Julia Mouton
email: julia.mouton@epf.fr

Mireille Turmine

Hélène Van Den Berghe

Jean Coudane

Prof M Turmine

Prof H Van Den Berghe

A new water

Keywords: polycarbobetaines, copper, complexation, isotherms, selectivity, reusable

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

Introduction

Currently, metals are highly studied in environmental management because of their depletion as resources and because of the increasingly common cases of pollution [START_REF] Reuter | Metal Recycling: Opportunities, Limits, Infrastructure; A Report of the Working Group on the Global Metal Flows to the International Resource Panel[END_REF]. Thus, their recovery and valorization are key themes of recent scientific studies concerning the environment. Copper ions are produced as waste in various chemical industries, such as smelting, mining, printed circuit board manufacturing, electroplating, wire drawing, copper polishing, and paint manufacturing [START_REF] Ali | Novel cross-linked polymers having pHresponsive amino acid residues for the removal of Cu2+ from aqueous solution at low concentrations[END_REF]. Various techniques, such as chemical precipitation, coagulation, solvent extraction, ultra-filtration, biological systems, electrolytic processes, reverse osmosis, oxidation with ozone/hydrogen peroxide, membrane filtration, ion exchange, photocatalytic degradation and adsorption, have been developed for the removal of metal contaminants from waste [START_REF] Ahmaruzzaman | Industrial wastes as low-cost potential adsorbents for the treatment of wastewater laden with heavy metals[END_REF]. Among these techniques, precipitation appears as the most attractive, primarily because of its economical operation. However, one disadvantage exists when treating multi-element samples because all metals are typically precipitated together in sludge without any possibility to valorize them. Selective precipitating agents, such as dithiocarbamate and their ramifications, 2,4,6-trimercapto-1,3,5-triazine, or dipropyl dithiophosphate, have been studied, but their high cost inhibits their use for different industrial applications [4]. Adsorption is also a well-studied and efficient technique that has been considered more often for metal ion selective treatment [START_REF] Chen | Adsorption of aqueous Cd2+, Pb2+, Cu2+ ions by nano-hydroxyapatite: Single-and multi-metal competitive adsorption study[END_REF]. Low-cost natural and synthetic adsorbents have already demonstrated good performances for copper removal from wastewaters [START_REF] Aydin | Removal of copper (II) from aqueous solution by adsorption onto low-cost adsorbents[END_REF][START_REF] Hsu | Experimental assessment of adsorption of Cu2+ and Ni2+ from aqueous solution by oyster shell powder[END_REF][START_REF] Kyzas | Copper and chromium(VI) removal by chitosan derivatives-Equilibrium and kinetic studies[END_REF][START_REF] Papageorgiou | Prediction of binary adsorption isotherms of Cu(2+), Cd(2+) and Pb(2+) on calcium alginate beads from single adsorption data[END_REF]. Hydrogels are also considered for these applications [START_REF] Muratalin | Preparation and characterization of microgels sensitive toward copper II ions[END_REF]. However, these techniques are less cost-effective compared with precipitation when considering the clogging/unclogging operational problems in the technology.

This research is devoted to the study of polybetaines as metal ion precipitating agents.

Polybetaines are specific polyampholytes that possess anionic and cationic groups on the same monomer unit. Among polybetaines, polysulfobetaines, polycarbobetaines and polyphosphobetaines are prominent [START_REF] Kudaibergenov | Polymeric Betaines: Synthesis, Characterization, and Application[END_REF]. This study investigates polycarbobetaines (PCBets)

and their interactions with copper to design chemical processes for copper valorization.

Cross-linked PCBet hydrogels have been tested for copper adsorption [START_REF] Ali | Novel cross-linked polymers having pHresponsive amino acid residues for the removal of Cu2+ from aqueous solution at low concentrations[END_REF][START_REF] Kudaibergenov | Novel macroporous amphoteric gels: Preparation and characterization[END_REF][START_REF] Noh | Synthesis, characterization, and stimuli-sensitive properties of novel polycarbobetaines[END_REF]; however, no literature exists about water-soluble PCBets for similar applications [START_REF] Tarannum | Advances in Synthesis and Applications of Sulfo and Carbo Analogues of Polybetaines: A Review[END_REF], although watersoluble polymers have been suggested to enhance filtration systems [START_REF] Geckeler | Removal of Hazardous Substances from Water Using Ultrafiltration in Conjunction with Soluble Polymers[END_REF][START_REF] Molinari | Selective separation of copper(II) and nickel(II) from aqueous systems by polymer assisted ultrafiltration[END_REF][START_REF] Rivas | Water-soluble functional polymers in conjunction with membranes to remove pollutant ions from aqueous solutions[END_REF].

Theorical background

The interactions of water-soluble polymers with metal ions are treated as surface phenomena as previously reported [START_REF] Moreno-Villoslada | Competition of Divalent Metal Ions with Monovalent Metal Ions on the Adsorption on Water-Soluble Polymers[END_REF]. The most appropriate method for assessing sorption capacity is the derivation of a whole sorption isotherm [START_REF] Wang | Equilibrium sorption isotherms for of Cu2+ on rice bran[END_REF]. Over the years, a wide variety of equilibrium isotherm models (Langmuir, Freundlich, Brunauer-Emmett-Teller, Redlich-Peterson, Dubinin-Radushkevich, Temkin, Toth, Koble-Corrigan, Sips, Khan, Hill, Flory-Huggins and Radke-Prausnitz isotherm), have been formulated [START_REF] Foo | Insights into the modeling of adsorption isotherm systems[END_REF]. This study considers some two parameter isotherms only (Langmuir, Freundlich and Dubinin-Radushkevich). All of the equations and parameters used to model the adsorption isotherms are reported in Table 1 where Ce and Qe are respectively the concentration of free copper at the equilibrium (mg L -1 ), and the amount of copper adsorbed (mg) per gram of polymer.

The Langmuir equation is valid for monolayer sorption onto a surface with a finite number of identical sites [START_REF] Choy | Langmuir Isotherm Models Applied to the Multicomponent Sorption of Acid Dyes from Effluent onto Activated Carbon[END_REF]. It assumes a homogeneous surface and no interaction between the adsorbed species on adjacent active sites [START_REF] Langmuir | The adsorption of gases on plane surfaces of glass, mica and platinium[END_REF]. From equation presented in Table 1, one can readily deduce that at low sorbate concentrations it effectively reduces a linear isotherm.

Alternatively, at high sorbate concentrations, it predicts a constant -monolayer -sorption capacity [START_REF] Ho | Equilibrium Isotherm Studies for the Sorption of Divalent Metal Ions onto Peat: Copper, Nickel and Lead Single Component Systems[END_REF]. A separation factor or equilibrium parameter (R L ) was defined from this model [START_REF] Hall | Pore-and Solid-Diffusion Kinetics in Fixed-Bed Adsorption under Constant-Pattern Conditions[END_REF], as :

ܴ = 1 1 + ܭ * ܥ -4 -
where C 0 is the highest initial concentration of adsorbate in solution (mg L -1 ).

Depending on its value, the isotherm shape can be interpreted as follows : R = 0 for the irreversible case, 0 < R < 1 for favorable equilibrium, R = 1 for the linear case, and R > 1 for unfavorable equilibrium [START_REF] Hall | Pore-and Solid-Diffusion Kinetics in Fixed-Bed Adsorption under Constant-Pattern Conditions[END_REF].

The Freundlich equation is characteristic of multilayer adsorption, with non-uniform distribution of adsorption site and affinities over the heterogeneous surface. The magnitude of exponent n gives an indication on the favorability of adsorption. It is generally stated that the values n in the range of 2-10 represent good, 1-2 moderately difficult and less than 1 poor adsorption characteristics [START_REF] Hadi | Equilibrium two-parameter isotherms of acid dyes sorption by activated carbons: Study of residual errors[END_REF]. The Freundlich expression is an exponential equation and therefore, assumes that as the adsorbate concentration increases, the concentration of adsorbate on the adsorbent surface also increases. To determine the maximum adsorption capacity (Qm), it is necessary to operate with constant initial concentration C0 and variable weights of adsorbent; thus ln Qm is the extrapolated value of ln q for C=C0 as previously reported [START_REF] Hamdaoui | Modeling of adsorption isotherms of phenol and chlorophenols onto granular activated carbon: Part I. Two-parameter models and equations allowing determination of thermodynamic parameters[END_REF].

Dubinin-Radushkevich isotherm is generally applied to express the adsorption mechanism with a Gaussian energy distribution onto a heterogeneous surface. The model has often successfully fitted high solute activities and the intermediate range of concentrations data well. The approach was usually applied to distinguish the physical and chemical adsorption of metal ions [START_REF] Foo | Insights into the modeling of adsorption isotherm systems[END_REF]. The mean free energy (E) can be calculated as follows [START_REF] Hobson | Physical adsorption isotherms extending from ultrahigh vacuum to vapor pressure[END_REF] :

ܧ = ቈ 1 ඥ2 × ܭ ோ
The magnitude of free energy is used for estimating the type of adsorption mechanism. When ranging from 8 to 16 kJ mol -1 , it indicates an ion-exchange process, beyond 16 kJ mol -1 a coordination reaction is expected, and below 8 kJ mol -1 physisorption takes place [START_REF] Copello | Adsorption of Cd(II) and Pb(II) onto a one step-synthesized polyampholyte: kinetics and equilibrium studies[END_REF]. One of the unique features of the Dubinin-Radushkevich isotherm model lies on the fact that it is temperature-dependent, which when adsorption data at different temperatures are plotted as a function of logarithm of amount adsorbed versus the square of potential energy, all suitable data will lie on the same curve, named as the characteristic curve [START_REF] Nguyen | The Dubinin-Radushkevich equation and the underlying microscopic adsorption description[END_REF][START_REF] Dubinin | The Potential Theory of Adsorption of Gases and Vapors for Adsorbents with Energetically Nonuniform Surfaces[END_REF].

