Marie E Vuillemin

Florentin Michaux

Aurélie Seiler

Michel Linder

Lionel Muniglia

Jordane Jasniewski
email: jordane.jasniewski@univ-lorraine.fr

Polysaccharides enzymatic modification to control the coacervation or the aggregation behavior: a thermodynamic study

Keywords: Complex coacervation, supramolecular structures, Isothermal titration calorimetry, gum Arabic, chitosan, interactions

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

Introduction

The balance of interactions between molecules or between molecules and solvent represents the driving force for the formation of self-assembled colloidal systems. In recent years, the compartmentalized environments of cells have been mimicked in bottom-up systems through the use of condensed polymeric phases [START_REF] Aumiller | RNA-Based Coacervates as a Model for Membraneless Organelles: Formation, Properties, and Interfacial Liposome Assembly[END_REF][START_REF] Crowe | Liquid-liquid phase separation in artificial cells[END_REF][START_REF] Jia | Rapid RNA Exchange in Aqueous Two-Phase System and Coacervate Droplets[END_REF][START_REF] Poudyal | Template-directed RNA polymerization and enhanced ribozyme catalysis inside membraneless compartments formed by coacervates[END_REF]. In these phases, oppositely charged polyelectrolytes condense into a polymer-rich phase due to their mutual electrostatic interactions. This process is called complex coacervation [START_REF] Yewdall | Physicochemical Characterization of Polymer-Stabilized Coacervate Protocells[END_REF]. Coacervates result from spontaneous mechanisms coming from interactions between building blocks as surfactants, proteins, polysaccharides [START_REF] Kötz | Self-assembled polyelectrolyte systems[END_REF][START_REF] Meyer | Modifications in stability and structure of whey protein-coated o/w emulsions by interacting chitosan and gum arabic mixed dispersions[END_REF]. Several types of interactions can exist to produce such complex structures as electrostatic, hydrophobic, Van der Waals [START_REF] Sahoo | Injectable network biomaterials via molecular or colloidal self-assembly[END_REF][START_REF] Whitesides | Molecular self-assembly and nanochemistry: A chemical strategy for the synthesis of nanostructures[END_REF]. These are described by their strength and their range [START_REF] Campbell | Effect of temperature on the intrinsic viscosity and conformation of chitosans in dilute HCl solution[END_REF][START_REF] Neel | Exploiting non-covalent π interactions for catalyst design[END_REF].

The understanding of the formation of supramolecular structures represents a key point to consider a possible control of the organization at the colloidal scale. Modifying the interactions in a given system to investigate their influence on the colloidal structure formed is therefore of major interest. There are two ways of modifying these interactions. The first one consists in controlling the physical-chemical properties of the medium like pH and/or ionic strength to modify electrostatic interactions for example. This strategy is well known and widely reported in the literature [START_REF] Butstraen | Preparation of microcapsules by complex coacervation of gum Arabic and chitosan[END_REF][START_REF] Coelho | Nanostructure of polysaccharide complexes[END_REF][START_REF] Espinosa-Andrews | Gum Arabic-Chitosan Complex Coacervation[END_REF][START_REF] Kizilay | Complexation and coacervation of polyelectrolytes with oppositely charged colloids[END_REF][START_REF] Vuillemin | Gum Arabic and chitosan self-assembly: Thermodynamic and mechanism aspects[END_REF]. To modify interactions between two molecules, a J o u r n a l P r e -p r o o f second possible strategy is to chemically modify the building blocks. This can lead to decrease, suppress or strengthen some potential interactions. Assemblies formed from biopolymers (protein, polysaccharides) appear to be good candidates for this purpose since the structures formed depend on the type and strength of the interaction. Since these molecules are often polyelectrolytes, electrostatic interactions frequently direct the formation of complexes, coacervates or aggregates. It is recognized that strong interactions between the building blocks mainly lead to the formation of aggregates while weaker interactions promote the formation of coacervates [START_REF] Comert | Liquid-liquid and liquid-solid phase separation in proteinpolyelectrolyte systems[END_REF]. Modifying the chemical structures of the molecules can change their interactions and lead to controlling the resulting colloidal structure. In this work, the influence of chemical modifications of polysaccharides on the supramolecular structure they spontaneously form in an aqueous phase was investigated. Therefore, Gum Arabic (GA) and chitosan (CN) were chosen for their ability to interact via electrostatic interactions in a given pH range and spontaneously form coacervates in water [START_REF] Vuillemin | Gum Arabic and chitosan self-assembly: Thermodynamic and mechanism aspects[END_REF].

CN is a polymer of glucosamine and N-acetyl glucosamine linked by β (1-4) glycosidic bonds [START_REF] Soni | Physicochemical, antimicrobial and antioxidant properties of chitosan/TEMPO biocomposite packaging J o u r n a l P r e -p r o o f films[END_REF]. It is derived from the deacetylation of chitin from crustacean shells.

The free amino groups of chitosan are protonated in solution, when the pH is below its pKa (i.e. between 6.3 and 7.0 [START_REF] Beppu | Influence of Calcification Solution on in vitro Chitosan Mineralization[END_REF]). In other words, CN is positively charged when protonated, which remains a property that few polysaccharides exhibit and it is the primary reason that led to the choice of CN in several studies on complex coacervation [START_REF] Avadi | Preparation and characterization of insulin nanoparticles using chitosan and Arabic gum with ionic gelation method[END_REF]Butstraen & Salauen, 2014;[START_REF] Espinosa-Andrews | Gum Arabic-Chitosan Complex Coacervation[END_REF], 2010;[START_REF] Maqbool | A combination of gum arabic and chitosan can control anthracnose caused by Colletotrichum musae and enhance the shelf-life of banana fruit[END_REF][START_REF] Roldan-Cruz | Electrical impedance spectroscopy for monitoring the gum Arabic-chitosan complexation process in bulk solution[END_REF][START_REF] Sakloetsakun | Chitosan-gum arabic polyelectrolyte complex films: physicochemical, mechanical and mucoadhesive properties[END_REF][START_REF] Tan | Polysaccharide-based nanoparticles by chitosan and gum arabic polyelectrolyte complexation as carriers for curcumin[END_REF]. Moreover, it has an extended conformation in acetic acid solution, so its charges are accessible [START_REF] Espinosa-Andrews | Gum Arabic-Chitosan Complex Coacervation[END_REF][START_REF] Schatz | Formation and Properties of Positively Charged Colloids Based on Polyelectrolyte Complexes of Biopolymers[END_REF]. Furthermore, it possesses attractive properties as this biopolymer is film-forming, biodegradable, biocompatible and exhibits antimicrobial and antioxidant activities [START_REF] Fathi | Nanoencapsulation of food ingredients using carbohydrate based delivery systems[END_REF][START_REF] Soni | Physicochemical, antimicrobial and antioxidant properties of chitosan/TEMPO biocomposite packaging J o u r n a l P r e -p r o o f films[END_REF][START_REF] Sonia | An overview of natural polymers for oral insulin delivery[END_REF].

J o u r n a l P r e -p r o o f 4 CN was chosen as the polycation for this study and GA was selected as the polyanion.

GA is a sticky exudate of air-solidified tree sap from the stems or branches of Acacia trees [START_REF] Avadi | Preparation and characterization of insulin nanoparticles using chitosan and Arabic gum with ionic gelation method[END_REF]. There are several species of Acacia. However, the two most exploited species are Acacia senegal and Acacia seyal, which essentially produce brittle gum [START_REF] Mcnamee | Emulsification and Microencapsulation Properties of Gum Arabic[END_REF]. GA contains three peculiar fractions: 85-90 wt. % of arabinogalactan, 10 wt. % of arabinogalactan-protein complex and 2 wt. % of one or two glycoproteins [START_REF] Lopez-Torrez | Acacia senegal vs. Acacia seyal gums -Part 1: Composition and structure of hyperbranched plant exudates[END_REF]. These fractions confer several different chemical groups to gum Arabic. It possesses amino groups due to its protein content. In addition, it contains carboxyl groups present on its glucuronic acid residues [START_REF] Padala | Adsorption of Gum Arabic, Egg White Protein, and Their Mixtures at the Oil-Water Interface in Limonene Oil-in-Water Emulsions[END_REF], which are responsible for the charge of GA in aqueous solution above its pKa of 3.6. GA is a hydrophilic polysaccharide that is highly soluble in water. It has a globular structure in aqueous solution (Moschakis et al., 2010;[START_REF] Renard | Structure of arabinogalactan-protein from Acacia gum: From porous ellipsoids to supramolecular architectures[END_REF]. Furthermore, it is used in various industrial applications as an additive in food formulation (E414) or for encapsulation [START_REF] Mosquera | Critical water activity and critical water content of freeze-dried strawberry powder as affected by maltodextrin and arabic gum[END_REF], for vectorization in the pharmaceutical industry [START_REF] Avadi | Preparation and characterization of insulin nanoparticles using chitosan and Arabic gum with ionic gelation method[END_REF]Bosnea et al., 2014[START_REF] Bosnea | Microencapsulated cells of Lactobacillus paracasei subsp. Paracasei in biopolymer complex coacervates and their function in a yogurt matrix[END_REF], as an adhesive in the cosmetic industry and for various other applications [START_REF] Verbeken | Exudate gums: Occurrence, production, and applications[END_REF].

Several authors have already studied the complex coacervation of GA and CN [START_REF] Butstraen | Preparation of microcapsules by complex coacervation of gum Arabic and chitosan[END_REF][START_REF] Coelho | Nanostructure of polysaccharide complexes[END_REF][START_REF] Espinosa-Andrews | Gum Arabic-Chitosan Complex Coacervation[END_REF][START_REF] Espinosa-Andrews | Determination of the gum Arabic-chitosan interactions by Fourier Transform Infrared Spectroscopy and characterization of the microstructure and rheological features of their coacervates[END_REF][START_REF] Tan | Polysaccharide-based nanoparticles by chitosan and gum arabic polyelectrolyte complexation as carriers for curcumin[END_REF].

The influence of several parameters such as pH, ionic strength, temperature, or polyelectrolyte concentration on GA and CN coacervation has been well characterized but there is still a need to deepen the knowledge on the thermodynamic pathways that govern this phenomenon. In recent studies, the mechanism of coacervate formation and the thermodynamic parameters driving the complex formation have been investigated [START_REF] Vuillemin | Gum Arabic and chitosan self-assembly: Thermodynamic and mechanism aspects[END_REF]. It was demonstrated that pH and temperature affected the thermodynamics of GA-CN interactions (especially at 45 °C and pH 5.5) and that coacervation was driven by enthalpy. The electrical charges arise from the presence of negatively charged carboxylic groups on GA and positively J o u r n a l P r e -p r o o f charged amine groups on CN, respectively. These chemical groups can be easily modified to induce (i) a decrease in electrical charge by replacing the chemical function with another chemical group and (ii) an addition of new functional groups that can introduce new possible interactions, thus obtaining different supramolecular structures. Some authors have previously attempted to modify chitosan by chemical methods to shed light on the impact of modification on coacervate formation with gum Arabic [START_REF] Huang | Complex Coacervation of O-Carboxymethylated Chitosan and Gum Arabic[END_REF][START_REF] Huang | Characterization of O-Carboxymethyl Chitosan -Gum Arabic Coacervates as a Function of Degree of Substitution[END_REF](Huang et al., , 2017)). They showed that modification of chitosan by N, O-carboxymethylation or O-carboxymethylation reduced electrostatic interactions while maintaining high coacervation yields. Other driving forces such as hydrophobic interaction could be considered for chitosan and gum Arabic to form coacervates, especially at high temperatures. This could initially lead to a deeper understanding of the mechanisms and forces driving complex formation. The modification of polymers could lead to different conditions of complex formation than the optimal ones obtained with native polymers and to the formation of different supramolecular structures (particles, aggregates, coacervates…). However, the modification route used in existing studies on the subject is chemical modification. To make this strategy green and to modify chitosan and gum Arabic using the same methodology, a sustainable way consists in an approach using the laccase-catalyzed oxidation reaction of ferulic acid. This approach has the advantage of being eco-acceptable, as it is implemented under mild, aqueous conditions and at room temperature. Besides, it is not extremely specific and can be implemented on a considerable number of substrates. However, this method presents some disadvantages, such as the fact that the reaction is difficult to control, the products can be grafted at various locations, and the structure of the products formed remains poorly understood. However, the carboxylic and amino groups present on polysaccharides have already been modified by this strategy.

