domain: spi.gproc chim sde

text:
Sher Ahmad

Wassim Sebai

Marie-Pierre Belleville

Nicolas Brun

Anne Galarneau

José Sanchez-Marcano
email: jose.sanchez-marcano@umontpellier.fr

Experimental and Modeling of Tetracycline degradation in water in a Flow-Through Enzymatic monolithic reactor

Keywords: Water treatment, Enzymatic silica monoliths, pharmaceuticals degradation, tetracycline degradation, Flow-Through-Reactor, Modelling, Scale-up

In this work, the laccase from Trametes versicolor was immobilized in highly porous silica monoliths (0.6 cm diameter, 0.5 cm length). These monoliths feature a unique homogeneous network of interconnected macropores (20 μm) with mesopores (20 nm) in the skeleton and a high specific surface area (330 m 2 /g). The enzymatic monoliths were applied to degrade tetracycline (TC) in model aqueous solutions (20 ppm). For this purpose, a tubular Flow-Through-Reactor (FTR) configuration with recycling was built. The TC degradation was improved with oxygen saturation, presence of degradation products and recirculation rate.

The TC depletion reachs 50% in the FTR and 90% in a stirred tank reactor (CSTR) using crushed monoliths.

These results indicate the importance of maintaining a high co-substrate concentration near active sites. A model coupling mass transfers with a Michaelis-Menten kinetics was applied to simulate the TC degradation in real wastewaters at actual TC concentration (2.8 10 -4 ppm). Simulation results show that industrial scale FTR reactor should be suitable to degrade 90% of TC in 5 h at a flow rate of 1 mL/min in a single passage flow configuration. Nevertheless, the process could certainly be further optimized in terms of laccase activity, oxygen supply near active sites and contact time.

Introduction

Pharmaceutical products (PPs) are found more and more frequently in the environment due to an increase in their consumption, because humains have a better access to medicines in conjunction with the aging of the population. Indeed, they can be found at very low concentrations (µg L -1 -mg L -1 ) in wastewaters [START_REF] Halling-Sørensen | Occurrence, fate and effects of pharmaceutical substances in the environment-A review[END_REF], rivers [START_REF] Burns | Temporal and spatial variation in pharmaceutical concentrations in an urban river system[END_REF], sediments [START_REF] Kerrigan | Sedimentary record of antibiotic accumulation in Minnesota Lakes[END_REF] and sea [START_REF] Björlenius | Pharmaceutical residues are widespread in Baltic Sea coastal and offshore waters -Screening for pharmaceuticals and modelling of environmental concentrations of carbamazepine[END_REF]. Moreover, traditional wastewater treatment plants (WWTP) are inefficient to completely remove PPs, [START_REF] Alvarino | Trends in organic micropollutants removal in secondary treatment of sewage[END_REF][START_REF] Tallarek | Occurrence and removal efficiency of pharmaceuticals in an urban wastewater treatment plant: Mass balance, fate and consumption assessment[END_REF][START_REF] Verlicchi | Occurrence of pharmaceutical compounds in urban wastewater: Removal, mass load and environmental risk after a secondary treatment-A review[END_REF]. Therefore, PPs remain in WWTP effluents and as a consequence they can be transferred to underground or surface waters, which are among the main sources of drinking water (Bruce et al., 2010;de Jongh et al., 2012). Several tertiary treatments have been proposed to improve the removal of PPs from wastewaters, they include advanced oxidation treatments [START_REF] Kanakaraju | Advanced oxidation process-mediated removal of pharmaceuticals from water: A review[END_REF][START_REF] Kıdak | Medium-high frequency ultrasound and ozone based advanced oxidation for amoxicillin removal in water[END_REF], physical adsorption [START_REF] Rajapaksha | 9 -Sorptive removal of pharmaceutical and personal care products from water and wastewater[END_REF][START_REF] Rocha | Recent advances on the development and application of magnetic activated carbon and char for the removal of pharmaceutical compounds from waters: A review[END_REF] or even enzymatic degradation.

Enzymatic degradation of PPs can be an alternative option among the other tertiary treatments named previously, since enzymes mediate biochemical reactions at a rapid rate under mild operating conditions (pH, temperature, solvents, and ionic strength). In particular, oxidoreductase enzymes such as laccases, tyrosinases and peroxidases have the ability to oxidize large variety of PPs like phenols, drugs, and hormones (Singh Arora and Kumar Sharma, 2010; [START_REF] Demarche | Harnessing the power of enzymes for environmental stewardship[END_REF][START_REF] De Cazes | Design and optimization of an enzymatic membrane reactor for tetracycline degradation[END_REF]. Nevertheless, enzymatic process can be expensive and unsustainable if enzymes are not recycled within the system.

Immobilization generally increases the stability of enzymes under reaction conditions while allowing their reuse and then reducing costs. [START_REF] Zhang | Enhanced Activity of Immobilized or Chemically Modified Enzymes[END_REF][START_REF] Ji | Direct immobilization of laccase on titania nanoparticles from crude enzyme extracts of P. ostreatus culture for micro-pollutant degradation[END_REF]. Several immobilization techniques like adsorption, entrapment and encapsulation have been applied for enzyme immobilization on solid supports however it is observed that covalent immobilization enhances enzymes stability and long-term process sustainability [START_REF] Zdarta | Developments in support materials for immobilization of oxidoreductases: A comprehensive review[END_REF]. Along with nature of enzymes and immobilization techniques, choosing suitable solid support is also very essential in enzymatic process. Inorganic support materials like silica, zirconia, active carbons have high mechanical strength and temperature stability therefore are explored for immobilization of enzymes [START_REF] Sadeghzadeh | Removal of bisphenol A in aqueous solution using magnetic cross-linked laccase aggregates from Trametes hirsuta[END_REF][START_REF] Bebić | Immobilization of laccase from Myceliophthora thermophila on functionalized silica nanoparticles: Optimization and application in lindane degradation[END_REF][START_REF] Zdarta | Mesostructured cellular foam silica materials for laccase immobilization and tetracycline removal: A comprehensive study[END_REF].

The application of enzymatic reactors for PPs degradation has been explored with different configurations like packed bed reactors [START_REF] Nguyen | Continuous adsorption and biotransformation of micropollutants by granular activated carbon-bound laccase in a packed-bed enzyme reactor[END_REF][START_REF] Bilal | Lignin peroxidase immobilization on Ca-alginate beads and its dye degradation performance in a packed bed reactor system[END_REF], fluidized bed reactors [START_REF] Lloret | Continuous operation of a fluidized bed reactor for the removal of estrogens by immobilized laccase on Eupergit supports[END_REF][START_REF] Piao | High efficiency biotransformation of bisphenol A in a fluidized bed reactor using stabilized laccase in porous silica[END_REF] as well as enzymatic membrane reactors (EMR) [START_REF] De Cazes | Design and optimization of an enzymatic membrane reactor for tetracycline degradation[END_REF][START_REF] Barrios-Estrada | Potentialities of active membranes with immobilized laccase for Bisphenol A degradation[END_REF]. EMRs have been widely studied for PPs degradation because they combine two functions in a single unit: filtration and enzymatic reaction (Sanchez Marcano and Tostsis, 2002;[START_REF] Parra Guardado | Effect of redox mediators in pharmaceuticals degradation by laccase: A comparative study[END_REF]. Nevertheless, EMRs can also present some drawbacks like membrane clogging or low reactivity because the amount of biocatalyst grafted on the separative layer of membranes is relatively low. [START_REF] Ji | Biocatalytic degradation of carbamazepine with immobilized laccase-mediator membrane hybrid reactor[END_REF] observed less than 10 % of carbamazepine degradation with laccase from Trametes versicolor in an EMR without using mediators. De [START_REF] De Cazes | Design and optimization of an enzymatic membrane reactor for tetracycline degradation[END_REF] found only 56 % of tetracycline (TC) degradation in 24 h with laccase from Trametes versicolor immobilized on ceramic membranes. Similarly, [START_REF] Barrios-Estrada | Potentialities of active membranes with immobilized laccase for Bisphenol A degradation[END_REF] found that only 33% of bisphenol-A was degraded in 24 h with an immobilized laccase on the same type of membranes. To overcome some of these disadvantages, meso-/macroporous monoliths have been recently applied for enzyme immobilization, the objective is to provide a big surface area to immobilize a large amount of enzymes, together with a macroporosity which allows low pressure drop while avoiding clogging [START_REF] Ahmad | Enzymatic monolithic reactors for micropollutants degradation[END_REF][START_REF] Biggelaar | Continuous Flow Mode Biocatalytic Transamination Using Macrocellular Silica Monoliths: Optimizing Support Functionalisation and Enzyme Grafting[END_REF][START_REF] Sebai | Biocatalytic Elimination of Pharmaceutics Found in Water With Hierarchical Silica Monoliths in Continuous Flow[END_REF]. Moreover, as far as substrates are forced to flow through the monolith porosity the probability of contact with the biocatalyst is enhanced while allowing a precise control of the contact time [START_REF] Parra Guardado | Effect of redox mediators in pharmaceuticals degradation by laccase: A comparative study[END_REF]. This configuration is called Flow-Trough-Reactor (FTR). Silica monoliths prepared by emulsion templating and presenting a large distribution of macropores (50 nm to 6 μm) have been recently used as supports in biocatalysis (Biggelaar et al., 2017[START_REF] Biggelaar | Continuous Flow Mode Biocatalytic Transamination Using Macrocellular Silica Monoliths: Optimizing Support Functionalisation and Enzyme Grafting[END_REF]. Another type of silica monoliths have been prepared by spinodal decomposition. They featured a narrow distribution of macropores and presented the advantage of an independent control of macropores (in the range 1 to 50 µm) and mesopores diameters (in the range 4 to 30 nm, with a corresponding to surface area of 800 to 200 m 2 g -1 , respectively) [START_REF] Fajula | Combining Phase Separation with Pseudomorphic Transformation for the Control of the Pore Architecture of Functional Materials: A Review[END_REF]Galarneau et al., 2016bGalarneau et al., , 2016a)).

There are few reported models of mass transport through monoliths. They consider the macroscopic structure and developed approaches for simulation of velocity fields, diffusion, and dispersion of chemical species within the porous structure [START_REF] Jungreuthmayer | The 3D pore structure and fluid dynamics simulation of macroporous monoliths: High permeability due to alternating channel width[END_REF][START_REF] Meyers | Network modeling of the convective flow and diffusion of molecules adsorbing in monoliths and in porous particles packed in a chromatographic column[END_REF][START_REF] Tallarek | Occurrence and removal efficiency of pharmaceuticals in an urban wastewater treatment plant: Mass balance, fate and consumption assessment[END_REF].

