The reactivity of the complexing anthracene-9-carboxylate ligand has been investigated with a niobium(IV) tetrachloride precursor (NbCl4·2THF) in isopropanol solvent. This resulted in the crystallization of a molecular assembly containing two distinct {Nb12O21} cores surrounded by multiple isopropanolate and anthracenoate ligands. The compound is formulated

illustrating the two different dodecameric oxo-clusters, for which the niobium(IV) precursor was oxidized in the niobium(V) state during the reactional process. The two distinct {Nb12O21} units mainly differs by the environment of the niobium centers, which exhibits unexpected fivefold coordination (square pyramid) for some of them, together with the classical six-fold coordination (octahedron) as usually found for niobium(V). In the crystallization process, the. IR spectroscopy was used to analyze the esterification reaction occurring between the anthracene acid an isopropanolate ligands responsible of the production of water used in the oxo-condensation of the niobium centers. 93 Nb Solid state NMR was tentatively used to assess the occurrence of the different niobium environments.

Introduction

Niobium is a d-block transition metal that is still studied, mainly under the oxide form or as a doping element in other metals, for catalytic purpose. [START_REF] Nowak | [END_REF][2][3][4][5][6] It exists with several oxidation states, principally Nb(III), Nb(IV) or Nb(V), the latter being the most currently used. The main reason is the stability of the oxidation state +5 in comparison with the two other ones (+3, +4) which require specific handling environment. Few examples of clusters with carboxylates ligands have been described and shows nuclearities ranging from the monomer up to the hexamer with the 4+ oxidation state. [7][8][9][10][11][12] Under its oxidation stable Nb(V) state, it has long been used in molecular chemistry for the synthesis of polynuclear oxo-clusters. Two main approaches have efficiently been performed and have led to two categories of well-known molecular families. The first one belongs to the polyoxometalates field, where niobium is surrounded by oxo ligands and feature the presence of "yl" terminal oxygen atoms preventing or strongly limiting the formation of extended networks. Nevertheless, numerous molecular polyoxometallic compounds have been described through the years, and are perfectly well represented and described in the recent review of Zhao et al.. [13] It is reported all the different architectures arising from the well-known [Nb6O19] 8-Lindqvist polyanion [14] up to the largest aggregate possessing 288 niobium centers. [15] Different strategies and classifications are used in the review to describe these molecular entities, generally based on isopolyoxoniobates or heteropolyoxoniobates, sometimes associated with heterometals allowing aggregation of complex molecular entities. The second family related to "hybrid poly-oxo clusters" consists of niobium centers surrounded by oxo-donor ligands from different organic ligands types (alkoxides, R-O -, or carboxylates, R-COO -) with a total absence of terminal "yl" oxygen atoms existing in classical polyoxometalates. In addition, for these polynuclear hybrid oxo clusters, bridging oxygen atoms can be found in the molecular structure. As far as we know, these hybrids niobium compounds are systematically molecular association and, extended networks have yet to be discovered. In molecular chemistry of hybrid niobium clusters, commercial precursors can be found and can be described as discrete Nb-centered mononuclear unit, precipitated by oxalic acid to give [A][NbO(C2O4)2(H2O)2]xH2O (A = NH4 + [16,17] or Cs + [18] ) or [A]3[NbO(C2O4)3]xH2O (A = NH4 + [19] or Rb + [20] ) salts, which can be further associated using pyridine-based molecules (phenanthroline, bipyridine, etc…) [21] or divalent transition metals complexes with [Zn(bpy)] 2+ , [22] [Co(terpy)] 2+ , [23] [(Fe-Zn)(bpy)] 2+ [24] (bpy = 2,2'-bipyridine, terpy = 2,2':6',2''-terpyridine) and rare-earth cations. [25] Other niobium oxalates have also been isolated in the presence of non-metallic divalent counter cations such as barium in [Ba2(H2O)5][NbO(C2O4)3][HC2O4]H2O [26] or with peroxide groups. [19] Several peroxo niobium centers can also be found in a series of complexes involving citrate, tartrate, ethylenediaminetetraacetate (edta) and propylenediaminetetraacetate (pdta) ligands. [27][28][29] To isolate molecules with more than one niobium centers, monomeric pentachloride NbCl5 or dimeric alkoxide Nb2(-OR)2(OR)8 niobium precursors have been used. NbCl5 reactivity was first studied using the benzoic acid ligand (HO2C-C6H5) and leads to the formation of two molecular dinuclear compounds, in which the niobium centers are bridged through the carboxylate arm of the monotopic benzoate molecules. In one case, the dimer [Nb2O(O2C-C6H5)2Cl6] [30] possesses one additional -oxo group bridging the two niobium centers, whereas it does not exist in the second molecule [Nb2(OEt)4(O2C-C6H5)2Cl4]. [31] Other monotopic benzoate derivatives (O2C-R where R = FC6H4, p-ClC6H4, p-IC6H4, p-MeC6H4) with modifications on the aromatic ring are also mentioned in the work of Brown et al. [31][32][33] In this work, addition of ethanol as a solvent in the initial solution led to the exchange of some chloride anions by ethoxy (OEt) groups, stabilizing the dimeric configuration without any central bridging -oxo atom. A related dinuclear complex was also identified with completely fluorosubstituted benzoate linker (O2C-C6F5) in a purely chloride salt [Nb2Cl8(O2C-C6F5)2]. [32] The second synthetical pathway is using metal alkoxides precursors (M(OR)n, where R= Me for methoxide, OEt for ethoxide, etc). This method has been more intensely studied, since these hybrid compounds are known for their utilization as precursors for the elaboration of ceramics [START_REF] Bradley | Alkoxo and Aryloxo Derivatives of Metals[END_REF][START_REF] Bradley | [END_REF][START_REF] Bradley | Metal Alkoxides[END_REF][START_REF] Chandler | [END_REF] and the substitution by a bidentate ligand can contribute to the modification of its properties. [38][39][40][41][42][43] Regarding the chemistry of molecular niobium alkoxides, many coordination complexes have been identified with various monocarboxylic acids and different niobium nuclearities. For instance, identification of dinuclear niobium complex [Nb2O(ONep)6(piv)2] (where ONep = OCH2CMe3 or neopentanoxy) in the presence of pivalic acid (Me3C-CO2H = Hpiv) was performed. This dimer is related to another one, [Nb2O(O2C-C6H5)2Cl6], [30] synthesized from the halide precursor and possessing two carboxylate arms from two pivalate ligands and one bridging -oxo group and chloride instead of neopentanoxy ligands. In the previous system, when using terbutylacetic acid (Me3C-CH2-CO2H = HtBAc) instead of pivalic acid, a higher condensed coordination complex [Nb4O4(ONep)8(tBAc)4] containing four niobium centers placed in a square configuration and connected to four -oxo groups has been isolated. This tetramer can be seen as the combination of the two previously described dinuclear compounds [Nb2O(Oalkoxide)3(carboxylate)2] linked together by two additional -oxo groups. Other research groups also come across this tetranuclear unit using other ligands. Hubert-Pfalzgraf et al. obtained the compound [Nb4O4(ac)4(O i Pr)8] [44] (with ac = acetate) and [Nb4O4(MAA)4(O i Pr)8] [45] (with MAA = methacrylate) starting from niobium isopropoxide. The use of niobium alkoxides is important for the formation of tetranuclear niobium coordination complexes. Indeed, the alkoxy groups, leaving as alcohols molecules when a carboxylate arm complex niobium centers, are involved in an esterification process slowly generating water molecules leading to the creation of -oxo-bridges between the niobium centers of the cluster. Other related compounds have been synthesized by using bis(hydroxymethyl)propionic acid (H2BHMP). In this system, the niobium dimers contain one -oxo bridge and are interacting with both the carboxylate and hydroxy functions of the ligand. [46] This interaction generates through the BHMP ligand, a tetranuclear species [(BHMP){Nb2O(OEt)5}2], without any additional -oxo groups linking the niobium atoms. Such an association of niobium carboxylate from the alkoxide route is also illustrated in a mixed bismuth-niobium moiety, [47] which has been synthesized using salicylic acid (2hydroxybenzoic acid) leading to a dinuclear [(Bi2Nb2O(OEt)2(sal)4(Hsal)4)] or tetranuclear [(Bi2Nb4O4(O i Pr)4(sal)4(Hsal)3)] (with sal = O2C-C6H4-2-O; Hsal = O2C-C6H4-2-OH) coordination complexes. Increasing clusters nuclearity in molecular chemistry is always a challenge. Lately, coordination modes behavior of niobium and tantalum was studied using catecholate ligands. [48] To date, the largest poly-oxo cluster in niobium carboxylate chemistry has been identified from a solvothermal reaction of niobium ethoxide with pivalic acid in acetonitrile. It led to the formation of an hexadecaniobate cluster [Nb16O28(OEt)12(piv)12]. [49] In our recent work on this topic, we have used several aromatic monocarboxylic and polycarboxylic acid ligands to complex niobium(IV) tetrachloride and niobium(V) ethoxide precursors, successfully increasing the nuclearity of the niobium centers in the final compounds up to the formation of an octameric {Nb8O12} moieties. [50][51][52] In this contribution, we report on the investigation of the reactivity of the anthracene-9carboxylic acid ligand with niobium(IV) tetrachloride NbCl4(THF)2 precursor as the niobium(IV) source. The reaction occurring in isopropanol solvent at 120°C leads to the formation of a molecular crystalline compound containing niobium(V) centers involved in two distinct {Nb12O21} cores. These two units are surrounded by isopropanolate and anthracenoate ligands