Indeed, in a study by [START_REF] Aljawish | Functionalization of chitosan by laccase-catalyzed oxidation of ferulic acid and ethyl ferulate under heterogeneous reaction conditions[END_REF], CN was functionalized by grafting ferulic acid (AF) oxidation products using Myceliophthora thermophila laccase. The authors showed J o u r n a l P r e -p r o o f that the functionalization of CN was achieved on C2 by forming a Schiff base (the formation of a C=N bond was proven by FTIR analyses). Grafting of these phenolic oxidation products onto the carboxyl group was also studied on GA [START_REF] Vuillemin | Polymer functionalization through an enzymatic process: Intermediate products characterization and their grafting onto gum Arabic[END_REF]. It was demonstrated that gum Arabic was grafted with oxidation products, close to the C6 of galactose units or close to the C5 of glucuronic acid units of gum Arabic. Furthermore, it was shown that the grafted products were mainly ferulic acid dimers, i.e. compounds with two aromatic rings, which could lead the modified polymers to establish new interactions such as π-π stacking or π-ion interactions. This strategy was implemented here to modify GA and CN. This paper aims to use these structure-modified polymers to contribute to the knowledge the formation of selfassembled colloidal systems. Thermodynamic parameters associated with the interactions were determined by Isothermal Titration Calorimetry (ITC). Measurements of Zeta potential and turbidity were compared to ITC results. These results were supported by microscopic observations to understand the mechanism of formation of the supramolecular structures.

Materials and Methods

Materials

Chitosan (CN) MMW (medium molecular weight, Mw 310 kDa verified by SEC-MALS experiments [START_REF] Aberkane | Laccasecatalysed oxidation of ferulic acid and ethyl ferulate in aqueous medium: A green procedure for the synthesis of new compounds[END_REF], deacetylation degree of 75%) was purchased from Sigma-Aldrich (France). Gum Arabic (GA) Instantgum AA from Acacia senegal (Mw 333 kDa determined by SEC-MALS experiments [START_REF] Vuillemin | Polymer functionalization through an enzymatic process: Intermediate products characterization and their grafting onto gum Arabic[END_REF]) was a gift from Nexira (France). Ferulic acid >99% (FA) was purchased from Sigma-Aldrich (France). Acetone, methanol and ethanol were purchased from Carlo Erba (France). Acetic acid, NaOH and ultrapure water were of analytical grade. The enzyme used for the modification of the polymers was commercialized as "Novozym® 51003" from Novozymes (Bagsvaerdt, Denmark). It is a J o u r n a l P r e -p r o o f fungal laccase from Myceliophthora thermophila, a polyphenol oxidase. The laccase stock activity was 32 038 ± 1 322 LAMU.g -1 .

Methods

Polymers modification

The functionalization of gum Arabic (GA) and chitosan (CN) was performed according to the standard procedure of our laboratory [START_REF] Aljawish | Functionalization of chitosan by laccase-catalyzed oxidation of ferulic acid and ethyl ferulate under heterogeneous reaction conditions[END_REF][START_REF] Vuillemin | Physicochemical characterizations of gum Arabic modified with oxidation products of ferulic acid[END_REF][START_REF] Vuillemin | Polymer functionalization through an enzymatic process: Intermediate products characterization and their grafting onto gum Arabic[END_REF].

For chitosan (CN), the method was adapted from Aljawish et al. [START_REF] Aljawish | Functionalization of chitosan by laccase-catalyzed oxidation of ferulic acid and ethyl ferulate under heterogeneous reaction conditions[END_REF]. For gum Arabic (GA), the method was the same as that used by Vuillemin et al. [START_REF] Vuillemin | Physicochemical characterizations of gum Arabic modified with oxidation products of ferulic acid[END_REF][START_REF] Vuillemin | Polymer functionalization through an enzymatic process: Intermediate products characterization and their grafting onto gum Arabic[END_REF]. Briefly, 1 g of polysaccharide was dispersed in 45 mL of phosphate buffer (50 mM, pH 7.5) and stirred at 4 °C overnight. Since the polysaccharides were not soluble in this buffer, functionalization was performed under heterogeneous conditions. The dispersion was stirred at 450 rpm for 1 h at 30 °C. 5 mL of ferulic acid dissolved in methanol (50 mM) was added to the batch. Once the temperature was stable, 13.5 UI.mL -1 of Novozym® 51003 was added to trigger the reaction. The batch was then stirred at 30 °C for 50 min.

For chitosan (CN), the reaction was stopped by filtration of the reaction medium under vacuum with Ministar-RC membranes (Sartorius, porosity 0.2 μm). The recovered functionalized chitosan (FCN) was washed with phosphate buffer pH 7.5, ultrapure water, methanol, ethanol and then acetone to remove all traces of substrates and enzyme on the chitosan derivative.

For gum Arabic (GA), the reaction was stopped by adding 150 mL of cold ethanol. The unfunctionalized GA precipitated. The resulting mixture of functionalized GA (FGA) and J o u r n a l P r e -p r o o f ungrafted FA oxidation products (OXP) soluble in the phosphate buffer/ethanol mixture and insoluble unfunctionalized GA was then centrifuged for 20 min at 12000 g with a Beckmann centrifuge (Beckman Coulter Inc., Villepinte, France). The precipitate contained unfunctionalized GA (verified by FTIR measurements) and some of the phosphate buffer salts, whereas the supernatant contained the functionalized GA and the non-GA-grafted ferulic acid oxidation products. Ethanol was then removed using a BUCHI R144 rotary evaporator (BUCHI SARL, Rungis, France) at a boiling temperature of 40 °C under 175 mbar. Some water in the buffer solution was also evaporated by decreasing the pressure to 72 mbar.

The functionalized polysaccharides were frozen and then lyophilized for 72 h. The resulting powders were stored in a desiccator until use.

Polymers purification

Native chitosan (NCN), functionalized chitosan (FCN), native gum Arabic (NGA) and functionalized gum Arabic (FGA) were dialyzed against acetic acid (1% v/v) and water, respectively. A high-grade quality regenerated membrane (MWCO 10 000 Da from Membrane Filtration Products Inc.) was used to remove excess salts and possibly ungrafted FA oxidation products (OXP). Dialysis was stopped when the conductivity outside the membrane was equal to the one of acetic acid (1% v/v) or ultrapure water. The samples were then centrifuged for 30 min at 12,000 g to remove insoluble material and air bubbles and then freeze-dried and stored at 4 °C. The molecular weights of the two polysaccharides were confirmed by SEC-MALS measurement after dialysis. Grafting of the oxidation products onto chitosan did not significantly alter its molecular weight (310 kDa). However, the functionalization of gum Arabic changed its molecular weight (333 kDa for NGA and 213 kDa for FGA) [START_REF] Vuillemin | Polymer functionalization through an enzymatic process: Intermediate products characterization and their grafting onto gum Arabic[END_REF].

J o u r n a l P r e -p r o o f

Polymers stock solutions preparation

Stock solutions of 0.02% w/v NCN and FCN (6.45x10 -4 mM) and 0.50% w/v NGA and FGA (1.49x10 -2 mM and 2.35x10 -2 mM, respectively) were prepared by dissolving purified NCN and FCN or NGA and FGA in acetic acid (1% v/v in ultrapure water). The solutions were stirred on a magnetic plate at 400 rpm overnight at 4 °C to ensure that the polymers were completely dissolved. The NCN and FCN solutions were prepared at 0.02% w/v (6.45x10 -4 mM), a concentration for which its viscosity was low and suitable for ITC experiments. The concentration of NGA and FGA was adjusted to 0.50% w/v (1.49x10 -2 mM and 2.35x10 -2 mM, respectively) based on preliminary measurements to optimize the concentrations for complete titration without signal saturation.

Polymers mixtures preparation

The polymer mixtures were prepared at 45 °C, pH 5.5 by adding acetic acid (1 N) or NaOH (5 N) as required, and different molar ratio of polymers. The required amount of polymer solutions was placed 15 min in a thermostatic degassing chamber (TA instruments, Waters SAS, France) to reach the target temperature and remove bubbles. The NGA or FGA solutions were then added to the NCN or FCN solutions.

Isothermal titration calorimetry (ITC)

ITC measurements were performed with an Affinity ITC equipped with a standard volume (965 µL) gold cell (TA instruments, Waters SAS, France) to measure the dissociation constant (Kd) and the enthalpic change due to the interaction between CN and GA. The entropy and the Gibbs free energy were obtained from the following equations: ΔG = -R.T.ln(1/Kd) (Eq. 1)

J o u r n a l P r e -p r o o f

10 T.ΔS = ΔH -ΔG (Eq. 2)
With ΔG: Gibbs free energy variation (kJ.mol -1 ), ΔH: enthalpy variation (kJ.mol -1 ), ΔS:

entropy variation (kJ.mol -1 .K -1 ), T: temperature (K) and R is the specific gas constant (8.3144598 J.mol -1 .K -1 ).

Ultrapure water was used in the reference cell. The injector microsyringe (320 µL), was loaded with NGA or FGA stock solution (0.50% w/v NGA or FGA (1.49x10 -2 mM and 2.35x10 - 2 mM, respectively) in 1.00% v/v acetic acid solution). 5 µL was sequentially injected 50 times every 300 s into the titration cell filled with NCN or FCN solution (0.02% w/v NCN or FCN (6.45x10 -4 mM) in 1.00% v/v acetic acid solution). At the end of the experiment, the molar ratio of the mixture in the cell was 8.7:1 for the mixture with NGA and 13.6:1 for the mixture with FGA. The polymer concentrations were chosen based on preliminary measurements to optimize the concentrations for complete titration without signal saturation. For each measurement, the pH, temperature and acetic acid solvent were the same in the cell and in the syringe to limit the dilution effect. The heat of dilution from the blank titration was subtracted from the raw data in all experiments. Data acquisition was performed using ITC Run software and data were analyzed with the NanoAnalyse software (TA instruments, Waters SAS, France). The "Independent" model was selected as the most robust model from the plot residuals to fit the obtained isotherms. All measurements were performed in triplicate.

