Stephanie Boutet-Mercey

François Perreau

Amélie Roux

Guillaume Clavé

Jean-Paul Pillot

Isabelle Schmitz-Afonso

David Touboul

Gregory G Mouille

Catherine Rameau

Francois-Didier Boyer
email: francois-didier.boyer@cnrs.fr

Stéphanie Boutet-Mercey

Grégory Mouille

François-Didier Boyer

François-Didier Boyer

Validated Method for Strigolactone Quantification by Ultra High-Performance Liquid Chromatography -Electrospray Ionisation Tandem Mass Spectrometry Using Novel Deuterium Labelled Standards

Keywords: Strigolactones, Plant hormones, Deuterium-labelled strigolactones, Pea, Ultra high-performance liquid chromatography-electrospray ionization tandem mass spectrometry

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

Introduction

Plant hormones are key signalling biomolecules acting at lower concentrations (µM to pM) to regulate numerous aspects of plant growth and development. In the past decades, structurally diverse plant hormones including auxins, cytokinins, abscisic acid, gibberellins, ethylene, brassinosteroids, jasmonates and salicylic acid have been identified [START_REF] Davies | The Plant Hormones: Their Nature, Occurrence, and Functions[END_REF].

Strigolactones (SLs), terpenoid-derived compounds are the most recent class of hormones identified in plants initially for their role in the control of shoot branching [START_REF] Gomez-Roldan | Strigolactone inhibition of shoot branching[END_REF][START_REF] Lopez-Obando | Strigolactone biosynthesis and signaling in plant development[END_REF][START_REF] Umehara | Inhibition of shoot branching by new terpenoid plant hormones[END_REF]). In addition, these compounds have long been known for their roles as allelochemicals in symbiotic and parasitic interactions [START_REF] Xie | The Strigolactone Story[END_REF]) in the rhizosphere. SLs are widely distributed in the plant kingdom. To date, more than 20 natural SLs [START_REF] Lopez-Obando | Strigolactone biosynthesis and signaling in plant development[END_REF][START_REF] Xie | Structural diversity of strigolactones and their distribution in the plant kingdom[END_REF]) have been isolated from plants including non-vascular plants, the moss Physcomitrella patens and liverworts and Charales [START_REF] Lopez-Obando | Strigolactone biosynthesis and signaling in plant development[END_REF]. The structural core of the SLs is a tricyclic lactone (ABC part) connected via an enol ether bridge to an α,β-unsaturated furanone moiety (D ring) (Figure 1). These compounds have one or two methyl groups on the A ring and one or more hydroxyl or acetoxyl groups in the A/B rings but always with the same C-D moiety.

Carlactone, biosynthesized from all-trans-β-carotene, is the common precursor of SLs [START_REF] Lopez-Obando | Strigolactone biosynthesis and signaling in plant development[END_REF] (Figure 1). SLs are highly hydrolysable carotenoid derivatives rapidly leading to inactive derivatives (formyl ABC tricyclic lactone and hydroxymethylbutenolide) in an aqueous medium. In order to study the functions of plant hormones, it is important to determine the concentrations of endogenous plant hormones [START_REF] Novak | Zooming In on Plant Hormone Analysis: Tissue-and Cell-Specific Approaches[END_REF]. Plant hormones are generally difficult to analyze because they occur in very small concentrations and abundant matrices can interfere with their detection. SL concentrations in root exudates or in planta have been measured at extremely low levels [START_REF] Sato | Confirmation and quantification of strigolactones, germination stimulants for root parasitic plants Striga and Orobanche, produced by cotton[END_REF] in comparison with the other plant hormones [START_REF] Davies | The Plant Hormones: Their Nature, Occurrence, and Functions[END_REF]) and they may be unstable during the purification process due to their sensitivity to hydrolysis in an aqueous medium [START_REF] Boyer | Structure-Activity Relationship Studies of Strigolactone-Related Molecules for Branching Inhibition in Garden Pea: Molecule Design for Shoot Branching[END_REF]. From cotton seedlings, strigol and strigyl acetate production are about 15 and 2 pg/plant/day [START_REF] Sato | Confirmation and quantification of strigolactones, germination stimulants for root parasitic plants Striga and Orobanche, produced by cotton[END_REF]). An analytical method for plant hormones requires an efficient extraction procedure and a highly sensitive and selective analytical system. Advances in liquid chromatography coupled to tandem mass spectrometry (HPLC-MS/MS) have facilitated the rapid identification of novel SLs [START_REF] Sato | Analysis of strigolactones, germination stimulants for Striga and Orobanche, by highperformance liquid chromatography/tandem mass spectrometry[END_REF]. SLs have been detected in various plants [START_REF] Xie | Structural diversity of strigolactones and their distribution in the plant kingdom[END_REF]. It has been discovered that the production of SLs are strongly enhanced in low phosphorus (P) culture conditions [START_REF] Yoneyama | How do nitrogen and phosphorus deficiencies affect strigolactone production and exudation?[END_REF] allowing their detection. Plants have developed this strategy to enhance the host recognition by symbiotic arbuscular mycorrhizal fungi from which plants benefit for their P nutrition.

Determining the accurate concentration of SLs in different parts of plants is extremely challenging. To date, SLs have been mostly quantified from different plants in root exudates (pea [START_REF] Braun | The pea TCP transcription factor PsBRC1 acts downstream of Strigolactones to control shoot branching[END_REF][START_REF] Foo | Strigolactones promote nodulation in pea[END_REF][START_REF] Foo | Strigolactones and the Regulation of Pea Symbioses in Response to Nitrate and Phosphate Deficiency[END_REF][START_REF] Pavan | Characterization of Low-Strigolactone Germplasm in Pea (Pisum sativum L.) Resistant to Crenate Broomrape (Orobanche crenata Forsk.)[END_REF], red clover, lettuce, wheat, marigold, chinese milk vetch, alfalfa, tomato [START_REF] Yoneyama | How do nitrogen and phosphorus deficiencies affect strigolactone production and exudation?[END_REF] and in some cases in root tissues (pea (Foo andDavies 2011, Foo, et al. 2013), tomato [START_REF] Visentin | Low levels of strigolactones in roots as a component of the systemic signal of drought stress in tomato[END_REF], sorghum [START_REF] Yoneyama | Extraction and Measurement of Strigolactones in Sorghum Roots[END_REF])) through direct analysis after concentration or through the use of an intermediate purification step with solid-phase extraction (SPE). In all cases, the authors used methods derived from the developed procedure by Yoneyama and coworkers [START_REF] Sato | Analysis of strigolactones, germination stimulants for Striga and Orobanche, by highperformance liquid chromatography/tandem mass spectrometry[END_REF]. But the latter method suffers from the absence of labelled standards that has hindered precise quantifications of SLs. Their method was based on the standard addition procedure taking into account the matrix effects due to the complex crude extracts without extensive validation in terms of linearity, trueness and precision as examples. Few isotope-labelled standards of low quality, i.e., only partially labelled (3a,4,4,5,5,6'-2 H 6 -5-deoxystrigol) [START_REF] Ueno | Preparation of multideuterated 5deoxystrigol for use as an internal standard for quantitative LC/MS[END_REF] or those containing a single 2 H or two 13 C (6'-2 H-orobanchol, 6'-2 H-orobanchyl acetate, 6'-2 H-fabacyl acetate, 13 C 2 -strigol) (Foo andDavies 2011, Proust, et al. 2011), or those available through laborious multi-step syntheses adapted from the syntheses of non-labelled SLs [START_REF] Zwanenburg | Synthesis of strigolactones, a strategic account[END_REF]) and limited to labelled 5-deoxystrigols (7'-2 H 3 -5-deoxystrigol, 7'-2 H 3, 13 C-5-deoxystrigol, 7'-13 C-5-deoxystrigol) and 2'-epimers [START_REF] Cheng | Synthesis of stable isotopically labelled 3-methylfuran-2(5H)-one and the corresponding strigolactones[END_REF] have been described.

We are especially interested in the SLs in pea, our plant model to study shoot branching [START_REF] Boyer | Structure-Activity Relationship Studies of Strigolactone-Related Molecules for Branching Inhibition in Garden Pea: Molecule Design for Shoot Branching[END_REF][START_REF] Gomez-Roldan | Strigolactone inhibition of shoot branching[END_REF]). Five SLs have been characterized in pea [START_REF] Xie | Structural diversity of strigolactones and their distribution in the plant kingdom[END_REF]) (Figure 1), belonging to the orobanchol-type SL with an α-oriented C ring. Only orobanchol (1), orobanchyl acetate (2) and fabacyl acetate (4) have already been quantified in root exudates [START_REF] Braun | The pea TCP transcription factor PsBRC1 acts downstream of Strigolactones to control shoot branching[END_REF][START_REF] De Saint Germain | An histidine covalent receptor and butenolide complex mediates strigolactone perception[END_REF][START_REF] Pavan | Characterization of Low-Strigolactone Germplasm in Pea (Pisum sativum L.) Resistant to Crenate Broomrape (Orobanche crenata Forsk.)[END_REF]) and root tissues (Foo andDavies 2011, Foo, et al. 2013). In [START_REF] Braun | The pea TCP transcription factor PsBRC1 acts downstream of Strigolactones to control shoot branching[END_REF][START_REF] De Saint Germain | An histidine covalent receptor and butenolide complex mediates strigolactone perception[END_REF][START_REF] Foo | Strigolactones promote nodulation in pea[END_REF][START_REF] Foo | Strigolactones and the Regulation of Pea Symbioses in Response to Nitrate and Phosphate Deficiency[END_REF]) the pea SLs were quantified using 6'-2 Horobanchol, 6'-2 H-orobanchyl acetate and 6'-2 H-fabacyl acetate. In [START_REF] Pavan | Characterization of Low-Strigolactone Germplasm in Pea (Pisum sativum L.) Resistant to Crenate Broomrape (Orobanche crenata Forsk.)[END_REF]) the quantifications were performed using 13 C 2 -strigol as an internal standard. In both cases, the SL concentration levels in low P culture conditions were found similar (0.1 to 10,000 pg.g FW -1 (fresh weight) depending on each natural SL).

