Céline Lamérand
email: lamerand.celine@gmail.com

Liudmila S Shirokova

Mathis Petit

Pascale Bénézeth

Jean-Luc Rols

Oleg S Pokrovsky

Kinetics and mechanisms of cyanobacterially induced precipitation of magnesium silicate

Keywords: magnesium, silicate, precipitation, cyanobacteria, mechanism, rate

Introduction

In the context of climate change and CO2 increase in the atmosphere, carbonate biomineralization interest has risen. The storage of this greenhouse gas in the form of carbonate rocks after dissolution of (ultra)mafic rocks is considered as an efficient remediation pathway for CO2 sequestration, which can be enhanced via the action of cyanobacteria [START_REF] Han | Precipitation of calcite induced by Synechocystis sp. PCC6803[END_REF][START_REF] Lamérand | Olivine dissolution and hydrous Mg carbonate and silicate precipitation in the presence of microbial consortium of photo-autotrophic and heterotrophic bacteria[END_REF][START_REF] Mccutcheon | A Greenhouse-Scale Photosynthetic Microbial Bioreactor for Carbon Sequestration in Magnesium Carbonate Minerals[END_REF][START_REF] Mccutcheon | Microbially accelerated carbonate mineral precipitation as a strategy for in situ carbon sequestration and rehabilitation of asbestos mine sites[END_REF]. Numerous experimental studies dealt with calcium or magnesium carbonate precipitation, in both abiotic and biotic conditions under low Si concentration in solution (e.g., [START_REF] Lee | Calcium carbonate formation by Synechococcus sp. strain PCC 8806 and Synechococcus sp. strain PCC 8807[END_REF][START_REF] Martinez | Effects of freshwater Synechococcus sp. cyanobacteria pH buffering on CaCO3 precipitation: Implications for CO2 sequestration[END_REF][START_REF] Mavromatis | Magnesium isotope fractionation during hydrous magnesium carbonate precipitation with and without cyanobacteria[END_REF][START_REF] Power | Microbially Mediated Mineral Carbonation: Roles of Phototrophy and Heterotrophy[END_REF][START_REF] Shirokova | Using Mg isotopes to trace cyanobacterially mediated magnesium carbonate precipitation in alkaline lakes[END_REF]. However, during Mg-bearing ultramafic mineral dissolution, high concentrations of Si, even beyond the saturation state with respect to amorphous silica, may arise [START_REF] Daval | Influence of amorphous silica layer formation on the dissolution rate of olivine at 90 °C and elevated pCO2[END_REF][START_REF] Lazaro | The kinetics of the olivine dissolution under the extreme conditions of nano-silica production[END_REF].

Biotically-induced formation of hydrous Mg carbonates and Mg silicate is well documented from field studies [START_REF] Del Buey | Microbially induced palygorskite-sepiolite authigenesis in modern hypersaline lakes (Central Spain)[END_REF][START_REF] Gérard | Key Role of Alphaproteobacteria and Cyanobacteria in the Formation of Stromatolites of Lake Dziani Dzaha (Mayotte, Western Indian Ocean)[END_REF][START_REF] Milesi | Formation of magnesium-smectite during lacustrine carbonates early diagenesis: Study case of the volcanic crater lake Dziani Dzaha (Mayotte -Indian Ocean)[END_REF]. In alkaline waters of the Mono Lake (California), Mg-silica precipitation is induced on the surface of cyanobacteria and exopolymeric substances (EPS) during microbial biofilm decaying [START_REF] Souza-Egipsy | Mg-silica precipitation in fossilization mechanisms of sand tufa endolithic microbial community, Mono Lake (California)[END_REF].

A hydrated amorphous Mg-Si phase was reported to occur in sediments of alkaline lakes [START_REF] Arp | Microbialite Formation in Seawater of Increased Alkalinity, Satonda Crater Lake, Indonesia[END_REF][START_REF] Benzerara | Nanotextures of aragonite in stromatolites from the quasi-marine Satonda crater lake, Indonesia[END_REF][START_REF] Pozo | Origin of kerolite and associated Mg clays in palustrine-lacustrine environments. The Esquivias deposit (Neogene Madrid Basin, Spain)[END_REF][START_REF] Spencer | Geochemistry of great Salt Lake, Utah II: Pleistocene-Holocene evolution[END_REF][START_REF] Tutolo | Experimental examination of the Mg-silicate-carbonate system at ambient temperature: Implications for alkaline chemical sedimentation and lacustrine carbonate formation[END_REF][START_REF] Wright | An abiotic model for the development of textures in some South Atlantic early Cretaceous lacustrine carbonates[END_REF][START_REF] Zeyen | Formation of low-T hydrated silicates in modern microbialites from Mexico and implications for microbial fossilization[END_REF]. In modern hypersaline lagoons, elevated pH due to cyanobacterial photosynthesis, combined with high Mg concentration due to hypersaline environments and high Si concentration, originated from diatom frustule dissolution, lead to preferential nucleation of Mg-Si phase on the EPS and sheaths of filamentous and coccoid cyanobacteria [START_REF] Burne | Stevensite in the modern thrombolites of Lake Clifton, Western Australia: A missing link in microbialite mineralization[END_REF][START_REF] Pace | Microbial and diagenetic steps leading to the mineralisation of Great Salt Lake microbialites[END_REF][START_REF] Pace | Formation of stromatolite lamina at the interface of oxygenic-anoxygenic photosynthesis[END_REF].

The mechanisms invoked in abiotic formation of Mg carbonate and silicate include primarily the pH control on Mg carbonate precipitation rate [START_REF] Mavromatis | Magnesium isotope fractionation during hydrous magnesium carbonate precipitation with and without cyanobacteria[END_REF][START_REF] Power | Biologically induced mineralization of dypingite by cyanobacteria from an alkaline wetland near Atlin, British Columbia, Canada[END_REF][START_REF] Shirokova | Using Mg isotopes to trace cyanobacterially mediated magnesium carbonate precipitation in alkaline lakes[END_REF] and Mg silicate structure [START_REF] Jones | Sepiolite and palygorskite[END_REF], notably production of layered 2:1 structure at high pH, and sepiolite-like products and amorphous silica at lower pH [START_REF] Darragi | Authigenic trioctahedral smectites controlling pH, alkalinity, silica and magnesium concentrations in alkaline lakes[END_REF][START_REF] Jones | Clay mineral diagenesis in lacustrine sediments[END_REF][START_REF] Tosca | Chemical controls on incipient Mg-silicate crystallization at 25 °C: Implications for early and late diagenesis[END_REF]. However, in contrast to numerous studies of the effects of pH, Si:Mg ratio and salinity of the parent solution on the nature of Mg-silicates precipitated in the inorganic systems (i.e., [START_REF] Arizaleta | A rate law for sepiolite growth at ambient temperatures and its implications for early lacustrine diagenesis[END_REF][START_REF] Kent | Mg 2+ removal in the system Mg 2+ -amorphous SiO2-H2O by adsorption and Mg-hydroxysilicate precipitation[END_REF][START_REF] Siffert | Synthèse d'une sépiolite à température ordinaire[END_REF][START_REF] Tosca | Chemical controls on incipient Mg-silicate crystallization at 25 °C: Implications for early and late diagenesis[END_REF][START_REF] Tutolo | Experimental examination of the Mg-silicate-carbonate system at ambient temperature: Implications for alkaline chemical sedimentation and lacustrine carbonate formation[END_REF][START_REF] Wollast | Experimental precipitation and genesis of sepiolite at earth-surface conditions[END_REF], only two very recent experimental studies of Mg silicate precipitation in the presence of cyanobacterial biofilm were conducted [START_REF] Moore | Biologically mediated silicification of marine cyanobacteria and implications for the Proterozoic fossil record[END_REF][START_REF] Moore | A new model for silicification of cyanobacteria in Proterozoic tidal flats[END_REF], although the rates and stoichiometry of reactions were not assessed.

As a working hypothesis, one can suggest, based on the bulk of available information from the field studies and laboratory experiments with inorganic systems [START_REF] Pace | Microbial and diagenetic steps leading to the mineralisation of Great Salt Lake microbialites[END_REF][START_REF] Pace | Formation of stromatolite lamina at the interface of oxygenic-anoxygenic photosynthesis[END_REF][START_REF] Tosca | Chemical controls on incipient Mg-silicate crystallization at 25 °C: Implications for early and late diagenesis[END_REF], and following a few available experimental studies with cyanobacteria [START_REF] Shirokova | Using Mg isotopes to trace cyanobacterially mediated magnesium carbonate precipitation in alkaline lakes[END_REF][START_REF] Moore | Biologically mediated silicification of marine cyanobacteria and implications for the Proterozoic fossil record[END_REF][START_REF] Moore | A new model for silicification of cyanobacteria in Proterozoic tidal flats[END_REF] that the photosynthetic activity of cyanobacteria could increase the pH to highly alkaline values therefore inducing the formation of hydrous Mg carbonates and silicates. Further, EPS secreted by cyanobacteria could act either as templates for Mg-silicate formation within the microbial mats [START_REF] Burne | Stevensite in the modern thrombolites of Lake Clifton, Western Australia: A missing link in microbialite mineralization[END_REF][START_REF] Del Buey | Microbially induced palygorskite-sepiolite authigenesis in modern hypersaline lakes (Central Spain)[END_REF][START_REF] Moore | Biologically mediated silicification of marine cyanobacteria and implications for the Proterozoic fossil record[END_REF][START_REF] Moore | A new model for silicification of cyanobacteria in Proterozoic tidal flats[END_REF][START_REF] Perri | Carbonate and silicate biomineralization in a hypersaline microbial mat (Mesaieed sabkha, Qatar): Roles of bacteria, extracellular polymeric substances and viruses[END_REF][START_REF] Souza-Egipsy | Mg-silica precipitation in fossilization mechanisms of sand tufa endolithic microbial community, Mono Lake (California)[END_REF][START_REF] Zeyen | Formation of low-T hydrated silicates in modern microbialites from Mexico and implications for microbial fossilization[END_REF] or as inhibitors of mineral precipitation due to complexation of their cation-binding groups with aqueous Mg 2+ ion and decreasing the solution saturation state with respect to the mineral end products.

Towards resolving the impact of cyanobacteria on authigenic Mg and Si hydrous mineral formation, we performed laboratory experiments with different cyanobacteria in carbonate-bearing solutions with variable Si : Mg ratios. For this, we used freshwater planktonic cyanobacteria because of their strong biomineralization capacity and we avoided too complex natural-like solutions (containing Ca, Fe, PO4 or SO4) to eliminate possible precipitation of Ca carbonate and phosphate, Fe silicate phases or to induce an inhibition of nucleation by some components (sulfate, phosphate). The reason for choosing this restricted aquatic environment was to assess the mechanisms of reaction and to quantify the rates of mineral precipitation in the Mg -CO2 -SiO2 -H2O system without interferences from other unknown factors and secondary phases. We aimed at quantifying the rates of solid phase precipitation and linking the nature and stoichiometry of Mg carbonates and silicates to the identity of various cyanobacteria and chemical composition of aqueous solution. The obtained results should allow better understanding of natural alkaline lake environments leading to the formation of authigenic Mg silicates and carbonates and may be useful for constraining optimal conditions for efficient CO2 sequestration during dissolution of primary Mg silicate rocks.

Material and methods

Culture of cyanobacteria

The biotic experiments were performed with three photoautotrophic cyanobacteria; each species was studied in a monoculture. The Synechococcus sp. PCC 7942 cyanobacterium was isolated from the surface of hydrous Mg carbonate stromatolites growing in the Salda Lake (Turkey) and used previously in mineral carbonation experiments in Si-free solutions (eg [START_REF] Shirokova | Using Mg isotopes to trace cyanobacterially mediated magnesium carbonate precipitation in alkaline lakes[END_REF]. Cells of Synechococcus are elongated and isolated; the thickness of surrounding EPS layer is between 20 and 30 nm (Fig. 1A). The Chroococcidiopsis sp. SAG 2023, obtained from the culture bank of the Experimental Phycology and Culture Collection of Algae (EPSAG, Germany), is commonly found in groups of two to eight cells, enclosed in a transparent sheath of 250 to 500 nm thick, rich in EPS which cover the cells (Fig. 1B). Cells of Aphanothece clathrata SAG 23.99, also obtained from the EPSAG, are spherical and covered with a thin (15-30 nm) EPS layer (Fig. 1C). The interest of using three different cyanobacteria was to test the impact of external cell envelopes and cell morphology on the rate of precipitation and the nature of formed solid products at similar initial solution composition and supersaturation degree. Each cyanobacterium was grown in the BG-11 (Sigma-Aldrich C3061) freshwater nutrient solution. Before the experiment inoculation, cells were rinsed twice in appropriate fresh culture media using centrifugation (~500 mL of solution for 1 g of wet biomass) at 2000 g to remove, as possible, the adsorbed organic matter, nutrients and metals accumulated from the growth media, and cell exudates from the surface.

