Alain Manceau
email: author.alain.manceau@univ-grenoble-alpes.fr

Comment on "New insights into the biomineralization of mercury selenide nanoparticles through stable isotope analysis in giant petrel tissues"

Keywords: Isotope fractionation, toxicology, bird, HgSe, mercury selenide, selenoprotein P

Some birds and cetaceans can demethylate the toxic methylmercury cysteinate (MeHgCys) complex into inert mercury sulfide (HgSe) through the formation of an intermediate tetrahedral selenolate complex with selenocysteine (Sec) residues (Hg(Sec)4). The nucleation of the HgSe biominerals involves the substitution of the Se ligand for the Sec residues, which is considered to occur in the form of multinuclear Hgx(Se,Sec)y clusters mediated by proteins. Queipo-Abad et al.

[1] isolated HgSe nanoparticles from the biological tissues of giant petrels and measured the massdependent fractionation of the 202 Hg isotope (δ 202 Hg). They concluded that the δ 202 Hg values of the HgSe nanoparticles from each tissue of individual petrels are specific to the HgSe species alone and that the Hg(Sec)4 → HgSe reaction occurs without fractionation of the 202 Hg isotope. We show

(1) that the HgSe nanoparticles are mixtures of MeHgCys, Hg(Sec)4, and HgSe, and therefore that the δ 202 Hg values are not species-specific, and (2) that the 202 Hg isotope is actually fractionated during the Hg(Sec)4 → HgSe reaction, and therefore that this isotope can be used to trace the Hg metabolic pathways between tissues in a single individual and in different animals.

It was shown recently that the Clark's grebe, giant petrels, long-finned pilot whales, and blue marlins detoxify the organic methylmercury cysteinate (MeHgCys) complex by the stepwise MeHgCys → Hg(Sec)4 → Hgx(Se,Sec)y → HgSe demethylation reaction [START_REF] Manceau | Demethylation of methylmercury in bird, fish, and earthworm[END_REF][START_REF] Manceau | In vivo formation of HgSe nanoparticles and Hg-tetraselenolate complex from methylmercury in seabird -Implications for the Hg-Se antagonism[END_REF][START_REF] Manceau | Chemical forms of mercury in pilot whales determined from species-averaged mercury isotope signatures[END_REF][START_REF] Manceau | The chemical forms of mercury in blue marlin billfish: Implications for human exposure[END_REF]. Hg(Sec)4 is a fourcoordinate selenocysteine complex and HgSe is tiemannite. The inorganic Hg(Sec)4 complex is likely bonded to selenoprotein P (SelP), for this protein was shown to be associated with Hg in Clark's grebe using double affinity chromatography high-performance liquid chromatography coupled to inductively coupled mass spectrometry (AF-HPLC-ICPMS) [START_REF] Manceau | Demethylation of methylmercury in bird, fish, and earthworm[END_REF]. SelP-bound Hg has been identified also in the plasma of Inuits [START_REF] Achouba | Plasma levels of seleniumcontaining proteins in Inuit adults from Nunavik[END_REF], and in the plasma of rats exposed simultaneously to sodium selenide (Na2SeO3) and mercury chloride (HgCl2) [START_REF] Sasakura | Biological interaction between transition metals (Ag, Cd and Hg), selenide/sulfide and selenoprotein P[END_REF][START_REF] Suzuki | Binding sites for the (Hg-Se) complex on selenoprotein P[END_REF][START_REF] Yoneda | Equimolar Hg-Se complex binds to selenoprotein P[END_REF]. HgSe nanoparticles (NPs) were imaged in liver, kidneys and muscle of giant petrels [START_REF] Manceau | In vivo formation of HgSe nanoparticles and Hg-tetraselenolate complex from methylmercury in seabird -Implications for the Hg-Se antagonism[END_REF], and their formation involves the replacement of selenocysteine ligands with selenide ligands, likely through the nucleation of multinuclear Hgx(Se,Sec)y clusters (y < 4x). The production of biominerals mediated by metalloproteins is common in nature [START_REF] Dove | Biomineralization[END_REF], and we show below that the results of Queipo-Abad et al. [START_REF] Queipo-Abad | New insights into the biomineralization of mercury selenide nanoparticles through stable isotope analysis in giant petrel tissues[END_REF] support the composite nature of the HgSe NPs, as observed for other biominerals [START_REF] Zhang | Conversion of the native 24-mer ferritin nanocage into its non-native 16-mer analogue by insertion of extra amino acid residues[END_REF][START_REF] Maity | Observation of gold sub-nanocluster nucleation within a crystalline protein cage[END_REF][START_REF] Pozzi | Chemistry at the protein-mineral interface in L-ferritin assists the assembly of a functional (μ 3 -oxo)Tris (μ 2 -peroxo) triiron(III) cluster[END_REF].

Queipo-Abad et al. [START_REF] Queipo-Abad | New insights into the biomineralization of mercury selenide nanoparticles through stable isotope analysis in giant petrel tissues[END_REF] isolated 37 HgSe NPs from the biological tissues of 11 giant petrels and measured the mass-dependent fractionation (MDF) of the 202 Hg isotope, expressed as δ 202 Hg [START_REF] Bergquist | Mass-dependent and -independent fractionation of Hg isotopes by photoreduction in aquatic systems[END_REF],

with the objective to characterize how MeHg is detoxified into insoluble HgSe. They observed that NPs from specific tissues in a single petrel had different δ 202 Hg signatures. They hypothesized that Hg(Sec)4 (denoted as Hg:Se (1:4)) and HgSe NPs have the same MDF in a given tissue, and therefore that the Hg(Sec)4 → HgSe reaction occurs without fractionation of the δ 202 Hg isotope in each tissue. Thus, they considered that a δ 202 Hg value measured on the extracted NPs is specific to the HgSe species alone, and that this species-specific δ 202 Hg value varies from one tissue to another.

This interpretation conflicts with two recent Hg isotope studies performed on 22 petrel tissues from 8 individuals and 89 whale tissues from 28 individuals [START_REF] Manceau | Chemical forms of mercury in pilot whales determined from species-averaged mercury isotope signatures[END_REF][START_REF] Manceau | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental Mercury Science[END_REF]. The δ 202 Hg value of a tissue (δ 202 Hgt) is the weighted sum of the MDF of each Hg species (Spi) expressed as follows:

( )

202 202 Hg Sp Sp t i i t i f  =  (1)
in which f(Spi)t is the fractional amount of species i in tissue t, and δ 202 Spi is the species-specific MDF. Through a mathematical inversion analysis of the petrel and whale isotopic data, Manceau et al. [START_REF] Manceau | Chemical forms of mercury in pilot whales determined from species-averaged mercury isotope signatures[END_REF][START_REF] Manceau | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental Mercury Science[END_REF] showed that the 22 δ 202 Hgt petrel values and the 89 δ 202 Hgt whale values, could be calculated with just three δ 202 Spi values, each associated with the MeHgCys, Hg(Sec)4, and HgSe species. Therefore, Equation 1 was reduced to We selected the five liver, four kidney, and two muscle tissues for which bulk δ 202 Hg ≈ δ 202 HgSe (Fig. 2A, R 2 = 0.993), and plotted their proportions of MeHg measured chemically by the authors (%MeHg) as a function of the total Hg concentration (Fig. 2B). The subset of samples does not include those with the highest Hg hepatic concentrations, but covers a large range of Hg concentration, from 0.56 ± 0.15 µg/g to 928.49 ± 73.31 µg/g, and includes the most diluted liver (1.11 ± 0.04 µg/g) and kidney (0.56 ± 0.15 µg/g) tissues. These two tissues have as much as 28.55 [1], and liver and muscle tissues of long-finned pilot whales studied by Bolea-Fernandez et al. [START_REF] Bolea-Fernandez | High-precision isotopic analysis sheds new light on mercury metabolism in long-finned pilot whales (Globicephala melas)[END_REF] with linear fit of the two independent sets of data (y = -0.134 + 0.57x, R 2 = 0.656). The outlier liver of petrel P-11 (δ 202 Hg = 0.32, δ 202 HgSe = 1.43) was omitted.

  MeHgCys = 2.69 ± 0.04 ‰, δ 202 Hg(Sec)4 = -1.37 ± 0.06 ‰, and δ 202 HgSe = 0.18 ± 0.02 ‰ for the petrel data, and δ 202 MeHgCys = 1.21‰, δ 202 Hg(Sec)4 = -1.37‰, and δ 202 HgSe = 0.00‰

± 2 .

 2 43 %MeHg and 27.34 ± 2.71 %MeHg, respectively. Another tissue from the subset of samples with bulk δ 202 Hg ≈ δ 202 HgSe has about 25.9 %MeHg for [Hg] = 35.8 µg/g, two other tissues have about 20 %MeHg for [Hg] ≈ 8 -60 µg/g, and one has 7.5 %MeHg for [Hg] = 36.6 µg/g. The last four tissues are also poor in Hg in comparison to those containing several hundreds µg/g Hg and for which bulk δ 202 Hg ≠ δ 202 HgSe. Clearly, the identity between bulk δ 202 Hg and δ 202 HgSe in tissues containing a high proportion of MeHg and low total Hg cannot be explained by a mixing of just two dominant Hg species, HgSe and Hg(Sec)4, and by the assumption that the HgSe NPs are pure HgSe. Therefore, not only are the HgSe NPs mixtures of MeHgCys, Hg(Sec)4, and HgSe, but we also conclude from this that the δ 202 Hg(Sec)4 = δ 202 HgSe hypothesis is incorrect.

Figure caption Fig. 1 .

 caption1 Figure caption

Fig. 2 .

 2 Fig. 2. A) δ202 Hg values measured on whole tissues of giant petrels and the corresponding isolated HgSe nanoparticles, for which bulk δ 202 Hg ≈ δ 202 HgSe, with linear fit (y = 0.007 + 1.04x, R 2 = 0.993). B) Molar fraction of MeHg to total Hg against the total weight concentration of Hg in the same tissues as in A).

Figure 2

for the whale data. The two sets of δ 202 Hg(Sec)4 and δ 202 HgSe values were remarkably similar, and the difference of δ 202 MeHgCys value was attributed primarily to metabolic differences between organisms in the rate of demethylation relative to new dietary MeHg intake and MeHg depuration, as giant petrels demethylate MeHg with extreme efficiency producing uncommonly large amounts of HgSe in liver, kidneys, brain, and even muscle.

