Marie E Vuillemin

Eric Husson
email: eric.husson@u-picardie.fr

Sylvain Laclef

Arash Jamali

Virginie Lambertyn

Serge Pilard

Dominique Cailleu

Catherine Sarazin
email: catherine.sarazin@u-picardie.fr

Improving the environmental compatibility of enzymatic synthesis of sugar-based surfactants using green reaction media

Keywords: readily biodegradable solvent, 2-methyltetrahydrofuran, 2-methyltetrahydrofuran-3-one, lipase, glucose laurate, PLS surface response design

INTRODUCTION

The development of innovative and environmentally friendly strategies for biomass transformation into market products falls within the scope of bioeconomy. Glycolipids represent a significant part of the global market from biorefinery as biocompatible and biobased molecules of interest exhibiting for some of them surfactant properties [START_REF] Ferrer | Comparative Surface Activities of Di-and Trisaccharide Fatty Acid Esters[END_REF]. These nonionic biosurfactants offer numerous applications as detergents and wetting agents in the industry [START_REF] Foley | Derivation and synthesis of renewable surfactants[END_REF]. Moreover, they can exhibit antimicrobial, antiviral or antitumoral properties [START_REF] Okabe | Disaccharide Esters Screened for Inhibition of Tumor Necrosis Factor-α Release Are New Anti-cancer Agents[END_REF] which make them suitable candidates for cosmetic, pharmaceutic, therapeutic formulations and biocontrol for crop protection [START_REF] Varvaresou | Biosurfactants in cosmetics and biopharmaceuticals[END_REF]. Sugar fatty acid esters can be obtained by the chemical esterification route which often does not meet environmental safety criteria in regard of ecocytotoxic organic solvents and hazardous chemical reagents use [START_REF] Anastas | Green Chemistry: Principles and Practice[END_REF]. Sugar fatty acid esters can also be produced microbiologically with some drawbacks such as stains selection regulation or numerous purification steps [START_REF] Healy | Microbial production of biosurfactants[END_REF]. As an alternative, enzymatic esterification reactions have been widely studied with lipases (EC 3.1.1.3) as biocatalysts due to the selectivity, the efficiency and the more sustainable reaction conditions in comparison to chemical synthesis since the pioneered works of Klibanov of lipase-catalyzed sugar-esters synthesis [START_REF] Ikeda | Lipase-Catalyzed acyiation of sugars solubilized in hydrophobic solvents by complexation[END_REF]. Enzymatic route offers regioselectivity, catalyst reusability, substrates versatility while decreasing by-products [START_REF] Gumel | Lipase mediated synthesis of sugar fatty acid esters[END_REF]. However, solubility of both sugar and acyl donor in the same medium still remains a constraint for the enzymatic catalysis limiting hydroxyl groups reactivity [START_REF] Yang | Enzymatic synthesis of sugar fatty acid esters in ionic liquids[END_REF]. In addition, rational solvent selection must be performed keeping in mind the need to maintain enzymatic activity and eco-cytotoxicity/biodegradability considerations [START_REF] Klibanov | Epilogue: Prospects and challenges of biocatalysis in organic media[END_REF][START_REF] Lozano | Enzymatic ester synthesis in ionic liquids[END_REF][START_REF] Husson | Chemoselectivity of the N,O-enzymatic acylation in organic media and in ionic liquids[END_REF]. Enzymatic synthesis of sugar esters has long been performed in organic solvents such as dimethyl sulfoxide, dimethyl formamide and pyridine, in which both sugars and fatty acids are soluble [START_REF] Maruyama | Enzymatic synthesis of sugar amino acid esters in organic solvents[END_REF]. However, these organic solvents do not answer the current environmental requirements and are incompatible with food, cosmetic and medical applications. For these reasons, enzymatic synthesis of sugar esters is classically implemented in 2-methyl-2-butanol (2M2B). Ways of improvement include the use of room-temperature ionic liquids, DMSO or more recently deep eutectic solvents [START_REF] Zhao | Enzymatic Synthesis of Glucose-Based Fatty Acid Esters in Bisolvent Systems Containing Ionic Liquids or Deep Eutectic Solvents[END_REF][START_REF] Flores | Influence of glucose solubility and dissolution rate on the kinetics of lipase catalyzed synthesis of glucose laurate in 2methyl 2-butanol[END_REF][START_REF] Grüninger | Enzymatic glycolipid surfactant synthesis from renewables[END_REF] but still raise toxicity questioning to hamper the development of carbohydrate biotransformation [START_REF] Alder | Updating and further expanding GSK's solvent sustainability guide[END_REF]. In this sustainability context, we propose an integrated alternative focusing on the replacement of these reaction media for enzymatic synthesis of sugar fatty acid esters from renewable bioresources. Coming from lignocellulosic biorefinery, glucose was selected as sugar substrate. The choice of the fatty acid from oil biorefinery, was based on the promising surfactant properties after grafting of lauryl chain on sugar moiety [START_REF] Méline | d-Xylose and l-arabinose laurate esters: Enzymatic synthesis, characterization and physico-chemical properties[END_REF]. Moreover, by using a free fatty acid as substrate instead of classical reactive vinyl (or ethy/methyl) acyl donor, the production of volatile acetaldehyde or primary alcohols as side-products is thought to be overcome. Novozym435®, the commercial immobilized on acrylic resins lipase from Candida antarctica, is the most widely used enzyme for industrial applications and specifically for esters synthesis [START_REF] Lee | Lipase-catalyzed synthesis of glucose fatty acid ester using ionic liquids mixtures[END_REF]. We selected this biocatalyst regarding its specificity towards the unbranched long chain fatty acids [START_REF] Soultani | Effect of acyl donor chain length and sugar/acyl donor molar ratio on enzymatic synthesis of fatty acid fructose esters[END_REF] and reusability. This present work aims to investigate the feasibility of lipase-catalyzed esterification of glucose by lauric acid in 2methyltetrahydrofuran (MeTHF), an agro-solvent known for its interesting properties in terms of safety, health, biodegradability, and environmental score [START_REF] Lei | Green-Solvent Selection for Acyl Buchwald-Hartwig Cross-Coupling of Amides (Transamidation)[END_REF][START_REF] Lei | Green Solvent Selection for Suzuki-Miyaura Coupling of Amides[END_REF] and 2methyltetrahydrofuran-3-one (MeTHF-3-one) a food-grade ingredient that is used in food industries to mimic flavors [START_REF] Rowe | Chemistry and Technology of Flavours and Fragrances[END_REF]. Optimal conditions were studied via a PLS-surface response design. Reaction performances and regioselectivity were compared to those obtained in the classically 2M2B used here as a reference. Finally, the stability of the lipase preparation in these two alternative solvents was evaluated as well as the structural integrity of the immobilization carrier. Biodegradability and eco-cytotoxicity considerations of these two alternative solvents are also reported to support the discussion.

EXPERIMENTAL SECTION

Chemicals and enzyme

Novozym435 ® (Candida antarctica lipase (CaLB) immobilized on acrylic resin formed by poly(methyl-methacrylate) cross-linked with divinylbenzene, EC 3.1.1.3, 5000 U/g), molecular sieves (4 Å) and anhydrous D-(+)-glucose (99%) were purchased from Sigma-Aldrich (USA). Lauric acid (98%) was supplied by Alfa Aesar (Thermo Fisher Scientific, Heysham). 2M2B, >99% and MeTHF, >99% were purchased from Sigma-Aldrich (USA). MeTHF-3-one, >97% was supplied by Alfa Aesar (Thermo Fisher Scientific, Heysham). HPLC grade trifluoroacetic acid (99%) was purchased from Sigma-Aldrich (Germany), HPLC grade acetonitrile from Merck (Darmstadt), and petroleum ether from Fisher Scientific (UK). Dimethyl sulfoxide d6 (99.80%) and Dimethyl sulfoxide (>99%) were obtained from VWR.

Biodegradability test

The biodegradability of the three organic compounds: 2M2B, MeTHF and MeTHF-3one was evaluated by Eurofins Ecotoxicologie France SAS (Maxéville, France) according to the OECD 301 F guideline. The three organic compounds tested met the validity criteria required for this test. The typical procedure consisted in determining the biodegradability of these organic compounds by microorganisms in an aquatic environment using a static test system. The tested mixture contained an inorganic medium, the organic compound as the only nominal source of carbon and energy with a theoretical oxygen demand of 100 mg.L -1 , and a mixed inoculation from an urban wastewater treatment plant (amount of suspended matter inferior to 30 mg.L -1 in the final mixture). O2 generated during microbial degradation was trapped and measured by the OXYTOP system. The O2 released was compared to the theoretical amount released and expressed as a percentage of degradation. The test standardly lasted 28 days, but could be extended if the biodegradation curve did not reach a plateau by day 28.

Fish Embryo Acute Toxicity Test

The acute or lethal toxicity of MeTHF and MeTHF-3-one on the embryonic stages of Danio rerio was evaluated by SGS SA (Rouen, France) according to OECD 236 guideline. The 96h LC50 corresponds to the concentration (mg.L -1 ) of the tested solvents leading to 50% mortality of Danio rerio eggs in 96 h. The two solvents tested have a respective 96h LC50 greater than 500 mg.L -1 . The safety data sheets for 2M2B report a 48h LC50 of 2430 mg.L -1 evaluated on embryonic stages of Leuciscus idus melanotus.

Sugar solubility

Determination of the solubility of glucose in 2M2B, MeTHF and MeTHF-3-one was performed by gravimetric analyses. 60 mg of glucose was introduced into 1 mL of each solvent. The saturated dispersion was stirred for 24h at 50°C. After centrifugation (13,000 g, 15 min, MiniSpin®, Eppendorf Austria GmbH), the supernatants were collected and evaporated under vacuum. The resulting amount of glucose was weighed (using a precision analytical balance, Sartorius Practum 224-1s, Germany) and used for concentration determination. All measurements were performed in triplicate and the solubility values were expressed as mean values with a standard deviation (±) in g.L -1 .

Enzymatic esterification

Enzymatic synthesis of glucose monoester was performed by esterification of glucose with lauric acid in 2M2B, 2-MeTHF or MeTHF-3-one catalyzed by Novozym435 ® .

Glucose concentration was fixed at 0.17 M all along the study. Acyl donor and enzyme loadings were described in the "Experimental design" section.

All reagents were stored under vacuum in a desiccator with saturated salt solutions (MgCl2) for at least 24h. Substrates, enzyme and solvents water activity (aw) was measured and controlled before use (0.30 ± 0.05). Molecular sieves of 4 Å were activated at 300°C for 24h then added to the reaction medium (100 g.L -1 ) according to literature data [START_REF] Zhao | Enzymatic Synthesis of Glucose-Based Fatty Acid Esters in Bisolvent Systems Containing Ionic Liquids or Deep Eutectic Solvents[END_REF][START_REF] Lin | Impacts of ionic liquids on enzymatic synthesis of glucose laurate and optimization with superior productivity by response surface methodology[END_REF][START_REF] Ren | Synthesis of some glucose-fatty acid esters by lipase from Candida antarctica and their emulsion functions[END_REF]. For the experimental design, reactions were performed in 1.5 mL Eppendorf tubes on a ThermoMixer C (Eppendorf, Hamburg, Germany) vortexed at 1000 rpm. For the other experiments, reactions were performed at a selected temperature at 350 rpm in 20 mL glass tubes with a Carousel 12 Plus Reaction Station TM (Radleys, United Kingdom) coupled to a Haake K35 thermoregulated bath (Thermo Fisher Scientific, Heysham) at 10°C to avoid evaporation. At the end of the reaction, the immobilized enzyme was removed by filtration. Control experiments were performed (one at 0°C, one without lauric acid, one without enzyme, and one with thermally inactivated enzyme). No product was detected in these control experiments.