Experimental setup

Abiotic batch experiments were conducted in 500 mL polycarbonate bottles placed in an IKA HS 260 orbital shaker (100 rpm) at 25 °C. Experiments were run in a 0.01 M NaHCO3 solution enriched with Si and Mg at different Si:Mg ratios. Although this bicarbonate concentration is five times higher than the contemporary seawater, it is consistent with high alkalinity environments of modern Mg carbonate formation such as the Salda Lake [START_REF] Shirokova | Using Mg isotopes to trace cyanobacterially mediated magnesium carbonate precipitation in alkaline lakes[END_REF]. Furthermore, our previous tests demonstrated a lack of Mg carbonate precipitation at lower concentrations of NaHCO3 in the initial solution, whereas at higher NaHCO3 concentrations, Mg silicate could not be formed. Magnesium was added from a MgCl2×6H2O (Alfa Aesar) concentrated solution; Si was added from a Na2SiO3×5H2O (Sigma Aldrich) concentrated solution. For each experiment, pH was fixed at four different values (8.5, 9.5, 10.5 and 11.5) using a NaOH solution.

Biotic experiments were conducted in duplicates in batch reactors (500 mL polycarbonate Erlenmeyer flasks closed with sterilized porous plugs (Bio-Silico®)), under continuous stirring via magnet stir bar and a constant light of 2500 lux (~40 µmol m -2 s -1 ). The abiotic controls included the same aqueous solution without cyanobacterial cells in which the pH was raised artificially via addition of NaOH to the level similar of that in the biotic runs. Another control included live but non photosynthesizing cells in the darkness with Si : Mg ratio of 0.52, but without artificial pH rise. All manipulations were performed in a sterile environment using a class A 100 laminar flow hood box. For the three cyanobacteria described before, experiments were run with 1 gwet L -1 of cyanobacterial biomass, in a carbonate-rich medium (0.01 M NaHCO3) at different Si:Mg ratios.

A 10% BG-11 growth medium was added to help promoting the development of cyanobacteria.

The chemical composition of all abiotic and biotic experiments is presented in Table 1.

Sampling and analyses

Aliquots of the experimental solutions were taken every day during the first week and afterwards according to the evolution of the physicochemical parameters, typically every 2-3 days, under sterile conditions. This allowed to follow the evolution of different biological (cyanobacterial biomass) and physicochemical (pH, Mg, Si, dissolved organic carbon (DOC), alkalinity) parameters and thus to characterize the bacterial development, evolution of solution saturation degree, and precipitation of secondary minerals.

After sampling, the aliquot of 13 mL of homogeneous suspension was subdivided into several portions. The 3 mL of unfiltered sample were used to measure pH and optical density. The remaining 10 mL of this volume was centrifuged (2000 g, 30 min) and the supernatant was filtered through a 0.45 µm filter. About 7 mL of the filtrate was used for the analysis of Si, DOC and alkalinity. The other part of the filtrate was acidified (pH = 2) using HNO3 for Mg analysis. The filtered samples were stored in the refrigerator (+ 4°C) pending analysis.

Cyanobacterial biomass was calculated via optical density measurement using a Helios Epsilon spectrophotometer at a wavelength of 680 nm (OD680nm), which corresponds to the absorption peak of chlorophyll-a, based on methodology developed for experiments with single cyanobacterial species [START_REF] Mavromatis | Magnesium isotope fractionation during hydrous magnesium carbonate precipitation with and without cyanobacteria[END_REF][START_REF] Shirokova | Using Mg isotopes to trace cyanobacterially mediated magnesium carbonate precipitation in alkaline lakes[END_REF][START_REF] Bundeleva | Experimental modeling of calcium carbonate precipitation by cyanobacterium Gloeocapsa sp[END_REF][START_REF] Grimm | The temporal evolution of carbon isotope compositions of calcite in the presence of cyanobacteria[END_REF]. This wavelength is preferred for the estimation of cyanobacterial biomass if the solution is turbid due to mineral particles [START_REF] Grimm | Enhancement of cyanobacterial growth by riverine particulate material[END_REF][START_REF] Myers | Improving accuracy of cell and chromophore concentration measurements using optical density[END_REF]. A linear relationship (r 2 > 0.90, p < 0.01) was obtained between cyanobacterial biomass and OD680nm using a set of cyanobacteria samples of different biomass concentrations; the uncertainty of this measurement was estimated as ±10%. The conversion factor from OD680nm to cyanobacterial dry biomass (gdry/L) was equal to 0.519 for Synechococcus sp., 0.332 for Chroococcidiopsis sp. and 0.371 for A. clathrata, and the ratio wet/dry biomass of all three species was 8 ± 0.5.

The pH was measured with a Mettler Toledo InLab Micro combined electrode calibrated with NIST buffers (4.01, 6.99 and 10.01), with an uncertainty of 0.01 units. Alkalinity was measured via standard HCl titration using an automatic Schott TitroLine alpha plus titrator, with an uncertainty of 1% and a detection limit of 5•10 -5 M. Dissolved inorganic carbon (DIC) concentration was calculated from pH and alkalinity. DOC concentration was determined using a Shimadzu TOC-VCSN analyzer with an uncertainty of 5% and a detection limit of 0.1 mg Corg L - 1 . Magnesium concentration was measured by flame atomic absorption spectroscopy using a Perkin Elmer AAnalyst 400 with an uncertainty of 1% and a detection limit of 1•10 -7 M. The total dissolved silica concentration was measured by colorimetry with the molybdate blue method using a colorimetric chain Technicon Autoanalyzer II, with an uncertainty of 2% and a detection limit of 3•10 -7 M.

The mineral saturation indices (SI) were calculated using the PHREEQC geochemical code [START_REF] Parkhurst | User's guide to PHREEQC (Version 2)-A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF], with its llnl.dat thermodynamic database modified to include new equilibrium constants of solid phases and aqueous species in the Ca-Mg-CO2-H2O system from [START_REF] Palmer | Potentiometric measurements of the first hydrolysis quotient of magnesium (II) to 250 °C and 5 molal ionic strength (NaCl)[END_REF], [START_REF] Millero | The dissociation of carbonic acid in NaCl solutions as a function of concentration and temperature[END_REF], [START_REF] Bénézeth | Experimental determination of the solubility product of magnesite at 50 to 200 °C[END_REF], [START_REF] Stefánsson | Carbonic acid ionization and the stability of sodium bicarbonate and carbonate ion pairs to 200 °C-A potentiometric and spectrophotometric study[END_REF][START_REF] Stefánsson | Potentiometric and spectrophotometric study of the stability of magnesium carbonate and bicarbonate ion pairs to 150 °C and aqueous inorganic carbon speciation and magnesite solubility[END_REF], [START_REF] Gautier | Hydromagnesite solubility product and growth kinetics in aqueous solution from 25 to 75 °C[END_REF][START_REF] Harrison | Solubility of the hydrated Mgcarbonates nesquehonite and dypingite from 5 to 35 °C: Implications for CO2 storage and the relative stability of Mg-carbonates[END_REF].

Microscopic observations and spectroscopic characterization of reaction products

For Transmission Electron Microscopy (TEM) observations, MilliQ-rinsed samples were treated in two different ways: (i) a drop of fresh suspension was deposited on a carbon-coated copper grid, and (ii) ultrathin slices were prepared from freshly sampled cells and precipitated minerals which were fixed with 2.5% glutaraldehyde / 2% paraformaldehyde in 0.1 M Cacodylate buffer (pH 7.2) at ambient temperature for 1.5 h, washed three times with 0.1 M Cacodylate, and subsequently soaked in 1% osmium tetroxide in 0.1 M Cacodylate buffer at ambient temperature for 1 h and finally washed three times with 0.1 M Cacodylate. The supernatant was removed and pellets were embedded in agarose, then samples were taken through stepwise ethanol dehydration (15 min at 25, 50, 70 and 90% ethanol at ambient temperature, and 3x30 min at 100% dry ethanol).

Samples were then embedded in epoxy resin and 80 nm thick ultrathin sections were processed with a Leica UCT ultramicrotome. Ultramicrocuts were mounted on Cu formvar-carbon grids, and electron stained with uranyl and lead citrate. Then samples were characterized using TEM with a JEOL JEM-1400 HC (40-120 kV) and a JEOL JEM-2100F (200 kV) coupled with a SDD Brucker EDS analyzer with an energy resolution of 127 eV. Fast Fourier Transform (FFT) analyses were performed to characterize possible crystallinity of samples.

Prior to characterization by Scanning Electron Microscopy (SEM), samples were centrifuged (100 g, 30 min) and the organic matter was removed from the solid phase by treating them in aqueous 10% H2O2 for a night, following the methodology elaborated for Mg-HCO3cyanobacteria system [START_REF] Shirokova | Using Mg isotopes to trace cyanobacterially mediated magnesium carbonate precipitation in alkaline lakes[END_REF]. The remaining solids were rinsed three times with MilliQ water and freeze-dried at -60 °C. They were observed after carbon film coated on the sample surface with a JEOL JSM 6360LV SEM coupled with a SDD PGT Sahara EDS analyzer operating at 30 kV.

The precipitated minerals and mineral-bacterial cell aggregates were also analyzed by Xray diffraction using a D8-Advance Bruker diffractometer (Cu-Ka1 radiation, step size 0.0142 (2 h), scan speed 10 s/step), and via FT-IR spectroscopy using a Perkin Elmer Spectrum 100 equipped with a ZnSeO4 crystal in the wave number range from 650 to 4000 cm -1 .

Results

Abiotic experiments without biomass and with non-photosynthesizing cells

Abiotic experiments were performed at three different Si : Mg ratios and four different pH values. At pH 8.5 and 9.5, no decrease of Si or Mg concentrations in solution occurred; presumably, there was no precipitation of Mg silicate (not shown). At pH=10.5, with 0.52 Si : Mg molar ratio, both Si and Mg concentrations decreased immediately by ca. 50%, then Mg concentrations remained stable at ~2 mM and Si concentrations decreased down to ~0.1 mM over 96 h of reaction time (Figs. S1A andB), indicating the precipitation of an Si, Mg-bearing phase. The evolution of Mg and Si concentrations was similar for 0.26 Si : Mg molar ratio. For the 1.73 Si : Mg molar ratio, Mg concentration decreased from 1.2 to 0.3 mM over 96 hrs. Silica concentrations decreased immediately from 2.1 to 1.6 mM then further decreased to 1 mM over 20 h and remained stable until the end of experiments. At pH 11.5, Si concentrations immediately decreased from 2.1 to 1.5 mM and then remained stable (Fig. S1B).

Aqueous solution Saturation Indices (SI) calculated with PHREEQC during experiments indicated a possible precipitation of a Mg silicate, sepiolite (Mg4Si6O15(OH)2•6H2O) given that at pH > 10.5 the SI of sepiolite decreased from 11…20 to -1…14 depending on experiments (Fig. S2 A-B). The precipitation was most likely to occur at pH 11.5. Magnesium hydroxide (brucite, Mg(OH)2) could also be formed but the decrease of its SI was rather low (0.5 -0.8 to 0.4 -0.5).

At pH < 10, the SI of both sepiolite and brucite were stable and low, indicating no possible precipitation of these minerals (not shown).