We show below that the Queipo-Abad et al. [START_REF] Queipo-Abad | New insights into the biomineralization of mercury selenide nanoparticles through stable isotope analysis in giant petrel tissues[END_REF] data can be reinterpreted in light of the findings of the recent petrel and whale studies [START_REF] Manceau | Chemical forms of mercury in pilot whales determined from species-averaged mercury isotope signatures[END_REF][START_REF] Manceau | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental Mercury Science[END_REF], which concluded that the three Hg species have a specific isotopic composition which is homogenized throughout the body. The demonstration is performed in two steps. We first provide evidence for a mixture of Hg species in the HgSe NPs, then we show that the δ 202 Hg(Sec)4 = δ 202 HgSe hypothesis is unsupported. Three (PGA01, PGA02, PGA03) of the six brain tissues studied by Queipo-Abad et al. [START_REF] Queipo-Abad | New insights into the biomineralization of mercury selenide nanoparticles through stable isotope analysis in giant petrel tissues[END_REF] had been analyzed previously for Hg speciation using X-ray absorption spectroscopy [START_REF] Manceau | In vivo formation of HgSe nanoparticles and Hg-tetraselenolate complex from methylmercury in seabird -Implications for the Hg-Se antagonism[END_REF]. PGA01 contains 40 ± 6% MeHgCys + 24 ± 15% Hg(Sec)4 + 36 ± 13% HgSe, PGA02 contains 83 ± 5% MeHgCys + 9 ± 9% Hg(Sec)4 + 8 ± 8% HgSe, and PGA03 contains 13 ± 5% MeHgCys + 16 ± 10% Hg(Sec)4 + 71 ± 8% HgSe. The high contrast in Hg speciation between the three samples correlates with the Hg concentration. As expected from a detoxification standpoint, the brain with the highest amount of Hg (PGA03: 13.23 ± 0.22 µg/g) has the highest proportion of HgSe (71 ± 8%) and the lowest proportion of MeHgCys (13 ± 5%) and, vice versa, the brain with the lowest amount of Hg (PGA02: 1.58 ± 0.02 µg/g) has the lowest proportion of HgSe (8 ± 8%) and the highest proportion of MeHgCys (83 ± 5%). When plotting δ 202 HgSe against total δ 202 Hg for the three brain tissues, one observes that the data are linearly correlated (Fig. 1A, R 2 = 0.998), which indicates that the NPs are not pure Hg species, but mixtures. This interpretation can be extended to all tissues by plotting δ 202 HgSe and total δ 202 Hg for all samples studied by Queipo-Abad et al. [START_REF] Queipo-Abad | New insights into the biomineralization of mercury selenide nanoparticles through stable isotope analysis in giant petrel tissues[END_REF] (Fig. 1B). The correlation is lower (R 2 = 0.796), but is higher within each tissue. Further, Bolea-Fernandez et al. [START_REF] Bolea-Fernandez | High-precision isotopic analysis sheds new light on mercury metabolism in long-finned pilot whales (Globicephala melas)[END_REF] extracted 7 HgSe NPs from the liver and muscle of long-finned pilot whales, and their data match well those of the giant petrels, which adds support to our interpretation.

Turning now our attention to the lack of fractionation of the δ 202 Hg isotope during the Hg(Sec)4 → HgSe reaction, Queipo-Abad et al. [START_REF] Queipo-Abad | New insights into the biomineralization of mercury selenide nanoparticles through stable isotope analysis in giant petrel tissues[END_REF] wrote: "In the giant petrel group exhibiting the highest Hg hepatic concentrations, there is a perfect match between the isotopic signature of THg (δ 202 Hgbulk) and those in HgSe NPs (δ 202 HgHgSe) for the liver, kidneys and muscles." Afterward the authors state that the bulk samples contain dominantly inorganic Hg in the forms of Hg(Sec)4 and HgSe, and they conclude from this and from bulk δ 202 Hg ≈ δ 202 HgSe that δ 202 Hg(Sec)4 = δ 202 HgSe.


Alain Manceau
email: authors.alain.manceau@univ-grenoble-alpes.fr

Romain Brossier

Sarah E Janssen

Brett A Poulin
email: bapoulin@ucdavis.edu

Implications for Environmental Mercury Science": Principles and limitations of source tracing and process tracing with stable isotope signatures

come   L'archive ouverte pluridisciplinaire

transmission electron microscopy. [START_REF] Manceau | In Vivo Formation of HgSe Nanoparticles and Hg-Tetraselenolate Complex from Methylmercury in Seabird -Implications for the Hg-Se Antagonism[END_REF] The tissues contained three dominant Hg species (Spi), a methylmercury cysteinate (MeHgCys) complex, a Hg-tetraselenolate (Hg(Sec)4) complex, and mercury selenide (HgSe) grains, and their fractions (f(MeHgCys), f(Hg(Sec)4), and f(HgSe)) in each tissue were known. The δ 202 Hg value of each tissue (δ 202 Hgt) is a weighted sum of the δ 202 Hg of each Hg species expressed as follows:

( )

202 202 3 δ Hg Sp δ Sp t i i t i f = =  (1)
in which δ 202 Spi are species-and tissue-specific δ 202 Hg values.

If each Hg species occurs in every tissue with a tissue-specific δ 202 Spi,t value, the system of 22 linear equations has 66 unknowns. Such a linear system is underdetermined and has infinitely many solutions. However, using an inversion algorithm, we found that the degree of freedom of the petrel system could be reduced from 44 (some of the tissues had only one or two species) to 3 with identical species-specific δ 202 Spi signatures between different tissues of the same bird and across the eight individuals. The δ 202 Hgt values of the 22 tissues were described as a weighted sum of three species- using δ 202 MeHgCys = 2.69 ± 0.04‰, δ 202 Hg(Sec)4 = -1.37 ± 0.06‰, and δ 202 HgSe = 0.18 ± 0.02‰.

Further, δ 202 Hg of MeHg (δ 202 MeHg) values were measured directly on a subset of 9 tissues using an established distillation and anion-exchange chromatography method. [START_REF] Rosera | Isolation of Methylmercury Using Distillation and Anion-Exchange Chromatography for Isotopic Analyses in Natural Matrices[END_REF] The majority of the MeHgspecific isotope measurements were in agreement with the calculated values (within the analytical uncertainty of 0.16‰). However, one feather had δ 202 MeHg = 3.05‰ which was attributed to differences in dietary sources or feeding ranges that were integrated over time in the bird. [START_REF] Manceau | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental Mercury Science[END_REF] Also, one muscle had a δ 202 MeHg = 0.90‰ that was anomalous and is discussed in detail below. While there may be limitations to MeHg-specific isotope measurements, all data were presented and discussed in Manceau et al. ( 2021) [START_REF] Manceau | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental Mercury Science[END_REF] for transparency. Wiederhold and Jiskra 4 comment on the validity of the data analysis approach and contend that the anomalous δ 202 MeHg outlier values can be explained using a closed-system Rayleigh fractionation model. We challenge these critiques below and detail a conceptual diagram that is appropriate for isotope analyses and interpretations for organisms with continuous dietary inputs, internal transformations of Hg, and depuration.

First, Wiederhold and Jiskra 4 comment that the petrel linear isotope system is mathematically underdetermined. Actually, the regression analysis was statistically robust and the 22 δ 202 Hgt values were recalculated within experimental uncertainty from the reduced linear system (Eq. 2). The observed linearity in the isotope system in the petrels is consistent with observed two-component isotope systems that documented MeHg demethylation to inorganic Hg in waterbirds [START_REF] Poulin | Isotope Fractionation from in Vivo Methylmercury Detoxification in Waterbirds[END_REF] and the brain of whales. [START_REF] Li | Selenium and Stable Mercury Isotopes Provide New Insights into Mercury Toxicokinetics in Pilot Whales[END_REF] Further, similar isotopic results were obtained in a companion article on long-finned pilot whales. [START_REF] Manceau | Chemical Forms of Mercury in Pilot Whales Determined from Species-Averaged Mercury Isotope Signatures[END_REF] In this companion study, the δ 202 Hgt values of 89 tissues (excluding blood) from 28 individuals reported in the independent studies of Bolea-Fernandez et al. Second, Wiederhold and Jiskra 4 point out the discrepancy between the calculated (2.69 ± 0.04‰) and two measured δ 202 MeHg values (3.05‰, 0.90‰). This was the first study to compare mathematical estimates of Hg isotope end-members based on Hg speciation from X-ray absorption spectroscopy with direct measurements of MeHg-specific isotope values. Given the complexities of each technique, perfect agreement between measured and calculated values was not expected. Giant petrels are wild animals that live across expansive ranges and have diverse prey items, and yet the measured δ 202 MeHg isotope values were overall well constrained. [START_REF] Manceau | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental Mercury Science[END_REF] We agree with Wiederhold and Jiskra 4 that the measured δ 202 MeHg value of 0.90‰ is puzzling, but it is anomalous compared to the other measurements.

Determining if this data point is truly an analytical anomaly is difficult, but the feather tissue from the same petrel containing 100% MeHgCys had a δ 202 MeHg value consistent with the mathematical estimate. Importantly, the muscle tissue with the anomalous δ 202 MeHg value of 0.90‰ had 40±8% Hg(Sec)4 and 60±8% HgSe, and the MeHgCys species was below the spectroscopic detection limit (<3%). Therefore, this tissue did not contribute to the calculation of δ 202 MeHgCys, as was detailed in the study. The holistic approach suggested by Wiederhold and Jiskra 4 holds merit as it highlights current gaps in the literature associated with Hg stable isotope applications in birds and mammals.

Specifically, further examination is warranted on the dietary variability of the MeHg pool in wildlife, [START_REF] Croizier | Foraging Plasticity Diversifies Mercury Exposure Sources and Bioaccumulation Patterns in the World's Largest Predatory Fish[END_REF] and the isotopic modeling of open systems (e.g., Figure 1) where multiple products are produced via steady-state reactions (e.g., demethylation and biomineralization) and select products are depurated (e.g., MeHg, Hg(Sec)4) and others accumulated (e.g., HgSe). These gaps, however, do not directly address any of the points made in an attempt to dismiss the findings of our study. The mechanistic examination of internal demethylation and biomineralization pathways could be overlooked by simplifying these findings to a mass balance within the organism, emphasizing the need for continued isotope research that applies complementary analytical methods (X-ray absorption spectroscopy, species-specific mercury stable isotope, and protein-level screening) 13 to advance mechanistic understanding and modeling. Furthermore, while the multi-dimensional nature of Hg isotopes (MDF and mass independent fractionation) is an important consideration in source tracking and assessing Hg isotope fractionation in vivo, as noted by Wiederhold and Jiskra, [START_REF] Wiederhold | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental 6 Mercury Science": Principles and Limitations of Source Tracing and Process Tracing with Stable Isotope Signatures[END_REF] there was no evidence that individual petrels had dissimilar exposures to Hg processing effects (e.g., photochemical demethylation noted by Δ 199 Hg) or differing atmospheric sources (noted by Δ 200 Hg). 14 Although much remains to be known on the internal cycling of Hg within mammals and birds, the petrels presented a unique opportunity to examine an organism where internal demethylation and biomineralization reactions are responsible for as much as a Disclaimer: Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.

  (2019)[START_REF] Bolea-Fernandez | High-Precision Isotopic Analysis Sheds New Light on Mercury Metabolism in Long-Finned Pilot Whales (Globicephala Melas)[END_REF] and Li et al. (2020)[START_REF] Li | Selenium and Stable Mercury Isotopes Provide New Insights into Mercury Toxicokinetics in Pilot Whales[END_REF] were analyzed by mathematical inversion. It was also found that three δ 202 Spi values were sufficient to describe the species-averaged δ 202 Hgt values between tissues of the same individual and across individuals sampled at different geographical locations and times. The highly consistent observations between the giant petrels and long-finned pilot whales indicate that the isotopic fractionation due to in vivo demethylation and biomineralization reactions occurs in diverse vertebrates, isotopic signatures of the chemical Hg species are homogenized within individuals, and localized isotope effects due to these reactions in tissues are not apparent.