Quantitative Analysis by HPLC

Glucose esters were quantified by HPLC (LC 20 AD, Shimadzu, Marne-la-Vallée, France) equipped with a UV detector at 214 nm and a light-scattering low temperature evaporative detector (ELSD-LT II, Shimadzu). The reaction medium was diluted in dimethyl sulfoxide (DMSO) to both stop the reaction and solubilize all the components. Samples were filtered through a 0.2 µm syringe filter and then injected for analysis. Separation was performed on a C18 amide Alltima HP (5µ 150mm x 2.1 mm) chromatography column (Grace Davison Discovery Sciences) with an Alltima HP Guard (7.5 x 2.1 mm), maintained at 30 °C. ELSD parameters were fixed at 40°C, sensitivity gain of 8 and He pressure of 3.8 bars. The elution method consisted of a gradient at a flow rate of 0.2 mL.min -1 : phase A (water/acetonitrile/TFA 30:70:0.3) and phase B (acetonitrile/TFA 100:0.3). The gradient consisted of a decrease from 100% to 70% of phase A for 15 min. Then for 10 min the proportion of phase A was maintained at 100%. The results were recorded and processed by a chromatogram processing software (LC solution, Shimadzu). For analytical extraction and purification, the reaction media were filtered. The permeate was placed under vacuum to evaporate the solvent. The resulting product contained residues of lauric acid and glucose ester. It was washed 3 times in petroleum ether, then centrifuged for 20 min at 13,500 rpm. The lauric acid was solubilized in petroleum ether and the pellet contained the purified GE.

The purified product was then characterized by LC-MS-MS and NMR analyses. The concentration of glucose ester was measured based on standard calibration with the purified product, from 1 to 5 g.L -1 . All HPLC analyses were performed in triplicate.

The yield of glucose ester was determined following the equation 1:

𝐺𝑙𝑢𝑐𝑜𝑠𝑒 𝑒𝑠𝑡𝑒𝑟 𝑦𝑖𝑒𝑙𝑑 (%) = 𝑔𝑙𝑢𝑐𝑜𝑠𝑒 𝑒𝑠𝑡𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑎𝑡 𝑡ℎ𝑒 𝑒𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 (𝑀) 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑔𝑙𝑢𝑐𝑜𝑠𝑒 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 (𝑀) 𝑥100 (Eq.1)

Liquid Chromatography-Mass Spectrometry (LC-MS-MS)

Separations were performed using an ACQUITY UPLC H-Class system coupled online with an ACQUITY PDA detector and a Synapt G2-Si QTOF high resolution hybrid mass spectrometer (Waters, Manchester, UK), equipped with an electrospray ionization (ESI) source and operated in the resolution mode (Resolution FWHM: 25,000). The reaction medium was diluted in dimethyl sulfoxide (DMSO). The samples were filtered through a 0.2 µm syringe filter and then 5 µL were injected.

Products were separated on a C18 amide Alltima HP (5µ 150mm x 2.1 mm) chromatography column (Grace Davison Discovery Sciences) with an Alltima HP Guard column (7.5 x 2.1 mm), maintained at 30 °C. The elution method was the same as that used for the HPLC experiments. Acquisitions were performed in the continuum mode. The scan range was 50-2000 at 0.2 s/scan. Data acquisition and processing were performed using MassLynx software (V4.1, Waters).

Nuclear magnetic resonance analyses (NMR)

The purified glucose ester was analyzed by NMR after solubilization in dimethyl sulfoxide-d6 (1.0% w/v) using a combination of 1D and 2D NMR techniques. The experiments were performed on a spectrometer (Bruker Avance III 600 MHz spectrometer) equipped with a 5 mm probe operating at 150.9128 MHz ( 13 C channel) and 600.1700 MHz ( 1 H channel) at 298 K. The 1 H, 13 C decoupled from 1 H, 1 H- 13 C HSQC, 1 H- 13 C HMBC and 1 H-1 H COSY diffusion NMR experiments were performed under standard conditions. NMR spectra were processed using Bruker's Topspin 3.6.2 windows processing software.

The 1 H and 13 C assignments for the glucose monoester, i.e. α/β-D-glucose-6-O-laurate, were as follows: 1 H NMR (600 MHz, DMSO-d6), δ 4.897 (d, J = 4.8 Hz, 1H, H1), 3.132 (m, 1H, H2), 3.430 (t, J = 9.3 Hz, H3), 3.036 (t, J = 9.7 Hz, 1H, H4), 3.763 (td, J = 8.4 Hz, 2H, H5), 4.258 (dd, J = 11.9 Hz, 2.4 Hz, 1H, H6a), 3.991 (dd, J = 11.7 Hz, 6.1 Hz 1H, H6b), 2.275 (td, 3.5 Hz, 2H, H8), 1.482 (m, 2H, H9), 1.246 (16H, lauric acid chain), 0.861 (t, J = 6.9 Hz, 3H, H18) and 13 The 1 H and 13 C NMR spectra can be found in SI (Figure S1). Furthermore, two chemical shifts for the H-6 indicated that as observed with the LC-MS-MS experiments, the esterification was performed on the α and β conformations of the Dglucose [START_REF] Roslund | Complete assignments of the 1H and 13C chemical shifts and JH,H coupling constants in NMR spectra of dglucopyranose and all d-glucopyranosyl-d-glucopyranosides[END_REF]. The structural elucidation of the product was in agreement with literature [START_REF] Ren | Synthesis of some glucose-fatty acid esters by lipase from Candida antarctica and their emulsion functions[END_REF][START_REF] Liang | Enzymatic Preparation of a Homologous Series of Long-Chain 6-O-Acylglucose Esters and Their Evaluation as Emulsifiers[END_REF]. In order to demonstrate the ester bond formed, HMBC experiments were performed (Figure 1). Protons on the C6 of the glucose ring interacted with the carbonyl carbon indicating the formation of an ester bond. This resulted in the chemical shift of the C8 to the higher field [START_REF] Ren | Synthesis of some glucose-fatty acid esters by lipase from Candida antarctica and their emulsion functions[END_REF]. Ultimately, the glucose ester produced was identified as α/β-D-glucose-6-O-laurate.

Experimental design

Optimization of factors affecting the enzymatic synthesis of glucose fatty acid ester (GE) was performed following an experimental design plan. Experiments were designed by Statgraphics Centurion XVIII ® software (FranceSTAT, Neuilly sur Seine, France) and MODDE 12 ® software (Umetrics, Umea, Sweeden). Four factors (i.e. X1: reaction temperature, X2: lauric acid/glucose molar ratio, X3: reaction duration and X4: enzyme concentration) were selected and the measured response was the yield of glucose ester production. The levels of the factors were chosen by considering several limitations. Regarding temperature, Novozym435 ® exhibits significant thermal stability, specially under high temperatures, even up to 100 °C [START_REF] Mei | Probing Water-Temperature Relationships for Lipase-Catalyzed Lactone Ring-Opening Polymerizations[END_REF]. However, in the literature, this reaction is often performed between 30 and 60°C [START_REF] Zhao | Enzymatic Synthesis of Glucose-Based Fatty Acid Esters in Bisolvent Systems Containing Ionic Liquids or Deep Eutectic Solvents[END_REF][START_REF] Lin | Impacts of ionic liquids on enzymatic synthesis of glucose laurate and optimization with superior productivity by response surface methodology[END_REF][START_REF] Ren | Synthesis of some glucose-fatty acid esters by lipase from Candida antarctica and their emulsion functions[END_REF][START_REF] Arcos | Quantitative enzymatic production of 6-Oacylglucose esters[END_REF]. It has been shown that better yields of monoester were achieved at 40°C by minimizing the impact of the subsequent esterification reaction leading to a by-product diester [START_REF] Arcos | Quantitative enzymatic production of 6-Oacylglucose esters[END_REF].

Among the three chosen solvents, the boiling point of MeTHF is the lowest (81°C).

Based on this knowledge and preliminary single-factor experiments (i.e. to obtain the optimum for one parameter at a time while keeping others constant), the upper level for temperature was 75°C and the lower level was set at 25°C, the room temperature.

The upper and lower levels for lauric acid/glucose molar ratio, enzyme concentration and experiment duration were first selected according to literature data [START_REF] Zhao | Enzymatic Synthesis of Glucose-Based Fatty Acid Esters in Bisolvent Systems Containing Ionic Liquids or Deep Eutectic Solvents[END_REF][START_REF] Lin | Impacts of ionic liquids on enzymatic synthesis of glucose laurate and optimization with superior productivity by response surface methodology[END_REF][START_REF] Ren | Synthesis of some glucose-fatty acid esters by lipase from Candida antarctica and their emulsion functions[END_REF][START_REF] Arcos | Quantitative enzymatic production of 6-Oacylglucose esters[END_REF][START_REF] Ganske | Optimization of lipase-catalyzed glucose fatty acid ester synthesis in a two-phase system containing ionic liquids and t-BuOH[END_REF][START_REF] Ganske | Lipase-Catalyzed Glucose Fatty Acid Ester Synthesis in Ionic Liquids[END_REF], then confirmed by preliminary single-factor experiments. For lauric acid/glucose molar ratio and enzyme concentration, the lower levels were 2:1 and 12.5 g.L -1 , respectively, and the upper levels were 6:1 and 37.5 g.L -1 , respectively. For the duration of the experiments, the lower level was 24 h and the upper level was 72 h, to avoid the formation of diester as a byproduct [START_REF] Zhao | Enzymatic Synthesis of Glucose-Based Fatty Acid Esters in Bisolvent Systems Containing Ionic Liquids or Deep Eutectic Solvents[END_REF][START_REF] Ren | Synthesis of some glucose-fatty acid esters by lipase from Candida antarctica and their emulsion functions[END_REF][START_REF] Arcos | Quantitative enzymatic production of 6-Oacylglucose esters[END_REF][START_REF] Ganske | Optimization of lipase-catalyzed glucose fatty acid ester synthesis in a two-phase system containing ionic liquids and t-BuOH[END_REF][START_REF] Ganske | Lipase-Catalyzed Glucose Fatty Acid Ester Synthesis in Ionic Liquids[END_REF][START_REF] Degn | Lipase-catalysed synthesis of glucose fatty acid esters in tert-butanol[END_REF]. The levels of the factors are shown in Table 1. After defining the factors and response of interest, an appropriate design type was selected. The design consisted of 2 4 factorial design and 2 central points. The experimental design matrix was generated and presented in Table 2. All experiments were performed in random order. Once the glucose ester yield data was collected, it was checked for outliers and model quality, after which it could be used for response surface optimization. This method uses a polynomial model including linear terms and all two factor interactions. The experimental design was subjected to goodness of fit routine using a first-and second-order equation by MODDE 12 ® software, as shown in Equation 2and Equation 3:

𝑌(𝐺𝐸) = 𝛽 0 + ∑ 𝛽 𝑖 𝑋 𝑖 + ∑ 𝛽 𝑖𝑗 𝑋 𝑖 𝑋 𝑗 4 𝑖,𝑗=1 4 𝑖=1 (Eq. 2) 𝑌(𝐺𝐸) = 𝛽 0 + ∑ 𝛽 𝑖 𝑋 𝑖 + ∑ 𝛽 𝑖𝑖 𝑋 𝑖 2 4 𝑖=1 + ∑ 𝛽 𝑖𝑗 𝑋 𝑖 𝑋 𝑗 4 𝑖,𝑗=1 4 𝑖=1 (Eq. 3)
Where 𝑌(𝐺𝐸) is the response variable for glucose ester yield, 𝑋 𝑖 are the independent factor variables, 𝛽 0 , 𝛽 𝑖 , 𝛽 𝑖𝑖 and 𝛽 𝑖𝑗 are the constant, linear, quadratic and two factor interaction coefficients, respectively. Each coefficient in the second-order equation was calculated and the possible interaction effects of the factors were obtained. The method used to fit the models was PLS supported by NIPALS algorithm [START_REF] Wold | PLS-regression: a basic tool of chemometrics[END_REF]. A Variable Importance to the Projection (VIP) was applied as part of the PLS calculations. The VIP is based on the weighted sum of square of the weights of the model factors, allowing hierarchical ordering of the independent variables [START_REF] Wold | PLS-regression: a basic tool of chemometrics[END_REF][START_REF] Araya-Farias | Wheat Bran Pretreatment by Room Temperature Ionic Liquid-Water Mixture: Optimization of Process Conditions by PLS-Surface Response Design[END_REF].