During control experiments in the presence of 1 gwet L -1 live but non photosynthesizing Synechococcus sp. cells in the darkness with Si : Mg molar ratio of 0.52, the biomass slightly decreased, the pH remained stable at 8.6 and no precipitation was observed. This could be explained by the absence of pH and supersaturation rise due to lack of photosynthesis. Indeed, artificial increase of pH from 8.6 to 10.7 (at 96 hours) via an addition of NaOH to the experimental suspension kept in the darkness yielded instantaneous precipitation of Si-Mg amorphous phase: concentrations of Mg and Si dropped from 4 to 0.3 mM and 2 to 1.2 mM, respectively (Fig. S3).

Biotic experiments with live photosynthesizing cyanobacteria

The concentration of cyanobacterial biomass increased during 10 days of the experiments from 0.1 to 0.4 gdry L -1 , 0.05 to 0.2 gdry L -1 and 0.08 to 0.1 gdry L -1 for Synechococcus sp., Chroococcidiopsis sp. and A. clathrata, respectively (Fig. 2 A Mg ratio (see Table S1). The DIC concentration was stable during the first 72 hrs, then it decreased from 11 to 6 mM for 0.26 Si : Mg ratio (Table S1) and from 12 to 7 mM for 0.52 Si : Mg ratio (Fig. 3 C), suggesting a formation of some Mg-bearing carbonate phase. For the 1.73 Si : Mg ratio, the evolution of Mg and Si concentration was similar (Figs. 3 D andE) but there was no decrease in DIC concentrations (Fig. 3

Rates and stoichiometry of Mg, Si and C removal in abiotic and biotic experiments

The Mg, Si and DIC apparent removal rates (Δi/∆t where i = Mg, Si or DIC) during t = first 100 hours in abiotic and biotic experiments were calculated following the approach of [START_REF] Arizaleta | A rate law for sepiolite growth at ambient temperatures and its implications for early lacustrine diagenesis[END_REF] for sepiolite precipitation and listed in Table 2. In biotic experiments, the ΔMg/∆t value in the presence of Synechococcus sp. and Chroococcidiopsis sp. were rather similar (~ 0.012 mmol L -1 h -1 for the Si : Mg ratio of 1.73 and ~0.033 mmol L -1 h -1 for other ratios). In abiotic experiments, the decrease of Mg was faster for the Si:Mg ratio of 1.73 (~ 0.041 mmol L -1 h -1 ) but similar for other ratios (~0.029 mmol L -1 h -1 ). The rates of Si removal (ΔSi/∆t) were similar in all biotic experiments regardless of Si : Mg ratios and in abiotic experiments with Si:Mg ratio of 0.52 and 0.26 (~0.011 mmol L -1 h -1 ). The ΔSi/∆t was 3 times lower in abiotic experiments with Si:Mg ratio of 1.73 (~0.003 mmol L -1 h -1 ). The ΔDIC/∆t was similar for biotic experiments with Synechococcus sp. and Chroococcidiopsis sp. (~0.016 mmol L -1 h -1 for Si:Mg ratio of 1.73, ~0.057 mmol L -1 h -1 for ratios of 0.52 and 0.26) and for abiotic experiments (~0.015 mmol L -1 h -1 for Si : Mg ratio of 1.73, ~0.053 mmol L -1 h -1 for ratios of 0.52 and 0.26). For biotic experiments in the presence of A. clathrata, the decrease of DIC during the first 100 hours of growth was the same for the three ratios (~0.011 mmol L -1 h -1 ).

The rates of Mg, Si and DIC concentration decrease over the first 100 hours of biotic and abiotic experiments have been reported as a function of SI of sepiolite (Fig. S4 A, B andC) and as a function of initial Si : Mg molar ratio in solution (Fig. S4 D, E andF). There was no correlation between these different parameters, showing no relation between the initial composition of aqueous solution and the apparent rates of solid phase precipitation. These rates were similar in biotic experiments with Synechococcus sp. and Chroococcidiopsis sp. and abiotic experiments, except for initial Si:Mg ratio of 1.73 where the abiotic ratios were 3 times smaller. Rates were smaller in biotic experiments in the presence of A. clathrata compared to other cyanobacteria, except with Si:Mg molar ratio of 1.73 where all ratios were similar.

In order to compare solid phase precipitation efficiency among cyanobacterial species, the apparent rates of element removal Δi/∆t (i = Mg, Si, DIC) in biotic experiments were normalized to dry biomass produced during the period of time t yielding efficient precipitation rate Ri (mmol h -1 gdry -1 ):

𝑅𝑖 = [𝑖] 𝑡 𝑏𝑖𝑜𝑚𝑎𝑠𝑠
where [i] is the change in element concentration over the t period and biomass is the change in the cyanobacterial biomass during the same period. Results are listed in Table 2 and illustrated for the period of maximal change in Mg, Si and DIC concentration (50 to 150 h) in the form of a histogram (Fig. 4). The silica removal rates (RSi) ranged from 0.02 to 0.2 mmol Si h -1 gdry -1 , and responded differently to Si: Mg variation for different cyanobacteria. The RSi decreased when Si : Mg ratio increased for Synechococcus sp. and Chroococcidiopsis sp. but it increased with Si : Mg ratio in the presence of A. clathrata (Fig. 4 A). Biomass-normalized Mg removal rates RMg were the highest for Chroococcidiopsis sp. and about similar for Synechococcus sp. and A. clathrata (Fig. 4 B). Finally, the RDIC values ranged from 0.3 to 1.0 mmol C h -1 gdry -1 for Synechococcus sp. and Chroococcidiopsis sp. at Si:Mg molar ratio of 0.26 and 0.52 and were zero to slightly negative for all three species at higher Si in solution (Fig. 4 C). There was no link (p < 0.05) between RSi, RMg and sepiolite saturation index of solution (not shown).

The carbon sequestration potential (CSP), calculated as the sum of inorganic and organic C retained in the solid mineral phase and newly produced biomass over the first 150 h of cell growth ranged from 0.1 to 0.9 mmol C L -1 day -1 for different species. It was an order of magnitude higher for Synechococcus sp. and Chroococcidiopsis sp. compared to A. clathrata at Si:Mg molar ratio < 1. The inorganic share of the CSP was 3 to 4 times lower at Si:Mg molar ratio > 1 for Synechococcus sp. and Chroococcidiopsis sp. and similar among all species (Fig. 5).

Microscopic observations

The SEM observations of the solid products formed in both abiotic and biotic experiments showed that the morphology of the precipitated phases depended on the experimental conditions. Inorganic experiments without bacteria yielded solid grains of irregular shape, ~100-200 µm in size, with rough, grainy surfaces (Figs. 6 A andB). After experiments with cyanobacteria Chroococcidiopsis sp., smaller particles (~30-50 µm) were obtained (Figs. 6 C andD). They were composed of aggregates of small spheres and/or spherical particles likely precipitated in the vicinity of the cells and forming a clump of amorphous Mg silicate. The phases obtained with Synechococcus sp. (Fig. 6 E) and A. clathrata (Fig. 6 F) had the same appearance as the ones obtained with Chroococcidiopsis sp., and overall, the SEM observations did not show differences between the phases obtained with different cyanobacteria.

In contrast, the TEM observations of cyanobacterial cells and associated newly-formed solid phases showed different precipitation features depending on the identity of cyanobacterium.

For Synechococcus sp., nanometric EPS-like filaments were observed around the cells, together with some solid products (indicated by a blue arrow in Fig. 7 A). For Chroococcidiopsis sp., the sheaths of the cells were covered with some electron-dense material which is likely to be a newlyformed product (Fig. 7 B). Residues of the sheaths, probably formed after cell death, were also present and covered with the same solid products. For A. clathrata, the inorganic precipitates around the cells were similar to Chroococcidiopsis sp. but they were not attached to the cells being likely unrelated to organic templating, although it is not excluded that some EPS or other components could have acted as nucleation sites (Fig. 7 C). Some solid amorphous phase was detectable inside the cells (Fig. 7 D). Overall, the amount of newly formed precipitates seemed to be more significant in Chroococcidiopsis sp. experiments. The FFT analyses of the solid phase on all samples showed a diffuse ring, indicating the absence of crystallized phases as illustrated for Chroococcidiopsis sp. cells in Fig. 7 E. A STEM map of a cell of Chroococcidiopsis sp. allowed characterizing the chemical composition of the cell, its environment and the newly formed solid phase (Fig. S5). The sheath of the cell contained Si, Mg and O, which was consistent with a newlyformed Si, Mg-rich phase such as amorphous Mg silicate. The white particle at the top left of the cell (Fig. S5 A) had the same composition as those at the sheath; it could be formed on the cell sheath and then detached into solution, or formed directly in the bulk solution due to overall pH rise.

Spectroscopic analyses

The XRD analysis of biotic products demonstrated the presence of poorly crystalline phase with broad peaks of brucite (Fig. 8 A). Note that, due to its poor crystallinity and small quantity, this brucite-like compound could not be identified by FFT analysis. The FTIR analyses were performed on samples from both abiotic and biotic experiments. Spectra of solid precipitates formed after 10 days of experiments with Synechococcus sp. are presented in Fig. 8 B. In all spectra, peaks indicating a hydrous Mg silicate phase such as amorphous analogue of sepiolite (Lazarevic et al., 2207;[START_REF] Tosca | Chemical controls on incipient Mg-silicate crystallization at 25 °C: Implications for early and late diagenesis[END_REF] were present: a peak around 3680 cm -1 for the vibrations of Mg-OH group; a large band at 3300-3400 cm -1 corresponding to the -OH vibration of water; a peak at 1640 cm -1 corresponding to the vibration of -OH groups of water in hydrogen-bonded silanols; a peak at 1000 cm -1 which could correspond to Si-O or C-O vibration, with a shoulder of low amplitude at 900 cm -1 corresponding to Si-H and/or Si-OH vibrations. In experiments with Si:Mg ratio of 0.52 and 0.26, another peak was observed at 1410 cm -1 , large and weak, corresponding to the vibration of CO3 2-groups.

The FTIR spectra from experiments with the other two cyanobacteria were similar to those described above; there was no difference between the phases formed in presence of the different cyanobacteria. Spectra of abiotic samples also showed similar peaks (Fig. 8 B). The organic stretch modes were not detected in samples from biotic experiments, presumably due to insufficient amount of organic material associated with Mg-Si precipitates.

Discussion

Role of cell sheaths and EPS in the precipitation of Mg silicate and carbonate

The evolution of Si, Mg and DIC concentrations during the experiments demonstrated that when Mg was initially in excess to Si (Si : Mg molar ratio of 0.52 and 0.26), and regardless of the degree of solution saturation with respect to amorphous silica, both Mg silicate and carbonate were formed, whereas when Si was initially in excess to Mg (Si : Mg ratio of 0.26), solely Mg silicate precipitated. Similarly, authigenic clay mineral formation in continental saline environments often include significant removal of Mg at high Si concentrations [START_REF] Calvo | Authigenic clay minerals in continental evaporitic environments[END_REF]. Regardless of chemical composition of aqueous solution, the newly-formed phases were amorphous from the TEM observations and associated FFT analyses. The FTIR spectra confirmed these results, indicating the presence of the Mg hydrous silicate and the Mg hydrous carbonates, whereas the XRD analyses did not show the presence of crystalline Mg silicates. The formation of amorphous Mg-rich silica phases has been recently demonstrated in experiments with cyanobacterial biofilm [START_REF] Moore | A new model for silicification of cyanobacteria in Proterozoic tidal flats[END_REF].

We hypothesize that the excess of Si in solution precluded the formation of Mg carbonate because of kinetic control on Mg and Si removal: the rate of Mg silicate formation is faster than that of the Mg carbonate. This possibility is consistent with the difference in DIC concentration evolution in solutions with various Si:Mg ratios. For the experiment with a Si:Mg ratio of 1.73, all Mg was precipitated with Si in the form of Mg silicate, or as individual semi-amorphous Mg hydroxide before Mg carbonate could be formed, whereas in solutions with 0.52 and 0.26 Si:Mg ratios, the lesser quantity of Si induced a slower Mg silicate precipitation rate which thus became closer to that of the Mg carbonate. At these conditions, both Mg silicate and Mg carbonate were able to precipitate. Note that there was little amount of the solid phase associated with A. clathrata which is consistent with lower rates and degree of Mg, Si concentration decrease during the experiments, probably because of a lower growth rate of this species, not capable to produce sufficient increase of solution pH and saturation index.

