Robin Guilhot
email: guilhoro@gmail.com

Simon Fellous

Joel E Cohen

Yeast facilitates the multiplication of Drosophila bacterial symbionts but has no effect on the form or parameters of Taylor's law

Interactions between microbial symbionts influence their demography and that of their hosts. Taylor's power law (TL)-a well-established relationship between population size mean and variance across space and time-may help to unveil the factors and processes that determine symbiont multiplications. Recent studies suggest pervasive interactions between symbionts in Drosophila melanogaster. We used this system to investigate theoretical predictions regarding the effects of interspecific interactions on TL parameters. We assayed twenty natural strains of bacteria in the presence and absence of a strain of yeast using an ecologically realistic set-up with D. melanogaster larvae reared in natural fruit. Yeast presence led to a small increase in bacterial cell numbers; bacterial strain identity largely affected yeast multiplication. The spatial version of TL held among bacterial and yeast populations with slopes of 2. However, contrary to theoretical prediction, the facilitation of bacterial symbionts by yeast had no detectable effect on TL's parameters. These results shed new light on the nature of D. melanogaster's symbiosis with yeast and bacteria. They further reveal the complexity of investigating TL with microorganisms.

Introduction

Animals and plants are often associated with several symbiotic microorganisms, the interactions of which affect the ecology and evolution of hosts and symbionts alike [START_REF] Ferrari | Bacterial symbionts in insects or the story of communities affecting communities[END_REF][START_REF] ´lvarez-Pe ´rez | Yeast-bacterium interactions: the next frontier in nectar research[END_REF]. Encompassing the diversity of symbiotic communities is challenging. However, experimental systems of modest complexity enable the investigation of microbial interactions, their mechanisms and consequences for host phenotypes [START_REF] Callens | Microbiota inoculum composition affects holobiont assembly and host growth in Daphnia[END_REF][START_REF] Gould | Microbiome interactions shape host fitness[END_REF] and the dynamics of microbial symbionts [START_REF] Sommer | Metabolic basis for mutualism between gut bacteria and its impact on the Drosophila melanogaster host[END_REF]. Conceptual tools developed by ecologists for the study of population and community ecology may be used to unveil general processes at play in symbiotic communities [START_REF] Ferrari | Bacterial symbionts in insects or the story of communities affecting communities[END_REF][START_REF] Brown | Metacommunity theory for transmission of heritable symbionts within insect communities[END_REF].

Among these tools, Taylor's law (TL) may be used to investigate the spatial and temporal distributions of symbiotic populations of microorganisms. TL asserts that the variance of population density (or size) is a power function of the mean population density (or size) across a set of replicate populations [START_REF] Taylor | Aggregation, variance and the mean[END_REF]. This power-law relationship is equivalent to a linear relationship on log-log coordinates: log(variance of population density) = a + b.log(mean of population density). A spatial TL holds when the mean and the variance are calculated over populations that differ in spatial location and the variance is a power function of the mean. A temporal TL holds when the mean and the variance are calculated over different points in time for each population and the variance is a power function of the mean. Many other variants of TL exist. TL holds in populations of various organisms [START_REF] Taylor | Taylor's power law: order and pattern in nature[END_REF] including bacteria [START_REF] Ramsayer | Taylor's Law holds in experimental bacterial populations but competition does not influence the slope[END_REF][START_REF] Kaltz | Bacterial microcosms obey Taylor's law: effects of abiotic and biotic stress and genetics on mean and variance of population density[END_REF] and can be of interest to address questions and resolve issues in conservation biology [START_REF] Beissinger | Population viability analysis[END_REF], epidemiology [START_REF] Krasnov | Abundance and distribution of fleas on desert rodents: linking Taylor's power law to ecological specialization and epidemiology[END_REF][START_REF] Cohen | Chagas disease vector control and Taylor's law[END_REF], human demography [START_REF] Xu | Evaluating multi-regional population projections with Taylor's law of mean-variance scaling and its generalisation[END_REF][START_REF] Cohen | Gompertz, Makeham, and Siler models explain Taylor's law in human mortality data[END_REF], fisheries [START_REF] Xu | Sequential analysis and design of fixed-precision sampling of Lake Kariba fishes using Taylor's power law[END_REF], forestry [START_REF] Cohen | Taylor's law and related allometric power laws in New Zealand mountain beech forests: the roles of space, time and environment[END_REF], and crop protection [START_REF] Do ¨ring | Taylor's power law and the stability of crop yields[END_REF]. Whether the relationship between population means and variances follows TL, and TL's parameters when TL holds, can shed limited light on the demographic processes in the populations studied [START_REF] Taylor | Taylor's power law: order and pattern in nature[END_REF][START_REF] Cohen | Every variance function, including Taylor's power law of fluctuation scaling, can be produced by any location-scale family of distributions with positive mean and variance[END_REF]. For example, a TL's slope b that falls in the range from 1 to 2 and differs from 2 may be due to interspecific interactions within ecological communities such as competition, predation or parasitism [START_REF] Taylor | Taylor's power law: order and pattern in nature[END_REF][START_REF] Johnson | Biological and statistical processes jointly drive population aggregation: using host-parasite interactions to understand Taylor's power law[END_REF].

In this study, we investigated whether the spatial TL describes well the relation of the variance to the mean of population density in experimental populations of bacterial symbionts of Drosophila; and tested whether the addition of another Drosophila symbiont-a yeast-in the system affects TL. Drosophila melanogaster larvae rely on both bacteria and yeast for larval development [START_REF] Becher | Yeast, not fruit volatiles mediate Drosophila melanogaster attraction, oviposition and development[END_REF][START_REF] Broderick | Gut-associated microbes of Drosophila melanogaster[END_REF]. However, although interactions between microorganisms associated with Drosophila flies have been reported [START_REF] Gould | Microbiome interactions shape host fitness[END_REF], the pervasiveness and nature of these relationships in the wild remains unclear [START_REF] Broderick | Gut-associated microbes of Drosophila melanogaster[END_REF][START_REF] Fischer | Metabolite exchange between microbiome members produces compounds that influence Drosophila behavior[END_REF]. So far, no study has investigated the nature of yeastbacteria interactions when associated to Drosophila larvae in ecologically realistic conditions using numerous strains freshly isolated from the wild. Our study hence aimed at describing and understanding the numerical effects of symbionts on each other in a context relevant to natural Drosophila biology. We assayed twenty bacterial strains in the absence and in the presence of a wild Hanseniaspora uvarum yeast strain. The growth of these microorganisms was studied in association with the larvae of a wild D. melanogaster population reared in natural fruit.

Material and methods

Biological material

We used twenty bacterial strains that were isolated from wild adult Drosophila and fruit homogenates collected in Montpellier (SF's garden and Montpellier SupAgro campus) and in Montferrier-sur-Lez (private property), southern France, except for three Acetobacter and Lactobacillus strains (Table 1). The yeast strain Hanseniaspora uvarum Dm6y (MN684824) was isolated from a wild D. melanogaster fly collected in Montpellier (SF's garden). Most of these microbial taxa had previously been identified as associated with Drosophila [START_REF] Chandler | Bacterial communities of diverse Drosophila species: ecological context of a host-microbe model system[END_REF][START_REF] Bellutti | Dietary yeast affects preference and performance in Drosophila suzukii[END_REF][START_REF] Staubach | Host species and environmental effects on bacterial communities associated with Drosophila in the laboratory and in the natural environment[END_REF]. All fieldisolated microorganisms were cultured a single time in the laboratory and stored in sterile Phosphate-Buffered Saline (PBS) solution (20% glycerol) at -80˚C until they were used in experiments. This ensured minimal adaptation to the laboratory of the tested microorganisms. The Drosophila melanogaster population was established from a few dozen wild individuals collected in Montferrier-sur-Lez (private property) about a year before the experiment. Conventionally reared flies had been maintained on a carrot-based laboratory medium (11.25 g.L -1 agar, 37.5 g.L -1 sugar, 15 g.L -1 corn meal, 37.5 g.L -1 dried carrot powder (Colin Ingredients SAS), 22.5 g.L -1 inactive dry yeast, 5 ml.L -1 propionic acid, 3.3 g.L -1 nipagin, 25 ethanol ml.L - 1 ). All biological samples were collected with the permission of the private owners and the Montpellier SupAgro administration.

Experimental design

Each experimental unit consisted of a halved grape berry that we first surface-sterilized following the procedure of Behar et al. [START_REF] Behar | Bringing back the fruit into fruit fly-bacteria interactions[END_REF] and embedded in jellified purified water (6 g.L -1 agar) in a small petri dish. This protocol ensures the removal of all microorganisms present at the surface of fruit, but does not eliminate possible microbial endophytes [START_REF] Compant | Endophytes of grapevine flowers, berries, and seeds: identification of cultivable bacteria, comparison with other plant parts, and visualization of niches of colonization[END_REF]. Our results hence reflect the biology of symbiotic bacteria and yeast in conditions comparable to those of the field, but not in sterile medium. We manually deposited fifteen fly eggs on each half grape berry. The eggs had been laid by groups of D. melanogaster females offered jellified grape juice plates (300 ml.L -1 grape juice, 6 g.L -1 agar) supplemented with cycloheximide (1 mg.L -1 ) to inhibit yeast growth, and chloramphenicol (10 mg.L -1 ) to inhibit bacterial growth. Repeated assays showed eggs produced in this manner are free of culturable microorganisms. After egg deposition, 10 5 cells of each bacterium were inoculated to fruit flesh either alone (seven replicates per bacterium, except for strain R6b that had six replicates) or together with 10 5 cells of the yeast strain (seven replicates per bacterium). The experiment was spread into seven blocks: one replicate of each bacterium × yeast combination was set up each day over seven days. Experimental units were incubated at 25˚C.

