Moulay Sahaka

Sawsan Amara

Jérôme Lecomte

Jean-David Rodier

Dominique Lafont

Pierre Villeneuve

Brigitte Gontero

PhD Frédéric Carrière
email: carriere@imm.cnrs.fr

Quantitative monitoring of galactolipid hydrolysis by pancreatic lipase-related protein 2 using thin layer chromatography and thymol-sulfuric acid derivatization

Keywords: Carbohydrate, Dietary lipids, Digestion, galactolipase, Glycolipids, gastrointestinal lipolysis

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

Quantitative monitoring of galactolipid hydrolysis by pancreatic lipase-related protein 2 using thin layer chromatography and thymol-sulfuric acid derivatization

Introduction

Galactolipids are the most abundant lipids on earth, where they are mainly found in thylakoid membranes of plant, algae, and cyanobacteria chloroplasts [1][2][START_REF] Awai | Thylakoid Development and Galactolipid Synthesis in Cyanobacteria[END_REF][START_REF] Gounaris | [END_REF]. They represent an important source of fatty acids for herbivores [5] and are the main source in Nature of the essential α-linolenic acid [6]. They also have several potential applications in pharmaceutical [7][8][9] and food industries [10][11][12][13]. For some applications such as in nutrition, galactolipids in their native form are sufficient, but for other applications like biofuels and surfactants, it would be necessary to transform them via acyl transfer reactions using chemical or lipasebased catalysis [14,15]. Here, we present a method for the qualitative and quantitative analysis of galactolipids and their galactosylated degradation products, using Thin-Layer-Chromatography (TLC) coupled to scanning densitometry.

Typically, lipid quantification by TLC comprises four steps: spotting samples on silica plate, separation by elution with a suitable solvent, revealing lipid spots or bands by derivatization and finally quantification by scanning densitometry at an appropriate wavelength. Many non-specific reagents can reveal a wide range of lipids simultaneously.

Among the most used and well-known methods, one can cite sulphuric acid treatment [START_REF] Skipski | Thin-layer chromatography of lipids[END_REF][17][18][19], sulphuric acid and potassium dichromate treatment [START_REF] Skipski | Thin-layer chromatography of lipids[END_REF]20,21], phosphoric acid and copper acetate treatment [18,22], and iodine vapor treatment [START_REF] Skipski | Thin-layer chromatography of lipids[END_REF][23][24][25]. These reagents cause cleavage, oxidation or dehydration reactions on acyl chains and produce brown to black spots of elemental carbon. The main weakness of these methods is their lack of sensitivity to saturated lipids. However, concerning polar lipids like phospholipids and galactolipids, one can use reagents that target their polar head irrespective of the degree of unsaturation of their acyl chains. Phosphorus reagents such as molybdenum blue are for instance widely used for the detection of phospholipids on TLC plates [26]. Similarly, it is possible to detect galactolipids based on the reactivity of their galactosyl groups.

The thymol-sulphuric acid reagent, developed by Hans Molisch in 1886, was initially used for the detection of carbohydrates. Molisch reported that a sugar solution turns carmine red when mixed with a solution of thymol with 15-20% alcohol and excess sulphuric acid. He also reported that the mixture turns purple when α-naphthol is used instead of thymol [27]. In 1888, Udranszky showed that furfuraldehyde reacts with the thymol-sulphuric acid reagent in the same way as sugars. He concluded that in the sugar detection, the coloured compound formation does not directly involve sugar but rather furfuraldehyde resulting from the dehydration of sugar by sulphuric acid and cyclization [START_REF] Udranszky | The furfuraldehyde color reaction[END_REF]. In 1954, Schmor performed the first quantitative analysis of glucose in blood samples using the thymol-sulfuric reagent and photometer. Schmor's objective was to find an alternative to anthrone reagent because the latter cannot be kept for more than one day due to a darkening phenomenon. He reported that the thymol-sulfuric acid reagent could be stored and used for several months [29]. In 1965, Sumu Adachi adapted it for the first time to TLC in order to perform a qualitative analysis of sugars in dairy products [30]. In 1970, Bukharov and Karneeva also adapted the reagent to TLC and used scanning densitometry to make it both a qualitative and quantitative method for sugar analysis [31]. Later, this reagent started to be used for the analysis of glycosylated molecules. For instance, in 1979, Racusen adapted it for the detection of glycoproteins separated by electrophoresis in acrylamide gel [32]. In 1999, Kim et al. used the thymolsulfuric acid reagent for the qualitative analysis of yeast glycolipids separated by TLC [33].

Several other reagents composed of thymol-like chromogens and excess acid were also used for the detection of carbohydrates. As examples of these chromogens we can cite cholic acid [34], resorcinol [35], naphthoresorcinol [36], benzidine [37], anthrone [38], and phenol [39]. The mechanism of action common to all these reagents can be separated into three steps. Firstly, osidic bonds are hydrolysed by the excess acid, particularly in the case of glycosylated molecules detection. Secondly, the released sugars are dehydrated and cyclized into furfural derivatives by acid catalysis [32,40]. Thirdly, the furfural derivatives are condensed with the chromogenic reagent, typically a phenolic compound, to give a three ring derivative (Figure 1) [41].

Pancreatic lipase-related protein 2 (PLRP2) is an enzyme expressed in the pancreas of several mammals such as mouse [42], human [43], guinea pig [44], rat [45], and coypu [46].

Its peculiarity compared to other pancreatic lipases is its combination of lipase, phospholipase A1 and galactolipase activities [47]. Among various physiological roles, one is to digest plant membrane lipids, mainly composed of galactolipids [6]. Most studies on the characterization of PLRP2 galactolipase activity were based on the analysis of the fatty acids released from galactolipids by the enzyme, but very few of these studies also examined other products released upon galactolipid hydrolysis. In 1996, Anderson et al. studied the in vitro digestion of tritium-labelled galactolipids ([ 3 H]-labelled fatty acids or galactose) by recombinant rat and guinea pig PLRP2s. Using TLC and radioactivity detection, they observed and quantified substrate consumption and production of free fatty acid, lyso-galactolipid and of a watersoluble compound [48]. In 2000, Sugawara and Miyazawa investigated digestion of MGDG and DGDG in the rat alimentary canal. Using HPLC combined with ELSD detector, they identified among the reaction products lysogalactolipids and two products corresponding to the full deacylation of galactolipids, mobogalactosyl glycerol (MGG) and digalactosyl glycerol (DGG) [49]. In 2010, Amara et al analyzed the action of recombinant human guinea pig PLRP2s on purified spinach leaf galactolipids using TLC and Fewster's copper acetate detection. They observed the consumption of galactolipids and the accumulation of lysogalactolipids and free fatty acids [50].

Here we investigated the use of thymol-sulfuric acid derivatization for the qualitative and quantitative analysis by TLC of galactolipids and galactosylated products resulting from their degradation. In a first part, we studied the influence of glycerol moiety, acyl chain length and number of galactose residues on the derivatization of galactolipid galactose residues by the thymol-sulfuric acid reagent. For this, we compared the detection by scanning densitometry of free galactose and galactolipids with different acyl chain lengths and polar head. In a second part, we used this detection method for the analysis of galactosylated products resulting from the hydrolysis of monogalactosyl di-octanoylglycerol (C8-MGDG) by PLRP2. Finally, we also used it to analyze natural galactolipids from spinach. This allowed validating a quantitative but non selective method for galactolipids and products resulting from their deacylation.

Materials and methods

Reagents

Galactolipids standards, monogalactosyl di-octanoylglycerol (C8-MGDG; M w =506 g/mole), monogalactosyl di-palmitoylglycerol (C16-MGDG), and digalactosyl dioctanoylglycerol (C8-DGDG), were synthesized as previously described [51][52][53]

Production and purification of rGPLRP2

Recombinant guinea pig PLRP2 (rGPLRP2) was produced in Aspergillus orizae and purified as previously described [44].

Preparation of the thymol-sulfuric acid reagent

One gram of thymol was dissolved in 190 mL of ethanol. Ten millilitres of 96% sulfuric acid were then added gradually. The reaction being exothermic, the operation was carried out in a cold water bath.

Standards preparation

Galactolipid standards were solubilized in chloroform:methanol mixture (2/1 v:v) and free galactose standard in water:ethanol mixture (1/5 v/v) before being applied on TLC plates.

Preparation of MGDG-bile salt micelles

Monogalactosyldioctanoylglycerol (C8-MGDG) was used to form mixed micelles with NaTDC. It was added in a buffer solution containing 65 mM NaTDC, 450 mM TES, 100 mM NaCl and 5 mM CaCl 2 and then dispersed using vortex followed by ultrasonic treatment for 8 min in a water bath at room temperature. NaTDC to C8-MGDG molar ratio was 1.33.

