Yanxia Hou

Nicole Jaffrezic-

Renault 1

Claude Martelet

Chaker Tlili

Aidong Zhang

Jean-Claude Pernollet

Loïc Briand

Gabriel Gomila4

Abdelhamid Errachid

Ludovic Salvagnac

Benoit Torbiéro

Pierre Temple-Boyer

Langmuir and Langmuir-Blodgett films of odorant binding protein/amphiphile study for odorant biosensor

We have investigated Langmuir and Langmuir-Blodgett films of odorant binding protein/amphiphile for elaboration of artificial olfactory system, namely odorant biosensor.

Under optimised experimental conditions the mixed monolayer at air/water interface is very stable and has been transferred efficiently onto functionalised gold supports by SAMs.

Electrochemical impedance spectroscopy and atomic force microscopy techniques were used to characterise mixed LB films for detecting one kind of corresponding odorant molecule isoamyl acetate.

Introduction

Many fundamental and central biological recognition and transduction processes occur at biological surfaces. To investigate molecular behaviour of biomolecules at biological surfaces, a well defined and flexible model system is required. Langmuir-Blodgett (LB) films and selfassembled monolayers (SAMs) are believed to be responsible for construction of such model system. LB and SAMs may provide the desired control on the order at the molecular level and thus should be considered desirable techniques for the construction of future organic materials [START_REF] Ulman | An Introduction to Ultrathin Organic Films: From Langmuir-Blodgett to Self-Assembly[END_REF].

So far extensive researches have been devoted to either simple systems of Langmuir-Blodgett films [START_REF] Petty | Langmuir-Blodgett films: An introduction[END_REF][START_REF] Bardosova | [END_REF](4) and self-assembled monolayers (5-7) or complex ones, which combine these two techniques with biocomponents. Particularly interesting, it has been noted that both of these techniques have potential application on elaboration of biosensors. Langmuir-Blodgett technique has already shown its efficiency for depositing well-defined films of enzymes (8)(9)(10)(11)(12)(13)(14)(15)(16) and antibodies (17)(18)(19) etc. for manufacturing biosensors and it presents numerous advantages such as: rapidity, reproducibility, controlling the amount of the biocomponents and preserving the activities and specific recognition properties of biomolecules. Since Nuzzo and Allara showed that SAMs of alkanethiolates on gold could be prepared from dilute solutions of dialkyl disulfides (20), the interest in SAMs rapidly expanded and the use of SAMs in various fields of research is rapidly growing. The recent developments show that SAMs play an important role in the construction of biosensors due to its advantages such as simplicity, adaptability, stability and the control over biomolecules surface orientation (21)(22)(23).

In this study, we try to elaborate very thin films in which odorant binding proteins (OBPs) were incorporated by the combination of LB films and SAMs; and on the other hand, we hope to perform a preliminary try for an artificial olfactory system, namely, to detect corresponding odorant molecules with such kind of thin films.

Odorant binding proteins, which are abundant low-mol. wt. soluble proteins secreted by the olfactory epithelium in the nasal mucus of vertebrates, play a carrier role for conveying odorants, which are commonly hydrophobic molecules, to their neuronal receptors through nasal mucus (24). In present report a novel rat odorant-binding protein variant OBP-1F was used for the preparation of mixed protein/amphiphile LB films. The experiments were performed in the subphase of phosphate buffer solution (PBS) at a pH of 7.5. Under the condition, OBP-1F is perfectly soluble and active. Fatty amine octadecylamine (ODA) was chose as amphiphile taking into account of the fact that at such a pH OBP-1F could be considerably better adsorbed onto positively charged lipid monolayers such as amine than neutral or negatively charged ones. Since the pKa of OBP-1F is 5.20, the protein molecules are mainly negatively charged at the pH of 7.5. In addition, the use of fatty amines will allow us to carry out a crossing-linking step between amine terminal moieties (coming from protein and fatty amine) and crossing-linking reagent glutaraldehyde molecules in order to improve the stability of the transferred LB films as reported by Sun (9) and Chovelon (15). Furthermore, the presence of the protein and counter-ions of hydrogen phosphate (25) in the subphase were expected to be favourable for stability of fatty amine monolayer at air/water interface.

It's known that the basic LB technique consists in elaborating a monolayer of the amphiphile molecules on the subphase containing soluble protein and then transferring the monolayer onto the solid substrates by vertical dipping procedure (Langmuir-Blodgett) or horizontal one (Langmuir-Schfäer). For obtaining high-quality LB films the properties of the monolayer especially stability was investigated here, since it plays a key role in obtaining stable and reproducible LB film, which is essential for biosensor application. And LB films of odorant binding protein/amphiphile, OBP-1F/ODA in our case, were deposited by most often used vertical dipping procedure onto gold substrates, which were functionalised with 1-octadecanethiol by SAMs to provide a biomimetic and hydrophobic surface favourable for the deposition of OBP-1F/ODA LB films.

Such mixed LB films system, on the one hand, may constitute interesting models for biological membranes and structures, may be used for studying and testing isolated functions; on the other hand, such structures may be used for designing artificial systems inspired by biological structures or functions and possessing analogous properties. Consequently, the obtained LB films were exposed to one of corresponding odorant molecule for elaboration of odorant biosensor.

In this study several techniques including scanning electron microscopy (SEM), atomic force microscopy (AFM) and electrochemical impedance spectroscopy (EIS) have been applied to characterise LB films for odorant biosensor. AFM is considered a powerful tool in microbiology (26)(27) and it is also an interesting tool for the characterization of LB films (10,16,(28)(29) in that it could give access to the molecular architecture. And a complementary technique EIS is a rapidly developing electrochemical technique for the characterization of biomaterialfunctionalised electrode surface and has potential application on biosensor, since it represents not only a suitable transduction technique to follow the interfacial interactions of biomolecules, but it also provides a very powerful method for the characterization of the structural features of the sensing interface and for explaining mechanisms of chemical processes occurring at the electrode/solution interface (30)(31)(32). The results obtained from AFM and EIS have been discussed.

Materials and Methods

Materials

OBP-1F (wt 18 538 Da) and its corresponding odorant molecule isoamyl acetate were both provided by Laboratory of Biochemistry and Structure of Proteins, INRA, France, and were used without any further purification. Octadecylamine (ODA) (purity > 99) and 1octadecanethiol (ODT) (purity 98%) were purchased from Fluka and Aldrich Chemical Company, respectively. Solvent chloroform (purity 99.8%) bought from Aldrich Chemical Company was applied for dissolving ODA at a concentration of 1 mg/ml. And ODT was dissolved at a concentration of 1 mM in ethanol (purity > 99.8%) obtained from Fluka for modification of gold substrate. Potassium dihydrogen phosphate, sodium chloride both from PROLABO, and sodium hydroxide from Aldrich Chemical Company were used to prepare 10 mM phosphate buffer solution (consisting of 100 mM NaCl, pH 7.5), which was used for both LB films preparation and impedance measurements. And all of them are analytical grade reagents (>99).

In present study, OBP-1F solutions were prepared freshly on the day of experiment by dissolving the protein in PBS (pH 7.5). Water used through all experiments was purified by an Elga system with a reverse osmosis, deionization, filtration and ultraviolet irradiation systems to have a resistance as high as 18.2 m.cm. Its pH was about 5.5, and its surface tension was 71.8 mN.m-1 at 25 °C in air.