These models, which are based on morphology and real structure of porous monoliths, require high computational techniques like image processing techniques as well as relatively large computing times and costs [START_REF] Jungreuthmayer | The 3D pore structure and fluid dynamics simulation of macroporous monoliths: High permeability due to alternating channel width[END_REF]. This assertion is especially true for large-scale geometries required for practical engineering problems. Furthermore, apart from the recent work of [START_REF] Ahmad | Enzymatic monolithic reactors for micropollutants degradation[END_REF], no work has been reported in the literature coupling a reaction kinetics with transport of species within macroporous silica monoliths for scale-up purposes.

The objective of this research work was to develop an original FTR configuration using enzymatic monoliths for PPs degradation and studying by modeling and simulation the possibility of scale-up of the FTR for this application. For this purpose the degradation of tetracycline (TC) was chosen as model molecule of PPs.

Firstly, a lab scale set up was used for stuying experimentally the enzymatic degradation of TC by laccase from Trametes versicolor covalently immobilized on silica monoliths with large homogeneous macropores (~20 µm) to assess low pressure drop and mesopores of 20 nm providing a large surface area (330 m 2 g -1 ) to improve enzymes immobilization. TC degradation tests were carried out in a tubular FTR configuration with recycling for the degradation of TC in aqueous solutions. Oxygen effect on TC degradation as well as the effect of TC degradation products on degradation rate were also studied. Morevoer, a previous reported model [START_REF] Ahmad | Enzymatic monolithic reactors for micropollutants degradation[END_REF] coupling mass transfers with an apparent Michaelis-Menten kinetics, determined under oxygen saturation conditions, was applied to simulate the TC degradation in real wastewaters having a TC concentration six orders of magnitude lower than oxygen concentration at saturation conditions at 25

°C. The developed model was employed to simulate the geometrical scale up of monoliths for the complete TC degradation at usual real wastewater concentrations at different flows. To our knowledge, it is the first time that such a study of TC degradation in a FTR has been carried out considering the influence of oxygen as co-substrate and the degradation products and performing the scale-up for an actual concentration of this antibiotic in wastewaters.

Materials and Methods

Silica monoliths with hierarchical porosity were synthesized and functionalized by amino groups for being used as solid support for covalent immobilization of laccases. The laccase-monoliths were then used for the removal of tetracycline (20 ppm) contained in water in batch and in flow through configuration. conditions using a recycling configuration. The effects of recirculation flow rate, of oxygen and presence of by-products were analyzed. Simulation was performed to evaluate the performance of laccase-monoliths for the removal of tetracycline in concentration (2.8 10 -4 ppm) found in real wastewaters and to determine the size of monoliths necessary to treat efficiently such effluents in flow condition using a single passage configuration.

Materials

Powdered commercial laccase (activity ≥ 0.5 U mg -1 according to the provider), Tetracycline (TC) (≥ 98.0%), Glutaraldehyde (GLU) (25% v/v) and 2,2'-azino-bis (3-ethylbenzothiazoline-6-sulphonic acid) (ABTS) were all purchased from Sigma-Aldrich, Germany.

Silica monoliths with macropores of 20 μm and mesopores of 20 nm diameters (specific surface area of 330 m 2 g -1 ) were prepared by a controlled sol-gel process with tetraethylorthosilicate (TEOS) and polyethylene glycol (PEO) 100 kDa in acidic solution (HNO3) followed by a basic treatment (NH4OH 0.1 M) at 100 °C for 24 h and then calcined at 550 °C for 8 h, according to a method previously reported (Galarneau et al., 2016a). The monoliths were then activated at 250 °C under vacuum for 4 h to remove water prior to be grafted with 3-aminopropylamine triethoxysilane (APTES) in ethanol under reflux (80 °C) overnight with an excess of 10 amino groups per nm 2 . The resulting NH2-monoliths presented a grafting density of 1.5 NH2/nm 2 corresponding to an amount of 0.8 mmol NH2/g and leading to a specific surface area of 197 m 2 g -1 . The NH2-monoliths were then cladded with a Teflon heat shrinkable gain at 180 °C for 2 h connected to stainlesssteel tubing [START_REF] Ahmad | Enzymatic monolithic reactors for micropollutants degradation[END_REF].

2.2

Activation of monoliths and laboratory scale setup for TC degradation in water Cladded NH2silica monoliths were activated with a GLU solution (4% v/v) prepared in citrate phosphate buffer (pH 7, 0.1 M). Then, monoliths were filled with 0.5 mL of laccase solution (5±1 U mL 1 ) prepared by solubilizing the necessary amount of commercial powder of enzyme in the same citrate phosphate buffer. After immobilization, the resulting laccase-monoliths were stored in the buffer solution (pH 7) at 4 °C. The measurement of the enzymatic activity of laccase immobilized in monoliths, was carried out through the oxidation of 1 mM solution of 2,2-azino-bis(3-ethylbenzothiazoline-6 sulfonic acid (ABTS) at pH = 4. For this purpose, the laccase-monoliths were crushed, and experiments were carried out in a stirred tank reactor (25-50 mL) under vigorous magnetic agitation at 200 rpm. The same methodology was applied for the determination of kinetic constants of TC degradation. Immobilization methods and activity determination of crushed laccase-monoliths were described in detail in a previous publication [START_REF] Ahmad | Enzymatic monolithic reactors for micropollutants degradation[END_REF].

A laboratory scale set up for TC degradation (Figure 1) was built by connecting in series three cladded enzymatic monoliths (0.6 cm diameter, 0.5 cm length, ~50 mg, each). A HPLC pump (Gibson model: 321, France) allowed recycling a TC solution in between the monoliths and a reservoir. A pressure transducer (Keller, 0-2 bar) was connected before the inlet of the monoliths to monitor the pressure changes during the process. The temperature of the feed tank was controlled placing the feed tank in a thermostatically controlled water bath. All the TC degradation tests were carried out in closed loop (complete recycling of the TC solution (20 ppm)) because of low TC degradation in a single pass. A TC solution (20 ppm, 30 mL) was flowed through the monoliths at three different flow rates (0.5, 1 and 5 mL min -1 ). In additional experiments, air was bubbled in the feed tank solution at the air flow rate of 30-40 mL min -1 to keep feed TC solution at oxygen saturation (measured with a DO meter VisiFerm RS-485 from Himlton (Switzerland)). Based on the macropore volumes of the monolith (3.41 mL/g), we can estimate the number of times (cycles) the solution passes through the FTR built with three enzymatic monoliths in series (~150 mg) (Table 1): The Reynolds number (Re) [START_REF] Bird | Transport phenomena[END_REF] was calculated from the well known following equation:

𝑅𝑒 = 𝜌𝑢𝑑 𝑚𝑎𝑐𝑟𝑜 𝜇 (1)
with  (0.997047 kg m -3 ) the density of water at 25 °C, μ (0.000891 kg m -1 s -1 ) the dynamic viscosity of water at 25 °C, dmacro (2 10 -5 m) the macropores mean diameter and u (m s -1 ) the velocity of water within macropores, u = V/A with V the flow rate (5 10 -4 to 5 10 -3 L min -1 ) and A the section of the monolith (6 mm diameter).

For comparison batch experiments were carried out with crushed laccase-monoliths (~150 mg) in a stirred tank reactor (CSTR 50 mL) containing 30 mL of TC solution (20 ppm) under magnetic stirring (200 rpm).

In this case also additional experiments were carried out by bubbling air in the solution at the air flow rate of 30-40 mL min -1 to maintain oxygen saturation conditions. Samples of 100 μL were taken from feed tank every 30 min to measure the TC concentration evolution during 24 h by HPLC-MS analysis (Waters 2695 separation module with micromass detector of Wythenshawe, Manchester, UK). Degradation products of TC Control experiments to study the effect of TC self-degradation or adsorption on the evolution of TC concentration were carried out with whole or crushed inactivated enzymatic monoliths. For this purpose, enzymatic monoliths were deactivated by heating in oven at 100 °C for 2 h.

3

Results and discussions

Effect of recirculation rate on TC degradation

Recirculation rate in enzymatic reactor is an important parameter for continuous flow. When the recirculation rate is slow, the reaction can behave a similarly as non-continuous processes [START_REF] Hamam | Structured and specialty lipids in continuous packed column reactors: comparison of production using one and two enzyme beds[END_REF] with a long contact time with the enzymes. When the recirculation rate is fast, more oxygen is is transported close to the enzymes, but the contact time is shorter. Therefore, it was important to determine the optimum flow rate for the TC degradation.

To study the effect of recirculation flow, TC degradation tests were carried out using the set up shown in Figure 1 at different recirculation flow rates from 0.5 to 5 mL min -1 at 25 °C. For each test, fresh laccasemonoliths were used with initial TC solution concentration of 20 ppm and total volume of TC solution of 30 mL. The decrease in relative TC concentration was similar for all flow-rates (Figure 2): 30% of the TC was degraded in the first five hours. For longer time reaction the degradation rate was extremely low (about 10% of additional degradation over the next 20 hours). This effect was observed with all tested recirculation flow rates, which means that there was no effect of flow rate on the TC degradation. This indicates that the transport of substrates inside the reactors containing the enzymes immobilized on silica monoliths is governed by the self-diffusion in the mesopores and not by external mass transfer in the macropores.

The increase of the flow rate did not allow modifying the hydrodynamic flow regime. Indeed, the Reynolds number (Re) calculated from Eq (1) for the recirculation flow rates of 0.5 to 5 mL min -1 was ranged between 0.01 -0.1 and corresponded for all flow rates to a laminar flow regime (limit of the laminar flow regime (Re ˂ 1) [START_REF] Bird | Transport phenomena[END_REF]. It can be concluded that the increase in flow rate was not high enough to further decrease possible external mass transfer resistances (decrease of boundary layers). In previous studies, Galarneau et al., (2016b) studied the dependence of flow rate on external mass transfer layer with silica monoliths for Diels-Alder reaction. They observed that productivity was increased by increasing the flow rate from 0.01 to 0.03 mL min -1 . In another study, [START_REF] Jatoi | Mass Transfer in Hierarchical Silica Monoliths Loaded With Pt in the Continuous-Flow Liquid-Phase Hydrogenation of p-Nitrophenol[END_REF] studied the effect of flow rate on external mass transfer limitations with silica monoliths loaded with Pt in the continuous-flow liquid-phase hydrogenation of p-Nitrophenol. They observed that for the range of flow rate from 1 to 8 mL min -1 there was no effect on mass transfer limitations in macropores and attributed this behavior to the relative "high" flow rate applied. From these results it can be concluded that boundary layer reduction and optimium flow depends on several other factors like reaction rate, substrate concentration, pore size and porosity, which needs to be optimized as well. There are other possible explanations to the low experimental TC degradation (Figure 2). For example, the process can be controlled by internal mass transfer, i. e. diffusion inside the mesopores, which can be considered as stagnant regions in monoliths. In fact, silica monoliths feature skeletons with a thickness in the range 5-15 μm, which contain an interconnected mesoporous structure where the major part of enzyme molecules is immobilized. Therefore, the most part of reactive sites inside the mesoporous structure are only accessible through the diffusion of substrates.