The microscopic observations showed a preferential precipitation of secondary phases in the vicinity of the cell surfaces (Fig. 7). In particular, an Si-Mg-O rich amorphous phase was formed in the EPS-rich sheath of Chroococcidiopsis sp. cells. These results are in favor of a heterogeneous nucleation process, where negatively-charged EPS attached to cyanobacterial cell wall or liberated to aqueous solution could serve as nucleation centers via complexing, first, Mg 2+ cations and then, attracting negatively charged silica anions as recently proposed in experimental study of [START_REF] Moore | Biologically mediated silicification of marine cyanobacteria and implications for the Proterozoic fossil record[END_REF][START_REF] Moore | A new model for silicification of cyanobacteria in Proterozoic tidal flats[END_REF]. Similar pathway of EPS-related Mg silicate mineral precipitation in and around cyanobacterial sheaths, and in the alveolar web of the EPS of biofilm were suggested for modern thrombolites in Lake Clifton, Australia [START_REF] Burne | Stevensite in the modern thrombolites of Lake Clifton, Western Australia: A missing link in microbialite mineralization[END_REF], microbial mat of the Qatar sabkha [START_REF] Perri | Carbonate and silicate biomineralization in a hypersaline microbial mat (Mesaieed sabkha, Qatar): Roles of bacteria, extracellular polymeric substances and viruses[END_REF] and for other microbial biofilms [START_REF] Decho | Microbial biofilms in intertidal systems: an overview[END_REF]. It is interesting to note that an amorphous early Mg-Si mineral phase in association with EPS has been interpreted as a precursor phase evolving to dolomite [START_REF] Bontognali | Dolomite formation within microbial mats in the coastal sabkha of Abu Dhabi (United Arab Emirates)[END_REF].

Although the composition and nature of the EPS produced by studied cyanobacteria are not known, we can hypothesize that their configuration does not allow the formation of a cationtrapping gel (i.e., [START_REF] Kühbeck | Evaluation of the nitroaldol reaction in the presence of metal ion-crosslinked alginates[END_REF] and thus they do not inhibit the precipitation. Noteworthy that contrasting behavior of various EPS with respect to mineral precipitation is consistent with a recent conclusion of [START_REF] Moore | A new model for silicification of cyanobacteria in Proterozoic tidal flats[END_REF] on the importance of cyanobacterial EPS in Mg-rich amorphous silica nucleation; the nature of EPS (for example, sulfate enrichment) was an important factor in solid phase precipitation. Moreover, even in the absence of cellular metabolism, cyanobacterial EPS mediated the precipitation of amorphous Si-Mg phase from silicaundersaturated seawater [START_REF] Moore | A new model for silicification of cyanobacteria in Proterozoic tidal flats[END_REF]. In this regard, chemical and structural composition of the EPS produced by different species is likely to be one of the primary factor controlling cyanobacterial bio-silicification (i.e., [START_REF] Golubic | Early cyanobacterial fossil record: Preservation, palaeoenvironments and identification[END_REF][START_REF] Sergeev | Proterozoic fossil cyanobacteria[END_REF][START_REF] Moore | A new model for silicification of cyanobacteria in Proterozoic tidal flats[END_REF].

Rate-controlling factors, possible mechanisms and implication for natural environments

The Synechococcus sp. cells placed in darkness did not demonstrate the biomass increase.

The absence of photosynthesis activity did not allow the pH to increase, and thus no precipitation occurred. Artificial increase of pH from 8.6 to 10.7 via adding NaOH in the solution led to the instantaneous precipitation of an Si-Mg amorphous phase, suggesting that photosyntheticallyinduced pH rise is the main factor responsible for mineral precipitation in these experiments. This result strongly corroborates recent finding of [START_REF] Moore | A new model for silicification of cyanobacteria in Proterozoic tidal flats[END_REF] on essential role of pH rise during Mg-rich amorphous Si precipitation from artificial seawater in the presence of cyanobacterial biofilms.

The apparent initial (0-100 h) rates of Mg, Si and IC removal were similar in the presence of Synechococcus sp. and Chroococcidiopsis sp. but sizably lower (by a factor of 4, 1.5 and 5 for Mg, Si and DIC, respectively) with A. clathrata. Consistent with these results, during biotic experiments, the removal rates of Mg and DIC normalized to produced biomass were the highest for Chroococcidiopsis sp. and the lowest for A. clathrata (see Fig. 4). This is also supported by TEM observations which showed less amounts of solid products associated to A. clathrata cells.

The thicker EPS layer around Chroococcidiopsis cells compared to A. clathrata cells could partially explain the higher precipitation in the presence of Chroococcidiopsis sp., whose EPS provided necessary nucleation sites and allowed a faster precipitation. This EPS layer could help forming a bulk solid phase which remained near the cells, whereas the thinner layer of EPS around Synechococcus cells favored the formation of smaller particles which then detached themselves from the cells. Note that the particles may have also been separated from the cells during the sample preparation. Despite a lack of thick EPS layer around Synechococcus cells, massive growth of this species allowed sizable pH increase necessary to produce high amount of precipitates. The magnitude of pH increase, which was smaller in the presence of A. clathrata cells compared to the other cyanobacteria, could explain the difference in precipitation rates, whereas the presence of a thick or thin EPS layer could be responsible for the formation of particles of different sizes and control their degree of attachment to the cells.

According to the SEM observations, the morphology of the solid phase was different in the experiments with and without cyanobacteria. The particles were significantly smaller in the presence of cyanobacteria, which can indicate some surface template control via active role of cell sheaths, providing the nucleation sites. At the same time, FTIR analyses demonstrated rather similar Mg-O-Si bonds in amorphous Mg-Si phases formed both with and without cyanobacteria, suggesting that the pH was the principal driver of solution supersaturation rise, necessary for the nucleation of Mg-Si hydrous amorphous phase.

The carbon sequestration potential was the highest for Synechococcus sp. and Chroococcidiopsis sp. and dominated (> 60 %) by newly produced biomass. The inorganic form of sequestered C was mostly pronounced for Synechococcus and Chroococcidiopsis species for Si:Mg ratios < 1 (up to 1 mmol C L -1 day -1 ). When Si : Mg ratio was higher than 1, less than 20 % of the sequestered carbon was under inorganic form. Therefore, a decrease of DIC uptake at high Si : Mg ratio is due to Si inhibition of Mg hydrous carbonate formation.

The main implication of obtained results is that, with increasing Si : Mg ratio in the aqueous solution, the rate of C sequestration (in the form of hydrous Mg carbonates) is reduced. This suggests that one has to avoid high Si concentration in the fluid used for cyanobacterially-induced biocarbonation of, for example, olivine leaching products. A possible way to increase the efficiency of bio-carbonation could be to use a double culture of some Si-concentrating organisms such as diatoms which may allow to decrease Si level in the fluid prior placing it in contact with cyanobacteria. At the same time, one has to be extremely careful when scaling up the experimental results to the whole ecosystems. For example, ignoring the heterotrophic bacteria and microbial consortium (i.e., Lamerand et al., 2020) may lead to underestimation of true biomineralization capacity. Intense culture conditions (constant stirring and light) used in the present study can modify the metabolism and biodiversity of microorganisms compared to natural environments. Moreover, because the dominant form of phototrophic organisms, notably in the Precambrian coastal environments Si-rich hot springs, or Mg-rich ultramafic work weathering sites is biofilm, this calls a need for experimental study of competitive biosilicification / biocarbonation in cyanobacterial mats following the pioneering works of [START_REF] Moore | Biologically mediated silicification of marine cyanobacteria and implications for the Proterozoic fossil record[END_REF][START_REF] Moore | A new model for silicification of cyanobacteria in Proterozoic tidal flats[END_REF].

Conclusions

The photosynthetic activity of the three cyanobacteria Synechococcus sp., Chroococcidiopsis sp. and Aphanothece clathrata led to the precipitation of hydrous Mg carbonate, and amorphous Mg silicate. Because the same phenomenon occurred in abiotic solutions at pH > 10.5, pH seemed to be the governing factor of precipitation during our experiments. However, amorphous Mg-Si minerals were preferentially formed within the bacterial sheaths rich in EPS indicating that these organic compounds could serve as nucleation centers and promote the precipitation. When Si was in excess, only a magnesium silicate, such as amorphous analogue of sepiolite, was formed, whereas when Mg was in excess in solution compared to Si, hydrous Mg carbonate such as hydromagnesite could also be formed. The C sequestration potential in the form of inorganic solid phase and newly produced biomass over 150 h of reaction was the highest at Si:Mg molar ratio < 1 in the presence of Synechococcus sp. and Chroococcidiopsis sp. and strongly decreased at Si:Mg ratio > 1. Thus, because Si strongly inhibited the efficiency of inorganic C removal in the presence of cyanobacteria, our recommendation for cyanobacterially-induced C sequestration would be to decrease the concentration of Si in aqueous solution, because under high Si : Mg ratio in water, carbonate minerals are less likely to form.

  ). The pH of the experimental solutions increased in all experiments. The evolution of pH was similar for experiments with Synechococcus sp. and Chroococcidiopsis sp., reaching a value of 11.8 at the end of experiment (Fig.2 B), whereas the increases of pH in experiments with A. clathrata was smaller, from 9.6 to 10.2. Note that Chroococcidiopsis sp. induced the pH increase similar to that by Synechococcus sp., but the biomass produced by the former specie was sizably lower. A possible reason for this difference could be enhanced EPS production by Chroococcidiopsis sp., having a common EPSrich capsule (Fig.1 B). Regardless of the morphological differences between species, the observed pH increase most likely occurred due to the photosynthetic activity of cyanobacteria as demonstrated by a correlation between the biomass and the pH (r² = 0.90, 0.78 and 0.45 at p < 0.01 for Chroococcidiopsis sp., Synechococcus sp., and A. clathrata, respectively) as shown in Fig.2C.The evolution of Mg, Si and DIC concentrations with time was similar in the experiments with Synechococcus sp. and Chroococcidiopsis sp.. For 0.52 Si : Mg ratio, ≥ 90% of Mg and Si were removed during the first ~150 h of experiments (Figs.3 A and B), suggesting the formation of a Si, Mg-rich solid phase. The evolution of Mg and Si concentration was similar for 0.26 Si :

  F). The evolution of these parameters was different with A. clathrata: Si and Mg concentrations decreased less rapidly (Figs. 3 A and B and 3 D and E) and DIC concentrations remained stable at 10 mM (Figs. 3 C and 3 F). The evolution of SI calculated with PHREEQC indicated the possible formation of sepiolite, brucite and Mg carbonates artinite (Mg2CO3(OH)2•3H2O), dypingite (Mg5(CO3)4(OH)2•5H2O) and hydromagnesite (Mg5(CO3)4(OH)2•4H2O) when Mg was in excess to Si (Si:Mg of 0.52, Fig. S2 C). In experiments with excess of Si (Si:Mg of 1.73), the formation of sepiolite and brucite was possible, whereas the artinite and hydromagnesite were unlikely to precipitate owing to their lower SI (Fig. S2 D).

Table 1 .

 1 Composition of initial fluids and identity of cyanobacteria used in the experiments.

	Experiment	NaHCO3, mM	Mg, mM	Si, mM	Initial Si:Mg mass ratio	Initial Si:Mg molar ratio	Cyanobacteria (1 gwet L -1 )
	Syn60.30	10	1.2	2.1	60:30	1.73	Synechococcus sp.
	Syn60.100	10	4.1	2.1	60:100	0.52	Synechococcus sp.
	Syn30.100	10	4.1	1.1	30:100	0.26	Synechococcus sp.
	Chr60.30	10	1.2	2.1	60:30	1.73	Chroococcidiopsis sp.
	Chr60.100	10	4.1	2.1	60:100	0.52	Chroococcidiopsis sp.
	Chr30.100	10	4.1	1.1	30:100	0.26	Chroococcidiopsis sp.
	Aph60.30	10	1.2	2.1	60:30	1.73	A. clathrata
	Aph60.100	10	4.1	2.1	60:100	0.52	A. clathrata
	Aph30.100	10	4.1	1.1	30:100	0.26	A. clathrata
	DSyn60.100*	10	4.1	2.1	60:100	0.52	Synechococcus sp.
	S60.30	10	1.2	2.1	60:30	1.73	-
	S60.100	10	4.1	2.1	60:100	0.52	-
	S30.100	10	4.1	1.1	30:100	0.26	-
	*experiment in the darkness

Table 2 .

