Antonio De Anchieta

Câmara Júnior

Pierre-André Maréchal

Raphaëlle Tourdot- Maréchal

Florence Husson
email: florence.husson@u-bourgogne.fr

Oxidative stress resistance during dehydration of three non-Saccharomyces wine yeast strains

Keywords: dehydration, ROS accumulation, lipid oxidation, Torulaspora delbrueckii, Metschnikowia pulcherrima, Lachancea thermotolerans

The development of new fermented foods and beverages requires more and more the use of new dehydrated yeasts species. In this context, the non-Saccharomyces (NS) yeasts Torulaspora delbrueckii, Metschnikowia pulcherrima and Lachancea thermotolerans are developed especially in winemaking as co-culture in the fermentation of the must or for the must bioprotection. However, during formulationdehydration the yeast cells are exposed to several stresses that reduce cellular activity. Among these, the oxidative stress induced by the stabilization processes has been described as one of the main causes of cell death. In this study, we analyzed the effects of growth medium associated with two dehydration kinetics on the accumulation of reactive oxygen species (ROS) and lipid peroxidation levels. The cultivability of tested yeast strains was dependent on growth and dehydration conditions. The L. thermotolerans strain was the most sensible to dehydration when growing in nutrient-poor media, which was characterized by high levels of ROS, lipid peroxidation and reduced cultivability. In contrast, this yeast was able to restore its cultivability when growing in nutrient-rich medium before dehydration. The other NS yeast strains acquired resistance to oxidative stress similarly but in a growthdehydration condition less dependent manner. These results showed that modulation of growing medium composition is a simple way to improve resistance to oxidative attack imposed by dehydration in NS yeasts. This was the first time that multiple quantitative and qualitative indicators of oxidative stress response in these three NS yeast strains were explored.

Introduction

Yeasts are microorganisms capable of surviving large periods with limited water content, i.e. in the dehydrated state; once in normal hydric conditions they reactivate their metabolism and restart their normal activity. This phenomenon is known as anhydrobiosis [START_REF] Rapoport | Application of anhydrobiosis and dehydration of yeasts for non-conventional biotechnological goals[END_REF]. Taking advantage of these characteristics, beverage industries have been interested in new species of yeasts stabilized in dry form (active dry yeasts: ADY), especially for winemaking. In general, for wine production the fermentation occurs spontaneously due to the indigenous flora present on grapes or in musts, which consists in different yeasts and bacteria species [START_REF] Liu | The contribution of indigenous non-Saccharomyces wine yeast to improved aromatic quality of Cabernet Sauvignon wines by spontaneous fermentation[END_REF]. Although this practice favors the development of the wine sensorial profiles, it remains difficult to control the microbial population development during winemaking, which represents a loss of quality and productivity.

Thus, the controlled application of ADY provides high quality products with guaranteed reproducibility [START_REF] Rodríguez-Porrata | Enhancing yeast cell viability after dehydration by modification of the lipid profile[END_REF].

During ADY formulation, yeast cells undergo multiple stresses which reduce their performance, e.g. cell starvation in the course of biomass propagation then heat and oxidation induced by the drying stages [START_REF] Beker | Conservation of yeasts by dehydration[END_REF][START_REF] Rapoport | Anhydrobiosis and dehydration of yeasts[END_REF][START_REF] Schmidt | Production, reactivation and nutrient requirements of active dried yeast in winemaking: theory and practice[END_REF]. In fact, cells subjected to hydric variations accumulate intracellular reactive oxygen species (ROS) as result of failure of their detoxification systems and the functionality of the proteins and enzymes responsible for the ROS elimination [START_REF] Dupont | Survival kit of Saccharomyces cerevisiae for anhydrobiosis[END_REF]. In addition, the modulation of energy metabolism, from fermentation to respiration, has a great influence on lipid stability to oxidation, especially during the biomass propagation (Gómez-Pastor, [START_REF] Gómez-Pastor | Improving yield of industrial biomass propagation by increasing the Trx2p dosage[END_REF].

In this context, depending on the composition of growing medium, yeasts activate biosynthesis of different compounds, like glutathione which plays part in their resistance to dehydration-oxidation [START_REF] Câmara | Improving total glutathione and trehalose contents in Saccharomyces cerevisiae cells to enhance their resistance to fluidized bed drying[END_REF][START_REF] François | Reserve carbohydrates metabolism in the yeast Saccharomyces cerevisiae[END_REF][START_REF] Lorenz | Enhanced incorporation yield of cysteine for glutathione overproduction by fed-batch fermentation of Saccharomyces cerevisiae[END_REF]. Glutathione (L-gamma-Glutamyl-L-Cysteinylglycine) is a tripeptide thiol antioxidant formed by the amino acids cysteine, glutamic acid and glycine. Its major role is to maintain the intracellular redox homeostasis by scavenging free radicals (ROS) in the cytosol and reducing the disulfide bonds produced during oxidative stress [START_REF] Grant | Role of the glutathione/glutaredoxin and thioredoxin systems in yeast growth and response to stress conditions[END_REF].

The stabilization processes of yeasts in dried form are mainly applied to the yeast Saccharomyces cerevisiae, which is undoubtedly the most studied eukaryotic microorganism. Numerous studies have elucidated the mechanisms of passive, genetic or adaptive response of S. cerevisiae during nutritional, osmotic, dehydration and oxidation stresses [START_REF] Câmara | Fisetin yeast-based bio-capsules via osmoporation: effects of process variables on the encapsulation efficiency and internalized fisetin content[END_REF][START_REF] Câmara | Improving total glutathione and trehalose contents in Saccharomyces cerevisiae cells to enhance their resistance to fluidized bed drying[END_REF][START_REF] Silva Pedrini | Osmoporation: A simple way to internalize hydrophilic molecules into yeast[END_REF][START_REF] Eleutherio | Oxidative stress and aging: Learning from yeast lessons[END_REF][START_REF] Garre | Oxidative stress responses and lipid peroxidation damage are induced during dehydration in the production of dry active wine yeasts[END_REF][START_REF] Nguyen | Understanding the responses of Saccharomyces cerevisiae yeast strain during dehydration processes using synchrotron infrared spectroscopy[END_REF]. Nevertheless, precise reports about these mechanisms associated to other yeast species are currently scarce. The non-Saccharomyces (NS) yeasts Torulaspora delbrueckii, Metschnikowia pulcherrima and Lachancea thermotolerans aroused great interest in recent years, especially for winemaking applications or bioprotection [START_REF] Benito | The impacts of Lachancea thermotolerans yeast strains on winemaking[END_REF](Benito, , 2018a;;[START_REF] Oro | Antimicrobial activity of Metschnikowia pulcherrima on wine yeasts[END_REF][START_REF] Simonin | Inoculation of Torulaspora delbrueckii as a bio-protection agent in winemaking[END_REF]. The NS yeasts are generally inoculated before the fermentation of must by S. cerevisiae yeast, or in a co-inoculation with S. cerevisiae [START_REF] Sadoudi | Yeast-yeast interactions revealed by aromatic profile analysis of Sauvignon Blanc wine fermented by single or coculture of non-Saccharomyces and Saccharomyces yeasts[END_REF].

These new practices in enology aim to improve the wines flavor complexity during winemaking [START_REF] Sadoudi | Yeast-yeast interactions revealed by aromatic profile analysis of Sauvignon Blanc wine fermented by single or coculture of non-Saccharomyces and Saccharomyces yeasts[END_REF][START_REF] Simonin | Inoculation of Torulaspora delbrueckii as a bio-protection agent in winemaking[END_REF]Suzzi et al., 2012).

Moreover, NS yeasts are able to inhibit growing of undesirable yeast species, reduce levels of ochratoxin-A -toxin possible carcinogenic to humans -and act as a biocontrol agent against spoilage fungi [START_REF] García | Non-Saccharomyces yeasts: biotechnological role for wine production[END_REF].

Despite NS yeasts importance in enology, their responses to oxidation during dehydration remain currently poorly characterized. Previous results demonstrated that NS yeasts strains T. delbrueckii, M. pulcherrima and L. thermotolerans produces large amounts of glutathione (mostly in the reduced form, GSH) when grown in nutrient-rich medium (GSM), which resulted in better resistance of the cells to dehydration [START_REF] Câmara | Dehydration stress responses of yeasts Torulaspora delbrueckii, Metschnikowia pulcherrima and Lachancea thermotolerans: Effects of glutathione and trehalose biosynthesis[END_REF].

In this study we focused on the dehydration-oxidation resistance of three wine yeast species T. delbrueckii, M. pulcherrima and L. thermotolerans cultivated and dehydrated in distinct conditions, and compared with S. cerevisiae, as reference. We implemented the oxidant probe dihydroethidium and the quantification of total lipid peroxidation in order to evaluate the impact of growing conditions and dehydration kinetics in cell oxidative response. Moreover, the accumulation of reactive oxygen species (ROS) in cells was analyzed by flow cytometry and by two-photon laser scanning microscopy. The findings presented in this work enhance the comprehension of the growing, dehydration and oxidative stress resistance of these three NS wine yeast strains. They may be useful to design a more efficient process for ADY production.

Materials and Methods

Yeast species and culture conditions

The strains used were S. cerevisiae CBS8066, T. delbrueckii CBS4865, M. pulcherrima CBS5833 and L. thermotolerans CBS6340. Two growing medium were used: GSM (nutrient-rich medium) and TSM (nutrient-poor medium) [START_REF] Câmara | Improving total glutathione and trehalose contents in Saccharomyces cerevisiae cells to enhance their resistance to fluidized bed drying[END_REF]. The determination of the population (count of colony forming units, CFU) was realized on YPDA medium [START_REF] Nguyen | Understanding the responses of Saccharomyces cerevisiae yeast strain during dehydration processes using synchrotron infrared spectroscopy[END_REF]. Three colonies previously isolated on YPDA were transferred to a 250 mL conical flask containing 50 mL of GSM (For 1 L: 30 g glucose, 30 g yeast extract, 0.6 g KH 2 PO 4 and 0.6 g cysteine) or TSM (30 g glucose, 3 g yeast extract, 0.6 g KH 2 PO 4 ). Cells were grown in a rotary shaker (Lab Therm, Kühner AG, Basel, Switzerland) at 140 rpm, 30°C for 24 h. Cells were harvested at stationary phase by centrifugation (5810 R, Eppendorf, Hamburg, Germany) at 20°C, 2,500×g for 5 min and washed twice in water-NaCl solution (0.9% w/v) and resuspended in the same physiological solution. The final cell density was 2×10 10 cells mL -1 . This procedure was repeated for each independent experiment (n = 3).

