P Hennebert

P Merdy

A Anderson

MINERAL NANOPARTICLES IN WASTE: POTENTIAL SOURCES (FRANCE 2014), OCCURRENCE IN SOME ENGINEERED NANOMATERIALS LEACHATES, IN MUNICIPAL SEWAGE SLUDGES AND IN MUNICIPAL LANDFILL SLUDGES

Environmental assessment of engineered nanoparticles (ENPs) and engineered nanomaterials (ENMs) suffer of lack of data on production, emission, transfer, behaviour, toxicity, ecotoxicity and fate in natural compartments. This paper tries to bring factual data of production, content and emission of elements in a colloidal form or in an ENP form from some products, and two waste potential sinks: municipal sewage sludge and non hazardous waste landfill sludge (part of the landfill leachate decanted and concentrated to produce a liquid or solid sludge). The study is focused on mineral ENPs, since routine analytical methods are not available for organic ENPs in complex waste leachates. The declaration of production and importation of ENPs in France in 2014 (400 000 t) include carbon black (> 100 000 t/y), organic pigments (1 250 -12 500 t/y), miscellaneous organic substances (575 -5 750 t/y) and mineral ENPs classified here as:

• Group #1/Major elements: ENPs substances of Si (> 100 000 t/y), Ca, Ti (10 000 -100 000 t/y), Al (2 000 -20 000 t/y), Mg (1 100 -11 000 t/y), Fe (300 -3 000 t/y), Mn (10 -100 t/y), P (1 -10 t/y): ubiquitous in total content, ubiquitous as a colloid fraction in leachate of waste;

• Group #2/Minor elements (with soluble substances not classified as ecotoxic in the CLP regulation): ENPs substances of Ba (21 -212 t/y), Bi, Cr(III) (2 -21 t/y), Sr (1 -10 t/y), Zr (0.1 -1 t/y), La (0.01 -0.1 t/y), Pd, Mo, W, Y, Au (< 1 kg/y);

• Group #3/Minor elements (with soluble substances classified as ecotoxic and hazard statement code H400, H410 and H411 in the CLP regulation): ENPs substances of Ce (1 000 -10 000 t/y), Cu, Zn (10 -100 t/y), Ni, Sb (1 -10 t/y), Ag, Co (0.1 -1 kg/y).

The most "engineered" ENPs are not declared in high quantities: carbon nanotubes (1 t -10 t), Fe(0) (10 kg -100 kg), Ag(0): 0.1 kg -1 kg, Au (quantity not declared). Colloidal form or ENPs of elements in leachates of 23 engineered nanomaterials have been found in a paint leachate (Si) and in three cosmetics leachates (Al, Si, and Zn). Two cosmetics have a colloidal or ENP Zn (Group # 3 element) fraction of 25 and 164 mg/kg. From 16 concrete samples, including two laboratory samples with nanosized TiO2, one demolition concrete released Ti in colloidal form at a concentration of 1 mg/kg. No ENP of Ag, Ce, Ti and Zn in 13 municipal sewage sludges have been found by TEM/EDS. From 10 sludges leachates from municipal landfill leachates, the colloidal fraction was very frequent for the elements of Group #3 Ni, Zn, Cu, Co and Sb (but with low mean concentration of 3.2, 2.0, 1.2, 0.6 and 0.5 mg/kg when present), and only one time on ten for Ag (with very low concentration of 0.04 mg/kg when present). Ce had 4 times on 10 a colloidal form but with a very low concentration of 0.01 mg/kg when present. From this limited research, it appears that cosmetics could be a source of ENPs, that sewage sludge could be a sink for Group #3 elements Ag and Ce since their total concentrations is significant (10 and 40 mg/kg respectively), that landfill sludges have moderate total concentration of Group #3 elements Ce, Cu, Zn, Ni, Sb, Ce, Ag and Co (46, 83, 26, 16, <10, <5 and <5 mg/kg, respectively). The concentrations of Cu, Zn and Ni are comparable to soils, composts or sediments. These concentrations could mean that the fluxes of these group #3 elements from the landfill cells (by the landfill leachate, and their build-up in landfill sludge) are low.

INTRODUCTION

Environmental assessment of engineered nanoparticles (ENPs) and engineered nanomaterials (ENMs) is an active field of research. Authors emphasized the lack of data on production, emission, transfer, behaviour, toxicity, ecotoxicity and fate in natural compartments. Are traditional waste facilities (waste water treatment plants, landfill) sinks? Will sewage sludge contaminate agricultural soils? Are NPs contained in landfills? Coming OECD papers present high quality overview of the situation, the current knowledge and gaps (OECD, 2015a to e). In absence of data, precautionary principle or attitude prevails and risks assessment (emission, transfer to a target, biological effect) are rated as potentially high. "Health and environmental hazards have been demonstrated for a variety of manufactured nanomaterials", however "Not all nanomaterials induce toxic effects. Some manufactured nanomaterials have already been in use for a long time (e.g. carbon black,TiO2) showing low toxicity. Therefore, the hypothesis that smaller means more reactive, and thus more toxic, cannot be substantiated by the published data. In this respect, nanomaterials are similar to normal chemicals/substances in that some may be toxic and some may not. As there is not yet a generally applicable paradigm for nanomaterial hazard identification, a case-by-case approach for the risk assessment of nanomaterials is still recommended." (SCENHIR, 2009;OECD, 2015aOECD, , 2015b)).

This paper try to bring factual data of production, content and emission of elements in a colloidal form or in an ENP form from some products, and two waste potential sinks: sewage sludge and landfill sludge. The speciation between these colloidal form and NP form is not done. An analytical method for elements (successive microfiltration and ultrafiltration, and elemental analysis) is available in waste leachate (Hennebert et al., 2013a). The study is focused on mineral ENPs, since routine analytical methods are not available for organic ENPs in complex waste leachates (Nowak et al. 2015). The analysis of colloidal or NP-sized organic substances has been proposed by HPLC for fullerenes [START_REF] Farré | First determination of C60 and C70 fullerenes and N-methylfulleropyrrolidine C60 on the suspended material of wastewater effluents by liquid chromatography hybrid quadrupole linear ion trap tandem mass spectrometry[END_REF], and by light transmission at a wavelength of 800 nm for single wall carbon nanotubes in synthetic leachate [START_REF] Lozano | Single-walled carbon nanotube behavior in representative mature leachate[END_REF]. For the other ENP of organic substances, a proposition could be, like for the elements, to analyse the content of the micro-and ultrafiltrate, and to calculate the colloidal fraction of the organic substance by difference.

Emission data of colloids or ENP allows to rank the elements and to further focus on ENPs that could be/are really emitted. Incineration bottom ash, sewage sludge and landfill sludge are important sinks to assess the fate of the ENPs of non hazardous ENMs as waste. These three sinks have each their specific legislation piece. In this paper, sewage sludge and landfill sludge are studied. Sewage sludge results of treatment of waste water and hard surface runoff water in municipal waste water treatment plant (WWTP). Landfill sludge results of treatment of the leachates of landfills that are mainly allowed to aerate and settle before discharge in WWTP or in streams or rivers, or that are concentrated (press-filtering, drying) and finally that produce a liquid or solid sludge.

The annual flow of raw material is estimated to 23 t/capita.year in France in 2012 (with crop residues, excavated materials, fossil fuel for the exported manufactured goods) [START_REF] Jdle | [END_REF]. The amount of waste is 5.3 t/c.y, and the amount of waste without mineral non hazardous waste (mainly excavated material) is 1.5 t/c.y. [START_REF] Medde | Bilan 2012 de la production de déchets en France[END_REF]. Municipal waste from households is about 0.6 t/c.y. For France (66 M inhabitants), the non mineral non hazardous waste (municipal waste) mass includes 33.7 Mt that are recycled, 19.8 Mt that are land filled, 16.2 Mt that are incinerated, and 0.8 Mt used in agriculture. The mass of non hazardous construction and demolition waste (without excavated material) is 30 Mt. The mass of municipal solid waste incinerator (MSWI) bottom ashes is 3.3 Mt, and of waste water treatment plant (WWTP) sewage sludge is 1 Mt (dry matter), including 700 kt recycled in agriculture [START_REF] Medde | Bilan 2012 de la production de déchets en France[END_REF]SYPREA, 2015). So the main potential "sinks" for ENPs and ENMs of non hazardous waste are landfill (rounded 20 Mt/y), recycled incinerator bottom ash for road construction (estimated rounded 2 Mt/y), and recycled sewage sludge for soil amendment (0.7 Mt/y). The amount of produced and imported ENPs in France in 2014 is 0.4 Mt/y (MEDDE, 2015), a significant quantity compared to waste fluxes.

MATERIAL AND METHODS

Engineered nanomaterials

The following materials have been studied: -8 products (claimed with nanoAg ( 0 The ENPs of the 4 cosmetics have been looked for (courtesy of O. Aguerre-Chariol, INERIS). The "Eye shadow #2" contains NP ZnO (mean size 30 nm) in an inorganic phase and NP TiO2 (mean size 80 nm) with plates containing Si, Al and K. The "Lipstick #3" contains NP ZnO (mean size 30 nm). The "Make-up #5" contains plates of Fe oxide (micrometric and nanometric), TiO2 (micrometric and nanometric) always associated with Si (indicating potential coating of TiO2 particles). The "Solar cream #6" contains TiO2 (mean size 50 nm) always associated with Al, Si (probably coating of TiO2) and Zn, and ZnO (estimated ZnO phase size 30 nm) in organic layers.

Paint additives were used following the supplier's instructions with 4 different bases (glycerophtalic, acrylic, solvent-based varnish or water-based varnish, applied on a wooden support (particle board), plaster (plate covered with a cardboard layer) or cement (roof tile) and exposed vertically outdoor for one month (May 2013, one rain occurring) in Aix-en-Provence (Figure 1). The wooden and cement plates were exposed to rain and the plaster plates were not exposed to rain. Other materials have not undergone aging.

Concretes products

Concretes manufactured at the laboratory with commercial white cement and with white cement containing TiO2 nanoparticles, and white concrete commercial products, a cinderblock, two concrete demolition samples and a concrete sample of Antiquity were studied in 2014. Various white cements were purchased from building material dealers. Three other cements with known nanoparticular TiO2 content (0%, 3%, and 10%) were provided by CEREGE in limited quantities. The concrete specimens were produced in the laboratory with a mixture of commercial sand and gravel, cement (350 kg per m 3 ) and 8% water (of the total weight of the mixture). After 28 days of curing, the specimens were removed from the moulds and followed the same treatment as other samples.

Three white cement outdoor products were purchased. A cinderblock with gray cement, a sample of demolition concrete of an industrial shed of about twenty years old in Aix-en-Provence (La Duranne) and the demolition concrete of an artisanal floor slab in Aix -in-Provence (Ardevie), and a sample of "Roman concrete" (mixture of sand, lime and crushed bricks, with stones of about 5 cm) from a piece fell at the foot of an aqueduct of Via Apia southeast of Rome in 2014 were studied.

Sewage sludges

Thirteen municipal sludge samples from various waste water treatment plants (WWTP) have been collected from various locations and stage of treatment in 2013 and used [START_REF] Gay | Substances émergentes dans les boues et composts de boues de stations d'épurations d'eau usées collectives -Caractérisation et évaluation des risques sanitaires[END_REF]. Two raw sludges were used. Three samples from the same station at different stages of processing (thickened sludge, dewatered sludge, composted dewatered sludge) or with different process (filter pressing, liming) were used.

