Pascale Goupil
email: pascale.goupil@univ-bpclermont.fr

Razik Benouaret

Olivia Charrier

Alexandra Ter Halle

Claire Richard

Boris Eyheraguibel

Denis Thiery

Gérard Ledoigt

Grape marc extract acts as elicitor of plant defence responses

Keywords: Elicitor, Grape marc, Pathogenesis related protein genes, Plant defence reactions, Tobacco

Plant protection based on novel alternative strategies is a major concern in agriculture to sustain pest management. The marc extract of red grape cultivars reveals plant defence inducer properties. Treatment with grape marc extract efficiently induced hypersensitive reaction-like lesions with cell death evidenced by Evans Blue staining of tobacco leaves. Examination of the infiltration zone and the surrounding areas under UV light revealed the accumulation of autofluorescent compounds. Both leaf infiltration and a foliar spray of the red grape extract on tobacco leaves induced defence gene expression. The PR1 and PR2 target genes were upregulated locally and systemically in tobacco plants following grape marc extract treatment. The grape extract elicited an array of plant defence responses making this natural compound a potential phytosanitary product with a challenging issue and a rather attractive option for sustainable agriculture and environmentally friendly practices.

Pathogen-derived metabolites (elicitors) are recognised by putative plant cell receptors and activate a complex network of signal transduction pathways and a variety of biochemical and molecular defence mechanisms. The signalling pathways mediated by microbial elicitors involve secondary signals such as salicylic acid (SA),

Introduction

Over the last few decades, there has been increasing concern about environmental pollution and damage to biodiversity as a result of the intensive use of chemical phytosanitary products. Significant research efforts have been expended to identify and develop newer and safer compounds modelled on natural systems. Currently, fast emerging natural phytosanitary products are known as plant defence inducers (PDIs). These compounds are capable of triggering plant immune responses [START_REF] Reglinski | Integration of induced resistance in crop production[END_REF]). The induction of the host plant defence system is a promising strategy to reduce pesticide use in conventional agricultural practices diminishing negative side effects on both the environment and human health [START_REF] Walling | Induced resistance: from the basic to the applied[END_REF][START_REF] Harm | Evaluation of chemical and natural resistance inducers against downy mildew (Plasmopara viticola) in grapevine[END_REF]. jasmonic acid (JA) and ethylene. These bioactive molecules can either act independently or in combination to orchestrate local and systemic induction of defence responses [START_REF] Shah | Plants under attack: systemic signals in defence[END_REF][START_REF] Yang | Signal perception and transduction in plant defence responses[END_REF]. Systemic acquired resistance (SAR) refers to a SA-dependant pathway and plays an important role in the ability of plants to defend themselves against pathogens [START_REF] Durrant | Systemic acquired resistance[END_REF][START_REF] Conrath | Systemic acquired resistance[END_REF]. After the formation of a necrotic lesion, either as part of the hypersensitive response (HR) or as a disease symptom, a local acquired response (LAR) is induced [START_REF] Costet | Relationship between localized acquired resistance (LAR) and the hypersensitive response (HR): HR is necessary for LAR to occur and salicylic acid is not sufficient to trigger LAR[END_REF]) and the SAR pathway is activated. LAR and SAR pathway activation results in the development of a broad-spectrum resistance associated with the modified expression of a large number of defence genes whose products may play crucial roles in restricting pathogen growth. Typical marker genes of SAR induction include the pathogenesis-related protein PR1 or other acidic PR proteins with antimicrobial activity [START_REF] Edreva | Pathogenesis-related proteins: research progress in the last 15 years[END_REF][START_REF] Van Loon | Significance of inducible defence-related proteins in infected plants[END_REF]. Tobacco is best studied model system for studying SAR induction.

Analysis of SAR in tobacco has resulted in a number of significant findings that have shed light on plant defence-induced transduction pathways. In tobacco, SAR activation results in a significant reduction of the disease symptoms caused by oomycetes Peronospora tabacina and Phytophthora parasitica, fungi Cercospora nicotianae, virus (tobacco mosaic virus and tobacco necrosis virus), and bacteria Pseudomonas syringae and Erwinia carotovora [START_REF] Vernooij | 2,6-dichloroisonicotinic acid-induced resistance to pathogens does not require the accumulation of salicylic acid[END_REF]. PDI-containing biomolecules can be used as stimulus-triggering factors to mobilize plant defence reactions.

These non-specific elicitors are structurally diverse compounds such as oligosaccharides, polysaccharides, peptides, proteins and lipids; most are derived from the plant or pathogen cell surface [START_REF] Walters | Induced resistance for plant disease control: maximizing the efficacy of resistance elicitors[END_REF][START_REF] Lyon | Agents that can elicit induced resistance[END_REF]). PDIs activate SAR and induce protection against a variety of pathogens for a large spectrum of vegetables and crops [START_REF] Reignault | Topical application of inducers for disease control[END_REF]. Identification of new PDIs could provide an exploitable perspective important for phytoimmunology, biotechnology and environment. There are several plant extracts that act as efficient PDIs. Extracts from Hedera [START_REF] Baysal | Extract of Hedera helix induces resistance on apple rootstock M26 similar to acibenzolar-S-methyl against fire blight (Erwinia amylovora)[END_REF], Laminaria [START_REF] Aziz | Laminarin elicits defence responses in grapevine and induces protection against Botritis cinerae and Plasmopara viticola[END_REF], Reynoutria [START_REF] Konstantinidou-Doltsinis | Efficacy of Milsana, a formulated plant extract of Reynoutria sachalinensis, against powdery mildew of tomato (Leveillula taurica)[END_REF], Trigonella [START_REF] Martinez | Elicitor made from Trigonella foenum graecum extracts for the treatment of plant pathogens, use thereof, and method for the production thereof[END_REF] and Rheum/Frangula [START_REF] Godard | Induction of defence mechanisms in grapevine leaves by emodin-and anthraquinone-rich extracts and their conferred resistance to downy mildew[END_REF]) have been shown to induce plant defence reactions and promote disease suppression in susceptible host plants. Trigonella extracts and Laminaria extracts are currently approved in France as PDIs.

In the present report, we investigate plant defence reactions mediated by red grape marc extract at the molecular level in tobacco. The uniqueness of this product is due to its origin and nature. Grape marc extract (GME) is a wine by-product that contains a large proportion of polyphenols with anthocyanins. GME has recently been shown to act as a pesticide photoprotector (Eyheraguibel et al. 2010). Differential PR gene expression and localised cell death following GME treatment suggest that GME may act as a PDI. This new class of phytosanitary bioproduct combining reduced input properties can make it an exceptionally well-suited product for environmentally oriented crop protection scheme.

Materials and Methods

Biological compounds and chemicals

Grape marc extract (GME) is a Vitis vinifera L. hydroalcoholic extract provided as red powder by Grap'Sud (Cruviers-Lascours, France, lot #08010). Pesticide (<1 ppm), heavy metal (<3ppm) and bacterial (E. coli-, Salmonella-and total coliforms-free) contents were certified by the company. The industrial process used to create the GME powder consists of an ethanol (<30% v/v) extraction of polyphenols followed by atomisation. The end product contains less than 100 ppm of residual solvent. GME is produced from marc of grapes harvested from several red wine varieties (e.g., Merlot, Syrah, Carignan, Grenache, Cabernet and Alicante) cultivated in Southern France. GME powder was prepared as a 1% aqueous solution and serially diluted 4-fold (0.25%), 8-fold (0.125%), 16-fold (0.0625%), 32-fold (0.0312%) and 64-fold (0.0156%) prior to use for dose-dependent experiments. Salicylic acid (SA), jasmonic acid (JA) and aminobutanoic acid (BABA) were purchased from Sigma (St. Quentin Fallavier, France).

Plant material and treatments

The biological activity of GME was assayed on 3-month old tobacco plants (Nicotiana tabacum L.) when plants had developed 20 to 22 fully expanding leaves (preflowering stage).

Tobacco plants were grown in the greenhouse under controlled conditions (22°C±5°C with a photoperiod of 16h of light). GME was applied as an aqueous solution to foliar tissue either by infiltration or as an aerosol spray.

Leaf infiltration was carried out on leaf blades using a plastic syringe. Routinely, 50 µl was infiltrated until the solution was spread across a leaf area of 1 to 2 cm 2 . Foliar spray treatments were administered by spraying onto the adaxial (upper) face of three leaves with a fine atomizer (2 ml per leaf). As a positive control, leaves were infiltrated or sprayed with 2 mM SA. For the negative control, the leaves were infiltrated or sprayed with ultrapure water. The compounds (GME, SA and BABA) were infiltrated into distinct areas on the same leaf for the examination of macroscopic symptoms under bright field or UV light (at 312 nm). Aerosol leaf sprays were monitored on the median leaves of a tobacco plant. The unsprayed leaves (USL) used to investigate the systemic defence reactions were those located immediately above and below the sprayed leaves (SL).

Cell death assay

Cell death in tobacco leaf tissue was monitored by Evans Blue staining [START_REF] Baker | An improved method for monitoring cell death in cell suspension and leaf disc assays using Evans Blue[END_REF]. Each assay was performed with 10 leaf disks (1 cm in diameter) punched out from each infiltrated area of the same leaf (triplicate on the same leaf). Leaf disks were incubated for 30 min in 0.25% Evans Blue (Sigma, France) at room temperature on a rotary shaker, rinsed extensively to remove the excess dye and ground in a tissue grind tube with 1 ml 1% SDS. The leaf extract was centrifuged for 20 min at 20000 xg and the supernatant was diluted 8-fold with water. The dye was quantified using spectrophotometry by monitoring the absorbance at 600 nm. Leaf tissues (200 mg) were grounded in liquid nitrogen and RNA extraction was performed using 1 ml Tri-reagent (Euromedex, France) according to manufacturer's instructions.

RNA isolation and quantification

Total RNA were cleaned up with 0.5U DNase I solution (Euromedex, France) solution containing 20U RNase inhibitor (Euromedex, France). RNA integrity was verified on a 1% agarose gel by detecting ribosomal RNAs.