And all glassware that was to come into contact with the samples, solutions, or solvents was washed well with sulfochromic solution, rinsed with copious amounts of ultrapure water and then acetone, dried for overnight in a sealed oven at 100 °C.

Gold substrate and monolayer preparation

Gold substrates were fabricated using standard silicon technologies. <100>-oriented, P-type (3-5 .cm) silicon wafers were thermally oxidised to grow a 800nm-thick field oxide. Then, a 30nm-thick titanium layer and a 300nm-thick gold layer were deposited by evaporation on top.

The formation of a monolayer on gold is strongly dependent on several factors, such as the cleanliness and structure of gold prior to modification. As known gold is a soft metal and is readily contaminated by organic and inorganic species during manual handling (33).

Consequently, the fabrication process and the pre-treatment, namely cleanliness, of gold substrates are considered very important.

In present study, the whole metallization process was optimised in order to keep a good adherence of the Ti/Au layer on the field oxide which reaching good properties of the gold upper surface. This optimisation has lead to the use of low deposition rates (around 1 nm/s) for both evaporation processes and to the suppression of the Ti/Au annealing process (250°C -20 min). In the same way, the cleaning procedure has been optimised as follows. The gold substrates were previously cleaned with acetone in an ultrasonic bath for 10 min and dried under a nitrogen flow, followed by immersion in 7:3 (v/v) H2SO4:H2O2 (piranha solution) for 1 min in order to get rid of inorganic and organic contaminants on the substrate surface. (Caution: piranha solution reacts violently with organic materials and should be handled very carefully.) They were subsequently rinsed thoroughly in absolute ethanol and finally dried under a N2 flow.

As soon as the cleaning procedure terminated, the substrates were put into 1 mM ODT solution for 21 h at room temperature to make their surfaces hydrophobic by the formation of SAMs. Excess thiol was removed from the surface by rinsing thoroughly the substrates with ethanol and finally substrates were dried under a N2 flow.

A contact angle meter (GBX scientific instruments, France) was utilised to measure the contact angle of gold surface. We found that the contact angle of the as-received gold substrate is 50° due to the presence of organic contaminant layer (34). However, after cleaning the substrates were hydrophilic with contact angle < 10° attributed to an organic contaminant-free surface, and they were hydrophobic after formation of well-ordered SAMs with contact angle 110°  3°.

Langmuir and Langmuir-Blodgett films

Langmuir experiments were performed with a Langmuir trough from NIMA (model 611) with an effective surface 30 cm  10 cm and the volume of the liquid phase 200 ml. Moreover, this system consists of two mobile barriers for compressing amphiphilic monolayer and a Wilhelmy balance which was equipped with a strip of chromatography paper suspended at the air-water interface for measuring interfacial pressure. The trough is equipped with a chamber for preventing external contamination.

The subphase temperature was controlled at 10  1 °C by a thermostatic system (Julabo-F25, France). The subphase was constituted of solution of OBP-1F in PBS at different concentration or in ultrapure water. Before each experiment, the surface of the subphase was cleaned by aspiration. And the surface was considered clean when the change in surface pressure of the subphase upon compression of the area was smaller than 0.1 mN m-1.

For elaboration of mixed layer of octadecylamine and protein OBP-1F, the protein solution in PBS was used as subphase and 22 μL ODA in chloroform was injected and spread onto interface by a micropipette of 50 μL. It was spread from the same distance above the subphase in all the experiments. After spreading, the monolayers were left for at least 15 min to allow the solvent to evaporate. Proteins were allowed to adsorb from the subphase to the air/water interface for about 3 h. The apparent molecular area was defined as the ratio of the film area to the number of ODA molecules. Then the compression of the film was initiated at a rate of 5 cm2 min-1 until the target surface pressure was reached. Relaxation time of 3 h was required before onset of dipping. While the surface pressure was kept constant and the monolayer was stable, the monolayer was transferred onto hydrophobic gold substrates by vertical dipping method at a rate of 3 mm min-1. After each upstroke, the substrate was maintained in air for 5 min to dry the films. The transfer ratio (TR) expressed as the ratio of area of film transferred onto the monolayer-coated area on the substrates was used as a parameter to characterize LB deposition. After transfer of LB films functionalized gold substrates were put into the vapour of glutaraldehyde for 15 min, in order to stabilize LB layers by cross-linking with this bifunctional reagent. Then they were used for the following experiments.

For understanding well formation of Langmuir and Langmuir-Blodgett films, first we have studied the properties of monolayer at air/water interface. For studying adsorption of protein to amphiphile monolayer at the air/water interface, surface pressure-time curves (referred to as π -t) were recorded with opened barrier, namely without compressing force. The characteristics of the monolayer on the surface were studied by measuring the changes in surface pressure upon compressing the monolayer at a given temperature, namely surface pressure-molecular area isotherm (referred to as π -A). The shape of the isotherm is characteristic of the molecules making up the film. The stability of the monolayer, which is an important parameter for the stability of the mixed LB films, can be demonstrated by molecular area ratio-time evolution isotherms at the target pressure (referred to as A/A0-t) or surface pressure-time evolution at a constant molecular area (referred to as Pb-t).

Volatile-odorant binding

After the deposition and stabilization of OBP-1F/ODALB films the substrates were put in a 2-L sealed glass chamber containing pure undiluted odorant isoamyl acetate (10 μL in the chamber) which evaporated freely for 16 h at room temperature. And the samples were then used for AFM and EIS measurements.

Langmuir-Blodgett films characterization

Scanning Electron Microscopy

SEM images were taken using a scanning electron microscope (Philips XL20). The accelerating voltage was 10 kV.

Atomic Force Microscopy

Atomic force microscopy measurements were obtained in air at room temperature using a Pico+ Microscope (Molecular Imaging). Tapping mode was used to avoid damaging the surface of the sample. Silicon rectangular cantilevers (NCL, Nanoworld) with normal spring constant of 48 N/m and resonance frequency of 190 kHz were used (typical curvature radius of the tip 10 nm).

Electrochemical Impedance Spectroscopy

Impedance measurements were performed in a conventional electrochemical cell containing a three-electrode system with an Electrochemical Interface SI 1287 and a Frequency Response Analyser 1255B from Solartron. A Pt plate and a saturated calomel electrode (SCE) were used as counter and reference electrode, respectively. And gold substrates with LB films deposition were working electrodes with effective surface 0.07 cm2. Impedance measurements were taken in absence of any redox probe in PBS at ambient temperature and in the frequency range from 50 mHz to 100 kHz, at the formal potential of -600 mV using alternate voltage of 10 mV.

Results and discussion

Adsorption of protein at air/water interface

The surface pressure vs time (π-t) curves presented in Figure1. were measured for the adsorption of protein at air/water interface at three different protein concentrations: 4, 6 and 10 mg L-1. And they were collected without compressing force.

It can be seen from the Figure1 increase of the surface pressure was strongly dependent on concentration of protein in the subphase and time. The higher concentration of protein OBP-1F, the higher surface pressure. It is in accordance with the kinetics of adsorption, for higher protein concentration, more protein molecules went up towards interface under the same adsorption time. In addition, from the figure we can find the surface pressure increases rapidly at the beginning and continues to increase but slowly after 250 min, which means there are still more protein molecules which are prone to penetrate lipid layers and adsorb at air/water interface. However, it took even more than 10 hours to reach adsorption equilibrium.