The significance of the models was checked by Analysis of Variance (ANOVA). The goodness of fit was expressed by the coefficient of determination R² and R²adj. Model significance was determined by F-test (p < 0.05). Optimization (best 𝑌(𝐺𝐸)) was determined by the PLS method. VIP charts and surface response plots were drawn using MODDE 12 ® software.

Enzyme stability in the solvents

Enzyme stability was determined by measuring glucose ester yields after prior incubation of the enzyme preparation in 2M2B, MeTHF and MeTHF-3-one. The enzyme was incubated in the solvents for 3, 6, 9, 12 or 15 days at 75°C. After centrifugation (10,000 rpm, MiniSpin®, Eppendorf Austria GmbH) the solvent (supernatant) was removed and the enzyme preparation was dried with a SpeedVac Concentrator (SPD121P, Thermo Savant, Waltham, MA, USA) for 45 min. Finally, the enzyme preparation was stored under vacuum (at a controlled aw of 0.37 ± 0.01) until use. Then, the glucose esterification reactions were performed under the optimal conditions determined by the experimental design (at 75°C for 72 h, with a lauric acid/glucose molar ratio of 6:1 and an enzyme concentration of 12.5 g.L -1 ). The reaction medium was analyzed by HPLC to determine the yield of ester production, as described previously.

Scanning electron microscopy

After incubation in the three distinct organic solvents (2M2B, MeTHF and MeTHF-3one) for up to 15 days at 75°C, the morphological properties of the Novozym435® carriers were observed with a high-resolution environmental scanning electron microscope (ESEM, FEI Quanta 200 FEG, OR, USA). A small amount of each sample was placed on double-sided carbon tape adhered to aluminum SEM stubs. The specimens were examined and the secondary electron images were acquired under lowvacuum mode at a pressure of 0.5 Torr using a pressure limiting apertures (PLA).

RESULTS AND DISCUSSION

The potential of an agro-solvent and a food grade ingredient used as solvents was investigated to challenge the 2M2B classically used for enzymatic synthesis of glycolipids. The rational selection was based on physicochemical and environmental considerations: biodegradability and cytotoxicity. Indeed, MeTHF constitutes a biobased solvent derived from biomass and MeTHF-3-one listed as generally recognized as safe (GRAS). The scheme of lipase-catalyzed esterification of glucose, main platform molecule from lignocellulosic biorefinery, with lauric acid leading to one product with interesting surfactant properties [START_REF] Méline | d-Xylose and l-arabinose laurate esters: Enzymatic synthesis, characterization and physico-chemical properties[END_REF] is presented in Figure 2. The challenge was to develop a system in which the substrate and the acyl donor would be available for the reaction despite opposite polarity while directing and maintaining the enzymatic activity towards ester synthesis.

Environmental considerations and substrate availability

Biodegradability and ecotoxicity of the three solvents were first compared according to OECD guideline. Concerning the fish embryo acute toxicity test, MeTHF and MeTHF-3-one both exhibited a respective 96 h LC50 superior to 500 mg.L -1 . Likewise, the Material Safety Data Sheets of 2M2B reported a 48h LC50 of 2430 mg.L -1 . These data, evaluated on embryonic stages of soft water fishes according to the OECD 236 guideline, suggested a similar ecotoxicity between these three organic compounds.

Then, the biodegradability kinetics for each solvent were presented in Figure 3A. The biodegradability rates after 28 days are provided in SI (Table S1). MeTHF and 2M2B exhibited similar biodegradation kinetic profiles with 5 and 6 % of biodegradability level after 28 days, respectively. In opposite, the biodegradation kinetic of MeTHF-3one showed a biodegradation level of 69 % at 28 days within a 10-day window in the 28-day period. This behavior allowed MeTHF-3-one to be classified as "readily biodegradable" in agreement with the OECD standards [START_REF] Liwarska-Bizukojc | Biodegradability and kinetics of the removal of new peralkylated imidazolium ionic liquids[END_REF]. Based on these environmental considerations, we therefore confirmed the suitability of selecting the MeTHF and MeTHF-3-one in this study to improve the sustainability of glycolipids synthesis by enzymatic route.

All enzymatic reaction systems implemented in this study included reactants, enzyme and organic solvents with controlled initial water activity (aw) for maintaining a sufficient hydration state of lipase for enzyme stability while favoring the synthesis activity [START_REF] Halling | What can we learn by studying enzymes in non-aqueous media?[END_REF]. Among the other numerous factors influencing performances of enzymatic esterification of glucose in organic solvent, the availability of the polar substrate and the acyl donor in the solvent constitute a key factor [START_REF] Ikeda | Lipase-Catalyzed acyiation of sugars solubilized in hydrophobic solvents by complexation[END_REF][START_REF] Yang | Enzymatic synthesis of sugar fatty acid esters in ionic liquids[END_REF][START_REF] Ren | Synthesis of some glucose-fatty acid esters by lipase from Candida antarctica and their emulsion functions[END_REF]. Lauric acid was soluble in all the three solvents with a solubility value much higher than 500 g.L -1 confirming the full availability of the acyl donor in the reaction systems. The solubility values of glucose were also determined and presented in Figure 3B as a function of the log P of each solvent where P constitutes the solvent partition coefficient between 1octanol and water. As log P of MeTHF-3-one was not available in literature, ACD/log P calculator procedure [START_REF] Petrauskas | ACD/Log P method description[END_REF] was used for establishing a relative comparison between the three solvents. The use of MeTHF-3-one with a log P of -0.65 led to a glucose solubility of 5.05 g.L -1 versus 1.15 and 0.85 g.L -1 in 2M2B (log P = 1.04) or MeTHF (log P = 0.85), respectively. Although the solubility of glucose in a solvent cannot be exclusively related to log P, the results are consistent with an increase of solubility with a decrease of log P. The selected alternative solvents meet thus the sustainability criteria while ensuring practical availability of both substrates, totally for acyl donors and partially for the sugar as well as in the reference organic solvent.

Feasibility of glucose ester synthesis in the neoteric solvents

Lipase-catalyzed esterification in organic solvent has been widely reported in the literature since 1990's [START_REF] Otto | Lipase-catalyzed synthesis of arylaliphatic esters of β-d(+)-glucose, n-alkyl-and arylglucosides and characterization of their surfactant properties[END_REF][START_REF] Coulon | Enzymatic synthesis of alkylglycoside fatty acid esters catalyzed by an immobilized lipase[END_REF][START_REF] Bousquet | Enzymatic synthesis of alphabutylglucoside lactate: a new alpha-hydroxy acid derivative[END_REF][START_REF] Bousquet | Enzymatic synthesis of unsaturated fatty acid glucoside esters for dermo-cosmetic applications[END_REF]. 2M2B has often been presented as a reference solvent for this type of reaction. It seems to receive a renewed interest in recent studies despite physicochemical properties far away from durability criteria [START_REF] Jocquel | An Integrated Enzymatic Approach to Produce Pentyl Xylosides and Glucose/Xylose Laurate Esters From Wheat Bran[END_REF][START_REF] Martinez-Garcia | Enzymatic Synthesis of Glucose-and Xylose Laurate Esters Using Different Acyl Donors, Higher Substrate Concentrations, and Membrane Assisted Solvent Recovery[END_REF]. Based on literature data, we firstly set the following experimental conditions as a basis for comparison with 2M2B: equimolar ratio of lauric acid/glucose, 100 g.L -1 of molecular sieve, 25 g.L -1 of Novozym435 ® , 50 °C, 350 rpm. After 72 h, the enzymatic reactions were stopped and the identification of the products was performed by LC-MS. After each reaction, NMR analysis of the recycled solvents was done. These results are presented in SI (see SI, Figure S2).

Regardless of the solvent, LC-MS-MS analyses revealed at least the presence of three distinct compounds after 72 h of reaction. In all cases, glucose laurate mono-ester (C18H34O7) is evidenced by RT = 2.88 -2.98 min, [M + Na] + m/z 385.220 and [MM+Na] + m/z 785.445 with the presence of both anomers [START_REF] Liang | Enzymatic Preparation of a Homologous Series of Long-Chain 6-O-Acylglucose Esters and Their Evaluation as Emulsifiers[END_REF] (Figure S3). 2021) reporting up to 4% of diester synthesized in 2M2B [START_REF] Martinez-Garcia | Enzymatic Synthesis of Glucose-and Xylose Laurate Esters Using Different Acyl Donors, Higher Substrate Concentrations, and Membrane Assisted Solvent Recovery[END_REF]. In MeTHF-3-one, the selectivity of the reaction seems exclusively orientated toward monoester synthesis. Indeed, only traces of diester product were detected. Total ion current (TIC) of the reaction of glucose with lauric acid in MeTHF-3-one showed however two additional impurity at 5.08 and 5.32 min (Figure S7). Mass spectra (molecular mass region) of these side products are presented in SI (Figure S8). These products are hardly detectable on NMR spectra. The structural characterization of these compounds as well as a proposed mechanism of formation in the reaction system have been described in SI (Figure S9). Briefly, these two compounds were obtained from lipase-catalyzed esterification of a derivative resulting from the ring opening of MeTHF-3-one, in the presence of lauric acid (Table S2 for details on the control experiments performed). Finally, the monoester compound was extracted and purified from the three distinct reaction media and then analyzed by NMR for structural elucidation (Figure 1). Complementary NMR data was presented in SI (Figure S1). NMR results confirmed the obtention of D-glucose-6-O-laurate in large amounts whatever the three distinct organic solvents used, showing also that the anomeric carbon of glucose remained not esterified. MeTHF and MeTHF-3-one were thus demonstrated for the first time as suitable alternative to 2M2B for lipase-catalyzed glucose esterification while preserving the regioselectivity i.e. the glucose-6monolaurate, the product of interest. We confirmed also that log P would be useful for predicting glucose solubility but failed to predict enzyme catalysis feasibility in our reaction systems as suggested in literature [START_REF] Kumar | Lipase catalysis in organic solvents: advantages and applications[END_REF].

