mucilage area, and 'Ɛ' is the residual term. The factor 'population' was treated as a fixed effect, whereas the factor 'accession' was treated as a random effect.

Following [START_REF] Brachi | Investigation of the geographical scale of adaptive phenological variation and its underlying genetics in Arabidopsis thaliana[END_REF], this model random term was tested with a likelihood ratio test (LRT) with and without this effect.

For both phenotypic traits, we estimated the percentage of variance observed among and within populations by running Model 1 with the VARCOMP procedure in SAS v9.4 (SAS Institute Inc., Cary, North Carolina, USA). In addition, we estimated genotypic values of the 166 natural populations by calculating least-square means (LSmeans) of each population after removing the 'accession(population)' term from Model 1. Spearman's correlation coefficients (Spearman's rho) were calculated to estimate relationships between genetic differences among populations (i.e. genotypic values) and abiotic ecological variables. Resulting p-values were corrected with an FDR (False Discovery Rate) correction.

GWA mapping with local score analysis

To detect QTLs associated with natural genetic variation of mucilage area related traits, we combined a GWA mapping approach with a local score approach. This combination of approaches already allowed the successful detection and cloning in A. thaliana of five QTLs in response to bacterial pathogens [START_REF] Aoun | A complex network of additive and epistatic quantitative trait loci underlies natural variation of Arabidopsis thaliana quantitative disease resistance to Ralstonia solanacearum under heat stress[END_REF][START_REF] Demirjian | Study of natural diversity in response to a key pathogenicity regulator of Ralstonia solanacearum reveals new susceptibility genes in Arabidopsis thaliana[END_REF][START_REF] Demirjian | Bacterial wilt susceptibility 1, an atypical NLR gene revealed through the natural genetic diversity of response of Arabidopsis thaliana to Ralstonia solanacearum type III effector mutants[END_REF]. First, we retrieved for each of the 166 populations the standardized allele frequencies for all the SNPs (see 'Plant material' subsection). These standardized allele frequencies result from raw allele frequencies after accounting for the scaled covariance of population allele frequencies, thereby making the phenotype-genotype associations robust to complex demographic histories and allowing decreasing drastically the rate of false positives [START_REF] Frachon | Adaptation to plant communities across the genome of Arabidopsis thaliana[END_REF][START_REF] Frachon | A genomic map of climate adaptation in Arabidopsis thaliana at a micro-geographic scale[END_REF]. Second, for each phenotypic trait, we estimated phenotype-genotype associations across the genome by calculating Spearman correlation coefficients between standardized allele frequencies of each SNP and genotypic values of the 166 populations, using the cor.test function implemented under the R environment. Third, we implemented a local score approach on the set of pvalues associated with the Spearman's rho values. The local score allows accumulating the statistical signals from contiguous genetic markers in order to detect significant genomic regions associated with phenotypic natural variation [START_REF] Fariello | Accounting for linkage disequilibrium in genome scans for selection without individual genotypes: The local score approach[END_REF]. Through the p-values in a given QTL region, the association signal will cumulate locally due to linkage disequilibrium between SNPs, which will then increase the local score [START_REF] Bonhomme | A local score approach improves GWAS resolution and detects minor QTL: application to Medicago truncatula quantitative disease resistance to multiple Aphanomyces euteiches isolates[END_REF]. The tuning parameter ξ was fixed at 2 and significant SNP-phenotype associations were identified by estimating a chromosome-wide significance threshold for each chromosome [START_REF] Bonhomme | A local score approach improves GWAS resolution and detects minor QTL: application to Medicago truncatula quantitative disease resistance to multiple Aphanomyces euteiches isolates[END_REF].

Based on a custom script described in [START_REF] Libourel | The genomic architecture of competitive response of Arabidopsis thaliana is highly flexible among plurispecific neighborhoods[END_REF], we retrieved all candidate genes underlying QTLs by selecting all genes inside the QTL regions and also the first gene upstream and the first gene downstream of these QTL regions. The TAIR 10 database (https://www.arabidopsis.org/) was used as a reference.

Transcriptomic profiling of candidate genes

The expression profiles of the candidate genes were extracted from A. thaliana tissue-specific seed development transcriptomic data [START_REF] Belmonte | Comprehensive developmental profiles of gene activity in regions and subregions of the Arabidopsis seed[END_REF]. Whole transcriptomic data set used to decipher the seed coat specific expression of the candidate genes are available in Supplementary Table S2.

Results

Large genetic variation among and within natural populations for seed mucilage

The sizes of adherent and non-adherent mucilage layers were measured for each accession belonging to the 166 populations (Fig. 1). Although the two different mucilages exist in 3D, their quantifications come from 2D images. Mucilage values are therefore expressed in surface unit. For adherent mucilage, measured values correspond to orthogonal projections, which are coherent with the reality since the seeds systematically lay down longitudinally in solution due to their oblong shape. Non-adherent mucilage spreads on a flat surface and is consequently already a 2D object before the imaging.

For both mucilage layers, a large phenotypic variation was observed among the 166 populations (Fig. 1). The ratio of the adherent-mucilage area compared to Col-0 largely differed among the accessions from south-west of France, with values ranging from 0.2 to more than 1.5 (Supplementary Fig. S1). In comparison, this ratio for the non-adherent mucilage shows less variability among populations, with values ranging from 0.5 to 1.5 (Supplementary Fig. S1). Note that the sizes of both mucilage layers are positively correlated with seed size (Supplementary Table S1).

On average, a large and significant genetic variation was detected among accessions within populations (Table 1), with the observation of populations presenting either no phenotypic or extensive variation among accessions for adherent or non-adherent mucilage areas (Fig. 2). Overall, while 32.3 % and 13.8 % of phenotypic variance for adherent and non-adherent mucilage areas result from a genetic differentiation among populations, respectively, a higher proportion of phenotypic variance result from a genetic differentiation among accessions within populations, i.e. 42.7% and 34.1% for adherent or non-adherent mucilage areas, respectively (Table 1).

No correlation was observed at the population level between the standardized adherent mucilage area and standardized non-adherent mucilage area (Spearman's rho 0.09, P = 0.27) (Fig. 3).

Relationship between mucilage and environmental abiotic parameters

Among population, variation of the standardized adherent mucilage area was significantly correlated with several abiotic parameters collected from their native sites [START_REF] Frachon | Adaptation to plant communities across the genome of Arabidopsis thaliana[END_REF][START_REF] Frachon | A genomic map of climate adaptation in Arabidopsis thaliana at a micro-geographic scale[END_REF]) (Fig. 4).

The area of the adherent mucilage was negatively correlated with the mean of annual temperatures (Fig. 4a) and the mean of coldest month temperatures (Fig. 4b) and positively correlated with the precipitation level during winter (Fig. 4c) and autumn (Fig. 4d). In contrast, variation of the non-adherent mucilage area among populations was not correlated with any climate variables (Table 2, Supplementary Table S1). In addition, while the adherent seed mucilage area was weakly associated with the total concentration in nitrogen (positive relationship) and pH (negative relationship) (Supplementary Table S1), no significant correlation was detected for the non-adherent mucilage with any of the variables describing the agronomic properties of the soil of the native sites (Table 2, Supplementary Table S1).