RNA concentrations were measured spectrophotometrically at 260 nm. Reverse transcription was performed using 1 µg of the total RNA and Euroscript Reverse Transcriptase (Eurogentec, France) according to manufacturer's instructions.

A quantitative assessment of mRNA levels was performed using an iCycler iQv3 (BIO-RAD). PCR reactions were prepared using the qPCR kit Mastermix for SYBR green (Eurogentec, France) according to the manufacturer's protocol. The cDNA concentration used produced a threshold cycle value (C T ) between 15 and 30 cycles. PCR efficiencies were calculated for each gene according to [START_REF] Pfaffl | A new mathematical model for relative quantification in real-time RT-PCR[END_REF]. The abundance of PR transcripts was normalised to the transcript abundance of the reference gene (EF-1 [START_REF] Peng | Expression of harpin Xoo in transgenic tobacco induces pathogen defense in the absence of hypersensitive cell death[END_REF] and set relative to control plants (ultrapure water infiltrated or sprayed) following the 2 -ΔΔCT method. The relative PR transcript quantities are the mean of at least three technical replicates. Two treated plants were pooled for each experiment and the results are presented as the means of duplicate experiments. Bars represent the mean values ± standard error (SE).

Primers were designated according to their availability in NCBI Genbank (http://www.ncbi.nlm.nih.gov/).

Primer sequences are listed in Table 1. Both forward and reverse primers cover a relatively short sequence (approximately 150 bp) and were designed with a GC percentage of approximately 60%. The specificity of amplicons was routinely verified by melt curve analysis at the end of each run and by 1.2% agarose gel electrophoresis after gene amplifications using 2.5U Taq DNA polymerase (Sigma, France) with the following thermal cycling program: 30 s at 95°C, 30 s at the Tm of the primers (Table 1) and 30 s at 72°C. Partial characterisation of the GME Total phenolic content was determined colorimetrically using the Folin-Ciocalteu reagent, as described by [START_REF] Emmons | Antioxydant activity and phenolic content of oat as affected by cultivar and location[END_REF]. Total anthocyanins were quantified using the pH differential method described by [START_REF] Muñoz-Espada | Anthocyanin quantification and radical scavenging capacity of Concord, Norton, and Marechal Foch grapes and wines[END_REF]. Soluble sugar content was determined by colorimetric method following the phenol sulphuric acid method as described by [START_REF] Dubois | Colorimetric method for determination of sugars and related substances[END_REF]. SA and JA detection were monitored using a Waters Alliance UPLC system containing the separation module E2695 and a Dual absorbance detector 2487. A 10 µL aliquot of GME, SA and JA were injected independently into the column Zorbax SB-CN (Agilent) with the following characteristics 4.6 mmx250 mm; 5 µm. The GME, SA and JA detection wavelength was set at 300 nm. The mobile phase consisted of 40% acetonitrile and 60% water acidified at pH 1.0 using 3% formic acid. The flow rate was set at 1 ml min -1 .

Statistical analysis

Student's test was used to compare the means of cell death for determination of significant differences between treated samples versus control. The data were written as the mean ±SE. Values were determined to be significant when p≤0.05.

Results

GME mediates local tissue injury Foliar infiltration of GME into tobacco plants resulted in the appearance

of localised HR-like lesions (brown zones). Figure 1 shows the abaxial face of tobacco leaves infiltrated with GME. Infiltration with 0.25% GME induced a chlorotic area by 2 days post-infiltration (dpi) (Fig. 1b). A brown desiccated area appeared within the infiltrated tissue zone at 4 dpi (Fig. 1c) and covered most of the infiltrated zone by 8 dpi (Fig. 1d). Examination of the leaf tissue by UV light ( 312 nm ) at 8 dpi revealed that the infiltrated area (ia), the surrounding zone (sz) and the necrotic zone (nz) displayed fluorescence indicative of phenolic compound accumulation while the rest of the leaf remained unaffected (Fig. 1f). The extent of macroscopic symptoms and the expansion of fluorescent leaf area were induced in a dose-dependent manner. As the concentration of GME decreased (from 0.25%/dilution 4-fold to 0.0156%/dilution 64-fold) a reduction of chlorosis within the infiltrated area was observed (Fig. 1e). When GME was infiltrated at low concentrations (0.0312%/dilution 32-fold or 0.0156%/dilution 64-fold), no macroscopic changes were detected in the infiltrated leaf tissues (Fig. 1e). The 0.0625% (dilution 16-fold) GME concentration induced a very faint chlorotic zone (Fig. 1e) with a scattered pattern of fluorescence (Fig. 1f). Higher GME concentrations (0.25%/dilution 4-fold or 0.125%/dilution 8-fold) produced the highest chlorotic changes and fluorescence in tobacco leaf tissues with an apparent large necrotic zone (nz) when the highest concentration was used (Fig. 1e). As expected, 2 mM SAinfiltrated tissues displayed both necrotic tissue and fluorescent areas (Figs. 1e,f). The 10 mM BABA-infiltrated tissues remained symptomless, similar to the water-infiltrated control tissues.

To determine the effect of leaf age on GME response, the macroscopic changes induced by GME infiltration were examined in young and old leaves. Figure 2 shows the HR-like lesions induced by 0.25% GME at 4 dpi on tobacco plants with 20 to 22 expanded leaves. Younger leaves turned slightly bright (Fig. 2a) while older leaves displayed shiny chlorotic areas (Fig. 2c). Intermediate symptoms were observed on middle leaves (Fig. 2b). The increasing severity of induced necrotic lesions with increasing leaf age suggests a greater sensitivity of mature leaves to GME.

Evans Blue is a stain used to determine cell viability as the non permeating pigment can only enter cells with damaged plasmalemma. The ability of GME to induce tissue injury was evaluated by measuring Evans Blue uptake in infiltrated foliar tissues (Fig. 3). Dye accumulation was measured over time from GME infiltrated mature tobacco plant leaves. As observed macroscopically in Figure 1, cell death evidenced by Evans Blue uptake was induced when GME was infiltrated at 0.25% (dilution 4-fold) or 0.125% (dilution 8-fold). At the highest GME concentrations applied, lower Evans Blue uptake was measured, which could be attributed to the large necrotic tissues creating a barrier for dye uptake. The 0.0625% (dilution 16-fold) GME treatment was slightly cell death-inducible. The amount of damaged cells increased relative to the GME treatment with the most significant cell death rate at 8 dpi. These data confirm the ability of GME to induce HR-like lesions including localised cell death.

GME induces local and systemic PR expression

Macroscopic evaluation showed that infiltration with GME induced necrotic lesions. We set out to investigate the GME capability to induce defence-related gene expression in tobacco plants. Transcript levels of defence-related genes coding for PR proteins following GME treatment were assessed by quantitative real-time PCR. The primer pairs designed for the tobacco PR1 and PR2

genes amplified a single isoform as documented by the iCycler melting curve and RT-PCR products electrophoresis analysis which resulted in a single product (data not shown). Total RNA was extracted from different regions of tobacco leaves treated with SA (positive control) or with GME (at 4 dpi, as described in Figure 4a). Leaf tissue was collected from infiltrated zones (IFZ), uninfiltrated zones (UFZ), uninfiltrated leaves (UFL), sprayed leaves (SL) and unsprayed leaves (USL).

As expected, SA treatment dramatically induced PR1 and PR2 gene overexpression in IFZ and UFZ. Only the uninfiltrated leaves were weakly responsive to the SA treatment. The foliar spray treatment also induced high levels of PR1 and PR2 gene expression in SL and USL.

High PR1 transcript accumulation was induced when GME (0.25%) was infiltrated or sprayed on tobacco plants.

All treated leaves (IFZ, UFZ, SL) and untreated leaves (UFL and USL) showed hundred times higher level of PR1 transcript accumulation compared with the negative control (Fig 4b transcript accumulation in untreated leaves (near 35%) relative to treated leaves regardless of experimental treatment (i.e., infiltration or spraying).

GME-efficiency threshold

The accumulation profile of PR1 and PR2 transcripts were analyzed locally and systemically in tobacco plants sprayed with concentrations of GME ranging from 0.25% to 0.0625% (Fig. 5). The GME doses ranging from 0.25% (dilution 4-fold) to 0.0625% (dilution 16-fold) achieved a high level of PR1 and PR2 transcript accumulation both in sprayed and unsprayed tobacco leaves. While the macroscopic effects of GME treatment were barely observed on tobacco leaves at 0.0625%, the induction of both PR target genes was Partial GME-characterisation GME was partially characterized for sugar, polyphenol and anthocyanin contents (Table 2). GME consists mostly of polyphenols (91.2%) including 3.5% anthocyanins. These flavonoidbased molecules are responsible for the red colour of the dry extract and the acidic (pH 4.3) nature of the aqueous solution when dissolved in water. GME contains low amounts of soluble sugars (0.9%). While SA and JA are known natural elicitors, these two compounds were not detected by UPLC in the polyphenolic-rich GME aqueous solution (Figure 6). amplitudes. -aminobutyric acid (BABA) known as a priming compound, was previously used as a positive control capable of developing HR-like lesions when applied at 10 mM on tobacco leaves [START_REF] Siegrist | -aminobutyric acid-mediated enhancement of resistance in tobacco to tobacco mosaic virus depends on the accumulation of salicylic acid[END_REF]. In our experiments, no symptoms were observed after BABA infiltration, which might be related to the concomitant action of several defence inducers (GME and SA).

The elicitor activity of GME was evidenced by (i) local injuries and biochemical changes and (ii) a systemic molecular response. GME induced microlesions and cell death when infiltrated into tobacco leaves. After infiltration, the surrounding leaf tissues spread out autofluorescent compounds and produced a local defence reaction with upregulated localised PR1 and PR2 transcript accumulation. These phenomena suggest that GME triggered primary processes resembling those initiated by microbes [START_REF] Dixon | Early events in the activation of plant defense responses[END_REF][START_REF] Hammerschmidt | Induced resistance: how do induced plants stop pathogens?[END_REF]).