[Nb12((3-O)3(-O)18(C15H9O2)8(O i Pr)10]•[Nb12(3-O)2(-O)19(C15H9O2)8(O i Pr)10]• 7(HO i Pr)
where the two distinct dodecamers are contained in the same crystal structure, but with different coordination environments for some niobium centers, which can be six-fold or five-fold oxo-surrounded. The compound was successfully characterized by single-crystal and powder X-ray diffraction and MAS 93 Nb NMR spectroscopy and some insights in its formation behavior is given with the help of infrared spectroscopy.

Results and discussion

Structure Description Compound [Nb12((3-O)3(2-O)18(C15H9O2)8(O i Pr)10]•[Nb12(3-O)2(2-O)19(C15H9O2)8(O i Pr)10]•7(HO i Pr)

The crystal structure consists of two distinct neutral closely related molecular moieties, which contain twelve niobium(V) centers linked through oxo groups and bonded to anthracene-9carboxylate and isopropanolate species, resulting in a {Nb12O21} core. They are named by the letter A for the entity [Nb12((3-O)3(2-O)18(C15H9O2)8(O i Pr)10] and letter B for the entity [Nb12(3-O)2(2-O)19(C15H9O2)8(O i Pr)10], related to the two crystallographically independent motifs. For a better understanding of the complex arrangement of the niobium atoms, both dodecameric cores A and B can be viewed as the association of two hexameric sub-units {Nb6Ox}. For molecule A, a first sub-unit (named A1) is built up from six octahedrally-coordinated niobium centers (Figure 1-left). There are two sets of two octahedral polyhedra [NbO6] sharing an edge of one 3-oxo and one 2-oxo bridges, which inserts two others octahedral polyhedra [NbO6] sharing one 2-oxo bridge. The three sets of niobium pairs are connected to each other either via three 2-oxo groups and one 3-oxo group, in order to generate a rectangular-like sub-unit {Nb6O8} (A1). For the second hexanuclear sub-unit (A2 -Figure 1-right), the arrangement of the Nb-centered polyhedra is similar, except that one of the two peripheral sets of niobium pairs is replaced by two niobium atoms sharing one 2-oxo bridge (instead of an edge-sharing mode as found in sub-unit A1). Indeed, for this particular niobium pair, one of the metal centers (Nb11) is five-fold coordinated with a distorted square pyramidal geometry. It results in a second type of rectangular-like sub-unit {Nb6O7} (A2). It is observed slightly shorter Nb-O bond lengths for the five-fold coordinated Nb11 atoms, with Nb11-O2 distances in the range 1.847(5)-1.891(6) Å, one Nb11-Oc (Oc = carboxyl oxygen) distance of 2.153(6) Å and one Nb11-OPr (OPr = oxygen from isopropanolate) distance of 1.828(6) Å. In comparison with the other Nb-O bond lengths of the octahedrally niobium atoms (Table S1), it is found that the Nb-OPr distance of the five-coordinated Nb11 atom is in the same range ( 1.86-1.83 Å). But the Nb-Oc and Nb-O2/3 distances for Nb11, are in the lower values of the Nb-O ranges ( 2.14-2.24 Å and  1.80-2.41 Å, respectively) reported in the eleven six-fold coordinated niobium centers of the molecule A. For the sub-unit {Nb6O8} (A1), one notes one untypical long Nb6-O73 distance of 2.409(5) Å, which strongly distorts the Nb6-centered octahedra environment. Such a geometrical distortion was sometimes reported in different niobium oxides, such as CaNb2O5 [53] (Rb,Cs)Sr2Nb3O10 [54] or LiNbUO6 [55] for instance. At the opposite, in the sub-unit {Nb6O7} (A2), the corresponding oxygen atom (O16see Figure 1-right) is much more remote with a Nb11•••O16 distance of 3.036(7) Å, which excludes the octahedral geometry of the Nb11 atom. This subtle Nb-O distance variation is at the origin of the differentiation between the two hexanuclear sub-units A1 and A2, with the occurrence of one unique five-fold coordination for Nb11 atoms in sub-unit A2. The arrangement of niobium-centered polyhedra is almost similar in molecule B, with the construction of two rectangular-like hexameric sub-units B1 & B2 of {Nb6O7} type, in the same fashion as the previously defined sub-units A2. There exists one pair of edge-sharing octahedrally coordinated niobium atoms, linked via two oxo bridges (one 2 and one 3 types) to a second pair of corner-sharing octahedrally coordinated niobium atoms. The latter is then connected via two oxo bridges (two 2 types) to a third pair of corner-sharing centered-niobium polyhedra (Figure 3). As found for the sub-unit A2, each hexamer contains one niobium atom (Nb15 & Nb22) exhibiting a square pyramidal environment, with the similar Nb-O2 bond lengths in the range 1.816( 13 The noticeable point is the occurrence of the five-fold coordination for some niobium centers, which is very uncommon in the poly-oxo niobium(V) carboxylates or poly-oxo niobate clusters. [13] Indeed, within the crystal structure, the two molecules A & B only differ by the number of such niobium atoms with square pyramidal geometries, for which apical Nb-O bonding interactions vary from 2.409 Å (sub-unit A1), up to Nb•••O of 2.841( 7) Å (sub-unit B2), 2.951( 7) Å (sub-unit B1) and 3.036( 7) Å (sub-unit A2). This metric difference allows to define two asymmetric units with the identical chemical formula of {Nb21O21} core. Even if the niobium square pyramidal environment is rarely encountered, it also appears in some dense niobium oxides such as Na5NbO5, [56] NaKLaNbO5, [57] KLa2NbO6 [58] or Ba4Nb2O9. [59] Interestingly, the identification of the {Nb21O21} poly-oxo cluster is second type of dodecameric core for the molecular niobium carboxylate family. Indeed, a previous work reported the formation of a dodecanuclear niobium(V) moiety {Nb12O20} [60] with similar isopropoxy species, but stabilized by means of embonate ligands (from embonic acidalso called pamoic acid), which consist of two 3-hydroxy-2-naphtoate fragments linked through a methylene group. The resulting association of the embonate molecules with niobium(V) gave rise to the isolation of dodecamer, which can be view as the as the stacking of two hexanuclear sub-units composed of three centered niobium atoms in pentagonal bipyramids connected to three peripheral octahedrally coordinated niobium atoms (Figure S4). Apart the similar features of the hexanuclear sub-units, the main difference lies on the variety of coordination environments for niobium, for which there exist mixed 6-and 7-fold environments in {Nb12O20} core [60] and mixed 5-and 6-fold environments in the present {Nb12O21} core. The analysis of the crystal structure reveals that all the twenty-four crystallographically nonequivalent niobium atoms exhibit a pentavalent oxidation state, as the typical Nb-O bond distance as well as the bond valence calculations (Table S2). Starting from a tetravalent niobium source (Nb IV Cl4(THF)2), the isolation of such a crystalline compound of {Nb12O21} cluster reflects of an oxidation reaction occurring during the synthetical process, and which may explain the very low value of crystallization yield. This might be due to a partial in-situ conversion of niobium(IV) into niobium(V), which gives rise to the formation of {Nb12O21}. Considering the synthetic conditions used to for the {Nb12O21} cluster, the oxidation process can only be initiated by the small quantity of water formed through the esterification process.