To measure microbial growth, we sampled fruit flesh after three days of incubation in a non-destructive fashion (analyses of D. melanogaster development will be described in a separate manuscript). Fruit flesh was sub-sampled by randomly inserting ten sterile pipette tips in the surface of each fruit, collecting approximately one twentieth of the flesh in total. Flesh samples were pooled per replicate, homogenized in 100 μl of sterile PBS solution and serially diluted. Cell counts were carried out by plating samples (serially diluted) on appropriate selective agar media (Table 1). For bacterial detection, we used Trypto-Casein-Soy (TCS) agar, Mannitol (MAN) agar or De Man, Rogosa and Sharpe (MRS) agar, all supplemented with cycloheximide (1 mg.L -1 ). For yeast detection, we used Yeast Extract-Peptone-Dextrose (YPD) agar supplemented with chloramphenicol (10 mg.L -1 ). Microbial colonies were counted after four days of incubation at species-specific temperatures and growth media (Table 1). We are confident the bacteria counted at the end of the experiment were those inoculated at the beginning for two reasons. First, we observed no bacterial growth in bacteria-free controls, which rules out the presence of culturable endophytes or contaminants in the fruits used for the experiment. Second, lack of cross-contaminations between bacterial treatments was attested by the systematic match between the morphology and metabolic abilities of the bacterial strains inoculated and those of the counted cells. These comparisons were possible because the majority of the strains used could be discriminated on the basis of the temperature and medium that enabled colony growth, as well as colony color, shape and transparency.

Statistical analyses

Analyses were split in two steps. First, we investigated the effects of each type of symbiont, and their interactions, on mean cell numbers. We hence tested whether yeast affected (i.e. increased or decreased) bacterial densities using microbial counts from each replicate as datapoints. We used a linear mixed model with the restricted maximum likelihood method. 'Modality' (i.e. presence / absence of yeast), 'bacterial strain' and their interaction were defined as fixed factors; 'experimental block' was defined as a random factor. A similar analysis was also carried out to test whether bacteria had variable effects on yeast growth. Finally, we tested whether numbers of yeast and bacterial cells correlated among bacterial treatments. To this end we used a symmetrical Major Axis regression with mean cell numbers per treatment as datapoints.

In a second stage, we investigated the variance of microbial cell numbers across twenty bacterial strains. We hence tested whether the relationship between the means and variances of bacterial densities followed a mixed-species power law (i.e. log 10 (variance) = a + b.log 10 (mean)) [START_REF] Taylor | Taylor's power law: order and pattern in nature[END_REF] and whether its parameters a and b depended on yeast presence. We hence used a linear model with 'modality', 'log 10 (mean)' and their interaction as fixed factors. We further included in early model formulations the quadratic term '[log 10 (mean)] 2 ' and its interaction with 'modality' to investigate deviation from a power law. A similar analysis was carried out to test whether the relationship between the means and variances of densities of the yeast Hanseniaspora uvarum in presence of the twenty bacterial strains followed a single-species power law (i.e. log 10 (variance) = a + b.log 10 (mean)). We used a linear model with 'log 10 (mean)' as a fixed factor. We further included in early model formulations the quadratic term '[log 10 (mean)] 2 ' to investigate deviation from a power law. We found no significant evidence of nonlinearity in the two relationships.

Analyzes were performed with JMP (SAS, 14.1) and R (3.6.2) (package lmodel2 [START_REF] Legendre | Package 'lmodel2'. Model II Regression[END_REF]).

Results

Interactive effects of symbionts on microbial numbers

Bacterial density was significantly greater on average across treatments in the presence of yeast than in its absence (F-test of modality main effect: F 1,229 = 4.02, p = 0.0463). However, the effect was modest-approximately 0.18 orders of magnitude (± 0.09 standard error of the mean)-compared to among-strain variation-approximately 3 orders of magnitude (F-test of bacterial treatment main effect: F 19,229 = 11.07, p < 0.0001) (Fig 1A). Even though some bacterial strains seemed to benefit more from yeast presence than others, the interaction between the presence / absence of the yeast and the bacterial strain identity was not significant (F-test of modality � bacterial treatment interaction: F 19,229 = 0.86, p = 0.6280) (Fig 1B).

Yeast density was significantly influenced by bacterial strain (F 19,110 = 4.12, p < 0.0001) (Fig 2A). Unfortunately, we did not measure yeast growth in the absence of bacteria. It was therefore not possible to assess whether bacteria had a generally beneficial or costly effect on the density of H. uvarum yeast. Overall, yeast and bacterial mean densities did not correlate significantly. Using Major Axis regression, the correlation coefficient between bacterial mean density and yeast mean density was 0.2198; the slope was 2.7458 (95% CI [-2.1477, 0.4665])

Relationships between cell numbers mean and variances

Overall, bacterial cell density across strains followed a spatial mixed-species TL. [log 10 (mean)] 2 main effect: F 1,34 = 0.06, p = 0.8152; F-test of modality � [log 10 (mean)] 2 interaction: F 1,34 = 3.06, p = 0.0891). The variance of bacterial density throughout the different microbial treatments therefore appeared to be consistent with a power function of mean bacterial density, as described by TL, and the parameters of TL were not affected by the presence of the yeast symbiont.

Taylor's law also appeared to hold in yeast densities (Fig 4). We found a linear positive relationship between log 10 (mean) and log 10 (variance) of yeast densities (F-test of log 10 (mean) interaction: F 1,18 = 251.49, p < 0.001). The slope of the regression was 1.828 (95% CI [0.3867, 3.2686]). The quadratic term '[log 10 (mean)] 2 ' was not significant (F-test of [log 10 (mean)] 2 : F 1,17 = 2.58, p = 0.1267).

Discussion

We demonstrated interactions between D. melanogaster's bacterial and yeast symbionts in ecologically realistic conditions. In our experiment, yeast presence slightly increased bacterial multiplication in fruit flesh infested with Drosophila larvae (Fig 1A). The twenty bacterial strains we tested also had variable effects on yeast multiplication (Fig 2). However, these interactive effects did not change the spatial Taylor's Law (TL): in the presence as in the absence of yeast, the variance of bacterial population density across the twenty bacterial strains was related to the mean of bacterial population density by a power law with a log-log slope indistinguishable from 2 (Fig 3). Interactions between microbial symbionts emerge as important factors affecting microbial dynamics [START_REF] Seth | Nutrient cross-feeding in the microbial world[END_REF][START_REF] Hassani | Microbial interactions within the plant holobiont[END_REF]. However, the nature and prevalence of interactions between symbionts of Drosophila larvae in ecologically realistic conditions had never been investigated to our knowledge, in particular with wild microbial strains. Our study hence sheds light on novel aspects of Drosophila symbiosis in the field. We found that yeast apparently benefited bacteria, although modestly (Fig 1). Besides, we observed large variation in Hanseniaspora uvarum yeast cell densities as a function of the strain of bacterium with which it shared the symbiotic environment (Fig 2A). These interactive effects may be the result of cross-feeding, which is wide-spread in symbiotic systems. It is well-established that yeast associated to Drosophila produce ethanol that is converted to acetic acid by Acetobacteraceae bacteria, a phenomenon referred to sourrot in farming [START_REF] Barata | New Insights into the ecological interaction between grape berry microorganisms and Drosophila flies during the development of sour rot[END_REF][START_REF] Rombaut | Invasive Drosophila suzukii facilitates Drosophila melanogaster infestation and sour rot outbreaks in the vineyards[END_REF]. A similar cross-feeding interaction occurs between Saccharomyces yeast and Lactobacillus bacteria [START_REF] Ponomarova | Yeast creates a niche for symbiotic lactic acid bacteria through nitrogen overflow[END_REF], which are among the most important bacterial members of the D. melanogaster microbiota [START_REF] Chandler | Bacterial communities of diverse Drosophila species: ecological context of a host-microbe model system[END_REF][START_REF] Staubach | Host species and environmental effects on bacterial communities associated with Drosophila in the laboratory and in the natural environment[END_REF]. In our experiment H. uvarum yeast may have provided nutrients to most of the bacterial strains, even though indirect effects through host physiology cannot be ruled out [START_REF] Sommer | Metabolic basis for mutualism between gut bacteria and its impact on the Drosophila melanogaster host[END_REF][START_REF] Ze ´le | Ecology and evolution of facilitation among symbionts[END_REF]. Independent of the mechanisms, interactive effects between symbionts may affect the dynamics of all partners at local and meta-population scales [START_REF] Gilbert | Dispersal of yeasts and bacteria by Drosophila in a temperate forest[END_REF][START_REF] Buser | Niche construction initiates the evolution of mutualistic interactions[END_REF].