Considering the fact that such a long adsorption time may result in evaporation of water in the subphase, and thereby may produce some unfavourable effect on accuracy of the experimental results, proteins were allowed to adsorb from the subphase for about 3 h. Curiously, comparing to the results of adsorption of protein Butyrylcholinesterase (BuChE) at air/water interface studied by Chovelon et al. (15), we found that the surface pressure in the case of OBP-1F is comparatively low. For example, at the protein concentration 4 mg L-1 for both of cases, the surface pressure is approximately 25 mN m-1 for BuChE with adsorption time 250 min, but only about 11 mN m-1 for OBP-1F.

Proteins follow a diffusion-controlled adsorption mechanism at air/water interface. They are prone to penetrate lipid layer and expose the hydrophobic part upon the air phase at air/water interface, which can proceed as long and as far as there is time and space at the interface available (35). Still more in the presence of ionic surfactant, such as ODA in this study, proteins can be adsorbed onto or between head of amphiphile molecules by electrostatic interaction. At low surface pressure while without compressing force, the diffusion-controlled adsorption plays a predominant role on increment of surface pressure. However, it's reported that the rate of desorption of proteins increases markedly with decreasing molecular weight at a given surface pressure (36). Consequently it's evident that surface pressure for BuChE (wt 444 000Da) is higher than that for OBP-1F (18 538Da). At low surface pressure, molecules of BuChE are more prone to spread at air/water interface. However for ODA-1F the rate of desorption of protein OBP-1F is so high in the subphase that the increase of surface pressure arises mainly from the adsorption of protein to amphiphile molecules.

Surface pressure-molecular area isotherms

Three different subphases: ultrapure water, PBS at pH 7.5 without and with OBP-1F were used to investigate the characteristics of the corresponding amphiphile monolayers. The surface pressure -molecular area isotherms of amphiphile molecules on ultrapure water were considered as a reference. For the mixed film, an apparent molecular area was defined as the ratio of the film area to the number of amphiphile molecules.

We can see from the Figure 2 ingredient in the subphase has apparent influence on the characteristics of the amphiphile monolayers at the air/water interface. For ultrapure water as subphase, extrapolated to 0 mN m-1 from the linear portion of the compression curve gives an apparent surface area of only15 Å2/ODA, which is lower than the theoretical value of 21 Å2/ODA. It results most likely from dissolution of one small part of ODA molecules in the water , which demonstrates the ODA monolayer at air/water interface is not very stable with ultrapure water as subphase.

However, for PBS as subphase the curve is considerably different from that with ultrapure water as subphase. We can see from Figure 2 at low surface pressure the molecular area of the ODA monolayer with the PBS as subphase was much larger than that of the ODA monolayer in ultrapure water. It is proposed that at the pH 7.5, the amino head of ODA should be positively charged, the addition of strongly bound multivalent counter-ions, such as PO43-in the PBS subphase have effect on the monolayer. Perhaps at low surface pressure aggregate of long-chain amine ODA around PO43-took place. On the other hand, extrapolated to 0 mN m-1 from the linear portion of the compression curve gives an apparent surface area of 20 Å2/ODA, which is very close to the theoretical value. We can conclude there is nearly no dissolution of ODA molecules under such condition. The increased stability of the monolayer as expected is attributed to the addition of counter ions in the subphase, which induced interaction between anions and positively charged amines.

In the case of PBS subphase with protein OBP-1F, the molecular area is still greater than without OBP-1F at low surface pressure, it means that interaction exists also between protein and ODA molecules, and some of protein molecules were inserted into the ODA monolayer.

As referred above while surface pressure lower than 10 mN m-1, protein not only can penetrate amphiphile monolayer and spread at air/water interface by diffusion, but also can be adsorbed onto or between head of amphiphile molecules by electrostatic interaction. And then with compression the increase of surface pressure leads to approach of protein and amphiphile molecules until phase transition of the monolayer from liquid to solid, between 30 and 40 mN m-1 in the Figure 2, in which a slight increase of the surface pressure is observed with the molecular area decreasing. Since protein molecules which spread at air/water interface are gradually expelled from the interface and pushed under the positively charged head of ODA molecules. As a consequence the surface pressure does not increase significantly. More importantly, in this range of surface pressure the monolayer includes a relative high amount of protein and it's desirable for the formation of Langmuir-Blodgett films and elaboration of biosensors. Since with the further compression, surface pressure above 40 mN m-1 in the figure, there is risk of overlapping of monolayer and even worse completely collapsing of the monolayer.

In the following study we investigated the stabilities of the monolayers in the transition state and tried to obtain the optimal target pressure for deposition of LB films.

The stability of the mixed protein/amphiphile monolayer

The stability of the mixed monolayer is very important because it is related to the transfer efficiency and the quality of deposited LB films. It can be examined by the evolution of surface pressure with time at a fixed molecular area, or the evolution of molecular area ratio with time at a fixed surface pressure, after the monolayer has been compressed.

As has been reported by our group ( 14), the pure amphiphile monolayer is unstable because of its partial solubility in the subphase. However, we can see from Figure 3 monolayers with different surface pressures (30,35,40, 43 mN m-1) and from Figure 4 monolayers with different concentration (4, 6, 10 mg L-1) of the proteins are all very stable. This confirms that the addition of protein in the subphase increases indeed the stability of the monolayer at air/water interface.

In fact, as has been referred by Miller (35) electrostatic interaction between protein and surfactant such as amphiphile induce formation of protein/surfactant complex which is much less soluble than the protein and also have the highest surface activity.

Another phenomenon can be seen from the figures is that the surface area of monolayer decreases quickly at the beginning and then decreases much more slowly with time. This initial loss in area depends on the rate at which the monolayer is compressed and is due to a structural rearrangement occurring within the layer. Meanwhile, the very slight decrease with time is likely result from the feeble solubilization of the mixed monolayer.

Transfer of Langmuir-Blodgett films

The stable mixed protein/amphiphile monolayer has been transferred from air/water interface onto solid substrates of gold under different target pressures (30,35,40, 43 mN m-1).

Since the gold substrates have been functionalised with SAMs of ODT to be hydrophobic, deposition of LB films should occur during the first downstroke of the substrate through air/water interface.

We found for target pressure 35 mN m-1, LB films can be transferred successfully onto gold substrates with good reproducibility and very high transfer ratio as shown in Figure 5, which corresponds to a Y-type deposition. Furthermore, it can be seen from Figure 4 that at target pressure (35 mN m-1) the stabilities of the monolayers seem have nothing to do with the concentration of protein, which is consistent with the fact that at this surface pressure the proteins were thoroughly expelled from the interface and were adsorbed onto the ODA monolayer. As a consequence, it has been chosen as target pressure for the following studies of AFM and EIS.

Scanning electron microscopy (SEM) was used to characterize LB films obtained at different target pressures at a submillimeter scale in order to obtain some general topographic information. As is shown in Figure 6 the structures appear to be heterogeneous with protein aggregates being dispersed at random on the samples and many defects presenting. The important rearrangements probably occur during and/or after the transfer. This kind of structure of LB films has been reported by Sommer et al. (29) and Chevelon et al. (16) with AFM. However, we could not identify the chemical nature of the different structures obtained by SEM.