Reaction optimization through model fitting and statistical analysis

The effects of temperature, lauric acid/glucose molar ratio, reaction duration and enzyme concentration were investigated and optimized in 2M2B, MeTHF and MeTHF-3-one using a design of experiments using PLS methodology to ensure a robust comparison. Table 2 shows the experimental design matrix of the three factors and glucose ester yields in 2M2B, MeTHF and MeTHF-3-one generated by PLS statistical tool. The results were fitted using Equation 2 and Equation 3. The factor coefficients are presented in Table 3. Analysis of variance (ANOVA) was performed and VIP charts were used to obtain the linear, quadratic and interaction effects on the response. The models F-values were between 33.496 and 41.445 and the p-values of "Prob>F" were always < 0.0001 (Table 3), which means that the models were considered highly significant at the 95% confidence level (p-value of "Prob>F" < 0.05 and F-value > 4). In other words, the models adequately predicted the glucose ester yields from the 4 factors. Furthermore, the coefficients of correlation (R²) ranged from 0.96 to 0.97. In general, quadratic models are considered as models with high degrees of correlation when their R² values are higher than 0.90 [START_REF] Araya-Farias | Wheat Bran Pretreatment by Room Temperature Ionic Liquid-Water Mixture: Optimization of Process Conditions by PLS-Surface Response Design[END_REF][START_REF] Cadoche | Assessment of size reduction as a preliminary step in the production of ethanol from lignocellulosic wastes[END_REF][START_REF] Myers | Response Surface Methodology: Process and Product Optimization Using Designed Experiments[END_REF]. The R²Adj values (0.93 to 0.94) were another proof of the significance of the models. The precision of the models and their reproducibility were assessed using the RSD% (Table 3), whose values went from 2.93 to 4.69% (less than 10%).

The effect of the factors on the glucose ester yield Y(GE) was studied using VIP charts (Figure E,F). For all solvents, the factors with the higher significant effects were temperature and lauric acid/glucose molar ratio (Figure E,F). However, for 2M2B and MeTHF-3-one, the quadratic terms did not have a significant influence over the reaction yields, whereas they all did for MeTHF at the 95% confidence level (Figure 4E). For all solvents, the interaction of temperature and lauric acid/glucose molar ratio was statistically significant according to the model, with a p-value < 0.05. The other interactions were not significant. Finally, the success of the predictions of the models were evaluated by comparing the experimental values of Y(GE) and the ones predicted by the model (Figure B,C). The experimental values were closely regrouped around the line of best fit, which means that the model was highly efficient to predict the experimental values. All statistical analyses proved the model was reliable and efficient. The predicted higher glucose ester yields were 38.1% in 2M2B, 35.3% in MeTHF and 56.7% in MeTHF-3-one (Figure 5). The obtained experimental glucose ester yields for these optimal conditions were in perfect agreement with those predicted (39.1 ± 2.0% for 2M2B, 37.0 ± 11.7% for MeTHF and 57.4 ± 2.4% for MeTHF-3-one). One can note that the lowest enzyme concentration was the optimal one as already reported [START_REF] Gutierrez-Lazaro | Effect of operating variables and kinetics of the lipase catalyzed transesterification of ethylene carbonate and glycerol[END_REF][START_REF] Jung | Enzymatic production of glycerol carbonate from by-product after biodiesel manufacturing process[END_REF][START_REF] Gao | Construction of biocatalytic colloidosome using lipase-containing dendritic mesoporous silica nanospheres for enhanced enzyme catalysis[END_REF][START_REF] Seo | Novel and highly efficient lipasecatalyzed esterification of formic acid with hexanol for waste gas reutilization[END_REF]. This may be due to the precipitation of produced ester in the enzyme microenvironment preventing the access of the substrate to the active site at higher enzyme concentration [START_REF] Tudorache | Biocatalytic alternative for bioglycerol conversion with alkyl carbonates via a lipase-linked magnetic nano-particles assisted process[END_REF][START_REF] Baek | Enzymatic Synthesis of Formate Ester through Immobilized Lipase and Its Reuse[END_REF][START_REF] Shin | Novel and Efficient Synthesis of Phenethyl Formate via Enzymatic Esterification of Formic Acid[END_REF]. In the three solvents, the highest substrate molar ratio was the optimal one likely to an equilibrium shift toward synthesis with higher content of the fatty acid [START_REF] Shin | Novel and Efficient Synthesis of Phenethyl Formate via Enzymatic Esterification of Formic Acid[END_REF]. Most certainly thanks to a decrease in viscosity, increasing the encounters between enzymes and substrates [START_REF] Shin | Novel and Efficient Synthesis of Phenethyl Formate via Enzymatic Esterification of Formic Acid[END_REF], the highest tested temperature of 75°C provide the best yields afforded by Novozym435 ® stability up to 80 to 100 ° C [START_REF] Mei | Probing Water-Temperature Relationships for Lipase-Catalyzed Lactone Ring-Opening Polymerizations[END_REF][START_REF] Poojari | Thermal stability of Candida antarctica lipase B immobilized on macroporous acrylic resin particles in organic media[END_REF]. The optimal conditions for obtaining the higher glucose ester yields were the same in all solvents, namely 75 °C lauric acid/glucose molar ratio of 6:1, during 72 h with an enzyme concentration of 12.5 g.L -1 and will be used for the following experiments. The highest glucose ester yields were similar in 2M2B and MeTHF, but significantly higher in MeTHF-3-one.

The excellent yields obtained in MeTHF-3-one seemed to be due to the solubility of glucose which was 5 fold higher than in the other two solvents (5.05 ± 0.35 g.L -1 in MeTHF-3-one versus 1.15 ± 0.07 g.L -1 and 0.85 ± 0.21 g.L -1 in 2M2B and MeTHF, respectively). Based on these results, MeTHF can be proposed as a biobased alternative to 2M2B preserving the performances. However, MeTHF-3-one, with its food grade label, was demonstrated as a better substitute to 2M2B by significantly improving the reaction efficiency in mild conditions.

Reaction kinetics

The reaction kinetics in the three solvents were then compared (Figure 6A). The kinetics in MeTHF and 2M2B were very similar and reached a plateau at 48h while the plateau in MeTHF-3-one was not reached at 72h. Initial rates (Figure 6B, Vi) in 2M2B or MeTHF were 16.3 ± 2.9 mmol.h -1 and 15.8 ± 0.5 mmol.h -1 , respectively and significantly higher in MeTHF-3-one (26.0 ± 2.8 mmol.h -1 ). The esterification yield at 72 h reached 79.4 ± 0.4 % for MeTHF-3-one versus 47.6 ± 0.8 % and 48.3 ± 0.8 % for 2M2B and MeTHF, respectively. These differences in initial rate and yield can be correlated to the greater solubility of glucose in MeTHF-3-one. Under our experimental conditions, MeTHF-3-one provided promising alternative with respect to enzymatic performances (eg. 1.7-yield increase) and regioselectivity towards monoesterification on the primary hydroxyl of glucose. Moreover, the yields obtained in MeTHF-3-one were competitive compared to the rates obtained previously and more recently in 2M2B [START_REF] Bousquet | Enzymatic synthesis of unsaturated fatty acid glucoside esters for dermo-cosmetic applications[END_REF][START_REF] Martinez-Garcia | Enzymatic Synthesis of Glucose-and Xylose Laurate Esters Using Different Acyl Donors, Higher Substrate Concentrations, and Membrane Assisted Solvent Recovery[END_REF].

Enzyme stability

Among the many advantages of immobilized enzymes, recycling and reuse of the biocatalyst enhances both the sustainability and the economic viability of the strategy [START_REF] Baek | Enzymatic Synthesis of Formate Ester through Immobilized Lipase and Its Reuse[END_REF][START_REF] Adlercreutz | Immobilisation and application of lipases in organic media[END_REF][START_REF] Tufiño | Synthesis with Immobilized Lipases and Downstream Processing of Ascorbyl Palmitate[END_REF][START_REF] Galonde | Reusability study of Novozym® 435 for the enzymatic synthesis of mannosyl myristate in pure ionic liquids[END_REF][START_REF] De Sousa | A practical approach to obtain high yield lipase-mediated synthesis of octyl caprylate with Novozym 435[END_REF][START_REF] Basso | Industrial applications of immobilized enzymes-A review[END_REF]. In this context, enzyme stability can be studied to assess the impact of incubation in an organic solvent on the structural integrity of enzyme carriers and activity. Such an approach provides valuable information on the possible reusability of biocatalysts [START_REF] Tufiño | Synthesis with Immobilized Lipases and Downstream Processing of Ascorbyl Palmitate[END_REF]. Stability studies on Novozym435 ® in 2M2B have already shown that the biocatalyst could be reused several times [START_REF] Tufiño | Synthesis with Immobilized Lipases and Downstream Processing of Ascorbyl Palmitate[END_REF][START_REF] Khare | Lipase-catalyzed production of a bioactive fatty amide derivative of 7,10-dihydroxy-8(E)-octadecenoic acid[END_REF][START_REF] Bousquet | Lipase-catalyzed αbutylglucoside lactate synthesis in organic solvent for dermo-cosmetic application[END_REF]. To our knowledge, no data on this enzymatic preparation in MeTHF or MeTHF-3-one have been reported in the literature. Novozym435 ® was first incubated in the three distinct solvents at 75°C and then used for esterification under the optimal conditions. The glucose laurate yields reached after varying incubation times of the enzyme in each solvent are shown in Figure 7. Likewise, SEM observations of the surface morphology of fresh Novozym435 ® versus Novozym435 ® after incubation in the solvents for 3 and 15 days (75°C) are presented in Figure 7. Whatever the solvents, an average of 63 % of enzyme activity was preserved after 3 days of incubation. The results also show that 15-days of incubation did not lead to a total deactivation of the enzyme preparation. However, the decrease in activity throughout the days of incubation was greater in MeTHF (from 48.3 ± 0.8 % to 3.6 ± 0.1 after 15 days of incubation) than in 2M2B (from 47.6 ± 0.8 to 10.5 ± 0.6 % after 15 days of incubation) or MeTHF-3-one (from 79.4 ± 0.4 to 16.9 ± 0.1 % after 15 days of incubation). The loss of activity after incubation in organic solvent could be due to the interaction of solvents with the surface of enzyme carriers, as previously reported in the literature [START_REF] José | Investigation of the causes of deactivation-degradation of the commercial biocatalyst Novozym® 435 in ethanol and ethanol-aqueous media[END_REF]. It may be emphasized that the enzymatic performance in MeTHF-3-one remained superior to that obtained in the reference organic solvent 2M2B regardless of the prior incubation duration. The stability of the enzyme preparation would therefore be improved in MeTHF-3-one compared to 2M2B or MeTHF. Regarding the morphological and textural properties of the enzyme carriers, no change was observed between the fresh enzyme preparation and those observed after 3 days of incubation regardless of the solvent (Figure 7). Indeed, carriers kept a spherical shape with a smooth surface before and after 3 days of incubation in each solvent. After 15 days of incubation, SEM observation of the enzyme carriers showed that MeTHF did not induce drastic structural changes (Figure 7). In opposite, prolonged incubation in 2M2B resulted in structural changes illustrated by the appearance of irregularities on the surface of the enzyme carriers. In the case of MeTHF-3-one, 15-days incubation resulted in an obvious change in the integrity of the carrier surface (Figure 7). This irregular deposit could be attributed to the adsorption of MeTHF-3-one onto the surface of the enzyme carrier. This therefore provides additional strong arguments for the substitution of 2M2B by MeTHF-3-one. The obvious adsorption of MeTHF-3-one on the surface of the enzyme carriers would minimize the deleterious effect on the lipase microenvironment and activity unlike other organic solvents used for this type of reaction [START_REF] Da Silva Corrêa | Lipase-Catalyzed Esterification of Geraniol and Citronellol for the Synthesis of Terpenic Esters[END_REF].