Landfill sludges

Ten sludges from 9 different locations in France were collected by the landfill managers (Feb-May 2015). There are 2 liquids, 2 intermediate (treated as solids) and 6 solid sludges, obtained by different process (decantation, centrifugation, filtration in press-filter after addition of FeCl3 and a polymer, evaporation with the heat of biogas electric generators). The liquid sludges are discharged in municipal WWTP, and the solid sludges are discharged in the landfill or incinerated. Some samples (concrete) were broken with a hammer, crushed to 10 mm with a jaw crusher, sieved to 4 mm, the refusal was ground by step to 4 mm and sieved. Other samples were shredded with a low speed laboratory shredder (Blik.fr) with 6 mm thick intersecting blades up to > 90% passing a 4 mm sieve. The water content was measured and the samples were leached 24 hours (EN 12457-2) with a net liquid/solid ratio of 10 l/kg of dry matter (DM). Leachates were successively microfiltered and ultrafiltered at 450 nm and ±3 nm (3 KDa) with a UF stirred cell (Millipore) with 200 MPa pressure. Membranes of cellulose nitrate for microfiltration (Whatman 0.45 µm, GE Life Science) and of regenerated cellulose for ultrafiltration (Ultracel® 3KDa, Millipore) were used. This method is statistically equivalent to tangential filtration with recirculation through regenerated cellulose membrane (Sartorius Sartocon 200 Hydrosart Slice) with a cut-off limit of 450 nm and 2 KDa, that was used in a previous study [START_REF] Hennebert | Experimental evidence of colloids and nanoparticles presence from 25 waste leachates[END_REF], but much more faster (result not shown). The expensive ultrafiltration membranes can be washed in deionized water and reused without contamination. The elements are measured by ICP in the microfiltrates (MF, leachable fraction, containing the colloidal/ENP fraction and the dissolved fraction) and in the ultrafiltrates (UF, assumed here to be the soluble fraction). The difference in content (MF-UF) is the colloidal or nanoparticulate fraction of this element in the leachate.

Size, Zeta potential (microfiltrates)

Sizes were first measured by Photon Correlation Spectroscopy (Zêtasizer, Malvern). This method is not fitted for waste due to polydispersity of the samples, and the assumptions leading to calculated size up to a few µm after a filtration at 0.45 µm (similar data in [START_REF] Hennebert | Experimental evidence of colloids and nanoparticles presence from 25 waste leachates[END_REF]. In 2015 measurements were done with Nanoparticle Track Analysis (Insight, Malvern). The zeta potential by was measured by Zêtasizer (Malvern).

Particle identification and composition

The presence of ENPs has been controlled in the raw cosmetic products. TEM-EDS identification and sizing of the cosmetics was done by dissolution in ethanol and TEM-EDS identification on a hydrophilised copper grid (Philips CM12, 120 kV, resolution of 0.5 nm, and Oxford X-max). About 15 pictures and dozens of EDS spectra were done per sample. NPs and elements association were looked for in three solid samples of concrete and sewage sludges by TEM-EDS (Oxford Instruments). For the same sample, several images are taken, an analysis area is chosen (rectangle with 1 -5 µm per side, with some area up to 25 µm -25 µm), and spectra of elements are obtained with relative concentrations in mass and number (excluding C and O) in the observed area. The method does not allow observing nanoparticles, but makes it possible to observe a frequency of occurrence and associations of elements in the analysis zone.

Aqua regia content of elements (sewage sludges and landfill sludges)

After pretreatment (EN 15002), the sludges were digested with aqua regia (ISO 13657). Quantitative analysis with standardized methods (ISO 11885, EN 16772) of the elements was used.

For landfill sludge, a screening method (AFNOR XP X30-489; Hennebert et al., 2012) with ICP was used for elements other than of 12 "heavy metals or metalloids" of the EU Landfill Decision 2003/33 (As Ba Cd Cr Cu Hg Mo Ni Pb Sb Se Zn). The loss on ignition (loss of mass at 550°C ± 25°C for 1 h, EN 15169) was used as estimation for organic matter in landfill sludges.

RESULTS AND DISCUSSION

Production and importation of engineered nanoparticles (ENPs) (France 2014)

The declaration of ENPs production and importation (> 100 g/year) is compulsory since 2013 in France. For the annual period closed in June 2014, 374 declarers introduced 10 417 declarations (including 6 418 declarations with CAS number) for 319 substances representing 397 131 tons (274 667 t produced and 122 464 t imported) [START_REF] Medde | Éléments issus des déclarations des substances à l'état nanoparticulaireexercice 2014[END_REF]. The name of a substance and the uses of the substance are published in the summary report if at least one declarer has not required not publishing it. The exact quantities are declared and recorded but are expressed in the report as mass intervals with an amplitude of 10 factor. The production sectors of the declarers are by decreasing frequency (sectors with >100 declarers, MEDDE, 2014): agriculture-forestry-fishing, formulation and preparations, others, machine and equipment fabrication, food industry, R&D, rubber industry, plastic industry, and fine chemicals. The declared chemicals are by decreasing frequency (chemicals with > 50 declarations, total 2 631 declarations): coatings-paints-solvents-thinners, cosmetics and personal care products, phytopharmaceuticals, fuels, polymers, others, pharmaceuticals, adhesives and sealants. The declared articles and products are by decreasing frequency (total 414 articles): rubber ( 99), mechanical, electrical and electronic machines (82), plastic products (56), vehicles (49), metallic articles (39), articles in stone, plaster coated with cardboard, cement, glass and ceramic (29), articles in paper ( 11), batteries and accumulators (3), articles in leather (1). The original report presents a Table with the amount of ENPs > 100 tons declared in 2014, and more detailed annexes. Here, the declarations were recalculated from the annexes and gathered by element (total content or leachable colloidal form, easy to analyse) and by organic substances (no routine method for analysis, i.e. black carbon, organic pigments, polymers and miscellaneous organic substances) (Table 1).

The total quantity of produced and imported ENPs is close to 400 000 t/y. The exported amount is not subtracted from that figure. The ENPs can be gathered by groups.

Proposition of classification

For organic ENPs, a first group of ENP is carbon black (> 100 000 t, > 25% of the declared exact quantity, 1 declaration). It is used in tires (reinforcing filler in tires and other rubber products), and as pigment in plastic, paint and ink. Composition of tires includes 21 % of carbon black and 20-35% of carbon black and silica [START_REF] Wik | Occurrence and effects of tire wear particles in the environment A critical review and an initial risk assessment[END_REF] and particles are emitted by tire wear and found in road runoff water and sedimented in road storm basin. Shredded tires mixed with crushed limestone (to be used as fill material in embankment) have not emitted particles during one year lysimeter study [START_REF] Hennebert | Assessing the environmental impact of shredded tires as embankment fill material[END_REF]. A second group are ENPs of synthetic organic pigments (referred in the colour index C.I. system; 102 declarations). The total of this group amounts 1 250 t -12 500 t, that is 0.31% -3.1 % of the declared exact quantity. These substances are not easily speciable in a NP form. A third group is nanosized plastic polymers (1 100 t -11 000 t). The last group is obtained by the merging of nanosized miscellaneous organic substances including pesticides, lactose, cellulose, paraffin and waxes (575 t -5 748 t). It includes 96 declarers.

For mineral ENPs, a first group (entitled #1 in the last column of Table 1) could be mineral ENPs of major elements Si, Ca, Ti, Mg, Fe, Mn, P with frequently natural analogues: silicon dioxide (including food additive E551), calcium carbonate, titanium dioxide (including E171), aluminium oxyhydroxides, iron oxyhydroxides, magnesium silicate (food additive), aluminosilicates (including E554), clay minerals, manganic pigments, calcium silicate, calcium hydrogenophosphate. The total of this group amounts 124 000 t -342 000 t, that is between 31% and 86% of the declared exact quantity. The NPs of this group will be difficult to differentiate from "unintentional" or "natural" colloids in waste leachates. They are mostly similar in composition to natural minerals, but intentionally different in cristallinity, size, shape, coating or purity, all characteristics not easy to measure when the ENPs are mixed with "natural" colloids in a waste stream.

The differentiation of the ENPs of minor elements can be based on ecotoxicity of soluble substances containing these elements. The ecotoxicity hazard of substances is defined in the European chemical legislation (CLP Regulation 2008). The substances must be tested with standardised laboratory tests (chronic, acute) at different concentrations in the biota culture media, the dose/response (a measured biological effect) relationship is statistically established and the results for acute tests are expressed as the concentration producing typically 50 % of response (EC50) on the population of tested organisms. The lowest EC50 concentration from a panel of laboratory ecotoxicological tests is used to assess the ecotoxicity of the substance. If it is below 1 mg/l, the substance is classified as acute ecotoxic with a hazard statement code (HSC) H400 (CLP 2008). For chronic ecotoxicity, longer-lasting tests are used and the result is based on the no observed effect concentration (NOEC), resulting, if lower than 1, 10 or 100 mg/l in a HSC with four levels from H410 to H413. Ecotoxicological data for the mineral substances or elements from a database (European Chemicals Agency ECHA, http://echa.europa.eu/fr/information-on-chemicals, INERIS Chemical Portal, www.ineris.fr/substances/fr/) were used here. The elements having substances with H400 and H410 hazard statement codes are Ce (EC50 not known, cerium trichloride), Cu (min EC50 = 0.041 mg/l, copper monochloride), Zn (0.032 mg/l, zinc dichloride), Ni (0.06 mg/l, nickel dichloride), Ag (EC50 not known, silver nitrate) and Co (0.334 mg/l, cobalt dichloride). These substances with HSC H400 and H410 are soluble in water. Sb has a hazard statement code H411 (generic entry in classification table for most Sb substances in the CLP).

Summary tables can be found in [START_REF] Hennebert | Assessment of four calculation methods proposed by the EC for waste hazard property HP 14 'Ecotoxic[END_REF]. The substances with the elements Ba, Cr(III), Sr Bi, Zr, La, Pd, Mo, Y and Au are not classified ecotoxic (ECHA portal, INERIS portal, Aug. 2015). It must be emphasized that the declared substances composing the ENPs of Ce, Cu, Zn, Ni, Sb, Ag and Co (Table 1) are not soluble at the ordinary pH of most waste. The couples ENP/ENM are: Ce/cerium dioxide, cerium hydroxide, cerium and iron isostearate, Cu/pigment, Zn/zinc oxide, Ni/organic complex, Sb/antimony pentoxide, Ag/Ag(0), Co/tricobalt tetraoxide. Their "chemical" ecotoxicity will probably be lower than the soluble substances mentioned above (EC50 > 1 mg/l). Their potential "physical" ecotoxicity (potential accumulation in the gastrointestinal tract, in the gills of the fishes, etc.) should also be considered. Their eventual specific "nanoecotoxicity" seems not to be known or published in a standardized and referenced system like REACH. The classification proposed here should be updated as soon as nanoecotoxicity data are available. The elements of Table 1 that are not in the first group could then be grouped by the ecotoxicity of their soluble species.

A second group could be ENPs of minor elements having non ecotoxic (non NP) soluble substances: Ba (21 -212 t/y), Bi, Cr(III) (2 -21 t/y), Sr (1 -10 t/y), Zr (0.1 -1 t/y), La (0.01 -0.1 t/y), Pd, Mo, W, Y, Au (< 1 kg/y).

