Shefaat Ullah

Marie Socha

Isabelle Fries

Stéphane Gibaud
email: stephane.gibaud@univ-lorraine.fr

Synthesis of S-nitrosoglutathione-alginate for prolonged delivery of nitric oxide in intestines

Keywords: nitric oxide, alginate, polymer, Crohn's disease, oral route

S-nitrosothiols are a class of NO-donors currently under investigation for the treatment of various diseases. In this study, we developed a novel NO-donor (Snitrosoglutathione-alginate, SNA) by cross-linking alginate with S-nitrosothiols, which can deliver NO in a sustained manner. This compound can be further evaluated for oral delivery to treat Crohn's disease. This new compound was prepared using a two-step procedure involving (I) linkage of reduced glutathione (GSH) to alginate and (II) postnitrosation with sodium nitrite (NaNO 2 ). The amount of linked thiol moieties for the possible nitrosation was calculated using Ellman's method, and the amount of NO abducted on the polymer was calculated using the Griess-Saville method. An ex vivo model (i.e., Ussing chamber) was used to investigate the permeation of this new NOdonor across the rat intestinal barrier. We obtained polymers with different numbers of abducted NOs (174 ± 21 µmol/g for SNA F1 and 468 ± 23 µmol/g for SNA F2) depending on the procedure used for nitrosation. In the ex vivo studies in the Ussing chamber, SNA F2 exhibited a sustained release for at least 10 h. The effect of pH on the stability of the new compound was also investigated, and the new compound was more stable at a mildly basic pH of 8.4 where 73% remained after 1 week. However, only 50% remained after 1 week at an acidic pH of 1.2. In the cytotoxicity studies (Caco2), this compound was non-toxic at concentrations of less than 200 µM.

Introduction

Nitric oxide (NO) is the smallest endogenous molecule and a free radical gas [START_REF] Saraiva | Nanocarriers for nitric oxide delivery[END_REF]. NO was first introduced in 1987 (Ignarro et al., 1981,Jerca et al.) and received medical and scientific interest in 1992 when it was nominated as molecule of the year [START_REF] Friedman | New biomaterials for the sustained release of nitric oxide: past, present and future[END_REF]. NO is produced by endothelial cells of the body and performs different physiological functions. NO is released as NO * or bound to -SH groups containing carrier molecules that stabilize its release [START_REF] Kilbourn | Nitric oxide: moving towards the clinic[END_REF][START_REF] Moncada | Nitric oxide: physiology, pathophysiology, and pharmacology[END_REF].

To evaluate the biological and physiological functions of NO, exogenous sources of NOdonors are required that can be used both as a drug and a research tool [START_REF] Saraiva | Nanocarriers for nitric oxide delivery[END_REF]. A variety of NO-donors are available and pharmacologically active as NO releasing drugs including S-nitrosothiols (RSNOs) [START_REF] Saraiva | Nanocarriers for nitric oxide delivery[END_REF]. In fact, due to the ability to act as NO-donors, RSNOs have attracted considerable attention. RSNOs are involved in different reactions in the bioregulatory system based on NO transfer (Singh et al., 1996b). RSNOs are more stable than NO, which has a very short in vivo half-life (0.1-15 s). For example, S-nitrosoglutathione (GSNO) decomposes in hours [START_REF] Josephy | Electron spin resonance spin trapping of thiyl, radicals from the decomposition of thionitrites[END_REF][START_REF] Shin | Microscopic modeling of NO and S-nitrosoglutathione kinetics and transport in human airways[END_REF]. Some RSNOs, such as GSNO and S-nitrosoalbumin, are endogenous NO carriers in mammals [START_REF] Seabra | Polynitrosated polyesters: preparation, characterization, and potential use for topical nitric oxide release[END_REF]. NO is covalently attached to a sulfur atom in the R-S-NO molecule and can be released via S-N bond cleavage to transfer NO to specific receptors for in vivo biological effects [START_REF] Seabra | Polynitrosated polyesters: preparation, characterization, and potential use for topical nitric oxide release[END_REF].

RSNO

RS * + NO * Chemically, RSNOs have been synthesized by reacting NO with compounds possessing thiol groups, such as reduced L-glutathione (GSH) and albumin. Then, these compounds are transported to the reaction sites where its corresponding biological effect is achieved by release of NO (Singh et al., 1996a). The most commonly used

RSNOs are GSNO and S-nitroso-N-acetyl-D,L-penicillamine (SNAP), which exhibit remarkable biological effects [e.g., GSNO offers significant protection to myocardial ischemia and SNAP is a powerful vasodilator [START_REF] Ignarro | Mechanism of vascular smooth muscle relaxation by organic nitrates, nitrites, nitroprusside and nitric oxide: evidence for the involvement of S-nitrosothiols as active intermediates[END_REF]].

Several other NO-donors, such as nitroglycerine and isosorbide dinitrate, rapidly release NO but can cause oxidative stress and frequent tolerance. Therefore, to minimize these problems, research is now focused on innovative NO-donors such as RSNOs and especially GSNO. GSNO is considered a lead molecule due to its safety (NO released is accompanied by GSH production). Therefore, GSNO is used to investigate the biochemical and pharmacological mechanisms of NO and its therapeutic applications [START_REF] Parent | A Complete Physicochemical Identity Card of S-nitrosoglutathione[END_REF]. S-nitroso compounds, such as GSNO, have a specific thiol coupling capability that allows it to store and carry NO and offers the advantage of preventing the development of tolerance in patients. These compounds can be used in different disease states, such as cardiovascular disorders [START_REF] Everett | S-Nitrosoglutathione improves haemodynamics in early-onset pre-eclampsia[END_REF], brain ischaemia [START_REF] Khan | Administration of S-nitrosoglutathione after traumatic brain injury protects the neurovascular unit and reduces secondary injury in a rat model of controlled cortical impact[END_REF], cancer [START_REF] Furuhashi | NO donor and MEK inhibitor synergistically inhibit proliferation and invasion of cancer cells[END_REF], cystic fibrosis [START_REF] Servetnyk | The effect of S-nitrosoglutathione and Lcysteine on chloride efflux from cystic fibrosis airway epithelial cells[END_REF], and wound healing [START_REF] Georgii | Topical S-nitrosoglutathione-releasing hydrogel improves healing of rat ischaemic wounds[END_REF]. In addition, these compounds may be beneficial in the treatment of Crohn's disease [START_REF] Savidge | Enteric glia regulate intestinal barrier function and inflammation via release of S-nitrosoglutathione[END_REF]. In the recent years, GSNO has been proposed for the treatment of Crohn's disease to prevent mucosal barrier failure.

Crohn's disease (CD) is a type of inflammatory bowel disease which is thought to be a result of inappropriate inflammatory response to intestinal microbes in a genetically susceptible host [START_REF] Abraham | Inflammatory Bowel Disease[END_REF]. It is an immune related disease in which the immune system attacks the GI tract. It is characterized by focal, asymmetric, transmural and occasionally granulomatous inflammation, which primarily affects the GI tract [START_REF] Thomsen | A comparison of budesonide and mesalamine for active Crohn's disease. International Budesonide-Mesalamine Study Group[END_REF]. In maximum cases the diagnosis reports in colonic and terminal ileal involvement [START_REF] Freeman | Long-term natural history of Crohn's disease[END_REF].

GSNO significantly promotes human intestinal mucosal barrier function, while this effect was lacking in patients with no inflammatory bowel disease conditions [START_REF] Savidge | Enteric glia regulate intestinal barrier function and inflammation via release of S-nitrosoglutathione[END_REF]. This result may be due to the enteric glial cell network being disrupted in noninflamed Crohn's disease intestinal mucosa. Therefore, the tissue GSNO concentration levels might be lower in these patients [START_REF] Shah | Synthesis and characterization of Snitrosoglutathione-oligosaccharide-chitosan as a nitric oxide donor[END_REF]. NO-donors can also be chemically linked to biopolymers, which can provide site specific delivery of NO [START_REF] Saraiva | Nanocarriers for nitric oxide delivery[END_REF]. Therefore, the aim of our study was to synthesize a new stable polymer by linking GSNO to alginate, and this compound was evaluated for the treatment of Crohn's disease.

Alginates are naturally occurring polyanionic polysaccharides and are isolated from brown seaweed [Phaeophyceae [START_REF] Tonnesen | Alginate in drug delivery systems[END_REF]], macrocystis pyrifera, laminaria hyperborea and ascophyllum nodsum [START_REF] Sosnik | Alginate Particles as Platform for Drug Delivery by the Oral Route: State-of-the-Art[END_REF]. Alginic acid, sodium and potassium alginates (ALG) are among the most extensively explored mucoadhesive biomaterials having good cytocompatibility, biocompatibility, biodegradation and chemical versatility, which make them suitable for further modifications to its properties [START_REF] Sosnik | Alginate Particles as Platform for Drug Delivery by the Oral Route: State-of-the-Art[END_REF]. In a study reported by Bernkop Schnurch et al, a mucoadhesive thiolated polymer was synthesized and characterized by linking alginate to cysteine.