Dehydration-rehydration conditions

After cell growing, yeasts were dried with two kinetics of dehydration (named KA and KB) composed of three steps that were characterized by their water activity, temperature and time (Fig. 1). The steps were carried out in different hermetically sealed boxes in which the activity of the water and temperature were in equilibrium.

The kinetics allowed to reproduce at laboratorial scale the dehydration processes applied in a previous work [START_REF] Câmara | Improving total glutathione and trehalose contents in Saccharomyces cerevisiae cells to enhance their resistance to fluidized bed drying[END_REF]. Procedure was carried out by disposing separately ten drops of 10 µL on two sterile polypropylene supports, resulting in 100 µL of cell suspension for each experiment. Cells were transferred to the first drying box until the end of the first cycle, then transferred to the second box, and to the third box until the end of the whole kinetics. After dehydration, cells were rehydrated with 5 mL of 0.9% water-NaCl solution at 38°C during 5 min. The final cell density was 4×10 8 cells mL -1 .

Determination of cell cultivability

After dehydration-rehydration cycle, cell suspension was serial diluted with the physiological solution (0.9% water-NaCl). Aliquots (10 µL) from 10 -3 to 10 -5 dilutions were spotted in triplicate separately onto YPDA plates. The plates were incubated at 30°C for 36 h and the number of colony forming unit (CFU) was counted. The control sample consisted of a fresh cells suspension of the same yeast strain kept at 25°C for 60 min or 90 min and not submitted to dehydration. Results were expressed as log of number of CFU per L.

Accumulation of ROS inside yeast cells

The oxidant probe dihydroethidium (DHE) was used to observe the ROS accumulation inside yeast cells. For the DHE stock solution, 10 mL ethanol was added to 25 mg DHE (Invitrogen Molecular Probes, Eugene, USA). The solution was stored at -20°C until analysis. After dehydration-rehydration, cell suspension was mixed with 4 µL of DHE stock solution. Flow cytometry analysis was carried out using a BD FACS Aria II flow cytometer analyzer (BD Biosciences, San Jose, USA) using excitation laser line at 488 nm and emission filter 585/42 nm. The DHE fluorescence signals were detected with at least 10,000 events in each analysis. Cells treated with H 2 O 2 (30% w/v) were used as positive control. Results were expressed as percentage of DHE positive stained cells.

Two-photon microscopic observations

Observations of ROS accumulation were carried out with a two-photon Nikon A1-MP laser-scanning microscope (Nikon, Tokyo, Japan). After rehydration, cell suspension was mixed with DHE, under the same conditions as for the flow cytometry analysis.

After 10 min in the dark, images were acquired with a ×60 (NA: 1.27) Plan Apochromat water-immersion objective (Nikon, Japan). Excitation was performed using an IR laser (Chameleon, Coherent) at 800 nm. Emission signals were collected on three detection channels : 2 (green): FF03-525/50-25 [500-550 nm]; 3 (orange) FF01-575/25-25 [563-588 nm] and, 4 (red) FF01-629/56-25 [601-657 nm]. The final cell density was 10 8 cell mL -1 .

Quantification of total lipid peroxidation

Total lipid peroxidation was measured with the malondialdehyde (MDA) colorimetric method [START_REF] Steels | Stress tolerance and membrane lipid unsaturation in Saccharomyces cerevisiae grown aerobically or anaerobically[END_REF]. The MDA (Alfa-Aesar, Karlsruhe, Germany) is one of the products of the oxidative attack (ROS) to polyunsaturated lipids present in cells. Its reaction with the thiobarbituric acid (TBA, Alfa-Aesar, Karlsruhe, Germany) forms a colorimetric compound proportional to the MDA concentration. Rehydrated cells were resuspended in distilled water and cell disruption was carried out using same protocol described above. The supernatant was collected and mixed with 0.1 mL 0.1M EDTA (Sigma-Aldrich, Missouri, USA) and 0.6 mL 1% (w/v) TBA in 0.05 M NaOH. After 15 min in water bath at 90 °C followed by 5 min on ice, the absorbance at 532 nm was measured.

Statistical analysis

The data mean values of three independent replicates (n = 3) and significant differences were analyzed by the Tukey HSD post-hoc test or by one-way analysis of variance (ANOVA) using the STATISTICA ® software (Statsoft, Tulsa, USA).

Results

Impact of growth and dehydration conditions on yeast cell cultivability is strain-dependent

Cell cultivability assays permitted to analyze the impact of growing conditions (GSM or TSM) and dehydration kinetics (KA or KB). No significant differences (between fresh and dried cells) were observed when yeasts were dehydrated with kinetics KA (Figs. 2A-D). Moreover, the values of cultivability obtained with fresh and dried yeasts cells T. delbrueckii, M. pulcherrima and L. thermotolerans grown in nutrientrich medium (GSM) were comparable with kinetics KB (Figs. 2B-D). S. cerevisiae cells grown in GSM showed a lower resistance when submitted to the same kinetics, with one log reduction (Fig. 2A) compared to fresh cells. T. delbrueckii and M. pulcherrima growth in nutrient-poor medium (TSM) had little effect on their cultivability, whatever the dehydration kinetics employed. S. cerevisiae and L. thermotolerans cells grown in nutrient-poor medium (TSM) showed two and three log reduction respectively after dehydration with kinetics KB, (Figs. 2A andD), compared to fresh cells. These results led us to infer that different growing conditions have a singular effect on yeast replication ability, especially for yeasts S. cerevisiae and L. thermotolerans. Moreover, the kinetics KA minimized the negative impact of dehydration on yeast cultivability, compared to kinetics KB, in the tested conditions.

Better nutritional environment correlates with reduction of ROS

The intracellular accumulation of ROS was evaluated with DHE as fluorescent indicator via two different techniques: two-photon laser scanning microscopy (qualitative) and flow cytometry (quantitative). DHE is a well-known ROS probe used to examine oxidative stress response in cells. The effects of growing medium and dehydration kinetics on the ROS accumulation were analyzed for each yeast strain.

S. cerevisiae yeast cultivated in nutrient-rich medium (GSM) showed slight ROS accumulation for fresh cells (Figs. 3A andAa). We observed cells with bigger vacuole (Fig. 3A) and a significant ROS accumulation when cells were dehydrated with kinetics KA (Fig. 3Aa). In addition, almost 30% of cells were ROS positive when submitted to the dehydration kinetics KB (p < 0.01). Cells grown in nutrient-poor medium (TSM) showed the highest ROS accumulation with approximately 80% of oxidized cells, even without dehydration stress (Figs. 3A andAa). These data suggest that this S. cerevisiae strain was particularly sensible to growing and dehydration conditions applied in this study. It was also interesting to note some 'green spots' in the cells grown in the nutrient-poor medium (Fig. 3A, white arrow).

The effect of growing and dehydration conditions to T. delbrueckii strain oxidative response was less evident in comparison with S. cerevisiae yeast. T. delbrueckii cells cultivated in nutrient-rich medium (GSM) showed almost no fluorescent signals in both microscopy and flow cytometry analyzes (Figs. 3B and Bb). In contrast, T. delbrueckii cells cultivated in nutrient-poor medium (TSM) showed a gradual accumulation of ROS (Figs. 3B and Bb). In this growing condition, the percentage of oxidized cells was significant different for both kinetics KA (46%) and kinetics KB (58%). These results correlated to microscopy analysis. Like previously observed, cells showed an increase of the 'green droplets' intensity when cultivated in nutrientpoor medium (TSM) and dehydrated with kinetics KB (Fig. 3B, white arrows), suggesting a stress response to nutrients availability and to the applied dehydration kinetics.

In the case of M. pulcherrima strain, when cells were cultivated in nutrient-rich medium (GSM) they exhibited insignificant accumulation of ROS before and after dehydration with kinetics KA, which represented 0.2% and 3% of oxidized cells, respectively (Figs. 3C andCc). However, a significant percentage of oxidized cells after kinetics KB (61%) was observed. When M. pulcherrima cells were cultivated in nutrient-poor medium (TSM), they showed an increase in ROS accumulation for both kinetics KA (25% of oxidized cells) and KB (65% of oxidized cells). We observed separated low intensity red spots in the M. pulcherrima cell membrane (Fig. 3C, white arrows). In addition, the yeast M. pulcherrima was the one that exhibited the 'green droplets' in greater quantity and intensity than the other yeasts even before dehydration and especially when cells were cultivated in nutrient-poor medium (TSM) (Fig. 3C). These results suggest a correlation between these 'green droplets' and nutritional stress.

Accumulation of ROS in the strain L. thermotolerans was most important and significant for cells dehydrated with kinetics KB for both grown cells in nutrient-rich or poor medium (GSM and TSM), with 48% and 78% of oxidized cells, respectively (Figs. 3D andDd). Moreover, L. thermotolerans yeast cells grown in nutrient-poor medium showed an important ROS accumulation before dehydration (29% of oxidized cells) and after kinetics KA (45% of oxidized cells). The presence of the same 'green spots' was observed for L. thermotolerans cells grown in nutrient-poor condition (Figs. 3D, white arrows), suggesting a stress response linked to the medium composition. The percentage of oxidized cells remained low (6% of oxidized cells) with L. thermotolerans cells grown in GSM and dehydrated with kinetics KA (Fig. 3Dd).

Can the regulation of medium composition protect lipids from oxidation?