Interactions between organisms and their density dependence may affect TL parameters, as genetics, ecology and other spatio-temporal factors do [START_REF] Kaltz | Bacterial microcosms obey Taylor's law: effects of abiotic and biotic stress and genetics on mean and variance of population density[END_REF][START_REF] Cohen | Taylor's law and related allometric power laws in New Zealand mountain beech forests: the roles of space, time and environment[END_REF][START_REF] Johnson | Biological and statistical processes jointly drive population aggregation: using host-parasite interactions to understand Taylor's power law[END_REF][START_REF] Anderson | Variability in the abundance of animal and plant-species[END_REF][START_REF] Mellin | Reef size and isolation determine the temporal stability of coral reef fish populations[END_REF][START_REF] Lagrue | Parasitism alters three power laws of scaling in a metazoan community: Taylor's law, density-mass allometry, and variance-mass allometry[END_REF]. Kilpatrick and Ives [START_REF] Kilpatrick | Species interactions can explain Taylor's power law for ecological time series[END_REF] predicted that the strength of competition between species would affect the slope in temporal versions of TL. A previous study tested a modification of this prediction (for spatial TL instead of temporal TL) with free-living bacteria that were grown alone or in competition in artificial environments of variable nutrient richness [START_REF] Ramsayer | Taylor's Law holds in experimental bacterial populations but competition does not influence the slope[END_REF]. Though competition did occur between the tested bacteria, it did not change the slopes of the spatial TL from 2, with or without competition. Here, we pursued this investigation further with symbiotic microorganisms in ecologically realistic conditions that facilitated each other rather than competed. Our results showed the variance of bacterial population density in different replicates related to the mean of bacterial population density by a power law consistent with TL. As in the case of competition [START_REF] Ramsayer | Taylor's Law holds in experimental bacterial populations but competition does not influence the slope[END_REF], we found no significant effect of facilitation on the form or parameters of a spatial TL (Fig 3). The discrepancy between our results for a spatial TL and Kilpatrick and Ives's predictions [START_REF] Kilpatrick | Species interactions can explain Taylor's power law for ecological time series[END_REF] about the effect of competition on a temporal TL highlights the importance of details in experimental tests of theoretical predictions and leaves open the challenge of finding experimental conditions suitable to test Kilpatrick and Ives's predictions.

Another theoretical study showed that biological replicate numbers affect the parameters of TL [START_REF] Giometto | Sample and population exponents of generalized Taylor's law[END_REF]. TL's slopes different from 2 may be undetectable if a stochastic multiplicative growth process in a Markovian environment (e.g. bacterial multiplication) is observed for a duration that exceeds the natural logarithm of the number of biological replicates. In our experiment, bacterial numbers were on average 1000 times that of the inoculum, implying that bacteria replicated at least ten times, since 2 10 � 1000. Cell divisions likely exceeded 10 cycles as most bacteria in a closed system quickly move from an exponential growth phase to a plateau phase, where bacterial density is constant because appearing cells and dying cells are in equal numbers. We had seven replicates, and a minimum of 10 replication cycles is larger than ln7 � 1.95. The slopes indistinguishable from 2 that were estimated in our study, and those of Ramsayer et al. [START_REF] Ramsayer | Taylor's Law holds in experimental bacterial populations but competition does not influence the slope[END_REF], may thus be statistical artifacts due to the long duration of the experiments relative to the number of replicates. Computing a power analysis with TL parameters from our study revealed the minimum number of replicates needed to observe a significant change in the intercept and the slope would have been approximately 100 and 20 000 respectively. Such replication is very challenging to reach in most experimental systems. It is possible that interspecific interactions such as competition [START_REF] Ramsayer | Taylor's Law holds in experimental bacterial populations but competition does not influence the slope[END_REF] and facilitation (the present study) affect TL's slope during relatively short time frames (e.g. a few hours for rapidly multiplying organisms such as bacteria). Unveiling the ecological mechanisms that alter TL's parameters using microbial organisms remains an exciting area of research that will necessitate experimental designs adapted to the specific demographic features of microorganisms.

Our study explored interactions between microbial symbionts associated with D. melanogaster larvae in ecologically realistic conditions. Screening twenty natural isolates of bacteria in the presence and absence of a freshly collected strain of the yeast H. uvarum, we found yeast had a small facilitative effect (Fig 1A ) that did not differ statistically among bacteria (Fig 1B). Bacterial identity had a large influence on yeast multiplication (Fig 2). These interactions between symbionts demonstrate this phenomenon in natural conditions in D. melanogaster, a key model system of symbiosis studies. Effects of yeast on bacteria did not significantly affect the parameters of the spatial version of TL. TL is a powerful means of investigating the forces that affect the demography of many species, however challenging it may be to use TL in microbial systems.

Fig 1 .Fig 2 .

 12 Fig 1. Effect of yeast on bacterial numbers in fruit across all bacterial strains (A) and by bacterial strain (B). Bacterial densities are expressed in log 10 of colony forming units (CFU) per milliliter of fruit homogenate. At the time of inoculation, log 10 bacterial density was 2.7 on the Y axis. Error bars indicate standard errors around the mean. https://doi.org/10.1371/journal.pone.0242692.g001

Fig 3 .

 3 Fig 3. holds in bacterial symbionts of Drosophila, and TL parameters are not significantly affected by yeast presence. Linear regressions of log 10 (variance) on log 10 (mean) for the densities of each bacterial strain in the absence (solid blue and the presence (dashed red line) of yeast. Each point corresponds to one bacterial strain in the absence (blue filled circle •) or presence (red open circle o) of yeast. https://doi.org/10.1371/journal.pone.0242692.g003

Fig 4 .

 4 Fig 4. law holds in a Hanseniaspora uvarum yeast strain in presence of different bacterial symbionts. Linear regression of log 10 (variance of yeast density) on log 10 (mean of yeast density). Each point corresponds to a yeast density for a given bacterial strain. https://doi.org/10.1371/journal.pone.0242692.g004

Table 1 . Bacterial strains used in this study.

 1

	Strain	Species	Origin	GenBank accession number	Type of agar plate	Temperature of incubation
	R3b	Gluconobacter sp.	Grape berry	not referenced	MRS	24˚C
	R6b	Staphylococcus sp.	Grape berry	not referenced	TCS	24˚C
	R8b	Gluconobacter thailandicus	Grape berry	not referenced	TCS	24˚C
	Dm2b	Yersinia sp.	D. melanogaster	not referenced	TCS	24˚C
	Dm5b	Gluconobacter sp.	D. melanogaster	not referenced	TCS	24˚C
	Dm6b	Escherichia coli	D. melanogaster	not referenced	TCS	24˚C
	Dm8b	Enterobacter sp.	D. melanogaster	not referenced	TCS	24˚C
	Dm10b	Erwinia sp.	D. melanogaster	not referenced	MRS	24˚C
	Dm11b	Enterobacteriaceae	D. melanogaster	not referenced	MRS	24˚C
	Ds3b	Serratia fonticola	D. suzukii	not referenced	TCS	24˚C
	Ds4b	Gluconobacter kondonii	D. suzukii	not referenced	TCS	24˚C
	Ds6b	Lelliottia jeotgali	D. suzukii	not referenced	TCS	24˚C
	Ds9b	Erwinia injecta	D. suzukii	not referenced	TCS	24˚C
	Ds10b	Lelliottia jeotgali	D. suzukii	not referenced	TCS	24˚C
	Ds25b	Lelliottia jeotgali	D. suzukii	not referenced	MRS	24˚C
	Ds27b	Serratia liquefaciens	D. suzukii	not referenced	MRS	24˚C
	Ds28b	Lelliottia sp.	D. suzukii	not referenced	MRS	24˚C
	Lp WJL	Lactobacillus plantarum	D. melanogaster	EU096230 [27]	MRS	35˚C
	Lb WJL	Lactobacillus brevis	D. melanogaster	EU096227 [27]	MRS	35˚C
	Ap WJL	Acetobacter pomorum	D. melanogaster	EU096229 [27]	MAN	35˚C

https://doi.org/10.1371/journal.pone.0242692.t001

  Fig 2B). Using OLS regression, the slope of yeast density as a function of bacterial density was 0.2847 (95% CI [-0.3411, 0.9104]) and the slope of bacterial density as a function of yeast density was 0.1697 (95% CI [-0.2033, 0.5426]), neither of which was significantly different from zero. Thus Major Axis regression and OLS regression consistently reported no correlation significantly different from zero.

https://doi.org/10.1371/journal.pone.0242692.g002 (

  The slopes of the regression of log(variance bacterial density) on log(mean bacterial density) was 2.044 (95% CI [1.7601, 2.3284]) in the absence of yeast, and 2.062 (95% CI [1.7742, 2.3489]) in its presence. The intercepts and the slopes of the two linear regressions were not significantly different (Ftest of modality main effect: F 1,36 = 1.56, p = 0.2199; F-test of modality � log 10 (mean) interaction: F 1,36 = 0.01, p = 0.9290) (Fig3). The coefficient of the quadratic term '[log 10 (mean)] 2 ' and the interaction of the quadratic term with yeast presence or absence were not significant (F-test of

PLOS ONE | https://doi.org/10.1371/journal.pone.0242692November 23, 2020

Acknowledgments

We thank Guido Favia and two anonymous reviewers for comments on an earlier version of the manuscript. We thank Edouard Jurkevitch and Elodie Vercken for insightful comments. We thank Anne Xue ´reb for her help at the beginning of the experiment and Laure Benoit for her help with the molecular identification of the bacterial strains.

This project was supported by French National Research Agency through the 'SWING' project (ANR-16-CE02-0015).

The dataset is available in the open data repository Zenodo (DOI: 10.5281/zenodo.3628674).

Author Contributions

Conceptualization: Robin Guilhot, Simon Fellous.