Since, alginate displays carboxylic acid groups; the sulfhydryl compound cysteine was easily covalently attached. This study showed that the thiolated alginate is a useful excipient for various drug delivery systems providing an improved stability and prolonged residence time [START_REF] Bernkop-Schnurch | Improvement in the mucoadhesive properties of alginate by the covalent attachment of cysteine[END_REF].

In a study reported by [START_REF] Katsumi | Development of polyethylene glycol-conjugated poly-S-nitrosated serum albumin, a novel S-Nitrosothiol for prolonged delivery of nitric oxide in the blood circulation in vivo[END_REF][START_REF] Katsumi | Development of polyethylene glycol-conjugated poly-S-nitrosated serum albumin, a novel S-Nitrosothiol for prolonged delivery of nitric oxide in the blood circulation in vivo[END_REF], an NO-donor was synthesized by conjugating SATA (N-Succinimidyl-S-acetylthioacetate) with polyethylene glycol and Bovine serum albumin. More recently, a new NO donor was synthesized by the nitrosation of chitosan-GSH conjugate (SNOC) with a good value of linked NO to the polymer backbone [START_REF] Shah | Synthesis and characterization of Snitrosoglutathione-oligosaccharide-chitosan as a nitric oxide donor[END_REF].

Materials and methods

Materials

GSH, sodium alginate, Ellman's reagent (5,5'-dithiobis (2-nitro benzoic acid), 2,3diaminonaphthalene (DAN), N-1-(naphthyl)ethylenediamine (NED), 1-ethyl-3-(3dimethylaminopropyl)carbodiimide (EDAC) and sulfanilamide were obtained from Sigma-Aldrich (Quentin Fallavier, France). Sodium nitrite and potassium hydrogen phosphate were purchased from Merck (Germany), and "total nitric oxide and nitrate/nitrite kit" was obtained from R&D systems (Abingdon, UK). All of the other chemicals were of analytical or chromatographic grade.

Synthesis and purification of the alginate-GSH conjugate

GSH was covalently attached to sodium alginate via amide bonds between the carboxylic acid groups of the polymer and the amine groups of GSH (Figure 1) according to a previously reported method [START_REF] Bernkop-Schnurch | Improvement in the mucoadhesive properties of alginate by the covalent attachment of cysteine[END_REF]. Briefly, 10 g of sodium alginate was hydrated in 1000 ml of ultrapure demineralized water. For the amidation reaction, the carboxylic groups of the polymer were activated by addition of EDAC in a final concentration of 50 mM. The reaction was allowed to continue for 45 min at room temperature. After this, 5 g of GSH (in a polymer to drug ratio of 2:1) was added, and the pH was adjusted to 4 with HCl (1 M). This mixture was continuously stirred at 200 rpm for 2 h at room temperature. The pH was readjusted to 6 with NaOH (1 M), and the reaction was allowed to continue for further 1 h.

The resulting conjugate was dialyzed in tubing (molecular weight cut-off 12 kDa) against

HCl (1 mM), twice against HCl (1 mM) containing 1% NaCl and at the end against HCl

(1 mM) for 5 h. The controls were prepared in the same way except using EDAC for the coupling reactions. The resulting conjugate was freeze-dried at -50 °C and 0.01 mbar followed by storage at 4 °C prior to use.

Determination of the thiol group content of various fractions of thiols after synthesis (Ellman's method)

The thiol group content was determined using previously reported methods with slight modifications [START_REF] Atyabi | Thiolated chitosan coated poly hydroxyethyl methacrylate nanoparticles: Synthesis and characterization[END_REF][START_REF] Yousefpour | Preparation and comparison of chitosan nanoparticles with different degrees of glutathione thiolation[END_REF]. Briefly, 0.5 mg of the polymer conjugate was hydrated in 0.35 ml of deionized ultrapure water for 30 min and diluted to a final volume of 1 ml by adding 0.65 ml of Tris-HCl. This solution was labelled solution-1.

Reduced thiols

To a 100 µl of solution-1, 900 µl of phosphate buffer (0.1 M, pH 7.5) and 200 µl of Ellman's reagent (0.3 mg/ml) were added and incubated for 15 min in the dark at room temperature. By using a UV-Visible spectrophotometer (Cary® 50 UV-Vis, Varian, Les Ulis, France), the absorbance was measured at 412 nm. The reduced thiol content was determined from a standard calibration curve of GSH with concentrations ranging from 3.25 to 32 µM.

Total thiols

For the determination of the total thiol groups either in their reduced form or disulfide bond form, 1 ml of NaBH 4 (4% solution w/v) was added to 1 ml of solution-1 followed by incubation at 37 °C for 1 h. Then, this solution was neutralized with 200 µl of HCl (5 M)

and allowed to incubate for further 10 min. After this, 1 ml of phosphate buffer (0.1 M, pH 7.5) was added followed by the addition of 200 µl of Ellman's reagent. The solution was incubated in the dark for 15 min at room temperature, and the amount of total thiol groups was determined using a UV-Visible spectrophotometer.

Thiol groups directly linked to alginate

The amount of thiol groups directly linked to alginate was determined using the same procedure that was mentioned above for the total thiol content. After a reducing pretreatment (1 ml of solution -1 with 1 ml of NaBH 4 ) and neutralization, the polymer was dialyzed in tubing (molecular weight cut-off 12 kDa) for 4 h in distilled water to remove the free thiol groups. The amount of free thiol groups were determined by subtracting the number of thiol groups directly linked to the polymer from the total number of thiol groups.

Nitrosation of alginate-GSH conjugates (SNA)

NO was linked to the alginate-GSH conjugate via nitrosation with NaNO 2 under acidic condition. Two methods were used to achieve nitrosation.

SNA-F1

Briefly, 100 mg of the alginate-GSH conjugates were hydrated in 45 ml of HCl (0.5 M)

for 30 min. To this alginate-GSH conjugate solution, 1 ml of a NaNO 2 solution (equimolar to the reduced thiols in alginate-GSH conjugates) in ultrapure water was added. The reaction was continued for 2 h at 4 °C in the dark and then neutralized with a NaOH solution (40% w/v).

SNA-F2

This method includes the nitrosation of all of the thiol groups (free and linked thiols)

present in the alginate-GSH conjugate. Prior to nitrosation of the polymer, the disulfide bonds were broken by the addition of NaBH 4 followed by treatment with HCl (5 M) to remove the unwanted NaBH 4 . Briefly, 100 mg of the alginate-GSH conjugate was wetted in 8.75 ml of ultrapure water and then diluted with 16.25 ml of Tris-HCl for 30 min to achieve complete hydration. To this solution, 5 ml of NaBH 4 (4% w/v) was added and incubated for 1 h at 37 °C to break the disulfide bonds (R-S-S-R) present between the thiol groups in the conjugate solution. After 1 h of incubation, the remaining NaBH 4 was eliminated by addition of 4 ml of HCl (5 M) followed by incubation for an additional 15 min under the same conditions. The final solution was cooled by adding 25 ml of HCl (0.5 M, 4 °C). To this solution, an equimolar NaNO 2 amount (equal to total amount of thiol groups present in the conjugate solution) was added, and this solution allowed to stand in the dark at 4°C for 2 h. The reaction was terminated by neutralizing it by adding NaOH (40% w/v). This solution was dialyzed for 4 h in ultrapure water using dialysis tubing (molecular weight cut-off 12 kDa) followed by storage at 4 °C in the dark prior to use.

Determination of the NO/NO 2 -concentration (Griess-Saville method)

An indirect method was employed for the determination of NO/NO 2 -, as described by Griess and Saville in 1958 [START_REF] Heines | Peter Griess-Discoverer of diazo compounds[END_REF][START_REF] Saville | A scheme for the colorimetric determination of microgram amounts of thiols[END_REF]. In a two-step diazotation reaction, dinitrogen trioxide (N 2 O 3 , NO derived agent) generated from the acid-catalysed formation of nitrous acid from nitrite reacts with sulfanilamide to produce a diazonium ion. Then, this diazonium ion couples to N-(1-naphthyl)ethylenediamine to form a chromophoric azo product that can be detected at 540 nm. The concentration of the NO moieties was calculated from standard calibration curves of NaNO 2 . The limit of detection for these analytical experiments was 0.25 µmol/l, the limit of quantification was 0.77 µmol/l and the correlation coefficient was 0.999. The method described by Saville [START_REF] Saville | A scheme for the colorimetric determination of microgram amounts of thiols[END_REF] is based on a pre-treatment using mercuric chloride (HgCl 2 ) that induces cleavage of RSNOs to form nitrite. This method allows for the determination of the total amount of NO/NO 2 -. For these experiments, the limit of detection was 0.11 µmol/l, the limit of quantification was 0.35 µmol/l and the correlation coefficient was 0.999.