The phenotypic variation of the standardized areas of seed mucilage layers is highly polygenic

GWA analyses revealed a polygenic architecture for both phenotypic traits, with the detection of 26 QTLs overlapping with 55 genes associated with the adherent mucilage area variability and 10 QTLs overlapping with 28 genes associated with the non-adherent mucilage area variability (Fig. 5, Supplementary Fig. S2 andS3). The 36 QTL(corresponding to 83 ORFs) encompass 512 SNPs distributed as following 198 in the coding regions (1 stop gained, 71 non synonymous mutations and 126 synonymous mutations), 177 in the non coding regions and 137 upstream or downstream of the genes (Supplementary Table S4). Among these 36 QTLs, none were simultaneously associated with both mucilage layers. Some of the 83 genes identified in our study are members of multigenic families containing already described myxospermy-related genes. For instance, we identified GA2OX4 (At1g47990) and SK19 (At2g03160) for adherent mucilage (Fig. 6a), and HB-2 (At4g16780), SK41 (At1g09840), CuAOα2 (At1g31690), and CuAOα3 (At1g31710) for non-adherent mucilage (Fig. 6b). These candidate genes belong to four independent multigenic families containing gene members of the seed mucilage toolbox: GA3OX4 (At1g80330; [START_REF] Kim | Contribution of gibberellins to the formation of Arabidopsis seed coat through starch degradation[END_REF]), SK11/SK12 (At4g34210/At4g34470; [START_REF] Li | Site-specific phosphorylation of TRANSPARENT TESTA GLABRA1 mediates carbon partitioning in Arabidopsis seeds[END_REF]), HB-25 (At5g65410; [START_REF] Bueso | ARABIDOPSIS THALIANA HOMEOBOX25 uncovers a role for gibberellins in seed longevity[END_REF]), and CuAOα1 (At1g31670, [START_REF] Fabrissin | Natural variation reveals a key role for rhamnogalacturonan I in seed outer mucilage and underlying genes[END_REF]). Interestingly, two multigenic families not previously shown to be involved in myxospermy were overrepresented, such as the cytochrome P450 (CYP450) family with six genes (At2g34500, At4g15380, At3g25180, At4g15393, At2g34490, At1g13110) and the serine carboxypeptidase like (SCPL) family with 4 genes (At2g23000, At2g22980, At2g22970, At2g22990) (Fig. 6). SFP2 (At5g27360) is a candidate gene identified within the most significant QTL associated with the adherent mucilage area (i.e. QTL 24, Fig. 5) with the presence of 42 significant SNPs within the gene (Supplementary Table S3). Interestingly, SFP2 is expressed during early seed development (Fig. 6a).

Worth of noting, QTL 24 also contains SFP1 (At5g27350), the tandem duplicated of SFP2, , but notexpressed during seed development (Fig. 6a). For the non-adherent mucilage area, an interesting polysaccharide-related candidate gene is At1g13130, an uncharacterized glycosyl hydrolase family 5 (GH5) found upstream of 22 significant SNPs within QTL 2 (Supplementary Table S3).

Discussion

In this study, we aimed at (i) exploring the natural genetic variation in myxospermy among 166 natural populations of A. thaliana collected at a regional scale, (ii) identifying the ecological factors acting as putative drivers of myxospermy, and (iii) applying recent methodologies in GWA mapping to unravel the underlying genetic architecture.

The whole study is based on seed mucilage area variation. The main concern was the slight positive correlation of seed size with the size of both mucilage layers, (Supplementary Table S1). If the cells keep a proportional size in the seed, it is coherent to have larger mucilage secretory cells in larger seeds (not evaluated here) and, consequently, more abundant mucilage. Nonetheless, extensive genetic variation was detected for both mucilage traits after correcting for the effect of seed size variation, using seed size as a covariate (Table 1).

Adherent and non-adherent mucilage have different polysaccharide compositions that provide structure only to the former (Macquet et al.,207b). According to the hypothesis that for an equal amount of synthetized mucilage, the adherent and non-adherent mucilage areas are dependent on their structural polysaccharide content, the two layers of mucilage should be negatively correlated. However, adherent mucilage and non-adherent mucilage were uncorrelated in this study (Fig. 3), thereby contradicting this hypothesis. Consequently, the differences of mucilage area among accessions could be due to differences in total amount of synthetized mucilage and/or in a change in mucilage composition. Then, in this scenario, we have more likely measured a value that depends on a combination of mucilage amount and composition. Note that the experimental estimation of the area of the non-adherent mucilage could also reflect its ability to spread on agarose, then its structure, rather than its total amount. Variation among populations in adherent mucilage is negatively correlated with temperature. It was recently demonstrated that the thick adherent seed mucilage of Lepidium perfoliatum, another myxospermous Brassicaceae species, prevents the germination under relatively low temperatures (10°C and 15°C) and increases germination under higher temperatures (25°C and 30°C), in particular in presence of a long light time exposure (Zhou et al., 2021). This suggests a role of the adherent mucilage in germination inhibition under cold temperature until optimal conditions are obtained. The observed correlation between adherent mucilage and temperature and precipitation could therefore be indirect, through germination regulation. Indeed, A. thaliana seed mucilage was often assumed to favor germination by providing more water to the embryo [START_REF] Arsovski | AtBXL1 encodes a bifunctional -D-Xylosidase/ -L-Arabinofuranosidase required for pectic arabinan modification in Arabidopsis mucilage secretory cells[END_REF][START_REF] Penfield | MYB61 is required for mucilage deposition and extrusion in the Arabidopsis seed coat[END_REF]. However, 24 h after imbibition, A. thaliana embryos contain less water in mucilaginous seeds than in nonmucilaginous seeds [START_REF] Saez-Aguayo | Local evolution of seed flotation in Arabidopsis[END_REF]. In L. perfoliatum, seed mucilage was shown to increase drastically germination in optimal water content whereas no impact was found under osmotic stress (Zhou et al., 2021). The absence of correlation between soil composition and non-adherent mucilage might be explained by the variability of soil composition at the micro-scale level and also because the non-adherent mucilage will rapidly spread into the ground during the first rain and thus could not influence directly the seed development. However, as the non-adherent mucilage spreads into the ground, it is surprising that physical parameters such as soil water holding capacity are not correlated.

Therefore, the fact that mucilage phenotypes are rather related to temperature and precipitation than soil properties suggests a possible physiological importance of seed mucilage during underground seedling development rather than during germination. Fewer QTLs were found for the non-adherent mucilage than for the adherent mucilage, which is in agreement with the estimates of inter-population variance (13.8 % for non-adherent mucilage compared to 32.3 % for adherent mucilage). In A. thaliana, about one hundred genes have previously been characterized for their implication in seed mucilage biosynthesis and release [START_REF] Viudes | Myxospermy evolution in brassicaceae: A highly complex and diverse trait with Arabidopsis as an uncommon model[END_REF]. Our GWA mapping approach allowed identifying 82 new candidate genes and only one belonging to the list of previously validated candidate genes. The 100 genes already characterized from previous studies together with our GWA analyses highlight the extreme polygenic character for mucilage traits. Our GWAS revealed genes inducing weak phenotypes more difficult to identify with regular genetic approach (reverse and forward genetics), which can in part explain the small overlap between the two sets of identified genes.

The gene already known for its implication in seed mucilage is CuAOα1 (At1g31670) encoding a copper amine oxidase that was identified with GWAS on non-adherent mucilage (Supplementary Fig. S3 and Table S3). The cuaoα1 mutant has less rhamnogalacturonan I (RGI) pectin domain in its non-adherent mucilage [START_REF] Fabrissin | Natural variation reveals a key role for rhamnogalacturonan I in seed outer mucilage and underlying genes[END_REF]. Interestingly, CuAOα1 has been identified with two other genes in a previous GWAS on mucilage, which did not reveal any other previously characterized myxospermyrelated genes [START_REF] Fabrissin | Natural variation reveals a key role for rhamnogalacturonan I in seed outer mucilage and underlying genes[END_REF][START_REF] Voiniciuc | Extensive natural variation in Arabidopsis seed mucilage structure[END_REF] illustrating the pionner capacity of this approach. In addition to its expression in seeds that is restricted to the seed coat, CuAOα1 is expressed in root tips [START_REF] Klepikova | A high resolution map of the Arabidopsis thaliana developmental transcriptome based on RNA-seq profiling[END_REF], especially in the lateral root cap and it is induced by high nitrate concentration [START_REF] Gifford | Cell-specific nitrogen responses mediate developmental plasticity[END_REF]. During the lateral root cap cell formation, a pectinaceous mucilage accumulates inside the sixth cell layer (c6) below the quiescent center to, in fine extrude root mucilage around the c7 [START_REF] Maeda | Identification of periplasmic root-cap mucilage in developing columella cells of Arabidopsis thaliana[END_REF]. Because CuAOα1 seems to be involved in RGI production in seed mucilage [START_REF] Fabrissin | Natural variation reveals a key role for rhamnogalacturonan I in seed outer mucilage and underlying genes[END_REF], it would be interesting to investigate as well its putative role in root mucilage. Although a putative link between root mucilage and non-adherent mucilage remains elusive, there is an interesting parallel between their developmental and ecological roles.