Data curation: Robin Guilhot, Joel E. Cohen.


J Dupré De Baubigny

P Perrin

N Pantoustier

T Salez

M Reyssat

C Monteux
email: cecile.monteux@espci.fr

Growth mechanism of polymer membranes obtained by Hbonding across immiscible liquid interfaces

Keywords: membrane, polymer, complexation, interface, hydrogen bond

Complexation of polymers at liquid interfaces is an emerging technique to produce all-liquid printable and self-healing devices and membranes. It is crucial to control the assembly process but the mechanisms at play remain unclear. Using two different reflectometric methods, we investigate the spontaneous growth of H-bonded PPO-PMAA (Poly Propylene oxide-Poly metacrylic acid) membranes at a flat liquid-liquid interface. We find that the membrane thickness h grows with time t as ht 1/2 , which is reminiscent of a diffusion-limited process. However, counter-intuitively, we observe that this process is faster as the PPO molar mass increases. We are able to rationalize these results with a model which considers the diffusion of the PPO chains within the growing membrane. The architecture of the latter is described as a gel-like porous network, with a pore size much smaller than the radius of the diffusing PPO chains, thus inducing entropic barriers that hinder the diffusion process. From the comparison between the experimental data and the result of the model, we extract some key piece of information about the microscopic structure of the membrane. This study opens the route toward the rational design of self-assembled membranes and capsules with optimal properties.

Complexation of polymers, surfactants or particles at immiscible liquid interfaces is an increasingly popular technique to produce all-liquid printable, reconfigurable and self-healing membranes, devices and capsules [START_REF] Forth | Building Reconfigurable Devices Using Complex Liquid-Fluid Interfaces[END_REF][START_REF] Capito | Self-Assembly of Large and Small Molecules into Hierarchically Ordered Sacs and Membranes[END_REF][START_REF] Forth | Reconfigurable Printed Liquids[END_REF][START_REF] Feng | Harnessing liquid-inliquid printing and micropatterned substrates to fabricate 3-dimensional all-liquid fluidic devices[END_REF][START_REF] Inostroza-Brito | Co-assembly, spatiotemporal control and morphogenesis of a hybrid protein-peptide system[END_REF][START_REF] Kim | One-Step Generation of Multifunctional Polyelectrolyte Microcapsules via Nanoscale Interfacial Complexation in Emulsion (NICE)[END_REF][START_REF] Hann | One-Step Generation of Cell-Encapsulating Compartments via Polyelectrolyte Complexation in an Aqueous Two Phase System[END_REF][START_REF] Xu | Interfacial Assembly and Jamming of Polyelectrolyte Surfactants: A Simple Route To Print Liquids in Low-Viscosity Solution[END_REF][START_REF] Monteillet | Complex coacervates formed across liquid interfaces: A self-consistent field analysis[END_REF][START_REF] Gunes | Tuneable thickness barriers for composite o/w and w/o capsules, films, and their decoration with particles[END_REF][START_REF] Xie | Interfacial rheological properties of self-assembling biopolymer microcapsules[END_REF][START_REF] Kim | pH-Induced Softening of Polyelectrolyte Microcapsules without Apparent Swelling[END_REF] . While layer-by-layer assembly of components at liquid interfaces enables to obtain good control over membrane thickness and composition [START_REF] Grigoriev | New Method for Fabrication of Loaded Micro-and Nanocontainers: Emulsion Encapsulation by Polyelectrolyte Layer-by-Layer Deposition on the Liquid Core[END_REF][START_REF] Tirilly | Interplay of Hydrogen Bonding and Hydrophobic Interactions to Control the Mechanical Properties of Polymer Multilayers at the Oil-Water Interface[END_REF][START_REF] Tirilly | Interfacial Rheology of Hydrogen-Bonded Polymer Multilayers Assembled at Liquid Interfaces: Influence of Anchoring Energy and Hydrophobic Interactions[END_REF] , an easier way to promote interfacial complexation is to dissolve the interacting species within two separate liquid phases, such as oil and water [START_REF] Kim | One-Step Generation of Multifunctional Polyelectrolyte Microcapsules via Nanoscale Interfacial Complexation in Emulsion (NICE)[END_REF][START_REF] Monteillet | Complex coacervates formed across liquid interfaces: A self-consistent field analysis[END_REF][START_REF] Gunes | Tuneable thickness barriers for composite o/w and w/o capsules, films, and their decoration with particles[END_REF][START_REF] Xie | Interfacial rheological properties of self-assembling biopolymer microcapsules[END_REF][START_REF] Kaufman | Soft microcapsules with highly plastic shells formed by interfacial polyelectrolytenanoparticle complexation[END_REF][START_REF] Kaufman | Single-step microfluidic fabrication of soft monodisperse polyelectrolyte microcapsules by interfacial complexation[END_REF][START_REF] Monteillet | Charge-driven co-assembly of polyelectrolytes across oil-water interfaces[END_REF][START_REF] Dupré De Baubigny | One-Step Fabrication of pH-Responsive Membranes and Microcapsules through Interfacial H-Bond Polymer Complexation[END_REF][START_REF] Steinschulte | Interface-enforced complexation between copolymer blocks[END_REF][START_REF] Bimendina | Polyelectrolyte Complexes on a Dimeric Interface[END_REF] or two partially-miscible phases [START_REF] Capito | Self-Assembly of Large and Small Molecules into Hierarchically Ordered Sacs and Membranes[END_REF][START_REF] Inostroza-Brito | Co-assembly, spatiotemporal control and morphogenesis of a hybrid protein-peptide system[END_REF][START_REF] Hann | AWE-somes: All Water Emulsion Bodies with Permeable Shells and Selective Compartments[END_REF][START_REF] Gazizov | Interpolymer complexes of some vinyl copolymers in a solution and on the boundary of two liquid phases[END_REF][START_REF] Michaels | POLYCATION-POLYANION COMPLEXES: PREPARATION AND PROPERTIES OF POLY-(VINYLBENZYLTRIMETHYLAMMONIUM) POLY-(STYRENESULFONATE)[END_REF] . As the components spontaneously diffuse towards the common interface, they selfassemble through non-covalent interactions such as electrostatic ones and form a membrane which grows over time up to micrometric thicknesses [START_REF] Gunes | Tuneable thickness barriers for composite o/w and w/o capsules, films, and their decoration with particles[END_REF][START_REF] Xie | Interfacial rheological properties of self-assembling biopolymer microcapsules[END_REF] . However, the current lack of understanding of the mechanisms at play during the assembly process hinders the development of membranes with controlled structure and properties. Indeed the very few experimental results available concerning the kinetics of growth of self-assembled membranes [START_REF] Gunes | Tuneable thickness barriers for composite o/w and w/o capsules, films, and their decoration with particles[END_REF][START_REF] Mendoza-Meinhardt | A fluidic device for the controlled formation and real-time monitoring of soft membranes self-assembled at liquid interfaces[END_REF] do not provide a clear microscopic picture of the assembly process. Capito et al. [START_REF] Capito | Self-Assembly of Large and Small Molecules into Hierarchically Ordered Sacs and Membranes[END_REF] who was the first to assemble membranes using peptides and polysaccharides of opposite charge, assumed that the membrane growth was controlled by the diffusion of small peptides through the growing peptide-polysaccharide membrane. However the results obtained later by Mendoza-Meinhardt et al. [START_REF] Mendoza-Meinhardt | A fluidic device for the controlled formation and real-time monitoring of soft membranes self-assembled at liquid interfaces[END_REF] for a similar system [START_REF] Inostroza-Brito | Co-assembly, spatiotemporal control and morphogenesis of a hybrid protein-peptide system[END_REF] where inconsistent with a diffusion-limited process. The diffusion of molecules in polymer networks has been the object of a large amount of theoretical and experimental studies, but is still an unsolved question [START_REF] Brochard | Polymer-polymer diffusion in melts[END_REF][START_REF] Masaro | Physical models of diffusion for polymer solutions, gels and solids[END_REF][START_REF] Deutsch | Interdiffusion and self-diffusion in polymer mixtures: A Monte Carlo study[END_REF][START_REF] Sillescu | Relation of interdiffusion and self-diffusion in polymer mixtures[END_REF][START_REF] Altenberger | On the theory of self-diffusion in a polymer gel[END_REF][START_REF] White | Diffusion in a crosslinked acrylamide polymer gel[END_REF][START_REF] Liu | Entropic trapping of macromolecules by mesoscopic periodic voids in a polymer hydrogel[END_REF][START_REF] Amsden | Solute Diffusion within Hydrogels. Mechanisms and Models[END_REF] . In this Letter, we use interferometric in-situ and ex-situ measurements to follow the thickness evolution of a model self-assembled interfacial membrane obtained from the H-bond complexation of poly(methacrylic acid) (PMAA) and poly(propylene oxide) (PPO) at a flat isopropylmyristate (IPM)-water interface. We showed recently that this system enables to obtain highly-stable oil-water emulsions using a simple rotor-stator emulsification technique [START_REF] Dupré De Baubigny | One-Step Fabrication of pH-Responsive Membranes and Microcapsules through Interfacial H-Bond Polymer Complexation[END_REF] . We measure the membrane thickness as a function of time during its spontaneous

IPM water

O HO CH 3 n O CH 3 n
complexation, for various polymer concentrations and molar masses. We find that the assembly process is diffusion-limited, controlled by both the PPO concentration and molar mass. Our measurements show that the diffusion coefficients of the PPO chains in the growing membrane are extremely slow. To account for these results, we suggest that the diffusion of free PPO chains within the growing membrane is hindered by entropic barriers due to the low mesh size of the polymer network. A minimal model including this assumption enables us to rationalize all the macroscopic data and extract some key information about the microscopic structure of the PPO-PMAA membrane thus opening the way towards its optimal design.