In the experiment, 20 µl of the sample was added to an acetic acid buffer (pH 2.5), and the final volume was adjusted to 200 µl. To determine the nitrite concentration, 40 µl of sulfanilamide (free nitrites) or sulfanilamide + HgCl 2 (total nitrites) was added and incubated in the dark for 3 min. After incubation, 10 µl of a NED solution was added and incubated in the dark for an additional 5 min. The nitrite concentration was determined using a microplate reader (Elx800 Absorbance Microplate Reader, BioTek, Colmar, France) or UV-Visible spectrophotometer (Cary ® 50 UV-Vis, Varian, Les Ulis, France).

The concentration of the RSNOs (GSNO and SNA) was calculated using the following formula:

RSNO's = (total nitrites) -(free nitrites)

Stability of SNA F1 and SNA F2 in the Ussing chamber

The effect of the experimental conditions (temperature and O 2 /CO 2 ) on the stability of SNA-F1 and SNA-F2 was investigated in one compartment of the Ussing chamber. A 100 µM SNA solution in Hank's buffer was added to the donor compartment of the Ussing chamber that was separated from the accepter compartment with a folded plastic film, consisting of several layers, to prevent flow to the other side. The temperature was adjusted to 37 °C, and the solution was circulated by a constant flow of O 2 /CO 2 . The samples were withdrawn at regular time intervals (i.e. 0, 0.5, 1, 1.5, 2, 3, 4, 5, 6, 7, 8, 9

and 10 h) and analysed using a UV-Visible spectrophotometer.

Effect of pH on the storage conditions of SNA F2

The SNA F2 solution was stored at 3 different pH values (acidic pH 1.2, neutral pH 7.0 and basic pH 8.4) and 4 °C in the dark. The effect of pH was assessed every 2 days by calculating the amount of free and linked nitrites. The results were compared to the initial concentration of SNA F2 and are expressed as percentages.

Intestinal permeation studies in the Ussing Chamber

Preparation of intestinal segments

Male Wistar rats (RjHan: WI, 400 g, 8-10 weeks old, males, Charles Rivers) were killed with 5% halothane in oxygen gas. Segments from the distal portion of the ileum were excised and washed in a beaker containing Krebs's bicarbonate Ringer (KBR, 4 °C) bubbled with an O 2 /CO 2 mixture to maintain membrane viability. These segments were flushed internally with approximately 20 ml of KBR and slid onto glass rods with tapered points wetted with KBR. These rods with intestinal segments were placed in buffer chilled in an ice bath at 3-4 °C. The serosa and muscularis externa were removed with tweezers according to the method described by [START_REF] Wolfe | Drug transfer across intact rat intestinal mucosa following surgical removal of serosa and muscularis externa[END_REF]. The segments were opened along the mesenteric border and mounted on the Ussing chamber (CHM8, 0.5 cm 2 WPI, Hitchin, UK) equipped with a water jacket (37 °C).

Permeability studies through the rat intestine (Ussing chamber)

Eight millilitres of the Hank's BSS buffer without phenol red was added to both the donor and acceptor compartments of the Ussing chamber with the intestinal segment mounted between them. The temperature was maintained at 37 °C, and the flow in both chambers was controlled by O 2 /CO 2 . The trans-mucosal electrical resistance (TER) was monitored by current electrodes (WPI, Hitchin, UK) fixed in the two compartments. The tissues having TER < 30 Ωcm 2 were discarded. A 100 µL sample was withdrawn from the donor and acceptor compartments at appropriate time intervals (starting from 0 to 6 h) to calculate the SNA concentrations using the Griess-Saville method. Nitrite and nitrate ions, which are degradation products of SNA, were measured by Griess and Saville method and by "Total nitric oxide and nitrate/nitrite kit" (R&D systems), respectively.

Cytotoxicity study on the Caco2 cell line

The cytotoxic activity of SNA F2 was tested on the Caco2 cell line. The Caco2 cell assay is based on the reduction of tetrazolium dye MTT by mitochondrial succinate dehydrogenase to its insoluble form (i.e., formazan), which has a purple colour and is quantified spectrophotometrically at 570 and 630 nm using a microplate reader. Briefly, the Caco2 cells at a concentration of 20.10 3 cells per well in 200 µl of medium were seeded into a 96-well culture plate. After 24 h, various concentrations of SNA F2 (1 to 200 µM) were added to the culture plates and incubated for an additional 24 h at 37 °C in a 5% CO 2 humidified atmosphere. Afterwards, 50 µl of MTT (5 mg/ml stock solution in HBSS buffer) was added to each well in the culture plate and incubated for an additional 3 h at 37 °C. The medium was removed, and the formazan crystals were dissolved in 50 µl of DMSO. Then, the culture plate was agitated for 5 min at 500 rpm. The absorbance was measured by a microplate reader. The experiments were carried out in triplicate, and the results are expressed as a percentage of viable cells compared to the control group.

Statistical data analysis

Statistical analyses were performed using the U-test with p<0.05 as the minimum level of significance [START_REF] Mann | On a test of whether one of two random variables is stochastically larger than the other[END_REF] or Kruskal-Wallis test.

Results and Discussion

Synthesis of alginate-glutathione conjugates

The lyophilized form of alginate-GSH consists of a creamy white powder with a fibrous structure. Alginate was the first anionic biodegradable polymer, and this polymer has been chemically modified by covalent attachment to thiol (-SH) groups by [START_REF] Bernkop-Schnurch | Improvement in the mucoadhesive properties of alginate by the covalent attachment of cysteine[END_REF]. Following this method, alginate was successfully attached to GSH, and the number of thiol groups attached to alginate was quantified using Ellman's method. The alginate-GSH conjugate contained 711 ± 24 µmol thiol groups per g of polymer, where 481 ± 22 µmol thiol groups were directly linked to the alginate polymer (Table 1). GSH consists of three amino acids i.e. glutamate, cysteine and glycine. Due to the sulfhydryl (-SH) groups in cysteine, GSH is present in a reduced form and chemically active for a variety of reactions (e.g., antioxidant reactions). GSH is also involved in direct reactions with a variety of oxidants in a non-enzymatic manner that results in its conversion to its dimeric oxidized form (i.e., glutathione disulfide, GSSG) [START_REF] Appenzeller-Herzog | Glutathione-and non-glutathione-based oxidant control in the endoplasmic reticulum[END_REF]. By breaking the disulfide bonds with NaBH 4 , all of the available thiol (-SH) groups can be converted to their reduced form (Table 2), which provides an opportunity to link a large quantity of NO molecules [e.g., the nitrosation process (Table 3)].

Nitrosation of the alginate-GSH conjugates

A pale yellowish solution was obtained after reacting the alginate-GSH conjugates with sodium nitrite (NaNO 2 ) under acidic conditions (pH 1.2). The nitrosation process was performed using two different methods, and the amount of NO adducts to thiol moieties using both methods are shown in Table 3. SNA F2 has a greater number of linked NO groups compared with SNA F1 because SNA F1 is based on the nitrosation of only the reduced thiols present in the alginate-GSH conjugates, which are smaller in number compared with SNA F2 based on the nitrosation of all of the thiol groups present in the conjugates. In SNA F2, the disulfide bonds between the -SH groups were reduced by the addition of NaBH 4 prior to nitrosation, which allows for the linkage of a larger number of thiol groups to NO. Comparatively, the amount of adducted NO to thiol groups was 174 ± 21 µmol per g in SNA F1. However, in SNA F2, this value was 468 ± 23 µmol per g, which suggests the suitability of SNA F2 for further experiments. Acidified NO - 2 can react with several functional groups on proteins, and two types of amino acid groups (i.e., cysteine and tryptophan) have been nitrosated in previous studies [START_REF] Katsumi | Physicochemical, tissue distribution, and vasodilation characteristics of nitrosated serum albumin: delivery of nitric oxide in vivo[END_REF]. The study also indicated that other amino acid residues including arginine, lysine, asparagine, glutamine and tyrosine can also be nitrosated but it can result in unstable products. The nitrosation of the cysteine thiol group in the alginate-GSH conjugate resulted in a high concentration of abducted NO and good stability under different experimental conditions (Figures 2 and3).

Stability of SNA F1 and SNA F2 in the Ussing Chamber

The stabilities of SNA F1 and SNA F2 were evaluated in the Ussing chamber without intestine at 37 °C in Hank's buffer with a constant O 2 /CO 2 flow. As shown in Figure 2, SNA F2 was more stable compared with SNA F1 and degraded into its degradation products after 10 h. This results may be due to the amount of abducted NO on each polymer because SNA F2 has a higher amount of NO, which makes it last longer. In our previous study [START_REF] Shah | Synthesis and characterization of Snitrosoglutathione-oligosaccharide-chitosan as a nitric oxide donor[END_REF], the stability of GSNO, which was studied at a basic pH of 7.4, was comparatively less stable than our new SNA compound.

Stability of SNA F2 under different storage conditions

The stability of SNA F2 was studied under various pH conditions at 4 °C in the dark for several weeks. The acidic condition consisted of 0.1 N HCl (pH 1.2), a neutral condition USP Phosphate buffer (pH 7.0), and a mildly basic conditions consisted of USP phosphate buffer at (pH 8.4). As shown in Figure 3, at an acidic pH of 1.2, only 55 ± 4.9% of SNA F2 remained after 1 week, and all of the SNA F2 degraded within 2 weeks.