In addition, by comparing the four images, we found that at higher surface pressure, in the cases of surface pressure 40 and 43 mN m-1, the structures of LB films on the gold substrates seem to be more homogeneous, and protein aggregates approach more and more each other. However, we found that under these high surface pressures, the transfer of LB films were not so effective. We think at the surface pressure 40 and 43 mN m-1, the phase transition of the monolayer from liquid to solid nearly terminates, the molecules are closely packed. In this state, the monolayer is much more sensitive and fragile to any external force, it's reasonable to assume the vertical passage of the solid substrate could damage probably the monolayer, resulting in overlapping or partly collapsing of the monolayer. And consequently the transfer ratios for LB films at these surface pressures are lower. After two layers of mixed OBP-1F/ODA LB films were efficiently transferred onto functionalized gold substrates at 35 mN m-1, an EIS measurement was employed to characterize such structures (gold/thiol/ODA/OBP LB films) before and after exposure to odorant molecules isoamyl acetate, since EIS is an effective method to investigate the interface properties of an electrode.

The magnitude (Z) and the phase angle theta versus frequency (Figure 7a andb, respectively) show the behaviour over the entire frequency range, while the linear Nyquist plot (Figure 7c) more clearly emphasizes the impedance changes at low frequency where the experimental results show best sensitivity to odorant molecules.

An understanding of the physical origin of the observed response can be enhanced using electrical circuit modelling, in which the electrical response of the physical interface is modelled using various discrete electrical circuit elements. However, no equivalent model can be guaranteed to be unique. The model that provides best fit to the data over the entire frequency range in our study is shown in Figure 7d. And the interpretation for each electrical circuit element (Rs, RSAMs, CSAMs, RLB, QLB) is described as follows:

Rs is solution resistance; RSAMs CSAMs RLB QLB At frequencies below A, the total impedance is dominated by the OBP-1F/ODA LB film and the impedance of QLB. At intermediate frequencies (between A and B) the impedance is dominately mainly by the CSAMs, which has no significant sensitivity to binding of odorant molecules. At the highest frequencies (greater than B) the impedance is mainly dependent on the bulk solution resistance, Rs, which is dependent upon the ionic strength of the buffer that is used.

Probably the binding of odorant molecules induced the rearrangement of the mixed OBP-1F/ODA LB film and/or local conformational change of OBP molecules.

The binding of ligand was shown to induce local conformational changes. The arrangement is therefore possible, leading to favourable contact between, a local conformational transition upon binding . It has been proved that OBP-1F undergoes a conformational change upon odorant binding (24). Odorant binding proteins share a conserved folding pattern, an eightstranded -barrel flanked by an -helix at the C-terminal end of the polypeptide chain. The barrel defines a central apolar cavity, called the calyx, whose role would be to bind and transport hydrophobic odorant molecules.

Atomic force microscopy (AFM)

Topographic images of the mixed OBP-1F/ODA LB films remain almost unchanged at the scale of the figures (10nm) after exposure to odorant. However we have identified a strong change in the AFM phase images after exposure to the odorant, see Figure 8. The white spots in topographic images are probably dust or salts remaining after the rinsing process. Indeed, before exposure to the odorant phase is almost uniform on the sample (variations of maximum of ~1º are observed), after exposure to the odorant a big contrast is observed between the aggregates and the rest of the sample, with phase variations of up to ~9º. Since phase atomic force microscopy images, by recording the phase lag existing between the signal that drives the cantilever and the cantilever oscillation output signal, provides information about surface properties such as elasticity, adhesion and friction, we can conclude that a modification in one (or all) of these mechanical properties has occurred on the aggregates. An obvious possibility would be to assume that the aggregates being more rich in OBPs have captured far more odorant molecules than the surrounding regions, thus changing the adhesion force of the tip on these regions. While this is a very reasonable possibility, that would confirm the EIS data, it seems difficult to correlate the observed changes with a possible molecular arrangement of the proteins since a lot of arrangements in this case are possible (16). Direct topographic information on the conformational changes would require almost atomic resolution that can not be achieved under the present experimental conditions.

Conclusion

In this paper, we have carefully studied both odorant binding protein/amphiphile Langmuir and Langmuir-Blodgett films and we have shown that stable mixed Langmuir films can form at air/water interface as well as can be efficiently transferred onto gold supports at target pressure 35 mN m-1 for construction of reproducible and stable mixed LB films. Such kind of thin films exhibits desirable application on elaboration of artificial olfactory system. With EIS and AFM techniques we found that such system can detect corresponding odorant molecules.

(need more results and information from EIS and AFM)In further study we expect to produce sensor array for fabricating electronic nose based on this kind of mixed LB films and to detect a series of airborne odorants not only one.

3. 5 . 3 . 5 . 1 .

 5351 Characterization of Langmuir-Blodgett (LB) films for odorant biosensor Electrochemical impedance spectroscopy (EIS) (Reference 37 38)

Figure 1 .

 1 Figure 1. Adsorption of OBP-1F at the air/water interface as a function of time.

Figure 2 .

 2 Figure 2. Surface pressure-area isotherms for amphiphile ODA with different subphases: water, phosphate buffer solution (10 mM, pH 7.5) without and with protein OBP-1F (4 mg L-1).

Figure 3 .

 3 Figure 3. Evolution of the molecular area for the mixed monolayer at constant surface pressures of 30, 35, 40, 43 mN m-1 (protein concentration 4 mg L-1).

Figure 4 .Figure 5 .

 45 Figure 4. Evolution of the surface pressure for the mixed monolayer at constant molecular area for the different concentrations of protein 4, 6, 10 mg Figure 5. Transfer ratio for 2 mixed proteinOBP-1F/ODA LB films deposition at target pressure 35 mN m-1.

Figure 6 .

 6 Figure 6. SEM images of two layers of mixed OBP-1F/ODA LB films transferred onto gold substrates at (a) 30 mN m-1, (b) 35 mN m-1, (c) 40 mN m-1 and 43 mN m-1.

Figure 7 .

 7 Figure 7. Bode diagrams are shown in (a) the magnitude (Z) versus frequency and (b) the phase angle theta versus frequency, in which the symbols represent experimental values and the straight lines represent fitting data: (□) before exposure to odorant molecules, (○) after exposure to odorant molecules. (c) Nyquist (Z" vs. Z') plots measured for gold/thiol/ODA/OBP LB films before (A) and after (B) exposure to odorant molecules. (d)The equivalent circuit model used to fit the impedance data, the electrical circuit elements were described in the text.

Figure 8 .

 8 Figure 8. AFM Topographic (a) and phase (a') images of a mixed ODA/OBP LB film transferred onto functionalized gold substrate at 35 mN m-1 before exposure to odorant. Scan area 2.1x2.1 m2. Vertical scale (a) 10 nm, (a') 1.3º. AFM Topographic (b) and phase (b') images of a mixed ODA/OBP LB film transferred onto functionalized gold substrate at 35 mN m-1 after exposure to odorant. Scan area 4x4 m2. Vertical scale (b) 10 nm, (b') 9.4º.

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3.

Figure 4 .

 4 Figure 4.

Figure 5 . 1 Figure 6 .

 516 Figure 5.

Figure 7 .Figure 8 .

 78 Figure 7.

Table 1 .

 1

	RS ( cm2)	RSAMS ( cm2)	CSAMS ()	RLB (M cm2) QLB (-1 cm-2)	n
	Before			1.18	1.86	0.93
	odorant
	After			0.025	2.35	0.95
	odorant

Acknowledgements

This work was financially supported by Concerted Action NMAC (N0 0220306) and the SPOT-NOSED Project (IST-2001-38739).