 2 Experimental changes of aqueous solution chemistry, rates and stoichiometry of precipitation reactions: ΔpH and Δbiomass for biotic experiments during the fastest precipitation period, quantity of DIC, Mg and Si removed during the different experiments, ratios of the removed DIC, Mg and Si, apparent rate of Mg (ΔMg/Δt), Si (ΔSi/Δt ) and DIC (ΔDIC/Δt) removal during first 100 h, and biomass-normalized maximal rate (R) of Mg, Si and DIC removal observed between 50 and 150 h of experiment.

Acknowledgements : Supports from grants BIOCASTRO of MITI (CNRS), SERPCARB (Carnot ISIFoR), BIOMIN (Interrvie INSU-CNRS) and CaBioCa of Défi (Action conjointe, MITI CNRS) are acknowledged. OP acknowledges support from the Tomsk State University Development Program "Priority-2030".

We are grateful to the Editor T. Bosak and two anonymous reviewers for their insightful and constructive remarks. The research was funded by projects CNRS MITI Origins 'BIO-CARB-STROM' and Défi 'CABIOCA'. OP acknowledges support from the Tomsk State University Development Program "Priority-2030". Fundings: CNRS MITI Origins 'BIO-CARB-STROM, CaBioCa of Défi (Action conjointe, MITI CNRS), TSU Development Program "Priority-2030"

All the data that support the findings of this study are presented in the tables of the main text and Supplemetary Information.

Data Availability Statement:

Experiment pH ΔpH

Δbiomass, gdry/L ΔDIC, mM

ΔMg, mM

ΔSi, mM ΔDIC/ΔMg (removed)

ΔSi/ΔMg (removed)

ΔSi/ΔDIC (removed)


Imane Haur
email: imane.haur@ec-nantes.fr

Olivier H Roux
email: olivier-h.roux@ec-nantes.fr

High-level Colored Time Petri Nets for true concurrency modeling in real-time software

Keywords: Multi-core execution, High-level Colored Time Petri Nets, Model-checking

The control of real-time systems often requires taking into account simultaneous access in true parallelism to shared resources. This is particularly the case for multi-core execution platforms. Timed automata or time Petri nets do not capture these features directly. We propose extending time Petri Nets with color and high-level functionality encompassing both timed multi-enableness of transitions and sequential pseudo code. We prove that the reachability problem is decidable for this model on which an on-the-fly TCTL model checking algorithm is efficiently implemented in the tool ROMÉO. We apply this approach to modeling a multi-core real time spinlock mechanism in order to check all possible execution paths and interleaving of service calls.

I. INTRODUCTION

For the lack of data structures, Petri nets are unsuitable for modeling systems where data affects the system's behavior. High-level Petri nets [START_REF] Hillah | Pn standardisation: A survey[END_REF] have been proposed for modeling scientific problems with complex structures allowing the description of both system data and control. The term High-level Petri net is then used for many Petri nets [START_REF] Kindler | A framework for the definition of variants of high-level petri nets[END_REF] such as Predicate/-Transition Nets, colored Petri nets, or hierarchical Petri nets. However, the common point is that they allow manipulating different types of expressions that use state variables. Input arcs are labeled with boolean expressions specifying conditions (guards or gates) that can also be associated with transitions. Arc annotations are expressions that can be associated with output arc. They can be viewed as computing systems that operate on shared data.

1) Petri nets: Petri nets are one of the many modeling languages used to describe distributed concurrent systems. A place can contain any number of tokens. A marking 𝑀 of a Petri Net is a vector representing the number of tokens of each place. A transition is enabled (it may fire) in 𝑀 if there are enough tokens in its input places for the consumptions to be possible. Firing a transition t in a marking 𝑀 consumes one token from each of its input places 𝑠, and produces one token in each of its output places 𝑠.

2) High-level Petri nets: The precondition (guard) and postcondition (update) over a set of variables (𝑋) are associated with transitions. A transition is enabled (it may fire) if there are enough tokens in its input places and if the guard is true. When the transition fires, the corresponding updates are executed, modifying the values of the variables. The variables take their values in a finite state (such as bounded integer or enumerate type...), guards are boolean expressions over 𝑋, and updates can be described as a sequence of imperative code expressed in a programming language but whose execution is atomic from the transition firing point of view.

3) Colored Petri nets: The colored extension of Petri nets allows the distinction between tokens.

Although the set 𝑋 of High-level Petri nets presented in the previous paragraph can be of arbitrarily complex type, places in colored Petri nets contain tokens of one type. This type noted 𝐶 is called the color set of the place.

An arc from a place to a transition (PT) specifies the color(s) that enabled the transition, and its firing will consume it. An arc from a transition to a place (TP) specifies the token color produced in that place by the firing of the transition. A particular color called any indicates in a PT arc that any color enabled the transition, and in a TP arc that the color consumed in the input place will be the one produced in the output place.

A marking M of a colored Petri Net represents not only the number of tokens in each place but also their respective colors. That is represented either by a multiset or by a matrix.

4) Time Petri Nets: Time Petri nets (TPN) extend Petri nets with temporal intervals associated with transitions, specifying firing delay ranges for the transitions. Assuming transition 𝑡 became last enabled at time 𝑑 and the endpoints of its firing interval are 𝛼 and 𝛽, then 𝑡 cannot fire earlier than 𝑑 + 𝛼 and must fire no later than 𝑑 + 𝛽 unless disabled by the firing of another transition. Firing a transition takes no time.

5) Colored Time Petri Nets: For real parallelism or with interleaving semantics of timed systems, the notion of multiple enableness is needed. It refers to the fact that a transition is enabled at least twice in the same state, which implies a dynamic number of timers. Multiple enableness in time Petri nets is a natural way for modeling paradigms like multiple servers and multiple instances of codes [START_REF] Boyer | Multiple enabledness of transitions in petri nets with time[END_REF].

For Colored Time Petri Nets, multiple enableness occurs when several combinations of colors enable a transition at a given time.

II. HIGH-LEVEL COLORED TIME PETRI NETS

Notations: The sets N, Q ≥0 , and R ≥0 are respectively the sets of natural, non-negative rational, and non-negative real numbers. An interval 𝐼 of R ≥0 is a Q-interval iff its left endpoint ↑ 𝐼 belongs to Q ≥0 and its right endpoint 𝐼 ↓ belongs to Q ≥0 ∪ {∞}. We denote by I (Q ≥0 ) the set of Q-intervals of R ≥0 . 𝐵 𝐴 stands for the set of mappings from 𝐴 to 𝐵. If 𝐴 is finite and | 𝐴| = 𝑛, an element of 𝐵 𝐴 is also a vector in 𝐵 𝑛 . The usual operators +, -, < and = are used on vectors of 𝐴 𝑛 with 𝐴 = N, Q, R and are the point-wise extensions of their counterparts in 𝐴.

A. Definition and semantics

We consider a Petri Nets model, which encompasses both color and High-level functionalities. We consider a set 𝐶 of colors. An arc is either associated with a color of 𝐶 or can take on a particular color called any. For the firing of a transition, all its arcs associated with the any color must match to instantiate any at the same color taken from 𝐶.

If several values of any allow its enabling, the transition is multi-enabled, and in this case, many clocks are associated with the transition, allowing several firing dates depending on the enabling date and the time interval.

We now give the formal definition. Definition 1 (High-level Colored Time Petri Net): A Highlevel Colored Time Petri Net (HCTPN) is a tuple N = (𝑃, 𝑇, 𝑋, 𝐶, pre, post, (𝑚 0 , 𝑥 0 ), 𝑔𝑢𝑎𝑟 𝑑, 𝑢 𝑝𝑑𝑎𝑡𝑒, 𝐼) where

• 𝑃 is a finite set of places,

• 𝑇 is a finite set of transitions such that 𝑇 ∩ 𝑃 = ∅, • 𝑋 is a finite set of variables taking their value in the finite set X (such as bounded integer), • 𝐶 is a finite set of colors and 𝐶 𝑎𝑛𝑦 = 𝐶 ∪ {𝑎𝑛𝑦}, • pre : 𝑃 × 𝑇 → N 𝐶 𝑎𝑛𝑦 is the backward incidence mapping, A transition 𝑡 ∈ 𝑇 is said to be enabled by a given marking 𝑚 ∈ N 𝑃×𝐶 in two cases depending on whether 𝑡 ∈ 𝑇 𝑎𝑛𝑦 or not: Finally, a transition 𝑡 ∈ 𝑇 is said to be enabled by a given marking 𝑚 ∈ N 𝑃×𝐶 and a valuation 𝑥 ∈ X 𝑋 if en(𝑚, 𝑡) = 𝑡𝑟𝑢𝑒 or colorSet any (𝑚, 𝑡) ≠ ∅ and 𝑔𝑢𝑎𝑟 𝑑 (𝑡, 𝑥) = 𝑡𝑟𝑢𝑒.

• post : 𝑃 × 𝑇 → N 𝐶 𝑎𝑛𝑦 is the forward incidence mapping, • 𝑔𝑢𝑎𝑟 𝑑 : 𝑇 × X 𝑋 → {𝑡𝑟𝑢𝑒, 𝑓 𝑎𝑙𝑠𝑒} is the guard function, • 𝑢 𝑝𝑑𝑎𝑡𝑒 : 𝑇 × X 𝑋 → X 𝑋 is the update function, • (𝑚 0 , 𝑥 0 ) ∈ N 𝑃×𝐶 × X 𝑋 →
• if 𝑡 ∈ 𝑇 𝑎𝑛𝑦 ,
We illustrate the enabling condition by two examples with two colors 𝐶 = {𝑏𝑙𝑢𝑒, 𝑟𝑒𝑑}. On the figures, a black arc means that it is associated with the color 𝑎𝑛𝑦. For the HCTPN given in Figure 1.a, the transition 𝑇 1 ∈ 𝑇 𝑎𝑛𝑦 .

We have pre(𝑇 1 ) =

𝑟 𝑒𝑑 𝑏𝑙𝑢𝑒 𝑎𝑛𝑦 𝑃 1 0 1 0 𝑃 2 1 0 0 𝑃 3 0 1 0
. The initial

marking is 𝑚 0 = 𝑟 𝑒𝑑 𝑏𝑙𝑢𝑒 𝑃 1 1 1 𝑃 2 1 0 𝑃 3 0 1
that enables the transition 𝑇 1 and en(𝑚 0 , 𝑇 1 ) = 𝑡𝑟𝑢𝑒. Now we consider the HCTPN given in Figure 1.b with the same initial marking 𝑚 0 but where the transition 𝑇 1 ∈ 𝑇 𝑎𝑛𝑦 since at least one arc (here two) is associated with the color 𝑎𝑛𝑦. We denote by newen((𝑚, 𝑥), 𝑡, 𝑐) the set of transitions that are newly enabled by the firing of 𝑡 from (𝑚, 𝑥) with the color

𝑃 1 𝑃 2 𝑃 3 𝑇 1 [2, 3] blue red blue •• • • 𝑃 1 𝑃 2 𝑃 3 𝑇 1 [2, 3] blue •• • • 1.a: 𝑇 1 ∈ 𝑇 𝑎𝑛𝑦 1.b: 𝑇 1 ∈ 𝑇 𝑎𝑛𝑦

We have pre(𝑇

1 ) = 𝑟 𝑒𝑑 𝑏𝑙𝑢𝑒 𝑎𝑛𝑦 𝑃 1 0 0 1 𝑃 2 0 0 1 𝑃 3 0 1
𝑐 (𝑐 = • if 𝑡 ∈ 𝑇 𝑎𝑛𝑦 ).
Let us go back to the HCTPN of Figure 1.a, the firing of

𝑇 1 ∈ 𝑇 𝑎𝑛𝑦 from 𝑚 0 leads to the marking 𝑚 1 = 𝑟 𝑒𝑑 𝑏𝑙𝑢𝑒 𝑃 1 1 0 𝑃 2 0 0 𝑃 3 0 0 . It is noted 𝑚 0 (𝑇 1 ,•)
----→ 𝑚 1 For the HCTPN of Figure 1.b, the firing of 𝑇 1 ∈ 𝑇 𝑎𝑛𝑦 is possible only for 𝑎𝑛𝑦 = 𝑟𝑒𝑑 and leads to the marking

𝑚 2 = 𝑟 𝑒𝑑 𝑏𝑙𝑢𝑒 𝑃 1 0 1 𝑃 2 0 0 𝑃 3 0 0 . It is noted 𝑚 0 (𝑇 1 ,𝑟 𝑒𝑑) -------→ 𝑚 2 .
2) Time behavior: For any 𝑡 ∈ 𝑇 𝑎𝑛𝑦 , 𝑣(𝑡, 𝑐) is the valuation of the clock associated with 𝑡 and the color 𝑐 ∈ 𝐶. i.e. it is the time elapsed since the transition 𝑡 has been newly enabled by 𝑚 with 𝑐 ∈ colorSet any (𝑚, 𝑡). For other transitions 𝑡 ∈ 𝑇 𝑎𝑛𝑦 , 𝑣(𝑡, •) is the valuation of the clock associated with 𝑡.