  Third, Wiederhold and Jiskra[START_REF] Wiederhold | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental 6 Mercury Science": Principles and Limitations of Source Tracing and Process Tracing with Stable Isotope Signatures[END_REF] propose that a Rayleigh fractionation model can explain the anomalous δ202 MeHg values quantified in the petrel tissues. Unfortunately, there are oversights with the proposed use of a Rayleigh fractionation model by Wiederhold and Jiskra[START_REF] Wiederhold | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental 6 Mercury Science": Principles and Limitations of Source Tracing and Process Tracing with Stable Isotope Signatures[END_REF] and they egregiously misrepresented the data inManceau et al. (2021).[START_REF] Manceau | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental Mercury Science[END_REF] The Rayleigh fractionation model assumes a closed system, where a finite reservoir of reactant (in this case MeHg) becomes enriched in heavier isotopes as lighter isotopes are demethylated to the product pool. In the birds, however, a constant influx of dietary MeHg replenishes the reactant pool, MeHg undergoes stepwise demethylation and biomineralization reactions, HgSe is accumulated, 2 both MeHg and Hg(Sec)4 are excreted in waste, and MeHg is depurated in feathers. Figure1presents a schematic representation of the birds, modified from Hayes (2002),[START_REF] Hayes | Practice and Principles of Isotopic Measurements in Organic Geochemistry (Revision 2)[END_REF] which serves as a model to guide isotope interpretations and applies to other vertebrates (e.g., marine mammals). The continuous input of MeHg and loss terms of MeHg and Hg(Sec)4 in the birds, as well as internal exchange between tissues via the circulatory system, likely explain the consistency in species-specific measurements of the MeHg isotope pool across different petrels; these processes are not accounted for in the Rayleigh fractionation model. If the finding of a steady-state isotopic fractionation of MeHgCys → Hg(Sec)4 demethylation and Hg(Sec)4 → HgSe biomineralization reactions in birds 1 and whales 7 is novel, its hypothesis is not. Bolea-Fernandez et al.(2019)[START_REF] Bolea-Fernandez | High-Precision Isotopic Analysis Sheds New Light on Mercury Metabolism in Long-Finned Pilot Whales (Globicephala Melas)[END_REF] observed that δ 202 Hg was uniform in the blood of 7 juvenile and 7 adult whales due to the homogenization of the isotopic composition of MeHg throughout the body. This interpretation is reinforced by a previous study showing a rapid shift in Hg isotope values of fish tissues to values of dietary MeHg replenishment.[START_REF] Kwon | Application of Mercury Isotopes for Tracing Trophic Transfer and Internal Distribution of Mercury in Marine Fish Feeding Experiments[END_REF] We assess the validity of a Rayleigh fractionation model in Figure2a, which presents the measured δ 202 MeHg values versus the fraction of MeHg in tissues (fMeHg) of the petrels from Manceau et al.(2021)[START_REF] Manceau | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental Mercury Science[END_REF] against the theoretical Rayleigh fractionation model from Wiederhold and Jiskra.[START_REF] Wiederhold | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental 6 Mercury Science": Principles and Limitations of Source Tracing and Process Tracing with Stable Isotope Signatures[END_REF] The measured δ 202 MeHg values are relatively invariant across a wide range of fMeHg (aside from the outlier of 0.90‰), and δ 202 MeHg values at low fMeHg never exceeded δ 202 MeHg at high fMeHg (counter to the Rayleigh fractionation model). These observations in petrels are consistent with the measurement of δ 202 MeHg values of other bird[START_REF] Poulin | Isotope Fractionation from in Vivo Methylmercury Detoxification in Waterbirds[END_REF] and seal tissues[START_REF] Perrot | Natural Hg Isotopic Composition of Different Hg Compounds in Mammal Tissues as a Proxy for in Vivo Breakdown of Toxic Methylmercury[END_REF] with varying degrees of demethylation (i.e., fMeHg), as shown in Figure2b. The observed invariance in measured δ 202 MeHg data across multiple organisms and studies is inconsistent with the use of a Rayleigh fractionation model in this system, seen by comparing the Wiederhold and Jiskra model (red line and data)[START_REF] Wiederhold | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental 6 Mercury Science": Principles and Limitations of Source Tracing and Process Tracing with Stable Isotope Signatures[END_REF] to the measurement of the petrels (Figure2a). the lack of transparency in the analysis provided by Wiederhold and Jiskra,[START_REF] Wiederhold | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental 6 Mercury Science": Principles and Limitations of Source Tracing and Process Tracing with Stable Isotope Signatures[END_REF] we can only surmise the source of the data points they present in their Figure1bthat reside precisely on the theoretical Rayleigh fractionation line. As such, Wiederhold and Jiskra 4 first used the anomalous δ 202 MeHg value of 0.90‰ as the starting pool (red diamond in Figure2a) and assigned a fMeHg = 1.0 despite the measured value of fMeHg ≤ 0.03. Next, from the "starting pool," the theoretical Rayleigh fractionation was modeled (shown as a solid line in Figure2a), and the other data points presented were assigned fictious fMeHg values (red circles in Figure2a). Simply, the measured paired δ 202 MeHg and fMeHg values from Manceau et al. (2021)[START_REF] Manceau | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental Mercury Science[END_REF] are not presented in Figure1bof Wiederhold and Jiskra.[START_REF] Wiederhold | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental 6 Mercury Science": Principles and Limitations of Source Tracing and Process Tracing with Stable Isotope Signatures[END_REF] Further, the range of δ 202 MeHg values presented by Wiederhold and Jiskra 4 were as high as ~4.0‰, exceeding those reported in Manceau et al. (2021),[START_REF] Manceau | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental Mercury Science[END_REF] and the source of the isotope enrichment factor used in their model simulation is incorrectly cited. While it is important to consider the potential of Rayleigh fractionation in environmental matrices, this model does not explain the outliers in the dataset and the consistency of δ 202 MeHg across multiple tissues. In summary, the premise that bulk δ 202 Hg signatures of giant petrel tissues can be described as weighted sums of three species-specific δ 202 MeHgCys, δ 202 Hg(Sec)4, and δ 202 HgSe values is fully validated using the direct measurements of δ 202 MeHg.[START_REF] Manceau | Mercury Isotope Fractionation by Internal Demethylation and Biomineralization Reactions in Seabirds: Implications for Environmental Mercury Science[END_REF]

Figure 1 .

 1 Figure 1. Schematic representation of an open isotope system with a continuous input of MeHg("reactant") to the bird, the bird being a "reaction chamber" where isotopic fractionation occurs due to stepwise demethylation and biomineralization reactions. HgSe accumulates in the bird ("accumulated product"), and MeHg and Hg(Sec)4 are excreted in waste and/or depurated (product streams). Modified fromHayes (2002).[START_REF] Hayes | Practice and Principles of Isotopic Measurements in Organic Geochemistry (Revision 2)[END_REF]

9

 9

Figure 2 .

 2 Figure 2. (a) Direct comparison of the measured δ 202 MeHg versus the fraction of MeHg (fMeHg) from Manceau et al. (2021) 1 for giant petrel tissues and the theoretical Rayleigh fractionation model presented by Wiederhold and Jiskra 4 (red line and data points, extracted using OriginLab (Version 2121b, OriginLab Corporation, Northampton, MA, USA). The source of the x and y data points from Wiederhold and Jiskra 4 is unknown. (b) Comparison of the measured δ 202 MeHg versus fMeHg for Clark's grebe (Aechmophorus clarkia) and Forster's tern (Sterna forsteri) tissues from Poulin et al. (2021) 5 and seal tissues from Perrot et al. (2016). 11 Measured data in plots a and b demonstrate that δ 202 MeHg is largely invariant across a wide range of fMeHg and is inconsistent with a Rayleigh fractionation model. For tissues from Manceau et al. (2021), 1 fMeHg = 0.03 was used for samples at the spectroscopic detection limit for MeHgCys.

Figure 2

 2 Figure 2

  3.5‰ difference in tissue δ 202 Hg values. This work opens the door for new methods and techniques to examine internal processing of Hg in biota. B. A. Demethylation of Methylmercury in Bird, Fish, and Earthworm. Environ. Sci. Technol. Tsui, M. T. K.; Blum, J. D.; Kwon, S. Y. Review of Stable Mercury Isotopes in Ecology and Biogeochemistry. Sci. Tot. Environ. 2020, 716, n° 135386.

	2021, 55, 1527-1534.
	(14)


Diego Chamorro
email: diego.chamorro@univ-evry.fr

Nicolas Meunier
email: nicolas.meunier@univ-evry.fr

Analysis of a nonlocal and nonlinear system for cell-cell communication

Keywords: Existence problems for PDEs: global existence, local existence, cell communication and cell polarization. MSC2020: 35A01, 35Q92, 92C37, 47J260

published or not. The documents may come

Analysis of a nonlocal and nonlinear system for cell-cell communication

Diego Chamorro, Nicolas

Introduction

Cell-cell communication is crucial for many biological phenomena such as cell migration, axon guidance e.g. some cell-cell communication does not require direct cell-cell contact and is mediated by a pheromone that diffuses into the intercellular space and binds to receptors on the cell surface. In this work we study two models, initially introduced in [START_REF] Muller | Mathematical and numerical studies of nonlinear and nonlocal models involved in biology[END_REF][START_REF] Muller | A predictive model for yeast cell polarization in pheromone gradients[END_REF][START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF][START_REF] Calvez | Non-linear analysis of a model for yeast cell communication[END_REF][START_REF] Calvez | Non-linear analysis of a tangential model for yeast cell communication[END_REF], that describe some aspects of single yeast cell polarization and cell-cell communication in the context of yeast mating. The first model describes the polarization of a single cell subjected to a pheromone concentration    ∂ t n(t, x) = D∂ xx n(t, x) + ∂ x χ * nH Sn (t, x), n(0, x) = n 0 (x).

(1.1)

Here χ, S, n : [0, +∞[×R -→ R are real valued functions, H is the Hilbert transform defined in the Fourier level by the formula H(f )(ξ) = -i ξ |ξ| f (ξ), D > 0 is the diffusion coefficient, n 0 : R -→ R and * is the usual convolution product.

Briefly, the main assumptions leading to (1.1) are the following: the cell is represented by the half-space H = R×]0, +∞[ and its membrane is the real line. We denote by n(t, x) the concentration of proteins (or markers) involved in the polarization process (e.g. Cdc42 protein) that are trapped at the boundary. The nonlocal nonlinearities in (1.1) describe positive feedback loops involved in the dynamics of polarization markers: molecular markers not only diffuse in the cytoplasm of the cell, but are also actively transported by molecular motors along the filaments of the cytoskeleton, whose dynamic organization is regulated by the markers themselves, by an external signal noted S. Moreover in (1.1) the function χ accounts for the mobility (depending on time and space) of the membrane markers to describe mobile surface polarity patch and for the yeast cell cycle [START_REF] Sieber | A focus on yeast mating: From pheromone signaling to cell-cell fusion[END_REF].

When χ and S are constant, it is known that (1.1) exhibits pattern formation (either a blowup or a convergence to a non-homogeneous steady state) under certain conditions, see [START_REF] Calvez | Modified Keller-Segel system and critical mass for the log interaction kernel[END_REF][START_REF] Calvez | Modèles et analyses mathématiques pour les mouvements collectifs de cellules[END_REF]. Our goal here is to extend the well-posedness study to the case where χ and S are functions of space and time.

Next, following ideas given in [START_REF] Muller | Mathematical and numerical studies of nonlinear and nonlocal models involved in biology[END_REF][START_REF] Calvez | Non-linear analysis of a model for yeast cell communication[END_REF][START_REF] Calvez | Non-linear analysis of a tangential model for yeast cell communication[END_REF], we study the well-posedness of the coupled system that allows to describe some aspects of cell-cell communication:

         ∂ t n 1 (t, x) = D 1 ∂ xx n 1 (t, x) + ∂ x χ 1 * n 1 H S 2 n 1 (t, x), ∂ t n 2 (t, x) = D 2 ∂ xx n 2 (t, x) + ∂ x χ 2 * n 2 H S 1 n 2 (t, x), n 1 (0, x) = n 1,0 (x)
and n 2 (0, x) = n 2,0 (x),

where n 1 , n 2 , χ 1 , χ 2 : [0, +∞[×R -→ R are real valued functions, D i > 0 are the diffusion coefficients, the functions S 1 , S 2 are given by the expression

S 1 (t, x) = K 1 * n 1 (t, x) and S 2 (t, x) = K 2 * n 2 (t, x), (1.3)
with K 1 , K 2 belonging to L 2 (R) and depending on the distance h between the two cell membranes see Figure 1.