To obtain insight into the mechanisms at play during the growth of polymer membranes at immiscible liquid interfaces we choose to work with two polymers that interact through hydrogen bonds, poly(methacrylic acid) (PMAA) as a H-bond donor and poly(propylene oxide) (PPO) as a H-bond acceptor (Figures 1a and1b). We dissolve both polymers in two immiscible phases, the PPO in isopropylmyristate (IPM) and the PMAA in water. When the two polymer phases are put into contact in a container a membrane instantaneously forms at the flat IPM/water interface (Figure 1c), which thickness time evolution is measured using two different interferometric methods. Briefly we perform an in situ measurement at a flat IPM/water interface using an optical spectrometer (Specim V8E) to measure the wavelength dependence of the light intensity reflected by the thin interfacial membrane and an ex-situ method consisting in removing the membrane from the liquid and leaving it on a glass slide and measuring its thickness with an optical interferometric profilometer (Microsurf 3D Fogale Nanotech). In Figure 2a, we quantitatively investigate the growth of the PPO-PMAA membrane at the flat IPM-water interface, for 3000 g/mol and 100 000 g/mol (Figure 2a) at varying weight fractions. In Figure 2b, we report on the membrane growth for several PPO and PMAA molar masses with a weight percentage fixed to 1 wt% for both polymers. For all the experimental conditions, the membrane thickness is found to scale with time as  , over up to four decades in time, which suggests the existence of an underlying diffusive process characterized by a diffusion coefficient that remains constant over the course of each experiment. These results differ from those presented in the works of Gazizov [START_REF] Gazizov | Interpolymer complexes of some vinyl copolymers in a solution and on the boundary of two liquid phases[END_REF] and Mendoza-Meinhardt [START_REF] Mendoza-Meinhardt | A fluidic device for the controlled formation and real-time monitoring of soft membranes self-assembled at liquid interfaces[END_REF] , where the thickness growth was observed to reach a plateau after a few minutes or hours respectively. The effective diffusion coefficients obtained from the experimental curves using a fit of the form , range between 10 -19 m 2 /s and 10 -17 m 2 /s depending on the experimental conditions (Table 1, third column). These values are six to eight orders of magnitude lower than the bulk diffusion coefficients expected for PPO in IPM or PMAA in water. Extremely low effective diffusion coefficients were also measured by Gunes et al. for a chitosan-phospholipid membrane growing at an oil-water interface [START_REF] Gunes | Tuneable thickness barriers for composite o/w and w/o capsules, films, and their decoration with particles[END_REF] . These low values of are inconsistent with the free diffusion of the polymers in the bulk phases and we therefore suggest that the diffusion of the polymers through the growing membrane controls the assembly process. Indeed, the H-bond complexation between PMAA and PPO at the water-IPM interface is fast, and a thin membrane is almost instantaneously formed as soon as the two phases are put in contact. To induce further membrane growth, it is then reasonable to expect that the PPO chains (respectively the PMAA chains) in the organic IPM phase (resp. the aqueous phase) have to diffuse through the membrane to reach the aqueous phase (resp. the organic phase) where they can complex with PMAA (resp. PPO). We further see from Figure 2a that increasing the bulk PMAA concentration in the aqueous phase does not influence the membrane growth, while increasing the PPO concentration in the organic phase leads to thicker membranes. Consistently, the values of the effective diffusion coefficient, , obtained from Figure 2a and reported in Table 1 (fourth column) increase with the bulk PPO concentration. This suggests that only the PPO chains diffuse through the PPO-PMAA membrane before reaching the aqueous phase. Consistently we find that the PMAA molar mass does not influence the growth of the membrane (Figure 2b). However increasing the PPO molar mass leads to thicker membranes and to a higher value of , 2) and (Equation 3) respectively, as obtained from Figure 2. The last column represents the pore size (nm) of the PPO-PMAA membrane estimated from Equation 5, using the best-fit parameters from the comparison between all the data and Equation 6.

which
To summarize, our experimental results show that the growth kinetics of the membrane is controlled by both the PPO molecular mass and concentration, and that the PMAA molar mass and concentration do not influence the process. These experimental results suggest that the membrane-assembly process is limited by the diffusion of the PPO chains through the growing membrane. In the following, we will therefore consider the diffusive transport of PPO chains through a membrane of thickness along the direction, and composed of complexed PMAA and PPO chains (Figure 3a). At the membrane-water interface ( ), the free PPO chains interact and complex with the free PMAA chains, leading to membrane growth. This implies a smaller free PPO concentration at the membrane-water interface with respect to the bulk value , and thus the existence of a gradient of PPO

with the volume of a PPO molecule.

z 0 h(t) PPO + IPM problem C PPO bulk C PPO (z) C PPO z= h ( t ) Membrane PMAA + H20
Equation 1 can be integrated, under the initial condition, and the solution reads: ,

( 1 shows that the diffusion coefficient of PPO decreases as the bulk PPO concentration increases.

To understand further the values and dependencies of the microscopic diffusion coefficient , we have to take into account the fact that the free PPO chains diffuse in a complex polymer network constituting the membrane. Moreover, another difficulty arises from the fact that the microscopic structure of the latter is unknown and remains an open question in the literature. Nevertheless, one can invoke a few minimal intuitive assumptions. First of all, the solid PPO-PMAA matrix of the membrane is likely to be a porous network, with a typical pore size that can a priori depend on the PPO concentration and molar mass. Secondly, we suggest that the polymer matrix is dense, i.e. is small compared to the Flory radius of the free PPO molecules. This implies the existence of entropic barriers hindering the diffusion of the PPO molecules within the membrane [START_REF] Liu | Entropic trapping of macromolecules by mesoscopic periodic voids in a polymer hydrogel[END_REF] . Thirdly, multiplying the Zimm bulk diffusion coefficient for real chains diluted in a good solvent, to an Arrhenius factor involving an elastic barrier of entropic origin, one arrives at the following expression:

, ( 4
)
with the monomeric Stokes-Einstein diffusion coefficient in a medium of viscosity , at temperature T K, and the Boltzmann constant. Finally, since the membrane is composed of short PPO chains bridging long PMAA chains together, we might expect that the typical pore size scales like the PPO size, , with a concentration-dependent correction prefactor accounting for the actual fraction of PPO crosslinkers. Assuming the corrective prefactor to be a power law with an unknown exponent , one gets the Ansatz:

       3 5 / 3 a C N a PPO bulk PPO , (5)
where is a dimensionless numerical constant. Combining Equations 4 and 5, we finally obtain the following prediction: By fitting the data of Table 1 to Equation 6, one gets a good agreement as shown in Figure 3b,obtained for . From the affine fit, we deduce the values: and 2.8 10 - 10 m 2 /s. The value of  calculated from Equation 5(Table 1, last column), using those best-fit parameters, ranges between 5 10 -10 and 6 10 -10 m for the large polymer chains (R F =25 10 -10 m), and is around 10 -10 m for the small polymer chains (R F =6.4 10 -10 m), which are all selfconsistently smaller than the PPO molecular size. We note that the values of are on the same order of magnitude as the persistence length expected for both polymers [START_REF] Hezaveh | Molecular dynamics simulation study of solvent effects on conformation and dynamics of polyethylene oxide and polypropylene oxide chains in water and in common organic solvents[END_REF][START_REF] Lee | Molecular Dynamics Studies of Polyethylene Oxide and Polyethylene Glycol: Hydrodynamic Radius and Shape Anisotropy[END_REF][START_REF] Mark | The Configuration of the Polyoxyethylene Chain[END_REF][START_REF] Ortiz | Entropic Elasticity of Single Polymer Chains of Poly(methacrylic acid) Measured by Atomic Force Microscopy[END_REF] . We also stress that the value of is comparable to the free diffusion coefficient of a monomer in a pure solvent, on the order of 10 -10 m 2 /s.

Increasing the bulk PPO concentration, and thus the crosslinking fraction in the membrane, leads to a decrease of the average pore size, as expected. Indeed, a NRM measurement (not shown) of the membrane composition showed that the polymer stoichiometry of the membrane is identical to the bulk stoichiometry. As such, increasing the PPO concentration in the IPM phase leads to an increased concentration of PPO in the membrane, and thus a decrease of the pore size. Furthermore, the above values of are consistent with a microscopic membrane architecture where the PPO chains act as macromolecular crosslinkers with PPO monomeric units sticking to the PMAA chains and bridging them together in a zip-like fashion. In this picture, the unbounded PPO units constitute the porous network of the membrane (see Figure 3c). To confirm independently these low values of , we invoke interfacial-rheology measurements [START_REF] Dupré De Baubigny | One-Step Fabrication of pH-Responsive Membranes and Microcapsules through Interfacial H-Bond Polymer Complexation[END_REF] previously reported in our group for the shear elastic modulus of these membranes, on the order of 13 MPa. The shear elastic modulus, which scales as G ≈kT/ 3 , enables us to estimate that  is on the order of 5 10 -10 m, consistently with the values obtained above. Altogether, from this minimal model, one is able to discuss the microscopic structure of the membrane based on simple macroscopic measurements only.