Effect of dissolved oxygen on TC degradation

Laccases, are multicopper oxidases that catalyze the oxidation of different aromatic compounds with molecular oxygen, which is concomitantly reduced to water [START_REF] Nyanhongo | Engineering Strategies for Successful Development of Functional Polymers Using Oxidative Enzymes[END_REF]. In order to study the importance of the amount of oxygen available for on TC degradation, the effect of the addition of air bubbling at a flow rate of 30-40 mL min -1 was first studied in a stirred tank reactor with crushed monoliths as discussed in section 2.2 (Figure 3). The concentration of the dissolved oxygen in the tank was measured during the experiments with and without air-bubbling (Figure 4). From Figure 3 it can be observed that for control tests (with inactivated laccase-monoliths) TC selfdegradation or adsorption (with or without air bubbling) was extremely low (less than 5%). Furthermore, For the crushed active monoliths without air bubbling the TC degradation was 55% in the first 6 h, while with air bubbling, the TC degradation reached more than 90% in the same period. These results show that oxygen seems to be a limiting substrate for TC degradation. The same conclusion was found in literature [START_REF] Ortner | Laccase mediated oxidation of industrial lignins: Is oxygen limiting?[END_REF] for the oxidation of lignins using laccase from Trametes villosa. The authors observed a rapid decrease of oxygen concentration at the beginning of the reaction, which limited the oxidation of phenolic compounds. They concluded that a sufficient oxygen concentration is necessary for an efficient laccase oxidation process. In the present work, when the solution was bubbled with air, the dissolved oxygen (DO) was always at saturation value (~8.3 mg L -1 at 25 °C) (Figure 4), which corresponded to an excess of 5.6 moles of oxygen/moles of TC. Without air bubbling, the DO concentration decreased of 20% in the first 6 h concomitantly with the decrease of TC concentration (55%) to reach a steady-state of DO concentration of 6 mg L -1 . The calculation of the number of moles of TC and oxygen (5.6 10 -6 and 1.35 10 -6 , respectively)

indicates that a final molar ratio O2/TC of 4, is therefore not enough to proceed to further oxidation. An excess of at least 5.6 O2 molecules per TC seems to be necessary for total TC oxidation.

The effect of air bubbling in the FTR was studied at two different flow rates (1 and 5 mL min -1 ) by bubbling air inside the feed tank (30-40 mL min -1 ) (Figure 5). Initial TC concentration for each experiment was 20 ppm, the total volume of solution in the tank was 30 mL and the total activity of the 3 cladded laccasemonoliths was 12 U. For the flow rate of 5 mL min -1 , the degradation of TC reached 50 % with air bubbling, while without air bubbling the TC degradation was only 30%. Similarly, for the flow rate of 1 mL min -1 , the TC degradation was 40 and 30% with and without air bubbling, respectively. With air bubbling, TC degradation was slightly higher in the case of the larger flow rate (5 mL min -1 ) due to the larger amount of oxygen provided to the reactor. Nevertheless, this TC degradation ratio remains low compared to 90% TC degradation reached in a stirred tank reactor with air bubbling (Figure 5). These results suggest the decrease of oxygen concentration near the bio-catalytic sites (which are located everywhere in the surface of monoliths, but mainly inside the mesoporosity), which limits the reaction rate and the conversion in flow. Furthermore, maintaining a high oxygen concentration or even an excess of oxygen in the solution of the reservoir is not enough to enhance the conversion; this emphasizes that the decrease of co-substrate (O2) concentration near catalytic sites certainly controls the process.

Effects of TC degradation products

Many different TC degradation products were identified using the same laccase immobilized in ceramic membranes [START_REF] De Cazes | Design and optimization of an enzymatic membrane reactor for tetracycline degradation[END_REF]. These products present a TC based structure like anhydrotetracycline, 4-epi-anhydrotetracycline, dehydrooxotetracycline and 12-dehydrotetracycline [START_REF] Llorca | Identification of new transformation products during enzymatic treatment of tetracycline and erythromycin antibiotics at laboratory scale by an on-line turbulent flow liquidchromatography coupled to a high resolution mass spectrometer LTQ-Orbitrap[END_REF]. In this study the main molecules identified by LC-HRMS (IDAEA-CSIC laboratory, Spain) during the degradation of TC are presented in Figure 6. One of the major products identified was dehydrooxytetracycline (m/z = 459.1396) obtained after hydroxylation of the tetracycline. Some other products were identified coming from the loss of functional groups (-OH, -NH2, etc.) of the tetracycline molecule. However the gap in the mass balance between initial tetracycline and the amount of identified by-products indicates that the majority of tetracycline was polymerized (as is usually the case for phenolic compounds), or completely oxydized into CO2 or transformed in other by-products, which were not detectable by LC-HRMS. As enzymes can be inhibited by the products of the enzymatic conversion [START_REF] Jing | Inhibition Performance of Lignocellulose Degradation Products on Industrial Cellulase Enzymes During Cellulose Hydrolysis[END_REF][START_REF] Barth | Effect of hydrolysis products on the enzymatic degradation of polyethylene terephthalate nanoparticles by a polyester hydrolase from Thermobifida fusca[END_REF], the effect of degradation products on TC degradation rate was studied with crushed monoliths and under vigorous stirring to avoid any influence of mass transport limitations. For this purpose, an initial TC degradation test was carried out for 24 h, with 50 mL of fresh TC solution (20 ppm) and a total enzymatic activity of crushed monoliths of 6 U. After 24 h of reaction (without air bubbling), the reaction mixture (unconverted TC + degradation products) was separated from the immobilized enzymes by filtration.

Afterward, a series of TC degradation experiments were carried out at different concentrations of degradation products by adding a volumetric fraction (50-90%) of the reaction mixture obtained from the initial experiment to a fresh TC solution (up to reach an initial TC concentration of 20 ppm) mixed with a fresh crushed enzymatic monolith (6 U). The results obtained are shown in Figure 7. It is important to notice that a lower degree of TC degradation (20%) was observed in comparison to the results (60% degradation)

reported in Figure 3. This is due to the use of a lower enzymatic monolith activity (6 U instead of 18 U) . The main result of this series of experiments is the enhancement of TC degradation with the increase of concentration of degradation products. Without degradation products and until a concentration of 60% of degradation products, TC degradation is around 20% in 24 hours, whereas for more than 70% of degradation products present in the solution, TC degradation reaches 40% after 24 hours. These results show that the TC degradation products formed did not inhibit the TC degradation, but in opposite they enhanced the conversion when their concentration reached a minimum level. A similar phenomenon has been recently observed by

Parra [START_REF] Parra Guardado | Effect of redox mediators in pharmaceuticals degradation by laccase: A comparative study[END_REF] and has been explained by the fact that some products of the reaction can act as redox mediators enhancing the enzyme reactivity and the conversion. This could also explain the higher degradation rate of TC (by a factor 2) obtained in the stirred tank reactor in comparison to the FTR, where degradation products do not spend enough time in the enzyme vicinity.

3.4

Application of a model for simulation of TC degradation in actual wastewater concentrations and proposed scale up of monolith system

In a previous work [START_REF] Ahmad | Enzymatic monolithic reactors for micropollutants degradation[END_REF] reported a computational fluid dynamics (CFD) model allowing computing TC degradation. The model was built coupling an apparent Michaelis-Menten reaction kinetics of immobilized enzymes, flow hydrodynamics within the FTR and a dynamic mass balance on feed tank. As far as this model was built in conditions of oxygen saturation conditions and with a reaction kinetics, which only takes into account TC and not oxygen, it should not be applied for TC concentrations as high as used in this work (20 ppm). Indeed it was demonstrated here that oxygen becomes rapidly a limiting substrate.

However, in some municipal wastewaters the TC concentration currently encountered is very low (2.8 10 -4 ppm) (Abejón et al., 2015b;[START_REF] Danner | Antibiotic pollution in surface fresh waters: Occurrence and effects[END_REF]. Thus, in this case and considering the saturation concentration of oxygen at 25 °C (8 mg L -1 ), the concentration of O2 (O2/TC = 4 10 5 ) would be five orders of magnitude larger than the requirement previously founded (O2/TC = 5.6). Under such conditions, we can consider that oxygen is no longer a limiting substrate and then the model can be reasonably applied for simulation purposes. The simulation of the TC degradation carried out with the FTR (3 monoliths in series of 0.6 cm diameter and 0.5 cm length, each) shows that for real TC concentration (2.8 10 -4 ppm) a removal of 90% of TC was reached in 5 h at a flow rate of 1 mL min -1 with a recycling configuration (see Figure 8). Taking account of these results, we then proposed to determine the minimum size of a monolith that allows a complete degradation of the TC in only one passage (without recycling) through the FTR. For this purpose, monoliths with different size (length and diameter ranged in between 5-50 cm and 1-20 cm, respectively) were considered for simulations (Table 2). When the size of monoliths is increased the conversion is enhanced. For example, for a single pass, when the monolith geometry was set at 5 cm length and 1 cm diameter only 0.5% of initial TC was converted. However, with monoliths of 50 cm length and 20 cm diameter it was possible to completely degrade TC at a flow rate of 1 mL min -1 . These theoretical "large-scale monoliths" were then used for further simulations. The TC conversion along the length of the optimum reactor (20 cm diameter, 50 cm length) was then studied at different flow rates (Figure 9). Complete TC degradation was achieved at the flow rate of 1 mL min -1 .

However, it can be observed that the rate of TC depletion in the last 15 cm was relatively low; this last section of the monolith is not enough efficient. For flow rates higher than 1 mL min -1 , TC conversion decreases up to 50% at a flow rate of 5 mL min -1 (Figure 9). In fact, the global process is controlled by reaction kinetics and at high flow rates, the residence time is too low to reach a complete TC depletion. However, as real wastewater fluxes are extremely high (600-70,000 m 3 /day) a compromise must be found between the level of TC depletion, the contact time and process effectiveness. For this, an optimization has to be carried out with multi-objective programming like Pareto optimality, this analysis has to be coupled with the cost of the process (Abejón et al., 2015b).