In short, in (1.2) -(1.3) the two cell membranes are modeled as two parallel lines separated by a distance h from each other (see Figure 1 below). On each cell membrane, proteins, with density n i (t, x), can either diffuse freely or be actively transported to regions with high concentrations of protein and external signal (S i ) produced by the opposite cell type. Moreover, space and time variable mobilities (χ i ) serve both as sites of pheromone secretion and pheromone perception. We refer to Section 2 for a detailed presentation of the model with biological motivations. The mathematical question which arises in this class of models is to determine which contribution will dominate, either aggregation by directed chemoattraction or dispersion by stochastic diffusion. Before stating our results, let us briefly comment the literature. When χ and S are constant, the problem (1.1) is similar to the Modified Keler-Segel (MKS) [START_REF] Calvez | Modified Keller-Segel system and critical mass for the log interaction kernel[END_REF]. In such a case, it was proved that above a critical value of χ or S some singularity occurs (so called blow-up). This dichotomy between concentration of the solution vs. self-similar decay is analogous to the classical Keller-Segel equation for chemotaxis [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF]. The aforementioned works deal with only one density (of proteins or attracting cells) and with constant coefficients χ and S.

In terms of modeling, the polarization of a cell occurs when a singularity appears in (1.1). In this case, the membrane markers concentrate in one small region, which is the signature of polarization. Similarly, a necessary condition for a dialogue between cells to take place (leading to the mating between the two cells) is the appearance of a singularity in (1.2). In this work, we perform a mathematical analysis of (1.1) and (1.2) which provides criteria to prevent cell polarization and dialogue between cells. These criteria depend on the biological ingredients of the model (χ i , S i , K i ) and correspond to the biological observations. This mathematical analysis is a first step towards a better understanding of the strategy that use yeast cells during pairing.

In the present work we first study local well-posedness for (1.1) for general functions χ ∈

L ∞ t (L 1 x ) and S ∈ L ∞ t (L ∞ x ).
For simplicity, we assume that D = 1 in (1.1). To do so, we consider mild solutions (via the Duhamel formula) in appropriate functional spaces, and although many different functional spaces can be considered, for simplicity, we work in Lebesgue and Sobolev spaces (see Section 3 below for precise definitions).

Theorem 1 (Local in time mild solutions for (1.1)). Assume n 0 ∈ L 2 (R), χ ∈ L ∞ t (L 1 x ) and S ∈ L ∞ t (L ∞ x ). Equation (1.1) admits a unique mild solution in L ∞ ([0, T 0 ], L 2 (R)), with T 0 > 0, of the form n(t, x) = h t * n 0 (x) + t 0 h t-s * ∂ x χ * nH Sn (s, x) ds, (1.4)
where h t is the one-dimensional heat kernel.

Next, we show that some local well-posedness results can be extended to the system (1.2) -(1.3). For simplicity, again, we assume that D i = 1 in (1.2). To do so, define the vectorial quantities

θ 0 (x) = n 1,0 (x) n 2,0 (x) and θ(t, x) = n 1 (t, x) n 2 (t, x) , (1.5)
S [ θ] (t, x) = χ 1 * n 1 H(S 2 n 1 ) χ 2 * n 2 H(S 1 n 2 ) (t, x). (1.6)
The system (1.2) can be rewritten as

     ∂ t θ(t, x) = ∂ xx θ(t, x) + ∂ x S [ θ] (t, x), θ(0, x) = θ 0 (x), (1.7)
whose integral (vectorial) representation is (via Duhamel's formula):

θ(t, x) = h t * θ 0 (x) + t 0 h t-s * ∂ x S [ θ] (s, x) ds. (1.8)
Theorem 2 (Local in time mild solutions for system (1.8)). Assume

θ 0 ∈ L 2 (R), χ 1 , χ 2 ∈ L ∞ t (L 1 x ) and K 1 , K 2 ∈ L 2 (R).
There exists a time T 0 > 0 such that equation (1.8) admits a mild solution θ(t, x) in L ∞ ([0, T 0 ], L 2 (R)).

Remark 1.1. In Theorem 1, the existence time T 0 depends on the L 2 norm of the initial data and of the L ∞ norm of S (see (4.32)). In Theorem 2, it depends on the L 2 norms of the initial data and the kernels K 1 , K 2 that define the functions S 1 , S 2 (see (4.35)).

To obtain global solutions we still look for mild solutions via a Banach-Picard fixed point algorithm. However, the space L ∞ t (L 2 x ) no longer seems to be suitable for this new purpose. Instead, we consider the functional space

E = L ∞ t (L 2 x ) ∩ L 2 t ( Ḣ1
x ) which captures different information using homogeneous Sobolev spaces (see Section 3 below for a precise definition of this space).

Theorem 3 (Global Existence for the bilinear equation (1.1)). Assume

n 0 ∈ L 2 (R), χ ∈ L 2 ([0, +∞[, L 2 (R)) and S ∈ L ∞ ([0, +∞[, L ∞ (R)). There exists a small positive constant C such that if n 0 L 2 χ L 2 t (L 2 x ) S L ∞ t (L ∞ x ) ≤ C, (1.9)
then system (1.1) admits a unique global mild solution in

L ∞ ([0, +∞], L 2 (R))∩L 2 ([0, +∞], Ḣ1 (R)) of the form n(t, x) = h t * n 0 (x) + t 0 h t-s * ∂ x χ * nH Sn (s, x) ds. (1.10)
Theorem 4 (Global existence for the trilinear system (1.2)). Assume

θ 0 ∈ L 2 (R), χ 1 , χ 2 ∈ L 2 ([0, +∞[, L 2 (R)) and K 1 , K 2 ∈ L 2 (R).
There exists a small positive constant C such that if

θ 0 2 L 2 ( χ 1 L 2 t (L 2 x ) K 1 L 2 + χ 2 L 2 t (L 2 x ) K 2 L 2 ) < C, (1.11)
then (1.2) has a unique global mild solution in the space

L ∞ ([0, +∞], L 2 (R))∩L 2 ([0, +∞], Ḣ1 (R)).
Let us make a few comments. In Theorems 1 and 2, we use a fixed point argument and if we consider very large initial data, then the existence time will be very small. In Theorems 3 and 4, although based on the same general arguments, no relation between the size of the initial data and the existence time is necessary. Thus, under these conditions global solutions can be considered, but only for small initial data. To the best of our knowledge, the existence of global in time mild solutions with large initial data is out of reach for the moment.

Apart from the two cases mentioned here, we leave the existence of mild solutions, and in particular blow-up criteria, for problems (1.1) and (1.2) - (1.3) as open problems and we refer to [START_REF] Calvez | Non-linear analysis of a tangential model for yeast cell communication[END_REF] for preliminary answers for classical solutions.

The present paper is organized as following. In Section 2, we recall the biological justifications of (1.1) and (1.2) -(1.3). In Section 3 we give some notations and results that will be used in the proofs. In Section 4 we prove Theorems 1 and 2. In Section 5 we prove Theorems 3 and 4.

Model construction

In this section we recall the biological justification of equation (1.1) in the context of yeast cell polarization. Next, we detail the construction of the coupled model (1.2) -(1.3).

A validated model for the polarization of yeast cells

Cell polarization is the process by which a cell transitions from a spherically symmetric shape to a state with a preferred axis [START_REF] Bagnat | Cell surface polarization during yeast mating[END_REF][START_REF] Nelson | Adaptation of core mechanisms to generate cell polarity[END_REF][START_REF] Muller | A predictive model for yeast cell polarization in pheromone gradients[END_REF][START_REF] Slaughter | Symmetry breaking in the life cycle of the budding yeast[END_REF][START_REF] Slaughter | Nonuniform membrane diffusion enables steady-state cell polarization via vesicular trafficking[END_REF]. The molecular pathways involved in cell polarization have been studied experimentally in some detail over the past decade, see [START_REF] Sieber | A focus on yeast mating: From pheromone signaling to cell-cell fusion[END_REF] for a recent review in the context of yeast cell mating. The results suggest that the small GTPase Cdc42 and its scaffold, the Bem1 protein, are involved during the polarization process. Moreover, it is known [START_REF] Wedlich-Soldner | Spontaneous cell polarization through actomyosin-based delivery of the Cdc42 gtpase[END_REF][START_REF] Muller | A predictive model for yeast cell polarization in pheromone gradients[END_REF], that the dynamics of Cdc42 depend on the dynamics of the cytoskeleton filaments. The cellular cytoskeleton is a network of long semi-flexible filaments made of protein subunits [START_REF] Phillips | Physical biology of the cell[END_REF]. These filaments (mainly actin or microtubules) act as roads along which motor proteins are able to perform biased ballistic movement and transport various molecules, in a process that consumes the chemical energy of adenosine triphosphate (ATP).

In [START_REF] Wedlich-Soldner | Spontaneous cell polarization through actomyosin-based delivery of the Cdc42 gtpase[END_REF], it has been proposed that yeast polarization relies on a positive feedback loop in the dynamics of polarity markers, which is mediated by the actin cytoskeleton. It has been shown that polarity markers (Cdc42 proteins) activate actin filament polymerization when adsorbed on the cell membrane, so that active transport along filaments pointing to Cdc42 -enriched regions is promoted in the cell cytoplasm. The positive feedback loop then results from the fact that the Cdc42 markers themselves are actively transported into the cytoplasm, and are thus preferentially directed to the high concentration regions.

According to the previous considerations, in [START_REF] Muller | Mathematical and numerical studies of nonlinear and nonlocal models involved in biology[END_REF][START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF][START_REF] Muller | A predictive model for yeast cell polarization in pheromone gradients[END_REF][START_REF] Calvez | Non-linear analysis of a model for yeast cell communication[END_REF], a model based on partial differential equation was used and studied to describe yeast polarization. Suppose the cell is represented by the half-space H = R×]0, +∞[ with x = (x, y) as spatial variable. Denote by ν(t, x) the concentration in the cytoplasm of proteins (or markers) involved in cell polarization, by n(t, x) the concentration of proteins trapped at the boundary and by c(t, x) the density of actin filaments in the cytoplasm of the cell. The model first introduced in [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF] and validated in [START_REF] Muller | A predictive model for yeast cell polarization in pheromone gradients[END_REF] is:

∂ t ν(t, x) = D∆ν(t, x) -χ∇ • (ν(t, x)∇c(t, x)) , t > 0 , x ∈ H , (2.12)
completed by the flux boundary condition:

D∂ y ν(t, x, 0) -χν(t, x, 0)∂ y c(t, x, 0) = -∂ t n(t, x) , x ∈ R , (2.13)
where χ > 0. Furthermore, the dynamic exchange of markers at the boundary occurs with an attachment rate k on and a detachment rate k off , hence the time evolution of n(t, x) is

∂ t n(t, x) = k on ν(t, x, 0) -k off µ(t, x) .
(2.14)

In this model, the advection, that is χ ∇c, describes the active transport of proteins along actin filaments. The nucleation of new filaments is assumed to occur at the cell boundary, under the combined action of molecules trapped on the boundary and external pheromone molecules with density S. After a dimensional analysis, the model that describes the density of actin filaments is

-∆c(t, x) = 0, x ∈ H , ∂ z c(t, x, 0) = S(t, x)µ(t, x), x ∈ R .
(2.15)

Heuristic justification of the use of the Hilbert transform in (1.1)

The Hilbert transform of a function f ∈ L p (R), 1 ≤ p < +∞ is defined as the Cauchy principle value integral [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF] H(f

)(x) = 1 π p.v. R f (y) x -y dy . (2.16)
Obviously, the integral (2.16) exists for almost all x. We define the function P by

P (x) = 1 π 1 x 2 + 1 ∀x ∈ R ,
and the Poisson kernel by

P ε (x) = 1 ε P x ε , (2.17)
with ε > 0. Since P L 1 (R) = P ε L 1 (R) = π, the family (P ε ) ε is an approximation of the unity. The conjugate Q y is defined by

∀(x, y) ∈ R × R + , Q y (x) = 1 y Q x y , (2.18)
with

Q(x) = 1 π x x 2 + 1 ∀x ∈ R .
As usual, S(R) is the Schwartz space function, defined as the space of all functions

f ∈ C ∞ (R) such that ∀(α, β) ∈ N 2 , sup x∈R |x α f (β) (x)| < +∞ ,
and S (R) denotes its dual space.