The high buffer concentration (450 mM TES) was selected from another study in order to limit pH variations due to the release of octanoic acid during the reaction (data not shown).

MGDG lipolysis

Lipolysis reactions were carried out by adding rGPLRP2 to the C8-MGDG-NaTDC micelles solution to obtain 50 nM enzyme concentration, at pH 8 and 37 °C. The total reaction volume was 200 µL. Aliquots of 25µL were collected at different reaction times and immediately mixed with 1 mL of a stop solution containing 200 mM HCl and 150 mM NaCl.

Extraction of reaction products

To extract the reaction products and residual substrate, 1.35 mL of chloroform:methanol mixture (2/1, v/v) was added to all stopped aliquots, mixed and shaken vigorously using vortex and the organic and aqueous phases were separated by centrifugation using a Sigma 3-18K centrifuge (1 min at 1,000g). Both phases were separated in different tubes and kept at -20 °C until the TLC analysis was performed.

TLC-densitometry analysis

The organic and aqueous phases were spotted on separate TLC silica gel Differences were considered significant at p≤0.05

Results and discussion

Choice of wavelength for analysis by scanning densitometry

Absorbance measurements at a wavelength of 510 nm were initially chosen because it corresponds to the maximum absorption of thymol-derivatized galactose [54]. However, we also tested whether absorbance at 366 nm, a wavelength commonly used for other types of staining, could be used for quantifying thymol-derivatized galactolipids.

The TLC separation of C8-MGDG and its derivatization by thymol was chosen as a model system to compare the two wavelengths. As shown in Figure 2A, absorbance at 510 nm was about 3.5 times higher than the absorbance at 366 nm, right after drying and staining the TLC plate (t=0). Absorbance at 510 nm then appeared to be unstable with time and decreased gradually with 10% loss of its intensity after 40 min, and 50% after 72 hr. Absorbance at 366 nm was lower but more stable. Indeed, its intensity decreased by 15% after 2 h 40 min but then remained constant until 72h at least. After stabilization of the absorbances, the intensity at 510 nm was about 2.1 times higher than the intensity at 366 nm. The color of the band also changed with time from red to gray color. Calibration curves for C8-MGDG could be established at both wavelengths (Figure 2B), showing the higher sensitivity of measurements at 510 nm.

However, when analyzing natural lipid extracts from plant leaves, measurements of absorbances at 366 nm had the advantage of reducing the interference of colored compounds present in lipid extract on galactolipid detection, as shown by the densitograms obtained at 366 and 510 nm (Figure 3). Previous TLC analysis of lipid extracts from plant leaves containing galactolipids and photosynthetic pigments have shown that some carotenoids, absorbing light maximally between 460 nm and 550 nm, are migrating close to MGDG [12].

One can assume that absorbance measurements at 366 nm allow reducing the detection of these compounds.

Another advantage of using 366 versus 510 nm was also noticed when we analyzed the lipolysis products of C8-MGDG that are soluble in water (see next section and Figure 5B).

When hydrolysis of C8-MGDG was performed in the presence of bile salts (NaTDC), the lysogalactolipid C8-MGMG was found to migrate very close to NaTDC. In the densitogram/chromatogram obtained at 510 nm (Figure 5C), the detected peaks of NaTDC and C8-MGMG overlapped, making the integration of C8-MGMG peak less accurate. The analysis at 366 nm allowed removing NaTDC detection and peak while preserving the absorbance of C8-MGMG. Since our present and future objectives were to analyze the hydrolysis of galactolipids from various sources in the presence of bile salts, we have chosen the wavelength of 366 nm for all quantitative analysis of galactolipids. The stability of absorbance at this wavelength also allows a better repeatability of measurements.

Influence of acyl chains length and galactose residue number on quantification with thymol-sulfuric acid reagent

The derivatization of free galactose, C8-MGDG, C16-MGDG and C8-DGDG by the thymol-sulfuric acid reagent were compared. All four compounds exhibited similar carmine red-coloured bands as illustrated in Figure 1. We then investigated the variations in absorbance with the amount of these compounds (Figure 4A and Table 1). For each compound, six different amounts ranging from 0.4 to 6.4 nmoles (0.2 to 3.3 µg for C8-MGDG) were spotted on TLC plates and revealed by derivatization after elution. The absorbances obtained with free galactose were on average 25 % higher than those obtained with MGDGs. This deviation however increased with the amounts of compounds whereas at the lowest amount tested (0.4 nmoles) absorbances for galactose and MGDGs were very close. The colour of the bands being identical, one can raise the hypothesis that the acidcatalyzed deglycosylation step [32] releasing galactose from MGDG is not complete during the derivatization process under the conditions used, giving less dye than with free galactose.

Therefore, the use of free galactose as a calibration standard for the quantification of galactolipids does not seem possible.

The calibration curves obtained with C8-MGDG and C16-MGDG are similar (Figure 4A and Table 1) with a normalized root-mean-square deviation of 10%. On average, the difference in absorbance of their derivatized compounds is not significant (Student's test, P>0.05) for five of the six amounts tested. Therefore, all derivatization steps including the deglycosylation appear identical for both MGDGs. The length of the saturated acyl chains does not affect significantly the yield of derivatization.

The absorbances obtained with C8-DGDG were increased two-fold compared to those obtained with MGDGs (Figures 4A andB), with a normalized root-mean-square deviation of 98%. We therefore assume that the acid-catalyzed deglycosylation of C8-DGDG occurs in the same proportion as with MGDGs and that the digalactosyl polar head is further hydrolysed into two molecules of galactose, resulting in identical dyes but twice the amounts obtained with MGDGs.

Qualitative analysis of MGDG enzymatic hydrolysis

C8-MGDG-NaTDC mixed micelles were incubated with rGPLRP2. Residual substrate and galactosylated compounds produced during the reaction were extracted, separated and quantified from samples collected in the course of the reaction. The organic and aqueous phases resulting from the extraction were separated and eluted on TLC plates with two different solvent mixtures consisting of chloroform/methanol/water (25/15/1.25, v/v/v) for compounds soluble in the organic phase samples and acetonitrile/water (85/15, v/v) for compounds present in the aqueous phase. In both cases, the separated compounds were derivatized with the thymol-sulfuric acid reagent. On TLC plates presented in Figure 5, the same extract volume was applied for all reaction times to follow the reaction progress.

Three thin bands were detected from the development of organic phase samples (Figure 5A). The middle band decreasing with time corresponds to residual C8-MGDG, as indicated by the reference standards present on the right side of the same plate. The faint lower band below C8-MGDG corresponds to a less hydrophobic compound increasing with time that we attributed to C8-MGMG. The faint upper band above C8-MGDG corresponds to a more hydrophobic compound, presumably containing additional acyl chains compared to C8-MGDG. We assume it is an acyl-MGDG, formed by the addition of one acyl chain to the primary hydroxyl group of MGDG galactosyl group (Figure 6). This type of acylation transfer reaction on galactolipids in the course of lipolysis was previously observed when using fungal and bacterial lipases in organic media [15,55,56] but not yet with rGPLRP2.

Three main bands were also detected from the development of the aqueous phase samples (Figure 5B). The lowest fuzzy band corresponding to the most hydrophilic product was coloured in red and its retention factor (Rf) was lower than that of free galactose (data not shown). It corresponds to the water-soluble compound previously observed by Andersson. et al using radiolabeled compounds [48] and we attributed this band to MGG. We have then confirmed this attribution using NMR (to be published). The fuzzy intermediate band above MGG had a green color, indicating that it did not contain galactose. Its intensity did not change during the reaction and it was attributed to NaTDC used to prepare mixed micelles. As mentioned in the previous section, the presence of NaTDC band close to that of C8-MGMG could interfere with the detection of this latter at 510 nm but densitometry at 366 nm allowed reducing this interference (Figure 5C). The last upper band had a red color and thus it corresponds to a compound containing galactose. Using organic phase samples eluted with the same polar solvent mixture, the retention factor (Rf) corresponding to this band was identical to that of C8-MGMG (data not shown). C8-MGMG thus partitioned between organic and aqueous phases. However, its quantification in both phases allowed to estimate that 96% of C8-MGMG remains in the aqueous phase after extraction. The higher affinity of C8-MGMG for the aqueous phase than for the organic phase probably results from its high hydrophilic/lipophilic balance (HLB=15.58). Nevertheless, the micellar solubilisation of C8-MGMG by NaTDC may also contribute to its larger partitioning towards the aqueous phase.