A third group could be constituted of minor elements with ecotoxic (non NP) soluble substances: Ce (1 000 -10 000 t/y), Cu, Zn (10 -100 t/y), Ni, Sb (1 -10 t/y), Ag, Co (0.1 -1 kg/y).

It can be noted that the most "new" engineered mineral NPs are not much declared: Carbon nanotubes: 1 t -10 t, Fe(0): 10 kg -100 kg, Ag(0): 0.1 kg -1 kg, Au: not declared.

Occurrence and monitoring in waste leachates

The elements of group #1 elements are ubiquitous in waste total content and colloidal form in leachate fraction (frequently found in 25 waste leachates; [START_REF] Hennebert | Experimental evidence of colloids and nanoparticles presence from 25 waste leachates[END_REF] The group #2 includes four elements monitored in routine in waste samples leachate, according to the landfill acceptance criteria (EU Council Decision 2003/33/EC). The allowed leachable concentrations of Ba, Cr(VI), Sb and Mo for inert waste landfill are 20, 0.5, 0.06 and 0.5 mg/kg respectively, and for non hazardous waste landfill are 100, 10, 0.7 and 10 mg/kg respectively. Those values were derived from a risk assessment [START_REF] Hjelmar | at landfills for inert waste, landfills for non-hazardous waste accepting stable, nonreactive hazardous waste and landfills for hazardous waste listed in Council Decision[END_REF]. Leachable chromium is considered to be Cr(VI) and not Cr(III), since the substances containing Cr(III) are not soluble in water at common pH of waste. The large quantity of cerium nanoparticulate compounds (1000 t -10 000 t) are declared to be used in many products, substances and processes, and the part that is used as additive to fuel to reduce emissions is not clear from the summarized report of declarations. Ce, Bi, Sr, Zr, La, Pd, W, Y and Au are not monitored in routine waste characterisation.

The group #3 includes well monitored Cu, Zn, Ni and Sb. Cu and Zn are ubiquitous in waste and with Co are essential human micronutrients. Cu, Zn, Ni and Sb are monitored in waste leachates for landfill acceptance. Their allowed leachable concentrations for inert waste landfill are 2, 4, 0.4 and 0.06 mg/kg respectively, and for non hazardous waste landfill are 50, 50, 10 and 0.7 mg/kg respectively. Ce, Ag and Co are not monitored in routine waste characterisation.

Emission of colloidal/NP elements by engineered nanomaterials (analysis of leachate)

Products

The potential emission of colloids and nanoparticles of different engineered materials or product coated with ENPs were estimated by crushing at 4 mm, leaching (EN 12457-2), settling for 15 minutes, filtering at 0.45 µm and 3 KDa, and analysing the claimed NP elements Ag, Al, Ce, Si, Ti, Zn.

Leachates of control pieces of wood, of plaster and of cement plate without paint and without additived paint had no colloidal fraction for these elements.

The size measured by PCS on the microfiltrate fraction (filtered at 450 nm) is indicative, since many calculated values are larger than the cut-off value of the filter. Another technique should be used for size measurements. The zeta potentials are all negative and greater than -30 mV and indicate a tendency to flocculation or co-flocculation (homo-or hetero-aggregation) in natural waters or sewage, as only particles with a zeta potential >|30 mV| are considered as electrostatically stable [START_REF] Hassellöv | Analysis and characterization of manufactured nanoparticles in aquatic environments[END_REF]. The colloidal concentrations of the elements present in the engineered NPs are presented in the table. Other elements showed no colloidal fraction, except the major elements Ca, Mg, K and Na (data not shown). Control leachate samples of wood, plaster with cardboard and cement plate without paint and with the four kinds of paint (acrylic, glycerophtalic, solvent-based lasure, waterbased lasure) without additive showed no colloidal fraction for the added elements. This contrasts with the findings of [START_REF] Kaegi | Release of silver nanoparticles from outdoor facades[END_REF] of emission of silver from paint during rain events, with a release of 0.5 mg/m² (30% of the applied Ag), and concentration in runoff water between 1 and 145 µg/l. From 24 samples (unweathered, unaged), there is presence of colloidal or manufactured nanoparticles in leachates of a glycero paint additived with SiO2 (24 mg Si/ kg), a lipstick (25 mg Zn/kg), a make-up formulation (33 mg Si/kg) and a solar cream (11 mg Al/kg, 33 mg Si/kg 19 mg Ti/kg, and 164 mg Zn/kg). The other ENMs or products don't emit significant concentration of elements in colloidal or ENP form. TiO2NPs have been found in facade and urban runoff water [START_REF] Kaegi | Synthetic TiO2 nanoparticle emission from exterior facades into the aquatic environment[END_REF]. The runoff rain water will flow to natural waters or to WWTP. The cosmetics form bathing water and personal wash water will to septic tank or to municipal waste water treatment plant (WWTP). The analysis of some water treatment sludges are presented later in the paper.

Concrete with and without TiO2 NPs (analysis of leachate)

The characteristics of concrete leachate and the contents of leachable and colloidal/NP Ti are presented in Table 3. The alkaline pH and slightly reducing redox potential (pe+pH) samples are classical for concrete, with a lower value for the Roman concrete sample, which has a pH close to the pH of limestone. The contents of leachable Ti are very low. The laboratory samples # 8 and 9 (unweathered, unaged) with cement containing 3% and 10% of ENP TiO2 have no significant release of colloidal or NP Ti. The demolition concrete of an industrial shed has a leachable colloidal or NP Ti content of 1 mg/kg, a low value. The Ti concentration in this solid sample was studied with electron microscopy probe. In total 15 spectra were recorded, focusing on particles visually different of the mass of the sample. The detection of elements in these records is presented at Table 4, by decreasing order of occurrence.

The matrix of this demolition concrete contains Ca, K, Si, Fe and Al (most frequently detected cations), and secondarily Na and Mg with accompanying anions (S Cl) and other elements present in lower concentration (Cr, Ti, Cu, Zn, Br and P) (Table 4). Titanium is present in 3 cases of 15 observations of the sample #14. It is each time associated with Al, Ca, Fe, K and Si, and two times out of three Cl, Mg and Na. It therefore seems associated with this sample matrix, and does not appear as a separate phase or ENP, at this scale of observation (rectangle of 1 -5 µm * 1 -5 µm).

Content and emission of colloidal/NP elements by municipal sludge and landfill sludge

WWTP sludges (total content)

Since there is no available method for detecting nanoparticles in solid, except a tedious search by microscopy, which is not accessible in routine, a method of selection of solid samples is tested here. The outliers of a Gaussian distribution (samples with higher total content of the element than expected if the population was normally distributed) will be looked for by microscopy.

The elements Ag, Ce, Ti and Zn were measured after aqua regia extraction of 13 sludges (Table 5).

The mean (± 1 SD) total concentration of Ag is 10 (± 5) mg/kg sludge DM. The common range of Ag for soils is 0.01 -5 mg/kg, with a selected average of 0.05 mg/kg [START_REF] Lindsay | The chemical equilibria in soils[END_REF], 200 times less than the mean concentration observed in the sludges. The high content in the sludge is hence not from soil origin. The anthropogenic accumulation factor (AMF) is the ratio of the mass extracted annually by mining and fossil fuel production divided by the mass released annually by crustal weathering and volcanic activity [START_REF] Klee | Elemental cycles: a status report on human or natural dominance[END_REF]Graedel 2004, cited in Sposito 2008). The AMF for Ag is 185. Is this high Ag level due to ENP? The declared ENP Ag(0) production and importation in France in 2014 is 0.1 kg -1 kg (Table 1). The imported goods and the accumulated goods including NPs are not taken into account in that figure. The production of sewage sludge in France is 1 million tons DM/year. The ratio gives a potential concentration of < 1 µg Ag/kg sludge DM. This calculation at the scale of one country may not apply to a particular sludge. [START_REF] Kaegi | Fate and transformation of silver nanoparticles in urban wastewater systems[END_REF] have found with spiking experiments that Ag-NP discharged to the wastewater stream will become sulfidized to various degrees in the sewer system and are efficiently transported to the WWTP, and will accumulate in the sewage sludge. It seems nevertheless that Ag in sewage sludge is not geogenic neither "NPgenic". The mean concentration of Ce is 40 (± 15) mg/kg sludge DM.

The most abundant of the rare earth elements is seldom monitored in environmental studies.

The concentration in the earth crust is 46 mg/kg (https://en.wikipedia.org/wiki/Cerium). As the annual production and importation in France of Ce is 1 000 t -10 000 t, but the use for fossil fuel is not clearly separated in the MEDDE report. The observed concentrations in the sludges seem to us quite high, and Ce should be monitored for an in-depth assessment. The mean concentration of Ti is [START_REF] Lindsay | The chemical equilibria in soils[END_REF] or 2 400 mg/kg [START_REF] Sposito | The chemistry of soils. 2nd edition[END_REF]). The AMF of Ti is 1. [START_REF] Kiser | Titanium Nanomaterial Removal and Release from Wastewater Treatment Plants[END_REF] have observed in 8 WWTPs around the United States much higher concentrations (1800 to 6400 mg Ti/kg DM, averaging 2800 ± 1500 mg Ti/kg DM). The mean concentration of Zn is 1 900 (± 1 000) mg/kg sludge DM, 20 times more than representative concentration in soil (50 mg/kg [START_REF] Lindsay | The chemical equilibria in soils[END_REF], 48 mg/kg [START_REF] Sposito | The chemistry of soils. 2nd edition[END_REF], range of 10 -100 mg/kg in soils without natural anomaly in France [START_REF] Baize | Teneurs totales en éléments traces métalliques dans les sols (France)[END_REF]). The AMF of Zn is 115. So Ag and Zn concentration in the sludges clearly originate from products rather than soil. These two elements have high AMF. The contents of Ag and Ce follow a normal (Gaussian) distribution, whereas the contents of Ti and Zn are not normally distributed (Shapiro-Wilk test, XLStat), with respectively 2 and 1 higher grade/outlier samples (samples #12 and #10, and sample #9, respectively). The sample #9 had also high content of Ag, Ce and Zn. The concentrations of the elements have been investigated in samples #9 and #10 with electron microscope with probe elements.

The mineral matrix of the sludge contains Ca Fe Si Al (most frequently detected cations) and secondarily Mg K Na (Table 6). The major anions P and S are probably part of organic compounds (C and O, unquantified) and are also present in almost all the observations of each sample. The matrix of the sludge thus appears as aggregates of aluminosilicate (probable earthy origin) and organic matter.

Separate phase of Ti, i.e. aggregated NP TiO2 areas, were not found (area of observations 25 microns per side). Ti is present in half of the cases (7 of 13 observations for the raw sludge and 4 of 9 observations for the digested sludge). When present, Ti is associated with Al, Ca, Fe, K, Mg, Na, P, S, Si (7 times 7) and Cl (5 times 7) for the raw sludge, and Al, Ca, Fe, K, P, S, Si (4 out of 4) and Cl (3 times 4) for the digested sludge. Concentrations (excluding C and O) are between 1 and 2%. Ti is therefore present in combination with the organo-mineral matrix sludge at this scale of observation. The origin of Ti may be soil, but it is not observed in all spectra of the same sample, and, as mentioned, soils have higher concentrations of Ti. Ti can also be anthropogenic (white paints, toothpaste, certain foods, sunscreen ...) and has been found as NP in façade runoff [START_REF] Kaegi | Synthetic TiO2 nanoparticle emission from exterior facades into the aquatic environment[END_REF]. This study cannot discriminate between pedologic and anthropogenic (nano or non nano) Ti. Zn was not detected by the probe (no peaks in the spectra) despite a high total concentration of almost 5 g/kg. Zn is scattered in the organo-mineral matrix of the sewage sludge, as an adsorbed phase on the reactive surfaces, and probably not as a particles in these two samples.