We describe here the synthesis of deuterium labelled orobanchyl and fabacyl acetates, according to fast and efficient procedures leading to standards following the requirements for MS quantification. These are standards with 3 deuteriums to avoid overlapping the isotopic patterns between the endogenous SLs and the labelled internal standards. In addition, we have developed a procedure for the extraction and purification of 1, 2, 4 from different pea genotypes. We propose a quantification method for orobanchol (1), for which no 3-deuterium labelled standard was available, using the synthetic SL GR24 (Figure 1) as internal standard.

Quantification of orobanchyl acetate (2) and fabacyl acetate (4) were performed using our corresponding deuterium labelled SLs. SL levels were measured and the results compared with those of the published procedure [START_REF] Foo | Strigolactones and the Regulation of Pea Symbioses in Response to Nitrate and Phosphate Deficiency[END_REF]. Our protocol provided a practical and validated method to study the levels of different endogenous SL plant hormones.

Experimental

Reagents and instruments for the SL standards preparation

Analytical thin-layer chromatography (TLC) was performed on plates precoated with silica gel layers. Compounds were visualized by illumination with a short wavelength UV lamp (i.e., λ = 254 nm), Nuclear magnetic resonance spectra ( 1 H; 13 C) were recorded respectively at [300; 75] MHz on a Bruker DPX 300 spectrometer. For the 1 H spectra, data are reported as follows: chemical shift, multiplicity (s = singlet, d = doublet, t = triplet, q = quartet, m = multiplet, bs = broad singulet), coupling constant in Hz and integration. IR spectra are reported in reciprocal centimeters (cm -1 ). Mass spectra (MS) and high-resolution mass spectra (HRMS) were acquired by electrospray ionization (ESI) coupled to a time-of-flight analyzer (Waters LCT Premier XE). Several chromatographic systems were used for the analytical experiments and the purification steps. Analytical UHPLC was performed on an Acquity

Waters UPLC system equipped with a PDA and a mass spectrometer detector. Semipreparative HPLC was performed on a Waters system equipped with 600E pump system, a

Waters 2767 sample manager, injector and collector and a waters PDA 2996 UV/Vis detector.

(±)-Orobanchol was obtained from OlChemIm Ltd (Olomouc, Czech Republic), 2 H 6 -acetic anhydride, 2 H 3 -iodomethane and the other reagents were purchased from Sigma-Aldrich Co (Saint Quentin Fallavier, France). (±)-GR24 (7) was prepared according to the [START_REF] Mangnus | Improved Synthesis of Strigol Analog GR24 and Evaluation of the Biological-Activity of Its Diastereomers[END_REF] procedure. (±)-Orobanchyl acetate (2), (±)-fabacol (3) and (±)-fabacyl acetate (4)

were synthesized from (±)-orobanchol by acetylation with acetic anhydride in pyridine and according to the [START_REF] Xie | Fabacyl acetate, a germination stimulant for root parasitic plants from Pisum sativum[END_REF]) procedure, respectively.

Deuterium labelled SL preparation and characterization

Preparation of (±)-7'-2 H 3 -GR24 (10)

Tricyclic lactone 9 [START_REF] Mangnus | Improved Synthesis of Strigol Analog GR24 and Evaluation of the Biological-Activity of Its Diastereomers[END_REF]) (112 mg, 0.55 mmol, 1 equiv), 6-2 H 3 -5-Bromo-3methyl-2(5H)-furanone (8) (see Supporting Information) (100 mg, 0.55 mmol, 1 equiv.), and anhydrous K 2 CO 3 (153 mg, 1.11 mmol, 2 equiv) were dissolved in anhydrous acetone (5 mL).

The reaction mixture was stirred for 18 h and concentrated under reduced pressure. The residue was dissolved in EtOAc (10 mL), filtered, evaporated under reduced pressure, and purified by chromatography on a silica gel (EtOAc/heptane 0/10 to 3/7 v/v) to give the desired product 10 (30 mg, 18%) and its 2'-epimer (37 mg, 22%) as white amorphous solids.

Plant materials and sample preparation

Pea hydroponic culture: The Pisum sativum L. mutant lines used were rms2-1 and rms4-1 derived from wild type (WT) Torsdag [START_REF] Beveridge | Branching mutant rms2 in Pisum sativumgrafting studies and endogenous indole-3-acetic acid levels[END_REF][START_REF] Beveridge | Branching in pea -Action of genes Rms3 and Rms4[END_REF]. The rms1-2T (Torsdag) mutant line was obtained by backcrossing the rms1-2 allele from the line WL5147 derived from Weitor [START_REF] Beveridge | The rms1 mutant of pea has elevated indole-3-acetic acid levels and reduced root-sap zeatin riboside content but increased branching controlled by graft-transmissible signal(s)[END_REF]) into the WT line Torsdag four times.

The basal nutrient solution (5 mM nitrate, 0.25 mM phosphate) was prepared by adding, in water (1,000 L), the following macronutrients: KH 2 PO 4 (0.25 mM), MgSO 4 .7H 2 O (0.25 mM), KNO 3 (4 mM), Ca(NO 3 ) 2 .4H 2 O (0.5 mM), NaCl (0.2 mM), Sequestrene ® (10 g) (Fe-EDTA solution), (NH 4 ) 2 MoO 4 (0.04 µM), H 3 BO 4 (24.3 µM), ZnSO 4 (3.48 µM) and CuSO 4 (1 µM).

The nutrient solution containing low phosphate concentration (5 mM nitrate, 0.05 mM phosphate), was prepared by adding, in water (1,000 L), the following macronutrients:

KH 2 PO 4 (0.05 mM), MgSO 4 .7H 2 O (0.25 mM), K 2 SO 4 (0.1 mM), KNO 3 (4 mM), Ca(NO 3 ) 2 .4H 2 O (0.5 mM), NaCl (0.2 mM), Sequestrene ® (10 g) (Fe-EDTA solution), (NH 4 ) 2 MoO 4 (0.04 µM), H 3 BO 4 (24.3 µM), ZnSO 4 (3.48 µM) and CuSO 4 (1 µM). Pea seeds were germinated in wet sand for 6 days. Germinated seeds were placed in premade holes in the lid (20 plants/lid, 20 mm diameter) of a hydroponic PVC opaque pot containing the hydroponic culture solution (16 L, pH 5.9). The plants were grown in a growth chamber at 22 °C/16 °C with a 16/8-h photoperiod at 350 µmol photons m -2 s -1 . The seedlings were grown hydroponically in basal nutrient medium (5 mM nitrate, 0.25 mM phosphate) for 9 days. Plants (rms1, rms2, rms4, WT) (4 pots) were then subjected for 10 days to a P-deficient nutrient medium (5 mM nitrate, 0.05 mM phosphate) except for one pot (WT) used as control to a basal nutrient medium (5 mM nitrate, 0.25 mM phosphate). Thereby the roots of 4 plants were pooled together to reach a sufficient amount to limit differences between plants.

Extraction and purification of natural SLs

The plant material was immediately frozen in liquid nitrogen after harvest and stored at -80 °C. It was then ground in liquid nitrogen using mortar and pestle to obtain a fine powder.

EtOAc (3 mL) containing stable isotope-labelled internal standards ( 2 H 3 -GR24 (10) (10 ng), 2 H 3 -fabacyl acetate (12) (10 ng) and 2 H 3 -orobanchyl acetate (11) (1 ng)) were added to a 1 galiquot of the fresh plant material for extraction: it was vigorously shaken for 30 s, sonicated for 1 min at 25 Hz, and shaken for 2 hours at 4 °C. After centrifugation at 1,000g (10 min, 4 °C), the supernatant (2.5 mL) was transferred into a glass tube. The pellet was re-extracted with EtOAc (3 mL), vigorously shaken for 30 s, sonicated (1 min; 25 Hz) and shaken overnight at 4 °C. Following centrifugation, the two supernatants were pooled and dried. The dry extract was dissolved in heptane/EtOAc (9/1 v/v, 2 mL) and percolated through a Upti clean Si-S 500 mg/3 mL Solid-phase extraction (SPE) column (Interchim, Montluçon, France) previously activated with heptane (2 mL) and pre-equilibrated with heptane/EtOAc (9/1 v/v, 2 mL). The SPE column was washed with heptane/EtOAc (75/25 v/v, 2 mL). The first fraction containing GR24 (7), fabacyl acetate (4) and orobanchyl acetate (3) was eluted with heptane/EtOAc (45/55 v/v, 2 mL). After another wash with heptane/EtOAc (40/60 v/v, 2 mL), the second fraction containing orobanchol was obtained by elution with heptane/EtOAc (17/83 v/v, 2 mL). Each fraction was evaporated to dryness, kept at -20 °C and finally dissolved in 100 µL of acetonitrile before LC-MS/MS analysis (Figure 3).