Approximate Zeta potential

The approximate Zeta potential of the polymers alone (0.02 w/v for NCN and FCN (6.45x10 -4 mM) and 0.50% w/v for NGA and FGA (1.49x10 -2 mM and 2.35x10 -2 mM, respectively)) and mixtures was calculated from electrophoretic mobility measurements at controlled temperature (45 °C) using a Zetasizer Nano-ZS instrument (Malvern Panalytical, J o u r n a l P r e -p r o o f UK) equipped with a He/Ne ion laser (= 532 nm). The measurements were collected on a detector at 173°. The instrument determined the Zeta potential with the Smoluchowski equation [START_REF] Jayme | Zeta potentials of gum arabic stabilised oil in water emulsions[END_REF]. All measurements were performed in triplicate.

Turbidity measurements

Turbidity of the mixtures was measured with a UV-visible spectrophotometer (Shimadzu 1280, Japan) at 660 nm. The transmittance (T) of the spectrophotometer was calibrated with 0.1 N acetic acid to 100% and the turbidity of the samples was reported as 100-T%. The relative turbidity () was calculated from the equation 3:

τ = - 1 𝐿 ln 𝐼 0 𝐼 𝑡 (Eq. 3)
Where L was the optical path length (1 cm) and It and I0 represented the transmitted light intensity and the incident light intensity, respectively. All measurements were made in triplicate.

Optical microscopy

The mixtures were observed by optical microscopy (Olympus BH2, Japan) at x40 magnification. They were observed as prepared, i.e. without dilution of the samples. The images were recorded with a 5 MPixel CMOS camera (Olympus) under the control of ToupView software. The micrographs presented are representative of the samples.

Statistical analysis

Values were expressed as mean ± standard deviation. Differences between mean values were evaluated using one-way analysis of variance (ANOVA), which was performed with J o u r n a l P r e -p r o o f

Excel for Statistical Analysis and XLSTAT. Differences were considered statistically significant when the pvalue was less than 0.05.

Results and discussion

Influence of polymer modification over their approximate Zeta potentials

Since coacervation is assumed to be primarily driven by electrostatic interactions, the influence of pH on polymer behavior was first investigated.

According to Table 1, native and modified polysaccharides were charged at pH 5.5 and 45 °C. Grafting of the oxidation products did not change the approximate Zeta potential of CN in acetic acid solution (17.4 ± 0.7 mV for NCN and 18.1 ± 0.2 mV for FCN). The grafted oxidation products were mainly ferulic acid dimers grafted onto the amine groups of chitosan [START_REF] Aljawish | Functionalization of chitosan by laccase-catalyzed oxidation of ferulic acid and ethyl ferulate under heterogeneous reaction conditions[END_REF][START_REF] Aberkane | Laccasecatalysed oxidation of ferulic acid and ethyl ferulate in aqueous medium: A green procedure for the synthesis of new compounds[END_REF]. With positive charges of the amine groups replaced by oxidation products, the charge of the polymer would be expected to decrease. The approximate Zeta potential of CN at pH 3.0 was different between NCN and FCN (71.3 ± 2.8 mV and 62.5 ± 0.4 mV at 25 °C, respectively) confirming the grafting. These results are consistent with those obtained by Aljawish et al. [START_REF] Aljawish | Functionalization of chitosan by laccase-catalyzed oxidation of ferulic acid and ethyl ferulate under heterogeneous reaction conditions[END_REF]. Nevertheless, at pH 3.0, neither NGA nor FGA were charged. Therefore, it was preferred to work at pH 5.5 so that each polymer was charged. The difference in Zeta potential observed between NCN and FCN at pH 3.0 was already small (12 %). Since at pH 5.5 the charge of NCN and FCN was lower, due to a pH close to their pKa (6.3), the influence of the grafting on the charge of the polymer was less marked and may enter into instrumental errors, explaining why no significant difference was observed.

Furthermore, at the selected temperature and pH (45 °C, pH 5.5), the hydrodynamic radius of chitosan in solution was higher than under standard conditions (room temperature) and the J o u r n a l P r e -p r o o f flexibility of the chains increased [START_REF] Campbell | Effect of temperature on the intrinsic viscosity and conformation of chitosans in dilute HCl solution[END_REF][START_REF] Fan | Formation mechanism of monodisperse, low molecular weight chitosan nanoparticles by ionic gelation technique[END_REF]. Thus, under the conditions studied (pH close to the pKa of chitosan and temperature of 45 °C), the lack of charge difference between NCN and FCN could also be attributed to chain flexibility.

Table 1: Approximate Zeta potential of native chitosan (NCN, 0.02% w/v), functionalized chitosan (FCN, 0.02% w/v), native gum Arabic (NGA, 0.50% w/v) and functionalized gum Arabic (FGA, 0.50% w/v) solutions in acetic acid 1.00% v/v at pH 5.5, 45 °C. "a, b, and c" mean that the statistical analysis showed no significant differences between the results. 1, the charge of FGA was twice as high as that of NGA. Indeed, the oxidation products were mainly composed of ferulic acid dimers and trimers that were potentially negatively charged causing an increase in the negative charge of the polysaccharide after grafting [START_REF] Vuillemin | Polymer functionalization through an enzymatic process: Intermediate products characterization and their grafting onto gum Arabic[END_REF].

Many authors have previously reported that charge stoichiometry between NGA and NCN (when all NCN charges were neutralized by all NGA charges) was achieved for a composition of five times more NGA than NCN [START_REF] Butstraen | Preparation of microcapsules by complex coacervation of gum Arabic and chitosan[END_REF][START_REF] Espinosa-Andrews | Determination of the gum Arabic-chitosan interactions by Fourier Transform Infrared Spectroscopy and characterization of the microstructure and rheological features of their coacervates[END_REF][START_REF] Espinosa-Andrews | Determination of the gum Arabic-chitosan interactions by Fourier Transform Infrared Spectroscopy and characterization of the microstructure and rheological features of their coacervates[END_REF][START_REF] Vuillemin | Gum Arabic and chitosan self-assembly: Thermodynamic and mechanism aspects[END_REF]. The results in Table 1 suggested that since the modification did not change the surface charge of chitosan, the introduction of FCN instead of NCN into the mixture should not change the stoichiometry. However, since the surface charge of FGA was twice that of NGA, it appeared that twice the amount of FGA compared to NGA would be required to neutralize chitosan charges and achieve stoichiometry.

J o u r n a l P r e -p r o o f

To confirm that more FGA than NGA was needed to achieve stoichiometry and to shed light on the thermodynamic pathways of complex formation, whether the polymer was modified or not, ITC measurements were performed.

Influence of the polymers modification on the thermodynamic parameters involved in the complexation process

The influence of the modification of gum Arabic and chitosan structures over the thermodynamic parameters involved in the interaction process was studied using ITC at pH 5.5 and 45 °C.

The corrected heat rates recorded after each gum Arabic injection were predominantly negative in all cases (Figure 1A). This means that the major contribution of the interaction was mainly exothermic and that the interactions were spontaneous. Each peak corresponded to the energy released by the interaction between chitosan and gum Arabic after each injection.

Initially, typical titration curves were observed. The intensities of the exothermic peaks decreased as a function of molar ratios until they stabilized. This stabilized sequence, related to a dilution phenomenon, corresponded to exothermic peaks. Furthermore, endothermic peaks were also observed at the end of the titration just after each exothermic peak. These phenomena were observed for all mixtures except the FGA/FCN mixture, for which no exothermic peaks were observed at the end of titration, only relatively high endothermic peaks. It is clear that the exothermic peaks are mainly related to the electrostatic interaction between the polymers [START_REF] Priftis | Weak protein complexes: Challenging to study but essential for life[END_REF][START_REF] Duhoranimana | Thermodynamic characterization of Gelatin-Sodium carboxymethyl cellulose complex coacervation encapsulating Conjugated Linoleic Acid (CLA)[END_REF][START_REF] Vuillemin | Gum Arabic and chitosan self-assembly: Thermodynamic and mechanism aspects[END_REF]. Solvent reorganization also accounts for a significant part of the exothermic signal, with extensive hydrogen bonding networks at the complex interface making the enthalpy change more favorable, even if J o u r n a l P r e -p r o o f counterbalanced by an entropy penalty (endothermic peaks) [START_REF] Turgeon | Protein-polysaccharide complexes and coacervates[END_REF]. The complexes were formed until the charges were neutralized. The endothermic peaks at the end of the titration could be related either to aggregation, reorganization within the system, or hydrophobic interactions (Kayitmazer, 2017;[START_REF] Weinbreck | Complexation of Whey Proteins with Carrageenan[END_REF]. Indeed, the endothermic peaks with the highest intensities were observed when Finally, the successive appearance of exothermic and endothermic signals is typical of ITC measurements on complex formation for many systems [START_REF] Turgeon | Protein-polysaccharide complexes and coacervates[END_REF]. It has been

shown to be induced either by other energetic contributions such as the release of water molecules and ions, by conformational changes of polysaccharides, by the aggregation of polymer complexes or to the different steps involved in coacervation [START_REF] Gonçalves | Binding of Oligoarginine to Membrane Lipids and Heparan Sulfate: Structural and Thermodynamic Characterization of a Cell-Penetrating Peptide[END_REF][START_REF] Harnsilawat | Characterization of βlactoglobulin-sodium alginate interactions in aqueous solutions: A calorimetry, light scattering, electrophoretic mobility and solubility study[END_REF][START_REF] Ziegler | Interaction of the Protein Transduction Domain of HIV-1 TAT with Heparan Sulfate: Binding Mechanism and Thermodynamic Parameters[END_REF][START_REF] Hadian | Isothermal titration calorimetric and spectroscopic studies of β-lactoglobulinwater-soluble fraction of Persian gum interaction in aqueous solution[END_REF]. The absence of exothermic peaks at the end of the FGA/FCN titration could mean that these phenomena did not occur in this system when gum Arabic was in excess, or be related to a reorganization phenomenon of the system.

The binding isotherms were fitted using an "Independent" model (Figure 1B). This model was used to determine the thermodynamic parameters involved in the interactions such as dissociation constant, interaction enthalpy and stoichiometry. The enthalpy upon molar ratio followed a typical titration curve. The interaction stoichiometry (n) corresponds to the inflection point of the sigmoid curve. As it changed upon mixture, the stoichiometry is clearly not the same. The enthalpy was also different upon mixture, which means that the thermodynamic parameters were also different for the conditions studied. The free energy and thermo-entropic contribution were calculated from Eq. 1 and Eq. 2. The results obtained are presented in Table 2. For all conditions, the ΔG was always negative, meaning that self-assembly was always spontaneous whether the polysaccharides were functionalized or not. The Kd values were always low (from 0.4 x 10 -8 to 17.7 x 10 -8 ) corresponding to strong interactions (an interaction is considered strong when the Kd value is less than 10 -6 (Qin & Gronenborn, 2014)). The Kd value was lower for the FGA/FCN mixture. This could mean that the strength of the interaction was greater when both polymers were functionalized. This could potentially be related to the increased charge of FGA in comparison to NGA and to the grafting of the oxidation product onto gum Arabic and chitosan, which could promote further interactions between them.

The ΔH values were negative in all cases, as were the TΔS values. The ΔH values, mainly related to electrostatic interactions [START_REF] Aberkane | Laccasecatalysed oxidation of ferulic acid and ethyl ferulate in aqueous medium: A green procedure for the synthesis of new compounds[END_REF]Kayitmazer, 2017;[START_REF] Vuillemin | Gum Arabic and chitosan self-assembly: Thermodynamic and mechanism aspects[END_REF], are considered favorable to interactions when they are negative (negative contribution to G, see Eq.1). On the contrary, the values of TΔS, related to hydrophobic interactions (Kayitmazer, 2017;[START_REF] Weinbreck | Complexation of Whey Proteins with Carrageenan[END_REF], are favorable to interactions when they are positive. In the given cases, ΔH values were always favorable to interactions (from -913.3 ± J o u r n a l P r e -p r o o f 14.7 kJ.mol -1 to -1331.5 ± 23.3 kJ.mol -1 ) and TΔS values were always unfavorable to interactions (from -872.2 ± 13.7 kJ.mol -1 to -1280.4 ± 24.5 kJ.mol -1 ).