Thermodynamic data such as adsorption energy can be obtained from Langmuir equation and can be calculated by using the following equation [START_REF] Kim | Arsenic Removal Using Mesoporous Alumina Prepared via a Templating Method[END_REF]:

ܭ = ݁ ି∆ீ ೌೞ బ ோ×் Table 1.
Equations and parameters used to model the adsorption isotherms of copper on PCEGC Name Equation Parameters

Langmuir Qe = Q ୫ * K * C ୣ 1 + K * C ୣ Qm = Maximum adsorption capacity (mg g -1 ) K L = Isotherm constant (L mg -1 ) Freunlich Qe = K * C ୣ ଵ/୬ K f = Isotherm constant (mg g -1 (L mg -1 ) 1/n ) n = adsorption intensity Dubinin- Radushkevich Qe = Q ୱ * exp [-Kdr * ε ଶ ] ε = RT * ln ൬1 + 1 C ୣ ൰ Q S = Maximum adsorption capacity (mg g -1 )
Kdr = Isotherm constant (mol 2 kJ -2 )

Materials and methods

Materials

Acetoacetic ester (AAE, ≥ 99%) and the initiator 2,2'azobisisobutyronitrile (AIBN, ≥ 98%)

were purchased from Aldrich and used without purification. Acrylic acid (AA, 99.5%) purchased from Aldrich was purified by distillation under vacuum. Glycine (GLY, grade for analysis) was obtained from Merck. Ammonia solution (NH

Analytical techniques

FTIR spectra were recorded using a Perkin-Elmer Spectrum 100 FTIR spectrometer using the attenuated-total-reflectance (ATR) method. 1 H-NMR spectra were recorded using a Bruker spectrometer (AMX300) operating at 300 MHz. Deuterium oxide (D 2 O) was used as the solvent. The chemical shifts are expressed in ppm with respect to tetramethylsilane (TMS).

The UV-vis spectra of the polymer solutions were recorded on a Hitachi U4001 spectrophotometer with a scan range of 220-900 nm. Free copper in the solutions was followed on a PHM250 ion analyzer equipped with a copper selective electrode (CSE) (Radiometer-ISE25Cu) and a sulfate reference electrode (Radiometer REF621). Figure 1 and Table 2 present the characteristics of the electrochemical method. The pH was also recorded using a PHM250 ion analyzer equipped with a glass electrode (Radiometer pHG301) and the same sulfate reference electrode.

Syntheses

The monomer was obtained by dropwise addition of a glycine 16% aqueous solution (37 mL) to acetoacetic ester (10 mL) under magnetic stirring for 3 hours. The solution was then left overnight to decant (12 hours), and the organic phase containing the key product ethyl-3glycinatocrotonate (EGC) was collected. The yield of the monomer was 98%, and its purity was tested by FTIR. The linear water-soluble polycarbobetaine named polycarboxyethyl-3glycinatocrotonate (PCEGC) was synthesized by radical polymerization, as previously reported [START_REF] Noh | Synthesis, characterization, and stimuli-sensitive properties of novel polycarbobetaines[END_REF][START_REF] Didukh | Synthesis and characterization of novel hydrophobically modified polybetaines as pour point depressants[END_REF], by mixing EGC, AA and the initiator AIBN (5 minutes). The polymer obtained in bulk was washed with an abundant amount of acetone and oven-dried at 70°C

(approximately 35% yield).

Scheme 1. Monomer and polymer syntheses.

C H 3 O O O CH 3 C H 3 O O O CH 3 H C H 3 O N O CH 3 O H O C H 3 O N O CH 3 H O H O -H 2 O C H 2 O OH C H 3 N O O CH 3 O OH OH O n NH 2 O H O C H 3 N O O CH 3 O OH OH O -10 -

Sorption experiments

The sorption experiments were performed by the batch method (10 mL) in a thermostated cell (25°C) with constant stirring, using PCEGC 0.5 g L -1 . The adsorption behavior of the copper ions (Cu(II)) was assayed over a range from 5.10 -5 to 5 10 -3 mol L -1 and over a pH range from 3 to 6. The ionic strength (I) was fixed at 0.3 mol L -1 with Na 2 SO 4 0.1 mol L -1 . The pH was adjusted with H 2 SO 4 or NaOH. Adsorption isotherms were determined by monitoring free copper decay in the solution and were fitted to the Langmuir, Freundlich and Dubinin-Radushkevich models. The amount of copper adsorbed in (mg/g) at equilibrium (Qe), defined as the soluble and precipitated fractions of PCEGC/Cu complex, was calculated from the mass balance of initial and final copper concentrations:

ܳ = ሺܥ -ܥ ሻ × ܸ ்௧ ܹ ாீ
where C 0 and Ce are respectively the initial concentration of copper and the concentration at the equilibrium, V Tot is the total volume of the solution (L) and W PCEGC is the weight of the PCEGC (g).

Some of the assays were used to test the selectivity of PCEGC toward copper, cobalt (Co 2+ ) and nickel (Ni 2+ ) using electrochemical measurements and UV-visible spectroscopy.

All of the experiments were conducted in triplicate under identical conditions.

The adsorption isotherms were determined by using non-linear regression analysis. The error function employed to evaluate the fit was the second order corrected Akaike information criterion (AICC) defined as follows [START_REF] Hadi | Equilibrium two-parameter isotherms of acid dyes sorption by activated carbons: Study of residual errors[END_REF][START_REF] Copello | Adsorption of Cd(II) and Pb(II) onto a one step-synthesized polyampholyte: kinetics and equilibrium studies[END_REF]:

ܥܫܣ = ܰ × ݊ܮ ൬ ܴܵܵ ܰ ൰ + 2ܲ ܥܫܣ = ܥܫܣ + 2ܲ × ሺܲ + 1ሻ ܰ -ܲ -1
where P is the number of parameters in the model (P=2), N the number of data points and RSS the residual squares sum.

-11 -AICc considers the size of the calibration data set and is therefore preferred over the original AIC [START_REF] Dai | Identification of sorption processes and parameters for radionuclide transport in fractured rock[END_REF].

Sorption/desorption cycles experiments

Sorption/desorption experiments were performed over 5 cycles by pH adjustments. A solution of copper 0.005 mol L -1 and PCEGC 0.5 g L -1 was prepared in Na 2 SO 4 0.1 mol L -1 at pH=5 (the final pH was 4.5). Desorption experiments were performed by decreasing the pH to 2, and the next adsorption cycle was obtained by pH adjustment back to 4.5. The pH was adjusted with H 2 SO 4 or NaOH. This operation was conducted four more times. The free copper in solution was followed by CSE and the concentration was corrected considering the volume increase due to H 2 SO 4 or NaOH addition. The adsorption-desorption behavior of the Cu(II) was assayed in the absence and in the presence of cobalt or nickel ions.

All of the experiments were conducted in triplicate under identical conditions. The main functional groups were identified by FTIR spectroscopy for PCEGC, including the O-H stretching vibration at 3124 cm -1 ; C-H stretching vibration at 2956 cm -1 ; and C=O stretching vibration at 1701 cm -1 , as previously reported for a similar structure (polycarboxyethyl-3-aminocrotonate) [START_REF] Didukh | Stimuli-sensitive behaviour of novel betaine-type polyampholytes[END_REF][START_REF] Kudaibergenov | Preparation and Characterization of Novel Polymeric Betaines Based on Aminocrotonates[END_REF]. The results are presented in Figure 1a and -12 -remained unchanged in the PCEGC/Cu complex. No carboxylate asymmetric stretching vibrations were recorded, while a symmetric vibration appeared at 1415 cm -1 . This tendency changes in the case of the PCEGC/Cu complex, with the appearance of a medium asymmetric stretching vibration at 1608 cm -1 , which is characteristic of copper carboxylates [START_REF] Cartechini | X-ray absorption investigations of copper resinate blackening in a XV century Italian painting[END_REF]. The difference between these two well-known characteristic bands in the PCEGC/Cu complex (193 cm -1 ) indicated that the coordination of the carboxylate groups occurred in a bridged form [START_REF] Karaliota | Synthesis and characterization of a binuclear coumarin-3-carboxylate copper(II) complex[END_REF].

Results and Discussion

The aqueous solution of PCEGC 0.5 g L -1 (I = 0.3 mol L -1 at pH = 5) exhibits an absorbance peak approximately 280 nm, which is specific to carboxylic groups (Figure 1b), while the copper solution absorbs at 816 nm. Figure 1b shows that the maximum absorption for the PCEGC/Cu complex shifted from 816 nm to 702 nm and indicated the coordination interactions between the species. This band is characteristic of square-pyramidal copper(II) complexes [START_REF] Cartechini | X-ray absorption investigations of copper resinate blackening in a XV century Italian painting[END_REF]. In addition, the specific peak of PCEGC at 280 nm was markedly increased in the presence of copper (from 0.04 to 0.87), confirming the interaction between copper and polycarbobetaine carboxylate groups. This band is assigned to the carboxylate-to-copper charge transfer (ligand-to-metal charge transfer) [START_REF] Karaliota | Synthesis and characterization of a binuclear coumarin-3-carboxylate copper(II) complex[END_REF].

Copper complexation

The mechanisms of copper uptake by PCEGC at pH 3.0, 4.0, 5.0, and 6.0 were explored by analyzing the results of the adsorption isotherms (the experimental and predicted isotherms are presented in figure 3). All the parameters extracted from the fitting models are in Table 3 for pH 4 to 6. The adsorption capacities at equilibrium (Qe) were low at pH 3.0 (around the detection limits); thus, this pH condition did not explain the binding mechanisms. As shown in Table 3, the difference between the corrected Akaike information criterion of the Freundlich or Dubinin-Radushkevich models and the Langmuir model was greater than nine, providing strong evidence favoring the Langmuir model as the best to describe the interaction between copper and PCEGC [START_REF] Burnham | AIC model selection and multimodel inference in behavioral ecology: some background, observations, and comparisons[END_REF]. The best fit to the Langmuir model translates the predominantly Considering the separation factor (R L ), adsorption can be considered unfavorable (R L 1), linear (R L = 1), favorable (1 > R L > 0) or irreversible (R L = 0) [START_REF] Kannan | Kinetics and mechanism of removal of methylene blue by adsorption on various carbons -a comparative study[END_REF]. R L calculated with the Langmuir model indicated the favorable adsorption of Cu(II) onto PCEGC (ranging from 0.2 to 0.02, depending on pH). This observation was confirmed by the values of n (>1) from the Freundlich model [START_REF] Kong | Adsorption of Pb(II) from aqueous solution using keratin waste -hide waste: Equilibrium, kinetic and thermodynamic modeling studies[END_REF]. The values of free energy change (∆ r G < 0) corroborated the spontaneity of the reaction. The maximum adsorption capacity (Qm) was substantially increased (from 15.5 to 264 mg g -1 ) as the pH increased (from 3 to 6) and appeared linearly correlated to this parameter (Figure 3).