Under specific growing-dehydration conditions, yeast cells accumulate ROS as result of failure of the enzymatic system or by contact of dehydrated cells with air [START_REF] Jamieson | Oxidative stress responses of the yeast Saccharomyces cerevisiae[END_REF]. In this case, lipids are particular sensible to ROS accumulation in cells, because of the double bonds in the unsaturated fatty acid chains of phospholipids [START_REF] Catalá | Lipid peroxidation of membrane phospholipids generates hydroxyalkenals and oxidized phospholipids active in physiological and/or pathological conditions[END_REF]. No significant difference for S. cerevisiae yeast grown in nutrient-rich medium (GSM) and dehydrated with kinetics KA was observed, but a strong increase in MDA content, approximately four times, was observed when applying kinetics KB (Fig. 4A). Despite the same behavior in nutrient-poor medium (TSM) and after dehydration with kinetics KA, MDA levels were three times higher than in nutrient-rich medium (GSM), as well as after kinetics KB. ROS attack to lipids of T. delbrueckii strain grown in nutrient-rich medium (GSM) seemed less strong. Indeed, we observed no significant differences between fresh cells and cells dehydrated with kinetics KA or KB (Fig. 4B). Even with higher MDA accumulation for fresh T. delbrueckii cells in nutrient-poor medium (TSM) and after kinetics KA, levels were reduced if compared with cells dehydrated with kinetics KB, which were almost eight times higher than fresh cells levels (Fig. 4B). M. pulcherrima and L. thermotolerans cells showed similar behavior after growth in nutrient-rich medium (GSM), with small amounts of MDA for fresh cells and after kinetics KA, and a strong increase (45 and 39 nmol respectively) after kinetics KB. For M. pulcherrima and L. thermotolerans, cells grown in nutrient-poor medium (TSM) showed a significant increase in MDA content for both kinetics KA or KB, from three to seven times higher (Fig. 4C) and from three to four times higher (Fig. 4D), respectively. These results led us to infer the existence of a strong correlation between cell oxidative stress (MDA and ROS) and growing-dehydration conditions performed in this study. All yeasts were less resistant to dehydration with kinetics KB, correlated with the stronger MDA accumulation.

Discussion

Dehydration-oxidation affects non-Saccharomyces yeast cultivability

Oxidation is one of the most important stresses to which yeast cells undergo during the production and dehydration processes. This stress plays an important part on reduction of cell performance after dehydration. Indeed, yeast cultivability can be directly affected by intracellular ROS accumulation. This comportment has been described as a major inductor or starting factor of numerous cell dysfunctions in a ROS concentration-dependent manner [START_REF] Ayer | Cellular redox homeostasis, reactive oxygen species and replicative ageing in Saccharomyces cerevisiae[END_REF][START_REF] Farrugia | Oxidative stress and programmed cell death in yeast[END_REF]. We observed that this behavior is not only dependent on the growing or dehydration conditions, but it was also correlated to intrinsic metabolic and physiological characteristics of each strain belonging to different genus and species.

One of the findings in common with all strains showed that cells growing in a nutrientrich medium (GSM) had the ability to replicate actively (Fig. 2), even after dehydration (kinetics KA). In fact, yeasts can adapt their energy metabolism to the nutrients available in the environment and growing conditions (Gómez-Pastor, Pérez-Torrado, [START_REF] Gómez-Pastor | Recent advances in yeast biomass production[END_REF]Wu, Liu, & Huang, 2013). The presence of essential amino acids, carbon and nitrogen in sufficient amounts is primordial to extend yeast life span [START_REF] Santos | Nitrogen and carbon source balance determines longevity, independently of fermentative or respiratory metabolism in the yeast Saccharomyces cerevisiae[END_REF]. Another factor is the presence of O 2 during the biomass formation that triggers the fermentative metabolism to change to respiration and enhances biomass propagation [START_REF] Gómez-Pastor | Recent advances in yeast biomass production[END_REF]. This could partially explain the ROS accumulation (Fig. 3) observed for T. delbrueckii and M. pulcherrima cells grown in nutrient-poor medium (TSM). M. pulcherrima yeast have a Crabtree-negative behavior [START_REF] Bely | Impact of mixed Torulaspora delbrueckii-Saccharomyces cerevisiae culture on high-sugar fermentation[END_REF][START_REF] Schnierda | Optimization of carbon and nitrogen medium components for biomass production using non-Saccharomyces wine yeasts[END_REF], unlike S. cerevisiae, T. delbrueckii, and L. thermotolerans yeasts. This particularity is an evolutionary benefit for Crabtree-positive yeast strains due to the higher rate of ATP production during aerobic fermentation [START_REF] Pfeiffer | An evolutionary perspective on the Crabtree effect[END_REF]. Furthermore, the ethanol produced by these strains can be latter used as an additional carbon source (diauxic shift), producing ATP via the tricarboxylic acid cycle and oxidative phosphorylation in the mitochondria [START_REF] Brauer | Homeostatic adjustment and metabolic remodeling in glucose-limited yeast cultures[END_REF], thus giving a fitness advantage to S. cerevisiae, T. delbrueckii, and L. thermotolerans yeasts [START_REF] Piškur | How did Saccharomyces evolve to become a good brewer[END_REF]. These findings suggest that a growth in nutrient-rich medium (GSM) allows an improvement of yeast robustness during and post-dehydration, especially after kinetics KA. The dehydration kinetics KA and KB, with different temperature and water activity gradients (Fig. 1), were chosen based on conditions usually employed in the ADY industry. Indeed, an increase of the dehydration gradient (kinetics KB) had a higher impact on yeasts, reducing cell cultivability (Figs. 2 and3), probably as a consequence of oxidation [START_REF] Espindola | The role of glutathione in yeast dehydration tolerance[END_REF][START_REF] França | The role of cytoplasmic catalase in dehydration tolerance of Saccharomyces cerevisiae[END_REF]. In addition, the yeasts T. delbrueckii and M. pulcherrima did not show the same behavior, as we did not observe significant differences in CFU analysis on all tested conditions. These results imply that these two yeasts strains were more resistant to dehydration than S. cerevisiae and L. thermotolerans.

GSSG:GSH ratio as oxidative homeostasis indicator for wine yeasts

Glutathione is one of the most important redox-scavenging molecules in cells.

Glutathione exists in two different forms, reduced (GSH) and oxidized (GSSG, glutathione disulfide) and the GSSG:GSH ratio indicates the physiological and oxidative state of the cell, where healthy cells have lower GSSG:GSH ratio. It was already demonstrated that GSSG:GSH ratio is an indicator of yeast S. cerevisiae oxidative and healthy state [START_REF] Espindola | The role of glutathione in yeast dehydration tolerance[END_REF][START_REF] França | The role of cytoplasmic catalase in dehydration tolerance of Saccharomyces cerevisiae[END_REF][START_REF] Rahman | Assay for quantitative determination of glutathione and glutathione disulfide levels using enzymatic recycling method[END_REF]. The synthesis of large amounts of glutathione correlates with the yeast oxidative stress resistance [START_REF] Câmara | Dehydration stress responses of yeasts Torulaspora delbrueckii, Metschnikowia pulcherrima and Lachancea thermotolerans: Effects of glutathione and trehalose biosynthesis[END_REF][START_REF] Câmara | Improving total glutathione and trehalose contents in Saccharomyces cerevisiae cells to enhance their resistance to fluidized bed drying[END_REF][START_REF] Espindola | The role of glutathione in yeast dehydration tolerance[END_REF][START_REF] Grant | Role of the glutathione/glutaredoxin and thioredoxin systems in yeast growth and response to stress conditions[END_REF][START_REF] Hatem | Glutathione is essential to preserve nuclear function and cell survival under oxidative stress[END_REF]. In fact, glutathione participates in several functions in cells, specially scavenging free radicals (ROS) present in the cytosol. In addition, better performance and viability are frequently associated to high glutathione content in cells [START_REF] Câmara | Improving total glutathione and trehalose contents in Saccharomyces cerevisiae cells to enhance their resistance to fluidized bed drying[END_REF][START_REF] Espindola | The role of glutathione in yeast dehydration tolerance[END_REF][START_REF] Hatem | Glutathione is essential to preserve nuclear function and cell survival under oxidative stress[END_REF].

Previous data [START_REF] Câmara | Dehydration stress responses of yeasts Torulaspora delbrueckii, Metschnikowia pulcherrima and Lachancea thermotolerans: Effects of glutathione and trehalose biosynthesis[END_REF], performed with the same NS strains under cultivation and dehydration conditions identically to the present study, showed a strong GSSG:GSH ratio (up to five times higher) for yeast L. thermotolerans when cells were grown in nutrient-poor medium (TSM) and dehydrated with kinetics KB, compared to fresh cells. Similarly for yeast T. delbrueckii (five times higher) and M. pulcherrima (two times higher). This behavior was also found when the T. delbrueckii and L. thermotolerans cells, grown in nutrient-poor medium (TSM), were dehydrated with kinetics KA, with GSSH:GSH ratio three times higher [START_REF] Câmara | Dehydration stress responses of yeasts Torulaspora delbrueckii, Metschnikowia pulcherrima and Lachancea thermotolerans: Effects of glutathione and trehalose biosynthesis[END_REF]. For all cells grown in nutrient-rich medium (GSM), no remarkable differences in the increase of GSSG:GSH ratio was observed. These findings suggest an adaptive nutritional response in function of the growing medium composition and strains.