It may also explain why the yield of the reaction is so low. In addition, electronic spectroscopy of the mother liquor after the crystallization process presents a broad transition around 460 nm, that could be attributed to niobium cation under the +IV oxidation state (Figure S5). [61] Infrared spectroscopy analysis The {Nb12O21} compound crystallization process was followed using infrared spectroscopy. Supernatant solutions were collected before crystallization, roughly 1 hour of mixing to homogenize the mixture, and after heating and crystallization of the {Nb12O21} compound (after approximately 5 days). On the IR spectrum (Figure 5) of supernatant solutions of the {Nb12O21} compound, we can observe several interesting areas. The 3000-2750 cm -1 range is related to the (C-H) stretching vibrations indicating the presence of the isopropyl groups with the typical asymmetric and symmetric band values at 2970 cm -1 and ~2882 cm -1 , respectively. The alkoxide ligands are also visible in the 1480-1370 cm -1 region (typical vibrations in the range 1470-1400 cm -1 and at 1373 cm -1 ) which is related to the (C-H) bending vibrations. The last domain range is between 1200-1000 cm -1 where we observed the (C-O) vibrations of linked isopropoxide ligand at characteristic peaks located at 1096 and 1057 cm -1 . The last broad vibrations in the molecule fingerprint at 529 cm -1 are related to the (Nb-O) stretching mode. All these vibrations are visible for the supernatant solutions as well as the crystallized compound and the niobium ethoxide precursor. The other interesting point is related to the formation of the {Nb12O21} compound. For this purpose, we can focus on the free ligand and compound spectra and see that the v(C=O) vibration of the free carboxylic acid function of the anthracene ligand located at 1678 cm -1 disappears to the advantage of those of bonded carboxylate arm, asym(COO) and sym(COO) centered at 1624 and 1538 cm -1 in the spectrum of the crystalline compound. Furthermore, the (Carom-H) vibrations of the anthracene ligands are also visible on both spectra above 3000 cm - 1 . The formation of an ester due to the esterification mentioned earlier could be confirmed by the presence of vibration in the range 1710 -1695 cm -1 . They appear to be extremely weak in this case and are not always observed in this kind of mixtures. [51] However, there is no evidence of the presence of the {Nb12} compound in the supernatant solutions. The (C=O) and (COO) vibrations are summarized in Table 2.

MAS 93 Nb NMR spectrum

Compound {Nb12O21} can be considered as a high nuclearity niobium cluster which involves two different {Nb12} cores, not as symmetrical as other ones reported. [50] Nevertheless, the fact that it contains two different coordination sites makes {Nb12O21} an interesting compound for recording an NMR spectrum. We have first tried to measure a spectrum using liquid NMR without success due to the insolubility of the compound despite the use of several deuterated solvents (DMF, DMSO, CDCl3, EtOD, MeOD). Consequently, we turn towards solid-state NMR. A MAS 93 Nb NMR spectrum was recorded at high magnetic field (18.8 T) and under MAS (magic-angle spinning, Figure 6). The ultra-wideline observed suggests an overlapping of different peaks that could not be resolved with the MAS. Such overlapping precludes an accurate fitting of the quadrupolar parameters but the maximum peak position is observed at -680 ppm, which is in line with the range of the chemical shift of 6-fold environment of the niobium(V). [62] Another signal is detected at -360 ppm deshielded compared to the main peak. The 93 Nb chemical shifts are known to be sensitive to the coordination number of the niobium, [62] and this signal may be attributed to the five-coordinated niobium. This conclusion is also supported by the other characterization methods of this article.

Conclusion

In summary, the reactivity of tetravalent niobium (under the pristine form NbCl4·2THF) with the anthracene-9-carboxylate ligand leads to the formation and crystallization of an unprecedented compound containing two dodecanuclear {Nb12O21} niobium(V) cores stabilized by anthracene-9-carboxylate and isopropanolates ligands. This polynuclear entity constitutes a second illustration with twelve niobium-oxo centers complexed by carboxylate ligands, involved in the {Nb12O21} together with the one stabilized by embonate ligands constituted of a {Nb12O20} moieties. [60] Single crystal X-ray diffraction reveals the presence of rare 5-fold coordinated niobium(V) centers with a distorted square pyramidal geometry and more classical 6-fold coordinated niobium(V) with octahedral geometries. MAS 93 Nb NMR spectrum suggest the presence of the 6-fold coordinated niobium(V) but despite their presence, the 5-fold coordinated niobium(V) centers were not clearly observed, possibly due to the overlapping by the large pic attributed to the octahedral niobium cations. Further work will focus in the comprehension about the formation of such large hybrid niobium cluster using monotopic carboxylic ligands, since the largest reported polynuclear moiety is currently the {Nb16O28} [49] . This opens new prospects to discover molecular systems with high nuclearity when compared with the wide variety and range of polyoxoniobates occurring with niobyl species. [13] Experimental Section

Reagents. The following reactants were used: niobium(IV)chloride (Nb(OCH2CH3)5, 99.95%, Sigma-Aldrich), anthracene-9-carboxylic acid (C15H10O2, 99% Sigma-Aldrich), anhydrous isopropanol (C3H8O, HO i Pr, 99.5%, Sigma-Aldrich). The chemical reactants were commercially available, were used without any further purification and manipulated in a glove box under argon to prevent any hydrolysis reaction of the niobium(IV) into niobium(IV or V)oxide. A mixture of 12 mg (0.05 mmol) NbCl4(THF)2, 50 mg (0.23 mmol) anthracene-9-carboxylic acid and isopropanol (1 mL, 13.1 mmol) was placed in a 2 mL glass tube, sealed with a phenolic cap, and placed in an oven at 120 °C. After 5 days orange parallelepiped blocks appeared. Compound {Nb12O21} was analyzed by optical microscope showing large parallelepiped-shape crystals up to 100 µm size, which are suitable for further SCXRD analyses (Figure S1). The resulting orange crystals filtered off, washed with isopropanol and dried at room temperature. Crystallization yield was 4.2 %Nb. Crystals of compound {Nb12O21} were found to be stable under air for several weeks and manipulated under ambient air, without any specific precaution.

Syntheses

Single-crystal X-ray diffraction X rays diffraction intensities of crystals of compound {Nb12O21} was collected from a synchrotron radiation facility (PROXIMA-2 A beamline, SOLEIL, France) by using a wavelength of  = 0.7293 Å at 100 K (under liquid nitrogen gas flow). The structural analysis shows high values for many thermal parameters related to the organic ligands. Due to this observation, most of the anthracenoate ligands were modelled using a rigid body approach. The aromatic part of the ligands was constructed and added to the model and refined by constraining the carbon atoms positions. Some restraints (DFIX, DANG and SIMU) were used on some isopropanolate ligands to stabilize the C-C and C-O distances that were heavily moving due to the high values for the thermal parameters of the carbon and oxygen atoms. The metal-oxo part of the compound was refined classically by successful difference Fourier syntheses, and refined by full-matrix least-squares on all F 2 data using SHELX program suites, implemented in the OLEX2 graphical tool. [63]

Figure 1 :

 1 Figure 1: Ball and stick (top), and polyhedral (bottom) representation of the two {Nb6}-containing subunits A1 (left) and A2 (right) in the molecular {Nb12O21} entity A, [Nb12((3-O)3(2-O)18(C15H9O2)8(O i Pr)10]. Purple circle: Nb centers in octahedral coordination [NbO6]; dark green circle: Nb11 center in square pyramidal coordination [NbO5]; green circles: 3-oxo; cyan circles: 2-oxo; orange circles: carbon atoms from bidentate carboxylate groups; red circles (bridging): oxygen atoms from carboxylate groups; red circles (terminal): oxygen atoms from isopropanolate species.

Figure 2 :

 2 Figure 2: (top) Schematic view of the assembly mode of the two {Nb6}-containing subunits A1 and A2 in order to construct the {Nb12O21} oxo cluster of the molecular entity A. The rectangular six membered planes of each {Nb6}-containing subunits A1 and A2 are arranged along a tilting angle of around 104°. (bottom) Representation of the {Nb12O21} oxo cluster (entity A) with surrounding 9-anthracenecarboxylate and isopropanolate species. Purple polyhedra: Nb center in octahedral coordination; dark green polyhedra: Nb center in square pyramidal coordination; green circles: 3-oxo; cyan circles: 2-oxo; orange circles: carbon atoms from bidentate carboxylate groups; red circles (bridging): oxygen atoms from carboxylate groups; red circles (terminal): oxygen atoms from isopropanolate species.