When applied as a foliar spray, GME induced PR1 and PR2 transcript accumulation on remote leaves. Unlike SA, GME treatment induced both target PR genes with high systemic amplitudes. These data strongly support that GME was perceived by tobacco cells as a PDI and subsequently activated SAR reactions throughout the entire plant. The synthesis of defence-related proteins is a critical step in the establishment of plant disease resistance. Most PR proteins possess antimicrobial activities in vitro and in vivo [START_REF] Van Loon | Significance of inducible defence-related proteins in infected plants[END_REF]). The accumulation of both PR1 and PR2 protein transcripts was used to monitor the enhanced defensive state conferred by pathogen-induced SAR [START_REF] Edreva | Pathogenesis-related proteins: research progress in the last 15 years[END_REF], and their ectopic overexpression increases resistance to plant pathogens [START_REF] Evans | Transgenic approaches to disease protection: applications of antifungal proteins[END_REF]. The ability of GME to induce antimicrobial protein transcripts in planta with high systemic amplitude strongly suggests a potential role as plant protector agent against microbes.

Our results showed variable immunity responses at the whole plant level with greater sensitivity to GME and increased production of HR-like microlesions in mature leaves. Young, rapidly growing leaves were less reactive to the GME elicitor molecules than mature leaves. Macroscopic changes associated with hypersensitive cell death were more developed in fully expanded leaves, entering their final developmental stage of senescence.

These specific leaf age events illustrated the documented interconnection between plant growth, development and defence (Develey-Rivière and Galiana 2007; [START_REF] Chung | New perspectives on plant defense responses through modulation of developmental pathways[END_REF]. Recent advances in plant immunity research have provided insights into the involvement of plant growth regulators such as ABA, auxins, gibberellins, cytokinins and brassinosteroids. These plant regulators orchestrate both agonistic and antagonistic links between defence and developmental pathways [START_REF] Bari | Role of plant hormones in plant defence responses[END_REF][START_REF] Vlot | Salicylic acid, a multifaceted hormone to combat disease[END_REF].

GME elicited a defence response in a dose-dependent manner as measured by PR transcript accumulation. The threshold activity was observed at 312 µg.ml -1 (dilution 16-fold). It is interesting to note that at this GME concentration, PR gene expression was upregulated but there was a limited effect on chlorosis and cell death.

This indicates that the PR-induced defence response was not proportional to the extent of cell death. Both defensive events were previously described as unrelated phenomena by [START_REF] Mercier | The algal polysaccharide carrageenans can act as an elicitor of plant defence[END_REF]. Laminarins commonly used as PDI are potent inducers of defence-related genes but are unable to induce HR-like lesions.

GME is a botanical extract containing a mixture of secondary plant metabolites, primarily polyphenolic compounds (≥90%) including anthocyanins. GME does not contain measurable SA or JA concentrations. Most classes of phenolic compounds have shown some involvement in defence, including hydroxybenzoic acids, free and conjugated hydroxycinnamic acids, flavonoids and stilbenes [START_REF] Grayer | A survey of antifungal compounds from higher plants, 1982-1993[END_REF]. GME should contain all these compounds, which have been associated with antimicrobial activity, the ability to form structural barriers, the regulation of cellular redox states and/or antioxidant protection [START_REF] Hammerschmidt | Phenols and the onset and expression of plant disease resistance[END_REF]. At this time, only a few biomolecules have been purified and identified as active components from plant extracts. Purified phenolic fractions from Rheum/Frangula extracts are anthraquinone-rich and capable of activating pathogen defense responses in grapevine leaves [START_REF] Godard | Induction of defence mechanisms in grapevine leaves by emodin-and anthraquinone-rich extracts and their conferred resistance to downy mildew[END_REF]. Oligomeric -1,3glucans (e.g., laminarins) and oligofucans from algae extracts can also act as PDIs [START_REF] Aziz | Laminarin elicits defence responses in grapevine and induces protection against Botritis cinerae and Plasmopara viticola[END_REF][START_REF] Klarzynski | Sulfated fucan oligosaccharides elicit defense responses in tobacco and local and systemic resistance against tobacco mosaic virus[END_REF]. The potential plant defence activity of polyphenolic or oligosaccharidic moiety of the GMEbioactive molecules is under investigation. Preparative HPLC will be used in further studies to fractionate the grape extracts and to individually test each fraction for SAR-inducing reactions. One could assume that grape extract activity would increase upon fractionation and purification. However, the combinatorial action of diverse biomolecules cannot be excluded.

PDIs are emerging as biomolecules with the potential for integration into plant protection strategies [START_REF] Walling | Induced resistance: from the basic to the applied[END_REF][START_REF] Vallad | Systemic acquired resistance and induced systemic resistance in conventional agriculture[END_REF]). In the present paper, we provide molecular evidence of PDI activity with the upregulation of PR genes in tobacco-treated leaves. GME was able to initiate plant defence reactions and therefore, should be classified as a PDIs. The uniqueness of this natural product is a result of its origin and nature since it is a high available by-product generated from food industry processing wastes. It contains natural compounds that are most likely not harmful to crops or to the environment. It has been patented as a new class of photoprotecting agent [START_REF] Ter Halle | Phytosanitary formulations containing plant extracts as photoprotectants for pesticides[END_REF]) with a previously demonstrated ability to reduce pesticide photodegradation (Eyheraguibel et al. 2010). The dual activity of GME (PDI and pesticide photoprotector) makes this plant product a promising alternative to other chemicals and a challenging issue for sustainable agriculture and green environmental approaches. Future projects will define the threshold defence reactions for agronomic crops of interest and identify the pathosystems (crop/pathogens) affected by these protection mechanisms. Molecular characterisation and analysis of the inducible crop resistance spectrum will potentially provide a better knowledge on GME mode of action.

Fig. 1 :

 1 Fig.1: Macroscopic symptoms induced in tobacco leaves by GME infiltration observed under bright field (a-e) and UV light (f) at 0 dpi (a), 2 dpi (b), 4 dpi (c) and 8 dpi (d-f). GME was infiltrated at 0.25% (a-d). (e,f): tobacco leaves were infiltrated with different GME concentrations: 0.25% (4X), 0.125% (8X), 0.0625% (16X), 0.0312% (32X), 0.0156% (64X), ultrapure water (control), 2 mM SA and 10 mM BABA. (d-f) the infiltrated area (ia) is delimited by the black line (...); nz, necrotic zone; sz, surrounding chlorotic zone. Bars, 1 cm.

Fig. 2 :

 2 Fig. 2: Macroscopic symptoms induced in tobacco leaves at the whole plant level. Leaves were infiltrated into several distinct areas with 0.25% GME and symptoms were observed 4 dpi on apical leaves (a), middle leaves (b) and basal leaves (c). Bars, 2 cm.

Fig. 4 :

 4 Fig. 4: PR transcript accumulation in tobacco leaves four days after 2 mM SA (positive control) or 0.25% GME infiltration or spraying. The amount of transcript encoding the PR1 (b.) and PR2 (c.) genes was quantified by real-time PCR in the infiltrated zone (IFZ), uninfiltrated zone (UFZ), uninfiltrated leaves (UFL), and after spraying in sprayed leaves (SL) and unsprayed leaves (USL) as described in (a.). Values are expressed relative to the control (water treatment) values. Two treated plants were pooled for each experiment, and the results are the means of duplicate experiments. Bars represent the mean values ± SE.

Fig. 5 :Fig. 6 :

 56 Fig. 5: PR transcript accumulation in tobacco leaves 4 days after GME spraying at different concentrations. The amount of transcript encoding PR1 (a.) and PR2 (b.) genes was quantified by real-time PCR in sprayed leaves (SL) and unsprayed leaves (USL). Values are expressed relative to the control (water treatment) values. Two treated plants were pooled for each experiment, and the results are the means of duplicate experiments. Bars represent the mean values ± SE.

Table 1 :

 1 Sequences of gene-specific primers used for real-time quantitative real-time RT-PCR

	Gene	Specific	Accession	5'primer	3'primer	Tm	PCR fragment
	family	class	number			(°C)	size (pb)
	PR1	PR1a, acidic	X12485.1	5'-TGCTAAGGCTGT-	5'-ACTGAACCCTAG-	58	143
		PR1		TGAGATGTGGGTC	CACATCCAACACG
	PR2	PR2a, acidic	M59443.1	5'-AGCTGTTGGAAA-	5'-GCTAAGATCCCT-	59	150
		glucanase		TGAAGTCTCTCC	GAATATGTTGCAG
	EF	Elongation	AF120093	5'-CCACAGACAAGC-	5'-TTCAGTGGTCAG-	65	149
		Factor-1		CCTCTCAGGCTCC	ACCAGTGGGACC
	PR, pathogenesis related protein

Acknowledgements This work had financial support provided by an ANR Ecophyto 2008-2011 project. We thank Céline Sac for her help in growing tobacco plants and Dominique Marcon for the tobacco leaf images.

Author manuscript published in Ecotoxicology (2012)


Céline Pélosi
email: celine.pelosi@versailles.inra.fr

François Chiron

Florence Dubs

Mickael Hedde

Jean-François Ponge

Sandrine Salmon

Daniel Cluzeau

Sylvie Nelieu

C Pelosi

J.-F Ponge

S Nélieu

A new method to measure allyl isothiocyanate (AITC) concentrations in mustard: comparison of AITC and commercial mustard solutions as earthworm extractants

Keywords: Earthworm sampling, Efficiency, Chemical extraction, Expellant

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

Introduction

Earthworms exert important agro-ecological functions (e.g. they influence organic matter dynamics and soil structure; [START_REF] Edwards | Biology and Ecoloy of Earthworms[END_REF][START_REF] Sims | Earthworms[END_REF] and are recognized as good biological indicators of soil quality and impacts of cultural practices [START_REF] Oberholzer | A novel method for soil quality in life cycle assessment using several soil indicators[END_REF][START_REF] Paoletti | The role of earthworms for assessment of sustainability and as bioindicators[END_REF]. These organisms are interesting, both for scientists studying the biological component of cultivated soils and for farmers concerned with monitoring the quality of their soils and assessing the effects of different cultural practices.