So, separate phase(s) containing Ti or Zn in outliers of a normal distribution have not been found, despite the relatively high total concentration (350 mg/kg and 5 g/kg, respectively).

Municipal landfill leachate sludges (analysis of total content and leachable content)

The main characteristics of the sludges are presented at Table 7. Water content, density, loss on ignition and the pH of the samples are very different. Characteristics are published in extenso due to variability between samples, and scarcity of data of this kind of waste. The samples are humid, excepted one dry powder with a very low density (sample 7). Organic matter contents of the solid samples, approximated by the loss on ignition, are between 11 and 63%. The pH varies between 3.2 and 12.4. Anions are measured by ionic chromatography in the leachates of solid samples and in filtered liquid samples. 45 elements have been measured.

The 27 elements with at least one value > LQ are presented at Table 7 by decreasing order of mean. The other 18 elements have all values lower than LQ (< 5 mg/kg unless otherwise specified): Ag, Au, Be, Bi, Cd (LQ 0.8 mg/kg), Co, Ga, Ge, Hf, Hg (LQ 0.3 mg/kg), In (LQ 10 mg/kg), Nb, Pd, Rh, Se, Ta, Te, W.

Elements of group #3 (highlighted) are present at low total concentrations, or below the LQ. Ce, 100 or 1000 times more used as ENP that the other "classical" elements Zn, Cu, Ni and Sb (EU Landfill Decision 2003/33) (Table 1), is not found. The "classical" elements are found at concentrations following the same order than the declared ENP annual quantities, reflecting probably their general use in non NP species in non hazardous products, rather than their ENP fraction. Ag and Co, declared at quantities < 1 kg, are not found.

Comparison with representative soil concentrations indicates higher values in the landfill sludges for B and Mo, equivalent values for Cr, Zn, Cu, Ni, Li, Pb, Sn and As, and lower values for Ba, Mn, Ti, V and Zr [START_REF] Lindsay | The chemical equilibria in soils[END_REF][START_REF] Sposito | The chemistry of soils. 2nd edition[END_REF][START_REF] Baize | Teneurs totales en éléments traces métalliques dans les sols (France)[END_REF]. Some representative mean concentrations of Zn, Cu and Ni in soils, composts and sediments of France are presented at Table 8. It seems that landfill sludges have content comparable those three environmental compartments. Concentrations of Sb, Ag and Co are not available. The solid samples have been leached, as well as the liquid samples 8 and 10. Liquid samples 1 and 6 have been used as they were. The leachates were microfiltered and ultrafiltered. The distribution of particles of microfiltrates of two samples is illustrated at Figure 2.

The concentrations are between 0.5 10 9 and 5 10 9 particles/ml, and the D50 and D90 are similar for the different samples at 92 nm and 147 nm (Table 9). It has been noticed that the visible absorbance of the microfiltrate is roughly correlated with the number of particles (result not shown). [START_REF] Baize | Pb, Se, Zn) dans les sols agricoles en France. Résultats d'une collecte de données à l'échelon national[END_REF] Compost from Household compostable waste [START_REF] Zdanevitch | Etude comparative de la qualité de composts et de digestats issus de la fraction fermentescible d'ordures ménagères collectée séparément ou en mélange[END_REF] Fluvial Sediments [START_REF] Padox | Qualité chimique des sédiments fluviaux en France, Synthèse des bases de données disponibles[END_REF] The concentrations of colloidal or NP fraction in the leachate are presented at Table 9. Au, Bi and La have been detected neither in the microfiltrate, nor in the ultrafiltrate (LQ 0.05 mg/kg solid, 0.005 mg/kg liquid). The leachable mean concentration of Pd is 0.09 mg/kg, and of Y is 0.06 mg/kg, without colloidal fraction. The major elements of group #1 have the highest colloidal fraction. Ti (group #1) has a maximal colloidal concentration of 5.5 mg/kg in a solid sample, and Ce (group #2) a maximal concentration of 10 mg/kg (in a liquid sample). The elements of group #3 (highlighted in the table) have low concentration of colloidal fraction, between 3.2 and 0.01 mg/kg (mean value when present).

The occurrence of elements in a colloidal or ENP form (referred below as the colloidal fraction) is presented at Table 10. The major elements of group #1 with low solubility oxides/hydroxides at neutral pH (Ti, Al, Fe and Mn, with the notable exception of Si) are totally or mostly colloidal, while cationic alkaline and alkaline-earth metals (Na, K, Mg, Ca, Sr and Ba) have high colloidal occurrence (between 9 and 6/10).

Elements of group #3 are highlighted in the table. Cu, Sb, Zn, Co and Ni are remarkably mainly in a colloidal form (between 10/10 and 8/10), with a ratio colloidal/leachable (when colloidal fraction is present) between 35 and 67% (mean value). Ce is 4 times on 10 in colloidal form, with a high colloidal/leachable fraction of 90% but with a very low concentration of 0.01 mg/kg, Table 9). Ag, the seventh element of the group, has a colloidal fraction of 100% of the leachable fraction (but in only one sample with at low concentration of 0.04 mg/kg, Table 9).

CONCLUSIONS

A classification is proposed, based on the production and importation amounts of ENPs in France in 2014 (400 000 t), and on CLP classification of ecotoxicity for the soluble forms of the elements: -Major elements (group #1): ENPs substances of Si (> 100 000 t/y), Ca, Ti (10 000 -100 000 t/y), Al (2 000 -000 t/y), Mg (1 100 -11 000 t/y), Fe (300 -3 000 t/y), Mn (10 -100 t/y), P (1 -10 t/y) : ubiquitous in total content, ubiquitous as a colloid in leachate fraction of waste -Minor elements (with soluble substances not classified as ecotoxic in the CLP regulation) (group #2): o With production and importation > 1 t/y: ENPs substances of Ba (21 -212 t/y), Bi, Cr(III) (2 -21 t/y), Sr (1 -10 t/y), o With production and importation < 1 t/y: ENPs substances of Zr (0.1 -1 t/y), La (0.01 -0.1 t/y), Pd, Mo, W, Y, Au (< 1 kg/y).

-Minor elements (with soluble substances classified as ecotoxic and hazard statement code H400, H410 and H411 in the CLP regulation) (group #3): ENPs substances of Ce (1 000 -10 000 t/y), Cu, Zn (10 -100 t/y), Ni, Sb (1 -10 t/y), Ag, Co (0.1 -1 kg/y). The most "engineered" ENPs are not declared in high quantities: carbon nanotubes (1 t -10 t), Fe(0) (10 kg -100 kg), Ag(0): 0.1 kg -1 kg, Au (quantity not declared).

Colloidal form or ENPs of elements have been looked for at laboratory in leachates (EN 12457-2) of 23 engineered nanomaterials and have been found in a paint leachate (Si) and in three cosmetics leachates (Al, Si, and Zn). Two cosmetics have a colloidal or ENP fraction of 25 and 164 mg/kg Zn, respectively. From 16 concrete samples, including two laboratory samples with nanosized TiO2, one demolition concrete released Ti in colloidal form at a concentration of 1 mg/kg. The total content of Ag, Ce, Ti and Zn in 13 municipal sewage sludges showed two outliers of Gaussian distribution of Ti and Zn, but no ENPs have been found by TEM/EDS. From 10 sludges leachates from municipal landfill leachates, the colloidal fraction was very frequent for the elements of group #3 Ni, Zn, Cu, Co and Sb (but with low mean concentration of 3.2, 2.0, 1.2, 0.6 and 0.5 mg/kg when present), and only one time on ten for Ag (with very low concentration of 0.04 mg/kg when present). Ce had 4 times on 10 a colloidal form but with a very low concentration of 0.01 mg/kg when present.

From this limited research, it appears that cosmetics could be a source of ENPs. Sewage sludge could be a sink for Ag and Ce since their total concentrations is significant (10 and 40 mg/kg respectively). Landfill sludges have moderate total concentration of Cu,Zn,Ni and Sb (46,83,26 and 16 mg/kg, respectively), and Ce, Ag and Co are not found. The concentrations of Cu, Zn and Ni are comparable to soils, composts or sediments. These concentrations could mean that the fluxes of these group #3 elements from the landfill cells (by the landfill leachate, and their build-up in landfill sludge) are low.

AKNOWLEDGEMENTS

The Ministère de l'Ecologie, du Développement Durable et de l'Energie (MEDDE, Paris) has funded this work (Program DRC01 and DRC54). The Centre d'Enseignement et de Recherche en Géosciences de l'Environnement (CEREGE, team Interfast, unité mixte CNRS -Aix Marseille Université) is acknowledged for the cement sample with nanoparticular TiO2. Mr Aguerre-Chariol is thanked for information about ENPs in the cosmetic raw samples. Pauline Molina, Ismahen Samaali, Camille Malburet and Samy Vidal are thanked for their nice contribution to the laboratory work.

Figure 1 :

 1 Figure 1 : Paints and varnished with ENPs additives, and paints containing ENPs on wood panel, plaster with cardboard, and cement plate

Figure 2 :

 2 Figure 2 : Granulometry of microfiltrate of landfill sludge by Nanoparticle Track Analyzer (left: solid sample 4 after leaching, 1.6 108 particles/ml dilution 10; right: liquid sample 10, 5.0 107 particles/ml dilution 2.5)

  ): dressings Mercurochrome® with Ag, flexible keyboard; claimed with TiO2: floor tiles (from construction material dealer), medical pen Senator Antibac Ref

	2645;	claimed	to	contain	NPs	without	further	information:	4	cosmetics;
	-6 paint additives (Nanobyk 3601, 3605, 3610, 3650, 3812, 3840 containing Al2O3, formulated
	Al2O3, SiO2, formulated SiO2, CeO2 and ZnO, all from BYK, www.byk.com) ending with 13
	paint/additive/support combination;
	-	3	paintings	(PhotoCAL	Masonry	(from	NANOFRANCE		Technologies,
	http://www.photocal.fr/photocatalyst_-_photocal.html); Stophotosan and Stocolor climasan from
	STO, http://www.sto.fr/).