0 is the initial valuation with ∀𝑡 ∈ 𝑇, ∀𝑐 ∈ 𝐶 ∪ {•}, 0(𝑡, 𝑐) = 0.

As an example, if we keep only the useful clocks, the initial valuation of the HCTPN of Figures 1, is

𝑣 0 = • 𝑟 𝑒𝑑 𝑏𝑙𝑢𝑒 𝑇 1 0
for Figure 1.a, and 𝑣 0 =

• 𝑟 𝑒𝑑 𝑏𝑙𝑢𝑒 𝑇 1 0 0 for Figure 1.b, A state of the net N is a tuple ((𝑚, 𝑥), 𝑣) in N 𝑃×𝐶 × X 𝑋 × R ≥0
𝑇 ×𝐶 , where: 𝑚 is a marking, 𝑥 is a variable valuation and 𝑣 is a valuation of the clocks.

Definition 2 (Semantics of a HCTPN): The semantics of a HCTPN is a timed transition system (𝑄, 𝑄 0 , →) where:

• 𝑄 ⊆ N 𝑃×𝐶 × X 𝑋 × R ≥0 𝑇 ×𝐶 • 𝑄 0 = ((𝑚 0 , 𝑥 0 ), 0) • →∈ 𝑄 × ((𝑇 × 𝐶 ∪ {•}) ∪ R ≥0 ) × 𝑄

B. Examples of HCTPN

We give two examples. The first one illustrates the highlevel functionalities, and the second illustrates the notion of color and multi-enableness.

a) Example 1: The HCTPN given in Figure 2 illustrates time behavior and high-level manipulation of variables. This HCTPN has only one color and a single variable 𝑐 𝑝𝑡. We assume that random(1,10) returns an integer between 1 and 10.

𝑃 1 T 1 [5, 7] 𝑐 𝑝𝑡 = 𝑟 𝑎𝑛𝑑𝑜𝑚(1, 10) 𝑃 2 𝑐 𝑝𝑡 <= 5 T 2 [2, 3] 𝑐 𝑝𝑡 = 0 𝑐 𝑝𝑡 > 5 T 3 [4, 6] 𝑐 𝑝𝑡 = 𝑓 (𝑐 𝑝𝑡) 𝑃3 • int cpt=0; int f(int x) { return 2 * x-1; }

Fig. 2. HCTPN illustrating high-level manipulation of variables

A marking is written by the matrix (|𝑃|, |𝐶 |). Since there is only one color, the marking is a vector and the initial marking is then

𝑚 0 = 𝑃 1 1 𝑃 2 0 𝑃 3 0
and enables the transition 𝑇 1 .

The valuations of the clocks are given by the matrix (here a vector) such that the initial valuation is

𝑣 0 = 𝑇 1 0 𝑇 2 0 𝑇 3 0
. Since the set of variables is 𝑋 = {𝑐 𝑝𝑡}, we note a state 𝑠 = (𝑚, 𝑐 𝑝𝑡, 𝑣). The initial state is 𝑞 0 = (𝑚 0 , 0, 𝑣 0 ). The transition 𝑇 1 can fire after elapsing 5 time units. We now consider the run where the random function called by the update of the firing of 𝑇 1 returned the value 7. Then the transition's guard 𝑇 2 is false, and the transition 𝑇 3 is enabled. We assume that the transition 𝑇 3 took 4.6 time units for this run. The firing of the transition 𝑇 3 executes the corresponding update and calls the function 𝑓 that returns 13. The corresponding run is as follows:

1 0 0 , 0, 0 0 0 5 - → 1 0 0 , 0, 5 0 0 (𝑇 1 ,•) -----→ 0 1 0 , 7, 0 0 0 4.6 --→ 0 1 0 , 7, 0 0 4.6 (𝑇 3 ,•) -----→ 0 0 1
, 13, 0 0 0 b) Atomicity : An update can be described as a sequence of imperative code expressed in a programming language such as C. This code is evaluated sequentially w.r.t. the semantics of the C language; however, its execution is considered atomic from the HCTPN point of view.

Hence, if 𝑥 and 𝑥 are respectively the values of the variables before and after the execution of the code of an update of a transition 𝑡 from 𝑥, the firing of 𝑡 leads atomically to 𝑥 = 𝑢 𝑝𝑑𝑎𝑡𝑒(𝑡, 𝑥).

c) Example 2: The model given in Figure 3 is a HCTPN with a set of two colors 𝐶 = {𝑟𝑒𝑑, 𝑏𝑙𝑢𝑒}. Several combinations of color usage, on guards and in updates, via the $𝑎𝑛𝑦 variable are presented.

𝑃 1 T 1 [8, 8] cpt[$any]==2 cpt[$any]=f($any,cpt) 𝑒𝑛𝑑 T 2 [5, 5] 𝑃 2 blue red • • 𝑃 3 • T 3 [6, 6] cpt[1]=2 blue typedef color {red = 0, blue = 1}; int[2] cpt = {2, 1}; int f(int color, int[2] c) { if (color == blue) { return c[color] + 1; } else if (color == red) { return c[color] * 2; } }

Fig. 3. HCTPN model illustrating colored multi-enableness

In the sequel a marking is written by the matrix (|𝑃|, |𝐶 |).

The initial marking is then 𝑚

0 = 𝑟 𝑒𝑑 𝑏𝑙𝑢𝑒 𝑃 1 1 0 𝑃 2 1 0 𝑃 3 0 1 𝑒𝑛𝑑 0 0
and enables the transitions 𝑇 1 , 𝑇 2 and 𝑇 3 . The variable 𝑐 𝑝𝑡 of the model is an array indexed by the color. Its initial value is

𝑥 0 = 𝑟 𝑒𝑑 𝑏𝑙𝑢𝑒 𝑐 𝑝𝑡 2 1
The valuations of the clocks are given by the matrix such that the initial valuation is

𝑣 0 = • 𝑟 𝑒𝑑 𝑏𝑙𝑢𝑒 𝑇 1 0 0 𝑇 2 0 𝑇 3 0
(We omit the insignificant values). We note a state 𝑠 = (𝑚, 𝑥, 𝑣). The initial state is 𝑞 0 = (𝑚 0 , 𝑥 0 , 𝑣 0 ). Since the time intervals are points, we have an unique run:

1 0 1 0 0 1 0 0 , 2 1 , 0 0 0 0 5 - → 1 0 1 0 0 1 0 0 , 2 1 , 5 0 5 5 (𝑇 2,•) ------→ 1 1 0 0 0 1 0 0 , 2 1 , 5 0 0 5 1 - → 1 1 0 0 0 1 0 0 , 2 1 , 6 0 0 6 (𝑇 3,•) ------→ 1 1 0 0 0 0 0 0 , 2 2 , 6 0 0 0 2 - → 1 1 0 0 0 0 0 0 , 2 2 , 8 2 0 0 (𝑇 1,𝑟 𝑒𝑑) --------→ 0 1 0 0 0 0 1 0 , 4 2 , 0 2 0 0 6 - → 0 1 0 0 0 0 1 0 , 4 2 , 0 8 0 0 (𝑇 1,𝑏𝑙𝑢𝑒) ---------→ 0 0 0 0 0 0 1 1 , 4 3 , 0 0 0 0
The time elapses from the initial marking until reaching date 5. 𝑇 2 is fired, and a blue token is dropped in the place 𝑃 1 . The clock of 𝑇 1 associated with the red color has reached the value 5. The clock of 𝑇 1 associated with the blue color cannot start yet because the guard is false for this color. At date 6, 𝑇 3 is fired, causing a change in the variable cpt that makes the guard of 𝑇 1 true for the blue color. The clock associated with the blue color for 𝑇 1 can therefore start. Both colors enable the transition 𝑇 1 , and the corresponding clocks give the time from the two enabling. After two more time units, 𝑇 1 is fired for the red color; at this moment, the clock of 𝑇 1 for the blue color has reached 2. Finally, after 6-time units, 𝑇 1 is fired for the blue color, ending the run.

C. Decidability and complexity

Let us recall that a High-level Colored Time Petri Net (HCTPN) is a tuple N = (𝑃, 𝑇, 𝑋, 𝐶, pre, post, (𝑚 0 , 𝑥 0 ), 𝑔𝑢𝑎𝑟 𝑑, 𝑢 𝑝𝑑𝑎𝑡𝑒, 𝐼) such that the set 𝐶 of colors is finite and 𝑋 is a finite set of variables taking their value in a finite set X.

Theorem 1: Reachability problem for bounded High-level Colored Time Petri Net is decidable Proof: From HCTPN semantics, a transition can be multienableded a maximum of |𝐶 | times at a given time. Hence, firing domains can be symbolically abstracted with state classes using difference bound matrix (DBM) over |𝐶 | × 𝑇 variables. As in [START_REF] Berthomieu | An enumerative approach for analyzing time petri nets[END_REF], [START_REF] Berthomieu | Modeling and verification of time dependent systems using time petri nets[END_REF], the number of DBM is finite. Moreover, the number of markings of a k-bounded Petri net is bounded by (𝑘 + 1) | 𝑃 | then the number of discrete states of a k-bounded HCTPN is bounded by (𝑘 + 1) | 𝑃 | × X 𝑋 . Hence, computable finite abstractions of the state space exist, and the reachability problem is decidable.

Temporal logics were introduced by Pnueli [START_REF] Pnueli | The temporal logic of programs[END_REF] as specification languages to express the behaviors of sequential and concurrent systems and TCTL (Timed Computation Tree Logic), introduced in [START_REF] Alur | Model-checking in dense realtime[END_REF], is a real-time extension of the branching-time temporal logic CTL (Computation Tree Logic).

We can prove, as in [START_REF] Boucheneb | TCTL model checking of time Petri nets[END_REF] for bounded Time Petri Nets, that the theoretical complexity of TCTL model-checking for bounded High-level Colored Time Petri Nets is PSPACEcomplete. However, as for Timed Automata and Time Petri Nets, no effective PSPACE algorithm exists in practice, and real implementations are with exponential algorithms.

Moreover, HCTPN can be extended with stopwatches allowing the modeling of preemptive scheduling as in [START_REF] Haur | Formal schedulability analysis based on multi-core rtos model[END_REF]. In the stopwatch setting, the reachability problem is undecidable but efficient semi-algorithms are implemented in ROMÉO [START_REF] Lime | Romeo: A parametric model-checker for petri nets with stopwatches[END_REF] that converges for almost all practical cases.

III. APPLICATION

The application chosen as an example is the modeling of the spinlocks mechanism present in the PowerPC MPC5643L dualcore microcontroller from NXP [START_REF] Semiconductor | [END_REF] and used to build critical sections for parallel program executions. This mechanism is based on a hardware unit, the SEMA4 for Semaphore Unit. For the software, this unit is materialized as an array of 16 registers implementing 16 locks. The exclusive access to the bus regulates the concurrent accesses to one of these registers. If a register contains the value 0, the lock is available, and it is possible to write to it. If the value contained is different from 0, the lock is occupied, and it is only possible to write the value 0 to it, and writing any other value has no effect. Therefore, getting a lock consists in writing a value different from 0 and releasing it consists in writing 0. Thus, using this unit requires respect of a protocol, and an example of implementation is given on page 1322 of [START_REF] Semiconductor | [END_REF]. Algorithm 1 reproduces it.