The common interest of all these people is to move towards sustainable agriculture, producing enough yields while limiting environmental damage.

In order to assess human impacts on soil biodiversity and soil invertebrate biomass, earthworms have been sampled by scientists for several decades [START_REF] Evans | Studies on the relationships between earthworms and soil fertility[END_REF] and more and more by the general public using standardized and simplified protocols (http://aceracre.ca/wp-content/uploads/2011/09/EMAN-MONITORING-BIODIVERSITY-IN-CANADIAN-FORESTS.pdf; http://ecobiosoil.univ-rennes1.fr/OPVT_accueil.php; http://observatoire-agricolebiodiversite.fr/sites/oab.mnhn.fr/files/upload/attached/oab_guide_utilisateur_2013_protocolevers-de-terre.pdf; http://www.opalexplorenature.org/soilsurvey).

Different "scientific methods" are used [START_REF] Valckx | Optimizing earthworm sampling in ecosystems[END_REF]. A common one consists of combining the application of a chemical expellant, which brings earthworms to the soil surface with hand-sorting of the underlying soil [START_REF] Bartlett | A critical review of current methods in earthworm ecology: from individuals to populations[END_REF]. Various chemicals can be used, like formalin (ISO 23611-1:2006;[START_REF] Raw | Estimating earthworm population by using formaline[END_REF], mustard powder [START_REF] Högger | Mustard flour instead of formalin for the extraction of earthworms in the field[END_REF][START_REF] Muramoto | Mustard powder vermifuge: an alternative to formalin expulsion method for earthworm sampling[END_REF] or different brands of commercial mustard [START_REF] Lawrence | A test of the "hot" mustard extraction method of sampling earthworms[END_REF][START_REF] Pelosi | Earthworm collection from agricultural fields: comparisons of selected expellants in presence/absence of hand-sorting[END_REF]. More recently, researchers have demonstrated the effectiveness of allyl isothiocyanate (AITC), an irritating molecule contained in commercial mustard, for sampling earthworms in cultivated areas [START_REF] Pelosi | Earthworm collection from agricultural fields: comparisons of selected expellants in presence/absence of hand-sorting[END_REF][START_REF] Zaborski | Allyl isothiocyanate: an alternative chemical expellant for sampling earthworms[END_REF].

Moreover, according to Čoja et al. (2008), while formalin may harm soil organisms, AITC has no undesirable side-effects on soil organisms.

In France, the participatory method of earthworm sampling consists of the application of commercial hot mustard Amora ® "Fine et Forte" solution on the soil (http://observatoireagricolebiodiversite.fr/sites/oab.mnhn.fr/files/upload/attached/oab_guide_utilisateur_2013_protocolevers-de-terre.pdf). However, due to commercial confidentiality, the concentration of AITC in this mustard is unknown, as for all other mustard brands used for earthworm sampling. To date, no study has compared the chemical properties of commercial mustard and AITC. The objectives of the present work are to compare (i) the irritating properties of two chemicals used for sampling earthworms and (ii) the efficiency of the two expellant solutions in terms of diversity, abundance and biomass of earthworms.

Materials and Methods

Chemical analysis of commercial hot mustard

Commercial hot mustard and chemicals

AITC concentration was analyzed in the commercial mustard Amora ® "Fine et Forte".

To assess variability in AITC concentration, ten 150 g jars from different batches and ten 150 g jars from the same batch were compared. AITC (synthetic grade, estimated 97.3%) was purchased from VWR and acetonitrile (HPLC-plus grade) from Carlo-Erba. LC-grade water (resistivity > 18.2 MOhm cm) was produced by a Maxima system (USF Elga, High Wycombe, UK).

Extraction procedure

To our knowledge, only one method has been published to measure AITC in mustard [START_REF] Hils | Bestimmung von Allylisothiocyanat in Speisesenfen und Meerrettichzubereitungen[END_REF]. This method uses water vapor distillation into an ammonia-holding receiver, the allyl thiourea from the AITC then being measured by spectrophotometry. It thus requires a specific distillation apparatus and allows quantification of a transformation product, but not AITC itself. We thus chose to develop a new method, based on AITC solvent extraction and HPLC quantification. According to the optimized method, 2.5 g of mustard were weighed in a 50 mL polypropylene tube (Falcon BD) and immediately suspended in 5 mL LC-grade water. Then, 7.5 mL of acetonitrile were added and the mixture was agitated for 10 min on an orbital shaker at 300 rpm (KS501 digital, Ika), sonicated for 30 min at a temperature below 30 °C and centrifuged (Allegra X-15R, Beckman Coulter) for 10 min at 1300 g and 20 °C. The supernatant was removed with a Pasteur pipette and collected in a 50 mL glass vial with a screw cap with PTFE coated seal and protected from light. The sample was extracted three times consecutively with water and acetonitrile as described above and all extracts were collected in the same vial. The final volume (35-37 mL) was estimated by weighing, the extract density being estimated at 0.871. A 2 mL aliquot was filtered with 0.45 µm hydrophilic PTFE filter (4 mm, Millex-LH, Millipore) prior to HPLC-UV analysis, collecting filtrate from the fifteenth droplet into the HPLC-injection vial. According to mustard spiking with AITC standard solution, extraction recovery was estimated as 98 ± 2%

(approximately 77%, 17% and 4% at each successive extraction cycle).

HPLC-UV analysis

HPLC-UV analyses were performed on a Dionex system, including an ASI100T autosampler, a P580 pump, a STH585 column oven and a UVD380S UV-photodiode array detector. Separation was done on a 125 x 2 mm 3 µm Nucleodur C18 HTec reversed-phase column (Macherey-Nagel ® , Düren, Germany) at 20 °C, using a gradient of water and acetonitrile at 0.3 mL min -1 flow rate (75/25 to 25/75 v/v in 20 min). Samples were kept at 15 °C and protected from light in the autosampler before being injected. The injection volume was 5 µL. Quantifications were carried out at 242 nm using external calibration. The limit of quantification was estimated at ca. 14 µg.g -1 and the linear range was above 2800 µg.g -1 (AITC/mustard). All analyzed mustard jars were extracted and analyzed in triplicate, with a standard deviation of less than 6%.

Earthworm sampling 2.2.1. Sites and crop systems

Field data were collected in spring 2012 in 22 cultivated fields located in Seine-et-Marne, east of Île-de-France. Among them, 11 fields were under conventional farming and 11 under organic farming. Soils were clay loamy (70% silt, 25% clay and 5% sand on average) at near-neutral pH (Appendix A).

The climate was temperate oceanic, with a mean annual rainfall of 640 mm and a mean annual temperature of 10.4 °C.

Conventional plowing at 25-30 cm depth was carried out in all fields, at a frequency ranging from every year to once every three years. The last plowing was done in 2010 or 2011, depending on the field. All fields were cultivated with winter wheat at the time of sampling. Samplings with AITC and commercial mustard were done on the same fields and on the same days.

Samplings with AITC

We use the procedure proposed by [START_REF] Pelosi | Earthworm collection from agricultural fields: comparisons of selected expellants in presence/absence of hand-sorting[END_REF] for AITC application. First, AITC was diluted with isopropanol (propan-2-ol, RPE grade, Carlo-Erba) to obtain a 5 g.L -1 solution [START_REF] Pelosi | Earthworm collection from agricultural fields: comparisons of selected expellants in presence/absence of hand-sorting[END_REF][START_REF] Zaborski | Allyl isothiocyanate: an alternative chemical expellant for sampling earthworms[END_REF]. This solution was then diluted with water to reach a concentration of 0.1 g.L -1 . After removing vegetation present at the ground surface, two applications of 3.2 L of the prepared AITC solution were applied to the soil at 10-min interval within a 40 x 40 cm metal frame. Taking into account the sample area, a jug with a spout has been used for the application of the AITC solution since a watering can with a rose would not have been appropriate. Emerging individuals were collected for 20 min and preserved in 4% formalin solution. Three replicates, spaced approximately 7-10 meters apart, were made in each field, at least 10 meters from the field edge (Table 1).

Samplings with commercial mustard

We use the procedure proposed by the French participatory method of earthworm sampling (http://observatoire-agricolebiodiversite.fr/sites/oab.mnhn.fr/files/upload/attached/oab_guide_utilisateur_2013_protocolevers-de-terre.pdf). After removing vegetation present at the ground surface, two applications of 10 L of a mustard solution were applied to the soil 15 min apart over 1 m 2 . A suspension in water was prepared by mixing two 150 g jars of commercial hot mustard (Amora ® 'Fine et Forte") in 10 L of water. All jars were from the same batch and had the same expiry date. A watering can with a rose was used to spread the solution evenly over the sampling surface.

Emerging individuals were collected for 30 min and preserved in 4% formalin solution. Three replicates, spaced approximately 7-10 meters apart, were made in each field, at least 10 meters from the field edge (Table 1).

Earthworm identification

All individuals (juveniles, sub-adults and adults) were counted and weighed with their gut content. Sub-adults and adults were identified at species level according to the identification key of [START_REF] Sims | Earthworms[END_REF]. Juveniles were attributed to species according to morphological characteristics and to the specific form they take in formalin in comparison with that of identified adults.

Statistical Analysis

Observations were first transformed to record earthworm abundance and biomass per square meter. Medians of the three replicates per field were calculated and used for statistical analysis. Data presented in the results section are means of these medians. AITC concentration of irritating chemical as well as earthworm diversity (species richness), abundance and biomass in each field recorded with the different chemicals were compared with Wilcoxon Mann-Whitney non-parametric tests because conditions for parametric tests were not fulfilled. We used the R software (R Development Core Team, 2005).

Results

1. Comparison of AITC concentrations

Chemical analysis of the ten jars of commercial hot mustard from the same batch revealed that concentrations of AITC were relatively uniform within the same batch. The mean of the 10 jars was 860 µg.g -1 with a standard deviation of 39 µg.g -1 , corresponding to a 5% standard deviation. Values ranged from 811 to 922 µg.g -1 .