The stability of SNA F2 did not improve at a neutral pH of 7.0 where 61 ± 2.8% remained after 1 week and 11 ± 2.9% remained after 2 weeks. As shown in Figure 3, a mildly basic pH of 8.4 was more effective at decreasing the decomposition of RSNO where 73 ± 9.3% remained after 1 week and 43 ± 8.2% remained after 2 weeks. In addition, small amounts were still present until the 3 rd week. The stability of RSNOs at various pH values was also evaluated in a previous study [START_REF] Parent | A Complete Physicochemical Identity Card of S-nitrosoglutathione[END_REF]. In this study, the GSNO stability was evaluated at an acidic condition (pH 0.3) and a basic condition (pH 7.4) at a storage temperature of 4 °C. Based on the previous results, GSNO should be stored at a basic pH to ensure better storage conditions. A recent study conducted on the stability of RSNOs (i.e., GSNO, SNAC and SN3MPA) suggested that RSNOs were more stable at mildly basic pH values (i.e., 8.4 to 8.8) after the exclusion of light at 4 °C, which are similar to our results [START_REF] Hornyak | Increased stability of S-nitrosothiol solutions via pH modulations[END_REF]. This new compound (i.e., SNA F2) is more stable than our previous compound S-nitrosoglutathione-oligosaccharidechitosan [SNOC , [START_REF] Shah | Synthesis and characterization of Snitrosoglutathione-oligosaccharide-chitosan as a nitric oxide donor[END_REF]].

Permeability studies in the Ussing chamber

The ex vivo drug permeation studies were performed in an Ussing chamber across rat intestine. For the ex vivo study, only SNA F2 was selected due to its higher nitrosation values (Table III) and improved stability (Figure 2) compared with SNA F1. GSNO was also studied through the rat intestine for use as a control. The GSNO concentration remained for 5 h in the Ussing chamber (Figures 4 and5) prior to being degraded into its two degradation products (i.e., nitrite (NO - 2 ) and nitrate (NO - 3 ) ions). The concentrations of the two degradation products were also calculated in the acceptor compartment, because these products were able to permeate the intestinal barriers.

Nitrite ions were released in the first hours of the experiment and found at the end of the experiments. These nitrite ions can be considered a sign of NO formation [START_REF] Schulz | Reevaluation of the Griess method for determining NO/NO2-in aqueous and protein-containing samples[END_REF][START_REF] Shah | Synthesis and characterization of Snitrosoglutathione-oligosaccharide-chitosan as a nitric oxide donor[END_REF]. Concentrations of nitrate ions, which can be considered the final products of RSNO degradation, increased throughout the experiments. Because equal amounts of medium were employed in both the donor and accepter compartments of the Ussing chamber, an equal distribution of nitrite and nitrate ions should be present in both the compartments, leading to a theoretical value of 50%. Nitrite and nitrate ions were believed to be inert end products of endogenous NO metabolism. However, recently reported results indicate that these products can be in vivo recycled and transformed back into NO, which represents an important unconventional source of NO outside of the classical L-arginine-NO synthase pathway [START_REF] Lundberg | The nitrate-nitrite-nitric oxide pathway in physiology and therapeutics[END_REF]. Based on the by [START_REF] Khan | S-nitrosoglutathione reduces oxidative injury and promotes mechanisms of neurorepair following traumatic brain injury in rats[END_REF], both nitrite and nitrate ions may be significant precursors of NO and can increase the NO concentration in brain.

SNA F2 was comparatively more stable than GSNO and finally degraded to nitrite and nitrate ions after 10 h (Figure 2). These results indicate that the conjugation of alginate to RSNOs can prolong the release of NO from RSNOs. This prolonged release of NO could have potential benefits for the treatment of various diseases. A number of studies have been performed to study the therapeutic benefits of RSNOs in different clinical settings. However, their therapeutic potential is limited due to the poor distribution to the targeted area and rapid release of NO. Therefore, for the development of a more effective NO delivery system, the release rate of NO from RSNOs and its tissue distribution must be properly controlled [START_REF] Katsumi | Development of polyethylene glycol-conjugated poly-S-nitrosated serum albumin, a novel S-Nitrosothiol for prolonged delivery of nitric oxide in the blood circulation in vivo[END_REF]. SNA F2 cannot cross the intestinal barriers but can deliver nitrite and nitrate ions to systemic circulation (Figures 6 and7).

In the treatment of Crohn's disease, a sustained release of NO is required in the colon, and the systematic absorption is considered a risk for cardiovascular side effects (e.g., hypotension).

Cytotoxicity study using the Caco2 cell line

Caco2 cells were incubated with specified concentrations of SNA F2 (Figure 8) under suitable conditions. The cells were observed before and after incubation with the drug.

The MTT test indicated that SNA F2 with less than 50 µM NO exhibited a viability of more than 100% and 100 µM SNA F2 exhibited 86% viability. However, SNA F2 with 200 µM NO was cytotoxic, and the viability of cells decreases to 38%.

Based on our results, alginate alone increased the viability of the cells even in the column "100 µM of NO". In addition, the alginate-GSH conjugate exhibits 100% viability in the column "50 µM of NO". However, its viability decreased at higher concentrations.

Therefore, based on this MTT assay, our polymer was not toxic at 10 µM NO and exhibits 100% metabolic activity in the cells after 24 h of incubation. Similar results were obtained in a study performed by Ramia [START_REF] Safar | Human monocyte response to Snitrosoglutathione-loaded nanoparticles: uptake, viability, and transcriptome[END_REF]Ronzani et al. (Ronzani et al., 2014,Safar et al., 2015), where empty nanoparticles consisting of Eudragit RL were seeded into the human THP-1 monocytic cell line, and a significant increase in the viability of the cells up to 40% was observed compared with the control. The cell viability increased due to an increase in the metabolic activity and growth of the cells after contact with the nanoparticles, and this phenomenon was related to an augmentation of the mitochondrial volume and activity. The mitochondrial activity may be related to cell proliferation, which leads to an increase in the mitochondrion number. The up regulation of genes involved in metabolism and proliferation may be related to this increased mitochondrial activity, which belongs to "transcription", "nucleus", "chromosomal part", "metabolic process" and "proliferation" [START_REF] Safar | Human monocyte response to Snitrosoglutathione-loaded nanoparticles: uptake, viability, and transcriptome[END_REF].

Conclusions

Based on our results, a novel polymeric NO-donor was successfully synthesized by linking alginate to GSH followed by a nitrosation step to link NO. The new donor was able to carry a significant amount of pharmacologically active NO with an interesting "extended release" effect. A stable NO-donor under physiologically acceptable pH conditions was achieved, and this donor has the potential for long-term therapeutic effects. Alginate is a good candidate for the long-term stability and controlled delivery of NO. The low toxicity of this new compound suggests future in vivo studies in animal models should be performed. This type of compound would be beneficial for the treatment of various types of diseases including Crohn's disease. The release of NO may differ under physiological conditions, especially due to the concentration of O 2 , which is necessary for tissue viability. This concentration may be lower in the colonic lumen.

We believe that our compound can be encapsulated in gastro-resistant soft gelatine capsules for oral delivery. Microparticles of SNA with chitosan can also be prepared [START_REF] Marcato | Development of a sustainedrelease system for nitric oxide delivery using alginate/chitosan nanoparticles[END_REF]. In addition, a polyelectrolyte complex of alginate with chitosan and Eudragit ® L100-55 can also be prepared [START_REF] Calija | pH-sensitive microparticles for oral drug delivery based on alginate/oligochitosan/Eudragit((R)) L100-55 "sandwich" polyelectrolyte complex[END_REF]. Animal models will be used in future to evaluate the in vivo potential of the new compound and to study its pharmacokinetics. Furthermore, this type of conjugation will be tried for other biodegradable polymers.

Acknowledgement

Figures :Figure 1 .

 1 Figures :

Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .

 2345678 Figure 2. Stability of SNA F1 (--), SNA F2 (--) in the Ussing chamber without rat intestine at 37°C (n=3, mean ± S.D.)

Table I . Amount of glutathione (SH groups) determined by Ellman's test (precursor of SNA F1)

 I

	Glutathione Fraction	Ellman's method (µmol/g)
	Reduced thiols	214 ± 10
	Total thiols	711 ± 24
	Directly linked thiols	481 ± 22

Table 2 . Amount of glutathione (SH groups) determined by Ellman's test (precursor of SNA F2)

 2

	Glutathione Fraction	Ellman's method (µmol/g)
	Reduced thiols	711 ± 24
	Total thiols	711 ± 24
	Directly linked thiols	481 ± 22

Table 3 . Amount of nitrites determined by the Griess and Saville tests

 3

		SNA F1	SNA F2
	Nitrites (µmol/g)	174 ± 21

SU Shah wishes to acknowledge funding support by the Higher Education Commission of Pakistan.