Table 1 :

 1 Quantities of ENPs manufactured and imported in France (declaration of 2014) by element(s) and by groups of organic substances

	Elements/s ubstances	Declar ations	Mass min t/y	Mass max t/y	Major substances	Proposed Group for elements
	Carbon Black	1	>100 000 >100 000 carbon black
	Si	38 >100 000 >100 000 SiO2 (including food additive E551)
	Ca	2	10 000	100 000 CaCO3
	Ti	1	10 000	100 000 TiO2 (including food additive E171)
	Al	4	2 010	20 100 boehmite AlOOH, Al2O3
	Organic Pigments	102	1 250	12 499 synthetic organic pigments
	Polymer	13	1 141	11 412	PVC 1 000-10 000 t, various esters 121-1 210 t, nitrile 10-100 t, methylmethacrylate 10-100 t, polyacrylates 0.1-1 t, PTFE 0.1-1 t, PS 0.1-1 kg, other N.D.
					Reacting mixture of cerium dioxide and zirconium dioxide (additive to fuel to reduce
	Ce	6	1 031	10 310	emissions, chemical synthesis) 10 000 t, cerium dioxide and iron dioxide isostearate
					11-110 t, CeO2 10-100 t, Ce(OH)4 10-100 t
	Si Mg	1	1 000	10 000 synthetic amorphous magnesium silicate MgO:XSiO2•H2O (including food additive)
	Organic substances	96	575	5 748 synthetic organic substances including pesticides
	Fe	8	302	3 022 FeOOH, Fe2O3, isostearate of iron oxide (including Fe(0): 10-100 kg)
	Si Al Mg	1	100	1 000	synthetic amorphous sodium aluminosilicate 14SiO2•Al2O3•Na2O•3H2O (including food additive E554)
	Si Al Na	1	100	1 000 sodium magnesium aluminium silicate
	Kaolinite	1	100	1 000 clay mineral Al2Si2O5(OH)4
	Ba	6	21	212 BaSO4, pigments
	Cu	3	10	101 Pigments
	Mn	3	10	101 Pigments
	Attapulgite	1	10	100 clay mineral (Mg,Al)2Si4O10(OH)•4(H2O)
	Zn	1	10	100 ZnO (main uses: adhesive, cosmetics, paints)
	Bi	3	2	21	bismuth iodide oxide, bismuth bromide oxide (main uses: paints, chemical preparations)
	Cr(III)	4	2	21	pigment, chromium iron oxide
	Si Ca	1	1	10	silicic acid, calcium salt
	CNT	2	1	10	carbon nanotubes, single and multi-walled
	Ca P	3	1	10	calcium hydrogenorthophosphate
	Ni	2	1	10	nickel, 5,5'-azobis-2,4,6(1H,3H,5H)-pyrimidinetrione complexes (use: chemical preparations)
	Sr	1	1	10	pigment
	Sb	1	1	10	diantimony pentoxide (main use: fabrication of plastic products)
	Zr	1	0.1	1	zirconium dioxide
	La P	1	0.0100	0.100 lanthanum phosphate
	Pd	1	0.0100	0.100 colloidal palladium
	Ag	1	0.0001	0.001 silver
	C	1	0.0001	0.001 carbon
	Co	1	0.0001	0.001 tricobalt tetraoxide
	Mo	1	0.0001	0.001 molybdenum
	W	2	0.0001	0.001 tungsten trioxide
	Y	1	0.0001	0.001 yttrium oxide
	Au	1	N.D.	N.D. gold nanoparticles
					Other complex substances with 2 or more elements; Al P, Ba Cr Cu, Ba Ti, Ca Sr P,
	Elements	16	14	144	Co Al, Cr Sb Ti, Li Ti, Mn Fe, Mo Si, Mo W P, Pb Cr Mo, S (n.d.), Si C, Si Li Mg Na,
					Sr Ti, Y Zr
	Total	334	227 695	476 953
	Declared
	exact
	quantity

Table 2 :

 2 Size, zeta potential and colloidal or nanoparticulate concentration of selected elements in leachate of nanomaterials

			Size	Zeta
	ENP	ENM/Product/Preparation	(PCS)	potential	Ag Al Ce Si Ti Zn
			nm	mV
	Ag	Keyboard	869	-27
		Wound dressing	256	-5.1
	Al2O3	Solvent-paint (lasure) on wood	205	-17		1
		Varnish on wood	203	-11
	Al2O3	Acrylic-paint on wood	268	-4.9			1
	formulated
		Water-paint (lasure) on wood	189	-14		5
		Solvent-paint (lasure) on wood	148	-13		1
		Varnish on wood	1201	-19
	CeO2	Glycero-paint on plaster with cardboard	432	-11		6
	SiO2	Glycero-paint on wood	9716	-24		24
	formulated
		Solvent-paint (lasure) on wood	1438	-12			1
		Varnish on wood	274	-7.2		3
	TiO2	Stoneware tiles	283	-24	2	3
		Medical pen	223	-7.1			7
		Photocatalytic indoor paint on plaster	356	-3.5		1
		Photocatalytic coating on masonry	361	-17	2	5
		Photocatalytic outdoor paint on cement sheet 3674	-2
		Eye shadow #2	422	-22	1	2
		Lipstick #3	114	-22		2	25
		Makeup #5	207	-14		33
		Solar cream #6	224	-6.9	11	33 19 164
	ZnO	Acrylic-paint on wood	770	-20
		Acrylic-paint on plaster with cardboard	394	-5	1
		Water-paint (lasure) on wood	594	-19
	Blank : < 0.1 mg/kg DM

Table 3 :

 3 pH, rédox status, conductivity, leachable Ti and colloidal or nanoparticulate leachable Ti from concrete products

	n°	Sample	pH pe+pH	CE (mS/cm)	Leachable Ti (mg/kg)	Leachable Colloidal or ENP Ti (mg/kg)
	1	Concrete with white CEM I 52.5 PRB 25 kg	12.7	12.2	7.38	0.020
	2	Concrete with white CEM I 52.5 Bostik 10 kg	12.8	12.7	9.52	0.016
	3	Concrete with white CEM I 52.5 Vicat 5 kg	12.8	12.6	7.62	0.021
	4	Concrete with white CEM I 52.5 Lafarge 5 kg	12.8	12.6	9.41	0.024	0.013
	5	Concrete with white CEM II 32.5 Lafarge 25 kg	12.8	12.7	8.80	0.018
	6	Concrete with grey CEMV/A 32.5 Lafarge 25 kg	12.6	11.9	4.98	0.029
	7 Concrete with cement CER 0% NP TiO2	12.8	12.2	8.85	0.018
	8 Concrete with cement CER 3% NP TiO2	12.7	12.2	6.16	0.012
	9 Concrete with cement CER 10% NP TiO2 12.9	12.3	8.93	0.034	0.013
	10 White concrete product -tiling	12.8	12.7	9.59	0.016
	11 White concrete product -balustrade	12.6	12.5	4.61	0.021
	12 White concrete product -edging	12.2	12.0	1.33	0.016
	13 Grey concrete product -cinderblock	12.1	12.3	2.21	0.015
	14	Demolition concrete from industrial structure	11.9	12.9	7.15	1.020	0.999
	15	Demolition concrete from artisanal outdoor slab	12.0	12.3	2.47	0.012
	16	Roman concrete from Via Apia Roma 2014	8.5	9.2	0.66	0.059	0.010

Table 4 :

 4 Presence and relative concentrations of elements in microscopy observation of concrete sample #14

	Element N observation /15 Mean concentration % Min Cc % Max Cc %
	Ca	15	36.23	0.40	93.95
	K	12	4.59	0.77	16.18
	Si	12	4.47	0.10	15.48
	S	12	2.53	0.11	7.95
	Cl	11	11.69	0.42	38.98
	Fe	9	46.77	1.46	84.32
	Al	8	2.73	0.45	5.86
	Na	8	10.14	1.94	30.72
	Mg	6	1.67	0.17	3.76
	Cr	6	9.33	0.88	14.30
	Ti	3	2.14	1.35	2.76
	Cu	2	42.04	29.69	54.39
	Zn	1	5.34	-	-
	Br	1	2.99	-	-
	P	1	0.10	-	-

Table 5 :

 5 Concentration of pseudo-total Ag, Ce, Ti and Zn in 13 municipal waste water treatment plant sludge

	Element concentration (mg/kg DM)	Ag	Ce	Ti	Zn
	Sludge 1	6	36	166	985
	Sludge 2	5	30	99	1877
	Sludge 3	7	63	198	845
	Sludge 4	8	36	86	2209
	Sludge 5	4	14	80	1145
	Sludge 6	9	30	50	2454
	Sludge 7	14	23	74	2159
	Raw Sludge #8	5	29	79	1644
	Raw sludge #9	18	62	232	4851*
	Thickened sludge #12	15	49	357*	2097
	Digested sludge #10	15	46	383*	2100
	Digested composted sludge #11	13	32	335	1875
	Solid sludge #13	8	53	74	800
	Mean (standard deviation)	10 (4.6)	39 (15)	170 (120)	1926 (1042)

*outliers of a Gaussian distribution

Table 6 :

 6 Presence and relative concentrations of elements in the microscopy observations of waste water sludge samples #9 and #10

		Raw sludge #9				Digested sludge #10
	Element	Presence/13 obs. Cc mean Cc min Cc max	Presence/9 obs. Cc mean Cc min Cc max
	Ca	13	30.71	9.90	57.85	9	8.07	4.47	14.09
	P	13	15.35	3.50	21.93	8	18.33	10.59	26.75
	S	13	8.60	2.11	42.15	8	4.72	0.90	10.11
	Fe	12	33.94	21.19	58.03	8	36.71	10.09	65.36
	Si	12	5.88	0.89	9.15	9	18.55	0.89	55.92
	Al	12	2.49	0.57	4.11	6	7.97	1.09	14.24
	Mg	12	1.48	0.92	2.11	3	2.84	1.81	4.46
	K	11	3.86	0.36	29.50	8	4.53	0.73	16.50
	Na	11	3.66	0.41	28.39	2	17.52	1.61	33.42
	Cl	8	5.93	0.12	38.60	6	6.65	0.25	33.63
	Ti	7	0.91	0.63	1.16	4	1.60	1.00	2.37
	Ni	1	1.96	-	-	1	8.36	-	-

Table 7 :

 7 Main characteristics and total concentration of 45 elements in 10 landfill sludge samples (mg/kg dry matter for solid samples, mg/kg raw material for liquid samples). Elements of group #3 are highlighted.