Minyoung Yun
email: minyoung.yun@ensam.eu

Clara Argerich
email: clara.argerich_martin@ensam.eu

Elias Cueto
email: ecueto@unizar.es

Jean Louis Duval
email: jean-louis.duval@esi-group.com

Francisco Chinesta
email: francisco.chinesta@ensam.eu

Nonlinear Regression Operating on Microstructures Described from Topological Data Analysis for the Real-Time Prediction of Effective Properties

Keywords: machine learning, data-driven mechanics, TDA, Code2Vect, nonlinear regression, effective properties, microstructures

come   L'archive ouverte pluridisciplinaire

Introduction

Recently, industry is experiencing a new revolution. In the past, product design, as well as their associated manufacturing processes, were based on the use of nominal models, nominal loadings (in their broadest sense), and a small amount of data for calibrating those models, with the product performance as a design target.

Very recently, predictions enabling real-time decision-making targeting zero defects in processing and zero unexpected faults in operation, were needed everywhere within the Internet of Things (IoT) paradigm, on the work-floor (smart processes), in the city (autonomous systems and smart-city), at the nation level (e.g., smart nation), etc., i.e., anywhere where engineering designs operate.

In those circumstances, the use of traditional simulation-based engineering (SBE) that was the major protagonist of 20th century engineering, is not anymore a valuable option due to three main reasons: (i) models become sometimes crude approximations of the observed reality; (ii) assimilating data enabling the continuous calibration of the models in operation remains difficult to perform under the stringent real-time constraint; and (iii) the real-time simulation of those extremely complex mathematical models needs alternative techniques to those commonly employed in traditional SBE.

It was at the beginning of the XXI century that two new revolutions in the domain of digital engineering emerged.

Model Order Reduction

Advances in applied mathematics, computer science (high-performance computing) and computational mechanics met to give rise to a diversity of Model Order Reduction (MOR) techniques [START_REF] Chinesta | Model Order Reduction Chapter in the Encyclopedia of Computational Mechanics[END_REF]. These techniques do not reduce or modify the model, they simply reduce the complexity of its resolution and thus transform a complex and time-consuming calculation, into a real-time response while maintaining precision. These new techniques have completely altered traditional approaches of simulation, optimization, inverse analysis, control and uncertainty propagation, all them operating under the stringent real-time constraint.

In a few words, when approximating the solution u(x, t) of a given Partial Differential Equation (PDE), the multipurpose finite element method assumes an approximation

u(x, t) = N ∑ i=1 U i (t)N i (x), (1)
where U i represents the value of the unknown field at node i and N i (x) is the associated shape function.

When N (the number of nodes) increases the solution process becomes cumbersome. POD-based model order reduction learns offline the most adequate (in a given sense) reduced approximation basis {φ 1 (x), • • • , φ R (x)}, and project the solution in it

u(x, t) ≈ R ∑ i=1 ξ i (t)φ i (x), (2)
where now, the complexity scales with R instead of N, with R N in general. The so-called Proper Generalized Decomposition (PGD from now on) goes a step forward and assume a general approximation

u(x, t) ≈ M ∑ i=1 T i (t)X i (x), (3)
where now both the space and time functions, X i (x) and T i (t) respectively, are computed during the solution process.

A particularly appealing extension of the just introduced space-time separated representation consists of the space-time-parameter separated representation leading to the a so-called computational vademecum that expresses the solution of a parametrized PDE from [START_REF] Chinesta | Parametric PGD based computational vademecum for efficient design, optimization and control[END_REF][START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations[END_REF]

u(x, t, µ 1 , . . . , µ Q ) ≈ M ∑ i=1 X i (x)T i (t) Q ∏ j=1 M j i (µ j ), (4)
where µ j , j = 1, . . . , Q, represent the model parameters. Once constructed off-line that parametric solution (4), it offers under very stringent real-time constraints-in the order of milliseconds-simulation, optimization, inverse analysis, uncertainty propagation and simulation-based control, to cite a few. Thus, at the beginning of the third millennium a real-time dialogue with physics no longer seemed to be the domain of the impossible. PGD-based techniques have been widely considered for the real-time simulation and decision-making in a variety of problems of industrial relevance. However, prior to use it, one must extract the parameters to be included as extra-coordinates in the problem statement, and then included in the parametric representation of its solution. In the case of morphological and topological descriptions, as considered later in the present work, the extraction of the adequate parametrization represents the most difficult task. Some attempts of combing PGD-based MOR and manifold learning [START_REF] Lee | Nonlinear Dimensionality Reduction[END_REF] were addressed in [START_REF] González | Thermodynamically consistent data-driven computational mechanics[END_REF][START_REF] González | Computational patient avatars for surgery planning[END_REF][START_REF] Lopez | A manifold learning approach for integrated computational materials engineering[END_REF][START_REF] Gonzalez | kPCA-based Parametric Solutions within the PGD Framework[END_REF].

Engineered Artificial Intelligence

Data bursts within engineering disciplines. For years, data was used in other areas where models were less developed or remained quite inaccurate. Data collected massively was successfully classified, cured, distilled, ... using artificial intelligence (AI) techniques. Thus, correlations between data ca be removed, proving that a certain simplicity remains hidden behind a rather apparent complexity. Data-driven modeling developed exponentially and advanced artificial intelligence techniques were developed, covering six major domains: (i) Multidimensional data visualization [START_REF] Maaten | Visualizing data using t-SNE[END_REF]; (ii) Data classification and clustering [START_REF] Cristianini | An Introduction to Support Vector Machines: And other Kernel-Based Learning Methods[END_REF][START_REF] Forgy | Cluster analysis of multivariate data: Efficiency versus interpretability of classification[END_REF]; (iii) Learning models from input/output pairs of data, with adequate techniques enabling real-time learning and able to operate in the low-data limit (e.g., sPGD [START_REF] Ibanez | A multi-dimensional data-driven sparse identification technique: The sparse Proper Generalized Decomposition[END_REF], Code2Vect [START_REF] Argerich | Code2vect: An efficient heterogenous data classifier and nonlinear regression technique[END_REF], iDMD [START_REF] Reille | Incremental dynamic mode decomposition: A reduced-model learner operating at the low-data limit[END_REF][START_REF] Schmid | Dynamic mode decomposition of numerical and experimental data[END_REF][START_REF] Williams | A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition[END_REF], NN [START_REF] Goodfellow | Deep Learning[END_REF], ThemodynML [START_REF] González | Thermodynamically consistent data-driven computational mechanics[END_REF][START_REF] Moya | Learning slosh dynamics by means of data[END_REF], ...); (iv) Knowledge extraction in order to identifying combined parameters and model richness/complexity, discovering hidden parameters, discarding useless parameters or even to extract governing equations; (v) Explaining for certifying; and (vi) Hybridizing physics and data for defining advanced and powerful Dynamic Data-Driven Application Systems, DDDAS [START_REF] Chinesta | Virtual, Digital and Hybrid Twins: A New Paradigm in Data-Based Engineering and Engineered Data[END_REF].

However, these data-driven models, when used in engineering and industry, were quickly confronted with three major and recurrent difficulties: (i) the need for a huge amount of data to make predictions accurate and reliable, knowing that data is synonymous with cost (acquisition and processing costs); (ii) the difficulty of explaining and interpreting predictions obtained by artificial intelligence; and (iii) related to the the latter, the difficulty of certifying engineering products.