  )-1.846(7) Å for Nb15 and 1.842(6)-1.992(13) Å for Nb22, Nb-Oc bond lengths of 2.155(11) Å (Nb15) and 2.162(10)Å (Nb22), and Nb-OPr bond lengths of 1.811(10) Å (Nb15) and 1.786(10)Å (Nb22). These two niobium centers Nb15 & Nb22 are considered as five-fold coordination since the nearest sixth oxygen atom which would complete the octahedral geometry is very far from the usual Nb-O bond lengths, with values of 2.951(7) Å (for Nb15•••O71) and 2.841(7) Å (for Nb22•••O58) (see Figure 3 for the location of the O71 & O58 atoms). For the other ten octahedrally coordinated niobium atoms, the Nb-O2/3 distances are in the range of 1.792(5)-2.122(5) Å, the Nb-Oc distances are in the range of 2.083(6)-2.234(8) Å, and the Nb-OPr distances are in the range of 1.833(6)-1.862(6) Å. So, the slight distortion of the Nb6-O73 bond length (resulting in an octahedral geometry) observed in the hexameric sub-unit A1 does not occur in the sub-units B1 or B2, which both contains one five-fold coordinated niobium atom.

Figure 3 :

 3 Figure 3: Ball and stick (top), and polyhedral (bottom) representation of the two {Nb6}-containing subunits B1 (left) and B2 (right) in the molecular {Nb12O21} entity B, [Nb12((3-O)2(2-O)19(C15H9O2)8(O i Pr)10]. Purple circles: Nb centers in octahedral coordination [NbO6]; dark green circles: Nb15 & Nb22 centers in square pyramidal coordination [NbO5]; green circles: 3-oxo; cyan circles: 2-oxo; orange circles: carbon atoms from bidentate carboxylate groups; red circles (bridging): oxygen atoms from carboxylate groups; red circles (terminal): oxygen atoms from isopropanolate species.

Figure 4 :

 4 Figure 4:(top) Schematic view of the assembly mode of the two {Nb6}-containing subunits B1 and B2 in order to construct the {Nb12O21} oxo cluster of the molecular entity B. The rectangular six membered planes of each {Nb6}-containing subunits B1 and B2 are arranged along a tilting angle of around 110°. (bottom) Representation of the {Nb12O21} oxo cluster (entity B) with surrounding 9-anthracenecarboxylate and isopropanolate species. Purple polyhedra: Nb center in octahedral coordination; dark green polyhedra: Nb centers in square pyramidal coordination; green circles: 3-oxo; cyan circles: 2-oxo; orange circles: carbon atoms from bidentate carboxylate groups; red circles (bridging): oxygen atoms from carboxylate groups; red circles (terminal): oxygen atoms from isopropanolate species.

Fig. 5 .

 5 Fig.5. Infrared spectroscopy analysis for compound {Nb12O21}. Infrared spectra of Nb(OEt)5 (blue line), anthracene-9-dicarboxylic acid (red line), supernatant after 1h mixing at room temperature (green line), supernatant after crystallization (purple line) and compound {Nb12O21} (black line). Top: full spectrum in the 4000-400 cm -1 range; bottom left: detailed region in the 950-1900 cm -1 range; bottom right: detailed region in the 1800-1600 cm -1 range.

Fig. 6 .

 6 Fig. 6. MAS NMR 93 Nb spectrum of compound {Nb12O21} recorded at 18.8 T and 0 = 62.5 kHz.

  Compound [[Nb12((3-O)3(2-O)18(C15H9O2)8(O i Pr)10]•[Nb12(3-O)2(2-O)19(C15H9O2)8-(O i Pr)10]•7(HO i Pr) (called {Nb12O21}):

Table 2 .

 2 Attribution of the ν(C=O) vibrations (from carboxylic acid function) and ν(COO) vibrations (from carboxylate function) for compound {Nb12O21} (in cm -1 ).

	(C=O)acid	asym(COO)	sym(COO)

Acknowledgements

The authors would like to thank Mrs. Nora Djelal, Laurence Burylo and Philippe Devaux for their assistances with synthesis, XRD powder patterns measurements (UCCS). Dr Clément Falaise (Université of Versailles Saint Quentin-en-Yvelines) is thanked for his help in the single-crystal XRD data collection (compound 1) at synchrotron SOLEIL (France). The "Fonds Européen de Développement Régional (FEDER)", "CNRS", "Région Hauts de France" and "Ministère de l'Education Nationale de l'Enseignement Supérieur et de la Recherche" are acknowledged for the funding of X-ray diffractometers from the Chevreul Institute platform. Keywords coordination complex, niobium(V), MAS 93 Nb NMR, poly-oxo clusters, single-crystal X-ray diffraction.

Data Availability Statement

Supporting Information for this article is available on the WWW under https://doi.org/10.1002/chem.2022XXX. The data associated with the findings of this study can be found in the online version (optical microscope photographs, powder XRD pattern, IR spectra, liquid 1 H NMR and solid state 93 Nb spectra, summary of the representation of polynuclear niobium-centered cores). Accession Codes Crystallographic data for the structural analysis has been deposited with the Cambridge Crystallographic Data Centre, CCDC No. 2280853 for 1. Copies of the data can be obtained free of charge on application to CCDC, 12 Union Road, Cambridge CB2 1EZ (fax: +44-1223-336-033; e-mail: data_request@ccdc.cam.ac.uk).

Data reduction was accomplished using SAINT V8.34a. [START_REF]Brucker Analytical X-ray Systems[END_REF] The substantial redundancy in data allowed a semi-empirical absorption correction (SADABS V2014/5) to be applied, on the basis of multiple measurements of equivalent reflections. [START_REF] Sheldrick | Brucker-Siemens Area detector Absorption and Other Correction[END_REF] The crystal data are given in Table 1. Supporting information is available in CIF format. CCDC numbers: 2280853 for 1, contain the supplementary crystallographic data for this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif. Powder X-ray diffraction X-ray powder diffraction was performed on Bruker D8 Advance diffractometer (LynxEye detector) in a Bragg-Brentano - mode using Cu-K radiation. Each powder pattern was recorded by using a classical glass specimen holder, within an angular range of 5-50° in 2, with steps of 0.02° and counting time of 0.5 s per step.

Infrared spectroscopy

Infrared spectrum was measured on Perkin Elmer Spectrum Two TM spectrometer between 4000 and 400 cm -1 , equipped with a diamond Attenuated Total Reflectance (ATR) accessory. No ATR correction was applied on the spectrum.

MAS 93 Nb NMR spectroscopy 93 Nb NMR spectra were recorded at 18.8 T (800 MHz for 1 H) using a narrow-bore Bruker BioSpsin spectrometer equipped with an AVANCE-III console and a 1.3 mm HX probe with a spinning speed of 62.5 kHz. A rotor synchronized solid echo (90--90 with  corresponding to one rotor period) was used with a pulse length of 4.5 s, a radiofrequency field amplitude of 55 kHz and a recycle delay of 0.25 s. The spectrum is the sum of 68752 scans. 93 Nb chemical shifts were externally referenced to NbCl5 (1 mol/L) at 0 ppm.

Conflict of Interests

The authors declare no conflict of interest.


Reina Dannaoui

Xiao-Ke Yang

Wei-Hua Huang

Irina Svir

Christian Amatore

Importance of diffusional constraints for the quantitative evaluation of calibration curves of enzymatic micro-and nanoelectrochemical sensors 

Keywords: Enzymatic nanoelectrode, calibration curve, Lineweaver-Burk plots, kinetic modelling, Michaelis-Menten related mechanisms

Importance of diffusional constraints for the quantitative evaluation of calibration curves of enzymatic micro-and nanoelectrochemical sensors

Reina Dannaoui, Xiao-Ke Yang, Wei-Hua Huang, Irina Svir, Christian Amatore, Alexander Oleinick

To cite this version:

Reina Dannaoui, Xiao-Ke Yang, Wei-Hua Huang, Irina Svir, Christian Amatore, et al.. Importance of diffusional constraints for the quantitative evaluation of calibration curves of enzymatic micro-and nanoelectrochemical sensors. Electrochimica Acta, 2023, 473, pp.143425. 10.26434/chemrxiv-2023-3n4rp. hal-04272699

Introduction

Enzyme modified electrodes allow to extend the limits of application of conventional electrochemical analytical sensors by making possible the detection of many nonelectrochemically active compounds and/or by increasing the selectivity and sensitivity of the electrochemical detection [START_REF] Yang | Enzyme-based ultrasensitive electrochemical biosensors[END_REF][START_REF] Pinyou | Enzyme-modified electrodes for biosensors and biofuel cells[END_REF][START_REF] Kucherenko | Advances in nanomaterial application in enzyme-based electrochemical biosensors: a review[END_REF][START_REF] Bartlett | Modelling Biosensor Responses[END_REF]. This permits the selective detection and monitoring of biologically active molecules in particular under complex in vivo conditions possibly involving several electroactive interferents.