When analyzing reaction products with such a high HLB, one has to check their recovery upon extraction by organic solvents. However, we previously observed that most MGMG produced from the hydrolysis of natural long-chain galactolipids can be recovered in the organic phase, with a good stoichiometry with respect to MGDG hydrolysis [12,50].

Quantitative analysis of MGDG enzymatic hydrolysis

All TLC detected galactosylated products were quantified using the absorbances at 366 nm of their derivatives formed upon acid-catalyzed reaction with thymol and a calibration curve established with various amounts of C8-MGDG (Figure 4), loaded on each TLC plate.

The volumes of organic and aqueous phases loaded on TLC plates were however adapted for the quantification. Several volumes/amounts of extract were tested and applied for each reaction time in order to obtain for all compounds an absorbance corresponding to the central part of the C8-MGDG calibration curve, corresponding to around 3 nmoles of compound loaded on the plate (1.5 µg for C8-MGDG). This allowed quantifying residual substrate and galactosylated products in the course of the reaction catalyzed by rGPLRP2 during two hours (Figure 7).

The quantification of C8-MGDG, C8-MGMG, acyl-MGDG and MGG also allowed estimating the conservation of total galactose, as well as the variation in the concentration of free octanoic acid released during the reaction using the following equations ( 1) and ( 2).

(

) 1
(2)

C8-MGDG was quickly hydrolyzed and had almost completely disappeared after 40 min of reaction. C8-MGMG, estimated from the sum of the amounts recovered in the aqueous and organic phases, initially accumulated up to an optimum at around 15 min and then slowly decreased over time (Figure 5). Andersson et al observed similar variations using radiolabelled galactolipids [48], while Amara et al did not observe MGMG consumption during the hydrolysis of natural galactolipids by rGPLRP2 [50]. Nevertheless, it should be noted that reaction was monitored for only 30 minutes in this latter case. One can deduce from these previous findings and our observations that rGPLRP2 can rapidly release one acyl chain out of the two present in galactolipids. Previous studies have shown that rGPLRP2 has a regioselective preference for the ester bond at the sn-1 position of glycerol in polar lipids [47,57]. Withers-Martinez et al have modelled a molecule of DGDG in the active site of the known 3D structure of a GPLRP2 chimera and the most favourable steric organization suggested that the ester bond at sn-2 position unlike the ester bond at sn-1 position could not be hydrolyzed by GPLRP2 [58]. More recent studies performed with synthetic acylglycerol substrates bearing non-hydrolysable bonds at sn-1 or sn-2 positions have however shown that rGPLRP2 can hydrolyze the ester bond at sn-2 position at a rate that is 10-fold lower compared to the hydrolysis of the ester bond at sn-1 position [59]. It is not known yet whether GPLRP2 displays a similar regioselective preference on galactolipids, but two hypotheses can be raised concerning the slow hydrolysis of C8-MGMG and the production of MGG observed during the enzymatic hydrolysis of C8-MGDG by rGPLRP2. After the rapid cleavage of the ester bond at sn-1 position of C8-MGDG, rGPLRP2 would either act directly on the generated 2-sn-MGMG or alternatively, an intramolecular acyl-chain migration from the sn-2 to sn-1 position could occur, thus generating 1-sn-MGMG that would be further cleaved by rGPLRP2 (Figure 6). Such an isomerisation could be favoured at pH 8 and has been shown in the case of lysophospholipids, thus explaining how a phospholipase A1 (PLA1) could release the two fatty acids present in a phospholipid molecule [60]. It is also worth noticing that a similar mechanism of sn-2 to sn-1 acyl chain migration was recently proposed to explain the role of PLRP2 in brain phospholipid remodelling [61].

Low amounts of acyl-MGDG (concentration < 1 mM) also initially accumulated and rapidly disappeared (Figure 7B). This was concomitant with the highest rate of octanoic acid release by rGPLRP2 at the beginning of the reaction and one can envision that the trapping of both C8-MGDG and octanoic acid in mixed micelles might favour the acyl-transferase activity of rGPLRP2, as observed with other lipases in organic media [15,55,56]. The transient production of acyl-MGDG by GPLRP2 was not detected or reported in previous studies.

MGG accumulated all along the reaction (Figure 7A) in the same way as the watersoluble product observed by Anderson et al [48]. The estimated total galactose concentration appeared to be constant all along the reaction (Figure 5A). A one-way ANOVA test on total galactose concentrations did not give any significant difference (P>0.05) between the different reaction times. Besides confirming the stoichiometry of the reaction based on product analysis, this conservation of galactose confirms that galactosyl group is revealed with the same intensity in C8-MGDG, C8-MGMG, MGG and acyl-MGDG, and by the way confirms that the number and position of acyl chains does not affect the identification of galactolipids by the thymol and sulfuric acid reagent.

The reaction scheme in Figure 6 takes into account all galactosylated products observed and quantified by TLC during the hydrolysis of C8-MGDG by rGPLRP2, considering that this enzyme preferentially hydrolyses ester bonds at the sn-1 position of glycerol or transiently formed on the galactosyl group. We did not envision the possibility that rGPLRP2 could also acylate galactosyl groups in MGMG and MGG to form acyl-MGMG and acyl-MGG, because no additional bands were visible on the TLC plates. Moreover these products, if produced, could migrate with the same Rfs as MGDG and MGMG, respectively, because of having equal numbers of acyl chains. The product variations with time shown in Figure 7 support this reaction scheme with very good stoichiometries. The amounts of octanoic acid released during the reaction could therefore be estimated with a good accuracy from the quantification of C8-MGMG and MGG. The specific activity of rGPLRP2 on C8-MGDG, expressed in µmoles of fatty acid released per min and per mg of enzyme (U/mg), was estimated from the release of octanoic acid during the first 10 minutes of the reaction.

The value obtained (1945 ± 116 U/mg; n=3) is in the same order of magnitude but 2.8-fold lower than the specific activity of rGPLRP2 reported by Amara et al (5420 ± 85) using the same substrate [50]. In this previous study, the activity was however deduced from the continuous titration of free fatty acids with the pH-stat technique. Stirring was more vigourous under these conditions and the fact that fatty acids were continuously neutralized to keep pH constant had certainly an impact on the reaction rate. Here, a high concentration of TES buffer (450 mM) was used to prevent large pH changes during the reaction but this may have also impacted the reaction rate.

Application to the analysis of natural galactolipids from plant leaves

TLC separation coupled to derivatization by thymol-sulfuric acid reagent and scanning densitometry was also applied to the quantitative analysis of natural galactolipids present in chloroplasts prepared from spinach leaves and galactosylated lipolysis products resulting from digestion by human pancreatic juice under conditions mimicking those found in the upper small intestine (Figure 8A). In the study previously published by Wattanakul et al., the respective amounts of MGDG and DGDG were estimated from calibration curves established with purified MGDG and DGDG as reference standards [12]. Here, we used only the calibration curve established with MGDG to re-estimate the amounts of MGDG and DGDG, as well as those of the lipolysis products MGMG and DGMG obtained after 90 minutes of incubation (Figure 8B). From the individual concentrations of all galactosylated products, we could estimate the equivalent concentration of galactose (1 galactose in MGDG and MGMG, 2 in DGDG and DGMG) before (t=0) and after digestion (t=90 min) and found that it was conserved (Figure 8B). It therefore seems that the derivatization of galactose by the thymolsulfuric acid reagent is similar in natural MGDG and DGDG, as observed with synthetic medium chain (C8) galactolipids (Figure 4A). Moreover, the quantification of MGMG and DGMG, leading to a conservation of galactose, suggests that this reaction is also similar in lysogalactolipids and is thus independent of the number of acyl chains on the glycerol backbone. The conservation of galactose in products recovered from the organic phase after lipid extraction indicates that no significant amounts of MGG nor DGG were formed under these conditions, This is in agreement with the amounts of fatty released in the course of this digestion experiments and previously reported [12]. We assume it is due to a slower lipolysis of natural long chain galactolipids under these conditions compared to C8-MGDG lipolysis by GPLRP2, as well as to a too short incubation time compared to digestion in the GI tract [49]. Nevertheless, these results pave the way to further quantitative analysis of natural galactolipids from various plants and vegetables using TLC and a simple derivatization method.