Theorem 5 (Stein [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF], Chapter III). The convergence holds

Q y → H in S as y → 0 .
The link between the Hilbert transform and the conjugate of the Poisson kernel explains why, in (2.15) the passage from the normal derivative of c at z = 0 to its tangential derivative at z = 0, involves the Hilbert transform.

Lemma 2.1. The solution of (2.15) is c(t, x, z) = - 1 π x ∈R log (x -x ) 2 + z 2 S(t, x )n(t, x ) dx , (2.19)
and it holds that

∂ x c(t, x, 0) = -H(Sn)(t, x) .
Proof. We first verify that such a function c is solution to (2.15). We compute

∂ x c(t, x, z) = - 1 π x ∈R x -x (x -x ) 2 + z 2 S(t, x )n(t, x ) dx = -(Q z * Sn(t, •)) (x) ,
where Q z is defined by (2.18), and

∂ xx c(t, x, z) = - 1 π x ∈R -(x -x ) 2 + z 2 ((x -x ) 2 + z 2 ) 2 S(t, x )n(t, x ) dx .
Furthermore, recalling the definition of the Poisson kernel (2.17) we have

∂ z c(t, x, z) = - 1 π x ∈R z (x -x ) 2 + z 2 S(t, x )n(t, x ) dx = -(P z * Sn(t, •)) (x) , and
∂ zz c(t, x, z) = - 1 π x ∈R (x -x ) 2 -z 2 ((x -x ) 2 + z 2 ) 2 S(t, x )n(t, x ) dx , hence ∆c(t, x, z) = 0 .
Since the Poisson kernel is an approximation of the unity, we deduce

lim z→0 ∂ z c(t, x, z) = ∂ z c(t, x, 0) = lim z→0 (-P z * Sn(t, •)) (x) = -S(t, x)n(t, x) .
Using Theorem 5, it follows that

lim z→0 ∂ x c(t, x, z) = ∂ x c(t, x, 0) = lim z→0 (-Q z * Sn(t, •)) (x) = -H(Sn)(t, x) .
Lemma 2.1 is thus proven.

In the supercritical case, i.e., numerical simulations [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF][START_REF] Calvez | Numerical simulation of the dynamics of molecular markers involved in cell polarisation[END_REF], suggest that the solution ν(t, x) of (2.12) -(2.15) concentrates on the boundary {y = 0}. Assuming that ν(t, x, y) = n(t, x)δ(y = 0), we can formally write the dynamics of n(t, x) as follows:

∂ t n(t, x) = ∂ xx n(t, x) + χ∂ x (n H(Sn)) (t, x) t > 0 , x ∈ R .
(2.20)

In this work we modify (2.20) into (1.1) in order to add a nonlocal effect.

Link between (1.1) and the MKS system

In the case where χ > 0 is a constant, there is a link between (2.20) and the so-called modified Keller-Segel system [START_REF] Calvez | Modèles et analyses mathématiques pour les mouvements collectifs de cellules[END_REF] (MKS):

     ∂ t n(t, x) = ∂ xx n(t, x) -χ∂ x (n(t, x)∂ x c(t, x)) t > 0 , x ∈ R , c(t, x) = K * n(t, x) t > 0 , x ∈ R , n(t = 0, x) = n 0 (x) x ∈ R , (2.21)
where * is the usual convolution product and

K(z) = - 1 π log |z| . Lemma 2.2. If c = K * n, then one has ∂ x c = -H(n).
Proof. For all ε > 0, introduce c ε by

∀x ∈ R , c ε (x) = |x-y|>ε K(y)c(x -y) dy .
We see that lim

ε→0 c ε (x) = c(x). Moreover we can compute its spatial derivative ∀x ∈ R , ∂ x c ε (x) = |x-y|>ε c(x -y) y dy , from which we deduce ∀x ∈ R , lim ε→0 ∂ x c ε (x) = - 1 π p.v. 1 x * c (x) = -H(c)(x) = ∂ x c(x) .
Using Lemma 2.2, we see that (2.21) rewrites as (1.1).

Boundary model (1.2) -(1.3) for cell-cell communication

In nature, the budding yeast, Saccharomyces cerevisiae, exists in either the diploid or haploid state with two possible types (a or α). Cells of both types secrete a certain pheromone (S a or S α ), see Figure 2, and carry a pheromone receptor to detect the pheromone produced by cells of the opposite type, [START_REF] Jackson | Courtship in s. cerevisiae: both cell types choose mating partners by responding to the strongest pheromone signal[END_REF]. Following [START_REF] Muller | Mathematical and numerical studies of nonlinear and nonlocal models involved in biology[END_REF][START_REF] Calvez | Non-linear analysis of a model for yeast cell communication[END_REF], we study a model where the release of extracellular pheromone (S a or S α ) depends on the concentration of Cdc42 protein at the membrane. Furthermore, in accordance with the biological literature, [START_REF] Madden | Cell polarity and morphogenesis in budding yeast[END_REF][START_REF] Bagnat | Cell surface polarization during yeast mating[END_REF][START_REF] Slaughter | Symmetry breaking in the life cycle of the budding yeast[END_REF][START_REF] Slaughter | Nonuniform membrane diffusion enables steady-state cell polarization via vesicular trafficking[END_REF][START_REF] Sieber | A focus on yeast mating: From pheromone signaling to cell-cell fusion[END_REF], we assume that the pheromone contributes to the nucleation of new filaments at the membrane of the opposite type cell, see Figure 2 below. This additional level of pairwise cell communication leads to the following rectification of the model (1.1): for the #i cell, we consider the advection field to be χ i H(S j ν i ) where S j is the concentration of pheromone produced by the cell of the opposite type j. Furthermore, to describe the biological observations, we assume that S j is produced by ν j , which means that the pheromone produced by the j-type cell is influenced by the activation level of the opposite cell. Finally, we incorporate a damping factor depending on the intercellular distance. The model consists of the following equations:

∂ t ν i (t, x) = ∂ xx ν i (t, x) + ∂ x ν i (t, x)H (S j (t, x, h)ν i (t, x)) , (2.22)
for (t, x) ∈ R + × R, i, j ∈ {1, 2} and j = i, with S j satisfying the elliptic equation

-∆S j + λS j = 0 , on Ψ , ∇S j • e j = -ν j , on Γ j , (2.23)
where e 1 = -e 2 denotes the unit outward normal vector to Γ j . The solution of (2.23) can be explicitly computed: for all t > 0 and x ∈ R, one has

S j (t, x, h) = 1 π R F ( (x -x ) 2 + h 2 ) ν j (t, x ) dx , (2.24)
where F is the Bessel potential

F (r) = 1 4π
+∞ 0 e -λz-r 2 4z z dz .

(2.25)

Notations

We gather here some notation and some material that will be used in the sequel.

• If f : [0, +∞[×R -→ R is a real valued measurable function, for 1 ≤ p, q ≤ +∞ we will denote by L p t (L q x ) the L p (in time)-L q (in space) Lebesgue space which is given by the condition

f L p t (L q x ) = +∞ 0 f (t, •) p L q dt 1 p < +∞,
with the usual modifications when p = +∞ or q = +∞.

• For t > 0, the heat kernel is given by the function

h t (x) = 1 √ 4πt e -|x| 2 4t
for which we have the estimates

∂ k x h t L p ≤ Ct -k+(1-1/p) 2 , (3.26)
with k ∈ N and 1 ≤ p ≤ +∞.

• The homogeneous Sobolev spaces Ḣ1 (R) is given as the set of tempered distributions (modulo polynomials) such that

f Ḣ1 (R) = ∂ x f L 2 R |ξ| 2 | f (ξ)| 2 dξ 1 2
< +∞.

• The homogeneous Sobolev space Ḣ-1 (R) can be defined by duality with respect to the space Ḣ1 (R), but it can also be characterized by the following condition in the Fourier variable

f Ḣ-1 (R) R |ξ| -2 | f (ξ)| 2 dξ 1 2
< +∞.

Local Solutions

We study here local in time solutions for the equations (1.1) and (1.2) and for this we will consider mild solutions (via the Duhamel formula) in suitable functional spaces. Although many different functional spaces can be considered, for the sake of simplicity we will work in the space L ∞ t (L 2 x ).

Proof of Theorem 1.

Here we study the existence of local Solutions for the bilinear system (1.1). Let us recall that the Banach-Picard fixed point theorem allows us to obtain a solution to equations of the form

n = e 0 + B(n, n), (4.27)
where e 0 is an initial data and B(•, •) is a bilinear application. Indeed, if we fix a Banach space (E, • E ) and if we have the estimates

e 0 E ≤ δ, B(n, n) E ≤ C B n E n E , (4.28)
then we only need to verify the condition

δ < 1 4C B , (4.29)
to obtain a unique solution of the equation (4.27). See [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF]Theorem 5.1] for a proof of this bilinear version of the Banach-Picard fixed point theorem.

In the particular case of equation (1.4) above -which is already in the form of equation ( 4.27)we will consider the Banach space E = L ∞ ([0, T ], L 2 (R)) endowed with the norm

f L ∞ t (L 2 x ) = sup 0<t<T f (t, •) L 2 ,
thus we first need to prove that we have the estimates (4.28) and then, if we have the condition (4.29), we will obtain a solution of equation (1.4).

Let us start with the initial data. If we denote by e 0 the quantity h t * n 0 (x), then by the Young inequalities for convolution we can write

h t * n 0 L 2 ≤ h t L 1 n 0 L 2 = n 0 L 2 ,
since h t L 1 = 1 and from this uniform in time estimate we obtain

h t * n 0 L ∞ t (L 2 x ) = sup 0<t<T h t * n 0 L 2 ≤ n 0 L 2 < +∞. (4.30)
For the second term of equation (1.4), let us write

B(n, n) = t 0 h t-s * ∂ x χ * nH Sn (s, x) ds,
it is easy to see that the quantity B(•, •) is bilinear and we have

B(n, ν) L ∞ t (L 2 x ) = sup 0<t<T t 0 h t-s * ∂ x χ * nH Sν (s, •) ds L 2
, and by the Young inequalities for convolution we can write

B(n, ν) L ∞ t (L 2 x ) ≤ sup 0<t<T t 0 ∂ x h t-s * χ * nH Sν (s, •) L 2 ds ≤ sup 0<t<T t 0 ∂ x h t-s L 2 χ * nH Sν (s, •) L 1 ds, since by (3.26) we have ∂ x h t-s L 2 ≤ C(t -s) -3 4
, by the Hölder and Young inequalities we can write

B(n, ν) L ∞ t (L 2 x ) ≤ C sup 0<t<T t 0 (t -s) -3 4 χ(s, •) L 1 n(s, •) L 2 H Sν (s, •) L 2 ds ≤ C sup 0<t<T t 0 (t -s) -3 4 χ(s, •) L 1 n(s, •) L 2 (Sν)(s, •) L 2 ds,
where in the last estimate we used the fact that the Hilbert transform H is a bounded operator in the Lebesgue space L 2 (R). Next we write

B(n, ν) L ∞ t (L 2 x ) ≤ C χ L ∞ t (L 1 x ) n L ∞ t (L 2 x ) S L ∞ t (L ∞ x ) ν L ∞ t (L 2 x ) sup 0<t<T t 0 (t -s) -3 4 ds ≤ CT 1 4 χ L ∞ t (L 1 x ) S L ∞ t (L ∞ x ) n L ∞ t (L 2 x ) ν L ∞ t (L 2
x ) . (4.31)

Now, with the estimates (4.30) and (4.31) if the following condition is satisfied

n 0 L 2 < 1 4CT 1 4 χ L ∞ t (L 1 x ) S L ∞ t (L ∞ x )
, we obtain a unique solution of the equation (1.10). Thus, as long as we have

T < 1 C n 0 L 2 χ L ∞ t (L 1 x ) S L ∞ t (L ∞ x ) 4 , (4.32)
we have the existence and the uniqueness of the solution of this problem.