RP-UHPLC setup and procedure

Separation was performed on a BEH C 18 column (2.1 × 100 mm, particle size 1.7 µm, Waters, Milford, MA, USA) at 40 °C, using an ACQUITY UPLC I-class system (Waters, Milford, MA, USA), equipped with an ACQUITY Sample Manager keeping the samples at 4°C and ACQUITY Binary Solvent Manager with a flow rate of 0.4 mL/min, for 1 µL injected. The mobile phase consisted of water (A) and acetonitrile (B). Elution was as follows: 5% B for 2 min, the gradient elution increased linearly to 50% B in 8 min, followed by a further linear increase to 100% B in 3 min, then 100% B for 3 min and the final gradient linear elution decreased to 5% B for 3 min.

Mass spectrometry conditions

SLs were detected online using a Waters Xevo TQ-S equipped with an electrospray (ESI) source and operated in positive ion mode. The source parameters for full scan MS and MRM mode were as follows: ion capillary voltage 1.5 kV, nebulizing gas 7 bar, source temperature 120 °C, drying gas 800 L/h, drying temperature 500 °C, and cone gas 250 L/h, optimized by infusion of GR24 solution. The mass spectrometer was operated in multiple reaction monitoring (MRM) mode for quantification. For each compound, quantification was based on the most intense MRM transition while the second most intense one was used for confirmation. All MRM conditions (precursor and ion products, cone voltage and collision energy) were optimized on each standard solution (Table 1).

Data were analyzed using MassLynx 4.1 software.

Preparation of standard solutions

The pure reference substances were dissolved at 1 mg/mL in acetone. Dilutions were prepared from this solution in order to establish a calibration curve in the concentration range of 0.2 -56.0 µg/L in acetonitrile for the different SLs (GR24 (7) from 4.0 to 56.0 µg/L, orobanchol

(1) from 0.2 to 3.0 µg/L, fabacyl acetate (4) from 2.0 to 30.0 µg/L and orobanchyl acetate (2) from 0.5 to 7.5 µg/L).

Method evaluation

The effectiveness of this analytical method was evaluated in terms of matrix effects, SPE recovery, sensitivity and linearity. Intra-and inter-day relative standard deviations (RSDs) of peak area ratio were calculated to determine the reproducibility of the method. Intra-day variation was determined by analyzing standard mixtures solution stored at 4 °C for 4 times within 1 day. Regarding inter-day variability, the standard mixtures solutions stored at -20 °C were analyzed 5 times for 6 months. The extraction recovery yields under SPE conditions were investigated on each SL standard, using blank matrix sample (rms1 root extract) spiked with standards. Sensitivity and linearity were evaluated via the limit of detection (LOD) and the correlation coefficients (R 2 ) of the calibration curves, respectively.

The UHPLC-ESI-MS/MS method described earlier was used to determine the response factors of GR24 (7), orobanchol (1), orobanchyl acetate (2) and fabacyl acetate (4).

The quantitative determination of 2, 4 and 7 was performed with an internal deuteriumlabelled stock solution at a concentration of 100 µg/L for 2 and 1000 µg/L for 4 and 7. 10 µL of this solution was added to each plant material before extraction.

For the quantification of orobanchol (1), a calibration curve (quantity of 1 = f (Peak area of 1/Peak area of GR24) was built using a series of calibration solutions of orobanchol (1) (0.2 µg/L to 3.0 µg/L)) containing the same concentration of reference standard GR24 (7) (50.0 µg/L). GR24 (7) (50.0 µg/L) was used as internal standard in the biological samples. The quantity of 1 in the sample was evaluated according to the calibration curve.

All values were expressed as the means ± standard deviation (SD). Due to a small sample size, the nonparametric method of Kruskal-Wallis was used with Dunnett's test for comparison to the control.

Results and Discussion

Preparation of deuterium labelled SLs

Three new labelled SLs, 7'-2 H 3 -GR24 ( 10), (±)-2 H 3 -orobanchyl acetate (11) and of (±)-2 H 3fabacyl acetate (12) were prepared by organic synthesis with deuterium labelled on the D ring for GR24 (10), on an acetate group for the orobanchyl acetate (11) and the fabacyl acetate (12), respectively (Figure 2). The deuterium labelled GR24 (10) synthesis involved the coupling between the deuterium labelled D-ring 8 and the ring ABC 9 obtained by the procedure of [START_REF] Mangnus | Improved Synthesis of Strigol Analog GR24 and Evaluation of the Biological-Activity of Its Diastereomers[END_REF]. Deuterium labelled bromo butenolide 8 was synthesized in 4 steps in a 25% overall yield (see Supporting Information in comparison with 4 steps and in a 15% overall yield for the synthesis of [START_REF] Cheng | Synthesis of stable isotopically labelled 3-methylfuran-2(5H)-one and the corresponding strigolactones[END_REF]). The coupling between the enol 13 and the bromo butenolide 12 was accomplished in acetone using an excess of potassium carbonate to give the awaited labelled GR24 ( 14) and its 2'-epimer in moderate yield after separation by silica gel chromatography.

The synthesis of the deuterium labelled orobanchyl acetate and fabacyl acetate using the procedure described for GR24 (10) was not convenient due to the laborious multi-step sequences [START_REF] Zwanenburg | Synthesis of strigolactones, a strategic account[END_REF]) to access to the ABC precursor before coupling with the labelled butenolide 8. Alternatively, the deuterium labelling was performed by acetylation of the commercially available orobanchol (1) using a mixture 2 H 6 -acetic anhydride/pyridine to furnish the labelled compound (11) in 96% yield. In the same manner, after epoxidation of 1 by m-CPBA as described by [START_REF] Xie | Fabacyl acetate, a germination stimulant for root parasitic plants from Pisum sativum[END_REF], the resultant fabacol (3) was acetylated to give the deuterium labelled 12 in a 60% overall yield after HPLC purification. Replacing acetic anhydride with ( 2 H 6 )-acetic anhydride provided a simple method for deuterated methyl group introduction. The ESI-MS spectra of 10-12 showed that their deuterium abundances were sufficient enough (up to 95%) for these compounds to be used as internal standards for quantification without overlapping with natural SLs in each mass spectrum.

Optimization of MRM transitions

Standard solutions of the 3 endogenous SLs (orobanchol (1), orobanchyl acetate (2), fabacyl acetate (4)), synthetic GR24 (7) and their deuterium-labelled analogues (10, 11, 12) were used to identify the appropriate precursor-to-product ion transitions and to ensure selectivity of the quantification method. First, appropriate precursor ions were selected and conditions optimized in full scan MS, limiting in-source fragmentation of the compounds. All the compounds were analyzed in both positive and negative electrospray (ESI + and ESI - respectively) ionization modes, but no intense enough signal was obtained in ESI -as in [START_REF] Sato | Analysis of strigolactones, germination stimulants for Striga and Orobanche, by highperformance liquid chromatography/tandem mass spectrometry[END_REF]. The protonated and sodium-adduct species ([M+H] + and [M+Na] + , respectively)

of SLs were the most prominent ions in ESI + MS full scan spectra (except for 2 and its deuterium-labelled analogue 11 which only produced [M+Na] + ). Atmospheric pressure chemical ionization mode was tested, but even when the MS spectrum contained mostly

[M+H] + , the sensitivity was significantly lower than that in the ESI mode (-40%), so ESI + was demonstrated as the most convenient mode. Mobile phases containing more acid or more sodium salt were tested but none gave a spectrum with only a single species as either [M+H] + or [M+Na] + , as previously obtained [START_REF] Sato | Analysis of strigolactones, germination stimulants for Striga and Orobanche, by highperformance liquid chromatography/tandem mass spectrometry[END_REF]. Both [M+H] + and [M+Na] + were therefore chosen as precursor ions for MRM. The MRM transitions were chosen in ESI mode as giving the maximal signal. Most of the product ions were selected as the most specific regarding the structure of the compounds (e.g., often losses of acetate group (-60 u nonlabelled analyte, -63 u deuterium-labelled analyte) or D ring (-97 u)) (Figure 4A). Twentyfour transitions for quantification (Q) and confirmation (C) were found (Table 1), among them only seven were already known in the literature [START_REF] Foo | Strigolactones promote nodulation in pea[END_REF][START_REF] Gomez-Roldan | Strigolactone inhibition of shoot branching[END_REF][START_REF] Kohlen | Strigolactones Are Transported through the Xylem and Play a Key Role in Shoot Architectural Response to Phosphate Deficiency in Nonarbuscular Mycorrhizal Host Arabidopsis[END_REF][START_REF] Pavan | Characterization of Low-Strigolactone Germplasm in Pea (Pisum sativum L.) Resistant to Crenate Broomrape (Orobanche crenata Forsk.)[END_REF][START_REF] Sato | Analysis of strigolactones, germination stimulants for Striga and Orobanche, by highperformance liquid chromatography/tandem mass spectrometry[END_REF]) and seventeen were newly found. MS/MS conditions were optimized to produce maximal signals for each compound (Table 1).