Towards Real-Time Decision Making

In summary, on one side models based on physics can be solved fast but, as discussed, in many engineering areas they remain poor approximations of the real components and systems. On the other hand, when approaching the problem from the data perspective, impressive amounts of data are sometimes needed (with the associated cost and technological difficulty of collecting them), to be processed in real time and then explained in order to certify both the designs and the decisions.

A possible winning option consists of merging both concepts and methodologies. The hybrid paradigm was born [START_REF] Chinesta | Virtual, Digital and Hybrid Twins: A New Paradigm in Data-Based Engineering and Engineered Data[END_REF][START_REF] González | Learning corrections for hyperelastic models from data[END_REF], associating in it two type of models: the first based on physics; the second being a completely new type of model, more pragmatic and phenomenological, based on data.

Real-time decision making in engineering design, manufacturing and predictive and operational maintenance, needs the evaluation of quantities of interest in almost real-time. The present work aims at proposing a technique able to determine under the stringent real-time constraints, effective properties of a complex microstructure by assimilating an image of it.

To conciliate accuracy and real-time constraints, the hybrid paradigm is retained: (i) the prediction engine will be trained offline from data coming form physics; then (ii) a non-linear regression, acting on some topological descriptors extracted from those images, will ensure a real-time evaluation of the effective properties (in the present case the homogenized thermal conductivity).

As previously discussed, dealing with shapes and topologies, the parametric description requires performant techniques able to express them in a compact and concise way. In this paper, we propose using Topological Data Analysis, TDA [START_REF] Wasserman | Topological data analysis[END_REF], for representing these rich topologies and morphologies, and then using the associated topological descriptors for generating accurate supervised classification and nonlinear regressions, enabling an almost real-time evaluation of the quantities of interest.

In the next section we will present the main methodologies used in the present study, that will be considered later for the training and then for the real-time evaluation of effective properties (homogenized thermal conductivity) of rich microstructures.

Methods

This section revisits the main methodologies that will be considered later for real-time classification and prediction of the effective thermal properties from collected images. For that purpose we will consider a rich enough training stage that consists of generating several microstructures whose effective thermal conductivity will be evaluated by using a standard linear homogenization technique, revisited in Section 2.1.

In order to associate the resulting homogenized conductivity tensor to each microstructure, the last must be described in a compact and concise way. For that purpose Topological Data Analysis and Principal Component Analysis (PCA) will be employed. Both are revisited in Sections 2.2 and 2.3, respectively.

The last step aims at performing a nonlinear regression to link the parameters extracted by the TDA to the thermal conductivity. The technique retained in our study is the so-called Code2Vect nonlinear regression, revisited in Section 2.4.

Linear Homogenization Procedure

Due to the microscopic nature of heterogeneity, a procedure is required for extracting the effective thermal conductivity. In what follows we proceed in the linear case, as was also the case in [START_REF] Lamari | Routes for Efficient Computational Homogenization of Non-Linear Materials Using the Proper Generalized Decomposition[END_REF], in a representative volume element Ω with a microstructure perfectly defined at that scale. The microscopic conductivity k(x) is known at every point x ∈ Ω.

The macroscopic temperature gradient G is defined from the space average

G = g(x) ≡ 1 |Ω| Ω g(x) dx, (5)
where g(x) represents the microscopic temperature gradient, i.e., g(x) = ∇T(x).

We define the localization tensor L(x) such that

g(x) = L(x) G. (6)
The microscopic heat flux q(x) follows the Fourier law

q(x) = -k(x) g(x), (7)
and its macroscopic counterpart Q reads

Q = q(x) = -k(x) g(x) = -k(x) L(x) G, (8)
from which the homogenized thermal conductivity reads

K = -k(x) L(x) . (9)
Thus, the calculation of the homogenized thermal conductivity tensor only needs the computation of the tensor L(x). The present work considers the simplest procedure that in the 2D case consists of solving two steady state thermal problems in

Ω ∇ • k(x) ∇T 1 (x) = 0 T 1 (x ∈ ∂Ω) = x, , (10)
and

∇ • k(x) ∇T 2 (x) = 0 T 2 (x ∈ ∂Ω) = y , (11)
whose solutions verify by construction

G 1 = ∇T 1 (x) T = (1, 0) G 2 = ∇T 2 (x) T = (0, 1) , ( 12
)
and whose gradients define the localization tensor columns

L(x) = ∇T 1 (x) ∇T 2 (x) , (13)
that allows calculating the effective thermal conductivity.

Remark 1. In the present work, since we are only interested in the effective thermal conduction along the y-direction, a single problem, problem [START_REF] Forgy | Cluster analysis of multivariate data: Efficiency versus interpretability of classification[END_REF], suffices for calculating the only component of interest, component K 22 .

Topological Data Analysis

Topological data analysis, TDA [START_REF] Wasserman | Topological data analysis[END_REF], is one of the most promising techniques in high-dimensional data analysis. In essence, TDA is a powerful tool to find the topology of data: if there are clusters, a manifold structure or even noise that is not relevant for the analysis.

For an intuitive description of the method consider the set of points depicted in Figure 1. In general, these points will live in high dimensional spaces, such that their intrinsic topology will not be visible at first glance. We then equip the set with a distance parameter r. By making r grow, different k-simplexes will appear. Remember that a 0-simplex is a point, a 1-simplex is an edge, a 2-simplex is a triangle, on so on. As r grows, holes appear (as the one defined by the edges between points A, B, C and D in Figure 1, for instance when r = d), and disappear for higher values of r (when r = √ 2d, the initial hole is covered by triangles ABC and ACD). Which is important in this discussion is that the overall structure of data is the one that persists for longer r values. Holes defined by noisy data are rapidly eliminated from the simple complex.

The value of r at which a hole appears, and then the one at which it disappears, defines a bar joining both, which characterizes the hole persistence. When collecting all the bars associated with all the holes appearing and then disappearing when r grows, the so-called persistence barcode results, the last representing compactly a given morphology.

An alternative consists of using a 2D representation, the so-called persistence diagram (PD), reporting in the x 1 -axis the value of r at which a hole appears, and on the x 2 -axis the value at which it disappears. Obviously, with the hole birth preceding its death, all the point are place on the upper domain defined by the bisector x 2 = x 1 , and any point (x 1 , x 2 ) remaining close to that bisector represents noise, a small scale, with the associated hole death following immediately its birth. Points far from the bisector represent the topology that persists.

The persistence barcode and the persistence diagram are two representations with a high physical content; however both representations can not be used for comparison purposes, because they are defined in a non-metric space where the calculation of distances for concluding on proximity has not sense.

To move to a more appropriate space making possible the calculation of distances, we first transform the persistence diagram according to (x 1 , x 2 ) → (y 1 = x 1 , y 2 = x 2x 1 ) and then apply on the last a convolution (usually with a Gaussian kernel) leading to the so-called persistence image (PI) y, the last defined in a vector space, y ∈ R D , that allows applying most of AI algorithms [START_REF] Adams | Persistence images: A stable vector representation of persistent homology[END_REF].

Principal Component Analysis

TDA is able to analyze a complex microstructure through its associated image, and to extract its relevant topological features in form of a persistence image, that can be viewed a matrix. However, using these matrix components is not the most compact and concise way of representing the microstructure, because it contains too many components that makes difficult using it for constructing regressions. Thus, in practice, a linear dimensionality reduction such as principal component analysis (PCA) can be applied for extracting the most representative modes of the persistence images and then to represent in a compact and concise way the microstructures by using the weight associated with the most important modes extracted.