-16 - The mean free energy of sorption (E) derived from the Dubinin-Radushkevich equation provides information on the type of sorption reaction [START_REF] Ho | Equilibrium Isotherm Studies for the Sorption of Divalent Metal Ions onto Peat: Copper, Nickel and Lead Single Component Systems[END_REF]. An energy range from 8 to 16 kJ mol -1 indicates an ion-exchange process, and a value above 16 kJ mol -1 indicates a coordination reaction [START_REF] Copello | Adsorption of Cd(II) and Pb(II) onto a one step-synthesized polyampholyte: kinetics and equilibrium studies[END_REF]. The results presented in Table 1 suggest that coordinative PCEGC. This conclusion can help to explain previous observations in which Qm increased as the pH increased. The functional groups most likely involved in the complexation reaction are carboxylic groups (COOH). Thus, a higher pH indicates that a larger amount of COOH groups is deprotonated to COO -, thus making them available to complex copper by dative bonds.

Table 4 shows the maximum adsorption capacities of various natural and synthetic materials recently studied for copper recovery from waste or PCEGC clearly appears as a promising agent for these applications, especially considering its potential regeneration for future use (as observed for similar PCBets in previous studies) [START_REF] Noh | Synthesis, characterization, and stimuli-sensitive properties of novel polycarbobetaines[END_REF].

-18 -

Copper selectivity

The influence of nickel (Ni 2+ ) and cobalt (Co 2+ ) on copper adsorption was investigated by electrochemical measurements (Figure 2). Copper solutions at 5.10 -4 mol L -1 (-215.6 < E Cu < -216.3 mV) were prepared in 0.1 mol L -1 Na 2 SO 4 at pH = 5 with several different concentrations of metallic ions (Ni 2+ or Co 2+ ). The ratio (R=concentration of metallic ion/concentration of copper) was fixed to 0, 0.5, 1, 2, 5 and 10 (the linearity of the electrode response was lost for higher ratios). Under these conditions, regardless of the metal and the ratio used, the potential of the electrode remained unchanged (with values between -215.6 and -216.3 mV). The potential was kept constant (ranging from -235.20 to -236.50 mV) with the addition of PCEGC (0.5 g L -1 ) to the same solutions, suggesting that Co 2+ and Ni 2+ were not competing for copper adsorption. The selectivity of PCEGC towards copper was confirmed for R=50 with cobalt UV-visible analyses (Cf. Figure 1). The spectra of PCEGC/Cu and absorbance. These assays were not possible with nickel because of the absorbance recovery with copper. In addition to these results, no desorption of copper occurred in favor of nickel (0 < R < 10) or cobalt complexation (0 < R < 50). This observation agreed with previous results about the selective complexation of copper by glycine groups [START_REF] Oraby | The selective leaching of copper from a gold-copper concentrate in glycine solutions[END_REF].

Sorption/Desorption cycles

The reusability of PCEGC was explored by pH adjustment at 2 and 4.5 respectively for desorption and adsorption. Figure 6 confirms that the efficiency of copper adsorption was maintained over 5 cycles of PCEGC reuse (91 ± 4 %). The selectivity toward copper in the presence of Ni(II) or Co(II) was also maintained in the same conditions (94 ± 3% and 96 ± 3% respectively).

Highlights :

-A new pH-sensitive polycarbobetaine was synthetized -Adsorption isotherms were built and modeled -High adsorption capacities were reached for copper at pH = 5 and pH = 6 -The selectivity toward copper was indicated in the presence of nickel and/or cobalt -The efficiency of copper adsorption was maintained over 5 cycles of polycarbobetaine reuse

Figure 1 .

 1 Figure 1. Calibration mean of the copper selective electrode in Na 2 SO 4 0.1 mol L -1 at pH

  Fit of Concatenated Data 95% Confidence Band of Concatenated Data 95% Prediction Band of Concatenated Data Potential (mV) Log(Cu2+) -8 -

3. 1 .

 1 CharacterizationCH 2 and CH 3 groups of the ethyl function were clearly identified by the presence of a quadruplet at 3.97 ppm and a triplet at 1.03 ppm, respectively, in 1 H NMR spectra of pure PCEGC. The other hydrogen atoms were observed over the expected range (from 1.43 to 2.54 ppm). No comparison could be performed with the 1 H NMR spectra of PCEGC/Cu complexes due to their insolubility in D 2 O.

Figure 2 .

 2 Figure 2. IR spectra of PCEGC and PCEGC/Cu complexes (a), and UV-visible spectra of PCEGC (0.5 g L -1 ), Cu(II) (0.0005 mol L -1 ) and/or Co(II) (0.025 mol L -1 ) in 0.1 mol L -1

Figure 3 .

 3 Figure 3. Adsorption isotherms of copper onto PCEGC at various pH values. Experimental

  the chelating groups on the polymer chains because the Langmuir equation assumes that the adsorbent surface is energetically homogeneous.

Figure 4 .

 4 Figure 4. Maximum adsorption capacity of PCEGC correlated to a) initial pH (R 2 = 0.98012); b) final pH (R 2 =0.97312).

Figure 5 .

 5 Figure 5. Selectivity of PCEGC toward Cu(II) (5 10 -4 mol L -1 ) in the presence of Co 2+ or Ni 2+ at varying concentrations in 0.1 mol L -1 Na 2 SO 4 at pH = 5 (R=[metallic ion]/[Cu(II)]).

  /Co were equivalent regarding the wavelength absorption maximum and the

Figure 6 .

 6 Figure 6. Copper adsorption efficiency over 5 cycles of sorption-desorption processes in the

  NiSO 4 . 6H 2 O) and the reagent grade solvent acetone were purchased from VWR. Cobalt sulfate (extra pure, CoSO 4 . 7H 2 O) was obtained from Acros Organics.

4

OH 30%, ACS) and sulfuric acid (H 2 SO 4 96%, for analysis) were obtained from Carlo-Erba. Analytical reagent grade, sodium sulfate hexahydrate (

Table 2 .

 2 Parameters of the electrochemical analytical method in Na 2 SO 4 0.1 mol L -1 at

	pH =5 (based on fourteen replicates)
	Parameters	Mean	Standard deviation
	Slope	29.3	0.2
	Intercept	-119.9	0.6
	Determination coefficient (R 2 )	0.9966	-
	Number of points	115	-
	Concentration range (mol L -1 )	5x10 -6 -1x10 -2	-
	Detection limit* (mg L -1 )	0.8	-
	* [32]

Table 3 .

 3 Isotherm model parameters for Cu(II) adsorption on PCEGC at different pH values.

	Models	Parameters	pH=4	pH=5	pH=6
	Langmuir	Qm (mg g -1 )	73.3±3.4	179±3	264±11
		b (L mg -1 )	0.011±0.001	0.14±0.01	0.49±0.08
		R 2		0.9960	0.9972	0.9881
		∆ r G (kJ mol -1 )	-16.0±3.4	-22.6±1.7	-25.6±4.4
	Freundlich	R L AIC C K f (mg g -1 (L mg -1 ) 1/n ) a)	0.228 10.4 2.87±1.08	0.021 28.3 37.6±7.8	0.031 34.7 96.8±16.7
		n		1.88±0.26	3.29±0.48	3.49±0.72
		R 2		0.9607	0.9322	0.9065
		a) ∆AIC C AIC C	b)	24.1 13.7	57.1 28.8	47.1 12.4
	Dubinin-Radushkevich	Q S (mg g -1 ) B D (mol 2 kJ -2 )	49.5±4.7 1.64 x 10 -5 ±5.78 x 10 -5	160±11 3.29 x 10 -6 ±9.95 x 10 -7	237±22 6.53 x 10 -7 ±2.79 x 10-7
		E (kJ mol -1 )	55.3±9.7	390±59	875±187
		R 2		0.9241	0.9446	0.8916
		a) ∆AIC C AIC C	b)	28.02 17.6	55.27 26.9	47.98 13.2
	a) Corrected Akaike information criterion [39]; b) Difference between one model AIC C and
	Langmuir AIC C

Table 4 .

 4 Previously reported adsorption capacities of Cu(II) onto various adsorbents.

	Origins	Adsorbents		Qm (mg g -1 ) pH T° (°C) References
	Natural	Shell of lentil		8.98	6	20	[6]
		Shell of wheat		7.39	6	20
		Shell of rice		1.85	6	20
		Banana pith		13.5	4.5	-	[42]
		Peath		12.7	4.5	25	[23]
		Oyster shell		49.3	2	30	[7]
		Calcium alginate beads	84.5	4.5	25	[43]
				12.1	4.8	22	[44]
		Lignin		22.9	5.5	20	[45]
		Chitosan		208	6	25	[8]
	Synthetic Nanohydroxyapatite	4.4	5.5	25	[5]
		Polyacrylic acid 3x10 6 g mol -1	369	6	25	[46]
		Polyethylene imine	333	6	25	[47]
		ASA-PGMA/SiO 2	a	37	5	20	[48]
		Amino functionalized silica gel	54	5	30	[49]
		PCEGC		264	6	25	This study
	a 5-aminosalicylic acid (ASA) grafted onto the poly(glycidyl methacrylate) (PGMA)

Conclusions

Based on the functionalization of polycarboxyethyl-3-aminocrotonate, novel pH sensitive polycarboxybetaines were successfully prepared with the goal of achieving great complexing properties with copper. The maximal adsorption capacities recorded were shown to be pHdependent, reaching a value of 264±11 mg of copper per g of polymer at pH=6. The present study also revealed the highly selective properties of PCEGC toward copper in the presence of cobalt, nickel or zinc. This study will be continued to determine the coordinative chemistry of the PCEGC/Cu complexes. The relevance of using PCEGC in environmental applications (such as metal recovery from wastes/wastewaters or environmental sensors) was demonstrated. Using a sustainable strategy based on its pH-sensitive properties, PCEGC reusability and the preservation of the copper adsorption efficiency and selectivity were also confirmed. The use of such materials for environmental applications is greatly encouraged. Further studies will be performed on copper recovery using PCEGC to design a full physicochemical process.