CONCLUSION

Using spectrometry and optical profilometry, we measure the growth kinetics of a PPO-PMAA membrane, resulting from H-bonding complexation at the water-Isopropyl Myristate interface. We find that the thickness of the membrane scales as the square root of time, consistently with a diffusive process. Systematic measurements for varying PPO and PMAA molar masses and concentrations lead us to the conclusion that the complexation process is limited by the diffusion of the PPO chains through the growing membrane. From the macroscopic growth curves we obtain the diffusion coefficient of the PPO chains in the membrane and find that it decreases with the PPO concentration and increases with the PPO molar mass. To rationalize these counter-intuitive observations, we model the process by considering that the growing membrane is a gel-like porous network, with a pore size smaller than the radius of the diffusing PPO chains, thus inducing entropic barriers which hinder the macromolecular diffusion. Moreover, we consider that the pore size of the membrane depends on the PPO concentration and molar mass. By fitting the experimental diffusion coefficients with the model prediction, we are able to deduce that the pore size of the growing membrane decreases with the PPO concentration and increases with the PPO molar mass, consistently with a membrane structure where the PMAA chains are bridged in a zip-like fashion by PPO molecules acting as macromolecular cross-linkers. Our model therefore enables us to discuss the microscopic structure of the PPO-PMAA membrane from macroscopic measurements only. This study opens the route toward a rational design of self-assembled membranes and capsules with optimal properties.

EXPERIMENTAL SECTION

The aqueous PMAA solutions are prepared by dissolving 0.001 to 1 wt% of PMAA in purified water from a milli-Q apparatus (Millipore) and the is adjusted to by adding drops of a HCl (Sigma-Aldrich) solution concentrated at Using SEC in THF we find that PPO 4000 has a number averaged molar mass M n =3000 and PDI = 1.97. We note that PPO with higher molar masses are not available with common chemical suppliers. As SEC is not suitable for short chains such as PPO 400, we analyzed it using GC-MS measurements. The measurements lead a molar mass of 360 g/mol. Despite the fact that this technique does not allow for the determination of a molecular weight distribution it can however can be reasonably assumed that M w = M n. We therefore assume that M w = M n = 360 g/mol.

The interfacial complexation and the membrane are obtained by putting the two polymer phases in contact (Figure 1c). The membrane thickness is measured using two different methods. The first method consists in an in situ measurement of the thickness of the membrane assembled at a flat IPM/water interface. Briefly, we use a reflected light microscope mounted with an optical spectrometer (specim V8E) connected to a camera. We focus white light on the oil-water interface where the membrane grows and the spectrometer provides the wavelength dependent intensity reflected by the membrane from which we deduce the membrane thickness. We note that this technique is not suitable for thicknesses below 300 nm, for which it is difficult to detect the interference fringes. The second method consists in removing the membrane from the liquid and leaving it on a glass slide and measuring its thickness ex situ, with an optical interferometric profilometer (Microsurf 3D Fogale Nanotech).

Figure 1 .

 1 Figure 1. Chemical formulas of a. Poly(Propylene Oxide) (PPO) and b. Poly(Methacrylic

Figure 2 .

 2 Figure 2. a. Thickness of the PPO-PMAA membrane at the IPM-water interface as a function of time,

  molecular concentration across the membrane. The molecular flux along of PPO chains diffusing through the membrane from the IPM phase to the water phase thus reads , with PPO m D the diffusion coefficient of the free PPO chains in the membrane.Assuming an almost instantaneous complexation at the membrane-water interface, we can further write , and consider the membrane growth as a relatively slow, diffusion-limited process with a quasi-steady concentration profile . The diffusive flux across the membrane therefore becomes , and the governing equation for the evolution of the membrane thickness is: ,

Table 1 .

 1 Effective and microscopic diffusion coefficients of PPO chains inside the membrane,

		wt %				
			x10 24	x10 -17 m 2 /s x10 -15 m 2 /s	nm
			molecules/m 3
	3000 g/mol	0.001 0.002	0.04	4.55	0.59
	N PPO = 52	0.01	0.02	0.14	1.70	0.55
		0.1	0.2	0.33	0.39	0.51
		1	2.0	1.4	0.17	0.48
	360 g/mol	1	17	0.15	0.11	0.12
	N PPO = 6

may seem counter-intuitive at first sight as diffusion coefficients of macromolecules are expected to decrease with molar mass.

(Equation

  Molar masses and polydispersity indexes (PDI) were measured on a Size Exclusion Chromatography (SEC, Viscotek GPC max VE 2001; TDA 302 triple detector array, system with triple detection using 0.2 M NaNO3 aqueous solution or THF containing 2% trimethylamine for PMAA and PPO samples respectively). Using SEC in aqueous solution, we measure for PMAA 100K a number averaged molar mass M n = 100000 g/mol and PDI around 2. PMAA 10K is purchased under the form of a sodium salt solution at a concentration of 30 wt%, which is then dialyzed against water at pH=3 for four weeks to obtain it under a PMAAH form and is then lyophilized. Using SEC in aqueous solution we measure 11000 g/mol and PDI = 1.82. The oil-based solution is prepared by dissolving 0.001 to 1 wt% of PPO in Isopropyl Myristate, IPM (Sigma-Aldrich). We use two PPO from Sigma-Aldrich with product names PPO 400 and PPO 4000, which are used without any further purification.

		1 mol/L, and measured with a
	-meter (	M 250 ion analyser, Meterlab, Radiometer Copenhagen). We used two PMAA

samples, sold as PMAA 100 K from Polysciences and PMAA 10K from Sigma Aldrich.

ACKNOWLEDGEMENTS

The authors acknowledge the ANR JCJC INTERPOL grant number ANR-12-JS08-0007 for funding. They also acknowledge Alesya Mikhailovskaya and Corentin Trégouët for fruitful discussions, Julie Brun and Charlotte Demonsant for their help with the thickness measurements, as well as Mohamed Hanafi for performing the SEC measurements of the polymers.


Patric Mantaropoulos

Florine Giraud
email: florine.giraud@lecnam.net

Brice Tremeac

Pascal Tobaly

Water boiling at low pressure: dynamics of growth and bursting of bubbles in a plate-type evaporator

Keywords: Refrigeration, Plate-type evaporators, Water, Bubbles, Low pressure, R-718

Water is a good candidate to be used as an eco-friendly refrigerant. However, its behaviour under subatmospheric pressure needs to be better understood to optimize the efficiency and sizing of compact evaporators. Previous studies conducted in a plate-type evaporator highlighted that after the bubbles burst, the formation and evaporation of a thin liquid film on the evaporator wall has a major contribution on heat transfer. This work thus aims to characterize the dynamics of vapour bubbles during nucleate boiling, and more specifically the projection height after the free surface breakup. Using previous high-speed visualizations, the bubbles' characteristics are computed thanks to an ad-hoc Python program. It is shown that the projection height is correlated with the growth velocity and the size of the bubble reached just before it bursts. The influence of relevant dimensionless numbers on the projection height is also studied to determine the underlying predominant forces.

INTRODUCTION

Many commonly used refrigerants are subject to restrictive standards due to their environmental impact. In the future, these restrictions are to be tightened. Alternatives must be found, and water could be one of them. However, using water in refrigeration installations requires to work at pressures close to the triple point (0.6 kPa). In these conditions, the hydrostatic pressure can no longer be neglected and thus induces an inhomogeneity in the boiling environment. In the case of a plate-type vertical evaporator, [START_REF] Giraud | Experimental study of water vaporization occurring inside a channel of a smooth plate-type heat exchanger at subatmospheric pressure[END_REF] highlighted that most of the heat transfer during boiling is due to the vaporisation of a thin liquid filmformed by growth and bursting of vapour bubbles.

Boiling and bubble growth have been theorized for over a century. From the first modelling of a bubble's radius by [START_REF] Rayleigh | VIII. On the pressure developed in a liquid during the collapse of a spherical cavity[END_REF] to the developments of [START_REF] Plesset | The Growth of Vapor Bubbles in Superheated Liquids[END_REF] and [START_REF] Forster | Growth of a Vapor Bubble in a Superheated Liquid[END_REF] to the general model proposed by [START_REF] Mikic | On bubble growth rates[END_REF] and the simulations of [START_REF] Robinson | The dynamics of spherical bubble growth[END_REF], among others, knowledge about bubbles behaviour in a variety of environments has not stopped extending. It is therefore well accepted that bubble growth is divided into three main regions, characterized by the phenomenon limiting the growth: the surface tension-, inertia-and diffusion-controlled growth regions. Yet, sub-atmospheric pressure boiling environments received little attention until now. Bubbles behave in a completely different way in these conditions: [START_REF] Van Stralen | Bubble growth rates in nucleate boiling of water at subatmospheric pressures[END_REF] brought to light that in pool boiling under sub-atmospheric pressure, bubbles grow mushroom-shaped and bigger than in atmospheric conditions. Most of the models developed in the past consider spherical or semi-spherical bubbles and do not consider the specificities of the boiling environment.

Thus, in order to study the parameters impacting the creation of the liquid film in a plate-type vertical evaporator, a characterization of the bubble growth and bursting dynamics based on previous experimental results is conducted. After a part dedicated to the presentation of the experimental bench, the data processing method is explained and the factors impacting the projection height of the droplets forming the liquid film are discussed.