For the 10 jars from different batches, AITC concentrations differed considerably between jars, with a standard deviation of 36%. Values ranged from 273 ± 11 to 1306 ± 13 µg.g -1 (mean ± standard deviation). We noticed that the concentration of AITC in the jars was higher when the indicated time before expiry of the mustard was longer (Fig. 1).

For the sampling with pure AITC (proposed by [START_REF] Zaborski | Allyl isothiocyanate: an alternative chemical expellant for sampling earthworms[END_REF][START_REF] Pelosi | Earthworm collection from agricultural fields: comparisons of selected expellants in presence/absence of hand-sorting[END_REF], 1 mL AITC was poured into 10 L of water. Since the density of the AITC was 1 g.cm - 3 , then 1 mL of AITC diluted in 10 L of water corresponded to a concentration of 0.1 g.L -1 .

For the sampling with commercial mustard (http://observatoire-agricolebiodiversite.fr/sites/oab.mnhn.fr/files/upload/attached/oab_guide_utilisateur_2013_protocolevers-de-terre.pdf) involving two 150 g jars of mustard diluted in 10 L of water, the average concentration of AITC was approximately 0.025 g.L -1 in the solution that was spread on the soil. However, this concentration could vary from 0.008 to 0.039 g.L -1 according to the analysis of ten jars of different batches with 171 to 305 days before expiry. The AITC solution, used in the scientific method of earthworm sampling, was thus about four times more concentrated in AITC than the mean of commercial mustard solutions used in the participatory method.

2. Comparison of chemical expellant efficiency

Mean abundances of total expelled earthworms over the 22 fields were 6.1 and 14.5 individuals per m 2 for mustard and AITC solutions respectively, but the difference was not significant (p=0.39) (Fig. 2a). Most values obtained with the AITC solution were higher than with the mustard solution but their variability was also higher (Fig. 2a). Mean biomasses of the total earthworm community over the 22 fields were 1.9 g.m -2 and 4.9 g.m -2 for mustard and AITC solutions, respectively. Again, this difference was not significant (p=0.18) (Fig. 2b).

Both chemical expellants allowed us to sample seven species of earthworms. Six species were common to mustard and AITC solutions: Lumbricus terrestris (Linnaeus, 1758), Aporrectodea longa (Ude, 1885), Aporrectodea giardi (Ribaucourt, 1901), Allolobophora chlorotica (Savigny, 1826), Aporrectodea rosea (Savigny, 1826) and Aporrectodea caliginosa (Savigny, 1826) (Fig. 3). The first three are anecic and the last three are endogeic species. Aporrectodea icterica (Savigny, 1826), an endogeic species, was only found with the AITC solution and Lumbricus castaneus (Savigny, 1826), an epigeic species, was only found with the mustard solution (Fig. 3). AITC appeared to be more efficient than mustard to bring the three dominant species to the soil surface (Fig. 3): for mustard and AITC solutions, L.

terrestris represented 21 and 27% of the total abundance, A. caliginosa represented 24 and 43% of the total abundance and A. chlorotica represented 27 and 20% of the total abundance, respectively.

Discussion

This study highlighted that AITC concentration was uniform among different jars of a given batch of commercial hot mustard (Amora® "Fine et Forte"), used in the participatory method for earthworm sampling and exported and therefore available worldwide. However, the AITC concentration in mustard jars varied five-fold, depending on their use by date. This can cause problems of standardization of the method since commercial mustards of the same brand may not have the same efficiency. Efficiency does not necessarily increase with AITC concentration (Čoja et al., 2008) but the concentration found in the mustard solution was low compared to the optimal AITC concentrations reported by [START_REF] Zaborski | Allyl isothiocyanate: an alternative chemical expellant for sampling earthworms[END_REF]. It is thus preferable to use jars with a long time before expiry. To get reliable and comparable results

between fields in a study, it is thus preferable to use mustard jars (i) with a long time before expiry for maximum efficiency and (ii) from the same batch for minimum variation in expellant efficiency.

Abundance of extracted individuals was quite low compared to data reported by Čoja et al. (2008) in a meadow or by [START_REF] Pelosi | Earthworm collection from agricultural fields: comparisons of selected expellants in presence/absence of hand-sorting[END_REF] in cultivated fields. This may be due first to the ploughing of the fields where earthworms were collected: in ploughed fields, endogeics are generally found in majority compared to anecics [START_REF] Chan | An overview of some tillage impacts on earthworm population abundance and diversity -implications for functioning in soils[END_REF], while chemical expellants are preferentially used to collect the latter ecological group [START_REF] Bouché | Lombriciens de France: Ecologie et Systématique[END_REF]. Secondly, the soil was perhaps slightly too dry for an optimal penetration of chemicals in soil. However, it could have been unnecessary to add water before doing the samples, since [START_REF] Eisenhauer | Efficiency of two widespread non-destructive extraction methods under dry soil conditions for different ecological earthworm groups[END_REF] explained that "surprisingly, the efficiency […] was not improved by beforehand water addition". The results might have been different if the soil had been wetter. Nevertheless, the values found in this study are in accordance with the data published by [START_REF] Bartlett | Inefficiency of mustard extraction technique for assessing size and structure of earthworm communities in UK pasture[END_REF] for mustard. Moreover, a hand sorting of earthworms at the end of the extraction time revealed that individuals were active (i.e. not curled in an inactive stage).

Both expellant solutions used the same compound, AITC, to extract earthworms but in one case it was pure and in the other case, it was only a component of a commercial product.

Although (i) a greater quantity of chemical is sprayed on the soil per square meter (Table 1) and (ii) the concentration of AITC was four times higher in the AITC solution, both solutions seemed to have almost the same efficiency to bring earthworms to the soil surface. While both expellant solutions retrieved the same number of species, we found a trend towards collecting more individuals of the three dominant species with the AITC solution. This could be due either to the more irritating character of the solution, given the difference in AITC concentration, or to the pure compound being more evenly spread in the water, thanks to early isopropanol dilution [START_REF] Pelosi | Earthworm collection from agricultural fields: comparisons of selected expellants in presence/absence of hand-sorting[END_REF]Zaborsky, 2003). Indeed, in the field, it was necessary to mix the mustard very thoroughly in water before application to prepare an adequate suspension of commercial mustard. Finally, commercial mustard"s components, other than AITC, may have participated to the efficiency of this expellant for earthworms.

We here compared advantages and drawbacks of two extraction methods that are currently used by scientists and citizens for earthworm sampling in France and elsewhere (Table 2). The area sampled with the mustard solution was 6.25 times larger than that sampled with AITC solution. Thus, the amount of material and water needed was proportionally higher.

Conclusion

We here described a new effective method to measure allyl isothiocyanate concentration in mustard. Testing commercial mustard jars from different batches and jars from the same batch, we found that AITC concentration in mustard varied five-fold, depending on its use by date. The comparison of the AITC concentration in two earthworm expellant solutions (diluted commercial mustard and pure AITC solution, used in a French participatory method and a scientific method respectively), revealed that AITC concentration was about 4 times higher in the AITC solution than in the mustard one. Moreover, AITC and commercial mustard solutions had the same earthworm extracting efficiency.

To improve the efficiency of earthworm samplings with commercial mustard (as in participatory method), it is thus recommended to use jars of commercial hot mustard Amora® "Fine et Forte" from the same batch and with a long time before expiry. It would also be useful to note the expiry date of jars used for earthworm sampling when using the participatory method.

Fig. 1 .Fig. 1 .Fig. 2 .Fig. 3 .

 1123 Fig. 1.

Table 1 .

 1 Characteristics of the two earthworm sampling methods.

	Sampling with AITC	Sampling with commercial mustard

download Table: Table 1.docx

Table 2 .

 2 Comparison of earthworm sampling using AITC or commercial mustard solutions Same expellant molecule (allyl isothyocianate) Additives: manufacturer' trade secret Easy access to chemical expellants Bought from a supplier Available in many shops, exported worldwide Amount of water to be transported in the field 6.4 L per replicate 20 L per replicate Time needed Approximately 35 min per person per replicate Approximately 50 min per person per replicate

	Sampling with AITC	Sampling with commercial mustard

Table2Click here to download Table: Table 2.docx

Acknowledgements

We would like to thank everybody who participated in earthworm sampling and more especially Lucile Toutous, Jodie Thénard, Félix Fraillon, Françoise Elsass and Nathalie Cheviron. Financial support was provided by the "Fédération Île-de-France de Recherche sur l'Environnement" (FIRE, FR3020) and the "Agence Nationale pour la Nature et la Biodiversité en Île-de-France" (Natureparif). We would also like to thank Marjolaine Deschamps for initial informations on AITC analysis and Alan Scaife for revising the English.

Appendix A. Site and soil characteristics of the 22 sampled agricultural field plots in Seine-et-Marne (all under winter wheat).

Plot name

Cropping system Clay (g kg -1 ) Silt (g kg -1 ) Sand (g kg -1 ) Organic matter (g kg -1 ) CaCO3 (g kg


Direct injection in organic SU8 nanowires and nanotubes for waveguiding properties investigation

Keywords: nanophotonics, sub-wavelength propagation, SU8, polymer, wetting-template, nanotube, nanowire, cut-back method

We report photonic concepts related to injection and sub-wavelength propagation in nanofibers (nanowires and nanotubes). These nanostructures are fabricated by the wetting template method leading to aspect ratio of over 250. At first, injection into nanowires and nanotubes of SU8, a photoresist used for integrated photonics, was successfully achieved by using polymer microlensed fibers with sub-micronic radius of curvature. Theoretical simulation by finite domain time-dependent (FDTD) method was used to determine the sub-wavelength propagation for nanowires and nanotubes and corroborate this coupling phenomena. The original confinment of energy density into SU8 nanotubes is highlighted. Finally, characterisation of propagation losses is reported by using a cut-back method transposed to such nanotubes and determined to range between 1 and 2 dB/mm. Both injection and cut-back method developed here are compatible with any sub-micronic structures. This work on SU8 nanofibers suggests broader perspectives for future nanophotonics.