Olivier Bernard
email: olivier.bernard@inria.fr

¢¡ ¤£

Georges Bastin
email: bastin@auto.ucl.ac.be

On the estimation of the pseudo-stoichiometric matrix for macroscopic mass balance modelling of biotechnological processes

Keywords: Modelling, Nonlinear systems, Biotechnology, Identification, Identifiability, Validation

In this paper we propose a methodology to determine the structure of the pseudo-stoichiometric coefficient matrix ¨in a macroscopic mass balance based model. The first step consists in estimating the minimal number of reactions that must be taken into account to represent the main mass transfer within the bioreactor. This provides the dimension of ¨. Then we discuss the identifiability of the components of ¨and we propose a method to estimate their values. Finally we present a method to select among a set of possible macroscopic reaction networks those which are in agreement with the available measurements. These methods are illustrated with 3 examples: real data of the growth and biotransformation of the filamentous fungi Pycnoporus cinnabarinus, real data of an anaerobic digester involving a bacterial consortium degrading a mixture of organic substrates and a process of lipase production from olive oil by Candida rugosa.

Introduction and motivation

For a long time the macroscopic perspective of modelling of biotechnological processes has proved to be efficient for solving many bioengineering problems [START_REF] Bailey | Biochemical engineering fundamentals[END_REF][START_REF] Bastin | On-line estimation and adaptive control of bioreactors[END_REF]. In this perspective the total cell mass in the reactor is considered as a "black-box" (see e.g. [START_REF] Stephanopoulos | Metabolic Engineering[END_REF], chapter 4) for the conversion of initial substrates into final products. The transformations are described by a small set of macroscopic (overall) reactions that lump together the many intracellular metabolic reactions of the various involved microbial species. Such macroscopic reaction networks represent the main mass transfers throughout the system and directly connect initial substrates to final products without describing the intracellular behaviour. On this basis, dynamical mass balance models can then be established. They rely on the category of the so-called "unstructured" models in the standard terminology reported e.g. in [START_REF] Bailey | Mathematical modeling and analysis in biochemical engineering: past accomplishments and future opportunities[END_REF]. The goal of macroscopic modelling is clearly to derive simple dynamical models which have proved of great interest in bioengineering for the design of on-line algorithms for process monitoring, control and optimisation [START_REF] Bastin | On-line estimation and adaptive control of bioreactors[END_REF][START_REF] Bastin | Nonlinear and adaptive control in biotechnology: a tutorial[END_REF]. This paper clearly relies on this macroscopic perspective and our goal is to describe an approach for the determination of the minimal number of macroscopic reactions that should be involved in a mass balanced model in order to ensure constant pseudo-stoichiometric coefficients and to represent the main mass transfers within the system.

To motivate our objectives, let us consider a simple biotechnological process which could be represented by the simplest reaction network made of a single overall reaction :

¢¡ £ ¤ ¥ §¦ ©¨¤ §¤ ¢ £ ¤ ¥ §¦ ¤ ! "¤
where

#¤ $ ! §%
denote the substrates and products, is the total biomass (possibly made of multiple microbial species) and ¨¤ , ¤ are the pseudo stoichiometric coefficients. In this case, as explained in [START_REF] Stephanopoulos | Metabolic Engineering[END_REF], the pseudo-stoichiometric coefficients ¨¤ and ¤ are exactly the yield coefficients that can be directly computed from experimental data.

The main problem with a single overall reaction is obviously that it is often not able to describe accurately the process all along a transient operation with the same constant yield coefficients. A classical and efficient way to overcome this difficulty, without relying on complicated intracellular metabolic descriptions is to consider a network of some macroscopic reactions which involve only the initial substrates and the final products (without the intracellular species) but which are able to describe the process accurately with constant stoichiometric coefficients.

Whenever more than one macroscopic reaction are considered it should be emphasised that there is no longer an equivalence between the yield coefficients and the pseudo-stoichiometric coefficients.

We are thus interested in macroscopic reaction networks achieving a tradeoff between simplicity for process monitoring and control applications, and accuracy in order to match available experimental data.

In many applications, especially for cultures involving only a single or a few microbial species, it is clear that a detailed description of the metabolic and biosynthetic pathways may be available. In such a case, a macroscopic reaction network may be simply obtained by network reduction, i.e. by lumping or aggregating elementary metabolic reactions together (for instance by using the technique of "elementary flux modes" as described in [START_REF] Provost | Dynamic metabolic modelling under the balanced growth condition[END_REF]). It must however be noticed that the lumping of reactions corresponding to competing pathways may induce the appearance of pseudo-stoichiometric coefficients that are a-priori unspecified and have to be calibrated from the experimental data. Hence the issue of the pseudo-stoichiometric parameter estimation that we address in this paper may be a relevant issue even in the case where the metabolism is perfectly known.

The following simple illustration clarifies this point. Consider a culture of E. coli with anaerobic production of both lactate ( ) and acetate (¡ ) from glucose (¢ ). It is obvious that there are two metabolic pathways that can be summarised by the two following reactions (stoichiometry in moles) :

¢ § £ ¢ § ¤ ¡
These two reactions have perfectly known stoichiometric coefficients (2 and 3). Suppose now that these two reactions are aggregated into a single reaction. It is clear that any lumped reaction of the form:

¢ § £ ¦¥ ¤ ¨ § © ¥ ¡
is valid for any value ¥ © . In fact ¥ represents the fraction of glucose going through the first pathway and § © ¥ the fraction going through the second pathway. The parameter ¥ can be interpreted as a pseudo-stoichiometric coefficient which is a priori unspecified but can be calibrated from the data if measurements of the three species (¢ $ $ ¡ ) in the culture medium are available. For instance, from the data reported in [START_REF] Ingram | Expression of different levels of ethanologenic enzymes from zymomonas mobilis in recombinant strains of escherichia coli[END_REF] with strain TC4 growing in anaerobic conditions we have ¥ ! £ .

Depending on the assumptions made to lump the metabolic pathways, several macroscopic networks can be obtained. For example some pathways may or may not be neglected leading to different macroscopic reaction networks. In the previous simple example this would correspond e.g. to choosing ¥ " #© and thus neglecting the acetate production. In section 4.2 we present a more complicated example where 3 possible lumped networks are a priori considered for the aerobic growth of the fungus Pycnoporus cinnabarinus.

It is worth noting that proposing a reduced macroscopic reaction network can be very difficult for some complex cases. For instance anaerobic wastewater treatment bioreactors involve more than 140 coexisting microbial species [START_REF] Delbès | Bacterial and archaeal 16s rdna and 16s rrna dynamics during an acetate crisis in an anaerobic digestor ecosystem[END_REF] and many different complex substrates whose proportion varies with time.

Once the ¢¡ macroscopic reactions have been assumed involving ¤£ biological or chemical species (microorganisms, substrates, metabolites, enzymes... ), the dynamical behaviour of the stirred tank bioreactor can be described by a general mass-balance model of the following form (see e.g. [START_REF] Bastin | On-line estimation and adaptive control of bioreactors[END_REF]):

¥ §¦ £ ¥ ©¨ § ¦ £ $ ¥ § ¦ £ ¤ ¦ £ Q § ¦ £ $ ¥ $ (1)
In this model, the vector

¦ £ § ¦ £ ¦ $ ¦ £ $ $ ¦ £ is made-up of
Q § ¦ £ $ ¥ .
The term § ¦ £ $ ¥ represents the biological and biochemical conversions in the re- actor (per unit of time) according to the underlying macroscopic reaction network. The § £ "! # ¢¡ matrix ¨is a constant pseudo stoichiometric coefficient matrix. The

term § ¦ £ $ ¥ § ¦ § ¦ £ $ ¥ $ § ¦ £ $ ¥ $ $ $ § ¦ £ $ ¥ is

¥

) such as temperature, light, aeration rate, etc. Matrix ¨is associated with the assumed macroscopic reaction network and plays a key role in the mass balance modelling. Each line of the matrix corresponds to one (bio)chemical species involved in the process. Each column of the matrix corresponds to a (bio) chemical reaction between some of the species. A positive entry % ¤ % means that the & th species is a product of the ' th reaction while a negative entry % ¤ % means that it is a reactant or a substrate of the reaction. If % ¤ % , the & th species is not involved in the ' th reaction.

The objective of this paper is to propose a method to guide the user in the identification of the entries of pseudo-stoichiometric matrix ¨. It is worth noting that the determination of matrix ¨is a problem equivalent to that of determining the structure of the reaction network. The usual approach dedicated to the determination of reaction networks relies on the linearisation of the dynamics around a reference solution [START_REF] Eiswirth | Mechanistic classification of chemical oscillators and the role of species[END_REF][START_REF] Chevalier | Toward a systematic determination of complex reaction mechanisms[END_REF]. Here, in the spirit of [START_REF] Chen | Structural identifiability of the yield coefficients in bioprocess models when the reaction rates are unknown[END_REF][START_REF] Bernard | Structural identification of nonlinear mathematical models for bioprocesses[END_REF], we use linear algebraic properties to exploit the structure of the system (equation ( 1)) and our arguments are not based on any linearisation. As a consequence we are not limited to steady state data and we can exploit all the available measurements, even when associated to transient states.

We will show how to use a set of available data consisting of measurements of ¦ £ , Q, and ¦ £ ¤ at sampling instants, to determine the size of the matrix ¨(ı.e. the number of reactions that must be taken into account) and to address the problem of the identification and validation of its coefficients.