Optimization of UHPLC-conditions

Separations were optimized using a solution containing a mixture of the three endogenous SLs and GR24 plus their isotopically-labelled standards by reversed-phase ultra-performance liquid chromatography on a UHPLC BEH C 18 column, followed by analysis by ESI-MS/MS.

In order to separate individual hormones, reduce the analysis time and avoid the co-elution with other compounds, the following gradient elution using an increasing acetonitrile content was chosen. The four SLs were successfully separated. The separation window time was less than 3 min and the total analysis was performed in 19 min. Typical total ion current chromatograms of 1, 2, 4 and 7 are shown in Figure 4. Our time analysis is intermediate between that of (Gomez-Roldan, et al. 2008) (50-min analysis time using HPLC) and the very fast analysis developed by (Foo and Davies 2011) (7-min analysis time using UHPLC). The latter method avoids the use of acid in the mobile phase as it hampers the sensitivity of SLs in electrospray mass spectrometry [START_REF] Sato | Analysis of strigolactones, germination stimulants for Striga and Orobanche, by highperformance liquid chromatography/tandem mass spectrometry[END_REF].

Solid-phase extraction

In order to purify the root extracts, we first tested the SPE protocol used for purification of SLs in extracts from root exudates by [START_REF] Kisugi | Strigone, isolation and identification as a natural strigolactone from Houttuynia cordata[END_REF]) which included silica solid phase extraction for purification with stepwise elution of 11 n-hexane/EtOAc (from 100/0 to 0/100 v/v) fractions. The process was however very time-consuming and required handling too many fractions. Only four of these fractions were worthwhile (e.g., containing SLs). Hence the protocol was simplified, the number of elution steps was reduced and the solvent composition modified. Hexane and heptane were compared as co-solvent of EtOAc for SPE elution. Heptane was finally selected because the signal of SLs was 20-30% higher in fractions using heptane than when using hexane. Some matrix could have been less soluble in heptane and therefore less in competition with SLs for ionization in the MS source. In addition, heptane is well known as being less toxic than hexane.

The SPE protocol was first reduced to its minimum with only 3 elution fractions (wash by 70/30, elution by 20/80 and elution of 100% EtOAc for the final step v/v) allowing the elution of all SLs in only one fraction (20/80), but matrix effect was high (until -80% for orobanchyl acetate ( 2)) and recoveries were very low (e.g., 40% for orobanchol (1)). Inversely, a 5fraction elution (wash by 75/25, elution by 45/55, wash by 40/60, elution by 17/83 v/v and final elution by 100% EtOAc) gave better recoveries and a less matrix effect, while giving an acceptable number of fractions to handle. For example if the wash fraction was 70/30 the matrix effect of fabacyl acetate (4) was the worst ≈ -77% and the recovery of GR24 (7) or orobanchyl acetate (2) were calamitous with the elution fraction 50/50 because they presented in 2 consecutive fractions (50/50 elution fraction and the 40/60 second wash fraction). The composition of each elution solvent for each fraction was fine-tuned so that each SL was not spread among several fractions but in only one. GR24 (7), orobanchyl acetate (2) and fabacyl acetate (4) were found only in the 45/55 fraction and orobanchol (1) in the 17/83 fraction.

Method performance

In this study, we took advantage of the pea ramosus1 (rms1) mutant which is SL-deficient (Gomez-Roldan, et al. 2008), in order to obtain an actual blank matrix sample. To evaluate the linearity of the method, rms1 root extract fraction 2 spiked with varying amounts of SL non-labelled standards (GR24 (7), orobanchyl acetate (2) and fabacyl acetate (4)) and fixed amounts of labelled standards ( 2 H 3 -GR24 (10) 100 µg/L, 2 H 3 -fabacyl acetate (12) 100 µg/L and 2 H 3 -orobanchyl acetate (12) 10 µg/L) were used to create isotopically-labelled internal calibration curves. Similarly, rms1 root extract fraction 4 spiked with varying amounts of standard orobanchol (1) and fixed amounts of its internal standard (GR24 (7) 100 µg/L) were used to create internal calibration curves. Four replicates were used for each level of concentration. Calibration curves were established plotting peak area ratio versus internal standard concentration.

Table 2 shows that calibration curves were linear in the concentration ranges selected for the selected hormones and the R 2 values were in the range 0.988-0.999. The limits of detection (LODs) of UHPLC-MS/MS for the different plant hormones were in the range 0.14-1.24 µg/L and the limits of quantification (LOQs) ranged from 0.46 to 16.46 µg/L. Here the intercept method was used, based on the calibration curve, as recommended in [START_REF] Kruve | Tutorial review on validation of liquid chromatography-mass spectrometry methods: Part I[END_REF]:

LOQ was calculated as 10 times the ratio of the standard deviation of the intercept divided by the slope of calibration curve. [START_REF] Sato | Analysis of strigolactones, germination stimulants for Striga and Orobanche, by highperformance liquid chromatography/tandem mass spectrometry[END_REF]) obtained a very close LOQ for LC-MS/MS of orobanchol (1) (1 µg/L) determined as a signal to noise ratio above 10. The reproducibility of the method was evaluated by analyzing the intra-and inter-day variations (Table 4). The RSDs of the peak area ratio were taken as the measure of the reproducibility. The intra-day variation was determined by analyzing the same standard mixtures four times within a day.

The RSDs of the peak area ratio of intra-day were between 3.1 and 5.5%. For the inter-day variation, the solution was examined 5 times within six months. The RSDs of the peak area ratio inter-day were 3.1-8.2%. Accuracies were calculated as the differences between the true (spiked) and calculated values, relative to the true (spiked) value. These values (< 20%) show the reliability of this method in time.

Three kinds of solutions were used for recovery and matrix effect evaluations: (A) a 100 µLstandard solution of SLs at 100 µg/L for GR24 (7) and fabacyl acetate (4), 10 µg/L for orobanchyl acetate (2) and 1 µg/L for orobanchol (1), (B) a 100 µL-extract of rms1 root where SLs were spiked at 100 µg/L for 7 and 4, 10 µg/L for 2 and 1 µg/L for 1 and (C) a 100 µL-extract of rms1 root previously spiked with SLs at 10 ng for 7 and 4, 1 ng for 2 and 0.1 ng for 1, before extraction. A and B therefore contained the same amount of SLs, and C could also contain the same amount under the assumption that the recovery of SLs from the sample treatment protocol was 100%. Recovery for a compound was evaluated as the peak area ratio of B/C and matrix effect was evaluated as the peak area ratio of B/A minus 1, and expressed in percentages. Every solution was prepared and analyzed in triplicate.

As estimated in Table 3, the recovery yields of the four plant hormones ranged from 57.3 to 77.0% which was considered reasonable for the SPE procedure. Losses could occur by resuspension, degradation, leak during percolation and wash on SPE cartridges, or too much retention on SPE phase. To test the degradation hypothesis of acetate strigolactones susceptible of turning into the corresponding alcohols, we extracted and analyzed rms1 root spiked with orobanchyl acetate (2) and fabacyl acetate (4), searching for orobanchol (1) and fabacol (3). Fabacol (3) fragmentation conditions (MRM transitions 385>254; 363>97) were optimized using the signal produced during fabacyl acetate (4) in-source dissociation. We found no peaks for orobanchol (1) and fabacol (3) transitions, demonstrating that there was no degradation of fabacyl acetate (4) and orobanchyl acetate (2) into the corresponding alcohols.

Stability of orobanchol (1) was also tested in all the solvents used during extraction and purification. After two days at 20 °C orobanchol (1) was stable in ethyl acetate, heptane and acetonitrile (data not shown), whereas a minor loss of 19% was observed in water, indicating that orobanchol degradation is negligible during sample treatment. So degradation was ruled out as a cause of losses. SPE percolation leak during percolation or wash steps were also ruled out since none of the SLs was detected in these fractions. One can therefore only suspect resuspension problems or too much SPE retention causing those below-100% recoveries.

However these recoveries do not impact at all the accuracy of the method thanks to the use of the isotopically-labelled standard, e.g., except for orobanchol (1).

The matrix effect ranged within a large scale between negative values for GR24 (7) and orobanchol (1) (-33.6, -23.5%) and positive values for fabacyl acetate (4) and orobanchyl acetate (2) (+1.8, +62.1%), suggesting that the matrix promotes ionization of acetatecontaining SLs. Likewise, the comparison of the matrix effect with or without SPE seems to confirm that the SPE step reduces the sample complexity with a significant value and allowed a gain of up to 40% in signal hence in sensitivity: the matrix effect ranged from -70 to -26%

without SPE (data not shown). As well as the below 100% recoveries, the medium matrix effect obtained using SPE does not impact the accuracy of the method thanks to the use of isotopically-labelled standards.

For orobanchol (1) no isotope-labelled standard was available. GR24 (7), a synthetic SL not occurring in the sample but eluting in the proximity of 1 (Figure 4b) (retention time 9.54 min versus 9.04 min) and possessing a similar interaction with the matrix (Table 3), was used as an internal standard.