	Sludge Sample	Solid						Liquid				Mean
		2	3	4	5	7	9	1	6	8	10	(if > LQ)
	Characteristics
	Water % w/w	68	65	83	81	16	55	96	96	72	70	70
	Bulk density g/cm3	0.64	0.63	1.07	1.34	0.09	0.38	1	0.98	1.15	1.15	0.84
	LOI % DM	8.9	63.4	19.8	8.6	11.2	14.2					21.0
	pH (leach./liquid)	12.2	3.2	8.1	8.4	9.7	12.4	8.2	7.9	5	9.3	8.4
	CE µS/cm (leach./liquid) 6910	6820 12090 2090	73600	9930 37100 5860 217000	53200	42460
	Parameter mg/kg
	Cl (leach./liquid)	1744 11238 6297	1686 222208	24325 3165 402 30929	41700	34369
	SO4 (leach./liquid)	666		50313 2738 105829	1329	6462 1481 90732	15948	30611
	Br (leach./liquid)	13	8	19	9	951	100	20		232	257	179
	F (leach./liquid)	216	9	111	5	132	10	6		51	61	67
	I (leach./liquid)	3		9	1	177	38			38	112	62
	Ca	100000 2500 37500 100000 6750	100000 150 1000 1000	6750	35565
	Na	375	1000 2500	2500 100000	1000	2500 1000 100000 100000	31088
	S	3240 39300 39300 7170	22000	4610	2660 642 77067	8900	20489
	K	1000	1000 6750	6750	37500	375	2500 375 37500	17500	11125
	Si	375	150	2500 37500	375	1000	150	150	375	150	4273
	Fe	6750 17500 2500	6750	150	375	13	150	150	375	3471
	Mg	1000	375	2500	2500	6750	1000	150	375	1000	2500	1815
	Al	150	150	1000	6750	13	150		35	35	35	924
	P	1000	1000 1000	1000	150	1000	150	150	375	2500	833
	B	35		15	15	375				150	1000	265
	Th					150						150
	Cr	90	299	213	58	22	56			21	21	98
	Sr	150	13	150	150	150	35		13	35	150	94
	Zn	38	251	161	153	9	39			8	8	83
	Ba	22		173	119	18	23				18	62
	Mn	35	13	150	150	35	35		35		35	61
	Cu	23	25	42	113		26					46
	Mo		17	72								44
	Ni	13	100	36	17	10	18			11	6	26
	Ti	35	35	13	35	13	13		13	35	35	25
	Li					35				13	13	20
	V		13		13	13	35			13		17
	Pb			12	21							17
	Zr	13	13	13	35	13				13		16
	Sb		19			12						16
	Sn	15	15	15								15
	As		27	14	6	7	7			10		12
	Ce											<10
	Ag											<5
	Co											<5
	Quantitative data for elements As Ba Cd Cr Cu Hg Mo Ni Pb Sb Se Zn (EN ISO 11885, EN 16772), semi-quantitative data for other elements. Blank
	value = <5 mg/kg. Other elements with all values lower than LQ (< 5 mg/kg unless otherwise specified): Au, Be, Bi, Cd (LQ 0.8 mg/kg), Ga, Ge, Hf,
	Hg (LQ 0.3 mg/kg), In (LQ 10 mg/kg), Nb, Pd, Rh, Se, Ta, Te, W.

Table 8 :

 8 Concentration of Zn, Cu and Ni in agricultural soils, in compost and in sediments

		Agricultural soils
	Compartment	(uncontaminated)

Table 9 :

 9 Colloidal or ENP leachable concentration of elements in 10 landfill sludge samples

	Colloidal or ENP Concentration Solid						Liquid				Mean
	Sample	2	3	4	5	7	9	1	6	8	10
	Concentration (10 9 particles/ml) 0.45 0.54 1.69 0.18	1.57 1.48	5.08 0.90 1.17	0.25	1.32
	D50 (nm)	96	81	96	75	97	102	83	94	102	91	92
	D90 (nm)	146	112	162	126	175	154	131	154	164	147	147
	Element (mg/kg)
	S	180			62 1900	290	30	87 37600 1000
	Na	30			220		70	50	45 33100 1000
	K	40			60		20	70	29 16500
	Fe	7 8600	11	33	162	4	0	1	79	30	893
	Ca	3600			84	152 1800	33	137	600	80	811
	Mg				16	510	2	27	56	970	140	246
	P	17	90	14	3	22	23	19	3	164	26	38
	Si					4	2	1		119		31
	Cr	0.1 12.4	6.3	0.6	14.6	0.6	1.5	0.3	9.3	6.0	5.2
	Ni	5.4	3.8	2.1	0.3	9.2	1.7	0.2	0.2	5.5	3.6	3.2
	Al	0.2	14	6	1	2	0	1	0.1	5	1	3
	Sr	5			0.4	5	8	0.2	0.4	4	0.4	3
	Zn	0.4 12.6	0.2	0.4	2.0	0.4	0.1	0.0	2.5	1.5	2.0
	Ti	1.2	5.5	2.1	0.4	2.6	0.6	1.2	0.5	3.2	0.6	1.8
	Mn	0.3		0.9	0.3	6.0	1.2	0.3	3.0	1.7	0.7	1.6
	Cu	10.6	0.1	0.0	0.1	0.2	0.9	0.0	0.0	0.1	0.2	1.2
	As	0.1	1.5		0.1	1.0	1.4	0.02	0.1	5.2	0.5	1.1
	Zr	0.1	1.2	0.7	0.1	3.6	0.3	0.3	0.1	1.7	0.7	0.9
	Co	0.8	0.1	0.2		1.9	0.7	0.1	0.1	0.6	0.6	0.6
	Sb	0.1	0.8	0.3 0.04	2.9	0.1	0.1	0.0	0.2	0.3	0.5
	Ba	0.7	0.1	0.0	0.3	0.5	1.9	0.1	0.1	0.3		0.4
	Se	0.03	0.1	0.1		0.6	0.1 0.005		0.5	0.7	0.3
	Mo		0.9		0.02	0.1	0.0 0.005		0.7	0.1	0.3
	W		0.4					0.01		0.3		0.2
	Ag					0.04						0.04
	Pb					0.006			0.003		0.1	0.03
	Ce		0.02				0.01 0.002		0.01		0.01
	Cd	0.01 0.01 0.003		0.02 0.01 0.001 0.001 0.01	0.01	0.01
	Hg					0.0005		0.0001			0.0009 0.0005
	Pd											-
	Y											-

Table 10 :

 10 Occurrence of colloidal or ENP fraction by element in 10 landfill sludge leachate samplesOccurrence as colloidal or ENP In case of occurrence, (Colloidal or ENP)/Leachable ratio

	Element	n/10	Mean	min max
	Ti	10	81%	53% 100%
	Cr	10	72%	40% 100%
	Zn	10	56%	24% 100%
	Zr		83%	40% 100%
	Co		67%	2% 100%
	Cu		65%	44% 83%
	Ni		47%	26% 73%
	Sb		35%	11% 76%
	As		30%	6% 58%
	Se		54%	20% 100%
	Mn		53%	15% 100%
	P		32%	16% 69%
	S		16%	1% 37%
	Sr		15%	5% 27%
	Mo		27%	3% 100%
	Ba		26%	4% 48%
	Mg		25%	7% 59%
	Ca		23%	7% 38%
	Fe		77%	5% 100%
	K		8%	2% 30%
	Ce		90%	61% 100%
	Cd		90%	60% 100%
	Al		68%	45% 96%
	Si		18%	2% 31%
	W		46%	8% 85%
	Pb		36%	9% 54%
	Hg		8%	5% 10%
	Na		5%	2%	9%
	Ag		100%	100% 100%
	Pd
	Y


Frédéric Y Bois
email: frederic.bois@utc.fr

Ghislaine Gayraud

Probabilistic generation of random networks taking into account information on motifs occurrence

Because of the huge number of graphs possible even with a small number of nodes, inference on network structure is known to be a challenging problem. Generating large random directed graphs with prescribed probabilities of occurrences of some meaningful patterns (motifs) is also difficult. We show how to generate such random graphs according to a formal probabilistic representation, using fast Markov chain Monte Carlo methods to sample them. As an illustration, we generate realistic graphs with several hundred nodes mimicking a gene transcription interaction network in Escherichia coli.

Introduction

Graph models are essential tools for understanding and modeling complex systems of interacting variables or agents [START_REF] Albert | Statistical mechanics of complex networks[END_REF]. The global features of social, telecommunication or biological networks can only be analyzed with correspondingly large graph models. Such models consist of bonds (edges) indicating relationships between agents (nodes), with parameters quantifying the strength of the bonds. Over the years, graph models have been extensively studied both in theory (see among others [START_REF] Lauritzen | Graphical models[END_REF] and in methodology [START_REF] Whittaker | Graphical models in applied multivariate statistics, Wiley series in probability and mathematical statistics: Probability and mathematical statistics[END_REF] in different scientific areas, such as statistical physics [START_REF] Gibbs | Elementary Principles in Statistical Mechanics[END_REF], genetics [START_REF] Wright | Correlation and causation[END_REF], economics [START_REF] Wold | Causality and econometrics[END_REF] or social sciences [START_REF] Blalock | Causal models in the Social Sciences[END_REF]. When there is no ambiguity about the links between nodes and their interactions strengths, complex systems are well described by deterministic networks. In the presence of ambiguity about the system, probabilistic graphs are usually considered. In that case, bonds represent stochastic links between nodes and their parameters specify conditional distributions. Probabilistic graphs offer a natural framework for statistical inference and knowledge integration. For example, in communication engineering, we may want to know what conditions the robustness of a network; in systems biology we may be interested in understanding how genes control each others.

The theory of random graphs was initiated by Erdös and Rényi in a series of papers [START_REF] Erdös | On random graphs[END_REF][START_REF] Erdös | On the evolution of random graphs[END_REF][START_REF] Erdös | On the strength of connectedness of a random graph[END_REF] in which they proposed to generate a random graph with a given number of labeled nodes by connecting every pair of nodes with probability p (p ∈ (0, 1)). The main goal of random graph theory has been to determine at which connection probability p a particular property of a graph will most likely emerge. One of the first properties studied by Erdös and Rényi is the appearance of given sub-graphs like cliques, triangles, etc. In some sense, the question they have addressed is relative to the structure/topology on a graph. In many settings, interest focuses first on the structure of a graph and the question arose of whether the random graphs proposed by Erdös and Rényi were able to display specific structures of real 1 complex networks. Over the past few years, there has been a growing interest in investigating and developing new tools and measures able to capture specific properties of real networks. Among such properties are the degree distribution, corresponding to the probability P (k) that a node in the network is connected with k other nodes [START_REF] Albert | Error and attack tolerance of complex networks[END_REF][START_REF] Leskovec | Kronecker graphs: an approach to modeling networks[END_REF], small-world properties [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] in which most nodes are not neighbours of one another, or the node clustering coefficient [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF].

Nevertheless, inference about network structures remains difficult because of the extremely large number of possible graphs, even with a modest number of nodes [START_REF] Markowetz | Inferring cellular networks -a review[END_REF]. Estimating the amount of data needed to recover the structure of a graph is also difficult, but it is clear at least in biology, that most of the current experimental designs are insufficiently powerful for that aim. In such a context, every bit of information counts. Still in biology, the relevant scientific literature indicates that all graphs are not equally plausible, some being a priori more likely than others. Accounting for prior knowledge is well formalized in Bayesian statistics [START_REF] Robert | The Bayesian Choice: from Decision-Theoretic Motivations to Computational Implementation[END_REF], but the probabilistic representation of such knowledge is still a question. [START_REF] Mukherjee | Network inference using informative priors[END_REF] have recently proposed a set of informative priors for network structure inference. More precisely, they have considered priors able to capture information relative to existence of edges, degree distribution or sparsity structure in Bayesian networks, i.e., acyclic directed graph models.

Network motifs are patterns (sub-graphs) that recur within a network much more often than expected for random graphs [START_REF] Milo | Network Motifs: Simple Building Blocks of Complex Networks[END_REF]. It has been shown that gene transcription regulation networks, for example in the bacteria Escherichia coli, contain a small set of network motifs (see Alon, 2007 and references therein), suggesting that such motifs are basic building blocks of transcription networks. An important aspect of network topology inference is therefore to include the probability of occurrence of such network motifs (see [START_REF] Janson | The large-scale organization of metabolic networks[END_REF] for an overview).

In this article, we address precisely that question. We extend the approach of [START_REF] Mukherjee | Network inference using informative priors[END_REF] by relaxing the acyclicity requirement which characterizes Bayesian networks and propose rigorous probabilistic representations of a priori information on pairwise links, degree node distribution, and network motifs. We use Markov chain Monte Carlo (MCMC) simulations to sample networks satisfying those joint distributions, for moderately large networks (several hundred to thousands of nodes). Such random networks can be used as priors for formal inference, after updating with data in a Bayesian framework. They can also be used for pure simulation purposes, e.g. for methods or software testing. We do not deal with data and associated likelihood (or "score") functions, but focus on probabilities. Our distributions, however, are entirely compatible with any score function and can be used for inference, in particular in a Bayesian framework. As an illustration of our results, we generate realistic graphs mimicking a gene transcription interaction network in E. coli. The weight of the various proposed priors is examined.