CONCLUSIONS

The substitution of 2M2B by the agro-solvent MeTHF or the food-grade ingredient MeTHF-3-one used as solvent allowed the development of a novel sustainable route for lipase-catalyzed synthesis of glucose laurate. These two neoteric solvents did not affect the regioselectivity classically observed in 2M2B toward the primary alcohol of glucose. The complete statistical assessment confirmed that the established model was adequate and accurately predicted the glucose ester yields within the range of reaction conditions. Enzymatic performances achieved in MeTHF were similar to those obtained in 2M2B while a significant improvement was evidenced in MeTHF-3-one. This latter, demonstrated as readily biodegradable, also increased the enzymatic stability of Novozym435® leading to consider its reusability. These results open a promising field in the design of sustainable enzymatic syntheses for biosurfactants production from renewable resources.

  The two other signals were identified as unreacted substrates: the residual glucose (RT = 1.89 min, [M + H] + m/z 181.073, [M +Na] + m/z 203.055) and the residual lauric acid (RT = 7.41 min, [M-H] -m/z = 199.171). Total and Reconstructed Ion Currents (TIC and RIC respectively, m/z 385.220) of the reaction media in 2M2B are presented in SI (Figure S4). The sodium adduct led to two fragments ions at m/z 345.220 and 327.217, issued respectively from subsequent losses of NaOH and H2O. Furthermore, the daughter ion at m/z 327.217 was consecutively fragmented by MS-MS, leading to a pyran fragment ion at m/z 127.041 (C6H7O3) due to the loss of a lauric acid molecule (C12H24O2). The MS-MS spectrum of the ion at m/z 327.217 is presented in SI (Figure S5) and a proposed fragmentation pattern of glucose laurate mono-ester sodium adduct is presented in SI (Figure S6). These data demonstrated the glucose laurate monoester formation in the three distinct solvents. In 2M2B and MeTHF, a minor compound can be detected (RT = 14.21 min, [M+ Na] + m/z 567.407, [M+NH4] + m/z 562.446, [MM + Na] + m/z 1111.821) suggesting the formation of traces of glucose dilaurate. Similar results were recently observed by Martinez-Garcia et al. (

Figure 1 .Figure 2 .

 12 Figure 1. Heteronuclear multiple bond correlation (HMBC) of the purified glucose ester. The horizontal and vertical axis indicate 1 H proton and 13 C chemical shift (ppm) respectively. The

Figure 3 .

 3 Figure 3. (A) Results from biodegradation test expressed in % for 2M2B (black crosses),

Figure 4 .Figure 5 .

 45 Figure 4. Correlation graphs between the predicted and observed experimental glucose

Figure 6 .

 6 Figure 6. (A) Conversion yield (%) of glucose and lauric acid into glucose monoester after

Figure 7 .

 7 Figure 7. Glucose laurate yield after enzyme incubation in the three distinct solvents

Figure 2

 2 Figure 2.

  C NMR (150 MHz, DMSO-d6), δ 92.838 (C1), 72.659 (C2), 73.359 (C3), 71.079 (C4), 69.836 (C5), 64.448 (C6), 175.108 (C7), 34.607 (C8), 32.1204, 29.462, 29.387, 29.229, 29.187, 29.029, 28.917, 25.282, 22.795 (C lauric acid chain), 14.506 (C18).

TABLE LEGENDS Table 1 .

 LEGENDS1 Experimental range of levels and coding of factors used to evaluate the glucose monoester yields.

Table 2 .

 2 Experimental design matrix of the three factors and glucose ester yields in 2M2B, MeTHF and MeTHF-3-one.

Table 3 .

 3 Regression coefficients and ANOVA of quadratic model.

	TABLES

Table 1 .

 1

	Factors	Coding	Range of levels
			Low	Middle	High
	Temperature (°C)	X1	25	50	75
	Lauric acid/glucose molar ratio	X2	2	4	6
	Reaction duration (h)	X3	24	48	72
	Enzyme concentration	X4	12.5	25	37.5

Table 3 .

 3

	Coefficients	For glucose ester yield in 2M2B	For glucose ester yield in MeTHF	For glucose ester yield in MeTHF-3-one
	Constant 𝛽 0	7.088	22.738	20.475
	Linear
	𝛽 1	-0.121	-0.335	-0.135
	𝛽 2	-3.744	-5.200	-5.453
	𝛽 3	-0.161	-0.408	-0.518
	𝛽 4	-0.077	-0.506	-0.285
	Quadratic
	𝛽 11	0.002	0.003	0.002
	𝛽 22	0.280	0.499	0.349
	𝛽 33	0.002	0.003	0.002
	𝛽 44	0.007	0.013	0.009
	Interaction
	𝛽 12	0.062	0.094	0.094
	𝛽 13	0.003	0.002	0.005
	𝛽 14	0.000	-0.002	-0.004
	𝛽 23	0.018	0.002	0.035
	𝛽 24	-0.023	-0.036	-0.030
	𝛽 34	-0.005	0.001	0.001
	Model F-value	41.445	40.938	33.496
	p-value Prob >F	< 0.0001	< 0.0001	< 0.0001
	R²	0.97	0.97	0.96
	R²adjusted	0.94	0.94	0.93
	RSD (%)	2.93	2.98	4.69

The authors declare no competing financial interest.

Funding

This research was supported by the Interreg France-Wallonie-Vlaanderen through the ValBran project with the support of the European Regional Development Fund (www.valbran.eu).

Supporting Information

Additional experimental details, results, figures and tables associated with (i) solvents environmental data, (ii) LC-MS-MS and NMR analyses of reaction media and purified endproduct (PDF).


Razvan Gabriel Iagar
email: razvan.iagar@urjc.es

Philippe Laurençot
email: philippe.laurencot@math.univ-toulouse.fr

Finite time extinction for a diffusion equation with spatially inhomogeneous strong absorption

Keywords: AMS Subject Classification 2010: 35B33, 35B40, 35K55, 35K65 porous medium equation, strong absorption, finite time extinction, inhomogeneous absorption

The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption

with m ≥ 1, q ∈ (0, 1) and σ > 0, is addressed. Introducing the critical exponent σ * := 2(1 -q)/(m -1) for m > 1 and σ * = ∞ for m = 1, extinction in finite time is known to take place for σ ∈ [0, σ * ) and an alternative proof is provided therein. When m > 1 and σ ≥ σ * , the occurrence of finite time extinction is proved for a specific class of initial conditions, thereby supplementing results on non-extinction that are available in that range of σ and showing their sharpness.

Introduction

The aim of this short note is to address the question of finite time extinction of bounded and non-negative solutions to the following diffusion equation with spatially inhomogeneous strong absorption

∂ t u = ∆u m -|x| σ u q , (t, x) ∈ (0, ∞) × R N , (1.1)
with initial condition

u(0) = u 0 ∈ L ∞ + (R N ) := z ∈ L ∞ (R N ) : z(x) ≥ 0 a.e. in R N , (1.2)
where m ≥ 1, 0 < q < 1, σ > 0.

(1.

3)

The dynamics of Eq. (1.1) features a double competition: on the one hand, the effects of the diffusion, which spreads the mass of the solutions as time advances, competes with the absorption term, which triggers a loss of mass, possibly leading to vanishing in finite time. On the other hand, the weight on the absorption term is likely to bring into play significant imbalances between the properties of the solutions in a neighborhood of the origin (where, at least formally, the absorption term is very small) and in outer regions, at uniform distance from the origin, where the weight |x| σ becomes very strong.

With respect to the first of the two competitions described above, the spatially homogeneous counterpart of Eq. (1.1) with σ = 0, that is

∂ t u -∆u m + u q = 0, m ≥ 1, (1.4)
has a well understood dynamics at least in some ranges of exponents. Indeed, the critical values for the absorption exponent q are q = m and q = 1, and its solutions have very different properties according to whether q > m, 1 < q ≤ m or 0 < q < 1. The former of these three ranges sees the diffusion term being sufficiently strong either to govern the dynamics, or to balance the effects of the absorption giving rise to profiles known as very singular solutions. Properties such as decay estimates for the solutions, construction of singular (or very singular) self-similar solutions and large time behavior of solutions in the sense of convergence to such self-similar profiles as t → ∞ are established in a number of works, see for example [16-19, 21, 22, 24] and references therein. While for q > m all compactly supported solutions have an algebraic time decay as t → ∞ and their supports expand reaching the whole space in the limit, a different situation occurs for m > 1 and q ∈ (1, m). In the latter range, the effect of the absorption is dominant and the expansion of the positivity region of the compactly supported solutions is limited, leading to the localization of supports, that is, there exists a large radius R > 0 not depending on time such that supp u(t) ⊆ B(0, R) for any t > 0. Solutions still present an algebraic decay as t → ∞, but self-similar solutions might become unbounded, presenting a specific growth at infinity [START_REF] Chaves | Optimal existence and uniqueness in a nonlinear diffusion-absorption equation with critical exponents[END_REF][START_REF] Mcleod | Solutions of a nonlinear ODE appearing in the theory of diffusion with absorption[END_REF] and delicate descriptions of the large time behavior in the form of matched asymptotics between "flat" solutions of the form K * t -1/(q-1) for some explicit constant K * > 0, and boundary layers appearing near the boundary of the localization ball B(0, R), have been established, see [START_REF] Chaves | Free boundary layer formation in nonlinear heat propagation[END_REF]. Working with spatially inhomogeneous absorption with general weights, Peletier and Tesei established in dimension N = 1 in [START_REF] Peletier | Diffusion in inhomogeneous media: localization and positivity[END_REF][START_REF] Peletier | Global bifurcation and attractivity of stationary solutions of a degenerate diffusion equation[END_REF] positivity of supports for q > m, conditions for localization of supports of the solutions for 1 < q < m and the existence of stationary compactly supported solutions also for 1 < q < m. The dynamics of both Eq. (1.4) and Eq. (1.1) seem to be by far more involved in the range m ≥ 1 and 0 < q < 1, also known as the strong absorption range due to the fact that the absorption term prevails. This dominance gives rise to two new mathematical phenomena not present for q ≥ 1. On the one hand, finite time extinction of (non-negative bounded) solutions occurs. This means that there exists a time T e ∈ (0, ∞) such that u(t) ≡ 0 for t ∈ (0, T e ) but u(T e ) ≡ 0, T e being thus called the extinction time of u. The finite time extinction stems from the ordinary differential equation ∂ t u = u q obtained by neglecting the diffusion, emphasizing thus the strength of the absorption, see [START_REF] Kalasnikov | The propagation of disturbances in problems of non-linear heat conduction with absorption[END_REF][START_REF] Kalashnikov | Dependence of properties of solutions of parabolic equations on unbounded domains on the behavior of coefficients at infinity[END_REF]. On the other hand, instantaneous shrinking of supports of solutions to Eq. (1.4) with bounded initial condition u 0 such that u 0 (x) → 0 as |x| → ∞ takes place, that is, for any non-negative initial condition u 0 ∈ L ∞ (R N ) such that u 0 (x) → 0 as |x| → ∞ and τ > 0, there is R(τ ) > 0 such that supp u(t) ⊆ B(0, R(τ )) for all t ≥ τ . This is once more due to the strength of the absorption term, which involves a very quick loss of mass, and has been proved in [START_REF] Evans | Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities[END_REF][START_REF] Kalashnikov | Dependence of properties of solutions of parabolic equations on unbounded domains on the behavior of coefficients at infinity[END_REF] in the semilinear case m = 1 and in [START_REF] Abdullaev | Instantaneous shrinking of the support of a solution of a nonlinear degenerate parabolic equation[END_REF] for m > 1. Finer properties of the dynamics of Eq. (1.4) in this range are still lacking in general. For example, a description of the extinction rates and behavior near the extinction time of the solutions to Eq. (1.4) seems to be only available when m+q = 2 in [START_REF] Galaktionov | Extinction for a quasilinear heat equation with absorption I. Technique of intersection comparison[END_REF], revealing a case of asymptotic simplification, and appears to be a complicated problem if m + q = 2.