The presence of cysteine in the nutrient-rich medium (GSM) allows cells to accumulate large amounts of intracellular glutathione [START_REF] Câmara | Dehydration stress responses of yeasts Torulaspora delbrueckii, Metschnikowia pulcherrima and Lachancea thermotolerans: Effects of glutathione and trehalose biosynthesis[END_REF]. In fact, cysteine is the most important amino acid for the synthesis of glutathione [START_REF] Alfafara | Effect of amino acids on glutathione production by Saccharomyces cerevisiae[END_REF]. Nevertheless, under physiological conditions, the free cysteine is practically inert and has low redox activity due to its relatively high pK a ~8.5 [START_REF] Netto | Reactive cysteine in proteins: Protein folding, antioxidant defense, redox signaling and more[END_REF]. Cells cultivated in nutrient-poor conditions (TSM), with a high GSSG:GSH ratio [START_REF] Câmara | Dehydration stress responses of yeasts Torulaspora delbrueckii, Metschnikowia pulcherrima and Lachancea thermotolerans: Effects of glutathione and trehalose biosynthesis[END_REF], were much more sensible to oxidation (Fig. 3), principally when submitted to a stronger dehydration step (kinetics KB). These findings suggest that growing in nutrient-poor conditions (TSM) may have affected yeast detoxification mechanisms. Moreover, the enzymatic glutathione recycle mechanism, one of the most important ROS elimination pathways in cells, mediated by both glutathione reductase and glutathione oxidase, may have been affected during cultivation in limited nutritional conditions (TSM) (Gamero-Sandemetrio, Paya-Tormo, Gómez-Pastor, Aranda, & Matallana, 2018; [START_REF] Schmidt | Production, reactivation and nutrient requirements of active dried yeast in winemaking: theory and practice[END_REF]. Indeed, glutathione can be mobilized as a source of nitrogen and sulfur under limited nutritional conditions, thus satisfying yeast growth requirements [START_REF] Mehdi | An important role for glutathione and γglutamyltranspeptidase in the supply of growth requirements during nitrogen starvation of the yeast Saccharomyces cerevisiae[END_REF]. This can partially explain why cells cultivated in nutrient-poor medium were more cultivable after dehydration with kinetics KA (Fig. 2). We observed that non-Saccharomyces yeasts (T. delbrueckii, M. pulcherrima and L. thermotolerans) accumulated much more GSSG after kinetics KB especially after growth in nutrientpoor medium [START_REF] Câmara | Dehydration stress responses of yeasts Torulaspora delbrueckii, Metschnikowia pulcherrima and Lachancea thermotolerans: Effects of glutathione and trehalose biosynthesis[END_REF], showing the impact of this drying condition on cell oxidative stress response. Indeed, temperature-time dehydration conditions have a crucial role during ADY production, as it can affect cell survival and stabilization post-dehydration [START_REF] Gamero-Sandemetrio | Antioxidant defense parameters as predictive biomarkers for fermentative capacity of active dried wine yeast[END_REF]. Yeast life occurs only in the presence of water and its removal involves severe consequences for the microbial survival. In particular, the level of water activity in the intracellular medium plays an important role in metabolic reactions and in the maintenance of cell structure [START_REF] Ball | Water as an active constituent in cell biology[END_REF]. Thus, during dehydration processes, the maintenance of cell cultivability is directly correlated with the level of activity of the water reached after dehydration and the intracellular redox environment [START_REF] Picazo | Mitochondria inheritance is a key factor for tolerance to dehydration in wine yeast production[END_REF]. The drying kinetics and dehydration levels determine the water transfer kinetics and the final water activity level of the cells. Thus, when the dehydration gradient is high, the transfer of water from the cell to the medium is greater and faster. Thus, drying amplitude and kinetics have an impact on cell stability [START_REF] Gervais | Osmotic mass transfer in the yeast Saccharomyces cerevisiae[END_REF][START_REF] Rodríguez-Porrata | Vitality enhancement of the rehydrated active dry wine yeast[END_REF], i.e. on the functionality of proteins, enzymes and cell structures.

Yeast oxidative stress response is affected by ROS levels during dehydration

It is well known that one of the reasons for cell death during dehydration processes is in part due to oxidative stress and accumulation of ROS inside cells [START_REF] Espindola | The role of glutathione in yeast dehydration tolerance[END_REF][START_REF] França | The role of cytoplasmic catalase in dehydration tolerance of Saccharomyces cerevisiae[END_REF]. During the biomass propagation, the oxidative stress resulting from aeration also leads to the accumulation of ROS [START_REF] Gibson | Yeast responses to stresses associated with industrial brewery handling[END_REF][START_REF] Gómez-Pastor | Improving yield of industrial biomass propagation by increasing the Trx2p dosage[END_REF]. In order to evaluate levels of intracellular ROS in yeasts, we used the oxidant indicator DHE (dihydroethidium). The DHE enters all cells and reacts with ion O 2 -to form a red fluorescent compound, 2-hydroxyethidium [START_REF] Zielonka | Detection of 2hydroxyethidium in cellular systems: a unique marker product of superoxide and hydroethidine[END_REF]. We analyzed the resulting metabolite 2-hydroxyethidium fluorescent signals using two methods: (i) qualitative with two-photon laser microscopy and (ii) quantitative by flow cytometry. A reduced number of oxidized cells when yeasts were cultivated in nutrient-rich medium (GSM) was observed (Fig. 3). These results correlated with lower GSSG:GSH ratio previously obtained [START_REF] Câmara | Dehydration stress responses of yeasts Torulaspora delbrueckii, Metschnikowia pulcherrima and Lachancea thermotolerans: Effects of glutathione and trehalose biosynthesis[END_REF], while cells growing in nutrient-poor medium (TSM) were more sensible to oxidation. In addition, dehydration via kinetics KB greatly affected the yeasts oxidation levels. In this dehydration condition, the intensity of the fluorescent signals detected by microscopy and the percentage of oxidized cells detected via flow cytometry were clearly higher, in comparison with results obtained using the kinetics KA (Fig. 3). Interestingly, cells growing in nutrient-poor medium showed irregular distribution of 'green droplets' during microscopic analysis. Probably, cells cultivated in this growing condition have produced more unsaturated lipids as consequence of nutritional limited environment [START_REF] Rodríguez-Porrata | Enhancing yeast cell viability after dehydration by modification of the lipid profile[END_REF].

Oleaginous yeasts, such as yeast M. pulcherrima, grown under limited nitrogen conditions and in the presence of large amounts of carbon accumulate lipids in response to nutritional stress [START_REF] Beopoulos | Yarrowia lipolytica as a model for bio-oil production[END_REF]. Another hypothesis supports that these cells were metabolically weaker, with lipids more susceptible to ROS attack leading to an enhancement of oxidation levels and intensity of detected fluorescent signals. It was previously shown that lipid droplets may arise during stressful cycles of dehydration-rehydration [START_REF] Wang | Lipid droplets, lipophagy, and beyond[END_REF]. The formation of these structures seems to be related to the aggregation and/or inactivation of proteins [START_REF] Jamieson | Oxidative stress responses of the yeast Saccharomyces cerevisiae[END_REF]. These data support the hypothesis that the formation of these lipid droplets are also related to the restructuration of the metabolic activity and to the reparation of the cell membrane structures [START_REF] Rapoport | Anhydrobiosis and dehydration of yeasts[END_REF]. This could explain the better resistance to dehydration-rehydration for yeasts with higher amount of these droplets, as observed for M. pulcherrima cells.

Impact of dehydration on lipid peroxidation levels

The lipids are one of the most important targets of oxidation damages induced by dehydration processes [START_REF] Wiseman | Vitamin D is a membrane antioxidant ability to inhibit irondependent lipid peroxidation in liposomes compared to cholesterol, ergosterol and tamoxifen and relevance to anticancer action[END_REF]. In order to evaluate the impact of dehydration kinetics, levels of lipid peroxidation were measured. We observed a correlation between a better nutritional growing environment (GSM) and the reduction of lipid peroxidation levels, mainly after kinetics KA (Fig. 4). These results also correlate with reduced ROS levels (Fig. 3) and a reduction of the GSSG:GSH ratio previously observed [START_REF] Câmara | Dehydration stress responses of yeasts Torulaspora delbrueckii, Metschnikowia pulcherrima and Lachancea thermotolerans: Effects of glutathione and trehalose biosynthesis[END_REF]. However, an enhancement of lipid peroxidation for cells dehydrated with kinetics KB was obtained. Indeed, lipids can be directly affected with dehydration time-temperature parameters. Modifications in the lipid composition of the plasma membrane, as quantity of sterols or changes in fatty acid unsaturation levels, has already been correlated with yeast sensitivity to environmental modifications [START_REF] Dupont | Nature of sterols affects plasma membrane behavior and yeast survival during dehydration[END_REF]. This could partially explain the observed improvement of oxidative attack to lipids. We also observed a nutritional effect to reduction of lipid peroxidation, especially for cells cultivated in nutrient-rich medium (GSM). Higher levels of reduced glutathione in the cytosol protect sulfhydryl proteins against oxidation and reduce the lipid peroxidation levels in yeast cells [START_REF] Fernandes | Oxidative stress response in eukaryotes: effect of glutathione, superoxide dismutase and catalase on adaptation to peroxide and menadione stresses in Saccharomyces cerevisiae[END_REF]. This observation is also supported by the fact that yeasts growing in nutrient-rich medium (GSM) exhibit lower ROS levels (Fig. 3) and a better replication activity (Fig. 2) after dehydration.

Conclusions

To the best of our knowledge, this is the first time that multiple quantitative and qualitative oxidative stress indicators have been analyzed simultaneously for the non-Saccharomyces yeasts T. delbrueckii, M. pulcherrima and L. thermotolerans during dehydration. We observed that cultivating yeasts in better nutritional environment 20 (GSM) led to a better cell cultivability after rehydration with lower ROS and lipid peroxidation levels. We also observed that dehydration kinetics KA led to a better post-dehydration cell survival. This fact is attributed mainly to the reduction of cell oxidative stress in this dehydration condition. These findings have proved that it is primordial to reduce the oxidative stress imposed to the cells during the formulation of the biomass and throughout the dehydration process in order to obtain new highquality dehydrated non-Saccharomyces yeasts. This can be achieved by increasing the intracellular glutathione levels or reducing GSSG:GSH ratio in yeasts. Future works should focus on the modulation of the composition of the growing medium in order to promote the production of new additive-free ADY, such as sorbitan monostearate and ascorbic acid.

Tukey's HSD was used to test significant difference with the corresponding control (Fresh) at *p < 0.05 or **p < 0.01. Scale bar = 10 µm.