Algorithm 1 Lock acquisition protocol. 𝑔𝑎𝑡𝑒 is one of the hardware registers of the SEMA4 unit.

𝑐𝑛 ← core_number ⊲ (1 .. N) do 𝑙𝑜𝑐𝑘 ← 𝑔𝑎𝑡𝑒 while 𝑙𝑜𝑐𝑘 ≠ 0 do 𝑔𝑎𝑡𝑒 ← 𝑐𝑛 𝑙𝑜𝑐𝑘 ← 𝑔𝑎𝑡𝑒 while 𝑙𝑜𝑐𝑘 ≠ 𝑐𝑛 A.

Modeling the spinlocks mechanism

The modeling takes advantage of the possibilities of the HCTPN. The hardware part, which by virtue of the exclusive access to the bus allows operations that are intrinsically atomic, is modeled using functions. To simplify the presentation, only one register of the SEMA4 unit, gate, is modeled but the model could just as well use an array to accurately model the hardware. The listing 1 shows this part of the model. gate is initialized to the UNLOCKED state (line 2) and is accessible through three functions. lock (line 4) mimics the behavior of the hardware by only allowing writing to gate if its value is UNLOCKED. The core number corresponding to a color and a color among N being coded by an integer from 0 to N-1, a core locks by writing 𝑐𝑜𝑙𝑜𝑟+1 in gate. unlock (line 10) simply writes the value UNLOCKED into gate. isLocked (line 14 returns 1 if the gate is locked, 0 otherwise). Finally, isLockedBy (line 22) returns 1 if core holds the lock and returns 0 otherwise.

Each function of the software is modeled by an HCTPN reproducing the control flow graph of the function. Two HCTPNs model the functions GetSpinLock and RelSpinLock (see Figure 4). GetSpinLock corresponds to the algorithm 1 and $any allows to represent on which core the function is executed. The call of a function modeled by a HCTPN is done by dropping a token of the core color in the initial place. Thus, "calling" the function GetSpinLock is performed by the update GetSpinLock[color] = 1 on a transition of the HCTPN of the calling function. This is identical to drawing an arc of the corresponding color between the transition and the initial place of GetSpinLock. The function return requires a synchronization. This one is implemented by a variable of type array and of size equal to the number of colors and indexed by the color, i.e. the core on which the function call is made. We have therefore for our two function models the two variables endOfGSL and endOfRSL, see listing 2.

B. Verification of the system

The spinlock model is completed by an application model. Two tasks, 𝜏 0 , running on core 0 (red color) and 𝜏 1 , running on core 1 (blue color), are modeled as shown in Figure 5. The task 𝜏 0 takes then releases the spinlock while the task 𝜏 1 has the possibility to take it, as 𝜏 0 does, or to reach the final state without taking the spinlock. We want to check that 𝜏 0 and 𝜏 1 cannot occupy simultaneously and respectively the places P 12 and P 22 by the CTL formula 𝐴 (¬(P 12 [0] == 1∧P 22 [START_REF] Hillah | Pn standardisation: A survey[END_REF] == 1)). Here P 12 [0] denotes the marking of P 12 for the red color and P 22 [START_REF] Hillah | Pn standardisation: A survey[END_REF] denotes the marking of P 22 for the blue color. ROMÉO answers true for this CTL formula.

IV. CONCLUSION

This paper has presented High-level Colored Time Petri Nets. This formalism allows to model complex systems and is well adapted to multi-core hardware and software modeling, as shown in the case study. The high-level features allow the modeling of the software, the timed transitions model the execution times, and the colors specify the hardware where the software is executed. A timed transition enabled by more than one color allows true concurrency modeling. The model checking of this formalism is implemented in the ROMÉO tool. Future work will focus on using this formalism for the certification of an OSEK and AUTOSAR compliant embedded OS.

  is the initial values 𝑚 0 of the marking and 𝑥 0 of the variables, • 𝐼 : 𝑇 → I (Q ≥0 ) is the static firing interval function, 1) Discrete behavior: For a marking 𝑚 ∈ N 𝑃×𝐶 , 𝑚( 𝑝) is a vector in N 𝐶 and 𝑚( 𝑝) [𝑐] represents a number of tokens of color 𝑐 ∈ 𝐶 in place 𝑝 ∈ 𝑃. A valuation of the set of variables 𝑋 is noted 𝑥 ∈ X 𝑋 . (𝑚, 𝑥) is a discrete state of HCTPN. a) Enabling of a transition: An arc pre( 𝑝, 𝑡) ∈ N 𝐶 𝑎𝑛𝑦 is a vector such that pre( 𝑝, 𝑡) [𝑐] is the number of token of color 𝑐 ∈ 𝐶 in place 𝑝 needed to enable the transition 𝑡 and pre( 𝑝, 𝑡) [𝑎𝑛𝑦] > 0 represents the fact that any color can enabled the transition. Let 𝑇 𝑎𝑛𝑦 ⊆ 𝑇 the set of transitions that can be enabled by 𝑎𝑛𝑦 color: i.e. 𝑇 𝑎𝑛𝑦 = {𝑡 ∈ 𝑇, ∃𝑝 ∈ 𝑃, s.t. pre( 𝑝, 𝑡) [𝑎𝑛𝑦] > 0}. Moreover we define the set 𝑇 𝑎𝑛𝑦 = 𝑇 \ 𝑇 𝑎𝑛𝑦 .

Fig. 1 .

 1 Fig. 1. Enabling transition

0 .

 0 The transition is enabled only if 𝑎𝑛𝑦 takes the 𝑟𝑒𝑑 value then colorSet any (𝑚 0 , 𝑇 1 ) = {𝑟𝑒𝑑}. If the place 𝑃2 had two tokens with one token per color, then the transition would be multienabled by the 2 colors leading to colorSet any (𝑚 0 , 𝑇 1 ) = {𝑏𝑙𝑢𝑒, 𝑟𝑒𝑑}. b) Firing of a transition: An arc post( 𝑝, 𝑡) ∈ N 𝐶 𝑎𝑛𝑦 is a vector such that post( 𝑝, 𝑡) [𝑐] is the number of token of color 𝑐 ∈ 𝐶 produced in place 𝑝 by the firing of the transition 𝑡 and post( 𝑝, 𝑡) [𝑎𝑛𝑦] gives the number of token produced in 𝑝 with the color 𝑐 ∈ colorSet any (𝑚, 𝑡) used for the firing of 𝑡. Firing an enabled transition 𝑡 ∈ 𝑇 𝑎𝑛𝑦 from (𝑚, 𝑥) such that en(𝑚, 𝑡) = 𝑡𝑟𝑢𝑒 and 𝑔𝑢𝑎𝑟 𝑑 (𝑡, 𝑥) = 𝑡𝑟𝑢𝑒 leads to a new marking 𝑚 defined by ∀𝑐 ∈ 𝐶, ∀𝑝 ∈ 𝑃, 𝑚 ( 𝑝) [𝑐] = 𝑚( 𝑝) [𝑐] -pre( 𝑝, 𝑡) [𝑐] +post( 𝑝, 𝑡) [𝑐] and a new valuation 𝑥 = 𝑢 𝑝𝑑𝑎𝑡𝑒(𝑡, 𝑥). This new marking is denoted 𝑚 = firing(𝑚, 𝑡, •) where • denote the fact that no any color has to be instantiated for this firing.

Firing

  an enabled transition 𝑡 ∈ 𝑇 𝑎𝑛𝑦 from (𝑚, 𝑥) with the any color 𝑐 𝑎 ∈ colorSet any (𝑚, 𝑡) leads to a new marking defined by ∀𝑐 ∈ 𝐶 \ {𝑐 𝑎 }, ∀𝑝 ∈ 𝑃, 𝑚 ( 𝑝) [𝑐] = 𝑚( 𝑝) [𝑐]pre( 𝑝, 𝑡) [𝑐] + post( 𝑝, 𝑡) [𝑐] and ∀𝑝 ∈ 𝑃, 𝑚 ( 𝑝) [𝑐 𝑎 ] = 𝑚( 𝑝) [𝑐 𝑎 ] -pre( 𝑝, 𝑡) [𝑐 𝑎 ] -pre( 𝑝, 𝑡) [𝑎𝑛𝑦] + post( 𝑝, 𝑡) [𝑐 𝑎 ] + post( 𝑝, 𝑡) [𝑎𝑛𝑦] and a new valuation 𝑥 = 𝑢 𝑝𝑑𝑎𝑡𝑒(𝑡, 𝑥). This new marking is denoted 𝑚 = firing(𝑚, 𝑡, 𝑐 𝑎 ).

  consists of two types of transitions: discrete transitions (firing 𝑡 from ((𝑚, 𝑥), 𝑣)) iff: * ((𝑚, 𝑥), 𝑣)(𝑡 ∈𝑇 𝑎𝑛𝑦 ,•) --------→ ((𝑚 , 𝑥 ), 𝑣 ) with • en(𝑚, 𝑡) = 𝑡𝑟𝑢𝑒 and 𝑣(𝑡) ∈ 𝐼 (𝑡), • 𝑚 = firing(𝑚, 𝑡, •) * ((𝑚, 𝑥), 𝑣) (𝑡 ∈𝑇 𝑎𝑛𝑦 ,𝑐) ---------→ ((𝑚 , 𝑥 ), 𝑣 ) with • 𝑐 ∈ colorSet any (𝑚,𝑡) and 𝑣(𝑡, 𝑐) ∈ 𝐼 (𝑡), • 𝑚 = firing(𝑚, 𝑡, 𝑐) * 𝑔𝑢𝑎𝑟 𝑑 (𝑡, 𝑥) = 𝑡𝑟𝑢𝑒 and 𝑥 = 𝑢 𝑝𝑑𝑎𝑡𝑒(𝑡, 𝑥) * ∀𝑡 s.t. en((𝑚 , 𝑡 ) = 𝑡𝑟𝑢𝑒) • 𝑣 (𝑡 , •) = 𝑣(𝑡 , •) if 𝑡 ∉ newen((𝑚, 𝑥), 𝑡, •), • 𝑣 (𝑡 , •) = 0 otherwise * ∀𝑡 and ∀𝑐 ∈ colorSet any (𝑚 , 𝑡 ) • 𝑣 (𝑡 , 𝑐) = 𝑣(𝑡 , 𝑐) if 𝑡 ∉ newen((𝑚, 𝑥), 𝑡, 𝑐), • 𝑣 (𝑡 , 𝑐) = 0 otherwise time transitions: ((𝑚, 𝑥), 𝑣) 𝑑 ∈R ≥0 -----→ ((𝑚, 𝑥), 𝑣 ), iff: * ∀𝑡 ∈ 𝑇 𝑎𝑛𝑦 s.t. en(𝑚, 𝑡) = 𝑡𝑟𝑢𝑒, • 𝑣 (𝑡, •) ≤ 𝐼 (𝑡) ↓ • 𝑣 (𝑡, •) = 𝑣(𝑡, •) + 𝑑 * ∀𝑡 ∈ 𝑇 𝑎𝑛𝑦 and ∀𝑐 ∈ colorSet any (𝑚, 𝑡), • 𝑣 (𝑡, 𝑐) ≤ 𝐼 (𝑡) ↓ • 𝑣 (𝑡, 𝑐) = 𝑣(𝑡, 𝑐) + 𝑑 We now illustrate the main features of HCTPN in an example. The guards are in green in the Figures and the update in purple.

Fig. 4 .Fig. 5 .