Declaration of interest

The authors have no conflict of interest.


Brice Portelenelle
email: brice.portelenelle@ensimag.grenoble-inp.fr

Jean-Pierre Croisille
email: jean-pierre.croisille@univ-lorraine.fr

AN EFFICIENT QUADRATURE RULE ON THE CUBED SPHERE

Keywords: Quadrature rule on the sphere -Cubed Sphere -Spherical Harmonic. Mathematics Subject Classification: 33C55, 41A55, 65D30, 65D32

come

An efficient quadrature rule on the cubed sphere

Introduction

In this paper we consider quadrature rules for functions defined on the sphere. Let S 2 be the unit sphere, and let x ∈ S 2 → f (x) be a regular function. A quadrature rule Q(f ) is defined by ( 1)

I(f ) = S 2
f (x)dσ(x)

P p=1 w p f (x p ) = Q(f ),
where x p ∈ S 2 are the nodes and w p are the weights. The quest for rules of the form (1) has been a longstanding topic of interest. The classical setup of the problem consists in finding a minimal number P of nodes x p with the associated weights w p for (1) to be as exact as possible. More precisely, the problem is to determine the location on the sphere of a minimal number of nodes for the largest number of Spherical Harmonics to be exactly integrated. Recent works on this topic include [START_REF] Sloan | Extremal systems of points and numerical integration on the sphere[END_REF][START_REF] Keiner | Fast evaluation of quadrature formulae on the sphere[END_REF][START_REF] Ahrens | Rotationally invariant quadratures for the sphere[END_REF][START_REF] Fornberg | On spherical harmonics based numerical quadrature over the surface of a sphere[END_REF]. We also refer to the review paper [START_REF] Hesse | Numerical Integration on the Sphere[END_REF] and the references therein.

Here, we are interested in a slightly different question. Over the past 20 years, the Cubed Sphere (see Fig. 1) has become a popular spherical grid among researchers dealing with mathematical or physical models. In particular, in numerical climatology, the Cubed Sphere serves for various numerical schemes for time-dependent climate models on the sphere [START_REF] Lauritzen | A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed sphere grid[END_REF][START_REF] Bao | A mass and momentum flux-form high-order discontinuous galerkin shallow water model on the cubed-sphere[END_REF][START_REF] Ullrich | High order finite-volume methods for the shallow-water equations on the sphere[END_REF]. In this context, accurately evaluating averaged quantities over the sphere such as mass, momentum, energy or total vorticity is particularly important. This is in particular the case for the finite difference scheme introduced in [START_REF] Croisille | Hermitian compact interpolation on the Cubed-Sphere grid[END_REF][START_REF] Croisille | Hermitian approximation of the spherical divergence on the Cubed-Sphere[END_REF]. This scheme uses discrete unknowns located at the nodes of the Cubed Sphere. For this scheme, it is therefore natural to have at hand a quadrature rule with nodes x p selected as the gripoints of the Cubed Sphere.

To fully determine such a rule, it remains to identify a set of suitable weights w p . A basic observation is that a particularly simple set of weights w p , described hereafter, provides a rule [START_REF] Ahrens | Rotationally invariant quadratures for the sphere[END_REF], which is exact for a proportion of 7/8 of all Spherical Harmonics. Furthermore, for the remaining 1/8 Spherical Harmonics not integrated exactly, fourth order accuracy convergence of the rule was Date: October 02, 2016. This work was performed during the master degree thesis of the first author at the Univ. Lorraine, Math. Dept., Institut Elie Cartan de Lorraine in Metz, France, during the second semester of the academic year 2015-2016. numerically obtained. This observation was the starting point of the present study. Building upon this first rule (called Q a ), we suggest a second rule Q b , which keeps the 7/8 property and appears to be remarkably accurate.

The outline of the paper is as follows. Section 2 gives the notation for the Spherical Harmonics and the Cubed Sphere. In Section 3, our first quadrature rule, called Q a , is introduced and the aforementioned 7/8 property is proved. This property is actually due to a set of combined symmetries shared by the Cubed Sphere and the Spherical Harmonics. In Section 4, we show how this property can serve to significantly enhance the selection of the weights associated with the nodes of the Cubed Sphere. This permits to define a second quadrature rule called Q b , which again uses the Cubed Sphere points as quadrature nodes. The weights w p are selected according to a specific least square problem. Numerical results show that the rule performs remarkably well compared to other spherical quadrature rules of the litterature.

Notation

Consider the reference

Cartesian frame R = (O, i, j, k) in R 3 . The unit sphere is S 2 = {x(x, y, z) ∈ R 3 /x 2 + y 2 + z 2 = 1}.
The Longitude-Latitude coordinate system is called

(λ, θ), with -π ≤ λ < π and -π/2 ≤ θ ≤ π/2. For n ≥ 0 and m with -n ≤ m ≤ n, the Spherical Harmonic Y m n : x ∈ S 2 → Y m n (x) ∈ C is defined by [9]: (2) Y m n (x) = (-1) m N |m| n P |m| n (sin θ)e imλ .
For 0 ≤ m ≤ n, the function x ∈ [-1, 1] → P m n (x) is the associated Legendre polynomial. It is defined in terms of the Legendre polynomial P n (x) = d n dx n (x 2 -1) n /(2 n n!) by the relation:

(3)

P m n (x) = (1 -x 2 ) m/2 d m dx m P n (x), 0 ≤ m ≤ n, x ∈ [-1, 1].
The Spherical Harmonics form an orthonormal basis of the space L 2 (S 2 ) and therefore ( 4)

I(Y m n ) = 0 if (n, m) = (0, 0), I(Y 0 0 ) = √ 4π. The normalisation constant N m n in (2) is (5) N m n = 2n + 1 4π (n -m)! (n + m)! 1 2
.

A quadrature rule is usually designed using the functions Y m n . The accuracy is measured by the order n such that the functions Y m n (x), |m| ≤ n, are integrated exactly. Our next ingredient is a particular grid of S 2 called the Cubed Sphere (see Fig. 1). The Cubed Sphere with parameter N consists of 6N 2 + 2 points located on six panels, called P k , (I) ≤ k ≤ (V I). These six panels can be matched to the six faces of the cube [-1, 1] 3 embedding the sphere. This fact is the reason of the terminology Cubed Sphere.

Each panel P k has a square like shape represented on Fig. 2. It is supplied with a coordinate system (ξ, η), whose coordinate lines are great circle sections. The coordinate lines ξ = 0, (resp. η = 0) represent the vertical (resp. horizontal) equatorial line at the center of P k . The function ϕ k is the bijective application defined by ( 6) For example for the panel P (I) , the point x(x, y, z) = ϕ (I) (ξ, η) satisfies the relations (7)

ϕ k : (ξ, η) ∈ [- π 4 , π 4 ] → x(ξ, η) ∈ P k .
         X = tan(ξ), Y = tan(η), X = y x , Y = z x , x 2 + y 2 + z 2 = 1.
Analog relations hold for the five other panels. On P k , the grid is defined by the (N + 1) 2 points (8)

s k i,j = ϕ k (i∆ξ, j∆η), -N/2 ≤ i, j ≤ N/2, where ∆ξ = ∆η = π/2N . 1 Finally the function (ξ, η) ∈ [-π 4 , π 4 ] → G(ξ, η) ∈ M 2 (R)
denotes the metric tensor defined by The function (ξ, η) → G(ξ, η) obviously does not depend on the panel. For, the panel P k is deduced from the panel P (I) by a fixed rotation of the cube, and ( 9) is invariant under this transformation. This metric tensor will serve in the next section. We refer to [START_REF] Ronchi | The Cubed Sphere: A new method for the solution of partial differential equations in spherical geometry[END_REF] for other properties of the Cubed Sphere.

3. A first quadrature rule 3.1. Derivation of the rule. In this section we introduce our first quadrature rule called Q a . It uses the points s k i,j of the Cubed Sphere as the nodes x p in (1). Let f : S 2 → R be a regular function defined on S 2 . Using the fact that the sphere S 2 is decomposed into the set of the six panels P k , (I) ≤ k ≤ (V I), we have ( 10)

I(f ) = S 2 f (x)dσ(x) = (V I) k=(I) P k f (x)dσ(x) I k (f )
.