Fig. 4

 4 Fig. 4 Lipid peroxidation measurements of yeasts S. cerevisiae, T. delbrueckii,

  Fig. 1

Acknowledgment

This work was supported by the Regional Council of Bourgogne -Franche-Comté, the "Fonds Européen de Développement Régional (FEDER)" and "Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)"-Brazil [grant number 200101/2014-5 to A. A. C. Jr.]. The authors are deeply grateful to the Plateau Technique DImaCell (Université Bourgogne Franche-Comté, Dijon, France) and to Camille Malnoury for technical assistance with the two-photon fluorescence imaging and to Raphaël Caire for the support with the enzymatic analysis.

Conflict of interest

The authors declare no financial or commercial conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals 484 performed by any of the authors.

485

Figure Legends

Fig. 1. Dehydration kinetics profiles applied to yeasts. Each kinetics, KA or KB, was carried out in three steps (time-temperature) using KNO 3 (step 1), NaBr (step 2) and CH 3 COOK (step 3) saturated solutions to control the water activity (a w ).


Jean Bertoin
email: jean.bertoin@upmc.fr

Vladas Sidoravicius
email: vladas@impa.br

The structure of typical clusters in large sparse random configurations

The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowski's coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors.

Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices.

In this work, we use the configuration model as the stochastic counterpart of Smoluchowski's coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to ∞. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors.

Introduction

The motivation for this work stems from a recent study of a deterministic model for coagulation with limited number of aggregations. Specifically, in [START_REF] Bertoin | Two solvable systems of coagulation equations with limited aggregations[END_REF], one considers particles that are determined by a pair of integers (a, k) where k ≥ 1 represents the size and a ≥ 0 the number of aggregations that the particle can perform. In the model called symmetric, coagulations

{(a, k), (a ′ , k ′ )} -→ (a + a ′ -2, k + k ′ ) occurs at rate aa ′ c t (a, k)c t (a ′ , k ′ ) ,
where c t (a, k) denotes the concentration of particles (a, k) at time t in the medium. Analytically, this means that the evolution of concentrations is governed by the following variation of Smoluchowski's coagulation equations (cf. the survey by Aldous [START_REF] Aldous | Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists[END_REF]):

d dt c t (a, k) = 1 2 a+1 a ′ =1 k-1 k ′ =1 a ′ (a -a ′ + 2)c t (a ′ , k ′ )c t (a -a ′ + 2, k -k ′ ) - ∞ a ′ =1 ∞ k ′ =1 aa ′ c t (a, k)c t (a ′ , k ′ ), (1)
where the first term in the right-hand side accounts for the creation of particles (a, k) as the result of coagulations of pairs {(a ′ , k ′ ), (a -a ′ + 2, k -k ′ )} and the second term for the disappearance of particles (a, k) after a coagulation with a particle (a ′ , k ′ ).

One of the main results in [START_REF] Bertoin | Two solvable systems of coagulation equations with limited aggregations[END_REF] is that under appropriate conditions on the initial data that we shall recall later on, the concentrations c t (a, k) have a limit as time t tends to infinity which is given by

c ∞ (a, k) = 1 {a=0} 1 k(k -1) ν * k (k -2) for a ∈ N and k ≥ 2 . (2)
Here, ν is a certain probability measure on N with ∞ n=0 nν(n) ≤ 1 that depends on the initial data, and ν * k = ν * • • • * ν denotes its k-th convolution power. The expression (2) bears a striking resemblance with a special case of the celebrated formula due to Dwass [START_REF] Dwass | The total progeny in a branching process[END_REF] who established that the total population T 2 (ν) generated by a (sub)-critical Galton-Watson branching process with reproduction law ν and started from two ancestors is given by

P(T 2 (ν) = k) = 2 k ν * k (k -2) , k ≥ 2 . (3)
This invites for a probabilistic explanation and provides the incentive for the present work.

Our approach for relating (2) to (3) stems from the fact that solutions to the classical Smoluchowski's coagulation equations (without restriction on the number of aggregations) appear as the hydrodynamical limit of certain stochastic coalescent models introduced by Marcus and Lushnikov. In some loose sense, the latter describe the microscopic random dynamics of the particle system when the macroscopic evolution is governed by Smoluchowski's coagulation equations. This important feature has been established rigorously by Norris [START_REF] Norris | Smoluchowski's coagulation equation: uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent[END_REF]. We also refer to Section 5.2.1 in [START_REF] Bertoin | Random Fragmentation and Coagulation Processes[END_REF] for an elementary approach in the special case of the multiplicative kernel, as the latter bears an obvious similarity with [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescence[END_REF]. On the other hand, it is well-known that the multiplicative coalescent is naturally related to the size of the connected components in the random graph model of Erdös and Rényi, see in particular the remarkable paper by Aldous [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescence[END_REF]. This leads us to consider an extension of the random graph model where the sequence of degrees of vertices is given, and which is known as the configuration model. Loosely speaking, the configuration model is constructed by an elementary stochastic algorithm which aims at producing a random graph on a set of vertices with pre-described degrees; in general the resulting graph is not simple, in the sense that there may exist loops and multiple edges. Typically, a certain number of stubs is appended to each vertex, and one joins pairwise the stubs uniformly at random to create edges between vertices. This induces a natural partition of the set of vertices into clusters, i.e. connected components.

Since its introduction independently by Bollobás [START_REF] Bollobás | A probabilistic proof of an asymptotic formula for the number of labelled regular graphs[END_REF] and Wormald [START_REF] Wormald | Some Problems in the Enumeration of Labelled Graphs[END_REF] (see also Bender and Canfield [3]), this model has been studied in the mathematical literature by many authors. We refer e.g. to [START_REF] Newman | The structure and function of complex networks[END_REF] for an interesting review of applications of this and other random graph models to some real life network systems. The main known results chiefly concern asymptotics when the number of vertices is large and the empirical measure of the degrees of vertices converges. In particular, Molloy and Reed [START_REF] Molloy | A critical point for random graphs with a given degree sequence[END_REF] have determined the critical parameter for the existence of a giant component; see also [START_REF] Molloy | The size of the giant component of a random graphs with a given degree sequence[END_REF] and [START_REF] Newman | Random graphs with arbitrary degree distributions and their applications[END_REF]. In different directions, van der Hofstad, Hooghiemstra and co-authors [START_REF] Van Den Esker | Universality for the distance in finite variance random graphs[END_REF][START_REF] Van Der Hofstad | Distances in random graphs with finite variance degrees[END_REF][START_REF] Van Der Hofstad | Distances in random graphs with finite mean and infinite variance degrees[END_REF] have made deep contributions to the study of distances between vertices in such random graphs, while Britton et al. [START_REF] Britton | Generating simple random graphs with prescribed degree distribution[END_REF] used the configuration model to produce large random simple graphs with pre-described asymptotic degree distribution.

If we neglect the appearance of multiple edges, loops or cycles which do not contribute to aggregation of clusters, the configuration model may serve as a stochastic counterpart to the deterministic evolution of concentrations in the variant (1) of Smoluchowski's coagulation equations. This leads us to investigate the size of typical clusters, and more generally their combinatorial structures. Roughly speaking, the main result of this work is a hydrodynamical limit theorem for the empirical distribution of the shapes of clusters rooted at a generic stub. The limit is expressed in terms of a pair of Galton-Watson trees which are connected by an extra edge between the two roots. In particular, this yields the probabilistic explanation of the formal similarity between the solution (2) and Dwass formula (3).

Let us now present some heuristics which are close to some of those that have already been used in the literature on configuration models to relate the latter to Galton-Watson processes; see in particular [START_REF] Molloy | A critical point for random graphs with a given degree sequence[END_REF] and [START_REF] Van Der Hofstad | Distances in random graphs with finite variance degrees[END_REF]. Imagine that we pick a stub uniformly at random; the degree of the vertex to which this stub is appended has then the size-biased law of the degree of a typical vertex. We then pick a second stub uniformly at random to create the first edge. Informally, when the number of vertices is large, the degree of the vertex to which the second stub is appended has again the size-biased law and is essentially independent of the first. These two vertices should be viewed as the ancestors of two growing populations, where, by induction, individuals beget independently and with a reproduction law given by the distribution of the outer degree of a size-biased vertex. When the reproduction law is critical or sub-critical, the Galton-Watson process eventually becomes extinct, and extinction occurs before any loop, multiple edge or cycle arises in that cluster of the configuration model. This suggests that the combinatorial structure of a typical (not too large) cluster could be described as a pair of independent Galton-Watson trees which are connected by an additional edge between the two roots. More precisely, the reproduction law should be given by the size-biased degree of a typical vertex, shifted by one unit, because the number of children corresponds to the outer-degree of the vertex.

The present work can be viewed as a companion to the recent paper [START_REF] Bertoin | A system of grabbing particles related to Galton-Watson trees[END_REF], in which we also identify in terms of certain Galton-Watson trees the limiting empirical distribution of random structures that appear in a toy model for polymerization. More precisely, we consider in [START_REF] Bertoin | A system of grabbing particles related to Galton-Watson trees[END_REF] a system of grabbing particles, where particles consist in monomers having a certain number of arms. Arms are activated successively uniformly at random, and each time an arm is activated, it grabs a particle uniformly at random amongst those which have not been previously grabbed and do not belong either to its own cluster. The main result of [START_REF] Bertoin | A system of grabbing particles related to Galton-Watson trees[END_REF] is that when the initial number of particles is large and the numbers of arms are given by i.i.d. random variables with mean less than 1, then the empirical distribution of the shapes of polymers is closed to that induced by a Galton-Watson tree with a single ancestor and reproduction law given by the distribution of the number of arms of a typical monomer.

The plan of this work is as follows. The next section is devoted to preliminaries on configuration models, the combinatorial structure of planar rooted trees, and Galton-Watson processes. The emphasis is put on planar structures and their codings by the sequence of degrees via breadth-first search. The main result on the empirical distribution of the structures of rooted clusters in large random configurations is stated in Section 3 and then proved by explicit first and second moments estimates. Finally Section 4 is devoted to some applications. We shall point at certain invariance properties of Galton-Watson trees under random re-rooting, and conclude by explaining the striking resemblance between the formulas (2) and (3).