Results

Graph Models for Networks

A graph model simply consists of nodes (vertices) connected by edges [START_REF] Wilson | Introduction to Graph Theory[END_REF] The nodes often represent physical entities (people, genes, servers...) and the edges represent links or dependencies between them ("is a friend", "controls", "is physically connected to", ...). Nodes can be assigned attributes (e.g., "on", "off") which can depend in turn on the attributes of the nodes to which they are connected. The edges may also have attributes influencing those of the nodes they connect. Edges may be "undirected" or "directed", the latter case (often noted by an arrow) introducing an asymmetry between the two nodes. For example, an arrow from node i to node j may indicate that i controls j, the reverse being not true. Directed edges can in turn be signed, indicating a positive or negative control, etc. Finally, graph models may have global features imposed to them. For example, we may impose no unconnected node. A commonly imposed feature is "acyclicity". In that case, the graph model cannot contain any path (succession of edges) linking any node to itself, and in particular no "auto-loop" edge from a node to itself. In the case of directed edges, paths are understood to follow the directions of the edges. A particular class of such acyclic directed graph models, Bayesian networks, has a clear probabilistic interpretation and is easily amenable to inference about network structure or parameters [START_REF] Neapolitan | Learning Bayesian Networks[END_REF] Since we are interested in generating general graphs, in particular those describing genetic regulatory systems, we consider directed graph which may be cyclic (see [START_REF] Jong | Modeling and simulation of genetic regulatory systems: a literature review[END_REF] for an overview of genetic regulatory networks modeling). More precisely, our graphs are composed of labeled nodes with directed edges. There can be two reverse edges between any two nodes and auto-loops are allowed.

Informative priors on networks

Let G be a graph described by a set V (G) = {v 1 , . . . , v n } of n vertices (n ≥ 2) and a set E(G) = {e i,j : (i, j) ∈ {1, . . . , n} × {1, . . . , n}} of directed edges and auto-loops. G may be described by its adjacency matrix A, a (n × n)-matrix whose generic term is given by a i,j = 1 if the edge e i,j exists 0 otherwise .

Incomplete a priori knowledge on such a graph can be described by a statistical distribution. Given n, the number of nodes of G, we propose to include three levels complexity in that distribution. It is only mandatory to define the first level, which does not reflect any specific structure except for the probability of presence of individual edges. We then refine it by including information on the degree distribution.The next step is to incorporate information related to the occurrences of sub-network motifs.

Priors on individual edges

As in the random graphs considered by Erdös and Rényi, prior knowledge on each individual edge can be conveniently modeled by a Bernoulli distribution, assigning probability p i,j to the existence of a directed edge from node i to node j, that is, e i,j ∼ B(p i,j ) for all (i, j) ∈ {1, . . . , n} × {1, . . . , n}. In this context, the adjacency matrix A related to a graph G with n nodes, becomes A = (e i,j ) {1≤i,j≤n} .

For a graph of a given size n, there are n 2 individual pairwise possible links in the cyclic case and at most n 2 -n in the acyclic case (auto-loops being ruled out by definition). Specifying the complete set of edges priors requires the definition of a n × n matrix P = (p i,j ) 1≤i,j≤n (with a null diagonal in the acyclic case). If the pairwise links are supposed to be independent, the probability distribution for the entire graph G is therefore

P Bern,G = n i,j=1 p ei,j i,j (1 -p i,j ) ei,j -1 .
(1)

There are various ways to choose or elicit values for the individual prior probabilities p i,j , which we will further discuss in our application but intuitively they are related to the weight of the prior evidence (e.g., p-values [START_REF] Bernard | Informative structure priors: joint learning of dynamic regulatory networks from multiple types of data[END_REF] we have on the existence of given edges. Other distributions could be used, such as a =multinomial if we had chosen to give a sign to the edges (to indicate positive or negative interactions, for example). But the principle would remain the same, and in the absence of precise prior information, a Bernoulli prior is probably all we can specify.

Priors on degrees' counts

The degree deg(v) of a vertex v is the total number of edges to which vertex v participates. The degree distribution of a graph G is a function

P (d) expressed in terms of |{v ∈ V (G) : deg(v) = d}|,
the total number of vertices having degree d. In many biological networks (Jeong et al, 2000), it appears that the degree distribution has a power-law tail, which means that P (d) ∝ d -γ , with γ > 0. Such networks are called scale-free [START_REF] Barabási | Emergence of scaling in random networks[END_REF]. Then, we define the probability distribution of a graph G as follows

P deg,G ∝ P Bern,G × n i=1    j∈{1,...,n}: j ei,j >0 e i,j    -γ
.

(2)

Since we do not impose that every node should be linked to another one, the degree distribution attributes implicitly a weight of one to any isolated node. It entails that the probability, with respect to degrees, of an empty graph (without any connection between its nodes) is one.

Here again, other distributions, even empirical (as defined by an histogram) and reflecting better the degree distribution of a given class of graphs, could be used if enough information was available.

Priors on motifs

One important local property of networks is the eventual occurrence of motifs. Motifs are defined here following [START_REF] Milo | Network Motifs: Simple Building Blocks of Complex Networks[END_REF], [START_REF] Alon | Network motifs: theory and experimental approaches[END_REF], Shen-Orr et al (2002), [START_REF] Milo | Superfamilies of designed and evolved networks[END_REF] and [START_REF] Kashtan | Efficient sampling algorithm for estimating sub-graph concentrations and detecting network motifs[END_REF], as over-represented sub-graphs compared to what is found in an Erdös and Rényi random graph. Some motifs have a notable importance in biological networks because they can carry out specific information-processing functions, and hence may help in understanding the global behavior of such networks (Masoudi-Nejad, 2012). For example, there are thirteen possible configurations for the relationships between three nodes (see Figure 1). Among the non-degenerate configurations, only the feed-forward loop (FFL, top row, first motif on the left, Figure 1) has been found in the transcriptional regulation network of E. coli [START_REF] Alon | Network motifs: theory and experimental approaches[END_REF]Shen-Orr et al, 2002;Mangan et al, 2003). No feed-back loop (FFB, top row, second motif on the left, Figure 1) has been observed in E. coli. The FFL is one of the most studied network motifs in transcription interactions. It corresponds to a directed sub-graph of three nodes (genes) such that one of them is regulated by the two others, which are linked. Given that each of the regulatory interaction can either an activation or a repression, there are eight sub-types of signed FFL, two of them occurring much more frequently than the other six in transcription networks (Mangan and Alon, 2003;[START_REF] Mangan | The incoherent feed-forward loop accelerates the response-time of the gal system of Escherichia coli[END_REF]. In this paper, we consider network motifs with three nodes; auto-loops are not taken into account in such sub-graphs. We define a motif distribution based on the proportion of FBL among all three-node loops. More precisely, for a graph G let us consider N 1 the number of FBL motifs and N 2 the number of FFL motifs and note they may be expressed in terms of (e i,j ) {1≤i,j≤n} as follows, N 1 = (i,j,k):i =j,k =j,i =k e i,j e j,k e k,i and

• • • d d s ' • • • d d ' • • • d d s ' • • • d d s • • • ' • • • ' E • • • d d ' E • • • d d s ' E • • • d d s d d • • • d d s d d © • • • d d s d d E • • • d d ' E © • • • d d ' E © d d s
N 2 = (i,j,k):i =j,k =j,i =k e i,j e j,k e i,k .
For a graph G with a total number N 1 + N 2 of motifs of type FBL and FFL, we place a beta-binomial probability with parameters u and v on N 1 , BB(N 1 |u, v, N 1 + N 2 ). The prior for graph G is then

P M otif,G ∝ P Bern,G × C N1 N1+N2 B(N 1 + u, N 2 + v) B(u, v) , (3)
where C N1 N1+N2 is a Binomial coefficient and B(•, •) denotes the Beta function. The choice of a Beta-Binomial distribution is justified by the fact that we may not know the exact proportions of FFL and FBL, but simply have observations about the numbers of such loops in some actual network we base our prior on. Obviously, as we will discuss later, other motifs could be tracked and entered in the definition of a graph probability, using a similar device.

Piecing together a global prior

In addition to the probabilities P Bern,G , P Deg,G and P M otif,G defined by (1), ( 2) and (3) respectively, we consider one more graph distribution P T otal,G which combines all informative priors independently. Therefore:

P T otal,G ∝ n i,j=1 p ei,j i,j (1 -p i,j ) ei,j -1 × n i=1    j∈{1,...,n}: j ei,j >0 e i,j    -γ × C N1 N1+N2 B(N 1 + u, N 2 + v) B(u, v) . ( 4
)
Application to a Biological Network A graph representation of the E. coli transcriptional regulatory network is shown on Figure 2. It contains 423 nodes, all connected, with 578 directed edges. That is actually only 0.32% of the number of possible edges, indicating that the network is sparse.

We report here the results for Alon's full size network (of 423 nodes). We investigate here which elements of our prior knowledge on E. coli regulatory network features are the most important to simulate realistic networks.

In a first set of simulations we assigned "vague" priors to the edge probabilities, setting them all to the same value (equal to 578/423 2 , i.e., 0.0032). With that prior, all connections are equally probable, and their expected number is equal to the one observed for E. coli. In E. coli the degree distribution follows approximately a power law with an exponent of 1.7 (see Figure 6), so we set γ to that value when the degree distribution was in effect. In addition, E. coli regulatory network is known to contains 42 FFLs and no FBL. To allow flexibility in the prior and allow some occurence of FBL motifs, we set equation (3) parameter u to 2 and parameter v to 50 in our simulations. That implies an expected proportion of only two percents FBLs.

In the second set of simulations we use the same priors for degree distribution and motifs frequencies, but used informative priors to individual edge probabilities: The edges reported by Alon et al., were assigned probability 0.95. Non-reported, therefore hypothetical, edges were assigned probability 0.00016. Together those probabilities lead again to 578 expected edges.

In all cases, three MCMC chains of 2 billions iterations were run independently. Convergence of the edge probabilities (according to Gelman and Rubin's criterion) was always attained after at most 1 billion iterations. The degree distribution (e.g., Figure 6) and motif frequencies (e.g., Figure 4) also converged within that time frame: Results from the three independent chains basically overlap, except in the case of rare events (with frequencies less that one in 10,000), where Monte Carlo sampling uncertainty becomes noticeable. We therefore discarded systematically the first billion simulations and base all the following results on the second billion. We optimized our MCMC sampling C code and simulations are rather fast. Running 2 billions iterations to generate a random graph with 423 nodes takes about 2 minutes on a Intel Core 2 Duo machine clocked at 2.13 GHz. It takes actually little more time to sample graphs with a thousand nodes. Overall, the time it takes to update all the elements of the adjacency matrix is approximately proportional to the number of its elements, and therefore proportional to the square of the number of nodes for the graph considered. In our implementation, memory requirements are simply proportional to the number of nodes and minimal. Figure 3 shows samples of networks generated using the above vague prior on individual edges. Used alone, that prior gives all edges the same probability and the resulting network has little structure, except that the expected number of edges is respected (Figure 3, panel A). The proportion of FBLs is 1/4, as expected in an unconstrained setting (there are two possible FBLs and six possible FFLs for each triplet of nodes). Adding the prior component on degree distribution imposes a major change in network shape. The number of edges is similar, but the structure becomes hierarchical (panel B). Placing a prior on the proportion of FBLs and FFLs, in addition to the prior on individual edges, has little visible impact on the structure (panel C), but the proportion of loops is now much lower and close to its expected value. Finally (panel D) putting the three priors together gives us again a hierarchical structure, but with the correct proportion of FBLs.