Using on P k the change of variables x = ϕ k (ξ, η) gives by the chain rule [START_REF] Simmonds | A Brief on Tensor Analysis[END_REF], [START_REF] Lauritzen | A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed sphere grid[END_REF]

I k (f ) = (ξ,η)∈[-π 4 ; π 4 ] 2 (f • ϕ k )(ξ, η) | det G(ξ, η)|dξdη.
We denote

f k = f • ϕ k . The integral I k (f ) is approximated by Q k (f ) defined by (12) Q k (f ) = ∆ξ∆η N/2 i,j=-N/2 c i,j g i,j f k (ξ i , η j ),
where the weights g i,j are given by (13)

g i,j = | det G(ξ i , η j )|.
Furthermore, the coefficients c i,j are defined as follows:

• The internal points (i, j) with -N/2 < i, j < N/2, displayed with circles on Fig 2 are counted with coefficient c i,j = 1. • The edge points (i, j) with i = ±N/2 and -N/2 < j < N/2 or (i, j) with j = ±N/2 and -N/2 < i < N/2, displayed as squares on Fig. 2, are multiplied by the coefficient c i,j = 1/2. • The four corner values (i, j) ∈ {(-N/2, -N/2), (-N/2, N/2), (N/2, -N/2), (N/2, N/2)} displayed as pentagons on Fig. 2 are multiplied by the coefficient c i,j = 1/3. This convention is natural since each edge point belongs to two neighbor panels. It is therefore counted twice whence the coefficient 1/2. Similarly, each corner point is counted three times, and this gives a coefficient 1/3 at the panel level. The rule Q a is finally defined by ( 14)

I(f ) Q a (f ) = (V I) k=(I) Q k (f ).
Remark 3.1. In one dimension, the trapezoidal rule, expressed for f (x) defined on I = [0, 1], is given by: (15)

1 0 f (x)dx 1 N 1 2 f (0) + N -1 1 f (i∆x) + 1 2 f (1) .
On the panel P k , the formula (11) can be interpretated as the tensor product of the rule (15) on [-π/4, π/4] × [-π/4, π/4] applied to the integrand in [START_REF] Lauritzen | A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed sphere grid[END_REF] except that the coefficient 1/3 is used at the corner points, instead of 1/4. In the case where nodes and/or weights have particular spherical symmetries, one expects some specific subset of the Spherical Harmonics to be integrated exactly. Building quadrature rules on the sphere using invariance of the nodes under a subgroup of SO 3 has been used by several authors. We refer to [START_REF] Mclaren | Optimal numerical integration on a sphere[END_REF] and the references therein. An important result in this direction is Sobolev's theorem, which states that a necessary and sufficient condition for a quadrature rule, invariant by a subgroup G of SO 3 (R), to be exact up to degree n is to be exact for all G-invariant Spherical Harmonics of degree ≤ n, [START_REF] Hesse | Numerical Integration on the Sphere[END_REF]. Our rules are not a direct consequence of this result and we will consider a different approach. In a first step, we identify without invoking the group of rotations of the cube, all Spherical Harmonics which are invariant under the rule Q a . Observe that the weights g i,j in (13) do not depend on the panel P k . Furthermore they satisfy the following invariance properties for -N/2 ≤ i, j ≤ N/2:

• Invariance by the rotation r of angle π/2, i.e:

(16)

g i,j = g -j,i ,
• Invariance by the symmetry s with respect to the first diagonal, i.e.:

(17) g i,j = g j,i .

Combining these two tranformations, it is easily seen that the coefficients g i,j also are symmetric with respect to the second diagonal and with respect to the coordinate lines i = 0 and j = 0, respectively. Otherwise stated, for -N/2 ≤ i, j ≤ N/2:

(18) g i,j = g -j,-i = g -i,j = g i,-j = g -i,-j .
For example for N = 8, the coefficients g i,j can be arranged as shown in Table 1, where the symmetry properties ( 16), ( 17) and ( 18) are represented with letters. For a Cubed Sphere with parameter N , the number of independent weights is given by the integer q N = (N + 2)(N + 4)/8. These weights correspond to the indices 0 ≤ j ≤ i ≤ N/2. They are displayed by boldface letters in Table 1. Table 1. Parameters of the weights in a typical panel P k in the Cubed Sphere with parameter N = 8. The independent parameters are displayed in boldface. The number of independent parameters is q N = (N +2)(N +4)/8. There is 15 independent parameters in this particular case.

Y m n (x) = (-1) m N |m| n P |m| n (sin θ)e imλ , n ≥ 0, -n ≤ m ≤ n
The exact integration property stated of Prop. 3.1 is based on the fact that non zero values of Y m n (x) have their exact opposite on another part of the Cubed Sphere. This property is due to the symmetries [START_REF] Ullrich | High order finite-volume methods for the shallow-water equations on the sphere[END_REF]17) of the weights g i,j , and to the following relations:

(21) Y m n (λ + π, θ) = (-1) m × Y m n (λ, θ), (22) Y m n (λ, -θ) = (-1) n+m × Y m n (λ, θ), (23) Y m n λ + p π m , θ = (-1) p × Y m n (λ, θ).
with n > 0, m ∈ {-n, . . . , n}\{0}, p ∈ Z, and (λ, θ) ∈ [-π, π[× -π 2 ; π 2 . Note that (21) is a particular case of (23).

We consider the four following cases. In each case, the approximate value

Q a (Y m n ) is compared to the exact integral I(Y m n ) in (4).
• Case n odd. In this case, if m is even, then the property (22) leads to (24)

Q (I) (Y m n ) = Q (II) (Y m n ) = Q (III) (Y m n ) = Q (IV ) (Y m n ) = 0,
due to opposite values of Y m n (λ, θ) for opposite values of θ, and to the properties g i,j = g i,-j of the weights. Moreover, due to the relation (22) and using the fact that the weights g i,j are identical on the six panels, we have ( 25)

Q (V ) (Y m n ) = -Q (V I) (Y m n ). This gives Q a (Y m n ) = 0. If m is odd, then the relation (21) leads to the relations (26) Q (I) (Y m n ) = -Q (III) (Y m n ), Q (II) (Y m n ) = -Q (IV ) (Y m n ).
We have used the relation g i,j = g -i,j and the fact that the weights g i,j are identical on each panel. Furthermore due to (21) and g i,j = g -i,-j (symmetry of the weights with respect to the center of each panel) we find

(27) Q (V ) (Y m n ) = Q (V I) (Y m n ) = 0.
Summing up over the six panels yields Q a (Y m n ) = 0 (exact value). • Case n even and m odd. In this case, n + m is odd, and then (22) shows that Q k (Y m n ) = 0 for the panels k = (I), (II), (III) and (IV ). This is due to the fact that the weights g i,j have the property g i,j = g i,-j . For the panels k = (V ) and k = (V I), we have Q k (Y m n ) = 0 due to (21) and g i,j = g -i,-j . Therefore Q a (Y m n ) = 0. • Case n even, m even and m ≡ 0 [START_REF] Croisille | Hermitian compact interpolation on the Cubed-Sphere grid[END_REF].

In this case, Q k (Y m n ) = 0 for the panels k = (V ) and k = (V I). This is due to (23) (take p = m 2 in this formula) and to the fact that the weights g i,j satisfy [START_REF] Ullrich | High order finite-volume methods for the shallow-water equations on the sphere[END_REF]. Moreover, again using (23) with p = m 2 , one has

Q (I) (Y m n ) = -Q (II) (Y m n ) and Q (III) (Y m n ) = -Q (IV ) (Y m n ). Note in addition that Q (I) (Y m n ) = Q (III) (Y m n ). This yields Q a (Y m n ) = 0.
• Case n even, m even and m ≡ 0 [START_REF] Croisille | Hermitian compact interpolation on the Cubed-Sphere grid[END_REF]. Note that Y 0 0 belongs to this case. In this case, (28)

Q (I) (Y m n ) = Q (II) (Y m n ) = Q (III) (Y m n ) = Q (IV ) (Y m n ).
This is due to the property (23). In addition, due to (22), ( 29)

Q (V ) (Y m n ) = Q (V I) (Y m n ),
In this case, a possible cancellation within the terms of the approximate integral Q a (Y m n ) could only occur between the contribution of panels (I), (II), (III) and (IV) on the one hand and the contribution of panels (V) and (VI) on the other hand. Such a cancellation does not occur in general. This explains why Q a (Y m n ) is not exact in this case.

Remark 3.2. The former three cases describe all together a proportion of (asymptotically) 7/8 of all Spherical Harmonics. The latter case corresponds to the remaining 1/8 Spherical Harmonics. 3.3. Numerical results. In this section we show numerical results obtained with the rule Q a . The functions f 1 , f 2 , f 3 and f 4 that are used to test numerical quadrature formula are the following, [START_REF] Fornberg | On spherical harmonics based numerical quadrature over the surface of a sphere[END_REF][START_REF] Beentjes | Quadrature on a spherical surface[END_REF].