The performance and limitations of these sensors are often pre-evaluated or post-evaluated by recording calibration curves that relate the current intensity of the sensor to the concentration of the target analyte in a series of bulk test solutions. This is perfectly fine whenever these sensors are intended to be used to study the compositions of other bulk solutions whose concentrations may vary slowly provided they remain homogeneous around the sensor. Indeed, when used under such conditions, the recorded calibration curves feature intrinsic characteristics of the sensors, viz. of the enzymatic conversion and the transport supply of the substrate to the sensor surface. Nowadays, highly miniaturized sensors are more often designed to detect and quantify transient releases of analytes under in vivo or ex vivo conditions [START_REF] Mcmahon | The efficiency of immobilised glutamate oxidase decreases with surface loading: an electrostatic effect, and reversal by a polycation significantly enhances biosensor sensitivity[END_REF][START_REF] Wang | Ultra-fast glutamate biosensor recordings in brain slices reveal complex single exocytosis transients[END_REF][START_REF] Yang | Quantitative Nano-Amperometric Measurement of Intravesicular Glutamate Content and of its Sub-Quantal Release by Living Neurons[END_REF]. Under these conditions the problem will be even more severe due to complex and intermittent diffusion patterns encountered in such systems.

In this work, we first wish to highlight the practical importance of these questions by considering a canonical Michaelis-Menten mechanism (Figure 1) to illustrate the drastic role of diffusional constraints on the responses of enzymatic sensors under stationary or quasistationary conditions and their analyses. Our analysis aims to analytically evaluate the consequences of these constraints. Note that similar attempts have already been published, although with different focus and format (see e.g. refs. [START_REF] Bartlett | Modelling Biosensor Responses[END_REF][START_REF] Kulys | Biosensor response at mixed enzyme kinetics and external diffusion limitation in case of substrate inhibition[END_REF][START_REF] Lyons | Mediated Electron Transfer at Redox Active Monolayers. Part 4: Kinetics of Redox Enzymes Coupled with Electron Mediators[END_REF]). The central question of the problem at hand is introduced in section 2 "Preliminary considerations…" and is treated quantitatively in the case of a simple Michaelis-Menten mechanism. However, stricto sensu, this is rarely applicable to describe the performance of electrochemical sensors which generally involve a cosubstrate. For this reason, our approach is then extended to treat ping pong bi-substrate enzymatic mechanisms with or without inhibition.

Preliminary considerations stressing the role of diffusional constraints.

Let us consider here that the system at hand involves a simple canonical Michaelis-Menten (MM) enzymatic mechanism (Scheme 1) converting a non-electroactive substrate , present at concentration in the analyzed bulk solution, into an electroactive product detected at a micro-or nanoelectrode surface on which a thin enzyme coating is deposited (Figure 1) [START_REF] Yang | Quantitative Nano-Amperometric Measurement of Intravesicular Glutamate Content and of its Sub-Quantal Release by Living Neurons[END_REF]. Note that by "thin" we mean that the thickness of the enzyme layer is extremely small relatively to the dimension of the electrode, therefore also vs. that of the diffusion layers which spontaneously establish in a few seconds at most in the enzyme-free solution surrounding the electrode [START_REF] Amatore | Electrochemistry at ultramicroelectrodes[END_REF][START_REF] Amatore | Electrochemical kinetics at microelectrodes. Part I. Quasi-reversible electron transfer at cylinders[END_REF][START_REF] Amatore | Equivalence Between Electrodes of Different Shapes: Between Myth and Reality[END_REF]. Therefore, one may rightly consider that most of the current flow stems from the electrolysis of the product generated in the enzyme layer without significant losses due to diffusion towards the bulk solution [START_REF] Kulys | Biosensor response at mixed enzyme kinetics and external diffusion limitation in case of substrate inhibition[END_REF].

On the other hand, owing to the flux conservation of the substrate one has at any time:

(

where is the diffusion flux of the substrate from the solution to the thin enzymatic layer where it is consumed with an overall rate of enzymatic conversion per unit of electrode surface, denoted . Denoting the immobilized enzyme concentration, the usual "Michaelis constant" and the mean substrate concentration within the enzyme layer (Scheme 1), one has:

(

for a simple canonical Michaelis-Menten enzymatic mechanism without product inhibition [13,[START_REF] Lescovac | Enzymatic kinetics[END_REF]. Note that represents the maximum conversion rate per unit of surface area, noted , achievable by the immobilized enzyme when .

[Fig. 1] Scheme 1. Canonical Michaelis-Menten (MM) mechanism with product electrochemical detection.

It is important to note that Eq. ( 2) involves the mean concentration of the species in the enzymatic layer and not that, , in the bulk solution [START_REF] Bartlett | Modelling Biosensor Responses[END_REF]. Actually, depends on and the current flow, indeed:

(

where is a function that depends on the diffusion profile of inside the diffusion layer, A is the global surface area of the enzyme layer, the number of electrons transferred per product molecule, is the Faraday constant. It follows that:

where is the inverse function of . Therefore, whenever one wants to relate the current to , as is performed in the usual practice, the following equation must be used:

(

instead of the oftenly used improper formulation [START_REF] Mcmahon | The efficiency of immobilised glutamate oxidase decreases with surface loading: an electrostatic effect, and reversal by a polycation significantly enhances biosensor sensitivity[END_REF][START_REF] Wang | Ultra-fast glutamate biosensor recordings in brain slices reveal complex single exocytosis transients[END_REF][START_REF] Yang | Quantitative Nano-Amperometric Measurement of Intravesicular Glutamate Content and of its Sub-Quantal Release by Living Neurons[END_REF][START_REF] Gonçalves | Penicillinasebased amperometric biosensor for penicillin G[END_REF][START_REF] Silva | An efficient poly(pyrroleviologen)-nitrite reductase biosensor for the mediated detection of nitrite[END_REF][START_REF] Enache | Electrochemical evaluation of Glutathione Stransferase kinetic parameters[END_REF][START_REF] Yuan | Creating a Low-Potential Redox Polymer for Efficient Electroenzymatic CO 2 Reduction[END_REF]: [START_REF] Wang | Ultra-fast glutamate biosensor recordings in brain slices reveal complex single exocytosis transients[END_REF] adopted within the context of analyses through Lineweaver-Burk plots, Eq.( 7):

However, Eq.( 5) establishes that the real relationship between current and is: [START_REF] Kulys | Biosensor response at mixed enzyme kinetics and external diffusion limitation in case of substrate inhibition[END_REF] which clearly confirms that contrary to what is considered in the usual practice, cannot vary linearly with even for an electrochemical enzymatic sensor relying on a canonical Michaelis-Menten mechanism, i.e., onto the simplest rate law given in Eqn. [START_REF] Pinyou | Enzyme-modified electrodes for biosensors and biofuel cells[END_REF].

Equations [START_REF] Yang | Quantitative Nano-Amperometric Measurement of Intravesicular Glutamate Content and of its Sub-Quantal Release by Living Neurons[END_REF][START_REF] Kulys | Biosensor response at mixed enzyme kinetics and external diffusion limitation in case of substrate inhibition[END_REF] highlight the critical role of the function whose exact structure depends on the system under the scrutiny. In fact, non-linear Lineweaver-Burk experimental plots as well as Hanes-Woolf ones were also reported for enzymatic/heterogeneous biosensors (see e.g. refs. [START_REF] Lyons | Heterogeneous redox catalysis at hydrated oxide layers[END_REF][START_REF] Jiao | Multifunctional Nanoprobe-Amplified Enzyme-Linked Immunosorbent Assay on Capillary: A Universal Platform for Simple, Rapid, and Ultrasensitive Dual-Mode Pathogen Detection Anal[END_REF]) being presumably due to the effect of function.