The identification of new putative candidate genes is also well supported by their spatio-temporal expression. Among the 83 candidates, 40 are expressed in the seed coat along the Col-0 seed development kinetics [START_REF] Belmonte | Comprehensive developmental profiles of gene activity in regions and subregions of the Arabidopsis seed[END_REF] (Fig. 6). Previously characterized functions, associated with seed development expression data, provide interesting elements for some candidates. Intriguingly, several candidate genes are related to the secondary metabolism. IMS2/MAM3 (At5g23020), with a role in glucosinolate biosynthesis [START_REF] Petersen | Changing substrate specificity and iteration of amino acid chain elongation in glucosinolate biosynthesis through targeted mutagenesis of Arabidopsis methylthioalkylmalate synthase 1[END_REF], is located upstream of 10 significant SNPs within the adherent mucilage QTL 21 (Supplementary Table S3) and expressed only at the mature green stage, just before seed drying (Fig. 5 and Fig. 6a). The CYP450 gene, CYP71A2 (At2g34490), contains the 17 significant SNPs of the non-adherent mucilage QTL 8 and shows a seed coat-specific expression increasing along seed development (Fig. 6b) and extending to endosperm at mature green stage (Supplementary Table S2). CYP450 genes can be implicated in seed suberin and cutin biosynthesis [START_REF] Renard | Identification of novel seed longevity genes related to oxidative stress and seed coat by genome-wide association studies and reverse genetics[END_REF] and are often implicated in secondary metabolite decoration [START_REF] Schuler | Functional genomics of P450s[END_REF]. SCPL11 (At2g22970), SNG1 (At2g22990), and SCPL10 (At2g23000), localized within non-adherent QTLs 6 and 7 (Fig. 5), have a seed coat specific expression. These three genes are serine carboxypeptidase-like proteins that belong to a large multigenic family implicated in secondary metabolism. SCPL19/SNG2 (At5g09640) has a role in sinapoylcholine formation in A. thaliana seeds [START_REF] Shirley | The sng2 mutant of Arabidopsis is defective in the gene encoding the serine carboxypeptidase-like protein sinapoylglucose:choline sinapoyltransferase[END_REF]. Although a direct link between A. thaliana seed mucilage and secondary metabolism remains elusive, a regulatory network was identified between seed mucilage, seed pigments and seed secondary metabolites [START_REF] Lepiniec | Genetics and biochemistry of seed flavonoids[END_REF][START_REF] Salem | Regulatory-associated protein of TOR (RAPTOR) alters the hormonal and metabolic composition of Arabidopsis seeds, controlling seed morphology, viability and germination potential[END_REF][START_REF] Viudes | Seed mucilage evolution: Diverse molecular mechanisms generate versatile ecological functions for particular environments[END_REF]. SFP1, the paralog of SFP2 that is located within QTL 24, encodes a monosaccharide transporter during leaves senescence [START_REF] Quirino | One of two tandem Arabidopsis genes homologous to monosaccharide transporters is senescence-associated[END_REF]. The involvement of SFP2 in sugar transport related to seed adherent mucilage polysaccharides remains to be demonstrated. At1g13130 found upstream of 22 significant SNPs is a GH5 whose substrate specificity is unknown. However, GH5 family members have glycosyl hydrolase activities on various substrates (http://www.cazy.org/GH5.html) including polysaccharides found in the mucilage secretory cells (MSCs) such as cellulose, xylan or mannan and might be implicated in mucilage polysaccharide hydrolysis. At1g13130 is expressed sharply at heart and linear cotyledon stages (Fig. 6a), co-occurring partially with cellulose fibrils formation since CESA5 (At5g09870) (main cellulose synthase involved in seed mucilage) is highly expressed at the linear cotyledon stage [START_REF] Sullivan | CESA5 is required for the synthesis of cellulose with a role in structuring the adherent mucilage of Arabidopsis seeds[END_REF].

Additionally, in other species than A. thaliana, seed and root mucilages were shown to change soil physical properties, which in turn influence water retention and water availability [START_REF] Ahmed | Drying of mucilage causes water repellency in the rhizosphere of maize: measurements and modelling[END_REF][START_REF] Deng | The effect of natural seed coatings of Capsella bursa-pastoris L. Medik. (shepherd's purse) on soil-water retention, stability and hydraulic conductivity[END_REF], and induce modification of the rhizospheric microbial communities [START_REF] Sasse | Feed your friends: Do plant exudates shape the root microbiome?[END_REF]. If the seed mucilage is also implicated in biotic interactions, it may explain the association of both seed mucilage layers with several candidate genes implicated in secondary metabolism identified in our GWAS, knowing that plant secondary metabolites play crucial roles during plant-microbes interactions [START_REF] Kessler | Plant secondary metabolite diversity and species interactions[END_REF].

Conclusion

The seed mucilage was highly variable among the Midi-Pyrénées regional populations for both layers.

The detected QTLs underlying this morphological variability, contain one previously characterized gene and 82 putative new seed mucilage actors. Considering the 100 genes already characterized in previous studies, this highlights the extreme polygenic character of this complex trait in A. thaliana. The candidate gene CuAOα1 (At1g31670) identified in our study for the non-adherent mucilage, was previously shown to be involved in non-adherent mucilage, thereby validating the significance of our approach. New candidate genes coming from our study await functional validation. The correlation of adherent seed mucilage area with temperature and precipitations seems to be important and the tempting indirect link of this correlation with germination will be interesting to validate through germination assays using seed with or without mucilage. Altogether this study shed light on putative new genetic actors and on the ecological role of seed mucilage. Zhou, Z., Xing, J., Zhao, J., Liu, L., Gu, L., Lan, H., 2021. The ecological roles of seed mucilage on germination of Lepidium perfoliatum, a desert herb with typical myxospermy in Xinjiang. Plant Growth Regul. 97, 185-201. https://doi.org/10.1007/s10725-021-00702-y

Table 1 Global statistics for the used GWAS model shows that variance is surprisingly more explained by morphological variability between the 424 accessions than between the 166 populations. F stands for a F-test, LTR for likelihood ratio test, P for the P-value and R 2 for coefficient of determination ("R squared")

Table 2 Spearman correlation across every populations between measured traits and their harvest site environmental parameters. Indicated values are correlation coefficients that are significant after FDR correction. Each dot corresponds to one population containing data from its accessions. rho is the spearman correlation coefficient. The Spearman correlation coefficient is indicated above each graph. The black line is the linear model with its confidence interval in grey. These four environmental parameters are the only ones to remain significant (P-value < 0.001) after FDR correction of Pvalues among the tested parameters (Supplementary Table S1). Each dot corresponds to one population containing data from its accessions. S1). Each dot corresponds to one population containing data from its accessions. Table S3 Distribution and number of significant SNPs in each QTLs. The table shows the number of SNPs identified with the GWAS performed on adherent mucilage (left) and non-adherent mucilage (right) and the ID of the gene overlapping with each SNP, or found within a 2 kb upstream and downstream region.

Fig. 1 .

 1 Fig. 1. Measurement of adherent (a) and non-adherent mucilage (b) of each accession belonging to the 166 natural populations and source picture examples illustrating the extreme area values. Each individual boxplot represents the measured area distribution (µm²) of the adherent mucilage within a population. Each population contains one to three accession. For each accession 50 to 100 seeds or 5 to 10 seeds are phenotyped for adherent and non-adherent mucilage respectively. Boxplots are ordered by their mean value. The images correspond to the extreme values to highlight the existing contrast. Measurement were realized on such images after vigorous shaking and ruthenium red staining or 24 h after dry seed deposition on a Toluidine blue agarose media for adherent and nonadherent mucilage respectively.

Fig. 2 .

 2 Fig. 2. The populations can contain either homogeneous or contrasted accessions for each mucilage layers. Example of adherent (a) and non-adherent (b) mucilage standardized area values for four population examples containing three homogeneous (left side) and three contrasted (right side) accessions, respectively.

Fig. 3 .

 3 Fig. 3. The two mucilage areas are not correlated at the population level. Scatter plots of the mean value of each population for the standardized adherent mucilage versus standardized non-adherent mucilage areas. The black line corresponds to the linear model with its confidence interval in grey.Each dot corresponds to one population containing data from its accessions. rho is the spearman correlation coefficient.

Fig. 4 .