Let us now turn our attention to Eq. (1.1) with exponents as in (1.3) and introduce the critical exponent

σ * :=        2(1 -q) m -1 > 0, if m > 1, ∞, if m = 1. (1.5)
It is established in [START_REF] Belaud | Long-time vanishing properties of solutions of some semilinear parabolic equations[END_REF][START_REF] Kondratiev | Asymptotic behaviour of solutions of some nonlinear parabolic or elliptic equations[END_REF] (m = 1) and in [START_REF] Belaud | Time-vanishing properties of solutions of some degenerate parabolic equations with strong absorption[END_REF][START_REF] Belaud | Long-time extinction of solutions of some semilinear parabolic equations[END_REF] (m > 1) that any solution to Eq. (1.1) posed in a bounded domain Ω ⊂ R N with homogeneous Neumann boundary condition vanishes in finite time provided 0 < σ < σ * (the analysis performed in the above mentioned references actually deals with more general weights instead of |x| σ ). A similar result is shown in [START_REF] Belaud | Abstract results on the finite extinction time property: application to a singular parabolic equation[END_REF][START_REF] Belaud | Long-time vanishing properties of solutions of some semilinear parabolic equations[END_REF] (m = 1) and in [START_REF] Belaud | Extinction in a finite time for solutions of a class of quasilinear parabolic equations[END_REF] (m > 1) for homogeneous Dirichlet boundary conditions. A direct consequence of the latter and the localization of supports established in [START_REF] Iagar | Self-similar shrinking of supports and nonextinction for a nonlinear diffusion equation with strong nonhomogeneous absorption[END_REF]Theorem 1.1] is that bounded and non-negative weak solutions to (1.1)-(1.2) vanish identically after a finite time when σ ∈ [0, σ * ). We shall recall this result in Theorem 1.2 below and provide an alternative proof, relying on integral inequalities and the well-known L 1 -L ∞ -regularizing effect of the heat equation or the porous medium equation in R N . In the complementary range m > 1 and σ ≥ σ * , fewer results seem to be available but finite time extinction is not a generic feature. Indeed, we recently proved in [START_REF] Iagar | Self-similar shrinking of supports and nonextinction for a nonlinear diffusion equation with strong nonhomogeneous absorption[END_REF] that, when m > 1, σ > σ * , and the initial condition u 0 is positive in a neighborhood of the origin, the positivity set

P(t) := {x ∈ R N : u(t, x) > 0}
of the solution u to (1.1)-(1.2) is non-empty and contains the origin for all t ≥ 0. Moreover, P(t) shrinks to {0} as t → ∞. We also identify in [START_REF] Iagar | Self-similar shrinking of supports and nonextinction for a nonlinear diffusion equation with strong nonhomogeneous absorption[END_REF] a class of initial conditions u 0 for which 0 ∈ P(t) for all t ≥ 0; that is, u(t) vanishes at the origin where the absorption is the weakest. It is then tempting to figure out whether finite time extinction could occur and the purpose of this note is to answer this question by the affirmative.

Main results. We deal in this note with the Cauchy problem (1.1) -(1.2) and first make precise the notion of solution we are using in the present work.

Definition 1.1. A non-negative weak solution to the Cauchy problem

(1.1)-(1.2) is a function u ∈ L ∞ + ((0, ∞) × R N ) (1.6a)
such that, for all T > 0, Nevertheless, we give a short and much simpler proof based on estimates on the L 1 norm of the solution at different positive times, which also works for the spatially homogeneous equation (1.4). Let us also stress here that, in the semilinear case m = 1, σ * = ∞ and all bounded and non-negative solutions vanish in finite time. However, things are a bit more complex in the range m > 1 and σ ≥ σ * . Indeed, for σ > σ * , it is proved in [START_REF] Iagar | Self-similar shrinking of supports and nonextinction for a nonlinear diffusion equation with strong nonhomogeneous absorption[END_REF]Theorem 1.3] that solutions to the Cauchy problem (1.1)-(1.2) with an initial condition u 0 which is positive in a small ball B(0, r) converge as t → ∞ to a unique non-zero self-similar solution and in particular do no longer vanish in finite time.

u m ∈ L 2 (0, T ), H 1 loc (R N ) (1.6b) and T 0 R N (u 0 -u)∂ t ζ + ∇u m • ∇ζ + |x| σ u q ζ dxds = 0 (1.6c) for all ζ ∈ C 1 c ([0, T ) × R N ).
We thus infer that we need to restrict the class of initial conditions u 0 for the finite time extinction to hold true. This is made precise in the following statement, which is the main result of this note.

Theorem 1.3. Let m > 1, 0 < q < 1 and σ ≥ σ * . Consider a, A and R > 0 such that a > σ/(1 -q) and A m-q R a(m-q)-σ-2 ≤ 1 am(am + N -2)

.

(1.7)

Let u 0 ∈ L ∞ + (R N ) be such that u 0 (x) ≤ A|x| a , for any x ∈ B(0, R), (1.8a) and u 0 ∞ ≤ AR a . (1.8b)
Then the solution u to (1.1)-(1.2) vanishes in finite time.

According to our previous results in [START_REF] Iagar | Self-similar shrinking of supports and nonextinction for a nonlinear diffusion equation with strong nonhomogeneous absorption[END_REF], the flatness condition (1.8a) as |x| → 0 is required in order to have finite time extinction. Moreover, in a forthcoming work [START_REF] Iagar | Eternal solutions for a quasilinear diffusion equation with strong nonhomogeneous absorption[END_REF] we will show that a limitation on u 0 ∞ such as the one in (1.8b) is also needed, as a selfsimilar solution with exponential time decay as t → ∞ and a dead core (that is, support located in an annulus far away from the origin) can be constructed for σ = σ * . We may thus conclude that our results are qualitatively sharp. A corollary of Theorem 1.3 which provides examples of initial conditions u 0 to which it applies is given in Section 3 after its proof.

2 Proof of Theorem 1.2

This section is dedicated to the proof of Theorem 1.2. Before the beginning of the proof, we recall here as a preliminary the precise statement of the well-posedness theorem for Eq. (1.1), which can be found in [START_REF] Iagar | Self-similar shrinking of supports and nonextinction for a nonlinear diffusion equation with strong nonhomogeneous absorption[END_REF]Theorem 1.1]. It includes properties such as instantaneous shrinking, localization of supports of bounded solutions and the comparison principle, that will be used in the sequel.

Theorem 2.1. For any m > 1, q ∈ (0, 1) and σ > 0, there is a unique non-negative weak solution to the Cauchy problem (1.1), (1.2) which satisfies

u(t) ∞ ≤ u 0 ∞ , t ≥ 0. (2.1)
In addition, it enjoys the properties of instantaneous shrinking and localization of the support; that is, for any t > 0, u(t) has compact support and, given τ > 0, there exists R = R(τ ) > 0, depending on u 0 and τ but not on t ∈ [τ, ∞), such that supp u(t) ⊆ B(0, R(τ )), for any t ≥ τ.

Also, the following comparison principle holds true: given u 0,i ∈ L ∞ + (R N ), i = 1, 2, such that u 0,1 ≤ u 0,2 in R N , the corresponding non-negative weak solutions u 1 and u 2 to (1.1), (1.2)

satisfy u 1 ≤ u 2 in (0, ∞) × R N .
For a proof, we refer the reader to [12, Sections 2 and 3], while similar results for the semilinear case m = 1 follow by a simple adaptation of the proofs. With these preparations, we are ready to prove our first main result. This is done by adapting an argument from [START_REF] Benachour | Extinction and decay estimates for viscous Hamilton-Jacobi equations in R N[END_REF].

Proof of Theorem 1.2. According to Theorem 2.1, we may assume without loss of generality that u 0 is compactly supported and that there exists R > 0 such that supp u(t) ⊂ B(0, R), for any t > 0.

(

Owing to the non-negativity of u, it follows from Eq. (1.1) that for any T > 0 and t ∈ (0, T ) we have

T t R N |x| σ u q (s, x) dx ds ≤ u(T ) 1 + T t R N |x| σ u q (s, x) dx ds ≤ u(t) 1 . (2.3) Pick now b ∈ R such that σ N + 1 -q < b < σ * N + 1 -q. ( 2.4)
We then infer from (2.3) that

u(t) 1 ≥ T t u(s) -b ∞ R N |x| σ u(s) b ∞ u q (s, x) dx ds ≥ T t u(s) -b ∞ R N
|x| σ u b+q (s, x) dx ds.

(2.5)

Next, on the one hand we deduce from (2.2), Hölder's inequality and the non-negativity of u that, for any s ∈ (t, T )

u(s) 1 = B(0,R) |x| σ/(b+q) u(s, x)|x| -σ/(b+q) dx ≤ B(0,R) |x| σ u b+q (s, x) dx 1/(b+q) B(0,R) |x| -σ/(b+q-1) dx (b+q-1)/(b+q) ≤ C R N |x| σ u b+q (s, x) dx 1/(b+q)
, since (2.4) guarantees that σ/(b + q -1) < N . We thus find that The adaptation for m = 1 is immediate as then θ = N/2 and the terms involving b just cancel in the previous calculation. We thus have N + 2bθ > N (b + q) and it follows from [27, Lemme 4.1] that u vanishes in finite time.

u(s) b+q 1 ≤ C R N |x| σ u b+q (s, x) dx . ( 2

Theorem 1 . 2 .

 12 Basic results of well-posedness of the Cauchy problem and of instantaneous shrinking and localization of supports of solutions in the framework of Definition 1.1 are established in [12, Theorem 1.1], the latter being restated as a preliminary fact at the beginning of Section 2. Let us next recall that finite time extinction always occurs if σ < σ * , see [4, Theorem 3.1] and [3, Theorem 3.16] (m = 1) and [6, Theorem 2.1] (m > 1). If m and q are as in (1.3), u 0 is as in (1.2) and σ < σ * , then the solution u to the Cauchy problem (1.1)-(1.2) vanishes in finite time. As already pointed out, Theorem 1.2 follows from the above mentioned references and the property of localization of supports of any solution to the Cauchy problem (1.1)-(1.2).

1 , ( 2 . 9 )

 129 t) bθ ds = 1 bθ + 1 (T -t) bθ+1 u(T ) b+q 1 ,which can be written in an equivalent form asu(T ) 1 ≤ C(T -t) -(bθ+1)/(b+q) u(t) (N +2bθ)/N (b+q)which holds true for any t ∈ (0, T ). Let us notice that, when m > 1,N + 2bθ -N (b + q) = N (1 -q) -N 2 (m -1) N (m -1) + 2 b = N N (m -1) + 2 [(1 -q)(N (m -1) + 2) -N (m -1)b] > 0, since we deduce from (2.4) that b < 1 -q + σ * N = (1 -q)(N (m -1) + 2) N (m -1).

  Combining the estimates (2.5), (2.6) and (2.7) gives, for any s ∈ (t, T ), that u(t) 1 ≥ C

							.6)
	On the other hand, since u is a subsolution to the porous medium equation if m > 1 or
	to the heat equation if m = 1, it follows from the well-known regularizing effect of the
	porous medium equation (see for example [28, Theorem 2.1]) if m > 1 or from the standard
	representation formula for the heat equation if m = 1 that
	u(s) ∞ ≤ C(s -t) -θ u(t)	2θ/N 1	,	s ≥ t,
	where θ = N/(N (m -1) + 2). Consequently, taking into account that b > 0, we have
	u(s) -b ∞ ≥ C(s -t) bθ u(t)	-2bθ/N 1	.	(2.7)
	t	T	(s -t) bθ u(t)	-2bθ/N 1	u(s) b+q 1	ds,
	or equivalently

T t (s -t) bθ u(s) b+q 1 ds ≤ C u(t) (N +2bθ)/N 1 .