INTRODUCTION

Sub-wavelength waveguides are the subject of numerous publications and applications 1 (Mach-Zehnder, add-drop filter, etc...). Due to their sub-wavelength dimensions, 1D-nanostructures are of particular interest in integrated optics by the ability to control their aspect ratio. Effective light waveguiding has been well investigated mainly in silica nanofibers while only few studies have been reported in organic nanofibers. The SU8 polymer was selected, because it is a widely-used epoxy based negative photoresist, and commonly used in integrated photonics, due to its excellent characteristics for micronic waveguides. 2 SU8 polymer presents large refractive index compared to most organics and polymers. This large refractive index enhances the confinment in light waveguiding. Further, the tubular aspect of nanotube give a new degree of modulation in photonics. In literature, Zhao et al. [START_REF] Zhao | Field and dispersion properties of subwavelength-diameter hollow optical fiber[END_REF] have shown that the tubular geometry of nanotube can uniquely confine part of the energy density in the lowest refractive index area, the core of the nanotube. This original confinment allows tuning of light propagation and light-matter interaction. In this present work, we detail keys for: low-cost production of SU8 polymer nanowires and nanotubes by wetting template method, experimental protocol to directly inject light into such nanostructures, numerical analysis of sub-wavelength propagation, and finally characterisation of optical losses for SU8 nanotubes.

SYNTHESIS

The wetting template method allows the creation of 1D nanostructures with high aspect ratio, diameter ranging from 50 nm to 250 nm and length up to 100 µm typically. This method achieves low-cost fabrication, fine control dimensions while providing the possibility of large-scale production. This method is based on the use of a previously nanostructured template, pores with nanometer dimensions, which allows to organise the growth of nanomaterials. This link between morphology and nanomaterial template allows control of size, shape and orientation of the nanofibers. In our study, AAO (Anodic Aluminium Oxide) membrane presents perfectly cylindrical pores arranged in the plane perpendicular to the axis of the pores, as visible on figure 1a. Wetting template method ensures wetting of AAO by the material in a liquid phase due to the surface energy of the alumina. Here, we used the SU8 photoresist in solvent. Additionally, the different viscosities of SU8 available in liquid phase are a central advantage for the template strategy. Elaboration of SU8 nanowires and nanotubes is selected by the difference of surface energy between the polymer in liquid phase and the alumina. High or low viscosity of solution leads to synthesis of nanowires or nanotubes respectively. Once the in-solution polymer penetrates over the entire thickness, a UV insolation (λ = 365 nm, 200mJ/cm 2 ) promotes cross-linking of the SU8 to get quite rigid nanofibers. A selective chemical etching by H 3 P O 4 solution, releases SU8 nanofibers from the AAO. Finally, SU8 nanofibers were dispersed into isopropyl alcohol or distilled water. A drop of the suspension of SU8 nanofibers was deposited onto a clean SiO 2 wafer. Figure 1b

EXPERIMENTAL SET-UP

In literature, the injection in light-guiding sub-micronic structures is based on two different mechanisms : by grating side couplers [START_REF] Saleh | Fundamentals of Photonics[END_REF] or through evanescent coupling between two photonic structures. [START_REF] Tong | Optical microfibers and nanofibers: A tutorial[END_REF] However, these approaches require micronic coupling distances and do not allow to fully optimise the quality of the injection in nanostructures nanoscale. In previous publications, [START_REF] Huby | Light propagation in single mode polymer nanotubes integrated on photonic circuits[END_REF][START_REF] Duval | Theoretical formulation to shape versatile propagation characteristics of threelayer-tubular waveguides: sub-wavelength and asymptotic study[END_REF] we have reported the single-mode behavior hybrid mode HE 11 of SU8 polymer nanowires and nanotubes by theoretical and experimental approach. Light injection was accomplished by evanescent coupling with SU8 sub-micronic structures. [START_REF] Huby | Light propagation in single mode polymer nanotubes integrated on photonic circuits[END_REF] Here, an innovative way to inject light into 1D nanostructures has been developed by using polymer microlens. The injection of the light is provided by single-mode (@630 nm) microlensed polymer fiber (Lovalite). In addition, microlens fibers are positioned by piezo-electric nano-positionner (Pi-611-SF) in order to precisely control their positioning with 10 nm step in the three directions and optimise light injection inside nanostructures. A top-view display system, figure 2, consisting of a monochrome CCD camera (AVT GX1920), with pixels size of 4.54 µm associated with x40 and x100 optical lenses (numerical aperture 0.90, working distance 1 mm) permits the simultaneous control and optimisation of the light injection in nanostructures. A beam-splitter element on optical path allows the spectral detection of nanostructures by spectrometer (OCEAN Optics USB2000).

SUB-WAVELENGTH PROPAGATION

Figure 3a displays a SEM image of a microlensed fiber which present at their end a curvature radius of 500 nm. The cylindrical geometry of the microlens has been chosen to coincide with the tubular symmetry of SU8 nanofibers. Thereby, the energy overlap integral between the injecting microlens and the SU8 nanofibers is maximised. Under laser excitation (675 nm), microlens polymer fibers exhibit a light lobe typically of a few micrometers in diameter. Figure 3b presents the approach phase of the microlensed fiber near the SU8 nanotube under visible light. After an adequate protocol of positioning optimised by the top-view display system, the microlens is brought thanks to the piezo control at few micrometers of the nanotube. On the figure 3b, when the microlens is located close to the SU8 nanotube, under laser excitation at 675 nm, a spot at the opposite end of the nanotube is visible. The presence of this spot, highlighted by 3D-view on figure 3c, is the proof of a significant injection into the SU8 nanotube. The absence of light scattering along the organic SU8 nanotube is an indicator of low roughness of the surface of the nanotube and the absence of morphological defects. An identical behavior has been determined for SU8 nanowires.

FDTD MODELISATION

The sub-wavelength propagation of nanowires and nanotubes has been investigated by a numerical 3D finite domain time-dependent (FDTD ) simulation and performed by Meep freeware. [START_REF] Oskooi | Meep: A flexible free-software package for electromagnetic simulations by the FDTD method[END_REF] FDTD simulation has already proved accuracy to study optical sub-wavelength propagation in nanostructures. [START_REF] Wang | Modeling endface output patterns of optical micro/nanofibers[END_REF][START_REF] Wang | Endface reflectivities of optical nanowires[END_REF] As determined by figure 1b,c in section 2, geometrical parameters of SU8 nanowires are set as an external diameter of 240 nm for SU8 nanowires and nanotubes, and an inner diameter of 120 nm for nanotubes. The model contains an Oy-polarised laser source at 675 nm (FWHM = 12nm) launched inside nanowires and nanotubes with refractive index of n SU8 = 1.56. Figures 4a and4b present orthogonal views (i.e. : along Oyz plan) of the optical power mode HE 11 inside the SU8 nanowires and nanotubes. It highlights the separate behavior of density energy confinment of both nanostructures. In case of nanowires, the maximum of light is confined into the SU8 nanofibers. In case of nanotubes, the tubular geometry allows a significant confinment of density power inside the hollow region, the maximum of light propagating in the wall of the nanotube with a extension of the density power of 50 nm in air. This makes nanotubes based devices sensitive to surface modification and offers great opportunities for near-field sensors.

In such sub-wavelength propagation regime, the optical coupling between the microlensed fiber and the SU8 nanofibers is critical and deserves a specific study. Figure 4c is a top-view of the power density of the microlensed fiber aligned in direction of free-standing SU8 nanotube. The presence of standing wave patterns along the axis of nanotube attest the coupling phenomena between SU8 nanotube and microlensed fibers. They are due to interference between propagating light and reflection at the output endfacet. This phenomena has been already established on waveguiding silica nanowires. [START_REF] Coillet | Near-field characterization of glass microfibers on a low-index substrate[END_REF] Projection of power density inside the nanotube along Ox on figure 4a corroborates the period of standing waves : P = λ dispersion of attenuation coefficient summarised on figure 5d, determined to range between 1 and 2 dB/mm. For comparison, SU8 microstructures exhibit losses of 17 dB/mm. [START_REF] Bêche | Single-mode TE00TM00 optical waveguides on SU-8 polymer[END_REF] It can be noted that plasmon nanowaveguides much investigated for sub-wavelength photonics present losses of 4000 dB/mm. [START_REF] Wang | Light propagation in curved silver nanowire plasmonic waveguides[END_REF] In previous work, 5 scattering losses of SU8 nanotubes had been estimated around 150 dB/mm. This higher value was allocated to the measurement process (i.e. : scattering), to a synthesis process not optimised, and to a different injection process of light inside SU8 nanotubes (evanescent coupling with microstructures).

CONCLUSION

In summary, direct injection into organic nanofibers (nanowires and nanotubes) has been demonstrated and confirmed their efficient waveguiding properties. A derivated cut-back method is proposed for attenuation coefficient estimation in such sub-micronic structures. In SU8 nanotubes, the attenuation coefficient is determined to range between 1 and 2 dB/mm. The sub-wavelength regime propagation in such SU8 nanotubes highlight the modulation of guiding light provided by tubular geometries in nanostructures. This study allows perspectives for SU8 nanotubes as building blocks in more complex systems or sub-wavelength optical probes for near-field interactions. This feature makes polymer SU8 nanotubes a good candidate as the basic element of highly integrated nanophotonics and sub-wavelength optical probes.

Figure 1

 1 Figure 1. a) SEM top-view image of AAO membrane with cylindrical nanopores. SEM image of b) SU8 nanotubes and c) SU8 nanowires elaborated by the wetting template method after the partial release from the AAO membrane.

Figure 2 .Figure 3

 23 Figure 2. Sketch of top-view display system to control and optimise the direct light injection into 1D nanostructures. A microlensed fiber is mounted on piezo-electric nano-positionner. CCD Camera or spectrometer allow optical analysis of coupling phenomena. Wafer Air Nanotube

Figure 5

 5 Figure 5. a) SEM image of a clean cleaved endface of SU8 nanotube, b) top-views of images during waveguiding of laser light at different lengths. c) Intensity of propagative spot for different propagation distances and d) distribution of losses determined to range between 1 and 2 dB/mm.