Proof of Theorem 2

Here, we study the existence of local solutions for the trilinear system (4.34). For the sake of completeness we state and proof a variant of the Banach-Picard fixed point theorem which will give us a suitable framework in order to study a trilinear problem.

e n+1 E ≤ δ + 8C B δ 3 = δ + (8C B δ 2 )δ ≤ 2δ,
and we have proven that for all n ≥ 1, the element e n belongs to B(0, 2δ).

We will prove now that we can obtain a solution e when n → +∞ and we study e n+1 -e n and using the trilinearity of the application B(•, •, •) we have: Unicity is granted as we are working in a Banach space.

In order to apply the previous theorem to the system (1.2) we use the vectorial quantities θ 0 (x), θ(t, x) and S [ θ] (t, x) defined by (1.5) and (1.6). Recall that if a(t, x) = a 1 (t, x) a 2 (t, x) is a vector, then we will say that a

∈ L ∞ t (L 2 x ) if a L ∞ t (L 2 x ) = (|a 1 | + |a 2 |) L ∞ t (L 2
x ) < +∞. Note in particular that we always have the estimate

a j L ∞ t (L 2 x ) ≤ a L ∞ t (L 2
x ) for all j = 1, 2.

In order to apply Theorem 6 to the previous problem, we will consider its integral representation (Duhamel's formula) given by (1.8):

θ(t, x) = h t * θ 0 (x) + t 0 h t-s * ∂ x S [ θ] (s, x) ds.
At this point, we remark that due to the definition of the terms S 1 and S 2 given in (1.3) we actually have

S [ θ] = χ 1 * n 1 H(S 2 n 1 ) χ 2 * n 2 H(S 1 n 2 ) = χ 1 * n 1 H (K 2 * n 2 )n 1 χ 2 * n 2 H (K 1 * n 1 )n 2 ,
which is of the form:

χ 1 * e 1 H (K 2 * f 2 )g 1 χ 2 * e 2 H (K 1 * f 1 )g 2 = B e, f , g),
where the quantity B(•, •, •) is trilinear and thus the whole integral in (1.8) is itself a trilinear application, thus this equation is of the form (4.34) and we are in the setting of the Theorem 6.

Proof of Theorem 2. Following Theorem 6, we will first prove the estimates

• for the initial data:

h t * θ 0 L ∞ t (L 2
x ) ≤ θ 0 L 2 , • for the trilinear term:

t 0 h t-s * ∂ x S [ θ] (s, •) ds L ∞ t (L 2 x ) ≤ C B θ 3 L ∞ t (L 2 x ) ,
where C B way depend on the kernels K 1 , K 2 and on the time variable.

Indeed, for the initial data we readily have

h t * θ 0 L ∞ t (L 2 x ) ≤ h t L 1 θ 0 L 2 = θ 0 L 2 .
For the trilinear term we write

t 0 h t-s * ∂ x S [ θ] (s, •) ds L ∞ t (L 2 x ) = sup 0<t<T t 0 ∂ x h t-s * S [ θ] (s, •) ds L 2 ≤ sup 0<t<T t 0 ∂ x h t-s L 2 S [ θ] (s, •) L 1 ds, and since ∂ x h t-s L 2 ≤ C(t -s) -3 4 by (3.26), we have t 0 h t-s * ∂ x S [ θ] (s, •) ds L ∞ t (L 2 x ) ≤ C sup 0<t<T t 0 (t -s) -3 4 χ 1 * n 1 H (K 2 * n 2 )n 1 (s, •) L 1 + χ 2 * n 2 H (K 1 * n 1 )n 2 (s, •) L 1 ds.
Next, by the Hölder and the Young inequalities it comes:

t 0 h t-s * ∂ x S [ θ] (s, •) ds L ∞ t (L 2 x ) ≤ C sup 0<t<T t 0 (t -s) -3 4 χ 1 (s, •) L 1 n 1 (s, •) L 2 H (K 2 * n 2 )n 1 (s, •) L 2 + χ 2 (s, •) L 1 n 2 (s, •) L 2 H (K 1 * n 1 )n 2 (s, •) L 2 ds,
we use now the boundedness of the Hilbert transform H in the Lebesgue spaces L 2 to write

t 0 h t-s * ∂ x S [ θ] (s, •) ds L ∞ t (L 2 x ) ≤ C sup 0<t<T t 0 (t -s) -3 4 χ 1 (s, •) L 1 n 1 (s, •) L 2 (K 2 * n 2 )n 1 (s, •) L 2 + χ 2 (s, •) L 1 n 2 (s, •) L 2 (K 1 * n 1 )n 2 (s, •) L 2 ds ≤ C sup 0<t<T t 0 (t -s) -3 4 χ 1 (s, •) L 1 n 1 (s, •) L 2 K 2 * n 2 (s, •) L ∞ n 1 (s, •) L 2 + χ 2 (s, •) L 1 n 2 (s, •) L 2 K 1 * n 1 (s, •) L ∞ n 2 (s, •) L 2 ds.
Applying the Young inequalities in the convolutions above we obtain

t 0 h t-s * ∂ x S [ θ] (s, •) ds L ∞ t (L 2 x ) ≤ C sup 0<t<T t 0 (t -s) -3 4 χ 1 (s, •) L 1 n 1 (s, •) L 2 K 2 L 2 n 2 (s, •) L 2 n 1 (s, •) L 2 + χ 2 (s, •) L 1 n 2 (s, •) L 2 K 1 L 2 n 1 (s, •) L 2 n 2 (s, •) L 2 ds ≤ C χ 1 L ∞ t (L 1 x ) n 1 2 L ∞ t (L 2 x ) n 2 L ∞ t (L 2 x ) K 2 L 2 + χ 2 L ∞ t (L 1 x ) n 2 2 L ∞ t (L 2 x ) n 1 L ∞ t (L 2 x ) K 1 L 2 × sup 0<t<T t 0 (t -s) -3 4 ds. Moreover, since n 1 L ∞ t (L 2 x ) ≤ θ L ∞ t (L 2 x ) and n 2 L ∞ t (L 2 x ) ≤ θ L ∞ t (L 2
x ) , we finally have

t 0 h t-s * ∂ x S [ θ] (s, •) ds L ∞ t (L 2 x ) ≤ CT 1 4 χ 1 L ∞ t (L 1 x ) K 1 L 2 + χ 2 L ∞ t (L 1 x ) K 2 L 2 θ 3 L ∞ t (L 2 x ) ,
where, as announced

C B = CT 1 4 ( χ 1 L ∞ t (L 1 x ) K 1 L 2 + χ 2 L ∞ t (L 1 x ) K 2 L 2 ).
Thus, to conclude, we only need to impose the condition

θ 0 2 L 2 < 1 12CT 1 4 ( χ 1 L ∞ t (L 1 x ) K 1 L 2 + χ 2 L ∞ t (L 1 x ) K 2 L 2 )
, which can be rewritten as

T < 1 C θ 0 2 L 2 χ 1 L ∞ t (L 1 x ) K 1 L 2 + χ 2 L ∞ t (L 1 x ) K 2 L 2 4 , (4.35)
and this achieves the proof.

Remark 4.1. It is worth noting here that the conditions (4.32) and (4.35) allow us to consider large initial data (n 0 or θ 0 ), large functions χ i and large kernels (S i or K i ) in the L 2 -sense. The price to pay is a corresponding potentially very small time of existence.

Remark 4.2. Other (and much more general) functional spaces can be considered here such as Morrey spaces or Besov/Triebel-Lizorkin spaces and we do not claim any optimality of our results.

The method displayed to prove Theorems 1 and 2 can not treat the case of simultaneous large data and large times of existence, and without an a priori information on the L ∞ t (L 2 x ) norm it seems hard to extend in time these local solutions. However, if we change the framework and we assume some smallness condition on the data, then it is possible to obtain global results as it is explained in the next section.

Global Solutions

We study now global in time solutions for the previous equations (1.1) and (1.2). To do so, we will consider a slightly different approach. Indeed, although we are still looking for mild solutions via a Banach-Picard fixed point algorithm, the space L ∞ t (L 2 x ) does not seem well suited for this purpose. Instead, we will consider a smaller functional space

E = L ∞ t (L 2 x ) ∩ L 2 t ( Ḣ1 x ) which captures more regularity information. This space L ∞ t (L 2 x ) ∩ L 2 t ( Ḣ1
x ) can be endowed with the norm

f E = • L ∞ t (L 2 x ) + • L 2 t ( Ḣ1 x ) = sup t>0 f (t, •) L 2 + +∞ 0 f (t, •) 2 Ḣ1 dt 1 2
.

(5.36)

In the next section, we first gather some classical inequalities on the norms

• E , • L ∞ t (L 2 x )
and

• L 2 t ( Ḣ1 x ) .

Useful estimates

Lemma 5.1.

If f ∈ L 2 (R) then h t * f ∈ L 2 ([0, +∞[, Ḣ1 (R))
and we have

h t * f L 2 t ( Ḣ1 x ) ≤ C f L 2 . Proof.
By the Plancherel formula and the Fubini theorem, we have

h t * f 2 L 2 t ( Ḣ1 x ) +∞ 0 R |ξ| 2 e -2t|ξ| 2 | f (ξ)| 2 dξ dt = R +∞ 0 |ξ| 2 e -2t|ξ| 2 | f (ξ)| 2 dt dξ. Set τ = 2t|ξ| 2 , then h t * f 2 L 2 t ( Ḣ1 x ) 1 2 R +∞ 0 e -τ | f (ξ)| 2 dτ dξ = 1 2 f 2 L 2 ,
from which the result follows.

Define

F (t, x) = t 0 h t-s * f (s, x) ds. Lemma 5.2. If f ∈ L 2 ([0, +∞[, L 2 (R 3
)), then we have

∂ xx F (•, •) L 2 t (L 2 x ) ≤ C f L 2 t (L 2 x ) .
This estimate is known as the maximal regularity of the heat kernel.