Conclusions

Biocatalytic reactors were built by grafting laccase into the mesopores of macroporous silica monoliths. The depletion of tetracycline (TC) (20 ppm) in aqueous solutions was carried out in a tubular flow through reactor (FTR) configuration (3 monoliths in series of 0.6 cm diameter and 0.5 cm length, each) with recycling in a reservoir tank. Only 30-40% of TC was degraded during the first 5 h and then the conversion slowed down to a threshold. The increase of flow rate from 0.5 to 5 mL min -1 did not change TC depletion showing that the phenomenon was not due to external mass transfer limitation, but rather to diffusion limitation of substrates inside the mesopores. Experiments performed in stirred tank reactors (CSTR) with crushed monoliths showed a higher TC depletion of 55% in 6 h, which was increased to 90% when the solution was bubbled with air. This showed that O2 is a limiting substrate for this reaction and that a molar excess of at least 5.6 O2/TC is necessary for a complete TC depletion. In FTR configuration, the addition of air-bubbling in the reservoir lead to an increase of TC depletion of only 10%, with a maximum of 50% at flow rate of 5 mL min -1 , this result indicates that oxygen limitations are enhanced in FTR configuration due to the mass transport limitations inside the mesopores. It was observed that the presence of by-products of degradation increases the TC depletion, this result could be explained by a possible the role of redox mediators of such by-products. In real municipal wastewater TC concentration is very low (2.8 10 -4 ppm) and O2 will be no more a limiting substrate as its excess of 5 orders of magnitude higher than TC. The modeling of FTR showed that in such conditions TC depletion should reach 90% in 5 h at a flow rate of 1 mL min -1 in a recirculating flow mode. The model allowed to predict that a monolith of 20 cm diameter and 50 cm length could be used for a total depletion of TC in a single pass (without recirculation) at a flow rate of 1 mL/min. However for higher flow rates (as the one encountered in real wastewater treatments) the decrease of the TC depletion was noticed. This is due by the fact that the depletion is governed by the reaction kinetic of the laccase. Further improvement on laccase activity could contribute to optimize the process. New monolithic configurations and processes have to be designed in order to feed oxygen continuously inside the porosity.

Declarations

Ethics approval and consent to participate

Not applicable

Consent for publication

Not applicable

  were separated and identified by means of liquid chromatography coupled to high resolution mass spectrometer QExactive (LC-HRMS) equipped with a HESI ionization source operating in negative or positive ionization mode, in separate injections. Data acquisition was performed in data dependent scan where the 10 most intense ions from full scan (m/z 50-600) were further fragmented with an isolation of 1.0 Da at a collision energy of 30 a.u. Data results were processed manually with Xcalibur 3.1 software (IDAEA-CSIC laboratory, Spain).

Figure 1 .

 1 Figure 1. Schematic draw of the laboratory scale set up for TC degradation with recycling. 1: reactor composed of three monoliths in series, 2: pressure transducer, 3: HPLC pump, 4: Stirred tank with TC solution, 5: air bubbling system, 6: thermocouple, 7: Thermostatic water bath.

Figure 2 .

 2 Figure 2. TC degradation at different flow rates (0.5-5 mL min -1 ). Temperature: 25°C; TC initial concentration of 20 ppm; total volume of TC solution of 30 mL, 15 U of enzymes in laccase-monoliths.

Figure 3 .

 3 Figure 3. Evolution of TC concentration with crushed laccase-monoliths in a stirred tank reactor, without (diamonds) and with (squares) air-bubbling. Control experiments with inactivated laccase-monoliths with (circles) and without (triangles) air bubbling. Initial TC concentration of 20 ppm in osmosed water (pH 6).Enzyme concentration of 18 U in the laccase-monoliths. Total volume 30 mL, temperature 25 °C.

Figure 4 .

 4 Figure 4. Evolution of dissolved oxygen (DO) concentration verse time in case of TC degradation test with crushed laccase-monoliths in a stirred tank reactor with (triangles) or without (diamonds) air bubbling. Initial TC concentration of 20 ppm in osmosed water (pH 6). Enzyme concentration of 18 U in the laccasemonoliths. Total volume 30 mL, temperature 25 °C.

Figure 5 .

 5 Figure 5. Evolution of TC concentration with laccase-monoliths under recycling flow at different flow rates without air bubbling (TC solution flow rates: circle, 1 mL min -1 ; triangle, 5 mL min -1 ) and with air bubbling (TC solution flow rates: diamonds, 1 mL min -1 ; squares, 5 mL min -1 ) at 25°C. Conditions: initial TC concentration of 20 ppm in osmosed water; total volume of TC solution of 30 mL; total activity of the laccasemonoliths of 12 U.

Figure 6 .

 6 Figure 6. Degradation products of tetracycline and possible reaction pathways using laccase from Trametes versicolor immobilized on silica monolith.

Figure 7 .

 7 Figure 7. Effect of the amount of TC degradation products on TC degradation in batch. Initial TC concentration of 20 ppm; crushed laccase-monolith (6 U); temperature 25 °C; reaction time of 24 h.

Figure 8 .

 8 Figure 8. Simulated degradation of TC at actual initial TC concentration (2.8 10 -4 ppm) found in wastewater in a FTR. Simulation conditions: flow rate (1 mL min -1 ), continuous recycled mode, Temperature 25 °C, 15 U of enzymatic activity in the monoliths.

Figure 9 .

 9 Figure 9. Relative tetracycline concentration along the length of the reactor from inlet to outlet at different TC flow rates from 1 to 5 mL min -1 . Reactor length: 50 cm, diameter: 20 cm; Inlet TC concentration: 2.8 10 - 4 ppm; Reaction rate: 4.4 10 -5 µmol L -1 min -1 .

Table 1 .

 1 Number of cycles through the laccase-monoliths reactor as a function of flow rate

	Flow rate	Number of cycles in 24h
	(mL/min)	(Vmacro: 3.41 mL/g)
	0.5	1409
	1	2818
	5	14090

Table 2 .

 2 Effect of the scale up of the silica monoliths (by increasing the length and diameter of the monoliths)

	towards TC degradation.
	Length	Diameter	Inlet	TC	Outlet	TC	TC conversion in a
	(cm)	(cm)	concentration		concentration		single pass
			(ppm)		(ppm)		(%)
			x10 -4		x10 -4
	0.15	0.06	2.8		2.79		0.3
	5	2	2.8		2.78		0.7
	10	4	2.8		2.68		4
	15	6	2.8		2.43		13
	20	8	2.8		2.01		28
	25	10	2.8		1.34		52
	50	20	2.8		0		100

Acknowledgements

. The authors aknowledge the ANR French agency, which founded the project DEMEMO under the number MUSE ANR-16-IDEX-0006. Mr. S. Ahmad acknowledges the Higher Education Commission, Pakistan for the PhD scholarship. The authors wish to acknowledge the support from the chemistry plateform of campus in Montpellier (Plateform MEA University of Montpellier), where SEM has been performed

Funding

This work was founded by The Agence Nationale de la Recherche (ANR, France) under the frame of the project MUSE ANR-16-IDEX-0006 project DEMEMO. The scholarship of Mr. S. Ahmad was provided by the Higher Education Commission, Pakistan

Availability of data and materials

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request, additional information and results are given in supplementary material.

Competing interests

The authors declare that they have no competing interests

Authors' contributions

Sher Ahmad and Wassim Sebai carried out the experimental work, synthesis of monoliths and enzymatic degradation of tetracycline, Marie-Pierre Belleville designed degradation experiments and contributed to the scientific discussion, Nicolas Brun, designed monoliths synthesis and contributed to the scientific discussion, Anne Galarneau and Jose Sanchez Marcano supervised all of the work, contributed with the discussions and wrote the whole manuscript. All of the authors revised the manuscript.


Emmanuel Pietriga
email: emmanuel.pietriga@inria.fr

Engineering Interactive Geospatial Visualizations for Cluster-Driven Ultra-high-resolution Wall Displays

Ultra-high-resolution wall-sized displays feature a very high pixel density over a large physical surface, typically covering a few square meters. They provide effective support for collaborative work sessions that involve the visualization of large, heterogeneous datasets. But the development of interactive visualizations for ultra-highresolution wall displays raises significant challenges. These range from the design of input techniques adapted to such surfaces, to the design of visualizations that effectively leverage their extreme display capacity. Challenges lie not only in the design but in the technical realization of these visualizations as well, as they run on computer clusters and thus require dedicated software frameworks for the distribution and synchronization of data and graphics. In this talk, I will essentially focus on challenges that relate to the engineering of interactive visualizations for cluster-driven wall displays, discussing different approaches that we explored over the last fourteen years to create geospatial visualizations and the associated multi-scale interaction techniques.

INTRODUCTION

Ultra-high-resolution wall displays have two key features, as their name suggests: a large display area, combined with a very high pixel density. On the most recent platforms, this density is getting close to human visual acuity when standing near (within arms' reach of) the display. For instance, the WILD-512K wall display depicted in Figure 1-a has a pixel density of more than 100 pixels/mm 2 for a surface area of more than 9 m 2 , amounting to a 61 440 × 17 280 pixel resolution in total.

Taking geospatial data visualization as an example, this display capacity means that compared to more conventional desktop display setups, the same spatial region can be visualized with a much higher level of detail, which can be of key importance as cartographic representations often feature many small elements and subtle visual cues. Put another way, this increased display capacity means that for a given level of detail, a much larger spatial region can be visualized.

This increased display capacity is not just an opportunity to show a more detailed rendering of one monolithic workspace, though. It also means that small-multiple visualizations, that juxtapose a series of related maps or charts to ease their comparison, can be rendered with a level of detail significantly higher than the typical thumbnail-like renderings shown on desktop displays [START_REF] Kamakshidasan | Comparative Visualization of Deep Water Asteroid Impacts on Ultra-high-resolution Wall Displays with Seawall[END_REF]. It also enables displaying data that do not directly belong to the map [START_REF] Pietriga | Exploratory visualization of astronomical data on ultra-high-resolution wall displays[END_REF], such as documents, tables, statistical charts, or node-link diagrams. Those can be juxtaposed with the map, or superimposed on top of it in regions of lesser importance, depending on the application and task at hand. Similarly, multiple locally-bounded interactive magnifications of the geographical region of interest can be displayed over the main map, possibly showing different layers of information (e.g., multispectral images [START_REF] Pietriga | Exploratory visualization of astronomical data on ultra-high-resolution wall displays[END_REF]) at different scales. Figure 1-b illustrates such an example: users work with multiple views giving complementary perspectives on the geographical region of interest at different levels of detail. They interact with these views using multitouch gestures performed either directly on the wall or indirectly on a handheld tablet [START_REF] Chapuis | Smarties: An Input System for Wall Display Development[END_REF].

Geospatial data visualization has been a key application area of our research on wall displays: interaction techniques for highprecision pointing [START_REF] Nancel | Mid-Air Pointing on Ultra-Walls[END_REF], for multi-scale navigation [START_REF] Nancel | Mid-Air Pan-and-Zoom on Wall-Sized Displays[END_REF], or more recently tangible interaction [START_REF] Courtoux | WallTokens: Surface Tangibles for Vertical Displays[END_REF]. But the amount, complexity and heterogeneity of geospatial data have also led us to research questions related to the engineering of interactive geospatial visualization early on, focusing on toolkits to display elaborate 2D multi-scale information spaces [START_REF] Pietriga | Rapid Development of User Interfaces on Cluster-Driven Wall Displays with JBricks[END_REF] as well as input management systems [START_REF] Chapuis | Smarties: An Input System for Wall Display Development[END_REF].