Note that it is quite rare for bioprocesses that all the involved variables are measured (sometimes it is even unclear which variables are involved). For this reason we will focus on the estimation of the submatrix of ¨associated with the available measurements ¦ ¢¡ .

We stress the fact that the methodology that we discuss is the first modelling stage. The second stage in the modelling, which is not discussed here, would consist in determining the reaction rates as functions of the state variables. This second problem is difficult and suffers as well from a lack of tools to assist the modeller. But this delicate step can be avoided for a large number of applications, where the knowledge of the mass balance (i.e matrix ¨) is sufficient to design controllers or observers [START_REF] Bastin | On-line estimation and adaptive control of bioreactors[END_REF].

The paper will address the three following problems: £ How many reactions (i.e. how many columns for matrix ¨) must be taken into account to reproduce the available data set ? £ Which reactions must be taken into account ? Which are the most plausible macroscopic reaction networks ? £ What are the values of the pseudo-stoichiometric coefficients ?

We will successively consider these three problems, without any a priori knowledge on the reaction rates § ¦ £ $ ¥ . The approaches will be illustrated with three examples of significant complexity: real data of the growth and biotransformation of the filamentous fungi Pycnoporus cinnabarinus, real data of an anaerobic digester involving a bacterial consortium degrading a mixture of organic substrates and a process of lipase production from olive oil by Candida rugosa.

2 Determination of the minimum number of reactions

Statement of the problem

In this section, we address the first problem, consisting in determining ¡ the min- imum number of reactions to explain the observed dynamics of the fermenter. We assume that we measure a subset

¦ ¢¡ of ¡ components of
¦ £ that are involved in the system (i.e. which present significant variations along time). Indeed the measurements of the other state components (denoted

¦ ¥¤

) may not be available, but we assume however that we measure more variables than the number of reactions:

¡ ¡ ¢¡ . If these components have a gaseous phase, we assume that the associated gaseous flow rates ¢ § ¦ ¡ $ ¦ ¤ $ ¥ are measured.

The equation associated with ¦ ¢¡ is thus:

¥ §¦ ¡ ¥ ©¨ § ¦ ¡ $ ¦ ¤ $ ¥ § ¦ ¡ ¤ ¦ ¡ ¢ § ¦ ¡ $ ¦ ¤ $ ¥ $ (2)
The matrices and ¢ are submatrices of ¨and Q, respectively, associated with ¦ ¡

. Note now that is a rectangular matrix with more rows than columns. In the expression of the mass balance model (2), only the term § ¦ ¡ $ ¦ ¤ $ ¥ needs to be mathematically expressed.

Theoretical determination

¦ ¡ § ¨ ¦ ¡ § ¨ "! § ¦ ¡ ¤ § $# ¦ ¡ § $# ¢ § ¦ ¡ § %# $ ¥ § $# ¥ # "! § ¦ £ § %# $ ¥ § $# ¥ # (3)
Let us denote:

& ¡ § ¨ ¦ ¡ § ¨ ¦ ¡ § ¨ ' ( "! § ¦ ¡ ¤ § $# ¦ ¡ § $# ¢ § ¦ ¡ § $# $ ¥ § $# ¥ # (4)
and

) ¡ § ¨ ( "! § ¦ £ § $# $ ¥ § $# ¥ # Equation (
3) can then be rewritten:

& ¡ § ¨ ) ¡ § ¨ (5)
The vector & ¡ § ¨ can be estimated along time from the available measurements.

The value of the integral in (4) can be computed e.g. with a trapeze approximation.

In order to improve the cleaning of the data (noise reduction and diminution of autocorrelation) it may be useful to apply any linear scalar filter (i.e. any combination of integration, differentiation and delay ) to Equation (2) leading to a linear relationship of the same type as [START_REF] Bastin | Nonlinear and adaptive control in biotechnology: a tutorial[END_REF]:

& § ¨ ) § ¨ (6)
where & § ¨ and ) § ¨ denote respectively the signal derived from & ¡ § ¨ and ) ¡ § ¨ after filtering. The moving average (3) that we have presented for sake of simplicity is of course only one example of such a filtering. Now the question of the dimension of matrix can be formulated as the determination of the dimension of the image of or in other words, of the dimension of the space where & § ¨ lives. Note that we assume to be a full rank matrix. Otherwise, it would mean that the same dynamical behaviour for & § ¨ could be obtained with a matrix of lower dimension.

Determining the dimension of the & § ¨ space is a classical problem in statistical analysis. It can be solved by a principal component analysis (see e.g. [START_REF] Johnson | Applied multivariate statistical analysis[END_REF]) that determines the dimension of the vector space spanned by the vectors % ¤ , rows of .

In order to reach this objective, we consider ¡ time instants ¨¦ $ $ £¢

(we choose ¡ ¡
). The way to select these time instants will be discussed in the sequel. We build then the

¡ ! ¤¡ matrix ¥ made of ¡ vectors & § ¨ at these time instants: ¥ § & § ¨¦ $ $ & § £¢
We will also consider the associated matrix of reaction rates, which is unknown:

¦ § ) § ¨¦ $ $ ) § £¢
We assume that matrix ¦ has full rank. It means that the reactions are independent (none of the reaction rates can be written as a linear combination of the others).

Property 1 For a matrix of rank ¡ , if

¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ ¥ ¦ ¥ . . . . . . ¥ $ . . . . . . ¦ ¨ § § § § § § § § § § § § § § § § § § § © with ¥ ¤ ! ¦ ¥ ¤ for & £ $ $ ¢¡ .
This property is a direct application of the singular decomposition theorem [START_REF] Horn | Matrix analysis[END_REF].

Since "! $# § § % &! $# § ¦ &! '# § % &! $# § ¢¡ , it provides the result.
Now from a theoretical point of view it is possible to determine the number of reactions in the macroscopic reaction network: it corresponds to the number of non zero singular values of ¥ ¥

. This theoretical approach must however be adapted in the real case where the available measurements are discrete data points perturbed by a noise.

Practical determination of the number of reactions

In the reality, the ideal case presented in the previous paragraph is perturbed for four main reasons:

£ The macroscopic reaction network that we are looking for results from lumping of chemical or biochemical reactions which can be very complex. The "true" matrix is probably a very large matrix. The reactions which are fast or of low magnitude can be considered as perturbations of a dominant reaction network. It is this central (perturbed) macroscopic reaction network that we want to estimate.

£ The measurements are corrupted by noise. This noise can be very important, especially for the measurement of biological quantities for which reliable sensors are rarely available.

£ The measurements are performed on a discrete basis. Moreover they are rarely all available exactly at the same time instant ¨¤ , and therefore they must be interpolated if we need a state estimate ¦ ¢¡ § ¨¤ at ¡ time instants ¨¤ in order to build vector ¥ .

£ In order to estimate & § ¨ in equation ( 4) we need to compute the approximate value of an integral. This may generate additional perturbations.

Data processing: interpolation and smoothing

The data collected on a biotechnological process often result from various sampling strategies carried out with various devices. As a consequence the data are seldom sampled simultaneously. In order to apply the proposed transformations vector ¥ has to be computed with values of the state variables at the same time instants ¨¤ .

A large number of tools are available in the literature to interpolate and smooth the data. We suggest here to use spline functions [START_REF] Boor | Applied mathematical sciences[END_REF] which will at the same time interpolate and smooth a signal. The trade-off between interpolation and smoothing can be chosen by the user.

In the sequel we assume therefore that the set of measurements is available at the (irregular) time instants # % (depending on the variable), and that after a smoothing and interpolation process all the variable estimates are available at the time instants ¨¤ .

We hypothesise that the estimates ¦ ¢¡ § ¨¤ are of reasonably good quality and in particular that the sampling frequency is in adequation with the time constants of the process.

Data normalisation

Conclusion for the determination of the minimal number of reactions

In the reality, the noises due to model approximations, measurement errors or interpolation perturb the analysis. Therefore in practice there are no zero eigenvalues for the matrix § ¥ ¥ . The question is then to determine the number of eigenvectors that must be taken into account in order to represent a reasonable approximation of the data & § ¨ . To solve this problem, let us remark that the eigenvalues ¥ ¤ of § correspond to the variance associated with the corresponding eigenvector (inertia axis) [START_REF] Johnson | Applied multivariate statistical analysis[END_REF].

The method will then consist in selecting the ¡ first principal axis which represent a total variance larger than a fixed threshold.

For instance, in the next example, we have fixed a threshold (depending on the information available on noise measurements) at 95% of the variance. This led to the selection of 6 axes, and therefore ¡ ¡ .

Remark: if % &! $# § § ¡ it means that &! '# § ¡
. In such a case we cannot estimate ¢¡ and measurements of additional variables are requested to apply the proposed method.

Finally, Figure 1 summarises the full procedure to compute the minimal number of reactions that are to be considered in order to reproduce an experimental data set.

Example 1: vanillin production by the filamentous fungi Pycnoporus cinnabarinus

We consider here the production of vanillin from vanillic acid by the filamentous fungus Pycnoporus cinnabarinus [START_REF] Falconnier | Vanillin as a product of ferulic acid biotransformation by the white-rot fungus Pycnoporus cinnabarinus I-937: identification of metabolic pathways[END_REF]. The species involved in this biotransformation process are the carbon sources (maltose and glucose), the nitrogen source (ammonium), oxygen, carbon dioxide, fungal biomass and phenolic compounds (vanillic acid, vanillin, vanillic alcohol and methoxyhydroquinone). This results from a complicated set of reactions [START_REF] Lesage-Meessen | A two-step bioconversion process for vanillin production from ferulic acid combining Aspergillus niger and Pycnoporus cinnabarinus[END_REF], most of them being ill known.

The process generally proceeds in two steps. In a first step (which generally lasts the first 3 days), the fungus uses the available substrates (nitrogen, maltose and glucose) to grow. The growth is aerobic, and therefore oxygen is consumed and CO produced.