Let us consider a vector y ∈ R D containing the different components of a persistence image. When considering a set of P microstructures, the associated PIs lead to y i , i = 1, . . . , P. If they are somehow correlated, there will be a linear transformation W defining the vector ξ ∈ R d , with d < D, which contains the still unknown latent variables, such that [START_REF] Lee | Nonlinear Dimensionality Reduction[END_REF]

y = Wξ. (14)
The transformation matrix W, D × d, satisfies the orthogonality condition W T W = I d , where I d represents the d × d identity matrix.

PCA proceeds by guaranteeing maximal preserved variance and de-correlation in the latent variable set ξ. Thus, the covariance matrix of ξ,

C ξξ = E{ΞΞ T }, (15)
will be diagonal. PCA will then extract the d uncorrelated latent variables from

C yy = E{YY T } = E{WΞΞ T W T } = WE{ΞΞ T }W T = WC ξξ W T , (16)
that pre-and post-multiplying by W T and W, respectively, reads

C ξξ = W T C yy W. ( 17
)
By factorizing the covariance matrix C yy , applying the singular value decomposition, SVD,

C yy = VΛV T , ( 18
)
and taking into account Equation [START_REF] Goodfellow | Deep Learning[END_REF], it results

C ξξ = W T VΛV T W, (19)
that holds when the d columns of W are taken collinear with d columns of V, i.e.,

W = VI D×d . ( 20
)

Code2Vect

Code2Vect [START_REF] Argerich | Code2vect: An efficient heterogenous data classifier and nonlinear regression technique[END_REF] maps data into a vector space where the distance between points is proportional to the difference of the QoI associated with those points, as sketched in Figure 2. We assume the available data consisting of P d-dimensional arrays, ξ i ∈ R d , with a QoI O i associated with each datum. The images, z i ∈ R q (q = 2 in our numerical implementation for the sake of visualization clarity), results from

z i = Wξ i , i = 1, . . . , P, (21)
that preserves the quantity of interest associated with is origin point ξ i , denoted by O i .

In order to place points such that distances scales with their QoI differences we enforce

(W(ξ i -ξ j )) • W(ξ i -ξ j )) = z i -z j 2 = |O i -O j |. (22)
Thus, there are P 2 2 -P relations to determine the q × d + P × q unknowns. Linear mappings are limited and do not allow proceeding in nonlinear settings. Thus, a better choice consists of a nonlinear mapping W(ξ), expressible as a general polynomial form.

Results

Model Training

Several microstructures, based on a population of holes (from now on called pores) with different sizes, shapes, location, and number of pores, distributed in the 2D square domain Ω, are created. Four of these microstructures are shown in Figure 3. They are equipped with a mesh on which finite element calculations will be done for computing the reference effective (homogenized) thermal conductivity, in particular the component K 22 of the homogenized conductivity tensor.

These meshes also serve to apply the TDA in order to obtain the persistence diagram (PD) and its associated persistence image (PI). As previously indicated, the last consists of a convolution applied on the former. Each persistence image defines a 20 × 20 matrix, or its vector counterpart y i ∈ R 400 .

Thus, TDA is able to analyze a complex microstructure through its image, and extract its relevant topological features in form of a persistence image, that can be viewed as a matrix. However, this matrix still contains too much information (its number of components, here 20 × 20) to perform classification or regression when not too much data is available (scarce-data limit). Obviously, large amounts of synthetic data can be produced by solving numerically thousands or even millions of thermal problems. However, in engineering cheap solutions are usually preferred, and in particular smart-data is preferred to its big counterpart. Efficiency seems a better option that brute force, and for this reason, here we prefer keeping the amount of data as reduced as possible, and compensate its absence by enhancing the amount of information that data contains. Thus, persistence images are still not the most compact way of representing the topological and morphological features of the analyzed microstructures. For improving the representation we apply a linear dimensionality reduction, the principal component analysis, for extracting the most representative modes of the persistence images. Thus, the weights of those PCA modes will constitute the compact and concise way to represent those microstructures.

From a practical viewpoint PCA allowed reducing from 400 = (20 × 20) the dimension of PI resulting from TDA, to 3 dimensions. Thus, each analyzed microstructure is concisely represented by 3 coordinates (the weights of the first three most relevant PCA modes) and each one has attached a QoI, the effective thermal conductivity K 22 obtained from a finite element simulation following the rationale described in Section 2.1. Now, the nonlinear regression relating the output, the QoI (the effective thermal conductivity in our case), with the parameters describing the microstructure, the three PCA weights, is performed by applying the Code2Vect nonlinear regression, summarized in Section 2.4.

As soon as the regression is constructed at the present training stage, it could be used online for predicting the conductivity of new microstructures.

Inferring Effective Properties

We prepared 5 samples, four of them were used in the training stage, represented in Figure 3, in which the pores volume fraction was kept constant (φ = 0.5) and the spatial distribution almost uniform.

The constructed nonlinear regression (based on the use of Code2Vect) described in the previous section, is now applied to the sample shown in Figure 4 where while keeping the same almost uniform pore distribution and the same volume fraction, hexagons and heptagons were randomly mixed. In this same figure, the solution of the thermal problem at the microscopic scale for obtaining the effective thermal conductivity that will serve as reference value, is also included. Finally, it also shows both the PD and the PI.

The PI, y ∈ R D , is then projected into the three retained orthonormal PCA modes to give the thee weights that constitute the data ξ ∈ R 3 (d = 3) to be processed by the nonlinear regression (based on the Code2Vect) that produces vector z ∈ R 2 (we enforce a 2D representation, q = 2, for the sake of clarity in the data visualization) z = W(ξ) ξ, [START_REF] Adams | Persistence images: A stable vector representation of persistent homology[END_REF] and then identify the set S(z) of data z i closest to z, from which the QoI, the effective thermal conductivity, is interpolated

O = ∑ i∈S (z) F (z, z i ) O i , (24)
with in the present case O ≡ K 22 and with radial bases as interpolation functions F (z, z i ).

Figure 5 places z with respect to its neighbors, where color scales with the target quantity, that is, with K 22 . The inferred value of the effective thermal conductivity K 22 using Equation (24) for the micorstructure depcited in Figure 4

Microstructures with Varying Shapes and Size Distribution

These first preliminary successful results were pushed forward by considering quite more complex microstructures. Thus, a total of 35 samples were generated, while varying other parameters, in particular pore size (following uniform and Gamma distributions) and pore shape (circular or 5 to 8 side polygons, randomly chosen). The volume fraction was kept constant (φ = 0.5). 34 samples were used in the training, keeping one, the one shown in Figure 6, for inferring the thermal conductivity and concluding on the ability of the proposed technique to infer accurately it. To check the prediction improvement with the sampling richness, the effective thermal conductivity in the microstructure shown in Figure 6 while considering different samplings in the training stage, from 13 to 35 microstructures, with the relative errors reported in Table 1.

Conclusions

The present study proves that effective properties can be associated with microstructures with complex morphological and topological features. For this purpose, those features are extracted by using TDA, post-compressed by using linear dimensionality reduction (PCA) which output represents the parameters employed by the nonlinear Code2Vect regression that finally assign a effective property (here the effective thermal conductivity) to a given microstructure.