Preliminaries 2.1 Pairings, configurations and clusters

The aim of this work is to relate random configuration models to Galton-Watson trees, and as the latter have a natural planar structure, we shall introduce the former in planar setting which is tailored for our purposes. In this direction, we should imagine particles as planar star-shaped objects consisting in a vertex to which a certain number of stubs are appended.

Formally, we consider some finite set V of vertices and a map d : V → N * where d(v) represents the degree of the vertex v, that is number of stubs attached to v. We denote by S = S(V, d) the set of stubs and shall suppose for the sake of simplicity that the total number of stubs

S := #S = v∈V d(v)
is even; otherwise we may always decide to add a new stub to some vertex (or to add a vertex with a single stub). We call a partition of S into S/2 pairs a pairing of stubs and write Π(S) for the set of pairings of stubs. We first point at the following elementary facts.

Lemma 1 (i) The cardinal of Π(S) is given by

#Π(S) = S! (S/2)! 2 -S/2 = S/2 i=1 (S -2i + 1) . (ii) Consider a partition of V into two subsets V 1 , V 2 such that S 1 := v∈V 1 d(v) and S 2 := v∈V 2 d(v) are even numbers. Set S 1 := S(V 1 , d) and S 2 := S(V 2 , d). Then the map (π 1 , π 2 ) -→ π 1 ⊔ π 2 is a bijection from Π(S 1 ) × Π(S 2 )
to the subset of Π(S) consisting in pairings π such that there are no pairs {s 1 , s 2 } in π formed by a stub s 1 attached to a vertex in V 1 and a stub s 2 attached to a vertex in V 1 .

Proof: Indeed, a generic pairing can be obtained by enumerating the stubs by {1, . . . , S} and then pairing the stubs according to the couples (1, 2), [START_REF] Bender | The asymptotic number of labelled graphs with given degree sequences[END_REF][START_REF] Bertoin | Random Fragmentation and Coagulation Processes[END_REF], . . . , (S -1, S). There are S! possible enumerations and the mapping is (S/2)!2 S/2 on 1, where (S/2)! accounts for the number of permutations of the S/2 couples (2i -1, 2i), and 2 S/2 for the number of ways S/2 unordered pairs can be ordered into couples. This establishes the first claim. The second is obvious.

We then form edges e = {v, v ′ } with v, v ′ ∈ V by joining the tips of pairs of stubs {s, s ′ }, where s (respectively, s ′ ) is appended to v (respectively, to v ′ ). We stress that an edge is unoriented, that it can be a loop (i.e. the two vertices v and v ′ defining an edge may coincide), and that the same edge may appear by joining different pairs of stubs. Each pairing of stubs π yields a configuration γ(π), that is the family of the S/2 edges induced by the pairing. Note that there may be multiple edges; the same edge is repeated in γ(π) as many times as it arises by joining different pairs of stubs in π. We also stress that the map π → γ(π) is not injective.

We view an edge which is not a loop as an elementary path connecting two different vertices, so a configuration γ(π) on (V, d) naturally induces a partition of V into connected components. Endowing a given connected component with the restriction of γ(π) to the set of edges formed by pairs of vertices in that component, we obtain a cluster.

Planar rooted trees and their structures

Lemma 1(ii) enables us to reduce the study of a given cluster to that of pairings π ∈ Π(S) such that the entire set of vertices V is connected for the configuration γ(π). We shall therefore focus on that case in this section. Recall that a cycle is a sequence of ℓ ≥ 3 distinct vertices, say v 1 , . . . , v ℓ , such that there exists an edge connecting v j and v j+1 for every j = 1, . . . , ℓ -1 and also an edge connecting v ℓ and v 1 .

A configuration γ(π) that connects V is called a tree if it contains no loops, no multiple edges, and no cycle. Note that this can occur only when S/2 = #V -1. Because particles (i.e. vertices and the stubs that are appended) can be viewed as planar objects, we may think of tree-configurations as planar structures, in the sense that they can be represented in the plane in such a way that edges are line segments which do no cross, by attributing lengths to the edges in an appropriate manner. Throughout this section, we assume that #V = k and that the configuration γ(π) is a tree; in particular γ(π) consists in k -1 edges and S = 2(k -1).

To describe precisely the shape, that is the combinatorial structure, of a tree, we need to specify an origin and an orientation. For this, we distinguish a stub s and call it the root. This stub is appended to a certain vertex v that we use as the origin. Distinguishing s also enables us to order all the stubs attached to v by deciding that the first stub is s and the next (if any) are ranked clockwise from that one. Further, for every vertex v ′ = v in that tree, we distinguish the stub appended to v ′ that points at the origin v. This provides a natural order on the set of stubs appended to any given vertex of the tree, and thus enables the use of breadth-first search to enumerate the vertices of the tree. Specifically, set s 1 = s and v 1 = v, define s i+1 as the stub that is paired with the i-th stub appended to v 1 for i = 1, . . . , d(v 1 ), and write v i+1 for vertex to which s i+1 is appended.

We should think of v 2 , . . . , v d(v 1 )+1 as the children of v 1 . The stub s 2 is chosen as the first of the stubs appended to v 2 , thus it is the unique stub pointing at the origin and the other stubs attached to v 2 are ranked clockwise from s 2 and point at the children of v 2 (i.e. the vertices at distance 2 from the origin v 1 and at distance 1 from v 2 ). We denote these d(v 2 ) -1 children by v d(v 1 )+2 , . . . , v d(v 1 )+d(v 2 )+1 , and continue with the next children v 3 , . . . , v d(v 1 )+1 of v 1 is an obvious way. Then we proceed with indexing the third generation of vertices, in the order which is naturally induced by the indexation of the second generation, and so on. See the figure below.

We write d i for the degree of the i-th vertex. We stress that for 2 ≤ i ≤ k, the outer-degree of v i , i.e. the number of stubs appended to v i that point away from the origin, is

d i -1. It is well-known that the sequence of degrees d = (d 1 , . . . , d k ) fulfills min{j ≥ 1 : d 1 + • • • + d j = 2(j -1)} = k , (4)
and characterizes a unique planar rooted tree structure. Conversely, any finite sequence d = (d 1 , . . . , d k ) such that (4) holds encodes a planar rooted tree structure with k vertices. We write D for the set of sequences d = (d 1 , . . . , d k ) which fulfill [START_REF] Bertoin | Random Fragmentation and Coagulation Processes[END_REF], where the lenght k ∈ N * is arbitrary, and think of the set D of sequences of degrees as the set of structures of planar rooted trees. We refer for instance to Section 6.2 in Pitman [START_REF] Pitman | Combinatorial Stochastic Processes[END_REF] for details.

We now summarize this discussion, introducing first some terminology for convenience. A bijection {1, . . . , k} → V can be represented as a sequence v = (v 1 , . . . , v k ) of distinct vertices and will be referred to as an enumeration of V. We also call a map ς : V → S that associates to each vertex v ∈ V a stub s ∈ S appended to that vertex a selection of stubs. For every pairing π ∈ Π(S) such that the configuration γ(π) on V is a tree and every choice of a distinguished stub s ∈ S, breadth first search yields a unique enumeration v = (v 1 , . . . , v k ) of V such that the sequence d(v) = (d(v 1 ), . . . , d(v k )) belongs to D (i.e. fulfills (4)), and a unique a selection of stubs ς. The map (π, s) -→ (v, ς) is bijective. More precisely, we recover the pairing π and the root stub s by first constructing the planar rooted tree structure associated to d = (d(v 1 ), . . . , d(v k )), and then placing the vertices v 1 , . . . , v k on this structure in the order induced by the breadth first search. The first stub appended to v 1 is s = ς(v), and for every i = 2, . . . , k, ς(v i ) is the stub appended to v i which points at the origin v 1 . This determines the pairing π.

In order to record this analysis, it is convenient to introduce the multinomial coefficient Otherwise (i.e. if d is not compatible), this number is 0.

M(V, d) := k ℓ 1 , . . . , ℓ j = k! ℓ 1 ! • • • ℓ j ! , (5)
We stress that all the rooted planar tree structures which are compatible with (V, d) are thus equally likely to occur if we choose the pair (π, s) ∈ Π(S) × S uniformly at random. In the same vein, it may be also interesting to point at the following simple formula, even though it will not be used in this paper .

Proposition 1 Suppose that #V = k and S = 2(k -1). The number of pairings π ∈ Π(S) for which the configuration γ(π) is a tree, is

(k -1)! v∈V d(v),
Proof: To establish the formula, we simply need to calculate the number of enumerations v of V for which the sequence (d(v 1 ), . . . , d(v k )) corresponds to some rooted planar tree structure.

Recall from the ballot theorem (see, e.g., Lemma 6.1 in [START_REF] Pitman | Combinatorial Stochastic Processes[END_REF]) that for each enumeration v, there is a unique cyclic permutation σ of {1, . . . , k} such that (d(v σ(1) ), . . . , d(v σ(k) )) fulfills ( 4). This shows that this number is (k -1)!.

Galton-Watson trees with two ancestors

We consider now a probability measure ν on N and associate to ν a measure on D by

GW ν 2 (d) = k i=1 ν(d i -1) , (6)
where d = (d 1 , . . . , d k ) denotes a generic rooted planar tree structure.

The measure GW ν 2 has a simple interpretation in terms of Galton-Watson branching processes, and is in fact a sub-probability. More precisely, consider a Galton-Watson process with reproduction law ν and started from two ancestors. The process can be represented on the upper-half plane, where the individuals at generation ℓ ∈ N lie on horizontal line y = ℓ, in an order consistent with that of their respective parents, so that the edges (line-segments) linking parents to children do not cross each other. We further connect the two ancestors by an additional edge, and distinguish the stub attached to the left-most ancestor that thus points at the right-most ancestor. This enables us to list individuals (vertices) by breadth first search just as in the preceding section. Observe that the degree of the left-most ancestor (i.e. the origin) is distributed as 1 + ξ where ξ is a random variable with law ν, whereas the outer-degrees of the other individuals (i.e. their numbers of children) are given by independent copies of ξ.

The event when the total population is finite has probability one if and only if the reproduction law ν is critical or subcritical, i.e.

i∈N iν(i) ≤ 1, and ν = δ 1 . Restricting our attention to this event, the structure of this planar rooted tree is a random variable in D which has distribution GW ν 2 and is defective in the supercritical case.