In a second step, the biosynthesis is triggered with addition of cellobiose 2 hours before continuous addition of vanillic acid. Then the fungus transforms the vanillic acid either in methoxyhydroquinone, or in vanillin. In this last case, vanillin can also be degraded into vanillic alcohol.

For illustration purpose, Figure 2 presents the typical evolution of some of the key variables during the fermentation. The figure presents also the splines used to smooth and interpolate the data in order to build the vector & § ¨ made of the 10 measured species. The data set consists in 9 experiments which have been resampled to get 4 time instant ¨¤ per day. Finally, 619 data points & § ¨¤ were considered.

Figure 3 represents the cumulated variance associated with the number of reactions.

Four reactions are sufficient to explain 80% of the observed variance. Five reactions explain 95% of the total variance. This analysis motivated the structure of the model presented in [START_REF] Bernard | Mass balance modelling of vanillin production from vanillic acid by cultures of the fungus Pycnoporus cinnabarinus in bioreactors[END_REF].

Example 2: an anaerobic digestion process for wastewater treatment

In this second example we study a more complicated case where the biotechnological process is an anaerobic digester used for wastewater treatment [START_REF] Bailey | Biochemical engineering fundamentals[END_REF]. The anaerobic process involves a complex consortium of bacteria degrading a mixture of substrates. Figure 4 presents a schematic overview of the degradation pathway from the set of macromolecules (proteins, carbohydrates, lipids, etc...) up to methane, carbon dioxide and hydrogen. Obviously this general overview lumps together a large number of simpler reactions involving single substrates. It turns out that more than 140 bacterial species can be found in the considered anaerobic digester [START_REF] Delbès | Bacterial and archaeal 16s rdna and 16s rrna dynamics during an acetate crisis in an anaerobic digestor ecosystem[END_REF].

The considered process for experimental data generation is a pilot-scale up-flow anaerobic fixed bed reactor processing raw industrial distillery wastewater [START_REF] Steyer | Evaluation of a four year experience with a fully instrumented anaerobic digestion process[END_REF].

The experimental results [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF] were obtained for a period of 70 days over a wide range of experimental conditions (see Figure 7). The available daily measurements consist in organic carbon measured by the so-called soluble chemical oxygen demand (COD), the total volatile fatty acids (VFA) the volatile suspended solids (VSS), the total alkalinity and the dissolved inorganic carbon. The data set also contains a series of measurements of CH and CO flow rates.

The proposed method was applied to the available data set, including periods of biomass inhibition by an excess of VFA [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]. The obtained variance distribution is represented in Figure 5. It is worth noting that despite the apparent complexity of the process, a reaction network involving only 1 reaction (and thus one biomass) represents 83.2% of the variability. With 2 reactions, 97.8% of the variability are represented, which justified the choice of the very simple model presented in [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF].

3 Validation of a macroscopic reaction network

Statement of the problem

In the previous section we have shown how to determine the number of reactions which must be considered in order to explain the available data. Let us now assume that a plausible reaction network, with this number of reactions, is postulated with the aim of describing the process. In this section, we shall now show how such a candidate reaction network can be validated from the data.

One additional difficulty in comparing a reaction network, via its stoichiometric matrix , with a set of data is that some pseudo-stoichiometric coefficients may be a priori unknown. We shall propose a method which will allow at the same time to test the validity of the macroscopic reaction network and to identify the missing pseudo-stoichiometric coefficients.

Finding the left kernel of the pseudo-stoichiometric matrix

Let us consider a vector ¢¡ ¤£ ¦¥ §¡ ©¨:

Assume moreover that is normalised such that one of its components In other words, we have:

& ¤ § ¨ £% ¥ ¤ % & % § ¨ (7)
This means that the & % are linked by a linear relation. The immediate idea that one can have is to check whether relationship [START_REF] Ingram | Expression of different levels of ethanologenic enzymes from zymomonas mobilis in recombinant strains of escherichia coli[END_REF] We will therefore select the vectors of the left kernel that imply only independent & % in [START_REF] Ingram | Expression of different levels of ethanologenic enzymes from zymomonas mobilis in recombinant strains of escherichia coli[END_REF].

It is worth noting that the vector involves three kinds of components:

(1) entries which are structurally zero (2) entries that have an a priori known non-zero value (either 1 for the normalising component, see above, or a known value related to the stoichiometry of the reaction network).

(3) entries which are unknown because they depend on unknown coefficients of the pseudo-stoichiometric matrix. These entries have to be estimated from the data.

Remark that Equation [START_REF] Ingram | Expression of different levels of ethanologenic enzymes from zymomonas mobilis in recombinant strains of escherichia coli[END_REF] states a conservation between the variables & ¤ . This conservation is directly connected to the notion of reaction invariants [START_REF] Fjeld | On a pitfall in stability analysis of chemical reactions[END_REF][START_REF] Asbjornsen | Response modes of continuous stirred tank reactors[END_REF].

Definition 1 We say that a set & ¤ & ¤
¡ is associated with a left kernel vector if % for all the indices ' ¢ & ¦ & ¤ . We say that is associated with the % ! ¡ submatrix which is the submatrix made of the rows & ¦ & ¤ of . Finally we call the vector obtained by removing all the zeros entries in . The dimension of (namely

%

) is called the regression dimension associated with and denoted ¥ § , the number of unknown components of is denoted ¥ ¤ § .

We have therefore , and has no zero component. Then,

£ ¤ ¡ ¤ ¡ & ¤ ¡ § ¨ .
Note that, due to the normalisation of , we have

¥ ¤ § ¥ § © .
Definition 2 We say that a left kernel vector is sound if its associated

¥ § ! ¡ matrix does not contain itself any % ! ¡ submatrix ( % ¥¤ ¥ § ) whose rank is not full or -equivalently -if £ ¥¤ §¦ § ¡ ¤£ ¦¥ §¦¡ ¨ ©¨.
Remark: For a sound vector we have

¥ § ¡ © Indeed, if it has % ¡
£ non zero entries, then its associated submatrix is a % ! ¢¡ submatrix whose left kernel is at least of dimension 2.

Example 3: a process of lipase production from olive oil by Candida rugosa

Let us consider the example of the competitive growth on two substrates [START_REF] Serra | A model for lipase production by Candida rugosa[END_REF] which could represent the production of lipase from olive oil by Candida rugosa. Here the microorganism is supposed to grow on two substrates that are produced by the hydrolysis of the primary organic substrate made of several molecules (mainly triglycerides). We assume the following 3-step reaction network:

£ Hydrolysis: % ¦ ¦ § % £ Growth on : % % § % ! £ Growth on : % ¡ % ¡¢ § % ¤£ % ¡¥ !
where ¦ is the primary substrate (olive oil, made of various compounds), (glyc- erol) and (fatty acids) are the secondary substrates, is the enzyme (lipase), the biomass (Candida rugosa), the dissolved oxygen and ! the dissolved CO . We assume that all the biochemical species are measured, except ¦ .

The associated pseudo-stoichiometric matrix ¨and the state vector ¦ £ are therefore:

¨ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ % ¦ © % % % ¡ % ¦£
Now the following vector belongs to the kernel of matrix

: ¦ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ ¤ ¡ ! ¤ ¢ ¤ £ % © ¦ ¨ § § § § § § § § § § § § § § § © We have ¥ § ¦ ¤ and ¥ ¤ § ¦ £ .
It is associated with the rank-2 submatrix ¦ and to the vector ¦ :

¦ ¢ £ £ £ £ £ ¤ % ¦£ © © % % ¡¥ ¦ § § § § § © $ ¦ ¢ £ £ £ £ £ ¤ ¤ ¤ ! ¤ ¢ ¤ £ % © ¦ § § § § § © which is
& £ § ¨ % & § ¨ % ¡¥ % % ¦£ & § ¨ (8)
Now the kernel of matrix is spanned by the 2 other sound vectors:

¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ ¤ ¥ ! ¤ ¦ ¤ £ % © ¦ § § § § § § § § § § § § § § § © $ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ % © ¤ § ¤ © ¤ ¤ £ % % ¦ § § § § § § § § § § § § § § § © Obviously, we have ¥ § ¤ , ¥ ¤ § £ , ¥ § , ¥ ¤ § ¤ ,
¥ £ ¥ § £ % S % & ¤ ¢ § ¨ £ % S ¥ % & ¤ ¢ § ¨ (9)
admits a unique minimum, of zero value, obtained for ¥ % % (for any ' ¢

S).

It is worth noting that minimising § ¥ is exactly a linear regression problem.

Validation of the kernel of with the available data

Now the validation will consist in verifying that § ¥ (Equation 9) can be correctly minimised, or in other words, that the regression between

£ % S % & ¤ ¢ and the & ¤ ¢ (' ¢ S) is significant.
This analysis must be performed on all the sound kernel vectors associated with the ' ¡ principal axis. We then sort the singular values so that ¥ ¦ ¥ £ ¥ % § " © ¥ £ ¥ ¨ § " © © ¦ . Let us recall that, in principle, ¥ £ ¥ ¨ § " © and ¥ £ ¥ % § " © © ¦ .

¥ ¤ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ § ¨¦ § £¢ & % § ¨¦ & % § £¢ . . . & % ¤ ! #" %$ § ¨¦ & % & ' (" )$ § £¢ ¦ § § § § § § § § § § § § § § §
We consider the following criterion (reminiscent to the conditioning number) which assesses the balance of the variance along the axis:

B § ¤ ¥ ¦ § ¤ ¥ £ § ¤
With this criterion, we can now order the kernel vectors as follows:

£ We first sort the kernel vectors

Identifiability of the pseudo-stoichiometric coefficients

The question that we want to discuss in this section is to determine whether it is possible to determine the set of pseudo-stoichiometric coefficients % ¤ from the values of ¤ identified from the set of regressions given by equations [START_REF] Ingram | Expression of different levels of ethanologenic enzymes from zymomonas mobilis in recombinant strains of escherichia coli[END_REF]. This identifiability property when the reaction rates § ¦ £ $ ¥ are unknown is referred to as C-identifiability in [START_REF] Chen | Structural identifiability of the yield coefficients in bioprocess models when the reaction rates are unknown[END_REF].