ACKNOWLEDGMENTS

The authors thank Jo Le Lannic and Francis Gouttefangeas for SEM characterization, Guillaume Raffy for FDTD study, Ludovic Frein and Steve Bouhier for helpful discussions and experimental assistance. J. Bigeon PhD is supported by the Lab-O-Mat french network between CNRS, Université de Nantes and Université de Rennes 1, and Bretagne and Pays de la Loire Regions.


Valentin Emiya

Antoine Bonnefoy

Laurent Daudet

Rémi Gribonval

Compressed sensing with unknown sensor permutation

Keywords: Inverse problem, sparsity, compressed sensing, dictionary learning, permutation, optimization, branch and bound

come

INTRODUCTION

Many studies have addressed the linear sparse estimation problem where an observed data y ∈ R M is modeled as y = Dx where D ∈ R M ×K , M ≤ K is the known dictionary or measurement matrix and the unknown representation x ∈ R K of the data in D is assumed to be sparse. Sparsity means that the number of non-zero elements in x, denoted by x 0 , is small compared to M . A good estimation [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF] of x from y and D is obtained by solving, for p = 1, arg min

x x p s.t. y = Dx, (1)
which is a convex relaxation of the NP-hard problem when p = 0. In many scenarios, the dictionary D is unknown. It can be learned from a collection of N examples given as the columns of a matrix Y ∈ R M ×N , each of them being sparse in the unknown dictionary. This task is known as dictionary learning [START_REF] Aharon | K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation[END_REF][START_REF] Yaghoobi | Dictionary learning for sparse approximations with the majorization method[END_REF][START_REF] Rubinstein | Dictionaries for sparse representation modeling[END_REF] and its principle can be formulated, for p = 0 or p = 1, as the problem arg min D,X X p s.t. Y = DX.

(

) 2
Dictionary learning is a difficult nonconvex problem. Adding prior knowledge is an option to make it simpler -yet still interesting in order to partially learn a dictionary from examples [START_REF] Rubinstein | Double sparsity: Learning sparse dictionaries for sparse signal approximation[END_REF][START_REF] Cevher | Greedy dictionary selection for sparse representation[END_REF][START_REF] Yaghoobi | Optimal dictionary selection using an overcomplete joint sparsity model[END_REF][START_REF] Bilen | Convex Optimization Approaches for Blind Sensor Calibration using Sparsity[END_REF]. This strategy may even lead to a convex problem, such as the estimation of gains in the measurement matrix [START_REF] Bilen | Convex Optimization Approaches for Blind Sensor Calibration using Sparsity[END_REF].

In this paper, we address a new problem, in the same philosophy, by considering the case where the order of the sensors has been lost during the sensing process: the dictionary is known up to a permutation of its rows. Such a scenario may be of interest in applications like bit-rate reduction in a channel that does not preserve data order. It may also happen in an experimental setup with a large number of sensorse.g., a microphone array -, due to some handcrafting mistakes in the wiring between sensors and A/D converters. We call this problem the sensor permutation problem 1 .

Formally, an unknown permutation matrix P is such that PY = DX. The related estimation problem addressed in this paper is arg min

P,X X 1 s.t. P ∈ P M PY = DX (3)
where P M is the set of M × M permutation matrices. Problem (3) is combinatorial since P M is discrete with size M !. Our contributions to overcome this complexity and solve the problem are:

• we study a convex relaxation of (3) by replacing P M by its convex hull and show that in practice, this strategy almost always fails to identify the true permutation (Section 2);

• we propose a branch-and-bound algorithm that converges to the solution of (3) by exploring P M cleverly (Section 3);

• we show experimentally that in many configurations, the proposed algorithm converges quickly, after exploring a small subset of P M ; we finally check that in the experiments, the solution of (3) is indeed the original permutation (Section 4).

RELAXED CONVEX PROBLEM

The only reason why problem (3) is not convex is that P M is not a convex set. A natural three-step strategy to obtain a tractable solution is: replace P M by its convex hull; solve the resulting convex problem; project the estimate back to P M . The convex hull of P M is the set B M of M × M bistochastic matrices, i.e., matrices with non-negative coefficients summing to one along each row and each column. The convex problem of interest is thus

B, X ≜ arg min B∈B M ,X X 1 s.t. BY = DX = arg min B,X X 1 s.t. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ B ∈ R M ×M + B1 M = 1 M B T 1 M = 1 M BY = DX (4)
where 1 M is an M -length vector filled with ones.

Since P M ⊂ B M , the estimate B is generally not a permutation matrix : it can be post-processed by the Hungarian algorithm [START_REF] Kuhn | The hungarian method for the assignment problem[END_REF] in order to obtain the closest permutation matrix P ≜ arg max P∈P M ⟨P, B⟩ where ⟨A, B⟩ ≜ ∑ i,j A (i, j) B (i, j).

By fixing P, the sparse representation is finally refined as X ≜ arg min

X X 1 s.t. PY = DX. ( 5
)
This strategy is summarized in Algorithm 1 where any convex optimization method is suitable to solve (4) and [START_REF] Rubinstein | Double sparsity: Learning sparse dictionaries for sparse signal approximation[END_REF].

Algorithm 1 Simple convex relaxation strategy Require: Y, D.

1: Estimate the bistochastic matrix B by solving (4).

2: Find the closest permutation P by applying the Hungarian algorithm to B. 3: Estimate the sparse matrix X by solving (5). 4: return P, X.

In practice, this strategy gives poor results : it almost never identifies the solution of problem [START_REF] Yaghoobi | Dictionary learning for sparse approximations with the majorization method[END_REF]. As an illustration, the average number of errors in the estimated permutation has been measured in a basic experimental setting, using uniformly-random permutations on M measures, a Gaussian M × 2M dictionnary D and N = 40 examples with K0 = 3 non-zero Gaussian components randomly located. An error is defined as a wrong sensor index out of the M possible positions in the permutation and the number of errors is averaged among 20 trials. As reported in Table 1 for 6 ≤ M ≤ 20, most of the permutations are far from being retrieved since the number of errors almost equals the number of elements in the permutations. Table 1. Average number of errors in the permutation estimated by the simple relaxation strategy (Algorithm 1) as a function of M .

A BRANCH-AND-BOUND ALGORITHM

Branch-and-bound principle

Branch and bound [START_REF] Lawler | Branch-and-bound methods: A survey[END_REF][START_REF] Boyd | Branch and bound methods[END_REF] is a clever way to explore a non-convex set and solve an optimization problem over this set. If the optimum is unique, it converges to the solution, with no theoretical speed guaranteei.e., the entire set is explored in the worst case. However, in many practical instances, a very significant speedup is obtained.

Let us consider the optimization problem over a discrete set

S o P (S o ) ∶ arg min x∈S o f (x) . (6)
We denote its solution by x * S o and the optimal value by f

* S o = f (x * S o ).
For any subset S ⊂ S o , we consider subproblem P (S) and we assume that one can efficiently compute a lower bound l P (S) and an upper bound u P (S) on the optimal value of P (S): Choose an active subset S ∈ A and remove it from A.

l P (S) ≤ f * S ≤ u P (S) . (7)

4:

Split S = ⋃ N n=1 Sn into S1, . . . , S N .

5:

Add new active subsets S1, . . . , S N in A.

6:

Compute lower bounds l P (S1) , . . . , l P (S N ) and upper bounds u P (S1) , . . . , u P (S N ).

7:

Update L (resp. U ) as the lowest lower bound (resp. lowest upper bound) among all the elements of A.

8:

Prune every subset S ∈ A such that l P (S) > U . 9: until L = U , i.e., A is reduced a singleton 10: return the solution x remaining in A = {{x}}.

Hence, the knowledge of lower and upper bounds can help to discard some subsets without exploring them exhaustively. A branch-andbound strategy uses these ingredients to create subsets (branching) and bound their optimal values in a iterative way until all the space but one element -the solution -has been pruned. The procedure is summarized in Algorithm 2. Starting from the full set S o , a list of active subsets is iteratively built. At each iteration, an active subset may be split into new active subsets, and some active subsets may be pruned using [START_REF] Bilen | Convex Optimization Approaches for Blind Sensor Calibration using Sparsity[END_REF], avoiding an exhaustive evaluation of the problem. In practice, designing such a procedure relies on the ability to:

• compute a lower bound l P (S), for instance by relaxing S to a convex set S ′ and by solving P (S ′ );

• compute an upper bound u P (S), for instance by projecting the solution of the relaxed problem back onto S to obtain a suboptimal feasible solution;

• exhibit a heuristic to choose which active subset to split (line 3 in Algorithm 2): a common heuristic is to select the active subset with lowest lower bound.

• exhibit a heuristic to split the selected subset (line 4 in Algorithm 2), usually resulting into subsets with balanced sizes.

Proposed algorithm

We propose a strategy to explore the set of permutations P M by constraining some elements to equal one in the permutation matrices.

An active subset of P M is fully characterized by such a constraint set. Fig. 1 illustrates the tree structure underlying the exploration of P3. Each active subset of P M is split by choosing a row in which there is no constrained element and by creating a subset with such a constraint for each possible position in this row. The proposed procedure is given in Algorithm 3 and its key steps are detailed below. Problems over subsets. An active subset of P M is fully characterized by a set of positions C ⊂ {1, . . . , M } 2 to constrain the related elements of P to equal one. Thus, the restriction of problem (3) to any subset of P M characterized by a constraint set C is defined as

P Y,D (C) ∶ arg min P,X X 1 s.t. ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ P ∈ P M P (i, j) = 1, ∀ (i, j) ∈ C PY = DX (9)
Note that the first two constraints in (9) are equivalent to fixing the lines and columns of P that contain one constraint from C and finding an unconstrained permutation matrix in P M -C . Select C ∈ A with lowest lower bound, remove it from A.