(30)

                                         f 1 (x, y, z) = 1 + x + y 2 + x 2 y + x 4 + y 5 + x 2 y 2 z 2 , f 2 (x, y, z) = 0.75e -(9x-2) 2 /4-(9y-2) 2 /4-(9z-2) 2 /4 +0.75e -(9x+1) 2 /49-(9y+1)/10-(9z+1)/10 +0.5e -(9x-7) 2 /4-(9y-3) 3 /4-(9z-5) 2 /4 -0.2e -(9x-4) 2 -(9y-7) 2 -(9z-5) 2 , f 3 (x, y, z) = (1 + tanh(-9x -9y + 9z))/9, f 4 (x, y, z) = (1 + sign(-9x -9y + 9z))/9.
The exact values are given by:

(31)

                   S 2 f 1 = 216π 35 , S 2 f 2 = 6.6961822200736179523 . . . , S 2 f 3 = 4π 9 , S 2 f 4 = 4π 9 .
The numerical results for several values of N (the total numbers of points for the discretisation of S 2 are precised) are displayed in Table 3. According to [START_REF] Fornberg | On spherical harmonics based numerical quadrature over the surface of a sphere[END_REF], we display the accuracy for the four test functions in (30), by retaining the worst case among 1000 random solid rotations operated on the argument (x, y, z) of these functions. On Fig. 3 the convergence rate is numerically evaluated as fourth order. The accuracy is excellent, compared to results reported elsewhere. For example, for a Cubed Sphere parameter N ≤ 12, which corresponds to a number of quadrature nodes on the sphere of 6N 2 + 2 ≤ 1000, the accuracy is of the same order as the one obtained with optimally selected quadrature nodes [START_REF] Fornberg | On spherical harmonics based numerical quadrature over the surface of a sphere[END_REF].

Remark 3.4. A proof of the fourth order accuracy of the rule Q a is not straightforward. In particular, it cannot be simply deduced from the accuracy of the trapezoidal rule in one dimension, since the rule Q a is not exactly a tensor product rule on each panel, see Remark 3.1.

4.

A second quadrature rule 4.1. General derivation. As shown in Prop. 3.1, the quadrature rule Q a in ( 14) is exact for a proportion of 7/8 of all Spherical Harmonics. However, it is not exact for the remaining 1/8 Spherical Harmonics. It is numerically observed to be fourth-order accurate, (see Fig. 3 and Remark 3.4). In this section we suggest a second quadrature rule, called Q b , whose nodes again are the points of the Cubed Sphere. It is designed as a perturbation of Q a . Similarly to Q a , the rule Q b is expressed as a sum of contributions of each panel P k (32) Table 3. Accuracy of the quadrature rule Q a for the test functions f 1 , f 2 , f 3 and f 4 in (30). The result corresponds to the worst case among 1000 randomly selected solid rotations operated on the argument (x, y, z) of the functions f 1 , f 2 , f 3 and f 4 . where the contribution of the panel

I(f ) Q b (f ) = (V I) k=(I) Q k,ε (f ), N Number of nodes |I(f 1 ) -Q a (f 1 )| |I(f 2 ) -Q a (f 2 )| |I(f 3 ) -Q a (f 3 )| |I(f 4 ) -Q a (f 4 )| 4
P k is (33) Q k,ε (f ) = ∆ξ∆η N/2 i,j=-N/2 c i,j (g i,j + ε i,j )f k (ξ i , η j ), = Q k (f ) + ∆ξ∆η N/2 i,j=-N/2 c i,j ε i,j f k (ξ i , η j ).
Therefore the quadrature rule Q b (f ) is seeked in the form of a perturbation of Q a as:

(34) Q b (f ) = Q a (f ) + ∆ξ∆η (V I) k=(I) N/2 i,j=-N/2 c i,j ε i,j f k (ξ i , η j ).
If the symmetries ( 16) and ( 17) are imposed to the weights ε i,j , then by Prop. 3.1, the rule Q b is exact for the same Spherical Harmonics than the rule Q a . Let us call I the set of indexes (i, j) such that 0 ≤ j ≤ i ≤ N/2. This set is represented in Table 1 in boldface letters. Recall that 12) can be expressed as

|I| = q N = (N + 2)(N + 4)/8. The term Q k (f ) in (
(35) Q k (f ) = ∆ξ∆η (i,j)∈I g i,j f k i,j .
where f k i,j denotes, for (i, j) ∈ I and (I)

≤ k ≤ (V I) (36) f k i,j = c i,j (i ,j )/g i ,j =gi,j f k (ξ i , η j ).
With this notation, (33) can be expressed as

(37) Q k,ε (f ) = ∆ξ∆η (i,j)∈I (g i,j + ε i,j ) f k i,j .
4.2. Determining ε i,j . The question is now to evaluate suitable values of the q N unknowns (ε i,j ) (i,j)∈I . We denote by

ψ 1 = Y 0 0 , ψ 2 = Y 0 2 , ψ 3 = Y 0 4 , ψ 4 = Y 4 4 , .
. . , the sequence of the Spherical Harmonics Y m n with n even and m ≥ 0, m = 0 [START_REF] Croisille | Hermitian compact interpolation on the Cubed-Sphere grid[END_REF]. This is the set of 1/8 of all Spherical Harmonics not integrated exactly by the rule Q a . The set of equations determining the values ε i,j is defined as

(38) Q b (ψ l ) = I(ψ l ), 1 ≤ l ≤ p N .
The integer p N , to be determined, is the number of Spherical Harmonics of the preceding form taken in account in (38). Using (34), equation ( 38) is recast in the form:

(39)

(i,j)∈I   ∆ξ∆η (V I) k=(I) ( ψl ) k i,j   ε i,j = I(ψ l ) -Q a (ψ l ), 1 ≤ l ≤ p N .
Lemma 4.1. For each l ≤ 0, (I) ≤ k ≤ (V I) and (i, j) ∈ I,

(40) I(ψ l ) -Q a (ψ l ) ∈ R and (41) ( ψl ) k i,j ∈ R. Proof. Let ψ l = Y m n with n = 2n , m = 4m , 0 ≤ m ≤ n. The imaginary part of Y m n (x) is called H m n (x). According to (2), H m n (x) is expressed as (42) H m n (λ, θ) = N m n P m n (sin θ) sin(mλ).

The imparity with respect to the longitude variable λ gives (43)

H m n (-λ, θ) = -H m n (λ, θ), and to the relation g -i,j = g i,j imposes that the discrete integral of H m n on the panel

P (k) with k = (I) is (44) Q k (H m n ) = 0.
Q (V ) (H n m ) = N/2 i,j=-N/2 c i,j g i,j (H m n ) (V ) (ξ i , η j ) = (i,j),j<-|i| c i,j g i,j (H m n ) (V ) (ξ i , η j ) + (i,j),i≥|j| c i,j g i,j (H m n ) (V ) (ξ i , η j ) + (i,j),j≥|i| c i,j g i,j (H m n ) (V ) (ξ i , η j ) + (i,j),i<-|j| c i,j g i,j (H m n ) (V ) (ξ i , η j )
The first term vanishes, due to (43) and the relation g -i,j = g i,j . The relations ( 22) and ( 23) imply that the three other terms also vanish. The argument is similar for the panel k = (V I). An analog argument shows that the imaginary part of ψk l,i,j ∈ R for all l ≥ 0, (i, j) ∈ I and (I) ≤ k ≤ (V I) also vanish. Therefore Q a (ψ l ) ∈ R and since I(ψ l ) ∈ R, (40) holds.

The set of relations (39) forms a linear system (46) Aε = b, Due to Lemma 4.1, the matrix A and the right-hand side b are real. The unknown vector is

(47) ε = (ε i,j ) (i,j)∈I ∈ R q N .
The integer p N is the number of Spherical Harmonics incorrectly integrated by the rule Q a , which are taken in account in (46). A relevant value of this integer has been determined numerically as follows. On Fig. 4 the error of the rule Q b is represented in function of the number of lines p N in (39) for Cubed Sphere parameters N = 4, N = 8 and N = 16, respectively. If we withdraw several pikes denoting a superconvergence effect, the curves can be considered as describing a monotonic increasing behaviour. In view of these results, we have selected the parameter value p N = N 2 /4 as a compromise.

With this choice, the matrix A in (46) is rectangular with

(48) A ∈ M p N ,q N (R), p N = N 2 /4, q N = (N + 2)(N + 4)/8.
For N ≥ 8, p N = N 2 /4 > q N N 2 /8, thus the system (46) is in general overdetermined. We solve (46) using the pseudo-inverse of Moore-Penrose [START_REF] Deuflhard | Numerical Analysis in Modern Scientific Computing[END_REF]Chap. 3,pp. 57sqq]. We have used the matlab routine pinv. Remark 4.2. Since the pseudo-inverse is used to solve the rectangular system (46), the identity (38) for the Spherical Harmonic ψ l , 1 ≤ l ≤ p N is not exactly satisfied, but only approximately. Therefore, strictly speaking, the rule Q b does not integrate all Spherical Harmonics up to a certain order. This is in contrast to the usual setting stated in [START_REF] Mclaren | Optimal numerical integration on a sphere[END_REF], [START_REF] Hesse | Numerical Integration on the Sphere[END_REF], [START_REF] Fornberg | On spherical harmonics based numerical quadrature over the surface of a sphere[END_REF]. However in practice, the accuracy of the rule Q b is remarkably good as will be seen in Section 4.5.