Therefore, it is the purpose of this work to precisely investigate this issue in detail in the following. This will be performed for the above MM canonical mechanism (Scheme 1) as well as for a series of other important enzymatic mechanisms of general interest for their involvement in bioelectroanalytical sensors. Note also that although they will be derived for a cylindrical micro-or nanosensor, the following results are readily transposable to other types of electrodes leading to steady state or quasi-steady state diffusional regime using previously established equivalence relationships [START_REF] Amatore | Electrochemistry at ultramicroelectrodes[END_REF][START_REF] Amatore | Equivalence Between Electrodes of Different Shapes: Between Myth and Reality[END_REF][START_REF] Oleinick | Efficient quasi-conformal map for simulation of diffusion at disk microelectrodes[END_REF]. As was noted at the end of the Introduction the above equations are obtained in the same spirit as was reported previously by, e.g., Bartlett et al [START_REF] Bartlett | Modelling Biosensor Responses[END_REF], Lyons [START_REF] Lyons | Mediated Electron Transfer at Redox Active Monolayers. Part 4: Kinetics of Redox Enzymes Coupled with Electron Mediators[END_REF] and Kulys [START_REF] Kulys | Biosensor response at mixed enzyme kinetics and external diffusion limitation in case of substrate inhibition[END_REF]. However, the results obtained below, are providing different insights so this study does not overlap with the previous ones and are rather to be seen as complementary.

Results and Discussion

To simplify the formulation of the models and of their ensuing key equations and focus on the real issues at hand, the two following assumptions are made: (i) the partition coefficients between the enzymatic layer and the solution are equal to unity for all considered species; (ii) diffusion coefficients of substrate and products in the solution are set equal to their average value, . It should be noted that using the proper values for partition coefficient and/or diffusivities may change the overall systems responses by introducing adequate multiplicative factors applying to , viz., can be viewed as being integrated in values. Yet, the general validity of the approach developed hereafter will not be affected since anyway value is not supposed to be known.

Finally, it worth to note that the flux balance analysis given below represents a usual practice for the analysis of the enzymatic systems (see. e.g. refs. [START_REF] Bartlett | Modelling Biosensor Responses[END_REF][START_REF] Kulys | Biosensor response at mixed enzyme kinetics and external diffusion limitation in case of substrate inhibition[END_REF]), however we provide all the equations for clarity and consistency with our final results vs. the approach we develop in this work.

Canonical Michaelis-Menten mechanism

Figure 2a displays schematically the concentration profiles of the substrate and product for a canonical MM mechanism in the real space and can be readily adapted to the more complex mechanisms which will be considered in the following. Since all these species diffuse in the solution their concentration profiles under steady state or quasi-steady state conditions are linear in the appropriate transformed space: for electrodes of radius equivalent to hemispherical ones [START_REF] Amatore | Equivalence Between Electrodes of Different Shapes: Between Myth and Reality[END_REF] or for cylindrical ones [START_REF] Amatore | Electrochemistry at ultramicroelectrodes[END_REF][START_REF] Amatore | Electrochemical kinetics at microelectrodes. Part I. Quasi-reversible electron transfer at cylinders[END_REF][START_REF] Amatore | Equivalence Between Electrodes of Different Shapes: Between Myth and Reality[END_REF]. As shown in Fig. 2b, the effect of diffusion can then be taken into account by considering steady or quasi-steady state diffusion layers of thickness given by: (

accordingly [START_REF] Amatore | Electrochemistry at ultramicroelectrodes[END_REF][START_REF] Amatore | Electrochemical kinetics at microelectrodes. Part I. Quasi-reversible electron transfer at cylinders[END_REF][START_REF] Amatore | Equivalence Between Electrodes of Different Shapes: Between Myth and Reality[END_REF], where is the time duration of each calibration step (note that for micro-and nanosensors, does not vary significantly as soon as is larger than a few seconds [START_REF] Amatore | Electrochemistry at ultramicroelectrodes[END_REF][START_REF] Amatore | Electrochemical kinetics at microelectrodes. Part I. Quasi-reversible electron transfer at cylinders[END_REF][START_REF] Amatore | Equivalence Between Electrodes of Different Shapes: Between Myth and Reality[END_REF], hence the term of "quasi-steady-state" applying to cylindrical diffusion when ).

[Fig. 2]

The flux at the conducting surface of the enzymatic electrode is null since is nonelectroactive substrate by definition. Conversely, the flux of towards the solution is (Fig. 2b): [START_REF] Amatore | Electrochemistry at ultramicroelectrodes[END_REF] where, for our purpose here, represents the value at , whose value is defined by the intersection of the two lines figuring the slopes of concentration profile on either side of the enzymatic layer in the conformal space (Fig. 2b). is much smaller than since its value is by definition smaller than that of the thickness of the thin enzymatic coating (see Fig. 2). Hence, Eq.( 10) can be readily simplified into:

For any electroactive species and generated by the enzymatic conversion of substrate and initially absent in the bulk solution, the conservation of fluxes imposes that:

(12a)
where since the substrate is not electroactive and , while where is the maximum concentration of P (Fig. 2). Hence, since , Eq. (12a) readily rewrites as:

(12b)
From Eqs.( 11) and (12b) it follows that: (13) which establishes that the function defined in Eq.( 4) is then: [START_REF] Lescovac | Enzymatic kinetics[END_REF] Note that Eq.( 14) is general and remains valid for any enzymatic mechanism since it stems only from (i) the diffusional supply of the substrate to the enzymatic layer and (ii) the electroactivity of the product at the electrode electroactive surface, both processes occurring in the enzyme-free sections of the electrochemical enzymatic sensor. Conversely, the very nature of the mechanism active in the thin enzyme coating is only acting through the exact analytical dependence of on whose value is related to and to the current intensity through Eq.( 13).

Let us then return to the canonical Michaelis-Menten mechanism. Incorporating Eq.( 14) into Eq.( 5) yields: [START_REF] Gonçalves | Penicillinasebased amperometric biosensor for penicillin G[END_REF] where and , being the maximum rate (in mol/m 2 /s units) of the sensor (note that for a canonical MM mechanism, as follows from Eq.( 5)). The dimensionless parameter compares reaction and diffusion rates for the system at hand so is, to some extent, related to the square root of the Thiele modulus [START_REF] Hill | An Introduction to Chemical Engineering and Reactor Design[END_REF]. Eq.( 15) then provides the correct relationship between the current and the substrate bulk concentration. Only this correct relationship is able to provide the kinetic data characterizing sensor under the present conditions. Special care should be taken whenever a calibrated enzymatic electrochemical sensor is aimed to be used under non-steady state conditions as will be shown in a following work [START_REF] Dannaoui | [END_REF].

[Fig. 3] Although Eq.( 15) is quadratic with respect to the current intensity, it is first order with respect to . This property allows to easily construct calibration curves providing as a function of the current for different values of which are shown in Fig. 3a.

Fig. 3b shows the same calibration curves plotted using a double reciprocal scaling. Although the calibration curves in Fig. 3a are strongly reminiscent of usual calibration plots, as hinted in the 'Preliminary consideration' section, those in Fig. 3b -which are the formal correct equivalents of Lineweaver-Burk plots -show that as soon as values are not negligible plots vs. are strongly curved except at large values where a linearity is apparently restored (Fig. 3b). Yet, as noted through comparing Fig. 3b to Fig. 3c (compare the dashed lines in Fig. 3c to the dotted lines in Fig. 3b) the theoretical limits predicted when given in Eq.( 16) require extremely large values to achieve linear behavior: [START_REF] Silva | An efficient poly(pyrroleviologen)-nitrite reductase biosensor for the mediated detection of nitrite[END_REF] This is a clear proof that the performance and limitations of enzymatic electrochemical sensors cannot be analyzed using approaches developed for homogeneous solutions even for the simplest MM case. This is a very important result of the above analysis. Indeed, to evaluate the performance and limitation of electrochemical bioenzymatic sensors one often uses double reciprocal plots expecting them to be linear for a well-behaving sensor and try to extract slopes (supposedly values) and intercepts (supposedly ) values to characterize their sensors [START_REF] Mcmahon | The efficiency of immobilised glutamate oxidase decreases with surface loading: an electrostatic effect, and reversal by a polycation significantly enhances biosensor sensitivity[END_REF][START_REF] Wang | Ultra-fast glutamate biosensor recordings in brain slices reveal complex single exocytosis transients[END_REF][START_REF] Yang | Quantitative Nano-Amperometric Measurement of Intravesicular Glutamate Content and of its Sub-Quantal Release by Living Neurons[END_REF][START_REF] Gonçalves | Penicillinasebased amperometric biosensor for penicillin G[END_REF][START_REF] Silva | An efficient poly(pyrroleviologen)-nitrite reductase biosensor for the mediated detection of nitrite[END_REF][START_REF] Enache | Electrochemical evaluation of Glutathione Stransferase kinetic parameters[END_REF][START_REF] Yuan | Creating a Low-Potential Redox Polymer for Efficient Electroenzymatic CO 2 Reduction[END_REF]. Furthermore, Fig. 3c shows that unless , this frequent but incorrect procedure yields intercepts of that are significantly smaller than unity which is a theoretical non-sense since by definition . Therefore, except when , i.e., when mass transport is excessively fast compared to enzymatic kinetics the usual double reciprocal diagrams provide misleading information. Note also that when , (black curves in Fis.3a-c), one has so that the corresponding situations are those in which the enzymatic system is almost inactive, the sensors being close to their saturation limit (compare Fig. 3a). Note that the same considerations would also apply to other types of (semi)reciprocal plots such as Hanes-Woolf or Eadie-Hofstee analyses [START_REF] Lescovac | Enzymatic kinetics[END_REF] if used to treat bioelectrochemical enzymatic sensor data.