Regarding the stoichiometry values (Table 2), the involvement of FGA may have resulted in a change in stoichiometry. For mixtures containing NGA, the stoichiometry was the same (2.7 ± 0.2 for NGA/NCN mixtures and 3.1 ± 0.2 for NGA/FCN mixtures). However, for mixtures containing FGA, the stoichiometry was different (3.4 ± 0.3 for FGA/NCN mixtures and 6.3 ± 0.6 for FGA/FCN mixtures). The difference between mixtures containing NGA and FGA was expected from the change in the approximate Zeta potential of gum Arabic after functionalization (Table 1). However, the difference in stoichiometry observed between the FGA/NCN and FGA/FCN mixtures (3.4 ± 0.3 and 6.3 ± 0.6, respectively) was surprising, because the approximate Zeta potential of chitosan was the same at pH 5.5 after functionalization. This result suggests that electrostatic interactions are not the only types of interactions between the two modified polymers. This could indicate that when both polymers were modified, potential hydrophobic interactions, π-π stacking, cation-π and anion-π induced by the grafting of phenolic compounds could lead to a change in complexation stoichiometry and an increase in polymer content. However, hydrophobic interactions would mainly lead to aggregation, as already studied by [START_REF] Kaibara | Characterizations of critical processes in liquid-liquid phase separation of the elastomeric protein-water system: Microscopic observations and light scattering measurements[END_REF] on the elastin/water system [START_REF] Kaibara | Characterizations of critical processes in liquid-liquid phase separation of the elastomeric protein-water system: Microscopic observations and light scattering measurements[END_REF]. While the ΔG was almost the same for all conditions examined, it was significantly more negative for FGA/FCN mixture (-51.1 ± 1.2 kJ.mol -1 ). Furthermore, this mixture had the lowest Kd (0.4x10 -8 ± 0.2x10 -8 ), which means that under these conditions the interaction between modified gum Arabic and modified chitosan was the most favorable.

The enthalpic and entropic parameters did not change whether chitosan was functionalized or not (Table 2). However, the functionalization of both polymers had an impact on them. ΔH varied from -965.2 ± 40.7 kJ.mol -1 for NGA/NCN mixtures to -1331.5 ± 23.3 kJ.mol -1 for J o u r n a l P r e -p r o o f FGA/FCN mixture while TΔS varied from -923.6 ± 40.9 kJ.mol -1 to -1280.4 ± 24.5 kJ.mol -1 , respectively. This means that the enthalpy contribution became more favorable to the interaction while the entropy contribution became less favorable when both polymers were modified. This makes sense, as the functionalization of gum Arabic, while increasing its hydrophobicity and/or the possibility for it to undergo π-π stacking, cation-π and anion-π interactions, increased its negative charges, likely leading to more electrostatic interaction.

A surprising result was obtained from the comparison of the enthalpic and entropic contributions of FGA/NCN and FGA/FCN. Since the charges of chitosan were the same whether it was functionalized or not, the observed difference could not be explained by an increase in electrostatic interaction. However, although the thermodynamic parameters did not indicate a significant role of hydrophobic interactions in the complexation process, the isotherms (Figure 1A) showed an endothermic contribution at the end of the titration. This could mean that in the case of FGA/FCN mixtures, there could be a competition between electrostatic interactions, which are strong at the beginning of the titration until stoichiometry is reached, and hydrophobic interactions, which are predominant at the end of the titration.

By measuring the approximate Zeta potential upon polymers molar ratio, the turbidity of the samples after mixing and with the support of optical microscopy, a more thorough understanding of the type of interactions that were occurring should be gained in order to elucidate the complexation mechanism.

J o u r n a l P r e -p r o o f 20

Characteristics of polymer mixtures upon molar ratio

The influence of the molar ratio on the approximate Zeta potential and turbidity of the samples was studied for each potential mixture (Figure 2). The experiments were performed at pH 5.5, 45 °C. Measurements and observations were performed 5 min after mixing the polymers solutions.

Figure 2: (A) Approximate Zeta potential (mV) and (B) turbidity (cm -1 ), of mixtures between native gum Arabic (NGA) or functionalized gum Arabic (FGA) and native chitosan (NCN) or functionalized chitosan (FCN) upon gum Arabic/chitosan molar ratio at pH 5.5 and 45 °C, 5 min after mixing.

When mixing gum Arabic and chitosan, the approximate Zeta potential decreased upon addition of gum Arabic (Figure 2A). This is because the charges of chitosan (NCN or FCN)

were progressively neutralized upon addition of FGA or NGA to the mixtures. The molar ratio

for which the approximate Zeta potential of the mixture was zero was not exactly the same as the stoichiometry obtained by ITC measurements (Table 2), but similar trends were reported for all mixtures except for the FGA/FCN mixture. The optimal molar ratio (for which charges were neutralized) was lower by Zeta potential measurements than by ITC measurements for FGA/FCN mixtures (3.27 ± 0.02 versus 6.3 ± 0.6, respectively). This means that the optimal molar ratio obtained by ITC for FGA/FCN interaction was not simply due to electrostatic interactions. This confirmed that other types of interactions such as π-π stacking, cation-π and anion-π were at work for this mixture at 45 °C.

Upon molar ratio, the turbidity of the mixtures increased up to a molar ratio of 4.9:1 for the mixtures containing NGA and 7.6 for mixtures containing FGA, then decreased slightly (Figure 2B). For molar ratio lower than these, the turbidity was different for the different mixtures, indicating the formation of various supramolecular structures (coacervates or aggregates of different size and shape). These differences may be related to the shape of the formed samples, as can be seen in Figure 3. samples appeared "homogeneously" turbid while aggregates were observed on others. Mixtures containing NGA were turbid while the presence of FGA seemed to lead to the formation of aggregates at certain molar ratios. In the photographs of the mixtures (Figure 3), it would appear that the mixtures become more turbid for NGA/NCN and NGA/FCN mixtures upon molar ratios. However, for mixtures containing FGA, the mixtures became turbid upon FGA addition until a molar ratio was reached when aggregation occurred (for ratio 3.1 for FGA/FCN mixtures and for the 6.1 ratio for FGA/NCN mixtures). To confirm the presence of coacervates and aggregates in the samples, the mixtures were observed by optical microscopy (Figure 4).

J o u r n a l P r e -p r o o f For molar ratio below 1.0, no supramolecular structures were observed for all mixtures.

When NGA was added to NCN or FCN up to a molar ratio of 2.9, a few coacervates appeared.

They became more numerous for a molar ratio of 4.9. Upon addition of NGA, coacervates tended to disappear (molar ratio 7.8).

For FGA/NCN mixtures (Figure 4), no supramolecular structures were observed for a molar ratio of 7.6. For a molar ratio of 7.6, a coexistence of two types of supramolecular structures was observed: there were both coacervates and aggregates. Above a molar ratio of 9.2, no more coacervates were observed, but the aggregates became larger and more numerous.

For FGA/FCN mixtures, no coacervates were observed at any molar ratio. However, aggregates started to appear for a molar ratio of 4.6, and they became larger until the molar ratio of 7.6 was reached. For a molar ratio of 9.2, it appears that the density of aggregates decreased. The latter result was positively correlated with the turbidity and macroscopic aspect of the mixtures (Figure 2 and Figure 3). This could be due to a reorganization within the different systems that would correlate with the endothermic events observed on ITC measurements above a molar ratio of 8 (Figure 1A). By linking the ITC measurements, the thermodynamic parameters obtained from fitting the raw data, to macroscopic observations, Zeta potential measurements, turbidity measurements and optical micrographs, a mechanism was proposed to explain what happened within the different systems.

Possible mechanism of the interaction between gum Arabic and chitosan

The interactions between NGA and NCN were electrostatic interactions between the carboxyl groups of gum Arabic and the amine groups of chitosan [START_REF] Butstraen | Preparation of microcapsules by complex coacervation of gum Arabic and chitosan[END_REF][START_REF] Espinosa-Andrews | Gum Arabic-Chitosan Complex Coacervation[END_REF], 2010;[START_REF] Vuillemin | Gum Arabic and chitosan self-assembly: Thermodynamic and mechanism aspects[END_REF]. The two polymers attracted each J o u r n a l P r e -p r o o f other due to their opposite charges, followed by the release of water and counterions. Then, upon addition of gum Arabic, the charges of the two polymers were balanced. Upon addition of gum Arabic beyond stoichiometry, reorganization of the complexes occurred until the complexes disintegrated to reach equilibrium in the systems (observed in previous studies on NCN and NGA [START_REF] Vuillemin | Gum Arabic and chitosan self-assembly: Thermodynamic and mechanism aspects[END_REF]).

Studies often agree that there is an initial step of soluble complexes formation prior to the actual coacervation step. Recent studies have focused on these complexes formed between oppositely charged polyelectrolytes as precursors to coacervates and multilayers [START_REF] Comert | The so-called critical condition for polyelectrolyte-colloid complex formation[END_REF]. The presence of these small precursors has been reported to increase the turbidity of samples, and it has been suggested that this is related to an increase in the size or number of complexes or soluble aggregates. In the optical micrographs, for the first few molar ratios, even though no supramolecular structures were visible, it could be assumed that soluble complexes or soluble aggregates were formed, due to the increased turbidity of the samples (Figure 2 and Figure 3).

For the NGA/NCN system, up to a molar ratio of 1.0, no complexes were visible (Figure 4). However, it was possible that supramolecular structures had formed, but they were too small to be seen by optical microscopy. This was confirmed by the turbidity of the mixtures (Figure 2) rising before the first coacervates could be observed by optical microscopy (Figure 4). When mixed beyond a ratio of 2.9, these two oppositely charged polysaccharides began to form coacervates (Figure 4). When more NGA was added to NCN, more charges of the polymers interacted. The stoichiometry value determined by ITC (2.7) was confirmed by optical microscopy (first supramolecular structures at 2.9), macroscopic observations (first turbid samples at 2.0) and Zeta potential (optimal ZP at 3.49). At ratio 7.8, when NGA was in excess, the complexes became smaller. To reach an equilibrium in the system, the GA could desorb J o u r n a l P r e -p r o o f and the coacervates could change. This assumption was consistent with the small endothermic event at the end of the titration (Figure 1). This could be explained by this reorganization within the system.

For the NGA/FCN system, the same phenomena were observed (Figure 4). The ITC results, macroscopic and microscopic observations as well as Zeta potential and turbidity indicated no change when replacing NCN with FCN. This means that the coacervation process was probably the same. Since the charges of FCN and NCN were the same at the chosen pH and temperature, the interactions between FCN and NGA were likely mainly electrostatic interactions. If another phenomenon occurred, such as hydrophobic interactions or π-anion interactions between NGA and the grafted oxidation products of FCN, the experiments did not reveal them.