Remark 4.3. As mentionned in Remark 3.3, the funtion Y 0 2 is exactly integrated by the rule Q a . Therefore this function could be withdrawn from the series of functions ψ l in (38). It corresponds to a null line in (46). However keeping it does not prevent the generalized inverse to be used to solve the linear system. Remark 4.4. An important question is whether or not the matrix A in (46) is full rank. This problem is open for the moment. Preliminary numerical experiments suggest that the matrix A may be not full rank for small values of the parameter N , which means for a coarse Cubed Sphere (N ≤ 12), and becomes full rank for larger values of N . This question requires further theoretical and numerical investigation. In practice however, we never observed so far numerical ill-conditionning effects of the matrix A when solving the linear system (46). We observe on Fig. 6 what is expected: when the Spherical Harmonic Y m0 n0 is not taken in account in the matrix A in (46), formulas Q a and Q b give indiscernable results. In the contrary, from the ) is slightly greater than with Q a (Y 16 24 ). This is probably due to the least square method, which solves "in the mean" (38) and gives a value of ε with a lack of accuracy for Y 16 40 for the particular value N = 16. Note that this phenomenon is unusual among all the Spherical Harmonics that have been tested. It does not impact the efficiency of the rule Q b , as will be seen in Section 4.5. In the contrary, we also observe the opposite behaviour of rule Q b compared to rule Q a : for a Spherical Harmonic Y m0 n0 not taken in account in (38), a more accurate value with Q b (Y m0 n0 ) than with Q a (Y m0 n0 ) can be obtained. Overall, it is observed that the rules Q a and Q b give similar results for the Spherical Harmonics not taken in account in the interpolation in (39), (or equivalently, in the matrix A in (46)). Furthermore, the rule Q b permits to integrate up to machine accuracy the Spherical Harmonics taken in account in (39). The greater is N , the more Spherical Harmonics can be taken in account in the matrix A. The former observation (for Spherical Harmonics not taken in account) is important in practice, since it indicates that the rule Q b preserves the good convergence properties of the rule Q a when N increases for all the functions Y n m with n ≥ n 0 , a fixed value.

4.5. Numerical results. In Table 4, numerical results obtained with the rule Q b (32) applied to the set of test functions (30) are reported. As in Table 3, the worst case among 1000 randomly selected solid rotations is retained.

According to results displayed in Table 4 and Fig. 7, the quadrature rule Q b is much more accurate than Q a . Indeed, for a very coarse Cubed Sphere with parameter N 6, the machine accuracy is reached ( 10 -15 ) for the functions f 1 , f 3 and f 4 . The remarkable accuracy for f 1 , f 3 and f 4 can be explained as follows. Since f 1 is a polynomial, only the first Spherical Harmonics are useful, and therefore when the parameter p N is such that p N ≥ 9, all Spherical Harmonics entering the decomposition of f 1 are integrated exactly by the rule Q b . The functions f 3 and f 4 illustrate the accuracy power of the rule Q b , which is due to the symmetry properties [START_REF] Ullrich | High order finite-volume methods for the shallow-water equations on the sphere[END_REF]17,18) combined with the good integration of Y 0 0 . Indeed, as mentioned in [START_REF] Beentjes | Quadrature on a spherical surface[END_REF], the Spherical Harmonics entering the decomposition of f 3 are limitated to Y 0 0 and Y m n with n odd. Thus it is sufficient for a rule to integrate exactly Y 0 0 since the Spherical Harmonics with an odd degree are exactly integrated thanks to the symmetries of the weights (7/8 property). The same observation holds for f 4 . Consequently, from the moment than p N ≥ 1, the rule Q b already gives much better results than the rule Q a simply because Y 0 0 is the first Spherical Harmonic taken in account in the matrix A.

In the case of f 2 , the improvement is less drastic. This is due to the fact that there are Spherical Harmonics entering the decomposition in Spherical Harmonics of f 2 , which are not taken in account in the matrix A in (46). Therefore the error decreases progressively. However the decreasing rate is remarkably fast: the slope shown in Fig. 7 is numerically evaluated as -13, instead of -4 for the rule Q a (see Fig. 3). Overall, as observed in Table 4, the level of accuracy obtained with the rule Q b are excellent when compared with quadrature rules using optimally selected set of nodes, [START_REF] Fornberg | On spherical harmonics based numerical quadrature over the surface of a sphere[END_REF]. Table 4. Accuracy of the quadrature rule Q b for the test functions f 1 , f 2 , f 3 and f 4 in (30). The machine accuracy is reached from a number of quadrature nodes as low as 218 for the functions f 1 , f 3 and f 4 . The results correspond to the worst case among 1000 randomly selected solid rotations applied to the arguments (x, y, z) of the functions f 1 , f 2 , f 3 and f 4 .

Conclusion

A new approach to numerical quadrature on the sphere is introduced. The quadrature nodes are the points of the Cubed Sphere. Two rules, called Q a and Q b , are introduced. Both of them satisfy the 7/8 property which states that 7/8 of all Spherical Harmonics are integrated exactly. The first rule Q a is an analog of the trapezoidal rule on the sphere. The second rule Q b , is an enhancement of the rule Q a , and provides a remarkable accuracy on a set of test functions [START_REF] Fornberg | On spherical harmonics based numerical quadrature over the surface of a sphere[END_REF][START_REF] Beentjes | Quadrature on a spherical surface[END_REF]. The rule Q b is efficient not only when applied to regular functions, but also when applied to irregular functions as the function f 4 in (30), which uses the sign function in its definition. Further studies are ongoing to better understand the mathematical properties of the rules Q a and Q b .

Figure 1 .

 1 Figure 1. The Cubed Sphere with a grid parameter of N = 16. The total number of gridpoints is 6 × N 2 + 2 = 1538 in this case.

Figure 2 . 3 . 2 .

 232 Figure 2. The points of a typical panel of the Cubed Sphere are classified in three categories: (i) Circles correspond to internal points; (ii) Squares correspond to edge points ; (iii) Pentagons correspond to corner points

  h l h e c b a b d g k g d b a

Remark 3 . 3 . 2 (

 332 One may wonder if the Spherical Harmonics of this latter category (the remaining 1/8 Spherical Harmonics) are effectively inexactly integrated by the rule Q a for all integers n even and m ≡ 0[START_REF] Croisille | Hermitian compact interpolation on the Cubed-Sphere grid[END_REF]. This question is unanswered for now. In fact, it happens that the function Y 0 (n, m) = (2, 0)) is exactly integrated by the rule Q a . Prop. 3.1 therefore only describes a sufficient condition for the exactness of the rule Q a .

Figure 3 .

 3 Figure 3. Quadrature error for the test functions f 1 , f 2 , f 3 and f 4 obtained with rule Q a in function of the total number 6N 2 + 2 nodes. Fourth-order accuracy is observed in the four cases. The worst case among 1000 randomly selected solid rotations on the argument (x, y, z) of the functions is displayed.

Furthermore

  the relation (23) with p = ±2m , m = 4m gives the relation (44) on the two panels k = (II) and k = (IV ). Next, (21) yields (44) for the panel k = (III). Consider now the panel k = (V ). The value Q (V ) (H n m ) can be decomposed in four terms each of them corresponding to an angular sector, in the form (45)

Figure 4 . 4 . 4 .

 444 Figure 4. Quadrature error in Log scale for the test functions f 1 , f 2 and f 3 in function of the number p N of the Spherical Harmonics taken in account in (38) defining the rule Q b .

Figure 5 .

 5 Figure 5. Maximum of the parameters |ε i,j | in (46) with respect to N , the parameter of the Cubed Sphere. The magnitude of (ε i,j ) (i,j)∈I behaves numerically as 1/N 2 .

Figure 6 .

 6 Figure 6. Integration errors in function of N , the parameter of the Cubed Sphere, for the three Spherical harmonics Y 8 16 , Y 4 24 and Y 16 40 . The LogLog scale is used. The thin curve with × symbols corresponds to the rule Q a . The bold curve with * symbols corresponds to the rule Q b .

N

  Number of nodes pN |I(f1) -Q b (f1)| |I(f2) -Q b (f2)| |I(f3) -Q b (f3)| |I(f4) -Q b (f4)|

Figure 7 .

 7 Figure 7. Quadrature error for the test functions f 1 , f 2 , f 3 and f 4 obtained with the rule Q b in function of the number 6N 2 + 2 of quadrature nodes on the sphere.

Table 2 .

 2 Number of quadrature nodes in function of the parameter N of the Cubed Sphere.

	Cubed Sphere parameter N	4	8	16	32	64
	Number of quadrature nodes in a panel (= (N + 1) 2 )	25 81	289 1089 4225
	Number of quad. nodes on S 2 (= 6N 2 + 2)	98 386 1538 6146 24578
	Number of independent weights q N = (N + 2)(N + 4)/8 6	15	45	153	561

Table 2

 2 displays typical values of the integer q N and of the number of quadrature nodes for a series of values of N . Under the assumptions (16-17) on the weights, the following statement holds:

	Proposition 3.1. The quadrature rule (14) associated with the nodes of the Cubed Sphere s k i,j and
	with a set of weights g k i,j satisfying (16-17) is exact for all Spherical Harmonics Y m n (x) such that
	• n odd,
	• n even and m not a multiple of 4.
	Proof. The Spherical Harmonic Y m n is given by (see (2)):
	(20)

We assume for simplicity that N is an even integer.