The answer to the C-identifiability question can be found in [START_REF] Chen | Structural identifiability of the yield coefficients in bioprocess models when the reaction rates are unknown[END_REF]. A version of this Theorem is recalled here in the considered framework of full rank matrices : We propose here a broader sufficient condition for the C-identifiability:

Theorem 2 Let be an

¡ ! ¡ full rank matrix with ¡ ¡ . The unknown element % ¤ %
of is identifiable if there exists a % ! ¡ full rank submatrix £ , with % ¢¡ , which does not contain any unknown element on its ' ¢ ¨¤ column such that the § % © ! ¢¡ submatrix of :

¢ £ ¤ £ §¦ ¤ ¦ ¨ §© is not full rank, i.e. % &! $# § ¤ % © , where ¦ ¤ is the & ¢ ¨¤ line of . Proof: see Appendix A.
Remark: This criterion, although it is more complicated than the one proposed in [START_REF] Chen | Structural identifiability of the yield coefficients in bioprocess models when the reaction rates are unknown[END_REF], allows to check the C-identifiability for each element of separately and not only for the columns.

Let us consider the following matrix :

¢ £ £ £ £ £ ¤ % ¦ ¦ © © % ¦ ¦ ¨ § § § § § © (10)
Theorem 1 states that the first column of is not C-identifiable, since it is not possible to find a £ ! £ submatrix £ which do not contain any unknown element in its first column. Now if we apply Theorem 2, we can use the following submatrices:

£ ¨ § © © $ §¦ § % ¦ ©
Then is of rank 1, and verifies the condition

% © £ &! $# § , it follows that % ¦ is C-identifiable.
It is now clear that % ¦ ¦ is not C-identifiable, otherwise the first column of would be C-identifiable.

Remark that the analysis of the kernel of matrix also provides a criterion to test the identifiability of the % ¤ %

. Even if this criterion is less convenient, it will give some hints on the practical identifiability, as we will see in the next Property.

Property 4 The pseudo-stoichiometric coefficient

% ¤ %

is C-identifiable if and only if it can be computed from a combination of sound kernel vectors.

In the previous example of equation [START_REF] Chevalier | Toward a systematic determination of complex reaction mechanisms[END_REF], the sound kernel basis of was § $ % ¦ $ © It follows that % ¦ is C-identifiable and that % ¦ ¦ is not C-identifiable.

Identification of the pseudo-stoichiometric coefficients and final validation

Now, once we know that the pseudo-stoichiometric coefficients are identifiable, we can estimate their value from experimental data using Property 4. For this, we will use the regression associated with the sound kernel basis of given by equation [START_REF] Ingram | Expression of different levels of ethanologenic enzymes from zymomonas mobilis in recombinant strains of escherichia coli[END_REF]. The statistical significance of the correlation will allow to test from the data whether the vectors ¤ are in the kernel of or not.

The final validation will consist in checking that the pseudo-stoichiometric coefficients are all positive. This test must be performed with regards to the uncertainty obtained from the linear regression [START_REF] Ingram | Expression of different levels of ethanologenic enzymes from zymomonas mobilis in recombinant strains of escherichia coli[END_REF]. Indeed, because of the uncertainty obtained on the estimates for the ¤ , the % ¤ may have a negative value, but with a confidence interval intersecting the positive domain.

The overall approach leading to the validation of an assumed macroscopic reaction network is summarised in Figure 6.

Improving the method

A fermentation is often composed of several phases. In each phase, some reactions are not triggered. Therefore it is generally possible to find time intervals

¤ $ ¤ © ¦ ¢¡
for which % for some ' . In the same way, the concentration of some compo- nents may remain constant during certain periods of time. This is for example the case in a reaction where the primary (associated with growth) and the secondary metabolisms are successively activated. During the first stage only growth takes place: no biotransformation appears since no precursor is added. During the secondary metabolism phase, the growth is inhibited and the microorganism concentrates on the bioproduction of a metabolite.

During these periods of time

¤ $ ¤

© ¦ ¢¡ , the system is then characterised by an index ' ¤£ such that % . System (1) is then equivalent to the following system:

¥ §¦ ¡ ¥ ©¨ ¥ ¥ § ¦ ¡ $ ¥ § ¦ ¡ ¤ ¦ ¡ ¢ § ¦ ¡ $ ¥ $ (11)
where the matrix ¥ is extracted from by removing the columns of corresponding to the index ' ¦£ .

Finally on these time intervals, the study of system (1) can be simplified by studying [START_REF] Chen | Structural identifiability of the yield coefficients in bioprocess models when the reaction rates are unknown[END_REF].

Application to the process of lipase production

Let us consider the example of the competitive growth on 2 substrates. Let us assume that substrates and can directly be obtained from another bioreactor where the enzyme has been purified and directly added to ¦ without the biomass. We will then consider such an experiment where the secondary substrates and are directly added. Therefore, for all these experiments we will have ¦ . The problem reduces thus to find the kernel of the submatrix ¥ obtained after removing the first column of :

¥ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ % % ¡ % ¦£ © © % % ¡¢ % % ¡¥ ¦ ¨ § § § § § § § § § § § § § § § ©
The kernel of ¥ is spanned by the following sound vectors:

¥ ¦ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ ¤ £ ¤ © ¦ ¨ § § § § § § § § § § § § § § § © $ ¥ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ ¤ ¥ ! ¤ ¦ ¤ § % ¦¢ © ¦ ¨ § § § § § § § § § § § § § § § © $ ¥ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ ¤ £ ¤ § © % ¤£ ¦ ¨ § § § § § § § § § § § § § § § © $ ¥ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ ¤ ! ¤ ¢ ¤ § % ¡¥ © ¦ ¨ § § § § § § § § § § § § § § § © The regression dimension are ¥ § ¥ ¦ £ $ ¥ ¤ § ¥ ¦ © and ¥ § ¥ ¤ ¤ $ ¥ ¤ § ¥ ¤ £ for & © .
Note that ¥ is associated with regressions of lower dimension than implying less unknown coefficients % ¤

. It will therefore provide more reliable results (with the same amount of data), which will be easier to validate.

Application to the anaerobic digestion process

Based on the results presented in paragraph 2.5, a reaction network relying on 2 main steps was assumed to summarise the main mass transfer within the digester. In the first reaction (acidification) the organic matter ¦ is degraded into VFA ( ) and CO by a consortium of acidogenic bacteria ( ¦ ). Then the VFA are degraded into CH and CO by methanogenic bacteria ( ). The reaction network is thus the following:

£ Acidogenesis (with reaction rate ¦ ):

% ¦ ¦ # ¦ % % ¡ £¢ (12)
£ Methanogenesis (with reaction rate ):

% # % £¢ % ¦ ¥¤ (13)
Let us denote by ¦ the total dissolved inorganic compounds (mainly CO and bicarbonate). The associated pseudo-stoichiometric matrix ¨and the state vector

¦ £

are therefore:

¨ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ © © % ¦ % % % % % ¦ ¦ ¨ § § § § § § § § § § § § § § § © $ ¦ £ ¢ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ¤ ¦ ¦ ¦ ¦ ¨ § ¦ ¨ § § § § § § § § § § § § § § § © Since ¦ and
(which represent a broad variety of species) are not measured, we will focus on the state ¦ ¡ associated with the submatrix :

¢ £ £ £ £ £ £ £ £ ¤ % ¦ % % % % % ¡ ¦ ¨ § § § § § § § § © $ ¦ ¡ ¢ £ £ £ £ £ £ £ £ ¤ ¦ ¦ ¦ ¨ § ¦ ¨ § § § § § § § § © $ ¦ ¤ ¢ £ ¤ ¦ ¦ ¨ §©
Theorem 1 shows that matrix is clearly not C-identifiable from the available data. Thus we normalised it, so that the rate of ¦ consumption and the rate of CH production are now unitary, leading to matrix ¥ :

¥ ¢ £ £ £ £ £ £ £ £ ¤ © ¤ ¤ ¤ § ¤ ¤ ¦ ¤ ¤ ¡ ¤ © ¦ § § § § § § § § © 21 Theorem 1 proves then that ¥ is C-identifiable, choosing £ ¢ £ ¤ © © ¦ ¨ §© . Thus the fractions ¤ ¤ $ ¤ § ¤ $ ¤ ¦ ¤ and ¤ ¡ ¤
can be identified. To estimate the pseudo-stoichiometric coefficients % ¦ to % ¡ biomass measurements are required.

This mass balance modelling was used in [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]. The methane whose solubility is low was assumed to stay at low constant concentration, and the methane gaseous flow rate was assumed to be directly related to the methane bacterial production rate.

Finally, in [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF] the kinetic rate modelling for ¦ and was performed by using a Monod type kinetics for the growth of acidogenic bacteria and Haldane kinetics for the methanogenesis. Then the kinetic parameter were estimated, leading to the results presented in Figure 7 for more than 70 days of experiments with various influent concentrations and various dilution rates. For more details see [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF].

Comparison between several macroscopic reaction networks

Statement of the problem

Once the number of reactions ¡ to be taken into account has been identified, the next step consists in selecting the set of reactions which are supposed to represent the main mass transfer in the fermenter. In general, several hypotheses can be stated with respect to the available knowledge.

We assume therefore that a set of plausible macroscopic reaction networks with associated pseudo-stoichiometric matrices ¤ are postulated by the user. It may e.g. be the result of automatic determination procedures, like those presented in [START_REF] Bogaerts | Sytematic generation of identifiable macroscopic reaction schemes[END_REF][START_REF] Bernard | Identification of reaction schemes for bioprocesses: determination of an incompletely known yield matrix[END_REF]. The aim of this section is to determine how to select among these hypotheses those who provide a pseudo-stoichiometric matrix in agreement with the available data. Remark however that, in most cases, is a small number since there are only a few possible macroscopic reaction networks.

The method consists therefore in testing each matrix ¤ by using the methodology exposed in Section (3.7) and then to select the models which pass the validation tests.