For the FGA/FCN system, no coacervation phenomena were clearly observed in photographs of the sample or in micrographs (Figure 3 and Figure 4). Up to a molar ratio of 3.1, the aspect of the sample became turbid but no supramolecular structures were visible by optical microscopy. This means that before this ratio, small complexes were formed. After a ratio of 4.6, aggregates appeared (Figure 4). The optimal molar ratio determined by Zeta potential measurements was consistent with the appearance of the turbidity (3.27) and the appearance of aggregates was consistent with the optimal molar ratio determined by ITC (6.3).

This means that the formation of soluble complexes was primarily driven by electrostatic interactions but the formation of aggregates would be driven by other forces. ITC parameters confirmed that the presence of FGA and FCN increased the contribution of the enthalpic J o u r n a l P r e -p r o o f

parameter. This could be due to the fact that both polymers are grafted with oxidation products, which leads to more π-π stacking, π-cation and π-anion interactions increasing the enthalpic contribution. The presence of exothermic hydrophobic interactions could not be excluded, but their effect on the titration isotherms would probably be concealed by all the other interactions involved in the aggregation phenomenon. However, at the end of titration, hydrophobic interactions between FGA and FCN would be predominant (Figure 1).

The increased charge of FGA in comparison to NGA could increase the strength of its interactions with chitosan. It is widely accepted that strong attractive properties promote aggregation [START_REF] Comert | Liquid-liquid and liquid-solid phase separation in proteinpolyelectrolyte systems[END_REF]. The polyelectrolyte structure also modulates the type of phase separation observed. Indeed, the structure of the modified polymers changed their steric hindrance, consequently modifying the accessibility of polysaccharides to each another. It was revealed in previous studies that the glass transition temperature of gum Arabic decreased upon functionalization, meaning that the grafting of phenolic compounds would have added to the steric hindrance of FGA and made its chains more flexible, allowing it to undergo more interactions [START_REF] Vuillemin | Physicochemical characterizations of gum Arabic modified with oxidation products of ferulic acid[END_REF][START_REF] Vuillemin | Polymer functionalization through an enzymatic process: Intermediate products characterization and their grafting onto gum Arabic[END_REF]. The increase in FGA charge, leading to stronger electrostatic interactions, combined with increase in steric hindrance and the increase in polymer flexibility would be responsible for the aggregation phenomenon observed for the FGA/FCN mixtures.

For the FGA/NCN system, the turbidity of the sample increased after the addition of FGA (Figure 2). A slight difference was observed at a molar ratio of 1.5, but the turbidity was higher at a ratio of 4.6. At a molar ratio of 6.1, aggregates appeared (Figure 3). The optimal molar ratio obtained by Zeta potential measurements (2.95) and ITC measurements (3.4) were almost the same, but no supramolecular structures were visible by optical microscopy until a molar ratio J o u r n a l P r e -p r o o f of 7.6. This means that before this ratio, soluble complexes were formed. However, at a ratio of 7.6, two distinct phenomena were observed by macroscopic observations. Coacervates and aggregates were present in the mixture. In the literature, the differentiation between the two mechanisms is often obscured by the common use of turbidity to detect these transitions [START_REF] Comert | Liquid-liquid and liquid-solid phase separation in proteinpolyelectrolyte systems[END_REF]. Both mechanisms induce an increase in turbidity, but microscopy is needed to know whether the turbidity arises from small soluble complexes, coacervation or aggregation. By adding more FGA to the mixture, only aggregates were visible in the optical micrographs.

A surprising result was the difference between the NGA/NCN mixture and the FGA/NCN mixture. Indeed, the ITC results (Table 2) were not significantly different for these mixtures and the stoichiometries were almost the same. However, turbidity showed the appearance of aggregates for the FGA/NCN mixtures as well as the appearance of coacervates for a molar ratio of 7.6 (Figure 3) which were confirmed by optical microscopy (Figure 4). Furthermore, while the stoichiometry obtained by ITC was almost the same for NGA/NCN and FGA/NCN mixtures, the stoichiometry obtained by approximate Zeta potential measurements was lower for the FGA/NCN mixtures. This could be explained by the fact that a lower amount of FGA was required to neutralize the charge of chitosan since FGA was more charged than NGA (Table 1). The fact that the charge stoichiometry and the one obtained by ITC were close (2.95 ± 0.02 and 3.4 ± 0.3, respectively) could explain why the two phenomena coexist for a molar ratio of 3.9 in the optical micrographs. At the chosen pH and temperature, due to their charges under these conditions, NGA/NCN and NGA/FCN exhibited typical « weakly charged polyelectrolytes » coacervation behavior and steps, mainly driven by electrostatic interactions (negative enthalpy), with the entropy of counterion release playing only a minor role [START_REF] Turgeon | Protein-polysaccharide complexes and coacervates[END_REF]. The fact that these mixtures exhibited the same behavior suggests that mainly electrostatic interactions occurred between NGA and NCN and NGA and FCN (Figure 5).

In the FGA/NCN system, several phenomena were observed. First, soluble complexes were formed. Then, coacervates appeared as well as aggregates (Figure 4). Then, the coacervates disappeared and only aggregates remained, meaning that stronger interactions were J o u r n a l P r e -p r o o f present [START_REF] Comert | Liquid-liquid and liquid-solid phase separation in proteinpolyelectrolyte systems[END_REF]. These were probably mainly more electrostatic interactions, due to the higher charge of FGA, but they could also be due to π-cation interaction between the amine groups of NCN and the phenol groups of the oxidation products grafted onto FGA (Figure 5).

Based on the fact that coacervation usually occurs when the interactions are weak and on the contrary, aggregation is a consequence of strong interactions, it might be surprising if coacervation occurred before aggregation or at the same time. However, the strength of the interactions should not be regarded as the sole strength of electrostatic and π-cation interactions between one FGA and one NCN but has the number of interactions in the system. Though some interactions are individually weak, they can become collectively important [START_REF] Biedermann | Complex Coacervation as a Novel Microencapsulation Technique to Improve Viability of Probiotics Under Different Stresses[END_REF][START_REF] Neel | Exploiting non-covalent π interactions for catalyst design[END_REF]. One explanation for the coexistence of coacervation and aggregation would be that the "collective importance" of the π-cation interactions between FGA and NCN was low at a given molar ratio, leading to coacervation, and that when the molar ratio increased, this "collective importance" increased with the amount of FGA aromatic rings, leading to the disappearance of coacervates in favor of aggregates. Another explanation could be based on the fact that chitosan opens the structure of FGA when they interact. This would lead to exposure of FGA phenolic groups from the grafted oxidation products to the solvent, which would decrease the affinity of the polymer for the aqueous solvent. This could lead to an aggregation phenomenon.

For the FGA/FCN system, soluble complexes appeared to have been formed at the beginning of the titration, but then only aggregates were visible. The strength of the interaction was higher than for the other systems (Table 2) and the enthalpy was the most favorable in this system. These stronger interactions could be due to FGA, as described for the FGA/NCN system, but other interactions were involved since the results for the FGA/FCN system were J o u r n a l P r e -p r o o f different from the ones for FGA/NCN systems. These interactions were not involved in the mixtures containing NCN or NGA/NCN systems, meaning that they were due to the modification of both FGA and FCN. These strong and numerous interactions would lead to aggregation instead of coacervation. The aggregation phenomenon could be related to the fact that both polymers were grafted with oxidation products, which leads to more π-π stacking and π-cation interaction, thus increasing the enthalpic contribution (Figure 5). The increase in FGA charge likely also contributed to enhanced electrostatic interactions. Similarly, the increase in steric hindrance and the increase in polymer flexibility induced by grafting of phenolic compounds onto the polymer is known to promote the aggregation phenomenon.

Conclusions

This study provided a better understanding of the impact of the interaction balance between two polymers on the formation of self-assembled structures. By comparing the interaction stoichiometry (ITC results) and charge stoichiometry (approximate Zeta potential results), it was shown that the interaction of chitosan with modified gum Arabic was not solely due to electrostatic interactions. The strength of the interaction (Kd measured by ITC) was higher for the mixture containing modified chitosan and modified gum Arabic, which meant that other interactions occurred between those two (new π-π stacking and π-cation interactions).

The balance between these different interactions (their nature but also their collective strength)

led not only to the formation of coacervates but also to the formation of aggregates (confirmed by turbidity measurements, macroscopic observation of the mixtures and optical microscopy).

By managing the physicochemical conditions of the medium and by judiciously choosing the polymers (modified or not) it is possible to control the structures formed. Moreover, with J o u r n a l P r e -p r o o f modified polymers, it is possible to form coacervates or aggregates under conditions different from the optimal conditions with native polymers. With native polymers, to control the structures formed, the physicochemical conditions of the medium must be controlled. When these structures are used in vivo or to mimic the compartmentalized environments of cells for example, the conditions of the medium are already fixed. The use of polymer modification seems to be a promising way to control the structures formed in a given medium for encapsulation purposes, for example.

Table captions

Table 1: Approximate Zeta potential of native chitosan (NCN, 0.02% w/v), functionalized chitosan (FCN, 0.02% w/v), native gum Arabic (NGA, 0.50% w/v) and functionalized gum Arabic (FGA, 0.50% w/v) solutions in acetic acid 1.00% v/v at pH 5.5, 45 °C. "a, b, and c" mean that the statistical analysis showed no significant differences between the results.

Table 2: Thermodynamic parameters (dissociation constant (Kd), stoichiometry (n), enthalpic contribution (ΔH), thermo-entropic contribution (TΔS) and Gibbs free energy (ΔG)) of binding between native gum Arabic (NGA)/functionalized gum Arabic (FGA) and native chitosan (NCN)/functionalized chitosan (FCN). "a, b and c" mean that the statistical analysis showed no significant differences between the results, for each column. Results with the same letter means there was no significant difference between them.

J o u r n a l P r e -p r o o f J o u r n a l P r e -p r o o f  Grafting phenolic compounds onto polymers changed interactions and objects.

 Polymer functionalization increased their ability to exhibit π interactions.  New interactions type, led to the formation of different objects.  In some conditions, coacervates and aggregates coexisted.

J o u r n a l P r e -p r o o f

Figure 1 :

 1 Figure 1: (A) Corrected heat rate upon time of mixtures between native gum Arabic (NGA) or functionalized gum Arabic (FGA) and native chitosan (NCN) or functionalized chitosan (FCN) at pH 5.5 at 45 °C. (B) Enthalpy and fit upon gum Arabic/chitosan molar ratio of mixtures between native gum Arabic or functionalized gum Arabic and native chitosan or functionalized chitosan at pH 5.5 at 45 °C. Grey points recorded at the beginning of the titration were excluded from the fit.

  r n a l P r e -p r o o f FCN was present in the mixtures. Both NGA and FGA had hydrophobic moieties from the protein fraction or grafted moieties, whereas NCN did not. Grafting of phenolic compounds onto chitosan provided hydrophobic moieties capable of exhibiting novel interactions with NGA and FGA.

  r n a l P r e -p r o o f

Figure 3 :

 3 Figure 3: Macroscopic observations of mixtures between native gum Arabic (NGA) or functionalized gum Arabic (FGA) and native chitosan (NCN) or functionalized chitosan (FCN) mixtures upon molar ratio at 45 °C, 5 min after mixing.