(

2.8)

Since u(s) 1 ≥ u(T ) 1 for s ∈ (t, T ) by (1.1), we further infer from (2.8) that C u(t)

Acknowledgements. The authors are partially supported by the Spanish project PID2020-115273GB-I00. R. G. I. wants to thank for the hospitality and support of Institut de Mathématiques de Toulouse where part of this work has been done.

3 Proof of Theorem 1.3 Throughout this section, we fix m > 1, q ∈ (0, 1) and σ ≥ σ * . Before starting the proof of Theorem 1.3, we need a preparatory result on the availability of suitable stationary supersolutions to Eq. (1.1). Lemma 3.1. Given a ≥ (σ + 2)/(m -q) and A > 0, R > 0 satisfying (1.7), the function

is a supersolution to Eq. (1.1) on (0, ∞) × B(0, R).

Proof. Setting l := a(m -q) -σ -2 ≥ 0, and fixing some x ∈ B(0, R), we obtain by direct calculation that

where the last inequality follows from (1.7).

We are now in a position to complete the proof of Theorem 1.3.

Proof of Theorem 1.3. It follows from (1.8a) that

and from (1.8b) and (2.1) that

and recalling that A and R satisfy (1.7), Lemma 3.1 and the comparison principle in Theorem 2.1 entail that

On the one hand, the previous inequality implies that, for (t, x) ∈ (0, ∞) × B(0, R),

On the other hand, for (t, x) ∈ (0, ∞) × (R N \ B(0, R)), we have |x| ≥ R, which gives, along with (2.1),

Introducing

we obtain from (3.1) and (3.2) that

Therefore, the comparison principle gives that u ≤ v in (0, ∞)×R N , where v is the solution to

with the same initial condition v(0) = u 0 in R N . Since (aq + σ)/a < 1 by the choice of a, v vanishes in finite time and so does u.

We provide more precise examples of initial data for which finite time extinction holds true when σ ≥ σ * in the following consequence of Theorem 1.3.

Corollary 3.2. Let m > 1, 0 < q < 1, σ ≥ σ * and consider a > σ/(1 -q) and u 0 ∈

for some A 0 > 0. Then there exists M > 0 depending only on m, a, N , q, σ and A 0 such that, if u 0 ∞ ≤ M , then the solution u to the Cauchy problem (1.1)-(1.2) vanishes in finite time.

Proof. We set K := [am(am + N -2)] 1/(m-q) , l := a(m -q) -σ -2 > 0, and choose A = A 0 , R = (KA 0 ) -(m-q)/l , M = A 0 R a .

Pick u 0 ∈ L ∞ + (R N ) satisfying (3.3) and u 0 ∞ ≤ M . Then A and R satisfy (1.7) and u 0 satisfies (1.8), so that we are in the hypothesis of Theorem 1.3. An application of this theorem ends the proof.


Zohaib Amjad Khan
email: zohaib.khan@l2s.centralesupelec.fr

Giuseppe Valenzise
email: giuseppe.valenzise@l2s.centralesupelec.fr

Aladine Chetouani
email: aladine.chetouani@univ-orleans.fr

Frédéric Dufaux
email: frederic.dufaux@l2s.centralesupelec.fr

Towards an Image Utility Assessment Framework for Machine Perception

Keywords: utility for machines, machine perception, task-based assessment, image utility assessment

In real-world applications, images and videos used in computer vision algorithms are often distorted due, e.g., to compression and transmission. As a result, they may lose relevant information content, or they may deviate significantly from the original data distribution used to train the machine task, rendering the visual content practically useless with respect to its initial purpose. Evaluating the utility of an image for machine tasks has received little attention so far in the literature. This concept of utility is substantially different from the visual quality typically used in image/video compression, as the latter is related to the perception of the human visual system. In this paper, we propose a definition of utility as the degree of confidence by which a machine task is able to take a decision. In this context, we propose a full-reference utility loss measure: we assume that the decision on the pristine image is correct (reference), and we measure the utility loss as the confidence reduction in the decision due to a noisy input with respect to this reference. We apply this general definition on two specific tasks, classification and object detection, and we study practical solutions to predict utility, as well as the ability of our utility measure to generalize across tasks.

I. INTRODUCTION

It is well known that the concept of visual quality for the human visual system (HVS) is different from the quality of an input image/video to perform a computer vision task [START_REF] Borowski | The notorious difficulty of comparing human and machine perception[END_REF] [2]. For example, a machine-based task may produce the same outcomes for two images with big differences in visual image quality and vice versa. In real world scenarios, the distortion introduced during acquisition and transmission impacts the utility of images for the machine tasks, which are generally trained on pristine examples [START_REF] Dodge | Human and dnn classification performance on images with quality distortions: A comparative study[END_REF]. Retraining the networks with distorted images in this case is not only time-consuming and unfeasible, but also an intrusive and a non-generalizable solution. Instead, having a measure of utility could be used as an optimization constraint by machine tasks [START_REF] Codevilla | Learned image compression for machine perception[END_REF] or as a more generalized and non-intrusive solution for improving the image utility and machine performance.

Existing image quality assessment methods have been designed by modeling human perception, and are typically trained on datasets with subjective mean opinion scores labels.

This work was funded by BPI France in the framework of the SMART-V2I project.

(a) PSNR (b) SSIM [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF] (c) OOD [START_REF] Liu | Energy-based out-of-distribution detection[END_REF] (d) Adversarial [START_REF] Martin | Inspecting adversarial examples using the fisher information[END_REF] Fig. 1: Histograms for different metrics with binary utility Image quality in this context is defined in terms of either the level of visual impairment in the image, or its perceptual quality compared to a pristine reference [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF]. This makes visual quality metrics essentially unsuitable to predict image utility for machines. Moreover, it is pertinent to mention here that the concept of machine image utility assessment is not to be confused with anomaly detection often used in relation to Deep Learning (DL) methods. Anomaly detection methods such as out-of-distribution (OOD) detection and adversarial image detection are concerned with special cases only, whereas the concept of image utility for machines is universal and applies to normal application scenarios as given by examples before. OOD images belong to the class of images that are different to the training set, whereas adversarial images are similar to training images but have intentional non-perceptual perturbations to create mis-classification errors. None of these two scenarios applies here. Fig. 1 illustrates how these other notions are unsuitable for assessing image utility with a simple example. For this example, a small set of images from imagenet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] validation data were selected, from which another set was generated by compressing them at different bitrates. Both these compressed and uncompressed images were then passed through a pretrained DL-based classification network. Each plot in the figure consists of two histograms, where the blue histogram represents those compressed images that are given the same labels as their uncompressed versions and the orange histogram shows the frequency of wrongly classified images. Each plot shows a representative method from among the quality assessment and anomaly detection class of methods [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF]- [START_REF] Martin | Inspecting adversarial examples using the fisher information[END_REF]. It can be seen from the plots that none of the existing notions capture the concept of image utility for machines as there is a large overlap between the two histograms in each plot. Therefore, in order to assess the image utility there is a need to first formally define this concept and then based on that to design an effective methodology for evaluating it. This work is an initial contribution in this regard as we present the notion of image utility for machines and a methodology for its evaluation followed by exploring its use and effectiveness.

II. RELATED WORK

Recently, some attention has been given in understanding the concept of machine perception and its differences from human perception. Generally, this is being done either by understanding how an input image affects the different parts of the network or by studying the properties of images which are suitable for the machines. For instance, in [START_REF] Borkar | Deepcorrect: Correcting dnn models against image distortions[END_REF], the authors have proposed to find the convolutional filters most affected by the distortions introduced during image acquisition and transmission, and then have proposed a correcting mechanism for those. On the other hand, in [START_REF] Dodge | Human and dnn classification performance on images with quality distortions: A comparative study[END_REF], the authors have studied the differences between human and Deep Neural Network (DNN) classification performance for images distorted by blur and noise. They have shown that humans outperform machine based classifiers even when they are retrained using distorted images. Besides these, some other works like [START_REF] Lin | Fidelity estimation improves noisy-image classification with pretrained networks[END_REF], [START_REF] Takagi | Quality recovery for image recognition[END_REF] have also proposed image correction and recovery methods to improve the outcomes of DL networks in case of distorted images without retraining. Another interesting work [START_REF] Jin | Just noticeable difference for deep machine vision[END_REF] has recently explored the concept of Just Noticeable Difference (JND) for machine vision, whereby they have shown that like for HVS, it is possible to find JND for machines also. However, all of these works focus on single machine task i.e. image classification and do not focus on evaluating the overall image utility.

With regards to machine utility assessment, the most relevant work has been done in [START_REF] Paul | Aqua: Analytical quality assessment for optimizing video analytics systems[END_REF] for videos, where authors have named it as analytical quality and have only considered the classification task. In that work, the authors have proposed to use a mean classifier opinion score (MCOS) instead of Mean Opinion Score (MOS) for training quality assessment networks. MCOS is evaluated using average of a combination of the class confidence and class rank from 5 different classifiers. Then, they have used different thresholds in different applications to filter out frames. However, in that work they only focus on image classification and then use the same MCOS for all other machine tasks. Contrary to that, in this work we will first consider image utility for machines in general, before providing a task-specific definition as done in the next section.

III. IMAGE UTILITY FOR MACHINES

For a specific machine task like image classification, face recognition, object detection etc., we can define image utility in terms of the confidence in prediction (no-reference) or change in prediction confidence from that of a pristine image prediction (full-reference). Based on that, we first define here image utility for machines in general terms below, followed by task-specific definition.

A. General Definition

Let's consider a pristine image I ′ and a distorted image I both of which have been passed through a DNN i for a specific task. Let's also consider S as the set of M possible outcomes for the task, which also contains the ground-truth for images I ′ and I i.e. S = {S 1 , S 2 , ..S M }. When an image is passed through i, the output is a prediction P such that P i (I) ∈ S along with a set of confidence values associated for each member of the outcomes set. If ϕ ′ i is the confidence value assigned to the ground-truth when I ′ is passed and ϕ i is the confidence value assigned to the ground-truth when I is passed, then utility U i for the DNN i can be defined as a function of ϕ ′ i and ϕ i as

U i (I) = f (ϕ i , ϕ ′ i ) (1)
If ϕ rec i is the recognition confidence threshold below which the prediction is rejected, we propose to define f (ϕ i , ϕ ′ i ) as

f (ϕ i , ϕ ′ i ) = ϕ i -ϕ rec i 1 -(ϕ ′ i -ϕ rec i ) (2)
The higher the value of this function, the higher the utility value. For the case where the distorted image gives the wrong prediction i.e. ϕ i < ϕ rec i , this becomes negative. If we have a set of possible DNNs for a specific task, then image utility for machines U can be defined as

U(I) = O(U ) (3)
where O is some kind of operator like mean, max or majority vote and U is a set of utility values of N different DNNs i.e.

U (I) = {U 1 (I), U 2 (I), .., U N (I)} (4)

B. Task-Specific Utility

It is possible to extend and simplify the general Eqs.(1-3) to any specific machine task, T . In the following, we show how this notion of image utility may be simplified for the two most common machine vision tasks of image classification and object detection.