However, plotting calibration curve as a function of would provide a linear curve in double reciprocal scale since in these case equations are equivalent to the classical Lineweaver-Burk relations. Yet, generally is not known, thus preventing construction of these simple to analyze calibration curves.

In fact, the dependence of with for non-negligible values of is obtained by combining Eqs.( 1), ( 2) and [START_REF] Amatore | Electrochemical kinetics at microelectrodes. Part I. Quasi-reversible electron transfer at cylinders[END_REF]. This yields a quadratic equation which is readily solved analytically to afford:

A set of these solutions is shown in Fig. 4 for the values already used in Fig. 3. All these plots display two interesting limiting behaviors (see the dotted lines in Fig. 4 for ):

(i) a tangent at whose equation is:

(18a)
and (ii) an asymptote with a unity slope at large values:

These two limits in Eqns.(18a,b) and the noticeable curvatures at intermediate values emphasize again that always differs from unless when , that is when the mass transport rate drastically outperforms that of the enzymatic activity as exemplified by the black curve ( ) in Fig. 4. Though, as said above, this corresponds to a highly inefficient enzymatic activity.

[Fig. 4]

Bisubstrate ping pong mechanisms (PP and PI)

In this section we would like to consider a bisubstrate ( and ) ping pong mechanism (Scheme 2) since it is one of the most frequently encountered situations in bioelectrochemical sensors. Two versions of ping pong mechanism will be considered: (i) one, denoted PP in the following, when as more often considered [START_REF] Clay | A Detailed Model of Electroenzymatic Glutamate Biosensors to Aid in Sensor Optimization and in Applications in Vivo[END_REF] or (ii) in the general case, noted PI since it may involve product inhibition, when , i.e. when the enzyme is inhibited by the intermediate product [START_REF] Lescovac | Enzymatic kinetics[END_REF] Ping Pong Bi Bi electrochemical mechanisms such as the one detailed in Scheme 2 [START_REF] Clay | A Detailed Model of Electroenzymatic Glutamate Biosensors to Aid in Sensor Optimization and in Applications in Vivo[END_REF] are often approximated by a canonical Michaelis-Menten MM mechanism (Scheme 1) [START_REF] Mcmahon | The efficiency of immobilised glutamate oxidase decreases with surface loading: an electrostatic effect, and reversal by a polycation significantly enhances biosensor sensitivity[END_REF][START_REF] Wang | Ultra-fast glutamate biosensor recordings in brain slices reveal complex single exocytosis transients[END_REF][START_REF] Yang | Quantitative Nano-Amperometric Measurement of Intravesicular Glutamate Content and of its Sub-Quantal Release by Living Neurons[END_REF].

However, this supposes that the first step of the Scheme 2 is globally irreversible, i.e., , and also that the second step is infinitely fast relative to the first one. Despite the fact that this approximation is often used we wish to separate in the following the situations depending on whether or not has a negligible kinetic role.

The rate law for PP mechanisms, i.e. in the absence of any product inhibition, is given by [13,[START_REF] Lescovac | Enzymatic kinetics[END_REF]: [START_REF] Lyons | Heterogeneous redox catalysis at hydrated oxide layers[END_REF] where:

, ( 20
)
Rate law [START_REF] Lyons | Heterogeneous redox catalysis at hydrated oxide layers[END_REF] involves again the interfacial concentrations of both substrates. However, as discussed above about oxidases when O 2 is the co-reactant while the electrochemical detection step involves the 2e-oxidation of the formed H 2 O 2 , concentration does not suffer significant variations, hence . We assume in the following that this is always the case.

As was noted before, Eqs.( 1), ( 3), ( 4), ( 13), ( 14) take into account the effect of diffusional transport and are independent of the exact mechanism occurring in the thin enzyme layer.

Thus, by performing the same derivations as in the MM case but using now the pertinent rate law in Eq.( 19) one readily obtains the equation describing calibration curves for PP mechanisms under the present electrochemical conditions: [START_REF] Oleinick | Efficient quasi-conformal map for simulation of diffusion at disk microelectrodes[END_REF] where:

, , (22)
The maximum current depends now on the second substrate concentration owing to the presence of the term in its denominator albeit the whole sequence remains kinetically controlled by the first irreversible step in Scheme 2 as evidenced by comparing Figs. 3 and5. Nonetheless, the second step may act by lowering the electrochemical current intensity which is now divided by the factor unless (i.e., when as evidenced by the above expression in Eq.( 22). This indirectly justifies the general use of analyses based on MM mechanistic framework using the kinetic characteristics of the first step although PP mechanisms involve a sequence of two enzymatic steps. Yet it is clear that using this MM pseudo equivalence leads to the determination of apparently smaller values.

More precisely, the smaller with respect to the second Michaelis constant , the larger is the decrease of the maximum current compared to that predicted assuming a canonical MM mechanism involving the same first step when . However, any Lineweaver-Burk plots established for sufficiently large values will have different significance than they would for a true MM mechanism. This is fully evident upon examining equation ( 23) that describes the linear asymptote of Eq.( 21) at large values (compare Eq.( 16) taking into account that ): [START_REF] Dannaoui | [END_REF] This caveat needs to be duly taken into account if the calibration results and their data extracted through Lineweaver-Burk plots need to be used for applications of the same sensor under different experimental conditions.

[Fig. 5]

In the case of PI mechanisms, i.e., with inhibition by the intermediate product , the rate law becomes: [START_REF] Clay | A Detailed Model of Electroenzymatic Glutamate Biosensors to Aid in Sensor Optimization and in Applications in Vivo[END_REF] where , and

Let us introduce: ,

Since we assumed that the diffusion coefficients of the substrate and intermediate product are equal, the substrate/product mass conservation equation affords: [START_REF] Herr | Electroosmotic Flow and Its Contribution to Iontophoretic Delivery[END_REF] since this stems from the substrate/product mass conservation equation. By substituting Eq.( 27) into (24), taking the limit at and converting rates into currents (as done when multiplying rates by ) one obtains a quadratic equation with respect to whose solution is:

(28)

The equation of calibration curves under PI mechanism is then given by: (

is structurally similar to Eqs.( 15) and ( 21), but incorporates additional kinetic terms representing the effect of the partial inhibition of the first step kinetics by the intermediate product . 15), ( 21) and ( 29)

Practical use of equations (

The above theoretical analyses had the merit of elucidating quantitatively through compact analytical formulations the important problem identified in the "Preliminary Considerations" section by evaluating the significant changes introduced by the necessary involvement of the diffusional transport of substrate and product(s) to and from the enzyme thin layer. This establishes that enzymatic kinetic formulations derived for homogeneous solutions need to be adapted to account for the performance of enzymatic electrochemical sensors. However, equations ( 15), ( 21) and (29) depend on a whole set of parameters whose values are generally unknown. Therefore, for practical reasons, it would be useful to provide an access to relationships that involve experimentally available information as much as possible.

The maximum sensor experimental current values can be evaluated, or at least reasonably approximated, from calibration curves relating current vs. substrate bulk concentration at high substrate concentrations under steady or quasi-steady state conditions. On the other hand, for a canonical homogeneous Michaelis-Menten mechanism is equal to the substrate concentration value at which the half-maximum activity of the enzyme is observed. Though, this is not the case for electrochemical enzymatic sensors.