Figure 4 :

 4 Figure 4: Optical micrographs of mixture between native gum Arabic (NGA) or functionalized gum Arabic (FGA) and native chitosan (NCN) or functionalized chitosan (FCN) mixtures upon different GA/CN molar ratios 5 min after mixing at 45 °C, pH 5.5.

  r n a l P r e -p r o o f

JFigure 5 :

 5 Figure 5: Schematic representation of the possible interactions between native gum Arabic (NGA) or functionalized gum Arabic (FGA) and native chitosan (NCN) or functionalized chitosan (FCN)

Figure 1 :

 1 Figure captions

Figure 2 :

 2 Figure 2:(A) Approximate Zeta potential (mV) and (B) turbidity (cm -1 ), of mixtures between native gum Arabic (NGA) or functionalized gum Arabic (FGA) and native chitosan (NCN) or functionalized chitosan (FCN) upon gum Arabic/chitosan molar ratio at pH 5.5 and 45 °C, 5 min after mixing.

Figure 3 :

 3 Figure 3: Macroscopic observations of mixtures between native gum Arabic (NGA) or functionalized gum Arabic (FGA) and native chitosan (NCN) or functionalized chitosan (FCN) mixtures upon molar ratio at 45 °C, 5 min after mixing.

Figure 4 :

 4 Figure 4: Optical micrographs of mixture between native gum Arabic (NGA) or functionalized gum Arabic (FGA) and native chitosan (NCN) or functionalized chitosan (FCN) mixtures upon different GA/CN molar ratios 5 min after mixing at 45 °C, pH 5.5.

Figure 5 :

 5 Figure 5: Schematic representation of the possible interactions between native gum Arabic (NGA) or functionalized gum Arabic (FGA) and native chitosan (NCN) or functionalized chitosan (FCN).

Figure 1 :

 1 Figure 1:

  Figure 2:

Figure 3 :

 3 Figure 3:

Figure 5 :

 5 Figure 5:

Table 2 :

 2 NCN)/functionalized chitosan (FCN). "a, b and c" mean that the statistical analysis showed no significant differences between the results, for each column. Results with the same letter means there was no significant difference between them.

	Mixtures	Kd.10	-08		n				ΔH (kJ.mol	-1 )	TΔS (kJ.mol	-1 )	ΔG (kJ.mol	-1 )
	NGA NCN	14.5 ± 0.6	a	2.7 ± 0.2	a	-965.2 ± 40.7	a	-923.6 ± 40.9	a	-41.6 ± 0.2	a
	NGA FCN	13.6 ± 5.3	a	3.1 ± 0.2	ab	-916.3 ± 48.8	a	-874.3 ± 47.7	a	-42.0 ± 1.1	a
	FGA NCN	17.7 ± 6.9	a	3.4 ± 0.3	b	-913.3 ± 14.7	a	-872.2 ± 13.7	a	-41.1 ± 1.0	a
	FGA FCN	0.4 ± 0.2	b	6.3 ± 0.6	c	-1331.5 ± 23.3	b	-1280.4 ± 24.5	b	-51.1 ± 1.2	b

Thermodynamic parameters (dissociation constant (Kd), stoichiometry (n), enthalpic contribution (ΔH), thermo-entropic contribution (TΔS) and Gibbs free energy (ΔG)) of binding between native gum Arabic (NGA)/functionalized gum Arabic (FGA) and native chitosan J o u r n a l P r e -p r o o f (

Table 1 :

 1

			Approximate Zeta Potential (mV)
	NCN 0.02% w/v in acetic acid	FCN 0.02% w/v in acetic acid	NGA 0.50% w/v in acetic acid	FGA 0.50% w/v in acetic acid
	17.4 ± 0.7	a	18.1 ± 0.2	a	-9.7 ± 0.7	b	-21.7 ± 1.1

c J o u r n a l P r e -p r o o f

Table 2 :

 2

	Mixtures	Kd.10	-08		n			ΔH (kJ.mol	-1 )	TΔS (kJ.mol	-1 )	ΔG (kJ.mol	-1 )
	NGA NCN	14.5 ± 0.6	a	2.7 ± 0.2	a	-965.2 ± 40.7	a	-923.6 ± 40.9	a	-41.6 ± 0.2	a
	NGA FCN	13.6 ± 5.3	a	3.1 ± 0.2	ab	-916.3 ± 48.8	a	-874.3 ± 47.7	a	-42.0 ± 1.1	a
	FGA NCN	17.7 ± 6.9

Acknowledgments

The authors acknowledge support of the LIBio by the "Impact Biomolecules" project of the "Lorraine Université d'Excellence" (Investissements d'avenir -ANR).

The authors acknowledge support of the CPER Agrovalor.

We would like to thank Nexira for kindly providing the gum Arabic used in this article.

We also thank Blandine Simard for the technical support.

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

J o u r n a l P r e -p r o o f


Jean-Yves Marion
email: jean-yves.marion@loria.fr

Romain Péchoux
email: romain.pechoux@loria.fr

A Characterization of NC k by First Order Functional Programs

This paper is part of a research on static analysis in order to predict program resources and belongs to the implicit computational complexity line of research. It presents intrinsic characterizations of the classes of functions, which are computable in NC k , that is by a uniform, poly-logarithmic depth and polynomial size family of circuits, using first order functional programs. Our characterizations are new in terms of first order functional programming language and extend the characterization of NC 1 in [9]. These characterizations are obtained using a complexity measure, the sup-interpretation, which gives upper bounds on the size of computed values and captures a lot of program schemas.

Introduction

Our work is related to machine independent characterizations of functional complexity classes initiated by Cobham's work [START_REF] Cobham | The intrinsic computational difficulty of functions[END_REF] and studied by the Implicit computational complexity (ICC) community, including safe recursion of Bellantoni and Cook [START_REF] Bellantoni | A new recursion-theoretic characterization of the polytime functions[END_REF], data tiering of Leivant [START_REF] Leivant | Predicative recurrence and computational complexity I: Word recurrence and poly-time[END_REF], linear type disciplines by Girard et al. [START_REF] Girard | Light linear logic[END_REF][START_REF] Girard | Bounded linear logic[END_REF], Lafont [START_REF] Lafont | Soft linear logic and polynomial time[END_REF], Baillot-Mogbil [START_REF] Baillot | Soft lambda-calculus: a language for polynomial time computation[END_REF], Gaboardi-Ronchi Della Rocca [START_REF] Gaboardi | A soft type assignment system for λcalculus[END_REF] and Hofmann [START_REF] Hofmann | Programming languages capturing complexity classes[END_REF] and studies on the complexity of imperative programs using matrix algebra by Kristiansen-Jones [START_REF] Kristiansen | The flow of data and the complexity of algorithms[END_REF] and Niggl-Wunderlich [START_REF] Niggl | Certifying polynomial time and linear/polynomial space for imperative programs[END_REF]. Traditional results of the ICC focus on capturing all functions of a complexity class and we should call this approach extensional whereas our approach, which tries to characterize a class of programs, which represents functions in some complexity classes, as large as possible, is rather intensional. In other words, we try to delineate a broad class of programs using a certain amount of resources.

Our approach relies on methods combining term rewriting systems and interpretation methods for proving complexity upper bounds by static analysis. It consists in assigning a function from real numbers to real numbers to some symbols of a program. Such an assignment is called a sup-interpretation if it satisfies some specific semantics properties introduced in [START_REF] Marion | Resource analysis by sup-interpretation[END_REF]. Basically, a sup-interpretation provides upper bounds on the size of computed values. Supinterpretation is a generalization of the notion of quasi-interpretation of [START_REF] Bonfante | Quasi-interpretations, a way to control resources[END_REF]. The problem of finding a quasi-interpretation or sup-interpretation of a given program, called synthesis problem, is crucial for potential applications of the method. It consists in automatically finding an interpretation of a program in order to determine an upper bound on its complexity. It was demonstrated in [START_REF] Amadio | Synthesis of max-plus quasi-interpretations[END_REF][START_REF] Bonfante | Synthesis of quasiinterpretations[END_REF] that the synthesis problem is decidable in exponential time for small classes of polynomials. Quasi-interpretations and sup-interpretations have already been used to capture the sets of functions computable in polynomial time and space [START_REF] Marion | Efficient first order functional program interpreter with time bound certifications[END_REF][START_REF] Bonfante | On lexicographic termination ordering with space bound certifications[END_REF] and capture a broad class of algorithms, including greedy algorithms and dynamic programming algorithms. Consequently, it is a challenge to study whether this approach can be adapted to characterize small parallel complexity classes. Parallel algorithms are difficult to design. Employing the supinterpretation method leads to delineate efficient parallel programs amenable to circuit computing. Designing parallel implementations of first order functional programs with interpretation methods for proving complexity bounds, might be thus viable in the near future.

A circuit C n is a directed acyclic graph built up from Boolean gates And, Or and Not. Each gate has an in-degree less or equal to two and an out-degree equal to one. A circuit has n input nodes and g(n) output nodes, where

g(n) = O(n c ), for some constant c ≥ 1. Thus, a circuit C n computes a function f n : {0, 1} n → {0, 1} g(n) . A family of circuits is a sequence of circuits C = (C n ) n , which computes a family of finite functions (f n ) n over {0, 1} * . A function f is computed by a family of circuits (C n ) n if the restriction of f to inputs of size n is computed by C n .
A uniformity condition ensures that there is a procedure which, given n, produces a description of the circuit C n . Such a condition is introduced to ensure that a family of circuits computes a reasonable function. All along, we shall consider U E * -uniform family of circuits defined in [START_REF] Ruzzo | On uniform circuit complexity[END_REF]. The complexity of a circuit depends on its depth (the longest path from an input to an output gate) and its size (the number of gates). The class NC k is the class of functions computable by a U E * -uniform family of circuits of size bounded by O(n d ), for some constant d, and depth bounded by O(log k (n)). Intuitively, it corresponds to the class of functions computed in poly-logarithmic time with a polynomial number of processors. Following [START_REF] Barrington | On uniformity within NC[END_REF], the main motivation in the introduction of such classes was the search for separation results: "NC 1 is the at the frontier where we obtain interesting separation results". NC 1 contains binary addition, substraction, prefix sum of associative operators. Buss [START_REF] Buss | The boolean formula value problem is in ALOGTIME[END_REF] has demonstrated that the evaluation of boolean formulas is a complete problem for NC 1 . A lot of natural algorithms belong to the distinct levels of the NC k hierarchy. In particular, the reachability problem in a graph or the search for a minimum covering tree in a graph are two problems in N C 2 .

In this paper, we define a restricted class of first order functional programs, called fraternal and arboreal programs, using the notion of sup-interpretation of [START_REF] Marion | Resource analysis by sup-interpretation[END_REF]. We demonstrate that functions, which are computable by these programs at some rank k, are exactly the functions computed in NC k . This result generalizes the characterization of NC 1 established in [START_REF] Bonfante | A characterization of alternating log time by first order functional programs[END_REF]. To our knowledge, these are the first results, which connect small parallel complexity classes and first order functional programs.

2 First order functional programs

Syntax of programs

We define a generic first order functional programming language. The vocabulary Σ = Var, Cns, Op, Fct is composed of four disjoint sets of symbols. The arity of a symbol is the number n of arguments that it takes. A program p consists in a vocabulary and a set of rules R defined by the following grammar: [START_REF] Huet | Confluent reductions: Abstract properties and applications to term rewriting systems[END_REF]. We will use the notation e to represent a sequence of expressions, that is e = e 1 , . . . , e n .