 4 Fig. 4. The adherent mucilage area is negatively correlated with temperature and positively with precipitation. Graphs show the relationship between the standardized adherent mucilage area and the mean annual temperature (a), mean coldest month temperature (b), winter precipitation (c) and autumn precipitation (d).The Spearman correlation coefficient is indicated above each graph. The black line is the linear model with its confidence interval in grey. These four environmental parameters are the only ones to remain significant (P-value < 0.001) after FDR correction of Pvalues among the tested parameters (Supplementary TableS1). Each dot corresponds to one population containing data from its accessions.

Fig. 5 .

 5 Fig. 5. Highlight of genome QTLs explaining the morphological variability of adherent and nonadherent seed mucilage standardized area. Manhattan plots representing the Lindey process value for each SNP among the genome of A. thaliana, highlights 26 QTLs for adherent (a) and 10 QTLs

Fig. 1 .Fig. 2 .

 12 Fig. 1. Measurement of adherent (a) and non-adherent mucilage (b) of each accession belonging to the 166 natural populations and source picture examples illustrating the extreme area values. Each individual boxplot represents the measured area distribution (µm²) of the adherent mucilage within a population. Each population contain one to three accession. For each accession 50 to 100 seeds or 5 to 10 seeds are phenotyped for adherent and non-adherent mucilage respectively. Boxplots are ordered by their mean value. The images correspond to the extreme values to highlight the existing contrast. Measurement were realized on such images after vigorous shaking and ruthenium red staining or 24 h after dry seed deposition on a Toluidine blue agarose media for adherent and non-adherent mucilage respectively.

Fig. 3 .Fig. 4 .

 34 Fig. 3. The two mucilage areas are not correlated at the population level. Scatter plots of the mean value of each population for standardized adherent mucilage versus non-adherent mucilage areas. The black line corresponds to the linear model with it confidence interval in grey. Each dot corresponds to one population containing data from its accessions. rho is the spearman correlation coefficient.

Fig. 5 .

 5 Fig. 5. Highlight of genome QTLs explaining the morphological variability of adherent and nonadherent seed mucilage layers. Manhattan plots representing the Lindey process value for each SNP among the genome of A. thaliana, highlights 26 QTLs for adherent (a) and 10 QTLs for non-adherent (b) seed mucilage. The threshold used for QTL selection is represented by the dotted line at Lindey value of 5. Everything below is considered as background noise.

  Fig. S1. Relatively low variation of the three parameters measured on the Col-0 standard included in each batch and repartition of the observed variability of the studied natural populations phenotypes after normalization with the internal Col-0 standard. (a) Boxplot showing mean values of seed area and adherent mucilage area of the Col-0 seeds used as a standard in the 44 individual experimental batches during the phenotyping of studied populations. The areas are expressed in µm². (b) Boxplot showing mean values of non-adherent mucilage areas of the Col-0 seeds used as a standard in the 98 batches of experiment during the phenotyping of studied populations. The areas are expressed in mm². (c) Violin plot showing mean values of seed, adherent mucilage, and non-adherent mucilage area normalized with the Col-standard (y axis) for each of the 424 accessions. Area values are expressed relatively to the mean area of Col-0 seeds (set to 1) included in each respective experimental batch. The x axis wideness of violin plot shows the relative frequency of accessions for the corresponding value displayed on the y axis.

Fig. S2 (

 S2 Fig. S2 (following).Zoom on significant QTLs related to the genetic variability of adherent mucilage. Each pic is delimited by the two vertical dot lines with the first and last significant SNP contain in it. Each SNP is ploted with its -log10 of the P-value and the colored line represent the Lindley process evolution.

Fig. S3 .

 S3 Fig. S3. Zoom on significant QTLs related to the genetic variability of non-adherent mucilage.Each pic is delimited by the two vertical dot lines with the first and last significant SNP contained in it. Each SNP is ploted with its -log10 of the P-value and the colored line represent the Lindley process evolution.

Table 1

 1 Global statistics for the used GWAS model shows that variance is surprisingly more explained by morphological variability between the 424 accessions than between the 166 populations. F stands for a F-test, LTR for likelihood ratio test, P for the P-value and R 2 for coefficient of determination ("R squared")

		Adherent mucilage area	Non adherent mucilage area
	Terms	F or LRT	P	R ²	F or LRT	P	R²
	Population	2,9 3,2E-14	32,3	1,82 1,1E-05	13,8
	Accession (Population)	30514,8 1,0E-16	42,7	642,3 1,0E-16	34,1
	Seed area	13811,8 1,0E-32	-	ne	ne	ne
	Residuals			25,0			52,1

Table 2

 2 Spearman correlation across every population between measured traits and their harvested site environmental parameters. Indicated values are correlation coefficients that are significant after FDR correction.

  About 50% of the 83 candidate genes from the GWAS are expressed in the seed coat during seed development. Fifty five and 28 selected genes are present within 26 and 10 QTLs highlighted by

	AGI code a	Number of SNPs Gene name	Preglobular	Globular	Heart	Linear cotyledon	Bending cotyledon	Mature green	AGI code b	Number of SNPs Gene name	Preglobular	Globular	Heart	Linear cotyledon	Bending cotyledon	Mature green
	At5g44080			1194 1158 898 1019 965 617	At2g34480		RPL18AB		947 761 654 832 924 532
	At5g22880	5	HTB2	736 807 217 184	94 112	At2g22970		SCPL11		675 361 258	92	56	40
	At4g15390	13		11	6	5 473 727 109	At2g34490	17	CYP71A2		17	52 154 101 527 463
	At3g21610			295 487 276 340 423 370	At1g09830		PUR2		283 362 361 231 238 172
	At5g26360		CCT3	374 349 244 257 282 136	At1g31670		CuAOα1		1	17	5	4	18 354
	At1g12270		HOP1	344 205 228 234 107 162	At2g22990		SNG1		245 299	81	55	22	43
	At4g35260		IDH1	326 193 319 216 234 196	At1g13130		GH5		35	21 172	66	6	18
	At5g27360	42	SFP2	213 301 276	94	42	27	At1g13120	22	GLE1		118	72 105	78 139 102
	At3g53110		LOS4	237 273 157	96 103 173	At1g09840	21	SK41		116	70	76	39	47	43
	At3g21630		CERK1	195	87 241	83 116 107	At4g16800				42	59 103	59	46	72
	At2g03150	14	RSA1	238 186	69	96 128 120	At2g23000		SCPL10		78	89	96	24	50	8
	At5g44070	14	PCS1	124 229 121 121 193 192	At1g31760		SWIB5		82	51	48	69	93	53
	At5g18200			142 124 223	93	75	83	At1g09850		XBCP3		40	51	79	45	37	62
	At5g22875			149 204 184 157 148 181	At1g31690	15	CuAOα2		18	17	15	14	22	39
	At3g15940	8	UGT	67	75	97 158 196 143	At1g73880	8	UGT89B1	16	35	14	14	14	26
	At5g55060			64	85 105	80	89 171	At1g13110		CYP71B7		22	29	18	34	28	32
	At2g39130	17		65	88 110 142	84	65	At1g31710		CuAOα3		8	16	6	3	16	27
	At5g23020		IMS2	4	9	9	9	89 126	At4g16790	6	HRGP		4	12	18	23	24	21
	At4g35240			45 119	37	28	51	92	At1g31770	5			10	11	4	7	6	23
	At3g53090	12	UPL7	26	48	29	32	31	96	At2g34500		CYP71A1		2	22	1	10	9	5
	At2g39110	9	PBL38	24	25	31	41	23	85	At4g16780		HB-2		3	1	8	9	17	6
	At1g47980			2	13	1	31	25	79	At3g26880				2	3	0	1	3	15
	At2g03140			60	63	36	17	67	73	At3g26870				8	2	2	9	13	10
	At3g25170	5	RALFL26	15	69	4	61	11	18	At2g22980	2	SCPL13		7	11	2	1	5	11
	At2g39140		SVR1	17	48	16	22	26	15	At1g73880	8	UGT89B1
	At1g12290		CC-NBS-LRR	16	45	22	22	32	48	At1g73875		CCR4E
	At2g39120	22	WTF9	45	9	35	27	20	25	At1g31772		DEFL
	At3g15950	5	NAI2	16	11	7	7	5	40	At1g73885
	At1g47990	4	GA2OX4	2	2	1	2	15	38
	At5g18210			6	18	19	14	34	12
	At3g15930			15	6	2	7	15	32	c	Seed stage	MSC morphology
	At4g35250	7	HCF244	13	16	18	17	30	10
	At1g12280 At2g39100	2	SUMM2	5 17	23 21	14 13	7 26	10 10	29 21			Preglobular
	At5g27350		SFP1	10	8	10	10	15	25
	At3g53100	3		22	25	19	10	9	11
	At3g25180		CYP82G1	4	3	7	6	15	23			Globular
	At4g15380	3	CYP75A4	0	6	7	23	15	17
	At3g21620	26		5	11	2	19	14	21
	At5g22870			0	3	8	4	8	19
	At5g22900		CHX3	10	1	5	1	1	17			Heart
	At5g22890	3	STOP2	2	11	2	3	4	15
	At5g22860			9	4	2	4	5	14
	At3g53080			5	1	2	10	9	12
	At5g26340		STP13	4	11	5	1	1	9			Linear
	At3g25165		RALFL25	3	11	2	1	6	10
	At4g15370		BARS1	2	2	6	3	10	3
	At1g48000		MYB112	1	1	5	3	3	9
	At2g03160		ASK19	6	6	1	5	3	7
	At5g44065											Bending
	At5g55056
	At3g15960
	At4g15393		CYP72A5
	At5g23027										Mature green
	At5g27370		DMP10
	Fig. 6.