Conclusion

Although the comparison of galactolipids and galactose derivatization by the thymolsulphuric acid reagent indicated that approximately 25% of galactosyl groups of galactolipids were not revealed by the derivatization reaction, further analysis on the comparison of C8-MGDG, C16-MGDG, C8-DGDG and hydrolysis products of C8-MGDG showed that the number and length of acyl chains, as well as the number of galactose residues in the polar head, do not affect significantly the derivatization of galactolipids by the thymol-sulfuric reagent. A single calibration curve established with C8-MGDG could therefore be used to quantify both the galactolipid substrate and all galactosylated reaction products (C8-MGMG and MGG) released in the course of C8-MGDG micelles incubation with rGPLRP2. Thus, a simple derivatization procedure based on Molisch's reagent for carbohydrates allowed developing a quantitative method for the analysis of galactolipids by TLC coupled to scanning densitometry. It is a very accessible method for the analysis of the most abundant acylglycerols on earth and for studying their digestion. Based on this method, we have shown that PRLP2 can release the two fatty acids present in MGDG. The overall reaction involving an intramolecular acyl chain migration prior to the release of the second chain remains to be investigated in more details, but these findings show that PLRP2 can contribute to the complete digestion of galactolipids and further intestinal absorption of their fatty acids.

PLRP2 could thus be the digestive enzyme that allows the full conversion of dietary galactolipids (MGDG and DGDG) into MGG and DGG, as observed in the distal part of the small intestine in rats [49]. Production of acyl-MGDG was also observed, meaning that rGPLRP2 like some microbial lipases can acylate the galactosyl group of galactolipids.

PLRP2 has therefore the capacity to both acylate and deacylate sugars as previously shown upon hydrolysis of mycobacterial glycolipid antigens, phosphatidylmyo-inositol mannosides (PIM) [62]. after hydrolysis (t=90min) by human pancreatic juice. The reaction was performed at pH 6 and 37 °C as described in [12]. From human pancreatic juice composition [63], it was estimated that the digestion mixture contained 250, 31 and 47 µg/mL (or 4.9, 0.6 and 0.5 µM) of human pancreatic lipase, PLRP2 and carboxylester hydrolase/bile salt-stimulated lipase, respectively, these two latter enzymes displaying galactolipase activities [63]. Panel A shows the TLC separation of compounds extracted using Folch method and derivatized using thymol-sulfuric acid reagent. Panel B shows the concentrations of galactolipids (MGDG, DGDG) and their lipolysis products (MGMG, DGMG) deduced from TLC scanning densitometry at 366 nm, as well as the sum of all these molecular species that showing the conservation of galactose between times 0 and 90 min. It indicates that no MGG or DGG were formed under these conditions.

  Moulay Sahaka, Sawsan Amara, Jérôme Lecomte, Jean-David Rodier, Dominique Lafont, Pierre Villeneuve, Brigitte Gontero, Frédéric Carrière To cite this version: Moulay Sahaka, Sawsan Amara, Jérôme Lecomte, Jean-David Rodier, Dominique Lafont, et al.. Quantitative monitoring of galactolipid hydrolysis by pancreatic lipase-related protein 2 using thin layer chromatography and thymol-sulfuric acid derivatization. Journal of Chromatography B Biomedical Sciences and Applications, 2021, 1173, pp.122674. 10.1016/j.jchromb.2021.122674. hal-03191084

Figure 1 :

 1 Figure 1: Scheme of the reaction of thymol-sulfuric acid reagent with carbohydrates, based

Figure 2 :

 2 Figure 2: Comparison of C8-MGDG absorbances at 510 nm and 366 nm after separation by TLC and derivatization with thymol. Panel A: Variation with time in the absorbances of C8-MGDG TLC band at 510 nm and 366 nm. Scanning densitometry was performed immediately after derivatization and plate drying, and was repeated at various times for 72 hours. Inserts show the color of the C8-MGDG band at times 0 and 20 min. Panel B: Calibration curves obtained from the separation on TLC plate of various amounts of C8-MGDG and scanning densitometry at either 510 nm or 366 nm.

Figure 3 :

 3 Figure 3: Chromatograms of polar lipids from spinach leave chloroplasts separated by TLC and revealed by derivatization with thymol. Chromatograms were obtained by scanning densitometry at either 510 nm or 366 nm. The scanned TLC lane is shown below the chromatograms, with arrows indicating the location of DGDG and MGDG bands. The chromatogram peaks corresponding to these compounds are colored in pink.

Figure 4 :

 4 Figure 4: Calibration curves established with free galactose, C8-MGDG, C16-MGDG and C8-DGDG (Panel A). Absorbances were measured at 366 nm after migration of various amounts (nmoles) of these compounds on TLC plates and derivatization using the thymolsulfuric acid reagent. Absorbances values are means ± SD (n=3). Panel B shows the C8-DGDG to C8-MGDG absorbance ratio as a function of the amounts (nmoles) of compounds.

Figure 6 :

 6 Figure 6: Reaction scheme for the hydrolysis of C8-MGDG catalyzed by rGPLRP2 at pH 8 and 37°C. Acyl-MGDG results from the rGPLRP2-catalyzed acylation of the galactose in C8-MGDG. 1-sn-MGMG and 2-sn-MGMG are monogalactosyl monooctanoylglycerol with one acyl chain at positions sn-1 and sn-2 of the glycerol backbone, respectively. The mechanism of intramolecular acyl chain migration catalyzed by a water molecule is indicated.

Figure 7 :

 7 Figure 7: Time-course variation in residual substrate and reaction products during the hydrolysis of C8-MGDG-NaTDC micelles by rGPLRP2. Reaction was performed at pH 8 and 37 °C. Enzyme concentration was 50 nM. Concentration values (mM) are means ± SD (n=3).

Figure 8 :

 8 Figure 8: TLC analysis of galactolipids from chloroplasts of spinach leaves before (t=0) and

Table 1 :

 1 Calibration curves obtained after fitting the variations in absorbance at 366 nm (y) with the amounts of galactolipids (x). A 4-parameter logistic regression appeared as the mostsuitable to fit the experimental data and the parameters were estimated using the following

	equation: y = d + [(a-d)/(1+(x/c) b ] or x = c[(a-d)/(y-d)-1] 1/b .
	Compound a	b	c	d	R 2
	Galactose	2933	1.31	2.17	1.19×10 5	0.9724
	C8-MGDG 6501	0.87	8.051	2.95×10 10	0.9884
	C16-MGDG 8414	0.81	8.50×10 7	2.08×10 5	0.9608
	C8-DGDG	19756	1.27	3.815	1.45×10 5	0.9888

Acknowledgements

This work received the financial support of Agence Nationale de la Recherche (France) in the framework of the GALACTOLIPASE project (ANR-09-CP2D-06-01). The PhD thesis of Moulay Sahaka was supported by a fellowship from Aix Marseille Université, France. We are grateful to Nathalie Barouh (CIRAD, Montpellier, France), Magali Arnaud and Nicolas Ritter (Gattefossé SAS, St Priest, France) for their technical assistance, as well as to Jutarat Wattanakul (Rajamangala University of Technology Krung Thep, Department of Food Science and Technology, Thailand) and David Gray (University of Nottingham, School of Biosciences, UK) for their fruitfull collaboration regarding the digestion of spinach chloroplast galactolipids.


F Salmon
email: salmon.fabien@yahoo.com

Relativity and quantum mechanics linked by two dimensionless numbers of order 10 1

Keywords: universe, vacuum, fundamental constants, speed of light, gravitation, pilot wave theory, theory of everything

The contemporary physical theories are based on fundamental physical constants. No theoretical framework provides their experimental values. Here, we present two formulas yielding the speed of light and the Planck constant according to the same vacuum properties and fundamental constants. In particular, they highly suggest that gravitational and electromagnetic waves must actually propagate in matter. Because there is no dense matter between high-mass bodies in the universe, the relations would mean that the spatial structure of the universe is at least fourdimensional. We then discuss the implications of this hypothetical feature on contemporary physics.

Introduction

In the mid-17th century, Isaac Newton published his law of gravitation in Philosophiae naturalis principia mathematica. Classical mechanics remained the mainstay of science until the end of the nineteenth century. The electromagnetism theory suggested that light had to propagate in a medium named luminiferous aether by scientists. In the 1880's, while most of them were confident in its existence, A. Michelson and E. Morley [START_REF] Michelson | On the Relative Motion of the Earth and the Luminiferous Ether[END_REF] carried out several experiments that challenged this hypothesis. In particular, they did not measure any "aether wind". Classical mechanics failed to explain this result and more and more scientists gave up the aether concept. It was the case of A. Einstein who contributed to the special and general theory of relativity with M. Grossmann, D. Hilbert, H. Lorentz and H. Poincaré (for the most famous ones). These theories did not need a tangible material such as the aether medium and made possible the observation of new physical phenomena. Based on redshift, the discovery of the expansion of the universe is one example [START_REF] Slipher | The Radial Velocity of the Andromeda Nebula[END_REF][START_REF] Friedman | Über die Krümmung des Raumes[END_REF]. Lemaître [START_REF] Lemaître | Un Univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques[END_REF] and then Hubble [START_REF] Hubble | A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae[END_REF] independently formulated a law that states that galaxies move away from each other at a speed proportional to their distance. At the same time, the development of quantum mechanics turned the perception of particles upside down.