Orobanchyl acetate (2), fabacyl acetate (4) and GR24 (7) were quantified using our deuterium labelled standards 11, 12 and 10 respectively. The use of deuterium-labelled 10, 11 and 12 as internal standards for the MS analysis appeared to be a more elegant method in order to avoid variations in sample preparation, injection and, in particular, ionization parameters.

Nevertheless, the presence of two precursor ion forms for the SLs (protonated form and sodium-adduct form) makes the quantification more complex and less accurate, as the amount of only one form is not always representative of the amount of the initial molecule. The results depended on the balance between ion precursor forms, which is a function of the ionization environment, e.g., the coeluting matrix (data not shown). To overcome this variation, two quantification modes are compared, in terms of RSD and accuracy for 7 and 4.

The first quantification mode (Table 5A) classically took into account either a protonated form or a sodium-adduct. Results based on one form or another were close but different (4-6% of variation between of two values). The second quantification mode (Table 5B) using the sum of areas for both parent ion forms was demonstrated to be more accurate with less variation e.g., lower RSD values.

The characteristics of the method shown above make it valid and appropriate for the analysis of SLs in their natural state.

Application to root tissue of pea analysis

To test its feasibility, this method was subsequently applied to the quantification of SLs in root tissues of pea stimulated by P starvation or not in WT and in the highly branched ramosus mutants (rms1, rms2, rms4) as done in [START_REF] Foo | Strigolactones and the Regulation of Pea Symbioses in Response to Nitrate and Phosphate Deficiency[END_REF]. The RMS1 gene encodes the CAROTENOID CLEAVAGE DIOXYGENASE8 (CCD8) enzyme implicated in the SL biosynthesis. In the SL-deficient rms1 mutant (Psccd8) used in the present study, the RMS1 gene is deleted [START_REF] Gomez-Roldan | Strigolactone inhibition of shoot branching[END_REF][START_REF] Sorefan | MAX4 and RMS1 are orthologous dioxygenaselike genes that regulate shoot branching in Arabidopsis and pea[END_REF]. The response mutant (rms4) contains a lesion in the F-box protein (RMS4) [START_REF] Johnson | Branching genes are conserved across species. Genes controlling a novel signal in pea are coregulated by other long-distance signals[END_REF]), which interacts with the SL receptor (RMS3) after SL perception [START_REF] De Saint Germain | An histidine covalent receptor and butenolide complex mediates strigolactone perception[END_REF]. The rms2 mutant also displays a high shoot branching and is affected in a shoot-to-root feedback signal which controls the expression of SL-biosynthesis genes [START_REF] Beveridge | Branching mutant rms2 in Pisum sativumgrafting studies and endogenous indole-3-acetic acid levels[END_REF][START_REF] Beveridge | Branching in pea -Action of genes Rms3 and Rms4[END_REF][START_REF] Foo | The branching gene RAMOSUS1 mediates interactions among two novel signals 464 and auxin in pea[END_REF]. As expected, no pea SL was detected in rms1 root tissue with LOD varying from 0.14 to 1.24 pg.g FW -1 according to each SL (Table 2 for 1-2, 4). This result confirmed the two reports that monitored SLs in rms1 mutant plants. In our analyses, only orobanchol (1), orobanchyl acetate (2), fabacyl acetate (4) [START_REF] Xie | Structural diversity of strigolactones and their distribution in the plant kingdom[END_REF] were detected in root tissue of rms2, rms4 mutant plants and WT plants. Only the levels of the two major SLs fabacyl acetate (4), and orobanchyl acetate (2) were sufficient to be quantified even in root tissue from WT roots grown hydroponically and starved of P. The effect of P starvation was studied in WT and led to 18-fold and 11-fold increases for fabacyl acetate (4), and orobanchyl acetate (2) respectively in accordance with over a 10-fold increase found in [START_REF] Foo | Strigolactones and the Regulation of Pea Symbioses in Response to Nitrate and Phosphate Deficiency[END_REF].

Fabacyl acetate (4) was detected as the major SL with concentrations varying from 3,194 ± 0.146 ng.g FW -1 and 3,559 ± 0.410 ng.g FW -1 for rms4 and rms2 mutant plants, respectively, to 8,640 ± 0.480 ng.g FW -1 for WT plants. Orobanchyl acetate (2) was detected as a minor SL with concentrations varying from 64 ± 6 pg.g FW -1 and 64 ± 14 pg.g FW -1 for rms4 and rms2 mutant plants, respectively, to 187 ± 14 pg.g FW -1 for WT plants (Figure 5). In [START_REF] Foo | Strigolactones and the Regulation of Pea Symbioses in Response to Nitrate and Phosphate Deficiency[END_REF], orobanchol (1) and fabacyl acetate (4) were measured as the major SLs. This discrepancy with our work in the major SL may be due to the culture conditions but also to the quantification method and the use of different deuterium labelled SL standards, however it is not due to degradation using our method, as previously demonstrated (see above). In Foo et al 2013, the SL concentration levels were found similar for rms2, rms4 mutant plants and WT plants (500 pg.g FW -1 for orobanchol (1) and fabacyl acetate (4) and 80 pg.g FW -1 for orobanchyl acetate (2)). In our work, reduced SL levels were found for the rms2 and rms4 mutants in comparison to WT. This result is consistent with the reduced RMS1 transcript level generally observed in the rms2 pea stem but not for rms4 where elevated RMS1 transcript levels resulting for feedback regulation are observed. Further studies should investigate in the same tissues both transcript levels of SL biosynthesis genes and SL levels for a better understanding of regulation of SL levels at the plant level.

This study reported the development of newly labelled SL standards and validation of the UHPLC-ESI-MS/MS method for the analysis of SLs with high sensitivity and selectivity.

The effectiveness of extraction and purification using SPE was evaluated and the effect of the matrix quantified. This procedure required about 1-g root samples and, in comparison with the quantification from exudates, avoided laborious extractions or uses of large SPE cartridges and consumptions of large quantities of organic solvents. Matrix effect was limited to over -34% thanks to SPE, whereas recoveries ranged from 57 to 77%, with little impact on sensitivity. The use of isotopically-labelled standards allowed to overcome these sub-100%

Figures and Tables

Figure 1. Natural SLs detected in pea [START_REF] Xie | Structural diversity of strigolactones and their distribution in the plant kingdom[END_REF], their biosynthetic precursor (+)-carlactone ( 6) and the synthetic SL GR24 (7).

Analyte

Recovery (%, ± SD) Matrix effect (%) GR24 ( 7 3. Recoveries calculated for the whole sample treatment process, and matrix effect for SLs. The value of matrix effect is positive when the presence of matrix promotes the signal and negative when it lowers the signal (n = 3).

Analyte

Intra-day n = 4 Inter-day n = 5 RSD (%) Accuracy (%) RSD (%) Accuracy (%) GR24 ( 7

Figure 2 .

 2 Figure 2. Synthesis of the labelled standards 10-12.

Figure 3 .

 3 Figure 3. Protocol of extraction and purification of SLs.

Figure 4 .

 4 Figure 4. (A) MS spectra of 2 H 3 -fabacyl acetate (12) and 2 H 3 -orobanchyl acetate (11), (B) Total ion current chromatogram from a UHPLC-MS/MS in MRM mode of a standard solution containing the four SLs non-labelled standards at 500 µg/L.

Figure 5 .

 5 Figure 5. Levels of fabacyl acetate (4) (A) and orobanchyl acetate (2) (B) detected in pea root tissues after 10-day phosphorus starvation. Nd = not detected. Mean ± SD of 4 biological replicates. *** denotes significant differences with the WT starved control as determined using the Kruskal-Wallis test with the Dunnett type of contrast (p < 0.001).

  LC-MS/MS parameters for each compound: Retention time (RT), characteristic parent and product ions for MRM transitions, cone voltage (CV), collision energy (CE) and use of the transition (Q: quantification, C: confirmation). a.[START_REF] Sato | Analysis of strigolactones, germination stimulants for Striga and Orobanche, by highperformance liquid chromatography/tandem mass spectrometry[END_REF] b.[START_REF] Pavan | Characterization of Low-Strigolactone Germplasm in Pea (Pisum sativum L.) Resistant to Crenate Broomrape (Orobanche crenata Forsk.)[END_REF][START_REF] Gomez-Roldan | Strigolactone inhibition of shoot branching[END_REF]) d.[START_REF] Kohlen | Strigolactones Are Transported through the Xylem and Play a Key Role in Shoot Architectural Response to Phosphate Deficiency in Nonarbuscular Mycorrhizal Host Arabidopsis[END_REF] e.[START_REF] Foo | Strigolactones promote nodulation in pea[END_REF]Response characteristics of the SL standards using UHPLC-ESI-MS/MS. Linearities with Y concentration of SL and x ratio of peaks area SL on internal standard (8 calibration levels, 4 replicates), LOQ and LOD of SLs in spiked blank matrix (rms1 pea extract).

  Relative standard deviation (RSD) and precision of the quantification for repeatability (intra-day with n = 4) and reproducibility (inter-day with n = 5) in standard sample. nd = not determined. Relative standard deviation (n = 3) and precision on two quantification modes for the compounds which have 2 ions in source (A) on each parent ion form [M + H] + and [M + Na] + , (B) on addition of area for the 2 parent ion forms.