(Values) T (Cns) v ::= c | c(v 1 , • • • , v n ) (Patterns) T (Var, Cns) p ::= c | x | c(p 1 , • • • , p n ) (Expressions) T (Var, Cns, Op, Fct) e ::= c | x | c(e 1 , • • • , e n ) | op(e 1 , • • • , e n ) | f(e 1 , • • • , e n ) (Rules) R r ::= f(p 1 , • • • , p n ) → e

Semantics

The domain of computation of a program p is the constructor algebra Values = T (Cns). Set Values * = Values∪{Err}, where Err is a special symbol associated to runtime errors. An operator op of arity n is interpreted by a function op from Values n to Values * . Operators are essentially basic partial functions like destructors or characteristic functions of predicates like =.

Set Values # = Values ∪ {Err, ⊥}, where ⊥ means that a program is nonterminating. Given a program p of vocabulary Var, Cns, Op, Fct and an expression e ∈ T (Cns, Op, Fct), the computation of e, noted e , is defined by e = w iff e * →w and w ∈ Values * , otherwise e = ⊥, where * → is the reflexive and transitive closure of the rewriting relation → induced by the rules of R. By definition, if no rule is applicable, then an error occurs and e = Err. A program of main function symbol f computes a partial function φ :

Values n → Values * defined by ∀u 1 , • • • , u n ∈ Values, φ(u 1 , • • • , u n ) = w iff f(u 1 , • • • , u n ) = w.
Definition 1 (Size). The size of an expression e is defined by |e| = 0, if e is a 0-arity symbol, and |b(e

1 , • • • , e n )| = i∈{1,...,n} |e i | + 1, if e = b(e 1 , • • • , e n ).
Example 1. Consider the following program which computes the logarithm function over binary numbers using the constructor symbols {0, 1, }:

f(x) → rev(log(x)) log(i(x)) → if(Msp(Fh(i(x)), Sh(i(x))), 0(log(Fh(i(x)))), 1(log(Fh(i(x))))) log( ) →
where if(u, v, w) is an operator, which outputs v or w depending on whether u is equal to 1( ) or 0( ), rev is an operator, which reverses a binary value given as input, Msp is an operator, which returns 1( ) if the leftmost |x| |y| bits of x (where ∀x, y ∈ N, x y = 0 if y > x and x -y otherwise) are equal to the empty word and returns 0( ) otherwise, Fh is an operator, which outputs the leftmost |x|/2 bits of x, and Sh is an operator, which outputs the rightmost |x|/2 bits of x.

The algorithm tests whether the number |x|/2 of leftmost bits is equal to the number |x|/2 of rightmost bits in the input x using the operator Msp. In this case, the last digit of the logarithm is a 0, otherwise it is a 1. Finally, the computation is performed by applying a recursive call over half of the input digits and the result is obtained by reversing the output, using the operator rev. For any binary value v, we have log(v) = u, where u is the value representing the binary logarithm of the input value v. For example, log(1(0(0(0( )))) = 1(0(0( ))).

Call-tree

We now describe the notion of call-tree which is a representation of a program state transition sequences induced by the rewrite relation → using a call-by-value strategy. In this paper, the notion of call-tree allows to control the successive function calls corresponding to a recursive rule. First, we define the notions of context and substitution. A context is an expression C[ 1 , • • • , r ] containing one occurrence of each i , with i new variables which do not appear in Σ. A substitution is a finite mapping from variables to T (Var, Cns, Op, Fct). The substitution of each i by an expression

d i in the context C[ 1 , • • • , r ] is noted C[d 1 , • • • , d r ].
A ground substitution σ is a mapping from variables to Values. Throughout the paper, we use the symbol σ and the word "substitution" to denote a ground substitution. The application of a substitution σ to an expression (or a sequence of expressions) e is noted eσ.

Definition 2. Suppose that we have a program

p. A state f, u 1 , • • • , u n of p is a tuple where f is a function symbol of arity n and u 1 , • • • , u n are values of Values * .
There is state transition, noted η 1 η 2 , between two states

η 1 = f, u 1 , . . . , u n and η 2 = g, v 1 , • • • , v m if there are a rule f(p 1 , • • • , p n ) → e of p, a sub- stitution σ, a context C[-] and expressions e 1 , • • • , e m such that ∀i ∈ {1, n} p i σ = u i , ∀j ∈ {1, m}, e j σ = v j and e = C[g(e 1 , • • • , e m )]
. We write * to denote the reflexive and transitive closure of . A call-tree of p of root f, u 1 , • • • , u n is the following tree:

the root is the node labeled by the state f,

u 1 , • • • , u n . -the nodes are labeled by states of {η | f, u 1 , • • • , u n * η},
there is an edge between two nodes η 1 and η 2 if there is a transition between both states which label the nodes (i.e. η 1 η 2 ).

A branch of the call-tree is a sequence of states of the call-tree

η 1 , • • • , η k such that η 1 η 2 . . . η k-1 η k .
Given a branch B of a call-tree, the depth of the branch depth(B) is the number of states in the branch, i.e. if B = η 1 , . . . , η i-1 , η i , then depth(B) = i.

Notice that a call-tree may be infinite if it corresponds a non-terminating program.

Fraternity

The fraternity is the main syntactic notion, we use in order to restrict the computational power of considered programs. Definition 3. Given a program p, the precedence ≥ Fct is defined on function symbols by f ≥ Fct g if there is a rule f(p) → C[g(e)] in p. The reflexive and transitive closure of ≥ Fct is also noted ≥ Fct . Define ≈ Fct by f ≈ Fct g iff f ≥ Fct g and g ≥ Fct f and define > Fct by f > Fct g iff f ≥ Fct g and not g ≥ Fct f. We extend the precedence to operators and constructor symbols by ∀f ∈ Fct, ∀b ∈

Cns ∪ Op, f > Fct b. Definition 4. Given a program p, an expression C[g 1 (e 1 ), . . . , g r (e r )] is a fra- ternity activated by f(p 1 , • • • , p n ) if: 1. f(p 1 , • • • , p n ) → C[g 1 (e 1 ), . . . , g r (e r )] is a rule of p, 2. For each i ∈ {1, r}, g i ≈ Fct f, 3. For every symbol b in the context C[ 1 , • • • , r ], f > Fct b.
Notice that a fraternity corresponds to a recursive rule.

Example 2. if(Msp(Fh(i(x)), Sh(i(x))), 0(log(Fh(i(x)))), 1(log(Fh(i(x))))) is the only fraternity in the program of example 1. It is activated by log(i(x)) by taking C[ 1 , 2 ] = if(Msp(Fh(i(x)), Sh(i(x))), 0( 1 ), 1( 2 )) since log ≈ Fct log.

3 Sup-interpretations A partial assignment I is monotonic if for each symbol b ∈ dom(I), we have ∀i ∈ {1, . . . , n} ,

∀X i , Y i ∈ R + , X i ≥ Y i ⇒ I(b)(X 1 , • • • , X n ) ≥ I(b)(Y 1 , • • • , Y n ).
A partial assignment I is polynomial if for each symbol b of dom(I), I(b) is a max-polynomial function ranging over R + . That is, I(b) = max(P 1 , . . . , P k ), with P j polynomials.

A partial assignment is additive if the assignment of each constructor symbol c of arity n is of the shape

I(c)(X 1 , • • • , X n ) = n i=1 X i + α c ,

Sup-interpretations

Definition 6. A sup-interpretation is a partial assignment θ which satisfies the three conditions below:

1. θ is a monotonic assignment. 2. For each v ∈ Values, θ * (v) ≥ |v| 3. For each symbol b ∈ dom(θ) of arity n and for each value v 1 , . . . ,

v n of Values, if b(v 1 , . . . , v n ) ∈ Values, then θ * (b(v 1 , . . . , v n )) ≥ θ * ( b(v 1 , . . . , v n ) )
A sup-interpretation is additive (respectively polynomial) if it is an additive (respectively polynomial) assignment.

Notice that if a sup-interpretation is an additive assignment then the second condition of the above definition is always satisfied. Intuitively, the supinterpretation is a special program interpretation which bounds from above the output size of the function denoted by the program, as demonstrated in the following lemma:

X) = θ(0)(X) = X + 1, θ( ) = 0, θ(Msp)(X, Y ) = X Y = max(X -Y, 0), θ(Fh)(X) = X/2 and θ(Sh)(X) = X/2.
We claim that θ is an additive and polynomial sup-interpretation. Indeed, all these functions are monotonic. Moreover, for every binary value v, we have θ * (v) = |v| since the sup-interpretation of a value is equal to its size. Finally, such an assignment satisfies the third condition of the above definition. In particular, we check that, for every binary values u and v, if Msp(u, v) ∈ Values then we have:

θ * (Msp(u, v)) = θ(Msp)(θ * (u), θ * (v)) By definition of assignments = θ * (u) θ * (v)
By definition of θ(Msp)

= |u| |v| Since ∀w ∈ Values, θ * (w) = |w| = | Msp(u, v) | By definition of Msp = θ * ( Msp(u, v) ) Since ∀w ∈ Values, θ * (w) = |w|
Notice that θ is a partial assignment since it is not defined on the symbols if and log. However, we can extend θ by θ(if)(X, Y, Z) = max(Y, Z) and θ(log)(X) = X in order to obtain a total, additive and polynomial sup-interpretation.

Arboreal and fraternal programs

In this section, we give two restrictions on programs (i) the arboreal condition which ensures that the number of successive recursive calls corresponding to a function symbol (in the depth of a call-tree) is bounded logarithmically by the input size (ii) and the fraternal condition which ensures that the size of each computed value is polynomially bounded by the input size.

Arboreal programs

An arboreal program is a program whose recursion depth (number of successive recursive calls) is bounded logarithmically by the input size. This logarithmic upper bound is obtained by ensuring that some complexity measure, corresponding to the combination of sup-interpretations and motononic and polynomial partial assignments, is divided by a fixed constant K > 1 at each recursive call.

Definition 7. A program p is arboreal iff there are a polynomial and additive sup-interpretation θ, a monotonic and polynomial partial assignment ω and a constant K > 1 such that for every fraternity C[g 1 (e 1 ), . . . , g r (e r )] activated by f(p), the following conditions are satisfied:

-For any substitution σ, ω(f)(θ * (pσ)) ≥ 1 -For any substitution σ and ∀i ∈ {1, . . . , r}, ω(f)(θ * (pσ)) ≥ K×ω(g i )(θ * (e i σ)) -There is no function symbol h ∈ e i such that h ≈ Fct f.