the GWAS for adherent (a) and non-adherent (b) mucilage layers, respectively. These are confronted to previously published transcriptomic data of the isolated seed coat during six seed developmental stages

[START_REF] Belmonte | Comprehensive developmental profiles of gene activity in regions and subregions of the Arabidopsis seed[END_REF]

. No significant values below the threshold of "45" are in grey, low expressed genes are in green, highly expressed genes are in red. Note that ten genes were not present in the tissue arrays. Number of significant SNPs found within the gene region are indicated, genes with no value were located downstream or upstream of significant SNPs. (c) Mucilage secretory cell (MSC) morphology for every seed developmental stage available in transcriptomic (adapted from

[START_REF] Francoz | Pectin demethylesterification generates platforms that anchor peroxidases to remodel plant cell wall domains[END_REF]

.

Acknowledgements

The authors are thankful to the Paul Sabatier-Toulouse 3 University and to the Centre National de la Recherche Scientifique (CNRS) for granting their work. This work was also supported by a grant by The Fédération de Recherche Agroscience Interactions Biodiversité (https://www.fraib.fr/fraib_eng/). We thank Werner Schweyer for his technical support during seed phenotyping.

for non-adherent (b) seed mucilage, respectively. The threshold used for QTL selection is represented by the dotted line at Lindey value of 5. Everything below is considered as background noise.

Fig. 6. About 50% of the 83 candidate genes from the GWAS are expressed in the seed coat during seed development. Fifty five and 28 selected genes are present within 26 and 10 QTLs highlighted by the GWAS for adherent (a) and non-adherent (b) mucilage layers, respectively. These are confronted to previously published transcriptomic data of the laser-captured seed coat during six seed developmental stages [START_REF] Belmonte | Comprehensive developmental profiles of gene activity in regions and subregions of the Arabidopsis seed[END_REF]. No significant values below the threshold of "45" are in grey, low expressed genes are in green, highly expressed genes are in red. Note that ten genes were not present in the tissue arrays. Number of significant SNPs found within the gene region are indicated, genes with no value were located downstream or upstream of significant SNPs. (c) Mucilage secretory cell (MSC) morphology for every seed developmental stage available in transcriptomic (adapted from [START_REF] Francoz | Pectin demethylesterification generates platforms that anchor peroxidases to remodel plant cell wall domains[END_REF]. Supplementary Fig. S2. Zoom on significant QTLs related to the genetic variability of adherent mucilage. Each peak is delimited by the two vertical dot lines with the first and last significant SNP contained in it. Each SNP is ploted with its -log10 of the P-value and the colored line represents the Lindley process evolution.

Supplementary

Supplementary Fig. S3. Zoom on significant QTLs related to the genetic variability of non-adherent mucilage. Each peak is delimited by the two vertical dot lines with the first and last significant SNP contained in it. Each SNP is ploted with its -log10 of the P-value and the colored line represents the Lindley process evolution.

Supplementary Table S1 Spearman correlation matrix across every population between measured traits and their harvested site environmental parameters. Indicated values in the upper half are spearman correlation coefficient and values in the down part are the coresponding p-values after an FDR correction. S2. Expression of every candidate genes in the different seed tissues during six seed developmental stages [START_REF] Belmonte | Comprehensive developmental profiles of gene activity in regions and subregions of the Arabidopsis seed[END_REF]. No significant values below the threshold of "45" are in grey, low expressed genes are in green, highly expressed genes are in red. Note that 10 genes had no spot ID in the tissue arrays.

Supplementary Table

Supplementary Table S3 Distribution and number of significant SNPs in each QTLs. The Table shows


Atsushi Takagi
email: atsushi.takagi@ntt.com

Carlo Bagnato
email: carlobagnato@gmail.com

Alejand Melendez-Calder

Nathanaël Jarrassé
email: jarrasse@isir.upmc.fr

Gowrishankar Ganesh
email: ganesh.gowrishankar@lirmm.fr

Member, IEEE Etienne Burdet
email: e.burdet@imperial.ac.uk

Alejandro Melendez-Calderon

Competition Increases the Effort Put Into a Physical Interaction Task

Keywords: Interaction, effort, competition, collaboration

Physical interaction can enhance motor learning, but it remains unclear what type of interaction is best suited to increasing the active effort put into a task, which should support learning. Here, we used the same interactive tracking task with different instructions to induce three training conditions: competition, collaboration, and self-improvement, where partners improve their own performance while interacting haptically with each other. The effort was gauged by measuring the total normalized muscle activity. Feedback of task performance and the haptic dynamics were identical in all three training conditions, so the effort needed to complete the task was the same. Only the instructions to 'compete with the partner', 'improve your and your partner's accuracy' and 'improve your accuracy' were different among the competition, collaboration, and self-improvement conditions, respectively. Despite having the same goal of maximizing self-performance during competition and self-improvement, participants exerted significantly more effort during competition, and their tracking accuracy was highest during competitive practice. Least effort was put into collaboration but tracking accuracy during collaboration was comparable to self-improvement. Our results suggest that interactive haptic competition can induce higher active drive or effort than either collaborative training or self-focused practice.

I. INTRODUCTION

H UMANS continually acquire new motor skills to physi- cally interact with the environment. The physical interaction and assistance provided by other people play a vital role in learning new motor skills like when learning to ride a bicycle. Recent evidence has revealed how haptic communication (mediated by touch) yields superior motor performance [START_REF] Beckers | Haptic human-human interaction does not improve individual visuomotor adaptation[END_REF], [START_REF] Takagi | Physically interacting individuals estimate the partner's goal to enhance their movements[END_REF] and can boost learning [START_REF] Ivanova | Interaction with a reactive partner improves learning in contrast to passive guidance[END_REF], but the trainee needs to take an active role in the task and not be passively guided by the partner. The effort put into a motor task, which may be quantified by the total muscle activity [START_REF] Reinkensmeyer | Slacking by the human motor system: Computational models and implications for robotic orthoses[END_REF], is a major factor in motor learning and improving the outcome of neurorehabilitation [START_REF] Lotze | Motor learning elicited by voluntary drive[END_REF], [START_REF] Hogan | Motions or muscles? Some behavioral factors underlying robotic assistance of motor recovery[END_REF], [START_REF] Kaelin-Lang | Role of voluntary drive in encoding an elementary motor memory[END_REF]. Thus, interactive practice may be beneficial to motor training and relearning [START_REF] Ivanova | For motion assistance humans prefer to rely on a robot rather than on an unpredictable human[END_REF], but the trainee should be exerting effort to reap its benefits.