Preparation of (±)-2 H 3 -orobanchyl acetate (11)

	MHz, CDCl 3 ) δ: 171.8 (Cq, C1'), 170.7 (Cq, C5'), 170.1 (Cq, C2), 150.4 (CH, C6'), 140.9
	(CH, C4a), 136.2 (Cq, C4'), 136.2 (Cq, C8a), 110.6 (Cq, C3), 99.7 (CH, C2'), 85.5 (CH,
	C8b), 82.9 (CH, C4), 45.3 (CH, C3a), 38.7 (CH 2 , C7), 28.0 (Cq, C8), 27.9 (CH 3 , C9), 27.5
	(CH 3 , C10), 23.7 (CH 2 , C5), 21.7 (Cq, C4c), 18.8 (CH 2 , C6), 10.7 (CH 3 , C7'). C4b was not
	found in the 13 C NMR spectrum. IR ν (film, cm -1 ): 1800, 1750, 1675, 1255, 1010, 795.
	HRMS (ESI): Calculated for C 21 H 21 D 3 O 7 Na [M + Na] + : m/z 414.1608. Found: m/z 414.1610.
	1 H NMR (300 MHz, CDCl 3 ) δ: 7.48 (m, 1H, HAr), 7.45 (s, 1H, H6'), 7.19-7.34 (m, 3H, HAr),
	6.95 (d, 1H, J = 1.3 Hz, H3'), 6.15 (d, 1H, J = 1.3 Hz, H2'), 5.92 (d, 1H, J = 7.7 Hz, H8b),
	3.92 (m, 1H, H3a), 3.41 (dd, 1H, J = 17.0 Hz, J = 9.4 Hz, H4α), 3.08 (dd, 1H, J = 17.0 Hz, J
	= 3.4 Hz, H4ß). 13 C NMR (75.5 MHz, CDCl 3 ) δ: 171.4 (Cq, C2), 170.4 (Cq, C5'), 151.2 (CH,
	C6'), 142.8 (CH, C8a), 141.2 (CH, C3'), 139.0 (CH, C4a), 136.1 (Cq, C4'), 130.2 (CH, CAr),
	127.7 (CH, CAr), 126.6 (CH, CAr), 125.3 (CH, CAr), 113.5 (Cq, C3), 100.8 (CH, C2'), 86.1
	(CH, C8b), 39.0 (CH, C3a), 37.5 (CH 2 , C4), 9.7 (Cq, C7'). IR ν (film, cm -1 ): 1786, 1733,
	1675. HRMS (ESI): Calculated for C 17 H 12 D 3 O 5 [M + H] + : m/z 302.1108. Found: m/z
	302.1115.
	To a solution of (±)-orobanchol 1 (1 mg, 2.89 mmol) in pyridine (0.2 mL), was added 2 H 6 -
	acetic anhydride (0.2 mL). The mixture was stirred overnight at room temperature. After
	several coevaporations with toluene, the residue was purified by preparative chromatography
	on silica gel with 40% EtOAc in heptane (v/v) to give 2 H 3 -orobanchyl acetate (11) (1.09 mg,
	96%). 1 H NMR (300 MHz, CDCl 3 ) δ: 7.46 (d, 1H, J = 2.9 Hz, H6'), 6.94 (t, 1H, J = 1.5 Hz,
	H3'), 6.15 (t, 1H, J = 1.5 Hz, H2'), 5.74 (s, 1H, H4), 5.61 (d, 1H, J = 7.8 Hz, H8b), 3.45 (ddd,
	1H, J = 7.3 Hz, J = 2.9 Hz, J = 2 Hz, H3a), 2.03 (t, 3H, J = 1.5 Hz, H7'), 1.96 (m, 2H, H5),
	1.70 (m, 2H, H6), 1.43 (m, 2H, H7), 1.16 (s, 3H, H9), 1.14 (s, 3H, H10). 13 C NMR (75.5

Preparation of (±)-2 H 3 -fabacyl acetate (12)

	(±)-Orobanchol (1) (1 mg, 2.9 µmol) was dissolved in CH 2 Cl 2 (200 µL), thereafter m-CPBA

(3 mg, 17 µmol) was added and the resulting reaction mixture was periodically stirred using a vortex every 4 h. The reaction was followed by RP-UHPLC (system A) until complete conversion after 12 h. The crude extract was purified by preparative TLC with a mixture of heptane/ EtOAc (4/6, v/v) giving epoxyorobanchol as white solid. R f (heptane/EtOAc, 4/6, v/v) 0.30. HRMS (ESI): m/z calc. for C 19 H 23 O 7 [M + H] + : 363.1444, found : m/z 363.1449.

The crude epoxyorobanchol was dissolved in a mixture of pyridine/ 2 H 6 -acetic anhydride (1/1, v/v, 400 µL) and the resulting reaction mixture was periodically stirred using a vortex every 2 h. The reaction was checked for completion by RP-UHPLC [BEH C18 column, 12.1 mm × 50 mm, particle size 1.7 µm) with 0.1% formic acid in CH 3 CN and 0.1% formic acid in water (pH 2.8) as eluents [5% CH 3 CN, followed by linear gradient from 5 to 100% of CH 3 CN (10 min) at a flow rate of 0.6 mL/min] until complete conversion after 6 h. The solvent was evaporated to dryness and the resulting residue was purified by semi-preparative RP-HPLC and then lyophilized giving (±)-3 H 2 -fabacyl acetate (12) as a white amorphous powder (700 µg, 2.2 µmol, 60% over the two steps). HRMS (ESI): m/z calc. for C 21 H 21 D 3 O 8 Na [M + Na] + : m/z 430.1557, found: m/z 430.1553.

Table 1

 1

	. LC-MS/MS parameters for compounds 7, 10, 1, 4, 12, 2 and 11: Retention time
	(RT), characteristic parent and product ions for MRM transitions, cone voltage (CV),
	collision energy (CE) and use of the transition (Q: quantification, C: confirmation).
	Table 2. Response characteristics of the SL standards using UHPLC-ESI-MS/MS. Linearities
	with Y concentration of SL and x ratio of peaks area SL on internal standard (8 calibration

levels, 4 replicates), LOQ and LOD of SLs in spiked blank matrix (rms1 pea extract).

Table 3 .

 3 Recoveries calculated for the whole sample treatment process, and matrix effect for SLs. The value of matrix effect is positive when the presence of matrix promotes the signal and negative when it lowers the signal (n = 3).

	Table 4. Relative standard deviation (RSD) and precision of the quantification for
	repeatability (intra-day with n = 4) and reproducibility (inter-day with n = 5) in standard
	sample. nd = not determined.
	Table 5. Relative standard deviation (n = 3) and precision on two quantification modes for the
	compounds which have 2 ions in source (A) on each parent ion form [M + H] + and [M + Na] + ,
	(B) on addition of area for the 2 parent ion forms.

H

recoveries and moderate matrix effect in order to obtain a satisfactory method accuracy (intraday 6-19%, inter-day 5-14%). For orobanchol (1), we found a SL analog, GR24 (7), which could also lead to accurate results. It was the first time an analytical method was fully validated for SLs. Finally, using this UHPLC-ESI-MS/MS method to quantify plant hormones, it may be possible to quantify SLs in other complex plant samples to understand their implication in numerous biological events.

Acknowledgements

We thank Frank Pelissier from CNRS-ICSN for the preparative HPLC of standards. We thank Rosemary Beau for her comments on the manuscript. We are grateful to the Institut National de la Recherche Agronomique (INRA), the Agence Nationale de la Recherche (contract ANR-12-BSV6-004-01), la Région Ile de France, le department des Yvelines and the Stream COST Action FA1206 for financial support. The Institut Jean-Pierre Bourgin benefits from the support of the Labex Saclay Plant Sciences-SPS (ANR-10-LABX-0040-SPS).


Theodore Aouad

Hugues Talbot

BINARY MORPHOLOGICAL NEURAL NETWORK

Keywords: Mathematical morphology, binary, deep learning, machine learning, image processing

In the last ten years, Convolutional Neural Networks (CNNs) have formed the basis of deep-learning architectures for most computer vision tasks. However, they are not necessarily optimal. For example, mathematical morphology is known to be better suited to deal with binary images. In this work, we create a morphological neural network that handles binary inputs and outputs. We propose their construction inspired by CNNs to formulate layers adapted to such images by replacing convolutions with erosions and dilations. We give explainable theoretical results on whether or not the resulting learned networks are indeed morphological operators. We present promising experimental results designed to learn basic binary operators, and we have made our code publicly available online.

INTRODUCTION

Convolutional Neural Networks (CNNs) constitute the basis of most deep-learning architectures. These can learn complex task-specific processes while requiring only input-output pairs, albeit sometimes in large numbers. Since their inception in the mid-1990s, they have achieved outstanding results in computer vision and have become the go-to technology for many computer vision tasks, provided enough annotated data is available.