EXPERIMENTAL BENCH

The evaporator

The main part of the bench is a plate-type vertical evaporator, illustrated in Fig. 1. It is made up of three vertical plates, forming two channels in which circulate the primary and secondary fluid (water). The plates are made of stainless steel, except for the front one, which is made of PMMA to visualise the flow in the channel, as shown on Fig. 2. The separation plate thickness is 6mm.

On the refrigerant side, the thickness of the channel through which the refrigerant flows (the confinement thickness) can be set at 2, 4 or 6 mm (depending on the O-ring and spacer selected). The outer plate is transparent (PMMA), which allows boiling phenomena to be visualised by means of a high-speed camera (Phantom VEO410). Acquisitions are made at 3000 frames per second with a resolution of 1280 x 800 and an exposure time of 191.8 µs.

The channel is fed from below by three 2 mm diameter tubes. The flow rate is set by means of a mass flow controller, which is controlled according to the desired liquid level in the evaporator. The water then leaves the evaporator as steam through a slit at its upper end.

The secondary fluid circulates in co-current at a flow rate of about 1 L/min. It enters the evaporator at the selected temperature by means of a thermostatic bath (set point between 11 and 28 °C). At the outlet, the measured temperature is between 10 and 21 °C. The channel thickness is set at 1 mm.

The bench

For the evaporator to operate continuously, the steam coming out of it is directed to a condenser. The condenser consists of a cylinder of 720 mm height and 300 mm inside diameter. A copper tubular exchanger (in which water circulates at a temperature imposed by a thermostatic bath) is placed in this container to condense the steam coming from the evaporator. This oversized exchanger allows the pressure to be imposed on the free surface in the evaporator. Thus, the experimental bench (Fig. 3) operates as a thermosiphon. Once in a liquid state, the water leaves the condenser and flows in a recirculation pump to compensate for pressure drops. Finally, the liquid circulates through an expansion valve before the evaporator to enter the evaporator in a two-phase flow.

K-type thermocouples are placed at the inlet and outlet of each exchanger in the system, for both circuits (primary and secondary). Pressure sensors are also placed at each component and at the inlet and outlet of the expansion valve. The working pressures are between 0.7 and 1.7 kPa.

IMAGE PROCESSING

Contours detection

A test campaign previously carried out on this bench [START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF] enabled video recordings of the bubbles formed in the evaporator, captured by the high-speed camera, thus making it possible to study their growth and bursting. To clear and analyse the captured videos, an image processing program was developed in Python language, part of the recordings could thus be analysed automatically.

This is done by cutting each video into a sequence of images. The images are then processed in such a way as to isolate the bubble: the background of the image is replaced by a black background, while the contours of the bubble are detected and traced in white (see Fig. 4, bottom). This process is made possible by the OpenCV for Python module.

Measurements

This pre-processing allows to compute the relevant characteristics of the bubbles' evolution. This study focuses on the very last instant of growth, before the bubble bursts: the parameters described below are therefore at their maximum, hence the index max. The apparent area, which informs about the size of the bubble, is computed as follows: for each column of pixels, the chord length of the bubble is computed by subtracting the coordinates of the ends of the bubble. The sum of all the chords gives the bubble's apparent area in pixels. The value is then converted in square meters. The equivalent radius is calculated as the radius of the circle with the same apparent area(see Eq. ( 1)).

𝑅 𝑚𝑎𝑥 𝑒𝑞 = √ 𝑆 𝑚𝑎𝑥 𝜋

Eq. ( 1)

Another important characteristic of the bubble growth is its growth velocity. The derivative of the apparent area with respect to time is the most direct way of describing the expansion of the bubble. It will then be referred to as the expansion velocity 𝑈 𝑓 , and its expression is detailed in Eq. ( 2).

𝑈 𝑓 = ( 𝑑𝑆 𝑑𝑡 ) 𝑡=𝑡 𝑏𝑢𝑟𝑠𝑡𝑖𝑛𝑔
Eq. ( 2)

Finally, the projection height 𝐻 𝑝 may be related to the deposited liquid film (see Fig. 6a). Indeed, during the evolution of the bubble, the wall of the channel is wetted by liquid displacement during growth, then by projection and droplet deposition after bursting. The projection height then corresponds to the maximum height of the liquid film. In this work, it is defined as follows: at each instant after the burst, the maximum pixel density of the image is calculated. That is, for each image, the number of white pixels above the free surface is counted line by line. The altitude of the line containing the most pixels corresponds to a height in pixels, the reference being the bottom of the image. The difference between this height and the liquid level 𝐻 𝑟 is used as the projection height. Figure 6b illustrates this definition. Although the contours detected by the program are probably those in contact with the PMMA plate, we will assume in this study that the calculated projection height and apparent area are close to those in contact with the stainless steel plate.

RESULTS AND DISCUSSION

Qualitative analysis

The processing of 146 exploitable videos enabled a preliminary qualitative analysis. Without considering the properties of the fluid, focusing on the bubble's dynamics and size, it appears that the projection height is linked to the state of the bubble before the free surface breaks up. Indeed, as illustrated on Fig. 7, the projection height is correlated to the maximum apparent area of the bubble at the time of burst and to its maximum expansion velocity at that same instant. In other words, the bigger the apparent area before bursting, the higher the projection. Likewise, the faster the growth at the time of free surface breakup, the higher the projection. This can be explained by the fact that the interface between the bubble and the liquid -which merges with the free interface at the time of bursting -turns into droplets at the time of rupture. The kinetic energy acquired by the interface during bubble's growth is then transmitted to the droplets.

This first insight into the influence of a bubble's size and velocity on the projection height can be further developed by integrating the thermo-physical properties of the working fluid and considering the different forces acting on the bubble at the time of bursting.

Forces and dimensionless numbers

During a bubble's growth, the predominance of certain forces over others determines its characteristics (size, shape, velocity…). In the present study, liquid inertia is compared to surface tension and viscous dissipation, as these forces seem to play a significant role in the bubble's evolution. For that purpose, dimensionless numbers are defined and their impact over projection height is investigated.

Weber number, defined in eq. ( 3), is used to evaluate the balance between inertia and surface tension. The characteristic velocity (𝑣 𝑚𝑎𝑥 + ) corresponds to the velocity of the interface between the liquid and the bubble at its highest point, at the time of burst. The values of We range between 10 2 and 10 4 . Since this number represents the ratio between inertia and surface tension, it appears that inertia is predominant in comparison with surface tension: the kinetic energy accumulated during growth surpasses the surface tension that keeps the free surface whole. Consequently, the free surface breaks.

On the other hand, the balance between inertia and viscous forces is commonly evaluated by Reynolds number. A modified Reynolds number (𝑅𝑒 𝑏 ) is therefore used for that purpose and defined in eq. ( 4). Like the previous result, it seems that liquid inertia predominates over viscous dissipation: 𝑅𝑒 𝑏 ranges between 10 4 and 10 5 . Thus, at the time of burst, viscous dissipation in the liquid is negligible.

𝑅𝑒 𝑏 = 𝜌 𝐿 𝑣 𝑚𝑎𝑥 + 𝑅 𝑚𝑎𝑥 𝑒𝑞 𝜇 𝐿
Eq. ( 4) Fig. 8 shows the influence of We and Reb on projection height. The projection height seems correlated to the Weber number: the greater the liquid's kinetic energy at the time of burst compared to surface tension, the higher the projection. In other words, the more "remaining" energy after the free-surface breakup, the more energy transferred to the droplets. Likewise, the projection height is correlated to the modified Reynolds number: the energy transferred to the droplets depends on the amount of energy transferred to the liquid by viscous dissipation (the less energy dissipated by viscosity, the more energy transferred to the droplets).

CONCLUSIONS

Video recordings of growth and bursting of water bubbles have been processed and analysed. In a subatmospheric pressure and narrow environment, the characteristics of the bubbles have been measured, amongst which the projection height, the apparent area and the expansion velocity. It has been brought to light that the projection height is correlated to the expansion velocity and the apparent area. Moreover, dimensionless numbers have been used to consider the force balance at the time of burst: the projection height seems to be also correlated with these numbers. In conclusion, the liquid's kinetic energy at the time of burst seems to have a major impact on the formation of the thin liquid film by projecting droplets on the exchanger wall.

Figure 1 :

 1 Figure 1: Photograph of the evaporator Figure 2: Scheme of the evaporator (from Giraud, 2016)

Figure 3 :

 3 Figure 3: Photograph of the experimental bench

Figure 4 :

 4 Figure 4: Example of image processing (top: raw, bottom: processed)

Figure 6a :

 6a Figure 6a: Illustration of the liquid film Figure 6b: calculation of the projection height (pixels)

Figure 7 :

 7 Figure 7: Projection height versus apparent area before bursting (left) and projection height versus expansion velocity (right)

  𝜌 𝐿 and 𝜎 are respectively the density and surface tension of the liquid.

Figure 8 :

 8 Figure 8: Projection height versus We (left) and versus Reb (right)

ACKNOWLEDGEMENTS

The authors are grateful to L. Fluro for performing the experimental measurements.