1) Image Classification: Let's consider the simplest example of image classification first with N classifiers and consider all images with wrong predicted class or f (ϕ i , ϕ ′ i ) < 0 as having no utility and those with positive values as 'useful'. Hence, the utility function for each individual network with

T = C becomes U (C) i (I) = 1, if P i (I) = S(I) 0, otherwise (5)
where P i (I) and S(I) are the predicted and the groundtruth classes respectively. The combined utility from all the networks in this case U (C) (I) can then be evaluated using some kind of operator like majority voting.

2) Object Detection: Let's consider another task of object detection where T = OD. The utility of an object detector lies in two aspects namely correct prediction of objects and their correct localization. Hence, each outcome in the set S in this case is a subset consisting of a label L and the localization box B i.e. S j = {L j , B j }. The ground-truth for an image may be one or multiple of these outcome subsets. For L, we apply the same condition of f (ϕ i , ϕ ′ i ) < 0 for no utility and vice versa. For correct localization we can use a threshold of 0.5 on its associated confidence score ϕ loc . Let's consider the simplest case where ground-truth consists of a single possible outcome. Hence, individual detector utility in this case becomes

U (OD) i (I) = 1, if P i (I) = S(I) and ϕ loc > 0.5 0, otherwise (6)
The combined image utility for object detection task can then be obtained the same way as for classification.

It is not feasible or in any way useful to evaluate these utility functions by passing the image through the inputting network of the task. The real usage of this comes if we know the utility of the image beforehand. Hence, there is a need to predict value of this utility function for each input image by having a utility assessment module. In the next section, we propose one such assessment method.

IV. MACHINE IMAGE UTILITY ASSESSMENT

For our proposed machine image utility assessment method, we consider a two-class formulation consisting of 'useful' and 'non-useful' labels as defined by the utility functions U (C) and U (OD) resulting from combining outputs from Eq.( 5) and ( 6). These labels are generated using a standard DNN like Inception-v3 [START_REF] Szegedy | Rethinking the inception architecture for computer vision[END_REF] for classification and a standard object detector like Faster R-CNN [START_REF] Ren | Faster R-CNN: Towards realtime object detection with region proposal networks[END_REF]. For classification, we also include here a threshold on the correct class softmax probability (CCSP) in the definition. Hence, the 'useful' label is given to an image whose predicted class is the same as the original image with a CCSP exceeding a threshold T H . On the other hand, any image which is wrongly classified by the DNN with a CCSP below a threshold T L is given a label 'non-useful'. Similarly, for object detection, we add an additional constraint of the Intersection over Union (IoU) value threshold. If at least one of the boxes in the image is detected with an IoU value of greater than and equal to 0.5, we consider it as an image with utility 1. On the other hand, all those images for which none of the boxes are detected with an IoU value of greater than or equal to 0.5 are considered as images with utility 0.

To understand the labeling process, let us consider an image I ′ that has an associated ground-truth S(I ′ ). In this work, we consider the specific case of JPEG image compression. In order to label the images, we first obtain the label of the uncompressed image P(I ′ ) using the labeling network. Then we compress the image using JPEG compression to get I and pass it through the same network to get its label P(I). For the classification task, if P(I ′ ) and P(I) are the same and also match the ground-truth label S(I ′ ) with CCSP exceeding T H , I is given a utility label of 1 where utility label belongs to one of the two classes in set {0, 1}. Here class 0 represents a 'non-useful' image and class 1 represents a 'useful' image. On the other hand, if P(I ′ ) = S(I ′ ) but P(I) is different with CCSP less than T L , I is given a utility label of 0. This way, we consider only the impact of distortion on utility. Similarly, for the object detection if at least one of the many predicted outcomes P(I ′ ) and P(I) are the same and also match the ground-truth label S(I ′ ) with confidence value and IoU both greater or equal to 0.5, it is given the utility value of 1. All other images with no right labels or having lower confidence scores or IoU values are given a utility label of 0. Finally, for combining the labels from multiple networks, the images assigned utility 1 from all networks are given the label 1, whereas all those with 0 assigned from all networks are considered as utility 0.

Once the labels are generated, we train a binary utility assessment network using these compressed images. In this work, we propose to use a baseline Resnet-18 network [START_REF] He | Deep residual learning for image recognition[END_REF] for utility assessment. The loss function used during training is the cross-entropy loss. The trained network can then be used to identify the useful and non-useful images from test images.

V. EXPERIMENTAL RESULTS

Besides finding out the accuracy of our assessment network for different tasks, we have evaluated our proposed utility assessment framework with focus on two questions (i) Can we use the utility evaluated for one network for all others in a single task? and (ii) Can we use the utility evaluated for one machine task for another (inter-task generalization)? We have considered two machine tasks here namely image classification and object detection.

For classification, we have used the Imagenet validation set [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] (50000 images) to generate the JPEG compressed images at 9 different compression ratios. For classification networks, we have used three standard networks namely the Inception-v3 [START_REF] Szegedy | Rethinking the inception architecture for computer vision[END_REF], VGG-19 [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF] and Densenet-121 [START_REF] Huang | Densely connected convolutional networks[END_REF]. For CCSP, the two thresholds T H and T L used were 0.9 and 0.1 respectively. Once the labels were generated, we divided the dataset into training and test set with a 80-20 ratio (around 200,000 training images and 50000 test images per class) ensuring that the images in the two sets do not overlap. In order to balance the images for the two utility classes, we also added images for utility 0 class using data augmentation techniques followed by relabeling. Finally the Resnet-18 was trained using Stochastic Gradient Descent method with a learning rate of 0.0001. Similarly, for object detection we have taken the COCO 2017 validation set [START_REF] Lin | Microsoft coco: Common objects in context[END_REF] (5000 images) and generated the JPEG compressed images at 9 different compression ratios from them. For carrying out object detection, we have selected three common models namely Faster-RCNN [START_REF] Ren | Faster R-CNN: Towards realtime object detection with region proposal networks[END_REF], SSD-300 [START_REF] Liu | SSD: Single shot multibox detector[END_REF] and SSDLite-320 [START_REF] Sandler | Mobilenetv2: Inverted residuals and linear bottlenecks[END_REF], [START_REF] Howard | Searching for mobilenetv3[END_REF]. The train-test ratio used was again 80-20 (around 20000 training images and 5000 test images per class) and images for utility 0 class added using data augmentation for class balancing. The same network, Resnet-18 was then trained for utility prediction using the same hyperparameters.

A. Results from individual and combined networks

Table I shows the accuracy performance of our proposed utility assessment network for different classification networks. We can see that there are slight differences in performance of the network when the classification network is changed for defining utility. For individual networks, Densenet-121 gives the best accuracy results of 91.2% followed by Inception with 87.5% and VGG-19 with 84.2%. We have also evaluated the results for a combined utility evaluated from all these networks using common predicted class. This resulted in a better training than the individual ones as we obtained the binary utility classification accuracy of 93.3%.

Similarly, Table II lists the accuracy of the utility assessment network for different object detectors. Here again, we can see that the combined utility defined as the common output from different object detectors gives the best classification accuracy of 92.8%. For prediction of utility for individual detectors, the classification accuracy is much worse in this case with the second best accuracy of only around 84.8% by the Faster R-CNN Resnet-50 method. The proposed method gives the worst performance for SSD-300 VGG-16 detector with classification accuracy of only 81.8%.

B. Intra-task generalization across machine types (networks)

One of the fundamental benefits of the utility assessment module is to be able to deploy it before any network to identify the useful inputs only and filter out the rest or enhance them before reuse. Ideally, any utility assessment method should be independent of the classification network and work well for all kinds of applications. To evaluate this generalization ability of our utility assessment network for the classification task, we train it for one classification network or a combination of them and then observe the change in performance for other networks after filtering out the images having 0 utility. These other networks selected are AlexNet [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], ResNext-50 [START_REF] Xie | Aggregated residual transformations for deep neural networks[END_REF] and EfficientNet-b7 [START_REF] Tan | Efficientnet: Rethinking model scaling for convolutional neural networks[END_REF]. Similarly, for evaluating the generalization for the object detection task, we train it for one object detector or a combination of them and then observe the change in performance for other detectors after replacing the images with 0 utility with the uncompressed ones to allow for mAP evaluation. The other object detectors used for this experiment are Mask R-CNN [START_REF] He | Mask R-CNN[END_REF], RetinaNet [START_REF] Lin | Focal loss for dense object detection[END_REF] and Faster R-CNN MobileNet-v3 [START_REF] Ren | Faster R-CNN: Towards realtime object detection with region proposal networks[END_REF], [START_REF] Howard | Searching for mobilenetv3[END_REF]. We used subsets of imagenet and COCO validation datasets with images compressed at bitrates other than used for training for classification and object detection respectively. Figure 2(a) shows how using the utility network impacts the performance of other classification networks with a clear improvement in accuracy in all the tested networks. This implies that our proposed network generalizes significantly well across different networks, although there is still a big room of improvement to get the ideal results. Similarly, Figure 2(b) illustrates the impact of using utility

C. Inter-task generalization across machine tasks

In order to evaluate whether the utility prediction network can be used globally when trained on one single task, we have also evaluated the inter-task generalization. To do this, we used the network trained for classification utility (from 3 combined networks) and evaluated its impact on object detection task and vice versa. Tables III and IV show the difference in accuracy and mAP values for classification and object detection tasks respectively. From the tables, we can clearly observe that only negligible difference in values (around 2% or less) is observed when utility training from one task is used to detect utility for another task. Hence, this shows that our proposed framework is generalizable across tasks.

VI. CONCLUSION

In this work, we have presented a formal specification and definition of a very important concept of image utility for machines for the very first time. We have further presented a simple strategy for predicting this machine image utility using training with outputs of multiple classification networks as well as of multiple object detectors. Experimental results have shown the effectiveness and generalization of our proposed method. Hence, such a method of predicting image utility for machines can not only be used in real-world applications to improve outcomes where images are affected by common distortions, but also for optimizing a machine task output for another subsequent machine task. This work can serve as an important first step in building more efficient machine utility assessment methods and for developing algorithms and networks optimized for machine perception. For future work, we also plan to extend this work to include more distortions.

Fig. 2 :

 2 Fig. 2: Impact of using the proposed utility assessment method trained for utility of (a) different classification networks on accuracy of test classification networks and of (b) different object detection networks on mAP of test detection networks

TABLE I :

 I Performance of utility assessment network trained for utility of different classification networks

	Classification Network for defining utility Accuracy
	Inception-v3	87.5%
	VGG-19	84.2%
	Densenet-121	91.2%
	Inception+VGG+Densenet	93.3%

TABLE II :

 II Performance of utility assessment network trained for utility of different object detection networks

	Object Detection Network for defining utility Accuracy
	Faster R-CNN Resnet-50	84.8%
	SSD-300 VGG-16	81.3%
	SSD-320 Mobilenet-v3	83.1%
	SSD-300 + SSD-320 + Faster R-CNN	92.8%

TABLE III :

 III Training Impact for Classification Accuracies

	Network	Classification trained OD trained
	AlexNet	46.8%	46.0%
	ResNext-50	66.3%	65.4%
	EfficientNet-B7	65.8%	65.9%

TABLE IV :

 IV Training Impact for Object Detection mAPs

	Network	OD trained Classification trained
	Mask R-CNN	0.128	0.126
	RetinaNet	0.128	0.125
	FR-CNN MobileNet	0.164	0.161
	network for the object detectors. Here, we can see that
	mean Average Precision (mAP) values (though low due to
	compression artifacts) increase when the utility network is
	used although this increase is smaller when utility assessment
	network is trained for multiple object detectors utility.