However, standard literature on image processing states that some tasks remain for which convolutions are not optimal. Mathematical Morphology (MM) [START_REF] Serra | Image analysis and mathematical morphology[END_REF] is one of these. For many applications, MM operators are more suitable than convolution-based methods, particularly when dealing with binary or discrete images. However, finding the right sequence of operations and the right structuring elements can be difficult and time-consuming depending on the problem at hand [START_REF] Soille | Morphological Image Analysis, principles and applications[END_REF]. Our objective is to mimic the way CNNs are built on convolutional filters and create a morphological network that can learn a compact sequence of operators together with their optimal parameters. Morphological networks can also be used conjointly with CNNs to learn the morphological operators that would be otherwise manually engineered, as in [START_REF] Shit | clDice -a Novel Topology-Preserving Loss Function for Tubular Structure Segmentation[END_REF].

Learning morphological operators is not new, whereas the trend of replacing the convolution of CNNs with morphological operations is recent. Some researchers have investigated the use of the max-plus algebra [START_REF] Mondal | Image restoration by learning morphological opening-closing network[END_REF][START_REF] Franchi | Deep morphological networks[END_REF], for example, to perform image filtering (de-raining and de-hazing) [START_REF] Mondal | Learning 2d morphological network for old document image binarization[END_REF]. Others have replaced the non-differentiable max / min operators by differentiable approximations, e.g. the adaptative morphological layer [START_REF] Shen | Deep morphological neural networks[END_REF], the PConv layer [START_REF] Masci | A learning framework for morphological operators using counter-harmonic mean[END_REF], or even the LMorph and SMorph layers [START_REF] Kirszenberg | Going beyond pconvolutions to learn grayscale morphological operators[END_REF]. All these methods were studied in the context of grey-scale morphology.

In this work, we seek to learn binary morphological operators from binary image inputs. End-to-end learning of these operators could be helpful in shape analysis. We first introduce the Binary Structuring Element (BiSE) neuron, which can learn erosion and dilation together with a structuring element. The BiSE neuron is built using convolution and benefits from the highly optimized implementations of this operation. By stacking multiple BiSE, we build a Binary Morphological Neural Network (BiMoNN). We give theoretical explainability of the BiSE such that each learned morphological operator can be recovered. Binarizing these networks can lead to faster and cost-efficient deep networks for inference [START_REF] Hubara | Binarized neural networks[END_REF][START_REF] Kim | Bitwise neural networks[END_REF][START_REF] Simons | A review of binarized neural networks[END_REF].

Our code is publicly available online at https:// github.com/TheodoreAouad/Bimonn_ICIP2022.

METHOD

Mathematical Morphology

Mathematical morphology [START_REF] Serra | Image analysis and mathematical morphology[END_REF] was created to study porous materials. It is based on set theory and is well suited to studying binary images. An image of dimension d (d " 2 for 2D images or d " 3 for 3D images) is seen as a subset of Z d : I Ă Z d . Morphological operations transform I based on a small structuring element S Ă Z d . The two fundamental operations are dilation and erosion.

ε S pIq " I a Š " č sP Š pI `sq, (1)
where Š is the symmetric of S with respect to the origin.

These operators thus defined are adjunct, i.e. @I, J, S, I Ď ε S pJq ðñ δ S pIq Ď J. All basic morphological operations are obtained from erosions and dilations. The opening is the application of erosion followed by its adjunct dilation, and the closing is the other way around. Here we consider any composition of dilations or erosions, extending to any sequence of openings or closings.

Erosions and dilations are similar to convolutions: the structuring elements can be interpreted as the kernel for the convolutional filter, and the sum is replaced by max or min operator. Therefore, we must establish a model that can learn the structuring element and operation type: erosion or dilation.

Binary Structuring Element Neuron

We now define the BiSE (Binary Structuring Element) neuron, which can learn both the operation and the structuring element. The BiSE neuron is built upon the convolution operation. First, we notice that the dilation and erosion can be exactly expressed using convolution. Note that in practice, for the erosion, we learn the symmetric ε Š such that ε Š pIq " I a S.

Proposition 1 (Morphological operators from convolution)

Let S Ă Z d be a binary structuring element and X Ă Z d be a binary image.

X ' S " ˆ1X f 1 S ě 1 ˙(3) X a S " ˆ1X f 1 S ě |S| ˙(4)
For the same structuring element S, the difference between dilation and erosion is determined by a scalar. Learning the operation is the same as learning this scalar.

To learn S Ă Z d , first we suppose that S is bounded: S Ď Ω with |Ω| ă `8. Let Ω be the grid bounded by some integer n, Ω " Z d X r´n, ns d . Similarly to [START_REF] Kim | Bitwise neural networks[END_REF], we define a relaxed weight W Ă R Ω , we apply a smooth increasing threshold function ξ such that ξpW piqq « 1 if i P S, else ξpW piqq « 0. We use the hyperbolic tangent:

ξpxq " 1 2 tanhpxq `1 2 (5)
We introduce the softplus function f `: x P R Þ Ñ lnp1 èxppxqq `0.5 to ensure that f `pxq ą 0.5.

Definition 2 (BiSE neuron) Let W P R Ω be a weight matrix, b P R a bias and p P R a scaling number. We define a BiSE (Binary Structuring Element) neuron as follow:

ϵ W,b,p : x P r0, 1s Z d Þ Ñ ξpppx f ξpW q ´f `pbqqq P r0, 1s Z d (6)
First, the weights are thresholded. Then we apply the morphological operation: we perform the convolution and subtract a bias. The bias is forbidden to become negative (in practice, f `pbq ą 0.5): we avoid the bias at 0, which leads to constant output and zero-grad zones. Finally, we have to threshold this result. Before thresholding, the result is multiplied by a scaling factor p.

The BiSE neuron can learn erosion, dilation, and the associated structuring element. The weights W learn the structuring element, and the bias b determines the operation. The scaling number p has two purposes. It determines how close to binary the output is, and if p ă 0, the output is inverted, so in theory, we could learn the complementation as well.

Our thresholding function, the hyperbolic tangent, is smooth to allow back-propagation; we do not deal with binary images. We thus introduce the concept of almost binary images, which are more flexible and are easier to handle than binary images. Definition 3 (Almost Binary Image) We say an image I P r0, 1s Z d is almost binary if there exists u ă v P r0, 1s such that IpZ d qXsu, vr" H. We denote this set Ipu, vq.

A pixel value of an almost binary image is either close to 0 or close to 1.

When is a BiSE neuron equivalent to dilation or erosion by a structuring element S? The following propositions allow us to perform a verification for a given structuring element. Given the weights W and the bias b, we can check if, in its current state, the BiSE neuron is a dilation or erosion by this structuring element.

Proposition 2 (Dilation / Erosion Equivalence) We assume the weights are thresholded: W P r0, 1s Ω . Given an almost binary input in Ipu, vq

• ϵ W,b,`8 is a dilation by S if and only if ÿ iPΩzS w i `u ÿ iPS w i ď b ă v min iPS w i (7)
• ϵ W,b,`8 is an erosion by S if and only if

max jPS ˆÿ iPSzj pw i q `u ¨wj ˙ď b ă v ÿ iPS W i (8)
If either of these expressions is fulfilled, we say that the BiSE neuron is activated.

If the weights and bias are correctly learned, the structuring element can be recovered by thresholding the weights for some value. The suitable threshold is given in proposition 3.

Proposition 3 (Linear Check) Let us assume the BiSE is activated for almost binary images Ipu, vq. Let b be the BiSE bias, let W be the normalized weights (W Ps0, 1r Ω ). Then there exists τ P R such that S " ti P Ω | W piq ě τ u.

• If the BiSE is a dilation τ " b v (9)
• If the BiSE is an erosion τ "

ř kPΩ W k ´b 1 ´u ( 10
)
To find out the learned structuring element and operation, we only need to check both thresholds. This can be done in Op|Ω|q operations.

Erosion and dilation are dual operations: applying a dilation is the same as applying an erosion to the background. This property allows us to recover the inequalities of one operation to get its dual. Finally, if the BiSE neuron is activated, then its output is almost binary. The notion of almost binary images is justified: we now deal with almost binary images instead of binary images.

BiMoNN

We can now define the Binary Morphological Neural Network (BiMoNN) as a composition of multiple BiSEs:

BiMoNN " ϵ L ˝... ˝ϵ1 (11)
If each BiSE is activated, the BiMoNN's inputs and outputs are almost binary. In theory, this framework can learn any sequence of dilations or erosions with any structuring element, including opening and closing.

We learn the BiMoNN using the classical deep learning framework. Given a loss L, we use a masked version of L. Borders act depending on our interpretation of the value outof-bounds pixels. To avoid this problem, we mask the borders of size kernel shape divided by two. Parameters are updated with the Adam [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] optimizer. We initialize the biases at f `p2q " 0.63. The weights W of the convolutions have kaiming uniform initialization [START_REF] Glorot | Understanding the difficulty of training deep feedforward neural networks[END_REF].

EXPERIMENTS

Datasets

We create a dataset of generated images that we call Diskorect. Each image is made of random-shaped and randomoriented rectangles and disks. Then, we add some random Bernoulli noise. Finally, complementation is applied half the time. The images dimensions are 50 ˆ50.

MNIST [START_REF] Deng | The mnist database of handwritten digit images for machine learning research[END_REF] is a dataset of 70000 handwritten digits of size 28 ˆ28. The images are grey-level. To be able to work with structuring elements of size p7, 7q, we reshape them to p50, 50q, with a cubic interpolation [START_REF] Keys | Cubic convolution interpolation for digital image processing[END_REF]. Then we threshold the image to recover binary images.