The proposed methodology will be presented through a real life case study: the modelling of the growth of the filamentous fungus Pycnoporus cinnabarinus

¢ £ £ £ £ £ £ £ £ ¤ % ¦ © £ % © ¦ ¨ § § § § § § § § © (18) ¢ £ £ £ £ £ £ £ £ ¤ % ¦ % % % % © © ¦ § § § § § § § § © ( 19
)
Using the method presented in section (3.7) we give in Table 1 the sound kernel vectors and the corresponding regressions which are associated with these three pseudo-stoichiometric matrices.

PS Matrix

Sound kernel basis of From Table 2 we conclude immediately that network 3 is invalidated by the data. The coefficients associated with networks 1 and 2 have the correct signs, and therefore only these two networks are in agreement with the data and will be kept. Note that it is not possible to distinguish between network 1 and network 2. However the parameters % and % in network 1 are not identifiable, and thus network 2 would be preferred. Nevertheless, if network 1 should be kept for some reasons, the value (of at least one) of the (unidentifiable) parameters % and % should be selected, in such a way that % £ % belongs to the confidence interval from Table 2.

Regressions B¡ £¢ ¤ ¤ ¦ ¢ ¦ ¦ ¦¥ § ¨¨¨¨¨© ¦ ¢ ¦ ¥ § ¨¨¨¨¨© ! " ¦ ¥ # %$ ¦ © ¦ " " " ¥ &# %$ ¦ © " B¡ £¢ ¦ ¦ ¤ ¥ B¡ £¢ ¦ ¤ ¥ ¢ ¦ '¥ § ¨¨¨¨¨© ¦ ¢ ¥ § ¨¨¨¨¨© " ¦ ¥ # %$ © ¦ " " " ¥ &# %$ © " B¡ £¢ ¦ ¤ ¥ B¡ £¢ ¤ ¥ ¢ ¦ ¥ § ¨¨¨¨¨© ¤ ¤ © ¤ § ¤ ¨ ¢ ¥ § ¨¨¨¨¨© ¤ ¦ ! ¤ ¤ " ¥ # %$ © ¦ " # ($ © " " ¦ ¥ )$ " # ($ © "

Conclusion

Modelling of bioprocesses is known to be a difficult issue since there does not exist universal validated laws on which the model can rely as in other fields like mechanics (fundamental equations of mechanics), electronics (ohm law), etc.

We have presented here only the first stage of the macroscopic modelling, i.e. the mass balance modelling involving the reaction network through matrix ¨. The second stage would now consist in estimating the reaction rates § ¦ ¡ with respect to the biochemical species in the system. This step is far from being trivial since the kinetics can be very sensitive to many factors, leading to high parametric uncertainties in the mathematical expressions. The reader can refer to [START_REF] Bernard | Mass balance modelling of vanillin production from vanillic acid by cultures of the fungus Pycnoporus cinnabarinus in bioreactors[END_REF][START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF] for details on this second step of kinetic determination, with example of model simulation and validation both for the vanillin production process (example 1) and for the anaerobic treatment plant (example 2).

The key point in the mass balance approach is to use linear algebra to uncouple the linear part of the model driven by matrix ¨from the nonlinear and unknown part of the model related to the microbial kinetics ( § ¦ ¡ ). It is worth noting that some algorithms aiming at e.g. process monitoring or controlling can be based only on the mass balance part [START_REF] Bastin | On-line estimation and adaptive control of bioreactors[END_REF]. After algebraic operation the effect of the unknown § ¦ ¡ is eliminated, limiting the uncertainty associated with variability of the bio- logical processes. However the resulting algorithms turn out to be very sensitive to the pseudo-stoichiometric matrix. Validation of this matrix and improvement of its identification is therefore a key issue for biotechnological processes.

¤ 7 % ¥ §¦ ¢ ¦ ¤ ¢ " ¤ ¢ ¡ ¦¥ ¤ # ¢ ¤ 8 ¢ ¤ ¨¤ 7 % ¥ §¦ ¢ ¤ ¢ " ¤ ¢ ¡ ¦¥ ¤ ¥
showing that the vector £ ¢ whose components are

¢ ¦ ¤ ¢ # § " 8 § " ¨¢ ¤ ¢ for % ¦ 2 2 2 ¤ % 9
¥ § ¡ is associated with the matrix £ obtained from £ by removing the § ©¨r aw. Now we can prove Theorem 2.

Proof of Theorem 2:

If 0 A@ CB ED 5F ¡ HG ¤ , then ! ¡ #" %$ '& PI ¨¤ Q0 3R
, there exists a kernel vector

¢ ¥ § © ¢ £ ¢ ¦ ¤ such that ¢ G ¥ . We have therefore ¢ £ £ ¢ ¦ ¤ ¦ ¤ ¥ .
Since £ is a TS VU ¡ full rank matrix with XW YU ¡ , then ! `" %$ '& P" ä ¥ R

, and thus ¢ ¦ ¤ cannot be zero.

If ¢ is not sound i.e. 5 ! b" %$ '& cI ¨0 d2

. We must then consider the sound vector £ ¢ associated with the submatrix

§ © £ £ ¦ ¤
, where £ £ is extracted from £ according to Property 5.

The sound vector £ ¢ , verifies:

£ ¢ £ £ £ ¢ ¦ ¤ ¦ ¤ ¥
Let us remark that it is a matrix equality, and let us consider the ¢ ¥¤ column of this matrix equation:

£ ¢ £ £ £ ¤ ¢ ¦ ¤ ¤ % ¥ where £ £ ¤ is the ¢ ¨¤ column of £ £ .
As we saw in Section 3.5, the coefficients of the sound kernel vector £ ¢ can be identified from a linear regression. Therefore, ¤ % can be computed as follows: Comparison between simulation results and measurements in a fixed bed anaerobic digester for the methane and CO gaseous flow rates, pH, COD, VFA and total inorganic carbon (from [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]).

  the concentra- tions of the various species inside the liquid medium. The term ¦ £ ¤ represents the influent concentrations. The matrix is the dilution rate matrix representing the hydraulics mechanisms (inflows and outflows and possible retention) associated with the various species in the reactor. The exchange of matter in gaseous form between the surrounding and the reaction medium is represented by the gaseous flow rate

  has rank ¢¡ . Since it is a symmetric matrix, it can be written:

  To avoid conditioning problems and to give the same weighting to all the state variables, we normalise each component & ¤ of the vector & as follows: average value of the & ¤ § ¥¤ for % © ¡ , and ¢ § & ¤ their standard deviation.

¤ is 1 :©

 1 ¤Now let us consider the scalar quantity & § ¨ . From equation (5), it satisfies at any time

  is in adequation with the data. This can be done by performing a linear regression between & ¤ and the & % . Nevertheless, we have to keep in mind that the & % are a priori not independent, since they may be related by other relationships associated with other left kernel vectors of . In particular, we have seen that &! $# § ¥ ¡ , and thus a regression (7) cannot involve more than ¡ © independent terms & % .

  sound since the 3 possible £ ! ¤ submatrices are of full rank. Thus & $ & and & £ are associated with ¦ , and related by the following relation:

3. 4 Property 2 A¤¡Property 3

 423 Regressions associated with a set of & ¤ vector associated with a set & ¤ & is sound if and only if the & ¤ are not related by any other linear relation. Proof: Indeed, it is clear that it is not possible to have another relation between ¢¡ © different & ¤, otherwise this relation would be associated with a second kernel vector ¡ , meaning then that the kernel of is at least of dimension 2, and thus would not be sound. Let us consider a sound kernel vector of , ¤£ ¦¥ ¨ § © . Moreover, let us denote by S the set of indices ' such that % is known. Then the following cost criterion: §

.

  So we perform a principal component analysis for the matrix

.Definition 3

 3 creasing index of associated variance balance B § ¤ The basis made of the first ¡ ¢¡ independent vectors ¤ is called the sound kernel basis.

for 6 ¡

 6 . If the § ©c omponent of ¢ ¦ or ¢ contains a zero, then we have the result. Otherwise, for ¢

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig.1. Scheme of the procedure to compute the minimal number of reactions that are to be considered in order to reproduce an experimental data set.

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Schematic overview of the anaerobic digestion reaction network.

  of £ ¥¤ §¦ § © §

	Let us integrate equation (2) between 2 time instants	¨	and	¨(	denotes the
	considered time window):

Table 2 .

 2 The confidence intervals for the parameters have been estimated using a Student distribution with a 5% threshold and the significance of the regression has been tested.

	B¡ £¢	¦	¤	¥ 10 32 40 65 A@
	B¡ £¢ ¤	¥ 87 92

Table 2

 2 Estimation of intervals for parameter values and significance of the regressions (threshold 5%) associated with each of the reaction networks (RN).

Acknowledgement: The authors are grateful to Benoit Chachuat for his comments and corrections. This work has been carried out with the support provided by the European commission, Information Society Technologies programme, Key action I Systems & Services for the Citizen, contract TELEMAC number IST-2000-28256. It also presents research results of the Belgian Programme on Inter-University Poles of Attraction initiated by the Belgian State, Prime Minister's office for Science, Technology and Culture. The scientific responsibility rests with its authors.

A real case study

We focus here on experimental phases were only aerobic growth of the fungus Pycnoporus cinnabarinus takes place. From a preliminary analysis of the available measurements, it turns out that 2 reactions are necessary to explain the observed data (representing 97% of the variance).

The aerobic growth of the fungal biomass ( ) from a carbon source (glucose ¢ and maltose § ) and a nitrogen source (¡ ) can a priori be reasonably represented by the 3 following reactions networks:

The fungus is growing on maltose, glucose and nitrogen, and it can transform maltose into glucose in a first step:

The fungus is growing only on glucose and nitrogen, and it transforms maltose into glucose in a first step:

£ Network 3:

The fungus can grow either on glucose and nitrogen or on maltose and nitrogen. In this second case glucose is produced.

The pseudo-stoichiometric matrices associated with ( 14), ( 15) and ( 16) are then respectively:

Appendix A: proof of Theorem 2

We first demonstrate the following Property.

¤ ¡ ).

Proof: if ¢ is not sound, it means that ! ¡ #" %$ '& )( " ¨¤ 10 32

. Therefore there exists at least 2 different vectors ¢