Figure 4 shows in more details how the number of feedback and feed-forward loops is influenced by the specification of prior knowledge in the context of a vague specification of individual edge probabilities. The hierarchical structure imposed by the degree distribution (prior "B") leads to a much lower number of loops, but without altering the ratio of FBLs to FFLs (25% in the case of prior "A" and "B"). In contrast, imposing a prior on the proportion of FBLs can later reduce both the ratio (4% in the case of prior "C" and 2% with prior "D") and the number of loops in the network.

If we now turn to networks simulated with an informative prior on individual edges (Figure 5), we see a striking difference with Figure 3. The structure of those networks, even if random, is quite close to the actual E. coli transcriptional regulation network (Figure 2). The difference between the networks with a prior on degrees (panels B and D) or without (panels A and C) is now more subtle. Actually the prior on individual edges is strong enough to impose a correct distribution of degrees, even if the degree distribution is not specified (see Figure 6). In that Figure, a deviation of the actual number of degrees from the power law, for high degrees, can be observed and is well simulated. A similar behavior of degree distribution can be found in [START_REF] Dobrin | Aggregation of topological motifs in the Escherichia coli transcriptional regulatory network[END_REF]. In Figure 5, the frequency of motifs is also controlled directly by the edges probabilities: The number of FBLs is about constant at 1.0 10 7 , for approximately 3 10 10 FFLs, hence a proportion of 0.05%. The differences between chains are small and all those results have a 5% CV.

Discussion

There are two important applications to the generation of semi-random graphs with known properties: i. Simulating actual networks for hypothesis testing, software bench-marking, statistical power calculations etc. (Van den Bulcke, 2006); ii. Assessing whether a graph is coherent with our prior knowledge in numerical data analytic methods such as Bayesian network modeling, Gaussian graphical methods etc. Such methods, in their naive implementation, are known to suffer badly from the curse of dimensionality. However, prior knowledge about network structure increases daily, at least for biological networks, and a proper accounting of such knowledge is our only hope to redeem the curse we face.

We have extended here the results presented in [START_REF] Mukherjee | Network inference using informative priors[END_REF] by including a flexible specification of edge probability, via Bernoulli priors, and by defining a prior on network motifs. In doing so we have dropped the commonly used distance penalty from a pre-specified reference network (whereby "distance" correspond to the number of differing edges between a proposed graph and the reference graph). Such a distance penalty is simple to specify and compute, but has several drawbacks: It is quite coarse and does not give a weight to the various edges, while in fact we may be more or less certain about some of them. Therefore it treats in the same way edges known to be absent or present, and edges for which the we do not know whether they are absent or present. Also, only one reference network is specified and this limits the number of questions we can ask. It is finally more an ad hoc penalty function than a proper distribution, although that may be seen as a purely technical argument. In any case, an edge by edge prior probability assignment is not much more difficult to specify, it can be simplified by giving default vague probability values if information is lacking, and can be quite powerful when information is available.

There are, however, cases where higher levels of structure are important. In fact, we can hypothesize that higher levels are more important than we usually suspect: That is the whole point of systems biology. Power law degree distribution is a well known characteristic of biological networks. The frequency of occurrence of particular network motifs is another point in case. Fascinating recent work has addressed the question of degree distribution [START_REF] Leskovec | Kronecker graphs: an approach to modeling networks[END_REF], but the problem remains for motifs since no direct sampling or generative method is available in that case. We have shown here how to use MCMC sampling to obtain the desired random graphs, even for realistically large number of nodes. An important point is that a sample of random graphs is much more informative than a single approximate, or even "best", estimate graph. With a single graph, all sense of uncertainty is lost, and only over-confidence is gained. Ensemble results are much more robust and useful; but they can be cumbersome and the question is how to best handle them.

Stochastic simulations also give insights about the relative weights of the various components of our prior knowledge. A first point is that prior knowledge about the probable number of edges, or at least about network sparsity, can strongly constrain the set of admissible networks. At least, that is the case when implemented in a form of Bernoulli priors on individual edges. Actually, more flexible priors (e.g., hierarchical) could be used instead of Bernoulli to allow more uncertainty about that expected number of edges. Specific knowledge about subsets of high probability, or conversely low probability, edges is also very informative. The degree distribution of a network may appear as a weak predictor of its structure since, for example, with a given number of nodes, two graphs with the same degree distribution may have completely different edge lists. However, we found that specifying a degree distribution has a visible impact on the network structure. Here again it would be easy to be more flexible about that distribution. Imposing a prior on the occurrence of specific motifs can be important in terms of functionality, but leads to more subtle modifications. Note however, that we imposed a distribution on the relative frequency of two loop motifs, rather than on there absolute number. That would be an easy extension, which could have profound consequences on the network structure. Overall, there are many variables with which the prior can play, and even potential conflicts between components of our prior knowledge, in particular if a strongly informative prior is placed on individual edges. Such conflicts can be hard to figure out without the help of simulations, because our intuition often fails in high dimension and in the case of graphs [START_REF] Helbing | Globally networked risks and how to respond[END_REF]. In that respect, the possibility to perform quickly billions of iterations for network of sizes commensurable to those of genomes is encouraging. A word of caution is in order here, however: With 423 nodes, a billion simulations correspond to 10 9 × 423 -2 , i.e. about 6000 full updates of the network. With 10 4 nodes we might have to go to trillions of simulations to get to convergence and this would currently take us a day and half of computation, although GPU computing, for example, could again increase speed.

To be more precise about the relative weight of the different priors would have required some evaluation of the number of possible graphs, or some form of enumeration of the number of different graphs sampled during stochastic simulations. However there are, for example 2 178929 different directed graphs with 423 nodes, and tracking the list of graph sampled would entailed considerable time and memory capacity.

Obviously, it would be interesting to extend those results to the important problem of statistical learning of the network structure from data acquired in large scale genotyping or phenotyping studies, for example. In a Bayesian context, that simply entails the computation of a data likelihood function, or its marginalisation. However, the models currently favored for that purpose: Gaussian graphical networks (Krumsiek et al, [START_REF] Mukherjee | Network inference using informative priors[END_REF], are either undirected or acyclic, respectively. Dealing with undirected graphs would be easy, but the acyclicity of Bayesian networks is quite restrictive. Loops motifs cannot exist formally in such models, unless they are made dynamic, and checking for acyclicity at every iteration imposes significant computational burden. The possibility of using hybrid models [START_REF] Silva | The hidden life of latent variables: Bayesian learning with mixed graph models[END_REF] is an interesting possibility to explore.

Materials and methods

Graph probabilities such as P T otal,G are only defined up to a multiplicative constant. In that case, a simple way to generate sample graphs according to that probability distribution is to use the Metropolis-Hasting sampler [START_REF] Casella | Monte Carlo Statistical Methods[END_REF]. From a current graph G, with total probability P T otal,G (eq. ( 4)), a graph G is proposed by first selecting two nodes, say v i and v j of G (v i may be equal to v j in case of auto-loops) and then by deciding on the presence of an edge from v i and v j by a random Bernoulli draw with edge probability p i,j . That amounts to sampling G from the Bernoulli prior on edges defined in eq. (1). The total probability P T otal, G of G is computed using eq. ( 4) and G is accepted with a probability equal to min(1, P T otal, G / P T otal,G ). In case of rejection, G is discarded, and G is again the current sample (that implies that the same graph can be drawn several times in succession). The procedure is iterated as many times as it is needed to reach convergence in probability to the target distribution sought. Convergence can be checked by running several simulation chains and computing Gelman and Rubin's R criterion [START_REF] Gelman | Inference from iterative simulation using multiple sequences (with discussion)[END_REF] on each element of the graphs' adjacency matrix, or by monitoring the degree distributions obtained, or the motifs probabilities in those independent chains.

A C language version of the algorithm has been implemented as a module of the free GNU MCSim software [START_REF] Bois | GNU MCSim: Bayesian statistical inference for SBML-coded systems biology models[END_REF] (http://www.gnu.org/software/mcsim). That software was used for all the simulations presented here. Graphs were produced with R, version 2.14 (R Development Core Team, 2011).

Figure 1 :

 1 Figure 1: The thirteen possible three-node motifs.

Figure 2 :

 2 Figure 2: E. coli transcriptional regulatory network, as reported in Shen-Orr et al (2002), with minor updates (see text).

Figure 3 :

 3 Figure 3: Transcriptional regulation networks generated using a vague prior on individual edges. Red dots: 423 genes in each network. Panel A: prior on individual edges only; B: prior on individual edges and degree distribution; C: prior on individual edges and the proportion of feed-back loops; D: all three priors together.

Figure 4 :

 4 Figure 4: Motifs frequencies in transcriptional regulation networks generated using a vague prior on individual edges. A: prior on individual edges only; B: prior on individual edges and degree distribution; C: prior on individual edges and the proportion of feed-back loops; D: all three priors together.

Figure 5 :

 5 Figure 5: Transcriptional regulation networks generated using informative priors on individual edges (on the basis of E. coli network). Red dots: 423 genes in each network. Prior on individual edges (A), on individual edges and degree distribution (B), on individual edges and the proportion of feed-back loops; D: all three priors together.

Figure 6 :

 6 Figure 6: Degree distribution in E. coli actual transcriptional regulation network (in red) and in Monte-Carlo sampled networks (in black). Informative distribution on individual edges. The dip for high degrees due to a deviation of reality from the power law assumption

  Transcriptional regulatory networks orchestrate the gene expression of cells. In such networks, the nodes are operons (one or more genes transcribed on the same mRNA template). Edges go from operons encoding a transcription factor to operons directly regulated by that factor.[START_REF] Shen-Orr | Network motifs in the transcriptional regulation network of Escherichia coli[END_REF] developed and applied motif-detection algorithms to the transcriptional regulation network of E. coli. They extracted data from the RegulonDB transcriptional database[START_REF] Salgado | RegulonDB v8.0: omics data sets, evolutionary conservation, regulatory phrases, cross-validated gold standards and more[END_REF], and enhanced them with additional transcription factors and interactions described in the literature. We used here the latest version of the dataset (version 1.1, made publicly available by Dr. U.Alon).

Acknowledgments

The research leading to these results has received funding from the scientific council of the Université de Technologie de Compiègne (project Prior-Motives) and the Innovative Medicines Initiative Joint Undertaking, under Grant Agreement number 115439 (StemBANCC), resources of which are composed of financial contribution from the European Union Seventh Framework Programme (FP7/2007-2013) and EFPIA companies in kind contribution. This publication reflects only the author's views and neither the IMI JU nor EFPIA nor the European Commission are liable for any use that may be made of the information contained therein.

Author Contributions

F.B. and G.G. designed and performed research, and wrote the paper.