 45 Fig. 4. The GetSpinLock and RelSpinLock function models

  and ∀𝑝 ∈ 𝑃 and ∀𝑐 ∈ 𝐶, 𝑚( 𝑝) [𝑐] ≥ pre( 𝑝, 𝑡) [𝑐]. We denote en(𝑚, 𝑡) ∈ {𝑡𝑟𝑢𝑒, 𝑓 𝑎𝑙𝑠𝑒}, the true value of this condition. • if 𝑡 ∈ 𝑇 𝑎𝑛𝑦 , and ∃𝑐 𝑎 ∈ 𝐶 such that ∀𝑝 ∈ 𝑃, 𝑚( 𝑝) [𝑐 𝑎 ] ≥ pre( 𝑝, 𝑡) [𝑎𝑛𝑦] and ∀𝑐 ∈ 𝐶, 𝑚( 𝑝) [𝑐] ≥ pre( 𝑝, 𝑡) [𝑐]. The corresponding set of color 𝑐 𝑎 is noted colorSet any (𝑚, 𝑡) ⊆ 𝐶

  Listing 2. Synchronization variables for the function return

	T 1	GetSpinLock	RelSpinLock
	[1, 1]
	isLocked(gate)
		T 2	T 8
		[1, 1]	[1, 1]
		! isLocked(gate)	unlock(gate);
	P 2	P 7
		T 3
		[1, 1]
		lock($any, gate)
	T 4
	[1, 1]	P 3
	isLockedBy($any, gate) == 0	T 5
		[1, 1]
		isLockedBy($any, gate) == 1
		endOfGSL[$any] = 1;

int[2] endOfGSL = {0, 0};

int[2] endOfRSL = {0, 0};


Yann Calbérac
email: yann.calberac@wanadoo.fr

Urban Dynamics and Urban Sustainability: An Impossible Marriage or a Marriage of Convenience?

Dealing with both urban dynamics and urban sustainability can appear as a paradox insofar as these terms seem radically opposed. But, obviously, thanks to the dynamics they created and from which they gained advantage, cities have been sustainable since they first appeared. This paper will demonstrate that it is no use trying to oppose dynamics and sustainability inasmuch as sustainability appears as the contemporary form of dynamics. Taking this major evolution in urban history into consideration will first enlighten the present context which promotes such innovations, then define the new kind of cities that are about to be produce. Last, it will focus on an epistemological reflection upon the social sciences and their ability to provide tools to analyse and explain what currently occurs in cities.

Introduction

Ordinary discourse usually opposes dynamics and sustainability, as if these two words were radically opposed: dynamics means what is moving (such as social relations, trade, cultural exchanges, etc), whereas sustainability deals with what is fixed, unchanging and immutable. Currently, the world is changing very quickly and one of the solutions to cope with the negative effects of this evolution is to try to fix everything and to stabilize the world as it was before.

In this paper, I would like to question the relations between dynamics and sustainability by wondering whether it is relevant (or not) to oppose them insofar as sustainability appears as the current form of dynamics. To answer this question, following the main topic of the conference, I will only focus on cities, which reveal the paradox of this false opposition. Indeed, cities have existed for a long time and they have kept growing and developing since that time, thanks to the various dynamics that historians or geographers have described. Today, to cope with huge changes, such as demographic growth, urban sprawl, globalization, etc. sustainability appears as the solution to improve the quality of the urban way of life.

I would like to demonstrate that sustainability is the current form of dynamics. So, I will first underline the fact that cities have always been dynamic and there is no use in trying to imagine a city without dynamics. Then, I will question the concept of sustainable cities as it emerges in connection with political and social demands. Last, I will question our social sciences disciplines: are they effective in analysing sustainable cities, this new object we are dealing with today?

Nothing is more sustainable than a city. Nothing is more dynamic than a city

In this first part of my paper, I would like to demonstrate that is no use trying to oppose dynamics and sustainability insofar as cities have always been being sustainable.

Dynamics have ensured the continued existence of cities.

Cities are long term objects. Indeed, depending on the civilization, cities have a long-standing existence. The Western-European urban framework stems from the Roman Empire. The Spanish and Portuguese conquerors founded numerous cities all over the world which still exist today. In China, cities were founded several millennia ago. Most of these cities still exist, and some of them have become some of the biggest in the world, such as Mexico City or Manila. The 21 st century is an urban one insofar as urbanization is the most importance mutation of our contemporary world. We can observe simultaneously the development cities of all-sizes and the diffusion of an urban way of living. Indeed, today, more than half of the world population lives in cities, and, according to Michel Lussault [START_REF] Lussault | L'homme spatial. La construction sociale de l'espace humain[END_REF], the world is now urban, i.e. the urban way of life has spread all over the world: even if people do not live in cities, their way of living is totally urban. The city is no longer specific to each civilisation: it is now a global object which has diffused everywhere. How to explain this success story? We can assume it is due to a variety of different dynamics, equally the dynamics of urban societies as well as the dynamics of the cities themselves.

Dynamics of the urban societies

Dynamics are at the heart and at the origin of the urban project. According to Jacques Lévy (Lévy, 1999 ;Lévy & Lussault, 2003), founding a city is a "geographical act": that means that people have an interest in living nearby in order to benefit from all the activities and functions produced by the cities, i.e. their dynamics. Lévy goes on to write that the city is made up of diversity and density: diversity of activities, people, services, amenities, and density of all these things. A city is a sort of gamble: people gamble that they will benefit from the copresence of all this diversity and therefore move to the city.

Cities appear as the place where various innovations have been created, such as political organisation (Athens), urbanism and urban planning (Rome during the Renaissance), modern trade or international relationships (Venice). Cities are a melting pot of innovation where intellectuals have always been inventing a new way of living all together. Today, cities still have this very important cultural function: they harbour the arts and entertainment of course, but also science and research. So, cities stem from all these kinds of dynamics; but they in turn project these dynamics outside the city.

Dynamics of cities

Social dynamics due to the co-presence of various inhabitants composing an urban society have geographical effects, insofar as they lead to dynamics that spatially affect cities and their environment. This environment tends today to encompass the whole world: the history of cities reveals this general process. The main mechanism explaining this influence is the accumulation of capital (economic of course, but also political, cultural…) and its redistribution all around.

Various authors (such as geographers, sociologists or historians) have demonstrated that cities, ever since they were founded, have been organizing the space surrounding them (Von Thünen) or an area they tend to polarize (Weber, Christaller). At the origin of these phenomena stands the process of capital accumulation: Braudel then Immanuel Wallerstein were the first to explain the fact that cities are the hubs of the world-system and that cities organize it. And according to Saskia Sassen (Sassen, 2001), cities and especially global cities (i.e. London, New York and Tokyo) are motors of the current globalisation: these cities concentrate wealth, skills, political and economic power, etc.

The emergence of the sustainable city

So, taking into account the fact that cities have always been sustainable, we have to analyse the current situation, which promotes the "sustainable city" as a political requirement all over the world. What is this sustainable city that people have to build all together? What does it look like? I want to assume in this second part of my paper that sustainability is the continuation of dynamics by other means.

From cities to urban monsters

The dynamics I have just described have taken a very long time to occur. The processes have been going faster during the second part of the 20 th century. The face of cities has deeply evolved since 1950s: we can assume that cities have to face with systemic crisis due to the acceleration of their own dynamics, especially demographic growth. Urban population has regularly grown because of demographic transition (especially in emerging countries) and rural exodus. Cities become bigger and bigger sometimes exceeding 10, 20 or even 30 million inhabitants (Tokyo: 47 million; NYC: 26 million; Mexico: 23 million; Manila: 20 million; Shangai: 18 million).

This augmentation of population leads to various problems that could make cities hellish and that authorities have to cope with. Even if the situation is different between developed and emerging country countries, the concentration of population has the same consequences everywhere. How to solve the issue of concentrating people? Two solutions exist to settle this issue. On the one hand, urban sprawl which leads to peri-urbanisation, and on the other hand, densification which leads to vertical urbanisation. These two processes bring about a major issue for the political authorities of these cities. First, how to cope with the social and spatial fragmentation that is the consequence of densification or urban sprawl? Then, how to administer cities that change so fast? Moreover, the increase of population and the densification of all kinds of networks have environmental consequences within cities: recycling waste, pollution, water, transportation, green spaces… All these issues which are not only relevant to questions of quality of life are also crucial in dealing with sanitary and social problems. So, the current urban crisis has various dimensions: social (due to social and spatial fragmentation), political (lack of governance) and environmental (decrease of quality of life). Cities seem to have reached a point of no return. Social demand mainly focuses on the environmental issue, but this demand also reveals the deep inequalities of urban societies. Indeed, whereas spatial justice is a field of research that has been studied by various geographers directly inspired by the philosopher John Rawls, some scientists are exploring a new field: environmental justice whose aim consists in linking the environmental and the political and social issues. The development of this field is the consequence of the social focus on the environmental issue to the detriment of the social and political ones. This is the cause and the consequence of the emergence of sustainable development as a global issue.

Sustainable development: a social and political demand

What is this sustainable development which acts as a backdrop to generate reflection upon the urban crisis? To put it into a nutshell, sustainable development is a new way of thinking about society and social relationships, a kind of paradigm which is spreading all over the world. It deals not only with environmental issues, even if public opinion mainly focuses on them. It stems from the realisation that the world is now reached a completed state and that humanity will have to cope with the lack of natural resources.

Of course, we can criticize this concept -various authors have done so. The main criticism consists in underlining the fact that sustainable development is a located paradigm. It was elaborated during the Cold War and can appear today as the continuation of that war by other means [START_REF] Brunel | Le développement durable[END_REF]; indeed, in the name of sustainable development, the great powers justify -or not -the legitimacy of groups or States to develop themselves. Moreover, this concept is deeply influenced by the Western conception of nature as it appeared at the end of the 19 th century: nature is under threat and needs to be preserved. What is amazing today is not the importance of this concept of sustainable development (i.e. the fact that both political authorities and the business world justify their actions in the name of sustainable development), but its diffusion all over the world. How is it possible for such a located concept to be integrated by such diverse populations?

We can assume that the world social demand explains this amazing diffusion, especially the environmental demand. That is why the implementation of sustainable development mainly focuses on solving environmental issues. Because of the acuteness of the current urban crisis, the city appears as the main arena to implement sustainable development, insofar as they both concentrate the main part of the world population and the various issues that need to be solved.

The city as a testing ground for our scientific disciplines

Implementing sustainable development in our way of living seems to provide a challenge. I would further suggest that sustainable development is also a challenge for all the scientists who have now to consider their own scientific concepts, methods and practices.

A political question

Sustainable development is a political question which invites us to think differently about the political dimension of our societies. For instance, the current urban crisis serves to reveal the inadequacies in governance and administration, while the environmental problems only reveal the inefficiency of politics. So, the priority is to solve political dysfunctions by imagining a new way of living together and founding a new social contract.

The city seems to be the best place to elaborate this new social contract. First, as I mentioned earlier, cities concentrate the main part of the world population. Furthermore, the scale of cities is appropriate to implement such reflexions upon the co-existence of people. The city becomes a sort of laboratory to work out these processes. Moreover, according to Colin Crouch [START_REF] Crouch | Post-Democracy[END_REF], we are now living in a post-democratic era: two major evolutions characterize the current period. On the one hand, economic power is increasingly more important than political power. On the other hand, supra-national organisations (such as the European Union, or the World Trade Organisation, etc.) are more powerful than States which are organised along the principles of representative democracy. Whereas citizens are generally subjected to top-down decisions, people in a sustainable city are invited to inverse this tendency and to take bottom-up action.

By promoting new forms of government (such as the direct participation of all citizens, etc.), sustainable development invites us to go back to the origin of cities, i.e. the place where political organisation was first codified. This is possible thanks to the specific dynamics of urban societies. The implementation of the sustainable city is the continuation of urban dynamics by other means. It is no longer any use trying to oppose them!

A scientific challenge

I would like now to focus on the ability of the social sciences to analyse this phenomenon and to take sustainable development into account as a major emerging paradigm. Whereas Western intellectual modernity relies on the strict separation between nature and culture, sustainable development invites us to change our point of view on the world and its recent changes. Scientists have therefore to follow this emerging path (Latour, 2004 a. and b.) to construct scientific objects that pair up both social and environmental dimensions as well as environmental and political ones.

Conclusion

To conclude, I would like to focus on these current processes: the world-wide diffusion and the assimilation of sustainable development, especially in cities, invite scientists to examine the conditions which make their knowledge efficient in understanding societies. Taking account with sustainable development, scientists have to realise that societies have deeply evolved and that the patterns they are accustomed to use for its analysis must be brought up to date. That's why sustainability appears as a paradigm: it is a huge work in progress for all of us.