Many experimenters carried out tests to challenge these theories. For now, special relativity and quantum mechanics passed these tests. General relativity could also be in accordance with observations. The perihelion precession of Mercury [START_REF] Einstein | The Foundation of the General Theory of Relativity[END_REF], the bending of light by the Sun [START_REF] Fomalont | Progress in measurements of the gravitational bending of radio waves using the vlba[END_REF][START_REF] Lebach | Measurement of the solar gravitational deflection of radio waves using very-long-baseline interferometry[END_REF][START_REF] Shapiro | Measurement of the solar gravitational deflection of radio waves using geodetic very-long-baseline interferometry data, 1979-1999[END_REF], gravitational redshift [START_REF] Holberg | Sirius B and the Measurement of the Gravitational Redshift[END_REF] and gravitational waves [START_REF] Weisberg | Gravitational waves from an orbiting pulsar[END_REF][START_REF] Taylor | A new test of general relativity -Gravitational radiation and the binary pulsar PSR 1913+16[END_REF] are successes of general relativity.

However, this theory needs dark matter [START_REF] Trimble | Existence and nature of dark matter in the Universe[END_REF] to agree with the galaxy rotation curves for instance [START_REF] Corbelli | The extended rotation curve and the dark matter halo of M33[END_REF]. Up to now, this hypothetical matter has never been observed. The accelerating expansion of the universe is another issue which is explained by dark energy [START_REF] Peebles | The cosmological constant and dark energy[END_REF]. This can be embedded in the equations of general relativity but its origin remains unknown. Linking all the physical processes, from the infinitely small (quantum mechanics) to the infinitely large (general relativity), remains an unsolved problem as well. Finally, the value of the fundamental physical constants on which contemporary physics is based, is not explained yet. Some efforts are being conducted to understand these physical problems. Scientists are trying to detect dark matter where it is supposed to be. Others are focusing on Modified Newtonian dynamics (MOND [START_REF] Milgrom | A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis[END_REF]) which corresponds to a theory where Newton's laws are modified to fit with the properties of galaxies. Theoretical physicists are also developing ideas to couple gravitation and the three other fundamental interactions (electromagnetic, weak and strong interactions). Such theories are named "theories of everything". The two most famous ones correspond to string theory and loop quantum gravity. Like in string theory, extra dimensions are thoroughly considered in many other theories. The first attempt consisted in Kaluza-Klein theory which adds a fifth dimension to spacetime [START_REF] Kaluza | Zum Unitätsproblem in der Physik[END_REF][START_REF] Klein | Quantentheorie und fünfdimensionale Relativitätstheorie[END_REF]. Similarly, a fifth dimension is considered in Randall-Sundrum models [START_REF] Randall | Large Mass Hierarchy from a Small Extra Dimension[END_REF][START_REF] Randall | An Alternative to Compactification[END_REF].

In this paper, we first present two relations which link some fundamental physical constants with vacuum properties of the universe. These formula, which derive from a dimensional analysis, highly suggest that gravitational and light waves propagate in matter, out of our field of view. Then, under the assumption that both relations do not result from two coincidences, we discuss about their consequences. In that respect, gravitation would actually correspond to the deformation of matter. A fourth spatial dimension would be necessary to explain the existence of this material. We then focus on the interpretation of some contemporary physical phenomena in the framework of the developed theory, especially quantum mechanics. Finally, we touch upon the variation of some fundamental constants and the fine-structure constant.

Results

The purpose of this section is to find a relation giving the speed of light. Special and general relativity theories assert that gravitational and light waves propagate in vacuum. However, common sense could make us think, like nineteenth-century scientists, that waves need a material to propagate. Despite the apparent demonstration that it is false (see introduction), we will still trust in this common sense and assume that waves do not travel in vacuum but in some hypothetical continuous matter.

One can observe that the speed of material waves is often related to the pressure and density of the medium by 𝑐 𝑤𝑎𝑣𝑒𝑠 ∝ √ 𝑃 𝜌 . Perfect gases, polytropes, incompressible liquids or elastic solids obey this relation (the pressure term is replaced by Young's modulus for a solid). If the continuum hypothesis holds for the universe, the velocity of gravitational and light waves could be given by a similar relation. For the calculation, we can use the critical density (vacuum density) 𝜌 𝑐 = 3𝐻 2 8𝜋𝐺 ∼ 9.2 × 10 -27 𝑘𝑔. 𝑚 -3 (with 𝐻 the Hubble constant and 𝐺 the gravitational constant). Considering that many measurements do not agree on the value of the Hubble constant, we assumed 𝐻 ∼ 70 𝑘𝑚. 𝑠 -1 . 𝑀𝑃𝑐 -1 because this corresponds to an average of the observations [START_REF] Riess | Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation for the Determination of the Hubble Constant and Stronger Evidence for Physics Beyond LambdaCDM[END_REF][START_REF]Planck 2013 results. I. Overview of products and scientific results[END_REF][START_REF] Hotokezaka | A Hubble constant measurement from superluminal motion of the jet in GW170817[END_REF][START_REF] Freedman | The Carnegie-Chicago Hubble Program. {VIII}. An Independent Determination of the Hubble Constant Based on the Tip of the Red Giant Branch[END_REF]. No known parameter can directly be selected for the pressure term 𝑃. We will thus try to find a parameter with the dimension of a pressure from the vacuum properties and physical fundamental constants. Tab. 1 Fundamental physical constants and vacuum properties [START_REF] Mohr | CODATA Recommended Values of the Fundamental Physical Constants[END_REF].

The temperature and the density should be selected since the calculation of a pressure often depends on both variables (perfect gas, liquid, ...). The vacuum permittivity is, by definition, a vacuum property, so it should participate. In the dimensional analysis, the electric charge of the vacuum permittivity can only be balanced by the elementary charge. Necessarily, both parameters will be associated such as 𝜀 0 𝑒 -2 . The same argument is applied to the temperature. So, the temperature and the Boltzmann constant are embedded in one variable 𝑘 𝐵 𝑇. The Planck constant takes part to the study because pressure depends on microscopic effects.

We can now manage the dimensional analysis with the following parameters: 𝑃, 𝜌 𝑐 , 𝜀 0 𝑒 -2 , 𝑘 𝐵 𝑇and ℎ. There are five physical quantities and three physical dimensions (length, time and mass). According to the Buckingham 𝜋 theorem, two dimensionless numbers can be constructed. The method consists in the arbitrary separation of the quantities into two groups. We consider both dimensionless numbers 𝛱 1 = 𝜌 𝑐 𝛼 (𝜀 0 𝑒 -2 ) 𝛽 (𝑘 𝐵 𝑇) 𝛾 𝑃 and 𝛱 2 = 𝜌 𝑐 𝛿 (𝜀 0 𝑒 -2 ) 𝜁 (𝑘 𝐵 𝑇) 𝜂 ℎ. The resolution of the linear systems yields

𝛱 1 = 𝑃 (𝜀 0 𝑒 -2 ) 3 (𝑘 𝐵 𝑇) 4
(1)

𝛱 2 = (𝜀 0 𝑒 -2 ) 5 2 ⁄ (𝑘 𝐵 𝑇) 2 ℎ √𝜌 𝑐 (2)
Relation (1) allows us to calculate √ 𝑃 𝜌 𝑐 and compare it with the speed of light. With 𝛱 1 = 10,

√ 𝑃 𝜌 = √10 (𝜀 0 𝑒 -2 ) 3 (𝑘 𝐵 𝑇) 4 𝜌 𝑐 ∼ 3.0 × 10 8 𝑚. 𝑠 -1 (3)
The inaccuracy on the Hubble constant prevents certifying that 𝛱 1 is exactly equal to ten which corresponds to 𝐻 ∼ 70.1 𝑘𝑚. 𝑠 -1 . 𝑀𝑃𝑐 -1 but its value should be between 8.5 and 11.5. The calculation of the second dimensionless number yields 𝛱 2 ∼ 21.7. Again, the uncertainties on the Hubble constant only allow us to know the order of magnitude of this number.