In this talk, I will essentially focus on challenges that relate to the engineering of interactive geospatial applications designed to run on cluster-driven wall displays. I will discuss different approaches that we explored over the last fourteen years to enable the development of interactive geospatial visualizations of dynamic data (Figure 1-c), making use of a variety of input devices and sensors: multi-touch frame, motion tracking equipment, tangible objects, handheld devices such as tablets and smartphones, augmented reality eyewear, highlighting some of the many challenges that remain to be addressed.

BIOGRAPHY

Emmanuel Pietriga is a Senior Research Scientist at Inria in France, where he leads the ILDA team. He is also Course Instructor at École Polytechnique where he teaches Data Visualization. His research focuses on the design and engineering of techniques for the visual representation and interactive manipulation of complex, heterogeneous data structures. His specific topics include interaction and visualization techniques for novel forms of displays (ultra-walls, AR, pen+touch surfaces), multi-scale user interfaces, visualization techniques for multivariate networks, and HCI aspects of Web browsing.

Figure 1 :

 1 Figure 1: (a) Visualization of an AIS marine traffic data feed on the WILD-512K ultra-wall. (b) Visualization of multiple data feeds at different scales, including topographic map, ortho-imagery, sea-level rise simulation, population density data, in a crisis management scenario on the WILDER ultra-wall. (c) Visualization of a live railroad traffic feed on the same platform.

ACKNOWLEDGMENTS

Projects discussed in this talk have been partially supported by ANR projects Digiscope (ANR-10-EQPX-26-01), MapMuxing (ANR-14-CE24-0011-02), and Continuum (ANR-21-ESRE-0030).


3.1) Method

3.2) Results

Low flow characterization and forecasting in a non-stationary context The XI th Scientific Assembly of the international Association of Hydrological Sciences The KGE values presented in table 1 show a small improvement with the SD model in the Meurthe case for the SS1. This efficiency increase with the SD model is also noted for the SS2 in the Meurthe case.

The figure 6 shows the result in validation mode of a calibration on naturalized flows.

2) A spatial pattern in low flows non-stationarity 1) Study context

This work is in line with two complementary Water managers that monitor hydrological extremes require low flow estimates (MAJORE) and forecasting tools (CIPRHES). This study aims to better understand the processes related to low flows space-time variability.

For that purpose, one of the main difficulties is to detect the respective signal of climate variability and human impacts.

3) Can a semi-distributed model improve low flow simulations ?

To better understand the low flows processes, the spatial resolution of low flow simulation and forecasting must be improved. We developed a new semidistributed version of the model PRESAGES (4) that integrates the main anthropogenic influences in the two main sub-basins presented above.

4) Main outcomes

This study rises the challenging question of identifying low flow trends and integrating non-stationarity processes into low flow modelling. The results show a spatial pattern in low flows trends in the French part of the Rhine-Meuse basin. The non-stationarity analysis will be used to identify the reference period for the modeling. For the PRESAGES-SD model, the results are promising. The model is still under development and other approaches will be tested to estimate the contribution of the intermediate catchment area. An extended test on a largest sample of basins will also be conducted.

  Two commonly modified nonparametric tests were used to detect nonstationary series: the modified Mann-Kendall test proposed by Hamed and Rao (1) and the Pettitt's test (2). Mean monthly minimum flows of the period 1971-2019 for 130 stations located in the Rhine-Meuse basin have been tested. Results show that there are 41 non-stationary series (i.e. 32%, for a rejection threshold at 5%), that are mainly located in the Vosges mountains. It appears that the station of the Moselle at Saint-Nabord is not stationary while the station of the Meurthe at Damelevières is stationary. One issue will be to identify factors explaining the non-stationarity of monthly minimum flows in the Vosges mountains. A semi-distributed model is under development to better understand processes generating low flows and to improve low flow modelling. Our research focuse s in particular on two different subbasins (fig. 1). The first one is the Meurthe at Damelevières, having an area of 2 281 km 2 and which is composed of 14 gauged subb a s i n s . T h e M e u r t h e a t Damelevières is influenced by the Vieux-Pré Dam used to release amounts of water for low-flow support. The second sub-basin is the Moselle at Saint-Nabord, which covers an area of 628 km 2 and is composed of 4 gauged sub-basins. Located in the upstream part of the Moselle basin, the river is subject to a little influence through intake for navigation.

Figure 2

 2 Figure 2 shows mean monthly minimum flows variability, with 1976 and 2003

H

  . El Khalfi, C. Delus, G. Drogue, S. Lebaut, L. Manceau, B. Grelier, D. François LOTERR UR 7304, Université de Lorraine, F-57000 Metz, France

Session 3 :

 3 Change in extreme droughts in the future P5 / IAHS2022-623 For the first version of the PRESAGES-SD model, the process follows two steps: applying the lumped version of PRESAGES model (fig. 4) to the subbasins as well as to the intermediate catchment area (ICA). The equation used is: The objective function used to calibrate the model is a combination of the KGE criterion which focuses on low flows. The PRESAGES-SD model is applied to observed and naturalized flow series. References: 1) Perrin et al., IAHS2022-665, Session 3 -Integrated chain for the hydrometeorological forecasting of low flows and droughts in France. The CIPRHES project. 2) Hamed K. H. and R. A. Rao., 1998, «A modified Mann-Kendall trend test for autocorrelated data», Journal of Hydrology, vol. 204, no 1-4. 3) Serinaldi F. and C. G. Kilsby., 2016, «The importance of prewhitening in change point analysis under persistence», Stochastic Environmental Research and Risk Assessment, p. 763-777, ISSN 1436-3240. 4) CEGUM, 2006, «PRESAGES : PREvision et Simulation pour l'Annonce et la Gestion des Etiages Sévères», rapport, Agence de l'Eau Rhin-Meuse, Université Paul Verlaine -Metz. The PRESAGES-SD model is applied to flows over the recent period (2001-2020). The period 2001-2010 is used for calibration and the period 2011-2020 for validation and vice versa. The results are preliminary and the PRESAGES-SD model will be improved in the future.

  projects MAJORE and CIPRHES (1) that focus on characterization and forecasting of low flows. The study area focuses on the Rhine-Meuse basin that covers 31 437 km², and includes the French part of the Meuse (7 766 km²) and Moselle (11 437 km²) basins, and those of some left bank tributaries of the river Rhine, including the Ill river. The French part of the Rhine-Meuse area has faced extreme low flows in recent summers causing difficulties in water resources management (fig. 2).


Dmitry Solnyshkov
email: dmitry.solnyshkov@uca.fr

Guillaume Malpuech

Love might be a second-order phase transition

Keywords:

The hypothesis of the human brain operation in vicinity of a critical point has been a matter of a hot debate in the recent years. The evidence for a possibility of a naturally occurring phase transition across this critical point is missing so far. Here we suggest that love might be an example of a second-order phase transition. We analyze three examples of love at first sight from classical literature, showing that they are compatible with the square root scaling of feelings with time predicted by the hypothesis. Two examples of love from liking also confirm the theory. Finally, we predict a critical value of liking ensuring the stability of relations. All results are also supported by the common knowledge expressed in proverbs.

Introduction

It is natural to try to use the well-developed methods of statistical physics when one is dealing with large numbers of entities exhibiting relatively simple interactions [START_REF] Kwapień | Physical approach to complex systems[END_REF]: financial markets (and economy in general), natural languages, and neural networks. Such studies have a long history, dating back at least to the dawn of the XX century [START_REF] Haret | Mecanique sociale[END_REF][START_REF] Rashevsky | Outline of a mathematical theory of human relations[END_REF]. Physical methods, in particular, the network science, can be applied to human-related problems at all levels [START_REF] Gosak | Network science of biological systems at different scales: A review[END_REF], up to the scale of "social physics" [START_REF] Perc | Statistical physics of human cooperation, Physics Love might be a second-order phase transition[END_REF]6]. Statistical approach has been used to treat artificial and natural neural networks almost since the very beginning of their studies. Indeed, a natural neural network represented by the human brain contains 10 11 neurons, which, while being smaller than the number of atoms in a typical macroscopic object 𝑁 𝐴 = 6×10 23 , still falls well into the domain of applicability of statistical physics [START_REF] Kinzel | Phase transitions of neural networks[END_REF]. Quite soon after the beginning of the studies of artificial neural networks, such as perceptrons, it was shown that the learning process of such networks was mathematically equivalent to a Monte-Carlo procedure to obtain thermal equilibrium [START_REF] Hopfield | Neural networks and physical systems with emergent collective computational abilities[END_REF][START_REF] Amit | Statistical mechanics of neural networks near saturation[END_REF].

The description of larger neural networks, such as the whole human brain or its major elements, such as the cortex, advanced thanks to experimental measurements of the electric activity of the brain. These measurements, as well as the studies of externally accessible behaviors, were showing the so-called 1∕𝑓 noise [START_REF] Gilden | 1/f noise in human cognition[END_REF]. This power law has become one of the major indications of the criticality of the brain's behavior [START_REF] Kello | The emergent coordination of cognitive function[END_REF]. Indeed, one of the leading hypotheses now is that the brain is operating close to the critical point of a second-order phase transition [START_REF] Kelso | A phase transition in human brain and behavior[END_REF][START_REF] Fraiman | Ising-like dynamics in large-scale functional brain networks[END_REF][START_REF] Chialvo | Emergent complex neural dynamics[END_REF][START_REF] Beggs | Being critical of criticality in the brain[END_REF][START_REF] Cocchi | Criticality in the brain: A synthesis of neurobiology, models and cognition[END_REF], which explains several groups of its properties: the presence of power-law scalings for many of its characteristics [START_REF] Gilden | 1/f noise in human cognition[END_REF]; the links between the corresponding scaling exponents, including those describing behavior [START_REF] Palva | Neuronal long-range temporal correlations and avalanche dynamics are correlated with behavioral scaling laws[END_REF]; the emergence of large-scale dynamical correlations [START_REF] Linkenkaer-Hansen | Long-range temporal correlations and scaling behavior in human brain oscillations[END_REF][START_REF] Haimovici | Brain organization into resting state networks emerges at criticality on a model of the human connectome[END_REF] between neurons (pattern formation) and the sensitivity to the state of individual neurons; the divergence of the susceptibility and the multitude of selfadapting reactions. The operation at the critical point allows the brain (and other biological systems) to achieve maximal efficiency, required for survival [START_REF] Moretti | Griffiths phases and the stretching of criticality in brain networks[END_REF]. It is ensured by the proper balance of excitation and inhibition [START_REF] De Arcangelis | Self-organized criticality model for brain plasticity[END_REF][START_REF] Poil | Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks[END_REF] On the other hand, the criticality is not universally accepted yet. One of the alternatives to the criticality is the selforganized bistability [START_REF] Buendía | Selforganized bistability and its possible relevance for brain dynamics[END_REF]. Power laws, and in particular, the 1∕𝑓 scaling, can emerge in networks even without criticality [START_REF] Bédard | Does the 1∕𝑓 frequency scaling of brain signals reflect self-organized critical states?[END_REF][START_REF] Touboul | Power-law statistics and universal scaling in the absence of criticality[END_REF]. Yet another argument against the criticality theory is that it implies the possibility to cross the transition point, and thus one can expect some evidence of such transitions. Ongoing studies are devoted to the type of the phase transition involved [START_REF] Fontenele | Criticality between cortical states[END_REF][START_REF] Korchinski | Criticality in spreading processes without timescale separation and the critical brain hypothesis[END_REF][START_REF] Santos | Topological phase transitions in functional brain networks[END_REF]. It is understood that the balance between excitation and inhibition should be one of the parameters allowing to cross the transition point. An artificially-induced transition to the subcritical regime and a return to criticality have already been observed in rats [START_REF] Ma | Cortical circuit dynamics are homeostatically tuned to criticality in vivo[END_REF], but no evidence of such transitions occurring naturally are known for human beings so far.