The procedure demonstrated its robustness and performance in the low-data limit, as well as its capacity to provide better predictions when considering larger training sets. It successfully combines physics-based data for learning purposes, with almost real-time inference based on the topological analysis of images.

Figure 1 .

 1 Figure 1. Illustrating TDA: (left) For r < d the four points (A, B, C and D) remain disconnected; (center) At r = d the hole ABCD appear from the four edges AB, BC, CD and DA; (right) The just created hole persist until r = √ 2d, value at which A connects with C and the two resulting triangles ABC and ACD cover the initial hole that disappears consequently.

Figure 2 .

 2 Figure 2. Input space ξ (left) and target vector space z (right).

Figure 3 .

 3 Figure 3. (a) Histogram of pores radius; (b) Pore shapes: Circle, Octagon, Heptagon and Hexagon.

  results K 22 (z) = 73.4 W/mK, very close to the reference value computed numerically from the temperature distribution shown also in Figure 4, of K 22,REF = 74 W/mK.

Figure 4 .

 4 Figure 4. (a.1) Histogram of the pores radius; (a.2) considered microstructure; (a.3) temperature field used for computing the effective thermal conductivity that will serve as reference for evaluating the regression performance; (a.4) persistence diagram; and (a.5) persistence image.

Figure 5 .

 5 Figure 5. Interpolation space z with color scaling with the values of the effective thermal conductivity K 22 .

Figure 7

 7 places the considered microstructure in the z-space where the thermal conductivity is interpolated, to infer the value of K 22 = 81 W/mK, for a reference value of K 22,REF = 78 W/mK.

Figure 6 .

 6 Figure 6. (a.1) Histogram of pores radius; (a.2) testing microstructure; (a.3) temperature field used for calculating the reference effective thermal conductivity; (a.4) persistence diagram; and (a.5) persistence image.

Figure 7 .

 7 Figure 7. Interpolation space z with color scaling with the values of the effective thermal conductivity K 22 .

Table 1 .

 1 Relative error in the effective conductivity prediction depending on the number of samples considered in the regression (training stage).

	Number of Samples Relative Error
	13	0.076
	16	0.056
	19	0.046
	35	0.037

Funding: This research received no external funding.

Acknowledgments:

The authors acknowledge the ESI Chairs at Arts et Métiers Institute of Technology and the University of Zaragoza, as well as the French ANR through the DataBEST project.

Conflicts of Interest:

The authors declare no conflict of interest.


Hassan Bazzi

Nicolas Baghdadi

Dino Ienco

Mehrez Zribi

Hatem Belhouchette

Potential of Sentinel-1 and Sentinel-2 data for Mapping Irrigated areas at plot scale

Irrigation plays a significant role in agricultural production in order to meet the global food requirement under changing climatic conditions. To fulfil the high demand of food with an ever-increasing global population, better planning of irrigation is required. Therefore, more focus is being set on the assessment of irrigation performance for improving water management in order to achieve higher water productivity and increase the agricultural water sustainability.

In the context of mapping irrigated areas, we propose an innovative approach to map irrigated areas using Sentinel-1 (S1) SAR (Synthetic Aperture Radar) and Sentinel-2 (S2) optical time series. Our proposed approach is based on the use of S1 and/or S2 time series combined with statistical and mathematical functions such as principal component analysis (PCA) and wavelet transformation (WT). The proposed approach was tested over the Catalonia region, Spain with a dataset containing 126 000 irrigated and 67 000 non-irrigated plots. The novelty of our study resides in eliminating the ambiguity between irrigation and rainfall by comparing between the SAR backscattering signal of each plot and that of the corresponding grid (10 km × 10 km).The potential of S2 images to classify irrigated areas by means of NDVI time series was also investigated in this study. Random forest (RF) and convolutional neural network (CNN) approaches were used to build up classification models using the PCA or WT parameters in three different scenarios: The first using only S1 data, the second using only S2 data, and the third using both S1 and S2 data.

The RF classifiers built using the PCA or WT on S1 time series perform well in mapping irrigated areas with an accuracy of 90.7% and 89.1% respectively. However, the CNN classification on the S1 temporal series produces a significant overall accuracy of 94.1%. Finally, the combined use of the SAR and optical data enhanced the accuracy of the RF classification but did not significantly change the overall accuracy of the CNN model.


Mohammed Merheb

Chadi Abdallah

Roger Moussa

Nicolas Baghdadi

Hydrological response characteristics and classification of Lebanese catchments

Hydrological response characteristics and classification of Lebanese catchments

Mohammed Merheb 1* , Chadi Abdallah 2 , Roger Moussa 3 , Nicolas Baghdadi 4

(1) Lebanese University, Faculty of Sciences 3, Tripoli, Lebanon.

(2) CNRS-L, Beirut, Lebanon.

(3) LISAH, Univ. Montpellier, INRA, IRD, SupAgro, Montpellier, France. when exist focus mostly on groundwater hydrology and water quality. Nevertheless. The Lebanese literature suggest that even though rainfall and physical characteristics are highly variable across the country, a certain regional pattern do exist. This is corroborated by the analysis of the Lebanese catchments temporal data. Indeed, mean annual runoff and runoff ratio across the country show regional tendencies with the highest values in the central part of mount Lebanon. Moreover, intra-annual variation of runoff also shows geographical pattern and can be divided into three main regimes: rainfall dominated, snow dominated and a mixed snow-rainfall regime. This is also true at the event scale, where unit maximum daily discharge and runoff depth and ratio, all show regional pattern with the highest values in the central part of Lebanon which receive the highest precipitation amount. In addition, the Lebanese catchments, and despite a high reference evapotranspiration, exhibit hydrological response characteristics that are similar to catchments in the wetter north-western Mediterranean than other catchments in the Eastern Mediterranean. Finally, the classification of the Lebanese catchments yielded five groups of catchments. This grouping is well in range with the regional pattern discussed in the literature and the data analysis of the Lebanese catchments.

*

  Corresponding author at: Mohammad Merheb, Lebanese University, Tripoli, Lebanon E-mail address: mohammadmerheb1987@gmail.com The Mediterranean region is very vulnerable to both water shortage and flooding. Hence, the study of the hydrological response characteristics of Mediterranean catchments is of a great importance especially that the region is one of the most densely populated on the planet. Numerous Mediterranean catchment hydrological studies exist in the literature, however, mostly concentrated in the European part of the Mediterranean, especially the north-western part of the basin. The eastern Mediterranean have fewer studies and this is indeed the case of Lebanon, a small mountainous country on the eastern shore of the Mediterranean. The objective of this work is to present a state of the art assessment of the hydrology of Lebanon in the largest Mediterranean context and to present a first classification attempt for the Lebanese catchments. Twenty-eight Lebanese basins where data are available are studied. Spatial data consist of Digital Elevation Model, land cover map, soil and geological maps of the country. Temporal hydrometeorological data exist at a monthly and daily timescale for the period 2001-2011. The methodology is threefold: (1) a synthesis of the previous hydrological studies in Lebanon, (2) an analysis of the hydrological response characteristics of the Lebanese catchments and a comparison with other Mediterranean catchments, and (3) the classification of the Lebanese catchments according to their physical and hydrological response characteristics. After extracting physical descriptors and runoff signatures for the available data, using Agglomerative Hierarchical Clustering. The results of the synthesis of previous studies in Lebanon assert that indeed hydrological studies in Lebanon are scarce and