One can wonder whether pure chance could build two dimensionless numbers of order 10 1 .

Given the great discrepancy between all the involved constants and properties (Tab. 1), one can check that any other dimensionless combinations of these variables always lead to huge or tiny orders of magnitude. The probability that two such relations between independent physical variables exist is almost zero. Like the fine-structure constant, these dimensionless numbers therefore mean that there are underlying phenomena. In the following, we thus assume that these relations are not two coincidences and discuss their possible implications.

Discussion

A tangible fourth spatial dimension

Before proceeding, we make some remarks about the current state of the art of gravitation. General relativity is based on the curvature of spacetime due to the presence of matter. Since human beings cannot depict the deformation of a three-dimensional space, literature often describes it through an analogy with the deformation of a two-dimensional space into a threedimensional one (Fig. 1). But this only corresponds to an illustration of the three-dimensional space warp in a four-dimensional one.

Fig. 1 Schematic representations of space deformation.

Left: With a two-dimensional analogy for space but not for the star (general relativity). Right: With the analogy for both (more consistent).

Besides, the universe's expansion shows that the observable universe does not contain any point that could be considered as the big-bang origin. Indeed, the Hubble-Lemaître law states that all points move away from Earth and there is no reason for our planet to be the center of the universe. To understand this process, scientists again use an analogy with two-dimensional spaces. For instance, if the curvature of the universe is positive, our living space can be understood as the surface of an inflating ball. All the points of the surface of an inflating ball move away from each other while the center does not belong to this surface. By extending this analogy to three dimensions, our living space would be a 3-sphere (sphere in three dimensions) embedded into a four-dimensional space. This was already Einstein's feeling after the publication of general relativity. He considered that the universe was a static 3-sphere [START_REF] Einstein | Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie[END_REF].

The relation 𝑐 𝑤𝑎𝑣𝑒𝑠 ∝ √ 𝑃 𝜌 derives from continuum mechanics. However, the critical density is too low to assume that vacuum is continuous so this relation should provide an aberrant result. Yet, relation [START_REF] Friedman | Über die Krümmung des Raumes[END_REF] gives the value of the speed of light with a density of a few atoms per cubic meter. How is it possible? One way of overcoming this argument is to add at least one spatial dimension to the universe. If we assume that our world seems to be in three dimensions but presents a fourth one, then the pressure 𝑃 should be given in 𝑁. 𝑚 -3 and the density in 𝑘𝑔. 𝑚 -4 . The density in 𝑘𝑔. 𝑚 -4 would then be given by

𝜌 𝑐 𝐿
with 𝐿 the thickness of our "threedimensional" home (in the direction of the fourth dimension). A very weak value of 𝐿 could lead to a great four-dimensional density (same argument for the pressure) and make continuum mechanics applicable. This would not change the proportional relation giving the speed of waves since only the ratio of both variables intervenes in this relation. Therefore, relation (3) would entail that the spatial dimension of the universe is at least four. The existence of a fourth spatial dimension could involve the reality of the analogy presented at the beginning of this section. Like Einstein's feeling, our living space would be the surface of a 4-ball which corresponds to a 3-sphere but the 4-ball would contain matter. The propagation of waves in vacuum would be a four-dimensional phenomenon. The surface would actually be a very thin volume (in four dimensions) so that we underestimate the number of dimensions.

The nature of gravitation

According to general relativity, gravitation corresponds to the deformation of spacetime induced by masses. Einsteins's theory does not concern the physical process of this deformation. But the presence of matter in the fourth dimension would seem to require the deformation to result from a physical phenomenon. According to the strong equivalence principle, gravitation corresponds to acceleration. In adhering to this principle, the force applied on high-mass bodies, which produces the deformation of the surface, must derive from an acceleration. The observations show that the expansion of the universe is accelerating. Gravitation might then be explained by the following process: due to the acceleration of the expansion of the universe, the high-mass bodies in the 3-sphere would be maintained against some matter that is located in the fourth dimension, out of our field of view. Then, bodies would locally deform the three-dimensional surface because of the acceleration. This deformation would thus stem from the same process as the deformation of a material supporting a mass on Earth (or in an accelerated frame of reference). The motion of a body in a gravitational field would correspond to the motion of a body, which experiences a vertical acceleration, on a deformed surface. In this theory, it is worth noting that gravitation would only exist in the 3-sphere and the four-dimensional material would not be affected by gravitation. Figure 2 displays this vision of the universe.

Fig. 2 Pentadimensional structure of the universe (four spatial and one temporal dimensions).

It is assumed that the space curvature is positive. On the left: The universe would be a four-dimensional ball so its expansion would correspond to a radial acceleration 𝑔. On the right: Physical deformation of the surface due to the radial acceleration. A light body would only experience a part of it, approximately 𝑎 = 𝐺𝑀 𝑟 2 according to Newton's law of universal gravitation.

Interpretation of some physical phenomena

This way of understanding the universe might explain several physical matters. First, the universe's expansion would be the result of some processes inside the four-dimensional ball. For instance, the interaction between particles could generate this expansion. This energy could correspond to the so-called dark energy. Its location, out of our three-dimensional world, could explain its mysterious origin. The asymmetry of matter and antimatter might also originate from this extra dimension.

Second, a physical explanation of gravitation is provided. With this definition, gravitation would not be a fundamental interaction. It would result from the acceleration of the universe's expansion and the other interactions. In particular, the bodies at the surface would interact with the particles in the fourth dimension. At macroscopic scale, this process would correspond to a support reaction. Gravitation would thus be intimately linked with electromagnetism.

Third, some effects predicted by general relativity could also be interpreted in the framework of this theory. The loss of energy due to gravitational waves could be the consequence of friction with the matter in the fourth dimension. It is worth noting that friction, to a certain extent, also exists in the current scientific models. The observations show that spacetime is expanded because all bodies are moving away from each other. Without friction between bodies and spacetime, the former would slide on the latter and would not follow its expansion. The Lense-Thirring effect could stem from the dragging of four-dimensional matter due to the rotation of high-mass bodies and friction.

Incompatibility with relativity?

Contrary to aether theories, we do not consider additional matter between us and stars radiating light. The propagation of waves is not achieved in a three-dimensional material but in a fourdimensional one that has the same density as the apparent 3D "vacuum". However, like these theories, there is an absolute frame of reference. This contradicts Einstein's feeling about the isotropy of the speed of light. Yet, in addition to relation [START_REF] Friedman | Über die Krümmung des Raumes[END_REF], it is well-known that vacuum presents some continuous properties similar to materials such as a non-zero permittivity or a stiffness to some extent (according to general relativity). Besides, Einstein admitted that general relativity required the existence of a particular form of aether without mechanical properties [START_REF] Einstein | Äther und Relativitätstheorie[END_REF]. Then, is it safe to conclude that a four-dimensional absolute frame of reference cannot exist?

Based on the isotropy of the speed of light, special relativity explains time dilation and length contraction. Can we ensure that it is not the other way around? Could time dilation and length contraction make the speed of light isotropic from our point of view? The isotropy would just stem from the nature of spacetime. Spacetime would be intimately linked to the properties of the four-dimensional material: speed of material waves (special relativity) and topology (general relativity). Then, from the absolute frame of reference, the speed of light would not be a fundamental constant (like material waves), but from the three-dimensional surface, time dilation and length contraction would always make us measure the same speed.

The relativity theories would then only be applicable in our three-dimensional visible world, where there is no apparent absolute frame of reference (3-sphere), but not outside it. The nature of time would thus be different in the four-dimensional material where there may be no speed limit. This could authorize information to travel infinitely fast within the material. This hypothetical interpretation authorizes light to propagate in matter without disagreeing with the isotropy of the speed of light on which relativity theories are based. In addition, it could explain the incompatibility between relativity and quantum mechanics regarding speed limits.

Quantum mechanics

A fourth dimension containing matter could lead to another interpretation of quantum mechanics. If gravitation results from a support reaction, this means that we are always subject to interactions with particles from the fourth dimension. Therefore, the apparent chance, on which quantum mechanics is based, could correspond to these interactions (which we cannot observe). Thus, from the three-dimensional surface point of view, a probabilistic theory would be needed to describe particles but not big enough bodies. Relation (2) could substantiate this point of view because it might be understood as

ℎ = 𝛱 2 √𝜌 𝑐 (𝜀 0 𝑒 -2 ) 5 2 ⁄ (𝑘 𝐵 𝑇) 2 (4)
In this case, the Planck constant (and quantum mechanics) would depend on the vacuum properties.