In order to analyze the duality of operators, we also test our network on the complementation of MNIST, which we call inverted MNIST.

Experiment description

We check the ability of a BiMoNN to learn basic morphological operators. We attempt to learn the erosion and dilation (table 1), as well as the opening (erosion then dilation) and closing (dilation then erosion) (table 2). The protocol is the same for each operation: the morphological operator is applied to the input image to create the target y i . To learn the erosion and dilation, we train a single BiSE neuron. To learn the opening and closing, we stack two BiSE neurons. We follow the DICE [START_REF] Lee R Dice | Measures of the amount of ecologic association between species[END_REF] of the target images vs. predicted images.

We also check if each BiSE neuron is activated.

For both MNIST and inverted MNIST, the structuring elements are of size 5 ˆ5 for the erosion and dilation; the transformations are too significant compared to the size of the digits otherwise.

The parameter p is fixed p " 4. For Diskorect, we use the Dice loss [START_REF] Milletari | V-net: Fully convolutional neural networks for volumetric medical image segmentation[END_REF] with learning rate 0.01 for the erosion / dilation and 0.001 for the opening / closing. For MNIST, we use the MSE loss with learning rate 0.1 for the erosion / dilation and 0.01 for the opening / closing.

DISCUSSION

Erosion and Dilation

For any optimizer or any initialization, on both MNIST and Diskorect, the dilation is learned perfectly for all structuring elements: after a few hundred iterations, the DICE is 1 for the validation set. The BiSE are all activated except for the cross. On the Diskorect dataset, the BiSE neurons are activated after a few thousand iterations for the disk and stick.

For the erosion, on Diskorect, we obtain similar results but with slower convergence. Sometimes the weights are darker than in the dilation case. This is not a problem: if the BiSE is activated, the weights do not need to be as high as 1. On MNIST, the learned disk is limited to its border, which is not surprising: given S and its border BS, the difference between I a S and I a BS is small and only visible on a dataset with small holes. While the DICE is not 1, the error is small at 0.003.

The perfect metric is reached before the BiSE neurons are activated. Indeed, it is possible for the inequality not to be fulfilled, while the BiSE is a dilation only for a specific dataset D Ĺ Z d . Could a necessary and sufficient condition be established for a specific dataset to know if BiSE is a dilation on this dataset? This question is left for future work.

Opening and Closing

On Diskorect, for both opening and closing, we learn perfectly the stick and cross, and almost all the BiSE are activated. However, the disk is more challenging to learn for opening (resp. the DICE is still high at 0.93 (resp. 0.96).

On MNIST, the opening is learned well for all structuring elements. However, the training on closing returns chaotic weights for the disk and cross. To explain this, we notice that closing with S has little effect on the image for this dataset. Therefore many weight combinations yield a similar transformation. See the disk closing: even with totally different structuring elements, the error is small at 0.002. For the inverted MNIST, the situation is almost the same, but for the dual operators: we will dive into that in the next paragraph.

We compare to SMorph and LMorph [START_REF] Kirszenberg | Going beyond pconvolutions to learn grayscale morphological operators[END_REF]. On both datasets, they achieve good results on erosion and dilation, with perfect metric and good structuring elements. However, they fail to converge on the opening and closing and present stability issues (because of an exponential in their formulation). This is not surprising as they are both built on a differentiable softmax function that is not suitable to deal with binary elements.

A few words on duality

In mathematical morphology, two operators δ and ε are dual if @X , δpXq " εpXq. This is the case for the couples (dilation, erosion) and (opening, closing). Let D be a dataset and D the set of its complementations. We formulate the hypothesis H: are the training of δ on D and the training of ε on D similar?

Let D be the Diskorect dataset. By construction, D " D and the training of dual operators should be the same. For the erosion and dilation, the results are not exactly the same. For the erosion, the structuring elements are darker, and the model takes longer to converge: from 10 times slower on Diskorect to a hundred times slower on MNIST. We still manage to learn both operations. On the other hand, the opening and closing do not behave the same on the closing, with dissimilar weights. These results contradict H. Now, let D be the MNIST dataset. The dilation behaves identically for both D and D, and the same can be said for the erosion, which goes against H. However, even though the learned weights seemed chaotic, the closing on D works similarly to the opening on D, which corroborates H. This similarity also happens for the opening on D and closing on D, being only different for the disk.

The case of MNIST and its invert suggests that there is a strong link between the learning of dual operators. However, they do not learn exactly the same. We leave the study of this link to future work.

CONCLUSION

We created a neural network built to operate on binary images. We successfully learn some morphological operations: results on dilations and erosions are perfect. Overall, we achieved good results on the opening and closing, only failing for the disk and more complicated operations. We also provide explainability results to understand the operation of each neuron.

We sometimes reach a perfect metric without the neurons being theoretically morphological operators. This raises the need for more relaxed inequalities depending on the dataset properties. Enforcing that each BiSE is activated (e.g., using a regularization loss) would allow us to binarize the entire network.

To complete our network, we should also learn the complementation and multiple filters (e.g., to learn the top-hat operators). The complementation can be done by learning the parameter P and allowing it to be negative. While the BiSE could theoretically complement its results, it was not yet demonstrated in practice: we leave it to future work. To fully simulate the CNNs' multiple-filter structure, we will incorporate intersection and union of filters.

These results demonstrate that we can learn simple morphological operators. In the near future, we will investigate its integration in more complex pipelines to leverage useful morphological information, with a view to construct highperforming binary deep networks.

Given almost binary inputs, the outputs of the BiSEL are also almost binary.

This follows the same logic as CNN. In CNN, at each layer, we have k filters. Each filter applies one convolution to each channel, then we apply a linear combination to all convoluted channels. In BiSEL, we have K filters (the number of LU I). For each filter, we apply a morphological operation to a channel, then we aggregate by taking the intersection or union of any channels. See figure 2.

The parameters are the BiSE weights tW n,k u n,k Ă R Ω , the BiSE biases tb n,k u n,k Ă R, the LU I parameters tβ n,k u n,k Ă R and the LU I biases tb l n,k u n,k Ă R. There are N Kp|Ω| `3q parameters.

Definition 1

 1 The dilation (') and erosion (a) of an image I Ă Z d by a structuring element S Ă Z d are defined as: δ S pIq " I ' S " ď sPS pI `sq

Fig. 1 .

 1 Fig. 1. Datasets example.

Fig. 2 .

 2 Fig. 2. Schema of BiSEL. Input x with 3 input channels. Output ϕpxq with 2 channels.

Table 1 .

 1 Results on erosion and dilation. DICE error (1 DICE)is presented for each case. ✓ indicates if the neuron is activated.

Erosion and dilation are dual operations, meaning @X, S P t0, 1u Ω , X ' S " X a S: applying a dilation is the same as applying an erosion to the background. This property allows us to recover the inequalities of the dilation using those of the erosions, and the other around. Let us show this in the binary case.

Let us consider two BiSE layers ϵ 1 W,b1 and ϵ 2 W,b2 sharing the same weights, but the first one being a dilation and the second one being an erosion. Let us denote u e , v e the bounds for the erosion and u d , v d the bounds for the dilation. We have:

" inf XPt0,1u,iPΩ

" inf XPt0,1u,iPΩ

" inf XPt0,1u,iPΩ

"

The same can be done for the other bound: u e " ř iPΩ w i ´vd .

B. BINARY STRUCTURING ELEMENT LAYER

One of CNNs' strengths is the ability to learn multiple filters per layer. We also want to be able to learn multiple filters.

In CNNs, each final channel is a sum of one filter per input channel. In our case, the final result is either a union or an intersection of the morphological operators (dilation or erosion). Therefore, we want a layer that can learn the union or intersection of any combination of inputs. Let us consider n binary images x 1 , ..., x n Ă Ω. Let C Ă rr1, nss. Then the intersection and union are given as:

As with the BiSE, we can use a single scalar to discriminate between the union or the intersection. To learn the set C, we can use a parameter β i for each image. This gives the following definition: Definition 4 (LU I) Let β " pβ 1 , ..., β c q P R c . Let ξ be a smooth increasing threshold. Let b P R `be a bias and p P R a scaling factor. We define the LU I (Layer Intersection Union) as a thresholded linear combination:

A LU I layer can learn any intersection or union of any number of almost binary inputs. We denote I " Ś n k"1 Ipu k , v k q the set of images with n almost binary channels. • LU I is an intersection by C if and only if

• LU I is a union by C if and only if

Like for the BiSE, if the LU I is properly learned (i.e. inequalities are respected), the set C can be found by thresholding the pβ i q i for a certain value. Moreover, if the inequalities are strict, if all the channels of an input image I are almost binary in Ipu, vq, then the output LU I β,b pIq is almost binary: LU I β,b pIq P Ipξpu ´bq, ξpv ´bqq.

We combine the BiSE neurons and the LU I to be able to learn the morphological operators, and aggregate them as unions or intersections.

Definition 5 (BiSEL) A BiSEL (BiSE Layer) is the combination of multiple BiSE and multiple LU I. Let pϵ n,k q be N ˚K BiSE and pLU I k q k be K LU I. Then we define a BiSEL as:

`ϵn,k px n q ˘nı ˙k P pZ d q K (26)