In this work, we suggest that love might be an example of a second-order phase transition occurring in the brain. We show that this hypothesis explains a lot of well-known properties of love. Analyzing several most famous literature examples and a private diary, we show that the intensity of feelings exhibits a universal scaling behavior, distinguishing two cases: love at first sight and love developing from liking or friendship (friends first), both being studied in psychology [START_REF] Barelds | Love at first sight or friends first? ties among partner personality trait similarity, relationship onset, relationship quality, and love[END_REF][START_REF] Hefner | From love at first sight to soul mate: The influence of romantic ideals in popular films on young people's beliefs about relationships[END_REF][START_REF] Vannier | Passion, connection, and destiny: How romantic expectations help predict satisfaction and commitment in young adults' dating relationships[END_REF].

Second-order phase transitions

The theory of second-order phase transitions developed by Landau [START_REF] Landau | Statistical Physics[END_REF] states that the thermodynamic potential 𝐹 (that we generalize to an abstract "cost function" minimized by the system) can be written in series expansion in powers of the order parameter 𝛼:

𝐹 (𝛼) = 𝐹 0 + ℎ𝛼 + 𝐴𝛼 2 + 𝐵𝛼 4 + … (1)
In general, ℎ corresponds to an external bias (e.g. applied field) that we will discuss later. A phase transition of the second order (or kind) occurs when the coefficient 𝐴 (depending on a certain parameter, usually temperature) changes sign, crossing zero. This is illustrated in Fig. 1(a), showing the typical shapes of 𝐹 for 𝐴 > 0 (black), 𝐴 = 0 (green), 𝐴 < 0 (red) for ℎ = 0. Since 𝐴 changes sign, it necessarily behaves as a first power of temperature (or reduced temperature 𝜖 = 𝑇 ∕𝑇 𝑐 , where 𝑇 𝑐 is the critical temperature, where the transition occurs): 𝐴 ∼ 𝜖. The minimum of 𝐹 is determined by 𝜕𝐹 ∕𝜕𝛼 = 0 giving a cubic equation

𝛼 3 + 𝐴 2𝐵 𝛼 + ℎ 4𝐵 = 0 (2)
For ℎ = 0, the minimum is located at 𝛼 = 0 for 𝐴 ≥ 0 and at 𝛼 ≠ 0 for 𝐴 < 0. Spontaneous symmetry breaking occurs at 𝐴 = 0. This theory has inspired a lot of activity in physics, because it turned out that such transitions are characterized by a universal scaling behavior. Independently of the nature of the system, their parameters exhibit power law dependencies on the dimensionless parameter (temperature). For example, the order parameter 𝛼 scales as a square root of temperature (below threshold): 𝛼 ∼ √ |𝜖|. Another important parameter, the susceptibility, diverges at the critical temperature as 𝜒 ∼ |𝜖| -1 . It means that the order parameter 𝛼 reacts strongly to any external perturbation (introduced via the term ℎ). This is one of the reasons why operating close to the critical point 𝑇 𝑐 (𝜖 = 0) is thought to increase the efficiency of the brain.

The second-order transitions are best illustrated by the Ising model of a ferromagnet, consisting of interacting spins (Fig. 1(b)). The fact that a neural network can be mapped to this model was discovered quite a long time ago [START_REF] Little | The existence of persistent states in the brain[END_REF]. The transition to the ordered phase usually occurs while decreasing the temperature. Above the critical temperature, the minimum of the generalized potential 𝐹 corresponds to zero order parameter 𝛼 = 0, and thus the system exhibits no order (random orientation of different spins). Because of this, the system is not sensitive to the orientation of a single spin. This is the subcritical regime. Below the critical temperature, the order parameter (average magnetization) is non-zero 𝛼 ≠ 0: all spins are pointing in the same direction (chosen randomly at the moment of spontaneous symmetry breaking). The sensitivity to a single spin is low: all spins are aligned, and a single spin flip does not perturb them all. This is the supercritical regime. Finally, exactly at the threshold the system exhibits long-range patterns and a high sensitivity to a single spin: flipping it starts flipping all other spins in the pattern. This is the critical regime. It is also characterized by divergent fluctuations of the order parameter ⟨Δ𝛼 2 ⟩ ∼ |𝜖| -1 (due to their low energy cost). For neural networks, where each spin is mapped to a neuron, this implies high sensitivity to a state of an individual neuron combined with long-range pattern formation, which corresponds to a high memory capacity and high adaptability of the system.

If one wishes to study the second-order phase transition process, and not the individual phases above and below it, the

a) b) c) 1 0 -1
Figure 1: Illustration of the general theory of second-order phase transitions. a) Generalized potential 𝐹 as a function of order parameter 𝛼. The system always seeks to minimize 𝐹 . The minimum is at 𝛼 = 0 "above" transition (subcritical behavior) and at 𝛼 ≠ 0 "below" transition (supercritical behavior). b) Ising model of a ferromagnetism illustrating the 3 regimes: subcritical (short correlations), critical (longrange dynamical correlations), supercritical (long-range static correlations). c) The order parameter 𝛼, growing as a square root beyond the transition, and the susceptibility 𝜒, diverging at the critical point, as a function of a dimensionless parameter (reduced temperature 𝜖).

reduced temperature 𝜖 has to vary with time, which becomes the new dimensionless parameter: 𝐴 ∼ -𝑡∕𝜏 (where 𝜏 is the characteristic transition time also called quench time). The variation of reduced temperature should be sufficiently slow, so that a finite-size system can reach equilibrium at each moment (we neglect the divergence of the relaxation time, known to play a strong role in the Kibble-Zurek mechanism of topological defect formation in the ordered phase [35,36]). Studying a time-dependent system, one can therefore expect the order parameter to be zero until the transition, to exhibit large fluctuations at the transition, and then to grow as a square root of time 𝛼 ∼ √ 𝑡.

Love at first sight

The key hypothesis of the present work is that love is a second-order phase transition occurring in the human brain under the influence of hormones, such as dopamine and serotonin [START_REF] Takahashi | Imaging the passionate stage of romantic love by dopamine dynamics[END_REF][START_REF] Boer | Love is more than just a kiss: a neurobiological perspective on love and affection[END_REF]. The brain switches from the normal operation in the critical regime to the supercritical regime because of the increase of the excitation (more dopamine) and reduction of the inhibition (less serotonin) [START_REF] Boer | Love is more than just a kiss: a neurobiological perspective on love and affection[END_REF]. The order parameter of the transition is the intensity of feelings: before the transition, the subject has no particular feelings, while after the transition the feelings are non-zero. The most direct consequence is that the order parameter should exhibit a universal power law behavior: the feelings should grow as a square root of time

𝛼 ∼ √ 𝑡 ( 3
)
for all individuals, in the absence of an external bias (ℎ = 0).

Here we spot the first indirect confirmation of our hypothesis. Indeed, a square root time dependence 𝛼 ∼ √ 𝑡 is characterized by an infinitely fast initial growth: 𝜕𝛼∕𝜕𝑡 ∼ 1∕ √ 𝑡, giving 𝜕𝛼∕𝜕𝑡 = ∞ at 𝑡 = 0. This infinitely fast growth of feelings is striking for the individual who feels it. This striking feeling is probably at the origin of the expression "love at first sight". It is also why the Roman god Cupid (or Amor), called Eros in Greek, carries a bow with arrows. The strike of Cupid's arrow represents the fast growth of feelings suggested by the model. Finally, in French "love at first sight" is translated as "le coup de foudre" (the strike of a lightning), confirming the hypothesis further. According to the prediction of psychological theories , "love at first sight" should be marked with a particularly high passion [START_REF] Zsok | What kind of love is love at first sight? an empirical investigation[END_REF], which agrees with this hypothesis.

Since it is impossible to analyze the human feelings directly during such uncontrollable phenomenon as "love at first sight", we have decided to perform a quantitative study of the intensity of feelings depicted in the literature. This indirect approach is similar to the one used to detect the 1∕𝑓 noise via measurement errors and reaction time in the seminal paper of D. L. Gilden [START_REF] Gilden | 1/f noise in human cognition[END_REF]. We have chosen three well-known books for this analysis: "Romeo and Juliet" (W. Shakespeare), "The Lily of the Valley" (H. de Balzac, see Appendix A), and "Martin Eden" (J. London). While all three books are fiction, we assume that the descriptions of feelings of main characters should be true enough, otherwise they would seem "wrong" to the reader and would not gain such popularity. Moreover, H. de Balzac is known for the knowledge of the human nature and for the attention to detail, whereas the book of J. London is considered as autobiographic and should also be true in its description of love.

From each of the books, we have chosen several consecutive phrases from which a reader can deduce the feelings of the main character, and submitted these phrases to a set of six to ten (different for different texts) female and male respondents aged from 40 to 83 years and unaware of the theoretical hypothesis being tested, in order to avoid subconscious or conscious bias. The respondents were asked to assess the intensity of feelings of the main character for each phrase using a scale from 0 to 10. We have normalized the maximal intensity of feelings to 1 for each book (the scaling only holds close to the transition point and cannot describe the evolution of feelings for longer times), and we assume that this intensity is achieved at 𝑡 = 𝜏 (discussed below). We assumed a constant time spacing between consecutive phrases: we suppose that while the speed of perception can be different, it cannot change over such a short period, and the speed of the narrative corresponds to the natural perception.