Proof. A general proof can be found in [START_REF] Lemarié-Rieusset | The Navier-Stokes Problem in the 21st Century[END_REF], but we give here the details for the sake of completness. Since we have

∂ xx F (t, x) = t 0 ∂ xx [h t-s ] * f (s, x) ds,
taking the Fourier transform in the space variable, with a change of variables in the time variable, it comes

F x [∂ xx F ] (t, ξ) = t 0 -|ξ| 2 e -(t-s)|ξ| 2 F x [f ] (s, ξ) d = t 0 -|ξ| 2 e -η|ξ| 2 F x [f ] (t -η, ξ) dη.
Setting f (t, x) = 0 if t < 0, we obtain

F x [∂ xx F ] (t, ξ) = +∞ 0 -|ξ| 2 e -η|ξ| 2 F x [f ] (t -η, ξ) dη,
taking now the Fourier transform in the time variable and by the Fubini theorem, we have

F t,x [∂ xx F ] (τ, ξ) = +∞ -∞ +∞ 0 -|ξ| 2 e -η|ξ| 2 F x [f ] (t -η, ξ) dη e -itτ dt = +∞ 0 -|ξ| 2 e -η|ξ| 2 +∞ -∞ F x [f ] (t -η, ξ)e -itτ dt dη = +∞ 0 -|ξ| 2 e -η|ξ| 2 e -iητ F t,x [f ] (τ, ξ) dη = -|ξ| 2 |ξ| 2 + iτ F t,x [f ] (τ, ξ).
We remark now that the symbol -|ξ| 2 |ξ| 2 +iτ is bounded, and thus we can write

∂ xx F L 2 t (L 2 x ) F t,x [∂ xx F ] L 2 t (L 2 x ) = -|ξ| 2 |ξ| 2 + iτ F t,x [f ] L 2 t (L 2 x ) ≤ C f L 2 t (L 2 x ) ,
which achieves the proof of the lemma.

Our last result concerns the norm • E defined in (5.36).

Proposition 5.1. If f ∈ L 2 [0, +∞[, Ḣ-1 (R) then we have the following estimate for the norm • E given by (5.36):

t 0 h t-s * f (s, •) ds E ≤ C f L 2 t ( Ḣ-1 x ) .
Proof. By definition of the norm • E we have

t 0 h t-s * f (s, •) ds E = t 0 h t-s * f (s, •) ds L ∞ t (L 2 x ) (a) + t 0 h t-s * f (s, •) ds L 2 t ( Ḣ1 x ) (b)
.

(5.37)

We start by the term (a) and we write

t 0 h t-s * f (s, •) ds L ∞ t (L 2 x ) = sup 0≤t≤T t 0 h t-s * f (s, •) ds L 2
.

By the L 2 -L 2 duality we obtain

t 0 h t-s * f (s, •) ds L 2 = sup φ L 2 ≤1 R t 0 h t-s * f (s, x) ds φ(x) dx = sup φ L 2 ≤1 R t 0 R h t-s (x -y)f (s, y)φ(x) dy ds dx = sup φ L 2 ≤1 t 0 R h t-s * φ(y)f (s, y) dy ds .
Now, using the Ḣ1 -Ḣ-1 duality we can write

t 0 h t-s * f (s, •) ds L 2 ≤ sup φ L 2 ≤1 t 0 h t-s * φ Ḣ1 f (s, •) Ḣ-1 ds ≤ f L 2 t ( Ḣ-1 x ) sup φ L 2 ≤1 h t-s * φ L 2 t ( Ḣ1 x ) ,
where we used the Cauchy-Schwarz inequality in the time variable. Now using Lemma 5.1, we have h t-s * φ L 2 t ( Ḣ1

x ) ≤ φ L 2 , hence, we obtain

t 0 h t-s * f (s, •)ds L 2 ≤ C f L 2 t ( Ḣ-1 x ) sup φ L 2 ≤1 φ L 2 ≤ C f L 2 t ( Ḣ-1 x ) ,
and from this uniform estimate in the time variable it follows

t 0 h t-s * f (s, •) ds L ∞ t (L 2 x ) ≤ C f L 2 t ( Ḣ-1 x ) .
(5.38)

We study now (b) and we write:

t 0 h t-s * f (s, •) ds L 2 t ( Ḣ1 x ) = t 0 h t-s * (-∆ x ) 1 2 f (s, •) ds L 2 t (L 2 x ) = t 0 h t-s * ∂ xx (-∆ x ) -1 2 f (s, •) ds L 2 t (L 2 x ) = ∂ xx t 0 h t-s * (-∆ x ) -1 2 f (s, •) ds L 2 t (L 2 x )
, where (-∆ x )

1 2 is the one dimensional operator of symbol |ξ|. We apply now the maximal regularity of the heat kernel given in Lemma 5.2 to obtain

∂ xx t 0 h t-s * (-∆ x ) -1 2 f (s, •) ds L 2 t (L 2 x ) ≤ C (-∆ x ) -1 2 f L 2 t (L 2 x )
, and we thus have

t 0 h t-s * f (s, •) ds L 2 t ( Ḣ1 x ) ≤ C f L 2 t ( Ḣ-1 x ) .
(5.39)

With estimates (5.38) -(5.39) and coming back to (5.37) we finally obtain:

t 0 h t-s * f (s, •) ds E ≤ C f L 2 t ( Ḣ-1 x ) ,
which ends the proof of the proposition.

Proof of Theorem 3.

With the results established in the previous sections, we can study the global existence for the bilinear system (1.1). Considering the formulation (1.10) above, we will apply the same arguments as in Theorem 1 (i.e. a fixed point argument) but using the functional space Let us start with the first estimate of (5.40). First, if n 0 ∈ L 2 (R), we get

E = L ∞ ([0, +∞], L 2 (R)) ∩ L 2 ([0, +∞], Ḣ1 (
h t * n 0 L 2 ≤ h t L 1 n 0 L 2 = n 0 L 2 .
Moreover, by Lemma 5.1 we have h t * n 0 L 2 t ( Ḣ1

x ) ≤ C n 0 L 2 and we can thus write

h t * n 0 E ≤ C 1 n 0 L 2 .
(5.42)

We study now the second estimate of (5.40) which is given by the quantity

B(n, n) E = t 0 h t-s * ∂ x χ * nH(Sn) (s, x) ds E . Applying Proposition 5.1 we have t 0 h t-s * ∂ x χ * nH(Sn) (s, x) ds E ≤ C ∂ x χ * nH(Sn) L 2 t ( Ḣ-1 x ) = C χ * nH(Sn) L 2 t (L 2
x ) . Let us study the last quantity above in the space variable. We first see that

χ * nH(Sn)(t, •) L 2 ≤ χ(t, •) L 2 n(t, •)H(Sn)(t, •) L 1 ≤ χ(t, •) L 2 n(t, •) L 2 H(Sn)(t, •) L 2 ,
hence, by the boundedness of the Hilbert transform in Lebesgue spaces together with the hypothesis S(t, •) ∈ L ∞ (R), we have

χ * nH(Sn)(t, •) L 2 ≤ C χ(t, •) L 2 n(t, •) L 2 S(t, •) L ∞ n(t, •) L 2 .
Taking the L 2 -norm in the time variable it yields

χ * nH(Sn) L 2 t (L 2 x ) ≤ C 2 χ L 2 t (L 2 x ) S L ∞ t (L ∞ x ) n L ∞ t (L 2 x ) n L ∞ t (L 2 x ) .
Consequently, reconstructing the norm • E given in (5.36), we get the estimate

t 0 h t-s * ∂ x χ * nH(Sn) (s, x)) ds E ≤ C B n E n E , (5.43)
where

C B = C 2 χ L 2 t (L 2 x ) S L ∞ t (L ∞ x )
. Now, from (5.42) and (5.43) if we have

C 1 n 0 L 2 < 1 4C 2 χ L 2 t (L 2 x ) S L ∞ t (L ∞ x )
, then the condition (5.41) is fulfilled (by choosing carefully the constant C given in (1.9)) and we can apply a fixed point argument to obtain a unique global solution since the time variable does not intervene in the estimates.

Proof of Theorem 4.

Here, we study global Solutions for the trilinear system (1.2). Following the idea given in formula (1.8) above, we introduce the term

S * [ θ] (t, x) = χ 1 * n 1 H(S 2 n 1 ) χ 2 * n 2 H(S 1 n 2 ) (t, x),
and with the variable θ defined in (1.5) the system (1.2) can be rewritten as

θ(t, x) = h t * θ 0 (x) + t 0 h t-s * ∂ x S * [ θ] (s, x)) ds (5.44)
We are going to apply Theorem 6 in the functional space

E = L ∞ ([0, +∞], L 2 (R)) ∩ L 2 ([0, +∞], Ḣ1 (R))
and for this we need to establish the required inequalities.

For the initial data satisfying the estimate (5.42) we have

h t * θ 0 L ∞ t (L 2 x ) ≤ θ 0 L 2 and h t * θ 0 L 2 t ( Ḣ1 x ) ≤ C θ 0 L 2 , hence h * θ 0 E ≤ C 1 θ 0 L 2 .
(5.45)

For the trilinear term we need to prove the following estimate

t 0 h t-s * ∂ x S * [ ] θ(s, •)) ds E ≤ C B θ 3 E , (5.46)
where C B may depend on the kernels K 1 , K 2 . By Proposition 5.1 we know that

t 0 h t-s * ∂ x S * [ ] θ(s, •)) ds E ≤ ∂ x S * [ ] θ L 2 t ( Ḣ-1 x ) ≤ S * [ ] θ L 2 t (L 2
x ) .

Moreover, by definition, we have

S * [ ] θ L 2 t (L 2 x ) ≤ χ 1 * n 1 H(S 2 n 1 ) L 2 t (L 2 x ) + χ 2 * n 2 H(S 1 n 2 ) L 2 t (L 2
x ) .

(5.47)

For 1 ≤ i, j, k ≤ 2, let us now study the term χ i * n i H(S j n k )(t, •) L 2 . By the Young and Hölder inequalities and by the boundedness of the Hilbert transform in Lebesgue spaces, we obtain

χ i * n i H(S j n k )(t, •) L 2 ≤ χ i (t, •) L 2 n i (t, •) L 2 H(S j n k )(t, •) L 2 ≤ C χ i (t, •) L 2 n i (t, •) L 2 S j (t, •)n k (t, •) L 2 ≤ C χ i (t, •) L 2 n i (t, •) L 2 (K j * n j )(t, •)n k (t, •) L 2 ,
where in the last line we used the fact that S j (t, •) = K j * n j (t, •). Hence, we have

χ i * n i (t, •)H(S j n k )(t, •) L 2 ≤ C χ i (t, •) L 2 n i (t, •) L 2 K j * n j (t, •) L ∞ n k (t, •) L 2 ≤ C χ i (t, •) L 2 n i (t, •) L 2 K j L 2 n j (t, •) L 2 n k (t, •) L 2 ,
where we used the Young inequalities in the convolution above. Taking this inequality to the power 2 and integrating in the time variable it comes:

+∞ 0 χ i * n i H(S j n k )(t, •) 2 L 2 dt 1 2 ≤ C K j L 2 n i L ∞ t (L 2 x ) n j L ∞ t (L 2 x ) n k L ∞ t (L 2 x ) +∞ 0 χ i (t, •) 2 L 2 dt 1 2
, and

χ i * n i H(S j n k ) L 2 t (L 2 x ) ≤ C χ i L 2 t (L 2 x ) K j L 2 n i L ∞ t (L 2 x ) n j L ∞ t (L 2 x ) n k L ∞ t (L 2 x ) ≤ C χ i L 2 t (L 2
x ) K j L 2 θ E θ E θ E , where we used the definition of the quantity θ and of the norm • E given in (5.36). Thus, coming back to (5.47) we obtain

S * [ ] θ L 2 t (L 2 x ) ≤ C( χ 1 L 2 t (L 2 x ) K 1 L 2 + χ 2 L 2 t (L 2 x ) K 2 L 2 ) θ E θ E θ E ,
which leads us to the wished inequality (5.46).