Example 4. In the program of example 1, there is one fraternity:

if(Msp(Fh(i(x)), Sh(i(x))), 0(log(Fh(i(x)))), 1(log(Fh(i(x)))))

activated by log(i(x)). Taking the additive and polynomial sup-interpretation θ of example 3, the polynomial partial assignment ω(log)(X) = X and the constant K = 2, we check that the program is arboreal:

ω(log)(θ * (i(x))) = X + 1 ≥ 2 × (X + 1)/2 = K × ω(log)(θ * (Fh(i(x)))) ≥ 1 Lemma 2.
Assume that p is an arboreal program. Then p is terminating. That is, for every function symbol f and for any values

u in Values, f(u) is in Values * . Moreover, for each branch B = f, u 1 , • • • , u n , . . . , g, v 1 , • • • , v m of
a calltree corresponding to one execution of p and such that f ≈ Fct g, we have:

depth(B) ≤ α × log(ω(f)(θ * (u 1 ), . . . , θ * (u n ))
), for some constant α Proof. Consider an arboreal program p, a call-tree of p and one of its branch of the shape f,

u 1 , • • • , u n g, v 1 , • • • , v m , with f ≈ Fct g.
We know, by definition of a call-tree, that there are a rule of the shape

f(p 1 , • • • , p n ) → C[g(e 1 , • • • , e m )]
and a substitution σ such that p i σ = u i and e j σ = v j . We obtain:

ω(f)(θ * (u 1 ), . . . , θ * (u n )) = ω(f)(θ * (p 1 σ), . . . , θ * (p n σ)) ≥ K × ω(g i )(θ * (e 1 σ), . . . , θ * (e m σ)) By definition 7 ≥ K × ω(g i )(θ * (v 1 ), . . . , θ * (v m ))
By lemma 1

Applying the same reasoning, we demonstrate, by induction on the depth of a branch, that for each branch

B = f, u 1 , • • • , u n . . . g, v 1 , • • • , v m , with f ≈ Fct g and depth(B) = i: ω(f)(θ * (u 1 ), . . . , θ * (u n )) ≥ K i × ω(g)(θ * (v 1 ), . . . , θ * (v m ))
Consequently, the depth is bounded by log K (ω(f)(θ * (u 1 ), . . . , θ * (u n ))), because of the first condition of definition 7, whenever f, u 1 , • • • , u n is the first state of the considered branch. It remains to combine this result with the equality log(x) = log K (x) log K (2) in order to obtain the required result. Since every branch corresponding to a recursive call has a bounded depth and we are considering confluent programs, the program is terminating.

θ(b)(X 1 , • • • , X m ) = max j∈J ( m i=1 α b i,j × X i + β b j )
where J is a finite set of indices.

In other words, a program is fraternal if every symbol in a context or in an argument of a fraternity admits an affinely bounded sup-interpretation.

Example 5. C[log(Fh(i(x))), log(Fh(i(x)))] is the only fraternity in the program of example 1, where C[ 1 , 2 ] = if(Msp(Fh(i(x)), Sh(i(x))), 0( 1 ), 1( 2)). Consequently, we have to check that the symbols if, Msp, Fh, Sh, 0 and 1 admit affine sup-interpretations. This is the case by taking the polynomial supinterpretation of example 3 and, consequently, the program is fraternal. Lemma 3. Given a sup-interpretation θ and a monotonic and polynomial partial assignment ω for which the program p is arboreal and fraternal, there is a polynomial P such that for each sequence of values u 1 , • • • , u n and for each function symbol f of arity n, we have:

| f(u 1 , • • • , u n ) | ≤ P (max i=1..n (|u i |))
Proof (Sketch). Lemma 2 states that the number of successive recursive calls occurring in the depth of the call-tree is logarithmically bounded by the input size. By definition 8, the contexts and arguments of a recursive call (fraternity) of a fraternal program are affinely bounded by the input size. Consequently, a logarithmic composition of affinely bounded functions remains polynomially bounded.

Characterizations of NC k and NC

Similarly to Buss' encoding [START_REF] Buss | The boolean formula value problem is in ALOGTIME[END_REF], we represent constructors and destructors (implicitly used in pattern matching definitions) by U E * -uniform circuits of constant depth and polynomial size. Given such an encoding code and a function φ : Values n → Values * computed by some program p, we define a function φ : {0, 1} * → {0, 1} * by ∀u ∈ Values n φ(code(u)) = code(φ(u)). A function φ of Values is computable in NC k relatively to the encoding code if and only if φ is computable by a U E * -uniform family of circuits in NC k . Now, we define a notion of rank in order to make a distinction between the levels of the NC k hierarchy: Example 6. In the program of example 1, the operators if, Fh, Sh, rev and Msp are of rank 0 since they all belong to N C 0 (Cf. [START_REF] Bloch | Function-algebraic characterizations of log and polylog parallel time[END_REF] for Fh, Sh and Msp) using an encoding code which returns the binary value in T ({0, 1, }) corresponding to a binary string in {0, 1} * . Consequently, we obtain that: Proof (Sketch). We can show, by induction on the rank and the precedence ≥ Fct , that an arboreal and fraternal program of rank k can be simulated by a U E * -uniform family of circuits of polynomial size and log k depth, using a discriminating circuit of constant depth which, given some inputs, picks the right rule to apply. The log k depth relies on a logarithmic number of log k-1 depth circuits compositions by lemma 2. The polynomial size is a consequence of lemma 3. Conversely, we use the characterization of Clote [START_REF] Clote | Sequential, machine-independent characterizations of the parallel complexity classes ALOGTIME[END_REF] to show the completeness. Clote's algebra is based on two recursion schemas called Concatenation Recursion on Notation (CRN) and Weak bounded Recursion on Notation (WBRN) defined over a function algebra from natural numbers to natural numbers when considering the following initial functions zero(x) = 0, s 0 (x) = 2 × x, s 1 (x) = 2 × x + 1, π n k (x 1 , • • • , x n ) = x k , |x| = log 2 (x + 1) , x#y = 2 |y|×|x| , bit(x, i) = x/2 i mod 2 and a function tree, which computes alternations of bitwise conjunctions and bitwise disjunctions. All Clote's initial functions can be simulated by operators of rank 1 since they are in AC 0 ( [START_REF] Clote | Sequential, machine-independent characterizations of the parallel complexity classes ALOGTIME[END_REF]). We show that a function of rank k in Clote's algebra can be simulated by an arboreal and fraternal program of rank k + 1, using divide-and-conquer algorithms (This requirement is needed because of the arboreal property). A difficulty to stress here is that Clote's rank differs from the notion of rank we use because of the function tree and the CRN schema.

rk(log) = ∇(log, if(Msp(Fh(i(x)), Sh(i(x))), 0(log(Fh(i(x)))), 1(log(Fh(i(x)))))) = max(rk(if) + 1, ∇(log, 0(log(Fh(i(x))))), ∇(log, 1(log(Fh(i(x)))))) = max(1, rk(0) + 1, rk(1) + 1, ∇(log, log(Fh(i(x))))) = max(1, ∇(log, Fh(i(x))) + 1) = 1

Comparison with previous works

This work extends the characterization of NC 1 in [START_REF] Bonfante | A characterization of alternating log time by first order functional programs[END_REF] to the NC k hierarchy. For that purpose, we have substituted the more general notion of fraternal program to the notion of explicitly fraternal of [START_REF] Bonfante | A characterization of alternating log time by first order functional programs[END_REF] and we have introduced a notion of rank. In the literature, there are many characterizations of NC 1 using several computational models, like ATM or functions algebra. Compton and Laflamme [START_REF] Compton | An algebra and a logic for N C[END_REF] gave a characterization of NC 1 based on finite functions. Bloch [START_REF] Bloch | Function-algebraic characterizations of log and polylog parallel time[END_REF] used ramified recurrence schema relying on a divide and conquer strategy to characterize NC 1 . Leivant and Marion [START_REF] Leivant | A characterization of alternating log time by ramified recurrence[END_REF] have established another characterization using ramified recurrence over a specific data structure, well balanced binary trees. We can show that our characterization strictly generalizes the ones of Bloch and Leivant-Marion since they both rely on divide-and-conquer strategies. The characterizations of the NC k classes by Clote [START_REF] Clote | Computational models and function algebras[END_REF], using a bounded recurrence schema à la Cobham, are distinct from our characterizations since they do not rely on a divide and conquer strategy. However, like Cobhams' work, Clote's WBRN schema requires an upper bound on the computed function which is removed from our characterizations. Other characterizations of NC are also provided in [START_REF] Leivant | A characterization of NC by tree recurrence[END_REF][START_REF] Bonfante | Towards an implicit characterization of N C k[END_REF]. All these purely syntactic characterizations capture a few algorithmic patterns. On the contrary, our work tries to delineate a broad class of algorithms using the semantics notion of sup-interpretation.

3. 1

 1 Monotonic, polynomial and additive partial assignments Definition 5. A partial assignment I is a partial mapping from the vocabulary Σ such that, for each symbol b of arity n in the domain of I, it yields a partial function I(b) : (R + ) n -→ R + , where R + is the set of non-negative real numbers. The domain of a partial assignment I is noted dom(I) and satisfies Cns ∪ Op ⊆ dom(I).

  with α c ≥ 1, whether n > 0, and I(c) = 0 otherwise.If each function symbol of a given expression e having m variables x 1 , • • • , x m belongs to dom(I) then, given m fresh variables X 1 , • • • , X m ranging over R + , we define the homomorphic extension I * (e) of the assignment I inductively by:1. If x i is a variable of Var, then I * (x i ) = X i 2.If b is a symbol of Σ of arity 0, then I * (b) = I(b). 3. If b is a symbol of arity n > 0 and e 1 , • • • , e n are expressions, then I * (b(e 1 , • • • , e n )) = I(b)(I * (e 1 ), . . . , I * (e n )) Given a sequence e = e 1 , • • • , e n , we will sometimes use the notation I * (e) to denote I * (e 1 ), . . . , I * (e n ).

Lemma 1 (Example 3 .

 13 [START_REF] Marion | Resource analysis by sup-interpretation[END_REF]). Given a sup-interpretation θ and an expression e defined over dom(θ), if e ∈ Values then we have | e | ≤ θ * ( e ) ≤ θ * (e) Consider the program of example 1. Define the additive assignment θ by θ(1)(

4. 2

 2 Fraternal programs Definition 8. A program p is fraternal if there is a polynomial and additive sup-interpretation θ such that for each fraternity C[g 1 (e 1 ), . . . , g r (e r )] activated by f(p 1 , • • • , p n ) and for each symbol b of arity m appearing in C or in e j , there are constants α b i,j , β b j ∈ R + satisfying:

Definition 9 .

 9 Given a program p composed by a vocabulary Σ, a set of rules R and an encoding code, the rank of a symbol b, rk(b), and the rank of a symbol b relatively to an expression e, ∇(b, e), are partial functions ranging over N and defined by induction over the precedence ≥ Fct : -If b is a variable or a constructor symbol then rk(b) = 0. -If b is an operator, which can be computed by a U E * -uniform family of circuits of polynomial size and depth bounded by log k relatively to the encoding code, then rk(b) = k. -If b is a function symbol we define its rank relatively to an expression e by: • If ∀b ∈ e, b > Fct b then ∇(b, e) = max b ∈e (rk(b )) • Otherwise ∃b ∈ e such that b ≈ Fct b and e = b (e 1 , • • • , e n ): * If b > Fct b then ∇(b, e) = max(rk(b ) + 1, ∇(b, e 1 ), . . . , ∇(b, e n )) * Otherwise b ≈ Fct b then ∇(b, e) = max(∇(b, e 1 ), . . . , ∇(b, e n )) + 1 -Finally, we define the rank of a function symbol b by: • rk(b) = max b(p)→e∈R (∇(b, e))where b ∈ e means that the symbol b appears in the expression e. The rank of a program is defined to be the highest rank of a symbol of a program.

  where x ∈ Var is a variable, c ∈ Cns is a constructor symbol, op ∈ Op is an operator, f ∈ Fct is a function symbol, p 1 , • • • , p n ∈ T (Var,Cns) are patterns and e 1 , • • • , e n ∈ T (Var, Cns, Op, Fct) are expressions. The program's main function symbol is the first function symbol in the program's list of rules. Throughout the paper, we only consider orthogonal programs having disjoint and linear rule patterns. Consequently, each program is confluent

  Theorem 1. A function φ from Values n to Values * is computed by a fraternal and arboreal program of rank k ≥ 1 (resp. k ∈ N) if and only if φ is computable in NC k (resp. NC).