The fitting parameter for each book was the moment at which the transition occurs, because it is not mentioned explicitly in these books. The results are shown in Fig. 2. The 3 sets of points correspond to the 3 books, and the magenta curve to the theoretical scaling 𝛼∕𝛼 0 = √ 𝑡∕𝜏. The single theoretical curve showing the predictions of the secondorder phase transition model is universally compatible with all 3 books. The error bars show the confidence intervals at 0.95 level.

Let us discuss these examples in more details, so that the reader can become convinced that the described feelings indeed grow as assessed by our respondents. Indeed, in Romeo and Juliet one finds the following phrases:

1. What lady's that, which doth enrich the hand of yonder knight? 2. O, she doth teach the torches to burn bright! 3. she hangs upon the cheek of night Like a rich jewel in an Ethiop's ear, beauty too rich for use 4. for earth too dear

The continuous growth of feelings can be easily noticed in this sequence, from the first phrase, merely noticing that the lady "enriches" the hand of a knight, to the last phrase, where Juliet is compared to a jewel "too dear for Earth" (which can be naturally taken as a maximum). Especially important is the fast initial increase illustrated by the second phrase.

It is interesting to note that the Romeo's behavior exhibits signatures of large fluctuations ⟨Δ𝛼 2 ⟩ ∼ |𝜖| -1 close to the transition point: before falling in love with Juliet, he loves others, although not to the same extent (not because they were worse than Juliet, but because he has not yet crossed the transition point). These short episodes also agree with the predictions of the theory.

From "Martin Eden", which has the advantage of being autobiographic, we have selected the following phrases:

1. He did not know how she was dressed, except that the dress was as wonderful as she 2. He likened her to a pale gold flower upon a slender stem. 3. No, she was a spirit, a divinity, a goddess 4. such sublimated beauty was not of the earth Again, a fast growth with saturation is clearly visible. The last phrase clearly expresses the maximal feelings possible ("not of the Earth", as in "Romeo and Juliet"), justifying taking it as 𝛼 𝑚𝑎𝑥 . It is also interesting to note that Jack London himself was interested in scientific analysis of the nature of love, which he discussed in the "Kempton-Wace Letters" [START_REF] London | The Kempton-Wace Letters[END_REF]. We can therefore expect his description of his own feelings to be quite precise. Moreover, in this book we even find a phrase, which indirectly confirms our hypothesis of second-order transition from criticality to super-criticality: "The phantasmagoria of his brain vanished at sight of her".

We note that fitting the whole set of points with a power law in a log-log scale gives a scaling exponent of 0.56±0.06, compatible with the theoretical prediction of 1∕2, which further confirms the hypothesis of the second-order phase transition. Finally, it is interesting to note that the red points, corresponding to the "Lily of the valley" by H. de Balzac fall almost exactly on the theoretical curve. If the theoretical hypothesis is correct, it confirms the reputation of H. de Balzac as a connoisseur of the human heart.

The hypothesis of second-order phase transition provides an explanation for other well-known features that characterize love, such as "love is blind". Indeed, the loss of sensitivity (the related physical parameter is the susceptibility 𝜒), associated with the transition to super-criticality, essentially makes the person "blind" to the defects of the beloved one, and also "deaf" to the warnings of parents and friends. Indeed, at the critical point the susceptibility diverges 𝜒 ∼ |𝜖| -1 , and therefore it strongly decreases when going away from the transition. This is also confirmed by recent psychological studies, demonstrating that falling in love typically involves cognitive biases [START_REF] Barelds | An assessment of positive illusions of the physical attractiveness of romantic partners[END_REF][START_REF] Fletcher | Through the eyes of love: reality and illusion in intimate relationships[END_REF]. Other examples of the proverbs explained by the hypothesis include "There is no difference between a wise man and a fool when they fall in love", and "When love is not madness, it is not love."

Most probably, contrary to the other aspects, for which the brain is operating in the critical regime, in the case of love the evolution has avoided the stabilization at the critical point with its maximal susceptibility 𝜒. Indeed, for evolution, the desire for procreation should overcome reason, as well as all other instincts, including the survival instinct. Before maturity, the brain is probably functioning subcritically in what concerns love, and when the maturity is reached, it goes supercritical. One can therefore expect a given person to be indifferent to potential sexual partners in childhood (low susceptibility), and to become indifferent to all partners except one after falling in love (equally low susceptibility). Between these two cases, while crossing the critical point, the behavior can become maximally rational, aimed at selection of the best possible partner (maximal susceptibility). This can be expected to show up as successive (or parallel) dating (without falling in love) of multiple partners appearing as optimal. Indeed, no one will buy a car or a house without trying out several ones, unless stroke by a sudden passion.

The region of rational behavior at the critical point, corresponding to extended correlations, falls beyond the validity of the Landau theory of phase transitions. While it seems impossible to estimate the width of this region for the transition parameters 𝜖 or 𝑡∕𝜏 a priori, because of the lack of knowledge concerning the microscopic brain functioning, we can estimate it from the requirement of rational behavior, or the absence of detectable feelings (as opposed to rationality). We can assume that the feelings at least above 0.1𝛼 𝑚𝑎𝑥 can be detected with our scale, and therefore the region of rational behavior should correspond to 𝛼 < 0.1𝛼 𝑚𝑎𝑥 , which gives 𝑡 < 0.01𝜏 for the transition parameter.

We stress once again that in the case of "love at first sight", the transition occurs inside the person and is caused by internal reasons. An individual is moving gradually across the transition with age. We see new faces all the time, but when the person becomes "ready" and crosses the transition point, the particular new face seen at this particular moment becomes the object of spontaneous symmetry breaking, which means that this effect is essentially random. The person could have seen somebody else (not completely disgusting) at this moment and would have fallen in love in that other person. This is confirmed by the saying "Beauty is in the eye of the beholder". Juliet seemed exceptional to Romeo not because she was exceptional, but because he has just crossed the transition point inside himself, that is, he has moved to an exceptional state himself.

The characteristic time of the transition 𝜏 cannot be determined from the books, because it is not indicated explicitly there. We have therefore analyzed several films based on these books, assuming that the director and the actors try to make the scene appear as natural as possible. We obtained the following characteristic times: 𝜏 = 14 s for "Martin Eden" (G. Battiato, 2011), 𝜏 = 100 s for "Romeo and Juliet" (F. Zeffirelli, 1968), and 𝜏 = 23 s for "Lily of the valley" (M. Cravenne, 1970).

Love from liking

We now consider the case of a non-zero external "bias", which is a situation that occurs more often than love at first sight. Indeed, one usually knows several persons better than all the others, and thus some positive feelings towards some of them can be present before the transition occurs. The generalized potential 𝐹 (𝛼) for non-zero ℎ is plotted in Fig. 3(a). The minimum of 𝐹 is now always at non-zero 𝛼. However, when the transition is crossed, this value of 𝛼 rapidly increases. In general, non-zero ℎ "broadens" the transition shown in Fig. 1(c). The main root of (2) reads

𝛼 = -2 ⋅ 3 2∕3 𝐴 + 3 1∕3 ( -9 √ 𝐵ℎ + √ 24𝐴 3 + 81𝐵ℎ 2 ) 2∕3 6 √ 𝐵 ( -9 √ 𝐵ℎ + √ 24𝐴 3 + 81𝐵ℎ 2 ) 1∕3 (4)
As previously, the parameter 𝐴 crosses zero and exhibits a linear dependence on time: -𝐴 ∼ 𝑡. The behavior of this solution 𝛼(𝑡) is shown in Fig. 3(b) for different values of ℎ. On the one hand, in this case the transition is, in general, slower, and it is more difficult for the individual to distinguish the moment when liking or friendship turns into love. On the other hand, the exact moment of the transition is characterized by the maximal derivative (infinite in the case of ℎ = 0). This rapid increase of feelings can hardly go unnoticed, and it is the moment when one understands that love has come.

To study this transition quantitatively, we have chosen another well-known book, "Jane Eyre" by Charlotte Brontë. This books is also quite autobiographic, and famous for the psychological details, that got its author the title of "first historian of the private consciousness". The delay between the initial acquaintance and love is now of several (about 45 days). The bias is clearly present ℎ ≠ 0, because Mr. Rochester is the only male person around Jane Eyre. We have selected several paragraphs from several chapters narrating the process of falling in love. The advantage of the detailed narrative is that it is possible to date these paragraphs quite precisely (see Appendix B for the phrases and dates).

The progressive change in the attitude towards Mr. Rochester is clearly visible in these paragraphs. One can also notice the strong change that occurs around Day 28, which we therefore choose as the origin of the transition 𝑡 = 0. As before, we normalize the intensity to the maximum, and the time scale to 𝜏 = 45 -28 = 17 days (the duration between the transition and the maximal intensity). The points are presented in Fig. 3(b) together with the theoretical curve. The model has a single fitting parameter ℎ, for which we find ℎ ≈ -0.68 ± 0.08 with a p-value 4 × 10 -5 , clearly allowing to reject the null hypothesis.

Finally, one of the respondents has managed to find a personal diary regularly kept over about 15 years in the 1970s. She has then analyzed one example of falling in love, described in detail over several months (137 days), using the same principle: for each day, the intensity of emotions was assessed on the scale from 0 to 10. We have then rescaled the intensity from 0 to 1, as in the other examples, and taken the day where the 0.5 value has been achieved as the moment 5: Love from liking. The intensity of feelings 𝛼(𝑡) for a personal diary (dots) and theory (line) with a fitting parameter ℎ (bias). The growth is faster than in the case of "Jane Eyre", due to a smaller liking ℎ. Here, |ℎ| < ℎ 𝑐 , and the relationship eventually breaks down. of transition, rescaling the time axis correspondingly (𝜏 = 90). The results of the analysis of the diary and the fitting are presented in Fig. 5. There are no error bars, since we could not ask several persons to analyze the diary because of its private nature. It is therefore difficult to estimate the uncertainty. The fitting procedure gives the bias ℎ ≈ -0.15± 0.05 (p-value 4 × 10 -3 ). This bias is smaller than in the case of "Jane Eyre", and the consequences of this difference will be discussed below.