A fourth dimension containing matter where the propagation of waves is possible might also give an explanation of wave-particle duality. Could the wave nature of particles stem from the propagation of waves in the four-dimensional material? At this scale, a particle could be affected by a wave created by itself and behave like this wave until we measure it and discover its position. The trajectory of particles would thus be perfectly defined, but we could only access to a probability of encountering them. It is worth noting that Bell's theorem does not avoid such a non-local hidden variable theory. This point of view is similar to De Broglie's idea [START_REF] De Broglie | Interpretation of quantum mechanics by the double solution theory[END_REF] and the De Broglie-Bohm theory [START_REF] Bohm | A Suggested Interpretation of the Quantum Theory in Terms of 'Hidden Variables' I[END_REF], [START_REF] Bohm | A Suggested Interpretation of the Quantum Theory in Terms of 'Hidden Variables' II[END_REF] which corresponds to a pilot wave theory [START_REF] Bush | Pilot-wave hydrodynamics[END_REF]. This kind of interpretation is supported by the Madelung equations [START_REF] Madelung | Eine anschauliche Deutung der Gleichung von Schrödinger[END_REF], [START_REF] Madelung | Quantentheorie in hydrodynamischer Form[END_REF]. Madelung noted the similarity between the Schrödinger equation and the Navier-Stokes equations. In particular, the quantum equation can be understood as a continuity equation for the probability density. It is worth noting that this interpretation is similar to the experiments on walking droplets achieved by Couder et al. [START_REF] Couder | Walking and orbiting droplets[END_REF] but with one additional dimension.

A minimization of the size of the universe

It is possible to minimize the size of the assumed structure of the universe (Fig. 2). The Hubble-Lemaître law states that the recessional velocity 𝑣 is proportional to the proper distance 𝐷: 𝑣 = 𝐻𝐷 with 𝐻 the Hubble constant. In the framework of the theory, the Hubble constant would not be truly a constant but we will take the actual Hubble constant for the minimization. Therefore, the recessional acceleration is given by 𝑎 = 𝐻 2 𝐷. We note 𝑅 the radius of the universe and 𝑔 the radial acceleration (Fig. 2). The same relation holds for both variables: 𝑔 = 𝐻 2 𝑅. The radius of the universe can then be deduced from an estimation of the radial acceleration. This acceleration must be greater than all the gravitational accelerations of the bodies in the universe. The Earth's acceleration is about 10 𝑚. 𝑠 -2 . This value, which should be far lower than the greatest acceleration, yields 𝑅 ∼ 2 × 10 20 light-years. This corresponds to 4.5 billion times more than the observable universe. This agrees with the apparent flatness of the universe from our point of view.

Variation of some fundamental constants?

The previous interpretations raise the question of the variation of some fundamental constants. The speed of light and the Planck constant could vary against time if relations (1) and ( 2) really describe physical phenomena. In this case, the definition of gravitation (Fig. 2) would also involve the variation of the gravitational constant. The latter would indeed depend on both the acceleration of the expansion and the pressure term 𝑃. For instance, the deformation of a solid depends on the applied forces and Young's modulus. Since both quantities should vary according to time, the gravitational constant would also vary.

The fine-structure constant is defined as 𝛼 = 𝑒 2 2𝜀 0 ℎc . This dimensionless number has been subject

to discussion about its value and its possible variation for decades. Indeed, its order of magnitude indicates that an underlying phenomenon must take place. Like A. Eddington, many scientists have tried to find a simple formula giving its experimental value without success.

Thanks to relations (3) and ( 4), we can substitute the speed of light and the Planck constant in the definition of the fine-structure constant. This yields

𝛼 -1 = 2√𝛱 1 𝛱 2 ∼ 137.035 (5)
Contrary to the Planck constant, the speed of light and the gravitational constant, the finestructure constant would truly be a constant. If the interpretation of quantum mechanics proposed in §3.5 holds, finding a simple formula for 𝛼 based on rational numbers and common functions seems impossible since 𝛱 2 could depend on parameters presenting random values.

Conclusion

From a dimensional analysis, we have found two relations linking the Planck constant and the speed of light with fundamental constants and properties of vacuum. These relations either genuinely describe physical phenomena or are just two coincidences. Given that the relation about the speed of light is based on a well-known continuum mechanics formula, this is rather probable that they are not a matter of pure chance. This is corroborated by the order of magnitude (10 1 ) of both dimensionless numbers (1) and ( 2) in spite of the great disparity between the involved fundamental constants and vacuum properties.

In this framework, these relations could involve the existence of at least one another spatial dimension. Based on Einstein's theories, the universe would be a 4-ball containing matter while our living world would correspond to its surface, a 3-sphere. The developed theory could explain the so-called propagation of waves in "vacuum", some astonishing aspects of quantum mechanics, dark energy and the nature of gravitation. The apparent contradiction with relativity can be overridden with another interpretation of Einstein's theories. In particular, the isotropy of the speed of light would derive from time dilation and length contraction which would be linked to the properties of the four-dimensional material. The speed of light would not be a constant from the hypothetical absolute frame of reference, but would appear as such from our point of view. Regarding quantum mechanics, we do not question the Schrödinger equation, only its interpretation. In particular, relation ( 4) is more consistent with pilot wave theories. The theoretical framework presented in the paper could also account for the potential variation of some fundamental physical constants. Finally, the two relations could lead to another interpretation of the fine-structure constant. In particular, they could provide clues on the underlying phenomenon inherent to this constant that scientists have tried to discover for decades.

Even if this study does not question the equations stemming from general relativity, we can however wonder whether four-dimensional mechanical deformations can lead to Newton's law of universal gravitation as a first approximation (without relativistic effects). This will be the subject matter of future work.

Table 1

 1 itemizes some fundamental constants and vacuum properties that could, a priori, be involved in the calculation.

	Temperature (Cosmic Microwave Background)	𝑇 ∼ 2.73 𝐾
	Density	𝜌 𝑐 ∼ 9.2 × 10 -27 𝑘𝑔. 𝑚 -3
	Boltzmann constant	𝑘 𝐵 ∼ 1.381 × 10 -23 𝑘𝑔. 𝑚 2 . 𝑠 -2 . 𝐾 -1
	Planck constant	ℎ ∼ 6.626 × 10 -34 𝑘𝑔. 𝑚 2 . 𝑠 -1
	Vacuum permittivity	𝜀 0 ∼ 8.854 × 10 -12 𝑚 -3 . 𝑘𝑔 -1 . 𝑠 4 . 𝐴 2
	Elementary charge	𝑒 ∼ 1.602 × 10 -19 𝐴. 𝑠
	Gravitational constant	𝐺 ∼ 6.674 × 10 -11 𝑚 3 . 𝑘𝑔 -1 . 𝑠 -2
	Speed of light	𝑐 ∼ 2.998 × 10 8 𝑚. 𝑠 -1


Sabah Boustila
email: boustila@unistra.fr

Dominique Bechmann
email: bechmann@unistra.fr

Antonio Capobianco
email: a.capobianco@unistra.fr

Effects of Stereo and Head Tracking on Distance Estimation, Presence, and Simulator Sickness using Wall Screen in Architectural Project Review

Keywords: H.5.1 [Information Interfaces and Presentation]: Multimedia Information Systems-Artificial, augmented, and virtual realities; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism -Virtual reality

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION AND RELATED WORK

Visual human system provides several cues for the perception of distances such as binocular disparity, accommodation and convergence, etc. In virtual environments (VE), the visual stimulus is perceived through several display devices that can distort distance perception cues. Our work takes place in an architectural project aiming to perform virtual visits of houses with clients using virtual reality (VR) tools. During virtual visits, clients are supposed to judge the size of rooms and living comfort to validate (or not) the house' mockup. Thus, they need to evaluate distances.

Previous works have found that distances are misperceived in VEs [START_REF] Piryankova | Egocentric distance perception in large screen immersive displays[END_REF][START_REF] Jones | The effects of virtual reality, augmented reality, and motion parallax on egocentric depth perception[END_REF]. For more realistic perception, VR systems can provide head tracking, stereoscopy, etc. The stereoscopy has prompted many researchers to study its influence on distance estimation. Piryankova et al. [START_REF] Piryankova | Egocentric distance perception in large screen immersive displays[END_REF] found that distances up to 2.5 are less underestimated with a stereoscopic display when using a large screen.