P 32 = 1 1 constraint 3 constraints P 13 = 1 2 constraints No constraint P 2 1 = 1 P 22 = 1 P 2 3 = 1 P 33 = 1 P 32 = 1 P 13 = 1 P 11 = 1 P 12 = 1 P 11 = 1 P12 = 1 P31 = 1 P31 = 1 P 33 = 1

4:

Choose splitting point (i0, j0) as the location of the greatest element in the bistochastic matrix obtained from l Y,D (C).

5:

Add C ∪ {(i0, j)} to A for each j such that P Y,D (C ∪ {(i0, j)}) is feasible.

6:

Compute the lower bound l Y,D (C ∪ {(i0, j)}) and the upper bound u Y,D (C ∪ {(i0, j)}) of each added constraint set.

7:

Update L (resp. U ) as the lowest lower bound (resp. lowest upper bound) among all the elements of A.

8:

Prune every S ∈ A such that l P (S) > U . 9: until L = U 10: return permutation matrix P and sparse matrix X related to the only element induced by the unique constraint set in A = {C}.

Lower and upper bounds. By relaxing P M to its convex hull, we define the following tractable lower bound on the solution over any subset constrained by C:

l Y,D (C) ∶ min B,X X 1 s.t. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ B ∈ R M ×M + B1 M = 1 M B T 1 M = 1 M B (i, j) = 1, ∀ (i, j) ∈ C BY = DX (10)
The upper bound u Y,D (C) is computed from the optimal bistochastic matrix obtained in [START_REF] Lawler | Branch-and-bound methods: A survey[END_REF] as follows : the closest permutation matrix is computed by the Hungarian algorithm applied in P M -C after removing all contrained lines and columns; the resulting permutation matrix P ∈ P M is used to solve (5); the upper bound is defined as the l1 norm X 1 of the solution. It may happen that if C is a set of contraints obtained from a parent set C ′ , u Y,D (C) may be greater than u Y,D (C ′ ) while the solution for u Y,D (C ′ ) does not violate constraints C. In this case, u Y,D (C) is set to u Y,D (C ′ ). Branching : choosing a new constraint. In order to build active subsets, a natural heuristic consists in choosing the maximum unconstrained element indexed by (i0, j0) ∉ C in the bistochastic matrix estimated in [START_REF] Lawler | Branch-and-bound methods: A survey[END_REF]. Indeed, it is likely that contraining this element to equal one will be a step toward the optimal solution, and that alternative subsets in which this element equals zero will be pruned early. Based on this heuristic, we explore constrained sets where the constraint is moved along line i02 of the permutation matrix (line 5 in Algorithm 3). One must ensure that the problem is feasible on the new set C ∪ {(i0, j)} by checking that (i ′ , j) , (i0, j ′ ) ∉ C, ∀i ′ , j ′ , i.e., the element (i0, j) in the permutation matrix is not induced to equal zero by an existing constraint in C.

EXPERIMENTS

The experiments rely on synthetic data generated randomly as follows : the M × K dictionary is composed of standard Gaussian i.i.d. entries; matrix X is composed by N K0-sparse columns, with standard Gaussian i.i.d. non-zero entries; a permutation matrix is drawn uniformly from P M ; the noiseless observation matrix is computed as Y = P T DX. Dimensions M, K, K0, N are detailed for each experiment.

The code is written in Matlab and uses the CVX toolbox [START_REF] Grant | CVX: Matlab software for disciplined convex programming, version 2.0 beta[END_REF] to solve the relaxed problems [START_REF] Lawler | Branch-and-bound methods: A survey[END_REF], an existing implementation of the Hungarian algorithm [START_REF] Börlin | Functions related to the assignment problem, version 1.0[END_REF], and the L1 magic toolbox [START_REF] Candès | l1-magic: Recovery of Sparse Signals via Convex Programming[END_REF] to solve [START_REF] Rubinstein | Double sparsity: Learning sparse dictionaries for sparse signal approximation[END_REF]. The code has been run on a 2.9GHz core. All the code and data are provided with this paper for reproducible research purposes3 .

Typical behavior

In order to illustrate how the algorithm behaves along iterations, a representative example is shown in Fig. 2 with M = 10, K = 20, K0 = 3 and N = 100. The algorithm converges to the true solution in 265 iterations, after exploring 1671 subproblems, i.e., about 0.05% of the size M ! = 10! ≈ 3.6⋅10 6 of P M . The list of active constraint sets grows in the first 10 iterations, the active sets C having 1 to 3 constraints. The lowest upper bound U in blue shows that the best candidate encountered is refined along the iterations and that the first one is not optimal. In the next 100 iterations, the exploration of the active constraint sets results in pruning most of the active sets obtained after splitting, their lower bound being greater than the lowest upper bound U . As a result, the list tends to shrink and the number of constraints does not exceed 4. At the very end, moving from 4 to 10 constraints, i.e., to the solution, is performed in very few iterations.

This example shows the benefits of the proposed approach: starting from the non-optimal permutation given by Algorithm 1, it explores the low-depth nodes of the tree (i.e., active sets with few constraints, see Fig. 2 bottom) which happens to be sufficient to discard most of the search space without exploring it and to converge to the true solution.

Global trends

Extensive experiments are conducted in order to depict the performance of the proposed algorithm, in terms of time complexityi.e., the number of subproblems explored before converging, averaged among several trials -and of recovery of the original permutation. Time complexity vs. sparsity and redundancy. Time complexity has been measured when fixing M = 8, N = 60 and changing the sparsity K0 and the redundancy M K. Average results over 10 trials are shown in Fig. 3. Convergence is typically achieved in less than 100 iterations for very sparse signals and slightly-redundant dictionaries -the set of permutations having 8! ≈ 40320 elements. Time complexity increases moderately with K0, more strongly with K. Time complexity vs. permutation size. The increase of the time complexity with the permutation size M is a key question since the As expected, the convergence needs to explore less subproblems when more examples are available. However, each subproblem demands more time, due to the size of the data, so that a tradeoff may be achieved by adjusting N . The quartiles represented in Fig. 5 also illustrate the distribution of actual time complexities which may deviate significantly from average or median results. Perfect permutation recovery. Finally, one may wonder whether the estimated permutation equals the original permutation or whether some error rates must be measured. Actually, the true original permutation is retrieved by our algorithm in all the experiments above.

CONCLUSIONS

We have defined the sensor permutation problem in compressed sensing and proposed a branch-and-bound algorithm to solve it. A number of experiments have demonstrated that sparsity is a good regularization to recover an unknown permutation: the proposed method based on non-convex optimization is tractable and gives perfect recovery results, while convex optimization fails in retrieving the original permutation.

This work opens a number of new directions. One may generalize the sensor permutation problem to a larger class of objective functions, including noise in the data fitting term or structured sparsity. Other discrete optimization problems in compressed sensing may be defined and addressed using branch-and-bound strategies: instead of a permutation matrix, one can consider an unknown diagonal matrix with diagonal elements in {-1, 1} to model unknown sensor polarization, or in {0, 1} to model unknown defective sensors. Time complexity of such brand-and-bound algorithms may be improved by investigating other heuristics and by adjusting the number of examples to find a balance between the time needed to exploit each subproblem and the time spent on exploring more subproblems. Early stopping may also be used to control the time complexity at the price of losing the perfect recovery guarantee.

8 ) 2

 82 By splitting the entire set S o = ⋃ N n=1 Sn into subsets S1, . . . , S N and defining their lowest upper bound U ≜ minn u P (Sn), we have ∀n, U < l P (Sn) ⇒ x * S o ∉ Sn. (Algorithm Generic branch-and-bound procedure Require: A problem P over a set S o . 1: Initialize list of active subsets A ← {S o } 2: repeat 3:

Fig. 1 .Algorithm 3 1 :

 131 Fig.1. Tree structure to explore P3 by adding constraints (white ones) on the permutation matrix. Some elements are consequently zeros (black), others are free (gray).

Fig. 2 .

 2 Fig. 2. Typical behavior of the proposed algorithm along the iterations. Top: lowest lower bound L, lowest upper bound U and area representing the range of all active lower bounds in A. Middle: number of active and pruned constraint sets in A. Bottom: histogram of the number of constraints among elements in A. size of P M grows as M ! ∼ √ 2πM M e M and since the branchand-bound procedure does not provide any theoretical convergence rate. Results are reported in Fig. 4 for M ∈ [6, 14], K = 20, K0 = 3 and N = 60. The time complexity increases from 90 to 17000 but the number of explored subproblems gets dramatically low compared to the size of P M (2 ⋅ 10 -5 % for M = 14). This result supports the idea that solving combinatorial problem (3) is not hopeless. Time complexity vs. number of examples. Time complexity has been measured when fixing M = 10, K = 20, K0 = 3, and changing the number of examples N . Median results and quartiles are represented in Fig. 5.As expected, the convergence needs to explore less subproblems when more examples are available. However, each subproblem demands more time, due to the size of the data, so that a tradeoff may be achieved by adjusting N . The quartiles represented in Fig.5also illustrate the distribution of actual time complexities which may deviate significantly from average or median results. Perfect permutation recovery. Finally, one may wonder whether the estimated permutation equals the original permutation or whether some error rates must be measured. Actually, the true original permutation is retrieved by our algorithm in all the experiments above.

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. Average number of explored subproblems as a function of δ = M K and ρ = K 0 M , on a logarithmic color scale.

Note that this problem is not a particular case of dictionary selection as defined in[START_REF] Cevher | Greedy dictionary selection for sparse representation[END_REF][START_REF] Yaghoobi | Optimal dictionary selection using an overcomplete joint sparsity model[END_REF], even if the search space is composed of a finite number of dictionaries.

Moving the constraint along column j 0 is likely to give similar results.

http://www.lif.univ-mrs.fr/ ˜vemiya/icassp2014/

This work was supported in part by Agence Nationale de la Recherche (ANR), project GRETA 12-BS02-004-01 and by the European Research Council, PLEASE project (ERC-StG-2011-277906). This project originated in Roscoff, at a workshop of the LASCARS consortium.