Eq. [START_REF] Gonçalves | Penicillinasebased amperometric biosensor for penicillin G[END_REF] shows that under electrochemical conditions the bulk substrate concentration , viz., that at which the sensor current intensity reaches its half-maximum value, , features not only the value of the immobilized enzyme but includes intrinsic enzymatic (viz., ) and transport properties of the system as shown by Eq.( 30):

(30)

Note that is obtained only in the limiting case when , that is when the transport of the substrate across the diffusion layer is infinitely fast compared to the enzymatic kinetics taking place at the surface of the electrode.

The explicit definition of in Eq.(30) allows introducing the dimensionless parameter :

(31) which plays a role equivalent in meaning to that of by comparing the enzyme kinetics to the mass transport rate, although this is now performed through involving only experimentally available data, viz. and as well as operational parameters ( , ).

Eqs.( 30) and (31) allows rewriting Eq.( 15) into Eq.( 32) which now only involves factors that can be experimentally accessible since they are readily evaluated from and experimental values:

(32)

Eq.( 32) provides a dimensionless "calibration curve" that depends only on and and can be plotted without any approximation for any given experimental value of whose value is itself available experimentally. A set of such "calibration curves" are shown in Fig. 6 in linear and reciprocal scales. Note that whatever is the value of , the renormalized calibration curves in Fig. 6a require a high experimental accuracy to be distinguished when .

[Fig. 6]

Along the same logic one may formulate dimensionless "calibration curves" equations for PP mechanism just in terms of the experimentally available parameters , and which are defined in Table 1. As before, this results in an equation with an identical structure to that reported for a MM mechanism in Eq.( 32):

(33)

It follows that in practice the MM and PP mechanisms cannot be distinguished on the basis of their calibration curves alone, although the significance of the parameters extracted from their quantitative processing differs profoundly at the level of enzymatic kinetics. This justifies that the PP mechanisms can be treated as simple MM mechanisms even if it is necessary to realize that the analysis now provides apparent values and values that differ from the effective and . Indeed, one has then and (see Eq. [START_REF] Hill | An Introduction to Chemical Engineering and Reactor Design[END_REF] or Table 1 for the definition of ).

[ predictable. This extends previous conclusions drawn for fully homogeneous situations [13,[START_REF] Lescovac | Enzymatic kinetics[END_REF][START_REF] Plesner | The apparent K m is a misleading kinetic indicator[END_REF] and further illustrates the added complexity brought by electrochemical enzyme micro-and nanosensors owing to convolution of diffusional transport terms with experimental currents. Indeed, upon increasing the mechanism complexity (viz., from MM to PP and then to PI) the operational parameters which govern the calibration curves quickly become complicated functions of primary parameters (see Table 1). Except for MM mechanism, these operational parameters depend on more than two primary parameters. It ensues that an infinite number of primary parameter combinations can lead to a same calibration curve. This is not a problem in itself whenever the only purpose of the calibration curves is to establish a correspondence between currents and substrate concentrations under precise experimental conditions involving steady or quasi-steady state diffusion transport from bulk solution. However, this becomes a debatable issue with significant consequences when sensors "calibrated" under these conditions are to be used under different conditions, i.e., for monitoring the same substrates under in vivo or ex vivo conditions [START_REF] Wang | Ultra-fast glutamate biosensor recordings in brain slices reveal complex single exocytosis transients[END_REF][START_REF] Yang | Quantitative Nano-Amperometric Measurement of Intravesicular Glutamate Content and of its Sub-Quantal Release by Living Neurons[END_REF][START_REF] Yang | Nanoelectrochemistry reveals how soluble Aβ42 oligomers alter vesicular storage and release of glutamate[END_REF][START_REF] Herr | Electroosmotic Flow and Its Contribution to Iontophoretic Delivery[END_REF].

Conclusion

In this work we established theoretically that Lineweaver-Burk analyses justly used for evaluating enzymatic kinetic properties when the enzymes are homogeneously dispersed in a bulk solution cannot be used straightforwardly for the analysis of electrochemical calibration curves recorded with enzyme-modified electrodes. Analytical equations extending Lineweaver-Burk analysis to the electrochemical conditions under (quasi-)steady state conditions were derived to allow proper analysis of enzymatic electrode calibration curves for Michaelis-Menten enzymatic mechanisms and ping pong bi-substrate ones with or without inhibition. More complex enzymatic mechanisms can be equally considered within the framework of the proposed approach. In addition, our analytical equations can be reformulated to fit the format of other reciprocal plots frequently used in enzymatic analysis, such as Hanes-Woolf and Eadie-Hofstee plots [START_REF] Lescovac | Enzymatic kinetics[END_REF].

Declaration of Competing Interest

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. with their linear regression segments are reproduced, while the gray dashed lines corresponds to the limits given by Eq.( 16). for several values labelled on each curve (identical to those in Fig. 3). The red dotted lines are the limiting cases given by (18a) and (18b) for . The black line ( ) is a line with a unit slope. The color code characterizing each value (see Eq.( 32)) is the same in (a) and (b). See Table 1 for the specific definitions of , and according to the MM or PP mechanism.

Figure captions:Fig. 1 .

 1 Figure captions:

Fig. 2 .

 2 Fig.2. Schematic representation of the concentration profiles of the involved species at

Fig. 3 .

 3 Fig.3. Canonical Michaelis-Menten mechanism. Theoretical calibration curves evaluated for

Fig. 4 .

 4 Fig.4. Canonical Michaelis-Menten mechanism. Variations of as a function of

Fig. 5 .

 5 Fig.5. Ping pong Bi Bi (PP) mechanism calibration curves evaluated based on Eq.(25) for

Fig. 6 .

 6 Fig.6. Simulated calibration curves plotted using normalized parameters based on

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .

 123456 Fig.1.

  . It should be noted that the possible

	and quantify through its 2e-oxidation involving dioxygen (O 2 ) as the electron acceptor and
	yielding H 2 O 2 as the final enzymatic product in Scheme 2. Indeed, H 2 O 2 electrochemical
	oxidation regenerates O 2 which automatically takes care of the above constraint (see
	additional inhibition by final product bottom equation in Scheme 2). For this reason, in order to simplify the following is generally not very efficient since under
	amperometric conditions presentation and focus on the issue of interest in this work we will only consider this concentration at the electrode surface vicinity is extremely
	small. ubiquitous case (Scheme 2).
	Scheme 2. Ping Pong Bi Bi electrochemical mechanisms referred hereinafter to as PP and PI
	depending on the reversibility of the first enzymatic step: PP when	, or PI for
	, i.e., when product may inhibit the first step. The global mechanism is detailed in
	the common situation encountered with oxidases when the final product electrochemical
	detection regenerates the co-substrate , i.e., generalizing the ubiquitous case where is
	O 2 and is H 2 O 2 ( =-2).
	For bioelectrochemical sensors developed to monitor the concentration of the main
	substrate of interest, viz., in Scheme 2, it is advisable to independently maintain the
	concentration of the co-substrate	as constant as possible, a constraint that may be
	difficult to fulfill. For this reason, many authors favor oxidases whenever possible to detect

Table 1 ]

 1 For the case of PI mechanisms the resulting equation has structure similar to those in Eqs.(32) and (33), see Eq.(34), but due to the kinetic competition between the direct (i.e.

	uninhibited) mechanism and its inhibition branch two additional competition parameters
	,	are involved in Eq.(29) although they are generally
	unknown:
					(34)
	Note that	, where	is a function of	defined in Eq.(35)
					(35)
	The structure of Eqns.(34) and (35) show that, although they are necessarily different from
	PP or MM mechanisms, PI mechanisms would be very hard to distinguish. Indeed, they
	exhibit very similar functional dependence on experimental parameters except for the
	kinetic influence of the additional parameters	and	whose values are difficultly

Table 1 .

 1 Definitions of parameters involved in Eqs.(32), (33) and (34) for each of the three mechanisms considered here.

	Enzymatic mechanism
	MM: Michaelis-Menten,
	Eq.(32)
	PP: Ping pong Bi Bi,
	Eq.(33)	(a)
	PI: Ping pong Bi Bi
	with inhibition by
	intermediate product,
	Eq.(34)	(b,c)

Acknowledgements

This work was supported in parts by CNRS, ENS -PSL University, Sorbonne University (UMR 8640 PASTEUR) in Paris and by the National Natural Science Foundation of China (NSFC) in China. The Chinese team acknowledges financial support from the NSFC (Grants 22090050, 22090051, and 21721005). All authors acknowledge the Sino-French IRP CNRS NanoBioCatEchem for the support. CA thanks as well Xiamen University for his Distinguished Visiting Professor position.