Discussion and conclusions

The cubic equation ( 2) can have 1 or 3 real roots, depending on the sign of the determinant

Δ = -4 ( 𝐴 2𝐵 ) 3 -27 ( ℎ 4𝐵 ) 2 (5)
(Δ < 0 -1 root, Δ > 0 -3 roots). For ℎ = 0, the determinant changes sign with 𝐴, with Δ > 0 corresponding to the supercritical phase. The 𝛼 = 0 root is unstable (maximum), but the two other roots (positive and negative) are stable. This may be the physical origin behind the mathematical description of love using the catastrophe theory [START_REF] Tesser | Aggression, love, conformity, and other social psychological catastrophes[END_REF]. In the framework of our hypothesis it means that love (𝛼 > 0) and hate (𝛼 < 0) are both stable solutions. Once fallen in love, it is difficult to become indifferent to the beloved person; but love can be relatively easily replaced by hate. This is in agreement with another proverb, "Love And Hate Are Just One Step Apart". The difficulty to get rid of love is also confirmed by the proverb "Old love and wood will burn as soon as they get the chance." However, according to recent studies, love can disappear naturally at the timescale of several years, which is probably explained by the phase transition (e.g. saturation of the brain's sensitivity to dopamine or just decrease of its level). The situation is different for a non-zero value of ℎ: while it is still possible to have 3 roots for a small value of |ℎ|, the situation changes when |ℎ| is increased. For |ℎ| higher than the critical value

ℎ 𝑐 = 2 3∕2 |𝐴 𝑚𝑎𝑥 | 3∕2 3 3∕2 𝐵 1∕2 (6)
where |𝐴 𝑚𝑎𝑥 | is the maximal achievable value of |𝐴| for a particular person, the equation ( 2) always has a single root and the potential (1) exhibits a single minimum. It means that when love is born from a strong liking or friendship, it does not cause a strong change in the intensity of feelings at the transition, but it cannot turn into hate or into indifference. This theoretical prediction is illustrated in the common knowledge by the saying "The happiest marriages are between best friends". This might be an important result for applied psychology in the field of couples therapy, which only relatively recently started to focus on the importance of love [START_REF] Willi | The significance of romantic love for marriage[END_REF] and on its consequences for successful relationships and marriage [START_REF] Vannier | Passion, connection, and destiny: How romantic expectations help predict satisfaction and commitment in young adults' dating relationships[END_REF].

For the normalization used in our work, 𝐵 = 1 and 𝐴 = (2∕3) 1∕3 , which gives ℎ 𝑐 ≈ 0.44. The value found for the case of Jane Eyre |ℎ 𝐽 𝐸 | ≈ 0.68 clearly exceeds this critical value, allowing to expect that her relationship with M. Rochester should be able to withstand any hardships. This turns out indeed to be the case in the book. On the other hand, the value found for the diary |ℎ 𝑑 | ≈ 0.15 is clearly below the critical value, allowing to expect that this relationship could disappear after a certain time. This indeed turned out to be the case after 1 year. We can therefore conclude that people who do not fall in love at first sight and feel a smooth increase of their feelings, so that they might be even unsure if they are in love, should not worry too much about it: while being difficult to distinguish from friendship, their relationship is actually more secure precisely because of this.

Indeed, the transition from friendship or liking to love is so smooth that it may even go unnoticed. However, the strength of the bonds is felt immediately if one loses the object of love. In this case one understands that the feeling was actually love from the intensity of the loss that one suffers. Recent studies in psychology confirm that "love at first sight" leads to a quicker romantic involvement [START_REF] Barelds | Love at first sight or friends first? ties among partner personality trait similarity, relationship onset, relationship quality, and love[END_REF], but the partners show more dissimilar personalities, confirming the interpretation of spontaneous symmetry breaking.

Universality implies that the same behavior should be observed for all individuals (but possibly with different scales, such as the time scale 𝜏). We stress that while the number of books that we have treated is quite small, it nevertheless provides a strong support for the hypothesis of universality, because we did not select these books. We have simply taken the ones providing a description of love at first sight and of slowly developing love that could be exploited, and the extracted behavior turned out to be in agreement with the hypothesis. We do not pretend, of course, that the present work provides a definitive proof of love being a secondorder transition, or of the criticality of brain's function. More extensive studies will be necessary, both based on the analysis of texts and on the measurements of the brain activity. These studies can probably be carried out with animals, who can probably also exhibit similar phase transitions during special periods because of the same evolutionary reasons. We also note that the way how the text is perceived by the reader changes with time. For example, the comparison with torches from "Romeo and Juliet" (phrase 2) seems probably much more emotional now than when it was written. It is also interesting to note how much can be really lost in translation (see Appendix A).

To conclude, we have shown that the hypothesis that love is a second-order phase transition occurring in human brain predicts a characteristic scaling of the intensity of the feelings over time after falling in love. It is confirmed qualitatively by the common wisdom expressed in proverbs and quantitatively by the analysis of examples of "love at first sight" and "love from liking" in literature. We note a possible existence of critical "liking" or friendly attitude, above which the relations might be more stable. A more direct confirmation by the measurements of non-critical brain behavior should be possible in the future. Other possibilities include the studies of records in blogs.

Acknowledgments

We acknowledge the support of the European Union's Horizon 2020 program, through a FET Open research and innovation action under the grant agreement No. 964770 (TopoLight), project ANR Labex GaNEXT (ANR-11-LABX-0014), and of the ANR program "Investissements d'Avenir" through the IDEX-ISITE initiative 16-IDEX-0001 (CAP [START_REF] Moretti | Griffiths phases and the stretching of criticality in brain networks[END_REF][START_REF] De Arcangelis | Self-organized criticality model for brain plasticity[END_REF][START_REF] Poil | Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks[END_REF][START_REF] Buendía | Selforganized bistability and its possible relevance for brain dynamics[END_REF][START_REF] Bédard | Does the 1∕𝑓 frequency scaling of brain signals reflect self-organized critical states?[END_REF][START_REF] Touboul | Power-law statistics and universal scaling in the absence of criticality[END_REF].

A. The Lily of the Valley

The selected phrases were:

1. Aussitôt je sentis un parfum de femme qui brilla dans mon âme comme y brilla depuis la poésie orientale. 2. Je regardai ma voisine, et fus plus ébloui par elle que je ne l'avais été par la fête ; elle devint toute ma fête. 3. Les plus légers détails de cette tête furent des amorces qui réveillèrent en moi des jouissances infinies 4. tout me fit perdre l'esprit In this case, we had to abandon four intermediate phrases (not shown here), which seemed too direct for modern "puritan" readers and were not qualified as manifestations of love. We have accounted for these phrases when placing the points along the time axis.

It is interesting to see how much can be lost in translation. Indeed, the English translation of phrase 2 reads: "I looked at my neighbor, and was more dazzled by that vision than I had been by the scene of the fete." The last part of the phrase ("she became all my fete") is simply absent, and without it the phrase looks much less positive and emotional. This is why we were using original phrases for our study.

B. Jane Eyre

We have selected the following paragraphs:

• (Day 1) dark face, with stern features and a heavy brow; his eyes and gathered eyebrows looked ireful and thwarted just now; The new face, too, was like a new picture introduced to the gallery of memory; and it was dissimilar to all the others hanging there: firstly, because it was masculine; and, secondly, because it was dark, strong, and stern. I had it still before me when I entered Hay, and slipped the letter into the post-office; I saw it as I walked fast down-hill all the way home.

• (Day 2) ...decisive nose, more remarkable for character than beauty; his full nostrils, denoting, I thought, choler; his grim mouth, chin, and jaw-yes, all three were very grim, and no mistake. neither tall nor graceful.

• (Day 17) not quite so stern-much less gloomy; great, dark eyes, and very fine eyes, too-not without a certain change in their depths sometimes, which, if it was not softness, reminded you, at least, of that feeling. I am sure most people would have thought him an ugly man; yet there was so much unconscious pride in his port; so much ease in his demeanour;

• (Day 22) I really possessed the power to amuse him, and that these evening conferences were sought as much for his pleasure as for my benefit. I had a keen delight in receiving the new ideas he offered, in imagining the new pictures he portrayed, and following him in thought through the new regions he disclosed, never startled or troubled by one noxious allusion.

• (Day 25) the friendly frankness, as correct as cordial, with which he treated me, drew me to him. I felt at times as if he were my relation rather than my master. So happy, so gratified did I become with this new interest. . .

• (Day 28) And was Mr. Rochester now ugly in my eyes? No, reader: gratitude, and many associations, all pleasurable and genial, made his face the object I best liked to see; his presence in a room was more cheering than the brightest fire. Yet I had not forgotten his faults; indeed, I could not, for he brought them frequently before me. He was proud, sardonic, harsh to inferiority of every description: in my secret soul I knew that his great kindness to me was balanced by unjust severity to many others. I believed he was naturally a man of better tendencies, higher principles, and purer tastes I thought there were excellent materials in him; I cannot deny that I grieved for his grief, whatever that was, and would have given much to assuage it.

• (Day 29) Till morning dawned I was tossed on a buoyant but unquiet sea, where billows of trouble rolled under surges of joy.

• (Day 45) "beauty is in the eye of the gazer." My master's colourless, olive face, square, massive brow, broad and jetty eyebrows, deep eyes, strong features, firm, grim mouth,-all energy, decision, will,-were not beautiful, according to rule; but they were more than beautiful to me; they were full of an interest, an influence that quite mastered me,-that took my feelings from my own power and fettered them in his. I had not intended to love him; the reader knows I had wrought hard to extirpate from my soul the germs of love there detected; and now, at the first renewed view of him, they spontaneously arrived, green and strong! He made me love him without looking at me. Did I forbid myself to think of him in any other light than as a paymaster? Blasphemy against nature! Every good, true, vigorous feeling I have gathers impulsively round him. while I breathe and think, I must love him."

We note that the days 17-25 can be attributed only approximately from the text.

Figure 2 :

 2 Figure 2:The intensity of feelings 𝛼 demonstrating universal scaling exponent 1∕2 as a function of the variable parameter (time 𝑡) for 3 books (black -"Romeo and Juliet", red -"The Lily of the Valley", blue -"Martin Eden"). Magenta curve -theory. The error bars indicate the confidence intervals. 𝛼 𝑚𝑎𝑥 is the maximal possible intensity of feelings, and 𝜏 is the characteristic time. In all cases, the intensity of feelings increases as a square root of time, in agreement with the theory.

Figure 3 :

 3 Figure 3: Love from liking. a) The generalized potential 𝐹 in presence of an external bias ℎ: the minimum is at 𝛼 ≠ 0 both above (before) and below threshold (after falling in love). Note the presence of a second (negative) minimum below threshold, meaning that love can turn to hate. b) The evolution of feelings with time 𝛼(𝑡) across the transition for different values of ℎ indicated in the figure legend. The growth is almost square root for a weak liking, but is much smoother for a stronger liking.

Figure 4 :

 4 Figure 4: Love from liking. The time dependence of the intensity of feelings 𝛼(𝑡) for "Jane Eyre" (dots) and theory (line) with a fitting parameter ℎ (bias). Error bars indicate the standard deviation. The growth of feelings is particularly smooth, as opposed to the case of "love at first sight", due to the strong liking ℎ. Here, |ℎ| > ℎ 𝑐 , indicating a particularly strong relationship.

Figure

  Figure5: Love from liking. The intensity of feelings 𝛼(𝑡) for a personal diary (dots) and theory (line) with a fitting parameter ℎ (bias). The growth is faster than in the case of "Jane Eyre", due to a smaller liking ℎ. Here, |ℎ| < ℎ 𝑐 , and the relationship eventually breaks down.