Interactive practice can be subdivided into collaboration, competition, or self-improvement [START_REF] Jarrasse | A framework to describe, analyze and generate interactive motor behaviors[END_REF]. Self-improvement is where partners improve their performance while interacting haptically with each other [START_REF] Jarrasse | A framework to describe, analyze and generate interactive motor behaviors[END_REF], [START_REF] Takagi | Flexible assimilation of human's target for versatile human-robot physical interaction[END_REF], which is different from solo training where the individuals practice alone and without interaction [START_REF] Ganesh | Two is better than one: Physical interactions improve motor performance in humans[END_REF]. Collaboration is commonly used in interactive practice literature (see [START_REF] Sawers | Perspectives on human-human sensorimotor interactions for the design of rehabilitation robots[END_REF] for a review). A diverse set of motor tasks have been examined in the collaborative practice literature. Examples include reaching [START_REF] Reed | Haptically linked dyads: Are two motor-control systems better than one?[END_REF], [START_REF] Takagi | Motion plan changes predictably in dyadic reaching[END_REF] and tracking a common target [START_REF] Takagi | Physically interacting individuals estimate the partner's goal to enhance their movements[END_REF], manipulating a common object [START_REF] Basdogan | An experimental study on the role of touch in shared virtual environments[END_REF], and tracing a common shape [START_REF] Kager | The effect of skill level matching in dyadic interaction on learning of a tracing task[END_REF]. Competition is less studied relative to collaboration, but it is usually fostered by presenting conflicting goals to each partner [START_REF] Groten | The role of haptic feedback for the integration of intentions in shared task execution[END_REF], [START_REF] Chackochan | Incomplete information about the partner affects the development of collaborative strategies in joint action[END_REF].

The goal of this study was to identify the haptic training condition (self-improvement, collaboration, and competition) that best enhances the trainees' effort during practice. Previous studies faced difficulty in comparing competition with other modes of training because the competitive task usually required more effort to begin with. To illustrate this point, imagine a trainer competing with a trainee by opposing their motion whilst reaching for a cup. This competitive task clearly requires more effort than when reaching alone or if the trainer assists the movement [START_REF] Groten | The role of haptic feedback for the integration of intentions in shared task execution[END_REF], [START_REF] Chackochan | Incomplete information about the partner affects the development of collaborative strategies in joint action[END_REF]. To have the same baseline effort needed to complete the task, the feedback provided to users and the haptic dynamics must be kept the same across all training conditions.

We propose a new interactive task wherein the three training conditions (self-improvement, collaboration, and competition) can be realized through instruction and without modifying the task or the target. This task consists of two partners following a common sinusoidal target trajectory whilst the hands are weakly repelled from each other like same-pole magnets. The repulsive First, the instructions were displayed for 5 seconds prior to every trial. After the 12 second trial, the accuracy of both partners was displayed.

torque could be overcome easily and was not strong enough to prevent the partner from reaching the target. Thus, the only way of maximizing tracking accuracy was to stay near the target while absorbing the force perturbations from the partner. Prior to a trial, partners are instructed to either "improve your accuracy" (self-improvement), "compete with the partner" (competition) or to "improve your and your partner's accuracy" (collaboration). These instructions were used to define the training condition in each trial. Since the feedback display, the type of feedback and the type of haptic interaction were identical across training conditions, only the instruction affects the total muscle activity. Therefore, we can assess which training condition is best suited to increasing the effort put into the task.

II. MATERIAL AND METHODS

A. Experimental Setup

The experiment and the procedure were approved by the Imperial College Research Ethics Committee (no. 15IC2470), with all participants providing written informed consent prior to participation. Sixteen participants (24 ± 2 years old denoting mean and standard error, all right-handed) were recruited in same-sex pairs, forming 4 male and 4 female dyads. Mixed-sex pairs were not tested as a difference in strength may have affected the results. Each individual's right wrist was strapped to a robotic interface that measured the angle and the torque at 1000 Hz [START_REF] Melendez-Calderon | Hi5: A versatile dual-wrist device to study human-human interaction and bimanual control[END_REF] (Fig. 1). Participants were seated side-by-side and separated by a curtain to prevent direct observation of the partner, which can influence the perception of the force [START_REF] Takagi | Facing the partner influences exchanges in force[END_REF]. They were also prohibited from verbally communicating with each other. They viewed the position of the target, their cursor and the partner's cursor on their individual monitor.

Participants' right wrists were attached to a robotic wrist interface [START_REF] Melendez-Calderon | Hi5: A versatile dual-wrist device to study human-human interaction and bimanual control[END_REF] which could be moved by flexing and extending the wrist. The angle of the wrist was picked up by a position sensor and was displayed as a vertical bar on a monitor. This cursor bar moved horizontally along an arc on-screen, moving left for flexion and right for extension. A circular target was displayed along the same arc.

Surface electromyograms (EMG) of the flexor carpi radialis and extensor carpi radialis longus were measured at 1000 Hz (g.GAMMASYS, g.tec). The envelope of the EMG activity was extracted by filtering the raw EMG data using a second order high-pass Butterworth filter with a cutoff frequency of 5 Hz. This signal was rectified then filtered using another second order low-pass Butterworth filter with a 5 Hz cutoff frequency.

The target position θ T was a sinusoid with a frequency of 0.25 Hz and an amplitude of 20 • . Each participant had to move their wrist to match their cursor position (henceforth denoted as θ 1 for the first partner and θ 2 for the second) with the target as accurately as possible. Each trial lasted 12 seconds. The mean tracking error e = θ T -θ 1 in degrees over a whole trial was calculated at the end of every trial for each participant (swapping θ 1 for θ 2 ), and normalized to a percentage accuracy measure a = 100 1 -e e 0 with e 0 = 10 • .

(

) 1
The accuracy was negatively and linearly related to the tracking error. While a can be negative if the tracking error is greater than 10 • , in practice the tracking error was always below this value. The accuracy of both individuals was displayed at the end of each trial on each participant's monitor (Fig. 1).

B. Training Conditions

Before the start of the main experiment, participants were instructed about the nature of the interactive task and the force they would experience through the following message on the monitor: "You will interact with the person next to you. You will both feel a repelling force field (like a magnet) when you are close to one another". The interaction instruction was displayed prior to each trial. In self-improvement, each participant was instructed to "improve your accuracy". During collaboration trials, a large banner with "collaborate" in green lettering was displayed with the instruction to "improve your and your partner's accuracy" (Fig. 1). Finally, in competition trials a red "compete" banner was displayed together with the instruction to "compete with your partner". The banner's color was changed to draw attention to the change in training condition. These instructions were not visible during the trial. No additional explanation was given beyond these written instructions onscreen to avoid biasing the participants' behavior. None of the participants asked for clarification beyond the given instructions.

Each dyad first completed 7 self-improvement trials and then experienced three blocks wherein each block was composed of 5 consecutive competitive trials and 5 consecutive collaborative trials. The ordering of the training conditions was counterbalanced such that four dyads (two male-male and two femalefemale) experienced the five competitive trials first, while the other dyads completed the five collaborative trials first in every block. Finally, every dyad completed 5 self-improvement trials in the final block.

C. Haptic Dynamics

In all three conditions, the first partner experienced a repulsive torque

τ θ = 2 sgn (Δθ) e -0.1|Δθ| -e -0.15|Δθ| Nm (2)
that is a function of the relative displacement Δθ = θ 1 -θ 2 between the first partner's position θ 1 and the second partner's wrist position θ 2 The second partner felt a repulsive torque in the opposite direction, i.e., -τ θ . The parameters of the repulsive torque were determined heuristically based on preliminary tests to avoid excessive force and fatigue. The repulsive torque was not strong enough to push the partner away to one side, such that completely blocking off a partner from reaching the target was impossible (Fig. 2). Since the repulsive torque was easy to overcome, partners often switched sides from being on the partner's left to going to their right and vice versa. These 'crosses' were counted throughout each trial and the number of crosses in a trial was used to quantify the behavior during each training condition.

D. Normalization of Muscle Activity

The filtered electromyograms of the wrist flexor and extensor muscles, measured in volts, was normalized to relate them to the torque generated by each muscle prior to the main experiment. The normalization procedure consisted of linearly regressing the activity of each muscle as a function of the torque produced by the muscle during an isometric contraction task [START_REF] Melendez-Calderon | Interpersonal strategies for disturbance attenuation during a rhythmic joint motor action[END_REF], and was carried out separately for each individual before the main experiment. Each participant produced a constant isometric flexion or extension torque of {1,2,3,4} Nm against the robotic interface whose position was locked in place by motors. Taking a wrist flexor muscle as an example, its activity u f was linearly regressed to estimate the flexion torque

τf = α f u f + β f > 0 (3)
where α f is the slope and β f the intercept parameter of the flexor torque function, respectively. A similar linear regression of the form τe = α e u e + β e > 0 was carried out for the wrist extensor muscle activity u e . The goodness of fit was calculated and averaged across the two muscles, which yielded a value of R 2 = 0.64 ± 0.02 for all participants. The estimated torque from each muscle was used to calculate the reciprocal activation τRA , the cocontraction τCC and the total muscle activity τ using the formulae

τRA = τf -τe τCC = 2 min (τ f , τe ) τ = τf + τe . ( 4
)
The total muscle activity is equivalent to the contributions from reciprocal activation and half the cocontraction, i.e., τ = τRA + 0.5 τCC .