Remark. In the case when ν is the Poisson distribution with parameter p ≤ 1, then it is easily checked that the law GW ν 2 also describes the law of the genealogical tree of a Galton-Watson process with reproduction law ν, started from a single ancestor, and conditioned to have size at least 2.

A limit theorem for typical rooted clusters

For each fixed integer n, we consider a set V n of n vertices and a function d n : V n → N * that specifies the number of stubs appended to each vertex. We introduce the empirical distribution of the number of stubs

µ n (i) := 1 n #{v ∈ V n : d n (v) = i} , i ∈ N * .
We write

S n := v∈Vn d n (v) = n ∞ i=1 iµ n (i)
for the total number of stubs, assuming for simplicity that this quantity is even. Our basic assumption is that the limit lim

n→∞ µ n (i) := µ(i) (7)
exists for every i ≥ 1, and that the average number of stubs

n -1 S n = ∞ i=1 iµ n (i) converges as n → ∞ to the first moment of µ, i.e. lim n→∞ ∞ i=1 iµ n (i) = ∞ i=1 iµ(i) := m < ∞ . (8)
We also denote by µ * the probability measure on N * which is obtained from µ by size-biased sampling, that is

µ * (i) := iµ(i) m , i ∈ N * .
A standard application of Scheffé's lemma shows that (7) can then be re-enforced to

lim n→∞ iµ n (i) n S n = µ * (i) in L 1 (N * ). (9)
Finally, we introduce the probability measure ν on N induced from µ * by the shift i → i -1 from N * to N, viz.

ν(i) = µ * (i + 1) , i ≥ 0 .
We write S n for the set of stubs appended to vertices in V n . We pick a pairing π ∈ Π(S n ) uniformly at random, and denote by Γ n := γ(π) the resulting random configuration on (V n , d n ). For every stub s ∈ V n , if the cluster of Γ n which contains s is a tree, then T s denotes the combinatorial structure which results from rooting that tree at the stub s (see Section 2.2), and otherwise, we decide that T s = ∅.

We are interested in the random variable

ρ n (d) := 1 S n #{s ∈ S n : T s = d} , d ∈ D
which counts the proportion of stubs s such that the cluster rooted at s induced by Γ n is a tree with structure d. Similarly, we write

ρ n (∅) := 1 S n #{s ∈ S n : T s = ∅}
for the proportion of stubs s such that the cluster containing s induced by Γ n is not a tree. The collection (ρ n (d) : d ∈ D) should thus be viewed as a variant of the empirical measure of tree-clusters. We now able to state our main asymptotic result on large random configurations.

Theorem 1 Assume that ( 7) and (8) hold. Then for every planar rooted tree configuration d ∈ D, the following limit holds in L 2 (P) :

lim n→∞ ρ n (d) = GW ν 2 (d) .
If we further suppose that

∞ i=1 i(i -2)µ(i) ≤ 0 , (10)
and also exclude the degenerate case when µ is the Dirac point mass at 2, then

lim n→∞ ρ n (∅) = 0 in L 1 (P).
The condition [START_REF] Van Den Esker | Universality for the distance in finite variance random graphs[END_REF] plays an important part for random configuration models. According to a well-known result due to Molloy and Reed [START_REF] Molloy | A critical point for random graphs with a given degree sequence[END_REF], when [START_REF] Van Den Esker | Universality for the distance in finite variance random graphs[END_REF] fails (assuming also some further technical conditions), then there is some constant c > 0 such that with probability one, the random configuration Γ n contains almost surely a cluster of size at least cn when n is sufficiently large. The size of this giant component is estimated in [START_REF] Molloy | The size of the giant component of a random graphs with a given degree sequence[END_REF]. At the opposite, when [START_REF] Van Den Esker | Universality for the distance in finite variance random graphs[END_REF] holds with a strict inequality (again assuming some further technical conditions), Molloy and Reed [START_REF] Molloy | A critical point for random graphs with a given degree sequence[END_REF] have shown that with probability one, the random configuration Γ n contains at most n 1/4 cycles and no cluster of size at least n 1/4 whenever n is sufficiently large. Note that in the critical case when ( 10) is an equality, Theorem 1 implies that the probability that there is a cluster of size at least εn tends to 0 for any ε > 0, because GW ν 2 is a probability measure on D.

The proof of Theorem 1 relies on asymptotics for the first and second moments of ρ n (d). We first state:

Lemma 3 We have lim n→∞ E(ρ n (d)) = GW ν 2 (d)
for every d ∈ D.

Proof: Let the structure d = (d 1 , . . . , d k ) have size k ≥ 2. We write V ′ for a generic subset of V n with k vertices and S ′ for the set of stubs in S n which are appended to vertices in V ′ . There are two cases.

If the unordered families of degrees {d(v ′ ) : v ′ ∈ V ′ } and {d i : 1 ≤ i ≤ k} do no coincide (recall that in such families, numbers are repeated according to their multiplicity), then there is no pairing of stubs for which the vertices of V ′ are those of a tree-cluster with structure d when properly rooted. We say that V ′ is bad.

Otherwise, we say that V ′ is good. Introduce the set G ′ of couples (s, π) ∈ S ′ × Π(S n ) such that the cluster rooted at s induced by the configuration γ(π) is a tree whose set of vertices coincides with V ′ and has structure d. The cardinal of G ′ can then be computed by combining Lemmas 1 and 2. Since #S ′ = 2(k -1), one gets

#G ′ = M(d) (S n -2(k -1))! (S n /2 -k + 1)! 2 -Sn/2+k-1 k i=1 d i , (11)
where M(d) denotes the multinomial coefficient

M(d) := k ℓ 1 , . . . , ℓ j = k! ℓ 1 ! • • • ℓ j ! ,
with j the number of different values in the sequence d and ℓ i the number of occurrences in d of the i-th value for 1 ≤ i ≤ j.

So it remains to estimate the number of good subsets V ′ with k vertices, and for this we use a probabilistic argument. We sample uniformly at random k vertices in V n , say, v 1 , . . . , v k , successively and without replacement. It should be plain from the hypothesis (7) that when n → ∞, the k-tuple of degrees (d n (v 1 ), . . . d n (v k )) converges in distribution to the k-tuple formed by i.i.d. variables with law µ; in particular the probability that (

d n (v 1 ), . . . , d n (v k )) = d tends to k i=1 µ(d i ) as n → ∞.
We readily deduce that the probability that the (unordered) family {d(v 1 ), . .

. , d(v k )} is good converges as n → ∞ to k! M(d) k i=1 µ(d i ) .
As there are n!/(n -k)! ∼ n k k-tuples of distinct vertices in V n and as the map that transforms a k-tuple into an unordered set is k! to 1, we conclude that the number of good subsets in V n is equivalent for large n to

n k M(d) k i=1 µ(d i ) . (12)
Recall from Lemma 1(i) that

# (S n × Π(S n )) = S n S n ! (S n /2)! 2 -Sn/2 ,
and that

d i µ(d i ) = mν(d i -1)
, by definition. Putting the pieces together, we find

E(ρ n (d)) ∼ n k S n (S n -2(k -1))!(S n /2)! S n !(S n /2 -k + 1)! 2 k-1 k i=1 (d i µ(d i )) ∼ n k S n S -2(k-1) n (S n /2) k-1 2 k-1 k i=1 (mν(d i -1)) = (nm) k S k n k i=1 ν(d i -1)
.

By ( 6) and ( 8), this completes the proof.

Lemma 3 essentially means that if we pick a stub s uniformly at random in S n and independently of the random configuration Γ n , then the conditional distribution of the combinatorial structure of the random cluster rooted at s given the event that this cluster is a tree, converges weakly as n → ∞ to the Galton-Watson law GW ν 2 . Theorem 1 is a much stronger statement that involves the empirical distribution of structures of clusters, and requires second moment estimates.

Lemma 4 We have lim n→∞ E((ρ n (d)) 2 ) = (GW ν 2 (d)) 2
for every d ∈ D.

Proof: The argument is similar to that of Lemma 3; in particular we shall use the same notation and terminology. We start from the expression

E((ρ n (d)) 2 ) = S -2 n E (#{(s ′ , s ′′ ) ∈ S n × S n : T s ′ = T s ′′ = d}) .
Let V ′ and V ′′ two generic subsets of V n , both with k vertices, and write S ′ (respectively, S ′′ ) for the set of stubs in S n which are appended to vertices in V ′ (respectively, V ′′ ). Note that for every stubs s ′ ∈ S ′ and s ′′ ∈ S ′′ , the identity T s ′ = T s ′′ = ∅ can occur only if V ′ and V ′′ are both good and, either coincide or are disjoint.

We first consider the situation when V ′ = V ′′ . Recall that for any s ′ ∈ S ′ , if T s ′ = d, then S ′ = S ′′ has exactly 2(k -1) stubs. It follows from the proof of Lemma 3 that the number of triplets (s ′ , s ′′ , π) ∈ S ′ × S ′′ × Π(S n ) such that the cluster rooted at s ′ induced by the configuration γ(π) is a tree whose set of vertices coincides with V ′ and T s ′ = T s ′′ = d, is bounded from above by

2(k -1)M(d) (S n -2(k -1))! (S n /2 -k + 1)! 2 -Sn/2+k-1 k i=1 d i ;
see [START_REF] Haas | Spinal partitions and invariance under re-rooting of continuum random trees[END_REF]. Multiplying this by the number of good subsets V ′ in V n , that is approximatively by [START_REF] Van Der Hofstad | Distances in random graphs with finite variance degrees[END_REF], we get a quantity which is small compared to

#(S n × S n × Π(S n )) = S 2 n S n ! (S n /2)! 2 -Sn/2
when n → ∞. We conclude that in the evaluation of E((ρ n (d)) 2 ), the contribution of pairs of stubs (s ′ , s ′′ ) that belong to the same cluster becomes asymptotically negligible.