As for head tracking, Jones et al. [START_REF] Jones | The effects of virtual reality, augmented reality, and motion parallax on egocentric depth perception[END_REF] found similar underestimations in VE using HMD for distances between 2 and 8 meters whether using head tracking or not.

Moreover, these two advanced features are expected to increase the sense of presence, in comparison to desktop tools, since the perception is more similar to the real world when experiencing depth and point of view changes during body and head movements/shifting [START_REF] Ijsselsteijn | Effects of stereoscopic presentation, image motion, and screen size on subjective and objective corroborative measures of presence[END_REF][START_REF] Lee | Effects of visual cues and sustained attention on spatial presence in virtual environments based on spatial and object distinction[END_REF]. Indeed, in [START_REF] Ijsselsteijn | Effects of stereoscopic presentation, image motion, and screen size on subjective and objective corroborative measures of presence[END_REF] participants reported a better sense of presence when seeing a rally racing on a monitor screen with the two factors. This is consistent with the result found in [START_REF] Lee | Effects of visual cues and sustained attention on spatial presence in virtual environments based on spatial and object distinction[END_REF]. Besides, most of previous works found that they affect the simulator sickness in VE [START_REF] Schild | Understanding user experience in stereoscopic 3d games[END_REF][START_REF] Keshavarz | Stereoscopic viewing enhances visually induced motion sickness but sound does not[END_REF]. In facet, the score of simulator sickness was significantly higher when participants watched video game using stereo [START_REF] Schild | Understanding user experience in stereoscopic 3d games[END_REF]. Same results were found by Keshavarz et al. [START_REF] Keshavarz | Stereoscopic viewing enhances visually induced motion sickness but sound does not[END_REF].

In this paper, we conducted an experiment to investigate whether the advantages of the stereo and the head tracking affect the perception of distances, presence and simulator sickness in VE.

EXPERIMENT DESIGN

We varied the stereo and the head tracking by two levels. This provided a 2 X 2 design: 1) Stereoscopic vision + head Tracking (ST) 2) Stereoscopic vision + no head Tracking (SnoT) 3) Monoscopic vision + head Tracking (MT) 4) Monoscopic vision + no head Tracking(MnoT)

In the virtual visit, the participant walks virtually in the scene while standing physically at a distance of 1.5 meters in front of the screen. In the head tracking conditions, participants were encouraged to move their body and head inside the tracking space. 40 voluntary students in our university participated in the experiment. Ten individuals per condition.

Apparatus and Stimuli

We used a rear-projected wall screen (3 m x 2.25 m) with a resolution of 1400 x 1050 and the Ninento Wiimote as interaction device. The platform is illustrated in figure 1.

Stereo was possible through active shutter glasses tracked using a Vicon tracking system based on 6 IR Bonita cameras (200Hz) providing a tracked space of 6 m 2 . The refresh rate was of 60Hz per eye. The virtual eyepoint height was set automatically corresponding to the subject's eye height. When the head tracking was not used, the experimenter calibrates this parameter before disabling the tracking system. Interocular distance was fixed to 6.3cm for all participants.

Our program is developed in C++ and is based on the VRJuggler library for device management.

Measurement

We used a questionnaire to evaluate the perception of distances. 1) Distance estimation: closed-ended questions with 4 response choices (in meter), to evaluate the length and width of the rooms. 2) Size perception and habitability: 7-point Likert type scale questions, to evaluate specific possibility of furnishing and the comfort in the house, respectively. 3) Task difficulty: a 7-point Likert type scale question, to evaluate the difficulty of the virtual visit and distance estimation task.

Furthermore, we measured the presence (PQ) [START_REF] Witmer | Measuring presence in virtual environments: A presence questionnaire[END_REF] and simulator sickness (SSQ) [START_REF] Kennedy | Configural scoring of simulator sickness, cybersickness and space adaptation syndrome: Similarities and differences? Virtual and Adaptive Environments: Appplications, Implications, and Human Perfoormance[END_REF]. At the end, we performed a debriefing.

Procedure

The experiment consists of virtual visits of houses at scale 1:1. Before starting the experiment, participant reads instructions and trains on the navigation technique in the VE.

The experiment starts, participant moves using an active guided navigation following a predefined path and uses buttons on wiimote to move forward, to stop, and to turn around himself, rotations up and down and going back were not possible. When the participant is inside a room, he hears a beep and he gets stuck with only the ability to around himself. The experimenter starts asking questions regarding distance perception. The participant answers orally. Afterwards, he unlocks the participant to visit the next room.

At the end, the participant answers the simulator sickness questionnaire, the presence questionnaire and gives a debriefing.

RESULTS AND DISCUSSION

In the analysis, for each group of question we calculated the mean of signed gap between the participant's answers and the veridical answers. For the statistical analysis, we performed a nonparametric Kruskal-Wallis rank sum test at a 5% significance level, and pairwise comparisons using the Nemeneyi post-hoc test.

Results revealed significant influence of stereo and head tracking on distance estimation (p-value = 0.011) and habitability (p-value < 0.05). Pairwise comparisons have shown significant differences between ST vs MT (p-value = 0.026, p-value = 0.001) and SnoT vs MT (p-value = 0.026, p-value = 0.015) for distance estimation and habitability respectively. We observe no significant difference between ST vs SnoT and MT vs MnoT. We confidently interpret this as an absence of effect for head tracking. Probably the amount of head motion was much smaller leading to no significant effects. Indeed, large screens provide display only on one screen in front of the view. Thus, even if participants were encouraged to move in the tracked space, they performed just small motions. Jones et al. [START_REF] Jones | The effects of virtual reality, augmented reality, and motion parallax on egocentric depth perception[END_REF] also found no significant effect of head tracking on distance estimations.

As for stereo, we found an effect on distance estimation and habitability only when the head tracking is used in ST and MT. Similarly, Willemsen et al. [START_REF] Willemsen | Effects of stereo viewing conditions on distance perception in virtual environments[END_REF] have found no significant effect of stereo on distance estimation when head tracking were not used.

Concerning task difficulty we found significant result (p-value = 0.016). Task difficulty was estimated as being average with a mean values around 4/7 (7 represent a difficult task). Pairwise comparisons revealed a significant difference only between SnoT and MnoT (p-value = 0.016). The task was more difficult when the two factors were removed (MnoT) with a mean of 4.3/7 while easier with stereo (SnoT) with a mean of 3.4/7. This means that the head tracking alone do not influence task difficulty while the stereo do only when the head tracking is removed. The task difficulty assesses the difficulty of the experiment in the VE and the difficulty of the distance estimation task. Thus, for head tracking, the result seems logical because it did not affect the estimation tasks. As for stereo, previous work found that the importance of the stereo depends on the difficulty of the task to perform in VE [START_REF] Miller | Some effects on depth-position and course-prediction judgments in 2-d and 3-d displays[END_REF].

Results revealed no significant effect of stereo and head tracking on size perception (p-value = 0.710) and presence (p-value = 0.576). In contrast to previous work that found improvement in presence when using stereo and head tracking [START_REF] Ijsselsteijn | Effects of stereoscopic presentation, image motion, and screen size on subjective and objective corroborative measures of presence[END_REF][START_REF] Lee | Effects of visual cues and sustained attention on spatial presence in virtual environments based on spatial and object distinction[END_REF], in our study, the presence was good in all conditions with means between 4.2 and 4.5 on 7 (7 is good). Probably, the influence of the two factors depends on the nature of the task. In our study, the participants visited houses and since the majority have already performed this task in real life, virtual visits seemed natural and probably led to a good sense of presence.

Furthermore, no significant effect of stereo and head tracking was found on simulator sickness (p-value = 0.144). This result is different from previous work that found significant influence of stereo in the context of videos and video games visualisation in VEs [START_REF] Schild | Understanding user experience in stereoscopic 3d games[END_REF][START_REF] Keshavarz | Stereoscopic viewing enhances visually induced motion sickness but sound does not[END_REF].

CONCLUSION

In this paper, we investigated the effect of stereo and head tracking on the estimation of distances, the presence and simulator sickness during virtual visits of houses using a wall screen. To evaluate distance perception we used a specific questionnaire to evaluate wall dimensions (distance estimation), the possibility of specific furnishing (size perception) and the leaving comfort (habitability). Results revealed significant effect of stereo when used with head tracking only on distance estimation and habitability. Besides, task difficulty was higher when the two factors were removed. No significant influence was found on presence and simulator sickness.

In future work we will investigate other estimation methods such as direct walking. Furthermore, adapted IPD will investigated in an additional experimental condition.

Figure 1 :

 1 Figure 1: The platform In Virtuo. Right figure shows an example of our environment displayed on the screen.