E. Statistical Analysis

The tracking error, the number of crosses, total muscle activity, reciprocal activation, and the cocontraction were calculated for each participant (or dyad in the case of crosses) every trial. The mean value in the first and last self-improvement, collaboration, and competition blocks, totaling 30 trials per participant, 5 trials coming from each block, corresponding to 10 trials from each training condition were used in the analysis. To counterbalance the effects of learning, the data from the first and last blocks were averaged per training condition. Anderson-Darling tests were conducted on each indicator (tracking error, number of crosses, total muscle activity, reciprocal activation and cocontraction) across all three training conditions to ensure that each indicator was normally distributed prior to further analysis. These tests showed that all indicators were normally distributed for every training condition (see Table I).

The pooled data was used to carry out a one-way repeated measures ANOVA that measured the impact of the training condition on each indicator. Multiple comparisons within each measure were corrected by using Tukey's method, while multiple comparisons across measures were corrected by the Holm-Bonferroni method.

III. RESULTS

We first examined the wrist trajectories during collaboration, competition, and self-improvement (Fig. 3). Partners tended to stick to one side of the target and stayed away from their partner during collaboration, while on competitive trials their positions tended to crossover. This was also evident in the torque sensor time-series as it remained positive or negative for a longer duration during collaboration (Fig. 3). While the torque fluctuated more during competition, the torque magnitude was not dependent on the training condition (Fig. 3). Hence, the magnitude of the force exerted by and against the partner was comparable between self-improvement, collaboration, and competition. A difference in behavior cannot be discerned from the torque magnitude alone.

We next examined how the tracking error, defined as the mean distance between the wrist and the target in a trial, changed with training and the training condition. The tracking error was calculated for each block consisting of five trials (Fig. 4). We compared the tracking error, averaged across the first and last blocks, in each training condition using a one-way repeated measures ANOVA with training condition as the factor (stats are summarized in Table II). The tracking error was significantly different across training conditions. Post-hoc comparisons using Tukey's HSD revealed a significant difference in the tracking error between competition and collaboration (p = 0.02), and between competition and self-improvement (p = 0.002), but not between collaboration and self-improvement. The tracking error was thus lowest during competition. Did partners purposefully avoid each other to realize collaboration, or was the tracking error higher in collaboration for another reason? To quantify the collaboration strategy, we calculated the number of crosses over a whole trial (Fig. 1, circles). The number of crosses was significantly affected by the training condition (Fig. 4). Tukey's HSD showed that the number of crosses was smaller during collaboration than in self-improvement (p = 0.03), and smaller in self-improvement than during competition (p = 0.006). Crosses were fewest during collaboration (45.1 ± 3.4 crosses, mean and SEM), greatest during competition (56.5 ± 3.3) and in-between during selfimprovement (49.9 ± 2.6).

Finally, we examined the total muscle activity in each training condition (Fig. 5). A normalization procedure prior to the main experiment was used to relate the envelope of the raw electromyograms to estimate the torque generated by each muscle. The total muscle activity was significantly dependent on the training condition. Post-hoc comparisons using Tukey's HSD revealed that the total muscle activity was larger during competition than in self-improvement (2.5 ± 0.4 Nm compared with 2.2 ± 0.4 Nm, p = 0.01), and bigger in selfimprovement than during collaboration (2.2 ± 0.4 Nm compared with 2.0 ± 0.3 Nm, p = 0.01). The total muscle activity can be split into reciprocal activation, which generates joint torque, or to cocontraction that increases joint stiffness [START_REF] Melendez-Calderon | Interpersonal strategies for disturbance attenuation during a rhythmic joint motor action[END_REF], [START_REF] Takagi | The dominant limb preferentially stabilizes posture in a bimanual task with physical coupling[END_REF]. The reciprocal activation was weakly dependent on the training condition, while the cocontraction was significantly influenced by it. Tukey's HSD showed no significant difference the reciprocal activation between any of the training conditions. However, the cocontraction was significantly larger during competition relative to both selfimprovement (p = 0.02) and collaboration (p < 0.001). The cocontraction during self-improvement was also bigger than during collaboration (p = 0.01). Thus, most of the increase in the total muscle activity came from the cocontraction.

We also looked for indications of a difference in motor learning between each training condition by carrying out a one-way ANOVA on the difference between the first and last block to see its dependence on the training condition. For this analysis, we excluded the self-improvement condition as its ordering of being the first and last block of the experiment for all dyads may bias the results. The reduction in the tracking error, total muscle activity, reciprocal activation and cocontraction were all comparable between collaboration, and competition (see Table III). However, the number of crosses reduced by a larger amount during competition in comparison to collaboration (p < 0.03). This difference likely comes from the larger number of crosses in the first competition block.

IV. DISCUSSION

The goal of this study was to identify the training condition that promoted the greatest amount of effort during interactive practice. We developed and tested a novel interactive task wherein the haptic dynamics and the feedback was kept the same whilst inducing collaboration, competition, and self-improvement through instruction alone. Individuals were most active during competition and least active during collaboration. Competition boasted the highest tracking accuracy and the largest muscle activity overall. The tracking accuracy was comparable between collaboration and self-improvement, but the total muscle activity and the cocontraction was greater during self-improvement [START_REF] Goršič | A multisession evaluation of a collaborative virtual environment for arm rehabilitation[END_REF]. Our finding that competition promotes active effort is consistent with previous reports of higher movement velocity during competitive training relative to both collaborative practice and self-improvement [START_REF] Goršič | Competitive and cooperative arm rehabilitation games played by a patient and unimpaired person: Effects on motivation and exercise intensity[END_REF].

There exist other ways to fairly compare the effort during collaboration and competition. For example, the feedback gain could be estimated in each condition to assess the strength of a participant's response to tracking errors [START_REF] Takagi | Reduced effort does not imply slacking: Responsiveness to error increases with robotic assistance[END_REF]. However, this method requires an accurate estimate of each person's force which is difficult to obtain during haptic interaction. Furthermore, the feedback gain methodology ignores cocontraction, which is critical to stabilizing the arm during movements and interactions [START_REF] Burdet | The central nervous system stabilizes unstable dynamics by learning optimal impedance[END_REF], [START_REF] Takagi | Analogous adaptations in speed, impulse and endpoint stiffness when learning a real and virtual insertion task with haptic feedback[END_REF]. Our analysis showed that the cocontraction increased during self-improvement relative to collaboration, and it was overall the largest during competition. The increase in the cocontraction was likely due to the instability of the repulsive torque field, particularly its changing sign that was evident in the torque time-series. Since the force switched directions rapidly and was difficult to predict, our participants stabilized their movement by coactivating their flexor and extensor muscles. Thus, an analysis based purely on the feedback gain could have missed the effort put into stabilizing the wrist.

Cocontraction is known to gradually decay with practice as motor learning progresses [START_REF] Thoroughman | Electromyographic correlates of learning an internal model of reaching movements[END_REF], [START_REF] Franklin | CNS learns stable, accurate, and efficient movements using a simple algorithm[END_REF]. Such a decay was observed in our experiment, but it reduced by comparable amounts between competition and collaboration. This suggests that the motor learning between these two conditions progressed at comparable rates. However, the number of crosses decreased by a larger margin during competition compared with collaboration. It remains unclear why the number of crosses was so high in the first competition block. One possibility is that participants may have explored a strategy to try and actively sabotage the partner by using the repulsive torque to prevent the partner from approaching the target. However, the repulsive torque field was not strong enough to accommodate such an antagonistic strategy, and so participants may have realized that the best way to compete was to outperform their partner by tracking the target as accurately as possible.