NOMENCLATURE

𝑅 𝑚𝑎𝑥 𝑒𝑞 equivalent radius (m) 𝑣 𝑚𝑎𝑥 + upper interface velocity (m×s -1 ) 𝑆 𝑚𝑎𝑥 apparent area (m²) 𝜌 𝐿 density of the liquid (kg×m -3 ) Uf expansion velocity (m²/s) 𝜇 𝐿 viscosity of the liquid (Pa×s) Hp projection height (cm) 𝜎 surface tension of the liquid (N×m -1 )


J Guilpart
email: jacques.guilpart@lecnam.net

Smart control of screw compressors in industrial conditions

Keywords: industrial application, screw compressor, capacity reduction, slide valve, variable speed, control

The compressors of a machinery room in food industries are often designed on the maximum refrigeration capacity required for the process, even if it is well known that this maximum requirement is effective a very scarce time fraction. Therefore, the compressors often run at partial charge. This situation lead to poor energy efficiency of compressors, and especially of screw compressors controlled with slide valves. To partially solve the problem, the use of variable speed devices are recommended, guessing important energy savings. This paper compares different control strategies of screw compressors in a machinery room of a dairy industry requiring 1 MW refrigeration and composed of 1 to 3 screw compressors of different size and models. It will demonstrate that a smart management of cascade and slide valve control delivers interesting results with a reduced investment cost. The interest of adding variable speed device on a selected compressor will be highlighted.

INTRODUCTION

The adjustment between cold production and needs is a key point when designing a refrigeration installation. Indeed, it is accepted that the design has to be based on the maximum reasonable need, knowing that the installation will work very few time at its maximum capacity. Answering this question needs on one hand to have a good knowledge of the refrigeration needs (chronogram, histogram), and on the other hand, to dispose of actuators allowing to adjust the cold production. In industrial conditions, the use of screw compressors with slide valve is often proposed. In addition, variable speed devices are sometime proposed, expecting energy savings. The drawing of the chronogram of refrigeration needs and the modeling of the performances of screw compressors controlled with slide valves and variable speed devices is therefore necessary to assess the performance of different control strategies.

DESCRIPTION OF THE INSTALLATION

Refrigeration needs

The studied installation is connected to a dairy process (sterilized milk, yoghurts and other flavored milk dessert). The process is based on the utilization of different equipments (pasteurizers, sterilizers, plate heat exchangers, blast coolers, …) fed with iced water. This iced water is produced with a MPG loop (-11°C / -6°C) in a plate heat exchanger with an adequate system preventing the exchanger's freezing. MPG is produced with an ammonia facility with a low pressure receiver feeding a plate heat exchanger (thermosiphon, cycle temperature : -15°C / +35°C). The MPG is also directly used in air coil heat exchangers located in storage facilities and in workshops. The refrigeration needs have been studied in detail with the production service and led to the following chronogram: The reading this data indicates that:  the maximum need of 992 kW is only called for a short time on Wednesday afternoon  most of the time, the refrigeration need is between 250 and 750 kW Therefore, the choice of the number of installed compressor and of their control mode is of great importance.

Capacity (kW) Frequency

Cumm.

[

Retained compressors models

For different reasons (including end-user requirements and habits), the use of GEA screw compressors is proposed. From this, different possibilities are studied:  use of one big compressor ( 1 MW) fulfilling all the requirements and adjusting its refrigeration capacity with a slide valve  use of two medium size compressors ( 2 x 500 kW)  use of three compressors (250 kW + 350 kW + 500 kW) Following table summarizes the main characteristics of these compressors (cycle temperature: -15°C / +35°C):

Table 2. Main characteristics of the selected GEA screw compressors

Studied control strategies

In order to adjust the production to the needs, different control strategies are studied and compared: Strategy 1 : only one big compressor (1MW) in use, with capacity control with slide valve to adjust the production to the needs. Strategy 2 : two medium compressors (2 x 500 kW) Strategy 2.1 : 1 st compressors with slide valve until 100%, then 1 st compressor 100% + 2 nd compressors with slide valve to adjust the production to the needs Strategy 2.2 : 1 st compressors with slide valve until 100%, then 1 st + 2 nd compressors at the same slide valve position Strategy 3 : three compressors (250 + 350 + 500 kW))

Strategy 3.1 : small compressor with slide valve until 100%, then small compressor 100% + medium compressors with slide valve to adjust the production to the needs, then small and medium compressors at 100% and big compressor with slide valve to adjust the production to the needs Strategy 3.2 : small compressors with slide valve until 100%, then small + medium compressors at the same slide valve position, then all compressors to the same slide valve positon. These strategies are completed with the use of variable speed devices, this device being installed on the biggest compressor for the strategy 3. The comparison of the performances of these different strategies requires the modeling of the performances of each compressor working with slide valves and variable speed devices.

Model

Ref

MODELING THE PERFORMANCES OF COMPRESSORS

COP vs slide valves position

Due to international recirculation of sucked gases (Moseman, 1999), the slide valve position has a direct and important impact on the coefficient of performance as shown Fig. 3. We propose to model this decrease by a power law inspired by pressure drop laws as followed :

𝑊 𝑊 𝑚𝑎𝑥 = 𝛼 ( 𝜙 𝜙 𝑚𝑎𝑥 ) 𝑛 + 𝛽 Eq. (1)
where : W is the electric power input of the compressor (kW),  is the refrigeration capacity (kW)  and  are calculated by forcing the function through two points :  one point coming from datasheet, for instance the ratio The exponent n is adjusted to minimize the sum of the squared differences between the datasheet values and the calculated values. Table 3 indicates the values of  and n obtained for the selected compressors. Fig. 4 shows the quality of the fitting, all coefficient of correlation being greater than 0.999.

Introducing a variable speed device

In order to better adjust the refrigeration capacity to the refrigeration needs, the implementation of variable speed device on compressor is often recommended. It is well known that these VSD allow energy savings. It is also well known that the efficiency of the electric motor somewhat decreases with the adopted frequency, and that a low limit of frequency has to be fixed in order to prevent lubrication concerns and motor warming-up.

VSD can be implemented in complement of slide valve action :  While VSD low limit is not reached, VSD is actioned in proportion of the needs  Below this limit, VSD is fixed to the minimum possible value, and the refrigeration production is adjusted to the need by acting the slide valve. For some compressors, the low limit of acceptable frequency could be lower, especially if a lubricant pump and an additional cooling system of the motor are installed. The maximum Freq could also be geater than 50 Hz. These two points would further improve the benefits of VSD.

EVALUATION OF DIFFERENT CONTROL STAGTEGIES

The approach presented above allows the assessment of the performances of compressors when their refrigeration capacity is adjusted by using slide valve and/or variable speed device.

The different control strategies introduced in paragraph 2.3 are compared in Fig. 6.

Evaluation of energy consumption

In order to assess the best strategy to be retained for the studied case, these results are applied to the chronogram presented paragraph 2.1. Results are presented in the table 4

CONCLUSION

For the chronogram corresponding to the studied application, it is confirmed that:  the fractionation of the refrigerating capacity led to lower energy consumptions : for the studied case, installing 2 or 3 compressors instead of a big one allows to reduce the energy consumption of 15 to 20%.  the use of variable speed devices (on the biggest compressor) led to substantial energy savings : for the studied case, it allows to reduce the energy consumption in between 5 and 15% depending on the number of installed compressors. For the studied case, the installation of 3 compressors with VSD on the biggest of them led to the lowest energy consumption. For instance, and compared to the lowest investment cost (only one big compressor without VSD), savings of 8.2 MWh / weekit is 410 MWh / year can give precious information on the payback time necessary to balance the incremental investment cost. It is important to mention that a different histogram of refrigeration needs would lead to somewhat different result. For instance, a histogram centered around high refrigeration needs would privilege the functioning at high charge, and therefore would disadvantage the use of small compressorgenerally presenting poorest COP at high charge than bigger compressors. Nevertheless, the approach presented in this paper would benefit to be used for many industrial configurations.

Figure 1 :

 1 Figure 1 : Chronogram of refrigeration needs for a typical production week

Figure 3 :

 3 Figure 3 : Impact of the slide valve position on a screw compressor performance

Freq

  As a first approach, we propose to model the consequence of both actions on the performances of compressors by moving the base line (linear behavior between 0 and 1) to a affine straight line defined = applied frequency (Hz) and  min =motor efficiency at Freqmin .

Fig. 5 Figure 5 :

 55 Fig.5illustrate the assessed performance of compressor WR-W2240S-28 with la limit of VSD fixed at 25 Hz (that is  / max = 50%) and with  min =90% at Freqmin = 10 Hz.

4. 1 .

 1 Evaluation of Coefficient of performances a-without VSD b-With VSD on the biggest compressor Figure 6 : COP of the different control strategies vs refrigeration capacity The observation of Fig. 6 indicates that :  The use of only one big compressor (strategy 1) delivers lowers COP, even if VSD is used in complement of slide valves  The use of two compressors (strategy 2) delivers better COPs, especially at low charge when using VSD  The use of 3 compressors delivers the best COPS on a wide range of refrigerating capacities, and especially at low charge. It also appears that the global energy consumption of the installation depends on the histogram of refrigeration needs.

Table 1 . Statistics on refrigeration needs Figure 2 : histogram of refrigeration needs

 1

Table 3 .  and n coefficients used in Eq. (1) for selected compressors Figure 4 : example of fitting for compressor WR-W2240S-28

 3

Table 4 . Weekly electric consumption of the different studied strategies

 4

		MWh / week
	Strategy 1	33.0
	Strategy 1 + VEV	27.5
	Strategy 2.1	27.4
	Strategy 2.2	26.7
	Strategy 2.2 + VEV	25.2
	Strategy 3.1	29.6
	Strategy 3.2	28.5
	Strategy 3.2 + VEV	24.8