Next we consider the situation when V ′ and V ′′ are good and disjoint. By calculations similar to those that yield [START_REF] Van Der Hofstad | Distances in random graphs with finite variance degrees[END_REF] in the proof of Lemma 3, we get that the number of good disjoint pairs of subsets (V ′ , V ′′ ) in V n is equivalent for large n to

n 2k M(d) 2 k i=1 µ(d i ) 2 .
It then follows from Lemmas 1 and 2 that the number of triplets (s ′ , s ′′ , π) ∈ S ′ × S ′′ × Π(S n ) such that T s ′ = T s ′′ = d and the stubs s ′ and s ′′ belong to disjoint clusters is close to

n 2k (S n -4(k -1))! (S n /2 -2k + 2)! 2 -Sn/2+2k-2 k i=1 d i µ(d i ) 2 .
Putting the pieces together yields the estimate

E((ρ n (d)) 2 ∼ n 2k S 2 n (S n -4(k -1))!(S n /2)! S n !(S n /2 -2k + 2)! 2 2k-2 k i=1 d i µ(d i ) 2 ∼ n 2k S 2 n S -4(k-1) n (S n /2) 2k-2 2 2k-2 k i=1 (mν(d i -1)) 2 ∼ k i=1 ν(d i -1) 2 .
By [START_REF] Bertoin | A system of grabbing particles related to Galton-Watson trees[END_REF], this shows our claim.

We are now able to establish Theorem 1.

Proof of Theorem 1: Combining Lemmas 3 and 4, we see that the variance of ρ n (d) tends to 0 as n → ∞, which establishes the first claim. Assume now further that [START_REF] Van Den Esker | Universality for the distance in finite variance random graphs[END_REF] holds and that µ = δ 2 . Equivalently, this means that the reproduction law ν of the Galton-Watson process is critical or sub-critical, and is not the Dirac mass at 1. So extinction occurs a.s. and

d∈D GW ν 2 (d) = 1 . As ρ n (∅) = 1 - d∈D ρ n (d) ,
Fatou lemma entails our second assertion.

Some applications

In this Section, we shall develop some consequences of our main result. Recall that Theorem 1 implies that if one selects a stub uniformly at random and independently of a large random configuration that fulfills the conditions there, then the structure of the cluster rooted at that stub has asymptotically the distribution GW ν 2 . This hints at an interesting property of invariance of such Galton-Watson trees under uniform random re-rooting. Recall the construction of the structure of a planar tree rooted at some stub as it has been presented in Section 2.2; Figure 2 below should explain better than words what is meant by re-rooting a rooted planar tree at some stub.

Corollary 1 Suppose that ν is a critical or subcritical probability measure on N with ν = δ 1 . Let D be a random rooted planar tree structure with distribution GW The tree on the right is the image of the tree on the left by re-rooting at the stub =⊲.

Vertices are labeled by breadth first order before re-rooting.

Proof: Re-rooting has no effect on the degree of a vertex, so we only need to verify the statement for the conditional law of the Galton-Watson genealogical tree with two ancestors given the unordered family of the degrees of vertices.

Fix some unordered family, say ∆, of k positive integers (with possible repetitions), which add up to 2(k -1) and such that ν(δ -1) > 0 for any integer δ in that family. Denote by D(∆) the subset of rooted planar tree structures corresponding to some ordering of ∆. We see from (6) that the conditional law GW ν 2 (• | D(∆)) is simply the uniform distribution on D(∆). Next consider the random configuration on a set k vertices with degree family ∆ that is induced by uniform random pairing, given that this configuration is a tree. Then root the configuration using some stub that is picked independently and uniformly at random. On the one hand, by construction, the law of the resulting combinatorial structure is obviously invariant by uniform random re-rooting. On the other hand, we see from Lemma 2 that it also coincides with the uniform distribution D(∆). This established our claim.

We also refer to the recent work by Haas et al. [START_REF] Haas | Spinal partitions and invariance under re-rooting of continuum random trees[END_REF] and references therein for a different property of invariance under uniform re-rooting for certain classes of random continuous trees.

It may be interesting to point also at the following avatar of Corollary 1. A planar rooted tree is said planted if the degree of the origin, i.e. of the vertex to which the root-stub is appended, is 1. In other words, the combinatorial structure d = (d 1 , . . . , d k ) fulfills d 1 = 1. So a planted Galton-Watson tree describes the genealogy of a population where individuals beget independently with the same reproduction law, except the ancestor who has exactly one child. An easy consequence of Corollary 1 is that in the critical or sub-critical case, the structure of a planted Galton-Watson tree is statistically invariant under re-rooting at a leaf (i.e. a vertex with degree 1) chosen uniformly at random.

We next turn our attention to some quantitative consequences of Theorem 1, denoting for every k ≥ 2 by C n (k) the number of clusters of size k in the random configuration Γ n , i.e. the number of distinct connected components with k vertices in the partition of V n induced by Γ n .

Corollary 2 Assume that (7), ( 8) and (10) hold, and exclude the case when µ = δ 2 . We have

lim n→∞ ∞ k=2 kE n -1 C n (k) - m k(k -1) ν * k (k -2) = 0 .
Proof: Let us introduce first for every k ≥ 2 the subset D k of D consisting of structures of rooted planar trees d = (d 1 , . . . , d k ) of lenght k, and recall that according to Dwass [START_REF] Dwass | The total progeny in a branching process[END_REF],

GW ν 2 (D k ) = 2 k ν * k (k -2) ,
where ν * k stands for the k-th convolution power of ν. As a tree of size k has exactly 2(k -1) stubs and D k is a finite set, we deduce from Theorem 1 that if we denote by τ n (k) the number of clusters which are trees of size k, then lim n→∞ 2(k -1)

S n τ n (k) = 2 k ν * k (k -2) ,
where the convergence takes place in L 2 (P) and for every k ≥ 2. Then we pick an arbitrary sequence of integers that tends to ∞, from which we can excerpt by a diagonal extraction procedure a subsequence such that with probability one,

lim n ∞ 2(k -1) S n τ n (k) = 2 k ν * k (k -2) for all k ≥ 2,
where the notation n ∞ means that n tends to infinity along that subsequence.

Then observe that for each n, there is the obvious inequality As the sequence of integers tending to infinity that we started from is arbitrary, this establishes our claim.

Corollary 2 provides the explanation for the asymptotic behavior (2) that motivated this work. Specifically, we know from Theorem 1 that when the requirements ( 7), ( 8) and ( 10) are fulfilled, then, roughly speaking, multiple edges, loops or cycles are rare. Roughly speaking, this means that almost all creation of edges correspond to aggregations of clusters and thus enables us to view the configuration model as a stochastic microscopic version of the terminal state of concentrations with a deterministic evolution governed by the variant (1) of Smoluchowski's coagulation equations. In [START_REF] Bertoin | Two solvable systems of coagulation equations with limited aggregations[END_REF], one assumes that initially all particles are monomers, i.e. consist in isolated vertices to which some stubs are appended. In the notation of the present work (beware that this differs from that in [START_REF] Bertoin | Two solvable systems of coagulation equations with limited aggregations[END_REF]!), the initial concentration of particles with i ≥ 1 stubs is m -1 µ(i), which is a finite measure on N * with unit first moment. In the framework of the random configuration model with n vertices, this corresponds to assuming that particles live in a volume mn and hence the the initial concentration of monomers with i stubs is given by

m -1 µ n (i) = 1 mn #{v ∈ V n : d n (v) = i} , k ∈ N * .
After the random pairing, the concentration of polymers with size k (i.e. clusters with k vertices) is then (mn) -1 C n (k) and Corollary 2 shows that lim n→∞

1 mn C n (k) := c ∞ (0, k) = 1 k(k -1) ν * k (k -2) .
One has thus recovered [START_REF] Aldous | Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists[END_REF].

We also note that Corollary 2 solves a problem that has been addressed in Section II.C of [START_REF] Newman | Random graphs with arbitrary degree distributions and their applications[END_REF] by analytic and numerical technics.

Figure 1 :

 1 Figure 1 : Enumeration by breadth-first search of the vertices of a planar tree rooted at the stub =⊲. The degree sequence is (3, 4, 1, 1, 3, 3, 1, 1, 3, 1, 1, 1, 1).

Lemma 2

 2 where j is the number of different values, say x 1 , . . . , x j , occurring in the family (d(v) : v ∈ V), and ℓ i the number of occurrences of the value x i in that family. For every structure d ∈ D, we say that d is compatible with (V, d) if there is at least an enumerationv = (v 1 , . . . , v k ) of V such that d = (d(v 1 ), . . . , d(v k )),that is if and only if the sequence d takes the same values with the same multiplicity as the family (d(v) : v ∈ V). The following statement should now be plain. Suppose that #V = k and S = 2(k -1). Fix a rooted planar tree structure d = (d 1 , . . . , d k ) ∈ D. If d is compatible with (V, d), then the number of pairs (π, s) ∈ Π(S) × S for which the configuration γ(π) is a tree with structure d when rooted at s equals M(V, d) v∈V d(v).

ν 2 .Figure 2 :

 22 Figure 2 : Two genealogical trees, both with two ancestors lying at the lowest level. The left-most ancestor serves as the origin, the root-stub pointing at the right-most ancestor.The tree on the right is the image of the tree on the left by re-rooting at the stub =⊲.Vertices are labeled by breadth first order before re-rooting.

k≥2 2 ( 2 (

 22 k -1)τ n (k) ≤ S n , D k ) = 1 ,since the reproduction law ν of the Galton-Watson process is critical or sub-critical and ν = δ 1 . A standard combination of Fatou and Scheffé lemmas entails that lim (k -2) = 0 .Next, note that τ n (k) ≤ C n (k) and k≥2 2(k -1)C n (k) ≤ S n as at least 2(k -1) distinct stubs are needed to connect k vertices. It follows that n (k)) , and we know from above that this quantity tends to 0 as n ∞. -1)S n C n (k) -2 k ν * k (k -2) = 0 ,and since by the assumption (8), S n /n → m, we have thus proved that lim -1) ν * k (k -2) = 0 .

Acknowledgment. We would like to thank Maria Eulalia Vares for stimulating discussions which have been at the origin of this work.