Did the repulsive haptic dynamics necessarily make all training modalities antagonistic, thereby making both self-improvement and collaboration competitive? While this possibility cannot be ruled out entirely, the difference in the number of crosses, the total muscle activity and in the cocontraction suggest that the participants did not consider all conditions to be equally competitive. When instructed to collaborate, our participants tried to track the target while disturbing the partner as little as possible by staying away from them and sticking to one side of the target. In contrast, the instruction to compete caused partners to increase cocontraction to stay as close to the target as possible, simultaneously tracking the target while preventing the competitor from approaching. One may argue that the partner's behavior could have been different in each condition, which would necessarily change the haptic dynamics of the task. However, the magnitude of the torque was similar in all conditions, so a difference in the partner's behavior alone cannot explain the differences between the training conditions.

Did the instruction to 'compete with your partner' confuse our participants such that they adopted different strategies during competitive training? To rephrase, did participants compete by aiming for a higher tracking accuracy than the partner, or did they resort to an antagonistic strategy wherein the repulsive torque field was used to destabilize the partner's movement or block them from reaching the target? Our analysis suggests that the participants focused on improving their tracking accuracy to compete with their partner. An antagonistic strategy would likely decrease the pair's tracking accuracy because at least one person is not focused on the tracking task. Since the tracking accuracy was overall greatest during competition, this supports the view that participants tried to achieve the highest possible tracking accuracy to compete with their partner.

Could participants have compromised between tracking error and effort in response to each condition? The analysis of the tracking error and the total muscle activity suggests otherwise as the tracking error during collaboration did not drastically decrease relative to self-improvement, even though the total muscle activity during collaboration was lower. Thus, the strategy undertaken during collaboration was not to relax and become lazy at tracking the target. Instead, partners actively tracked the target while staying away from the partner to prevent the repulsive torque from disturbing the partner's movement as much as possible.

The analysis of the kinematics and the EMG showed that the effort put into competition was significantly higher than during collaboration and self-improvement. Our results suggest that competitive practice could be a suitable method in increasing the active drive during physical training.

Fig. 1 .

 1 Fig. 1. Experimental setup. The target position θ T (yellow), one's own cursor position θ 1 (solid bar) and the partner's position θ 2 (unfilled bar) were visible throughout the experiment. The right panel shows the experimental protocol.First, the instructions were displayed for 5 seconds prior to every trial. After the 12 second trial, the accuracy of both partners was displayed.

Fig. 2 .

 2 Fig.2. Repulsive torque was centered on each person's wrist position θ 1 and θ 2 , which disturbed the partner's movement but was not strong enough to prevent them from approaching.

Fig. 3 .

 3 Fig. 3. (Left and middle) Time-series of the wrist position and the torque (from a torque sensor) from a sample collaboration and competition trial from the same representative dyad. Partners switched sides more often during competition, which is more noticeable on the torque sensor. During collaboration the measured torque remained positive or negative for longer. (Right) Time-series of the torque magnitude (absolute value of the torque sensor readings) from the repulsive torque field during the first and last blocks from all dyads, which was comparable across the three training conditions.

Fig. 4 .

 4 Fig.4. (Top) tracking error, when averaged across the first and last blocks, was the lowest during competition, outperforming both self-improvement and collaboration conditions. Filled circle is the mean and the bar denotes standard error of the mean. (Bottom) number of crosses was smallest during collaboration and highest during competition, with self-improvement in between these two.

Fig. 5 .

 5 Fig. 5. (Left) total muscle activity reduced with block number and was greatest during competition, lowest in collaboration and in-between for self-improvement. (Middle and right panels) total muscle activity separated into reciprocal activation and cocontraction, plotted as a function of the block number. Both were larger during competition relative to collaboration.

TABLE I SUMMARY

 I OF ANDERSON-DARLING TESTS OF NORMALITY

TABLE II SUMMARY

 II OF ONE-WAY REPEATED MEASURES ANOVA TO EXAMINE THE EFFECT OF TRAINING CONDITION

TABLE III SUMMARY

 III OF ONE-WAY REPEATED MEASURES ANOVA TO EXAMINE DIFFERENCES IN MOTOR LEARNING

This work was supported in part by EU under Grants FP7 PEOPLE-ITN-317488 CONTEST, H2020 ICT-644727 COGIMON, FETOPEN-829186 PH-CODING, PEOPLE-ITN-861166 INTU-ITIVE, and ICT-871803 CONBOTS, and in part by the U.K. EPSRC under Grants EP/NO29003/1 MOTION and EP/R026092/1 FAIR-SPACE. (Corresponding authors: Atsushi Takagi; Etienne Burdet.) This work involved human subjects or animals in its research. Approval of all ethical and experimental procedures and protocols was granted by the Imperial College Research Ethics Committee under No. 15IC2470, and performed in line with the Declaration of Helsinki. National de la Recherché Scientifique (CNRS), and is currently


Yakoub Salhi

Michael Sioutis

A Decomposition Framework for Inconsistency Handling in Qualitative Spatial and Temporal Reasoning

 for a recent survey.

Motivation. In [START_REF] Salhi | A Decomposition Framework for Inconsistency Handling in Qualitative Spatial and Temporal Reasoning[END_REF], we study the decomposition of an inconsistent constraint network into consistent subnetworks under, possible, mandatory constraints. To illustrate the interest of such a decomposition, we provide a simple example described in Figure 1. The QCN depicted in the top part of the figure corresponds to a description of an inconsistent plan. Further, we assume that the constraint Task A {before} Task B is mandatory. To handle inconsistency, this plan can be transformed into a decomposition of two consistent plans, depicted in the bottom part of the figure; this decomposition can be used, e.g., to capture the fact that Task C must be performed twice. More generally, network decomposition can be involved in inconsistency handling in several ways: it can be used to identify potential contexts that explain the presence of inconsistent information; it can also be used to restore consistency through a compromise between the components of a decomposition, e.g., by using belief merging [START_REF] Condotta | A Syntactical Approach to Qualitative Constraint Networks Merging[END_REF]; in addition, QCN decomposition can be used as the basis for defining inconsistency measures.

Contributions.

We summarize the contributions of [START_REF] Salhi | A Decomposition Framework for Inconsistency Handling in Qualitative Spatial and Temporal Reasoning[END_REF] as follows. First, we propose a theoretical study of a problem that consists in decomposing an inconsistent QCN into a bounded number of consistent QCNs that may satisfy a specified part in the original QCN; intuitively, the required common part corresponds to the constraints that are considered necessary, if any. To this end, we provide upper bounds for the minimum number of components in a decomposition as well as computational complexity results. Secondly, we provide two methods for solving our decomposition

Figure 1 A

 1 Figure 1 A decomposition of an inconsistent qualitative constraint network (QCN) into consistent subnetworks (components).

Jakob Suchan, Mehul Bhatt, and Srikrishna Varadarajan. Commonsense visual sensemaking for autonomous driving -On generalised neurosymbolic online abduction integrating vision and semantics. Artif. Intell., 299:103522, 2021.

Koji Tanaka, Francesco Berto, Edwin D. Mares, and Francesco Paoli, editors. Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science. Springer, 2013.

Matthias Thimm. On the expressivity of inconsistency measures. Artif. Intell., 234:120-151, 2016.

16:2

A Decomposition Framework for Inconsistency Handling in QSTR problem. The first method corresponds to a greedy constraint-based algorithm, a variant of which involves the use of spanning trees; the basic idea of this variant is that any acyclic constraint graph in QSTR is consistent, and such a graph can be used as a starting point for building consistent components. The second method corresponds to a SAT-based encoding; every model of this encoding is used to construct a valid decomposition. Thirdly, we consider two optimization versions of the initial decomposition problem that focus on minimizing the number of components and maximizing the similarity between components, respectively. The similarity between two QCNs is quantified by the number of common non-universal constraints; the interest in maximizing the similarity lies mainly in the fact that it reduces the number of constraints that allow each component to be distinguished from the rest. Of course, our previous methods are adapted to tackle these optimization versions, too. Additionally, we introduce two inconsistency measures based on QCN decomposition, which can be seen as counterparts of measures for propositional KBs introduced in [11, 1], and show that they satisfy several desired properties in the literature. Finally, we provide implementations of our methods for computing decompositions and experimentally evaluate them using different metrics. The work was partially funded by the Agence Nationale de la Recherche (ANR) for the "Hybrid AI" project that is tied to the chair of Dr. Sioutis, and the I-SITE program of excellence of Université de Montpellier that complements the ANR funding.

ACM Subject Classification