To conclude, with (5.45) and ( 5.46) at our disposal, following Theorem 6, if we have the condition θ 0 2

L 2 < 1 12C( χ 1 L 2 t (L 2 x ) K 1 L 2 + χ 2 L 2 t (L 2 x ) K 2 L 2 )
, we obtain a unique global solution for the problem (5.44).

Conclusion

The present paper considers two models to describe yeast cell polarization. In the first model a single cell in a pheromone field is considered. In the second model a system of two coupled equations is considered to describe the cell-cell communication for yeast cells of opposite types. The models include the pheromone concentration (pheromone secretion and pheromone perception) and also a non-constant and nonlocal mobility coefficient, namely the χ function, which describes the mobility of membrane polarity patch and the cell cycle [START_REF] Martin | Molecular mechanisms of chemotropism and cell fusion in unicellular fungi[END_REF][START_REF] Sieber | A focus on yeast mating: From pheromone signaling to cell-cell fusion[END_REF]. We provided conditions under which global existence and uniqueness of solutions occur for these two models. Physiologically, yeast cells mate when encountering a partner at close enough range in rich environments. Here, it was shown that even at close enough range and rich enough environment (large values of K i L 2 ) cells may not mate when the mobility is low or when one of the cells is leaving the cycle window during which it can mate. Indeed, in such a case global existence might hold true for the solution of the models studied here and then there will be neither polarization nor dialogue. This is in good agreement with biological observations showing that polarization of mating partners towards each other requires the spatial decoding of pheromone gradients, relies on mobile polarity patches and is only possible during a certain period of the cell cycle [START_REF] Martin | Molecular mechanisms of chemotropism and cell fusion in unicellular fungi[END_REF].

We believe this work is the first step towards modelling and understanding the key parameters invoved during the polarization of mating partners towards each other. This represents an important challenge. Future works would be to extend the results of this paper to find conditions for which blow-up of the solution occurs and then to the case of two-dimensional geometry with free boundary.

Figure 1 :

 1 Figure 1: Tangential model for the communication between two cells

Figure 2 :

 2 Figure 2: Model for yeast cell communication. On the left, yeast cells of both types secrete some pheromone (a or α) and bear a pheromone receptor to detect the pheromone produced by the cells of the opposite type.On the middle and on the right a two-dimensional model of protein dynamics inside each cell. The middle panel shows a cell, the right a more detailed view. Actin is polymerized into short filaments, that interact with each other and these are bundled together to form actin cables (which form the cytoskeleton) that cross the cell. The nucleation of filaments is proportional to both the local density of Cdc42 (the proteins that are transported by the cell cytoskeleton in each cell) and to the concentration of pheromone.

e

  n+1 -e n E = (e 0 + B(e n , e n , e n )) -(e 0 + B(e n-1 , e n-1 , e n-1 )) E = B(e n , e n , e n ) -B(e n-1 , e n-1 , e n-1 ) E = B(e n -e n-1 , e n , e n ) + B(e n-1 , e n -e n-1 , e n-1 ) +B(e n-1 , e n , e n -e n-1 ) E ≤ B(e n -e n-1 , e n , e n ) E + B(e n-1 , e n -e n-1 , e n-1 ) E + B(e n-1 , e n , e n -e n-1 ) E , using now the boundedness of the application B(•, •, •, ) we can write e n+1 -e n E ≤ C B e n -e n-1 E e n E e n E + C B e n-1 E e n -e n-1 E e n-1 E +C n e n-1 E e n E e n -e n-1 E , and since we have e n-1 E ≤ 2δ and e n E ≤ 2δ, we obtain e n+1 -e n E ≤ 12C B δ 2 e n -e n-1 E , thus, by iteration we have e n+1 -e n E ≤ 12C B δ 2 n e 1 -e 0 E , but since 12C B δ 2 < 1, we have lim n→+∞ e n+1 -e n E = 0, and thus the sequence (e n ) n∈N converges to a point e in E which satisfies the equation (4.34).

  Setting e 0 = h t * n 0 (x) and B(n, n) =

			t
				h t-s * ∂ x χ * nH Sn	(s, x) ds, we need to
	prove:		0
	e 0 E ≤ δ	and	B(n, n) E ≤ C B n E n E .	(5.40)
	Moreover, if we have the condition
			δ <	1 4C B	,	(5.41)
	then we obtain a unique solution of the problem (1.10) in the functional space E.

R)) endowed with the norm • E given in

(5.36)

. Thus, we have to prove the estimates (4.28) and (4.29) to obtain a solution. If the constants involved in these estimates do not depend on the time variable, we will obtain global solutions in time.


Manon Erguy
email: manon.erguy@cea.fr

Data visualization

Simple correlation

Cross-correlation

Auto-correlation of piezometry Auto-correlation of temperature  Study is being extended to other rain events. The first results on the October-November 2019 and June 2015 episodes do not necessarily show a faster temperature response but always a temperature peak appearing before the piezometry peak. The temperature signal could be integrated into a model as a precursor of groundwater flooding.

For some piezometers, no temperature variations are observed: sensor problem, rainfall/aquifer temperature delta too low, or bad position of the sensor in a non transmissive zone  A study is currently underway with the installation of several sensors at different depths within several piezometers for:  Verify the impact of censor depth on temperature data  Identify potential water inflows and link them to karst structures.

  Fig. 1. Simplified geology of the Cadarache site and hydrological monitoring network Fig. 2. Cross-correlogram example

Fig

  Fig. 5. Cross-correlations and hypothesis on the aquifer behavior

Fig

  Fig. 6. Future piezometer equipment diagram


Germán Leiva
email: leiva@lri.fr

Nolwenn Maudet
email: nolwenn@iis.u-tokyo.ac.jp

Michel Beaudouin-Lafon

Towards Collaborative Prototyping Tools for Interaction Design ABSTRACT

Keywords: Human-centered computing → Interface design prototyping, Systems Designer-developer collaboration, Video prototyping, Interactive prototyping

Current prototyping tools poorly support the co-creation of novel interactions beyond standard widgets and touch-based interactions. We present two collaborative tools that let designers quickly prototype non-standard interactions. M supports an enhanced video prototyping process while E supports design by enaction through multiple interconnected viewpoints. We believe that such prototyping tools facilitate collaboration across the boundaries of communities of practice and enable the creation of truly evolutionary prototypes.

INTRODUCTION

Cooper et al. define interaction design as "the practice of designing interactive digital products, environments, systems, and services" [START_REF] Cooper | About face 3: the essentials of interaction design[END_REF]. Interaction design borrows methods and techniques from other disciplines, such as product design and soware engineering. Usually, these multiple responsibilities are distributed between designers and developers.

During the interaction design process, designers and developers articulate their work using various design artifacts such as sketches [START_REF] Buxton | Sketching User Experiences: geing the design right and the right design[END_REF][START_REF] Greenberg | Sketching User Experiences: The Workbook[END_REF], storyboards [START_REF] Newman | Sitemaps, Storyboards, and Specifications: A Sketch of Web Site Design Practice[END_REF], and computational proxies [START_REF] Brown | Collaborative Events and Shared Artefacts: Agile Interaction Designers and Developers Working Toward Common Aims[END_REF]. These interaction design artifacts can generally be seen as boundary objects [START_REF] Leigh | The Structure of Ill-structured Solutions: Boundary Objects and Heterogeneous Distributed Problem Solving[END_REF][START_REF] Leigh | Institutional Ecology, 'Translations' and Boundary Objects: Amateurs and Professionals in Berkeley's Museum of Vertebrate Zoology[END_REF]: shared artifacts used by dierent communities of practice to satisfy their information needs. The interpretive flexibility of boundary objects allows collaboration to proceed in the absence of consensus and standards.

We are interested in how these artifacts are collaboratively co-created [START_REF] Sanders | Co-creation and the new landscapes of design[END_REF], especially when designers and developers need to co-create non-standard interactive system, i.e. systems that include interactions that go beyond established design paerns. Within the past few years, over 40 commercial prototyping tools have emerged 1 , and a 2015 survey of 4,000 designers found that 53% of them use such tools [START_REF] Vinh | The Tools Designers Are Using Today[END_REF].

1 hp://www.cooper.com/prototyping-tools These commercial tools focus on supporting remote communication between design and development, assisting the extraction of design information for implementation, and helping to quickly prototype standard interactions.

Few tools, however, support the co-creation of the design artifacts, for example through collaborative graphical authoring soware 2 . Usually, prototyping tools are used by a single community of 2 hps://www.figma.com/ practice [START_REF] Wenger | Communities of practice: Learning, meaning, and identity[END_REF], ignoring the back-and-forth between design and development. This results in focusing on a single viewpoint, either the designer's, e.g., visual representations, or the developer's, e.g., symbolic representations. Moreover, while current boundary objects such as sketches, storyboard, wireframes, and mock-ups excel at supporting the exploration of visual appearance, they poorly support the exploration of interaction behavior [START_REF] Myers | How designers design and program interactive behaviors[END_REF]. How can we provide beer collaborative tools supporting boundary objects for the manipulation of dynamic and continuous interactions?

VIDEO AS A RICH MEDIUM FOR EXPLORING INTERACTION

Figure 1: An overview of the M video prototyping system. Here, two designers, a wizard and a user-actor, collaborate side-by-side. The UserCam captures the user-actor enacting user inputs in the actual context, i.e. using his phone at the oice. The WizardCam captures the paper prototype and the Canvas captures the wizard's digital sketches.

Video prototyping [START_REF] Mackay | Video Prototyping: a technique for developing hypermedia systems[END_REF][START_REF] Mackay | Using video to support interaction design[END_REF][START_REF] Mackay | Video brainstorming and prototyping: techniques for participatory design[END_REF] combines paper prototyping [START_REF] Snyder | Paper prototyping: The fast and easy way to design and refine user interfaces[END_REF] with the Wizard-of-Oz technique [START_REF] Greenberg | Sketching User Experiences: The Workbook[END_REF] to create video artifacts that materialize the interaction design. Video artifacts can range from a simple and inexpensive recording of a traditional paper prototyping session [START_REF] Reig | Prototyping for tiny fingers[END_REF] to a complex and expensive movie production [START_REF] Tognazzini | The "Starfire" video prototype project[END_REF]. Here, we are focusing on low-cost video prototypes in the early stages of the design. Such videos make it diicult to create dynamic transformations that re-shape or modify visual elements, such as re-sizing or stretching elements in response to continuous user input. Moreover, introducing changes in the prototype creates inconsistencies with previously recorded scenes, requiring re-shooting and/or post-production editing.

We created M [START_REF] Leiva | Montage: A Video Prototyping System to Reduce Re-Shooting and Increase Re-Usability[END_REF], a new video prototyping tool to help a team of designers create video artifacts with minimal post-production editing and re-shooting. M is composed of a central device -the Canvas -connected to two mobile devices with video streaming and recording capabilities -the UserCam and the WizardCam (Figure 1). These devices, typically phones or tablets, are used as remote cameras, providing an inexpensive mobile movie studio. These cameras stream to the central Canvas, either in parallel or independently, the context of use in which the user interacts with the prototype and the prototyped user interface itself. The Canvas organizes the video clips into a grid of storylines. M lets designers compose shots using automatic green-screen replacement, whereby a video representing the interface replaces a green area of another video representing the use scenario 3 . Designers can also augment the interface or the scenario with digital sketches drawn directly on the Canvas, without the need of re-shooting (Figure 2). Videos are backed up in a cloud server, allowing new remote cameras to join and expand the current video prototype with new scenes. M provides multiple viewpoints that let designers collaborate simultaneously or independently. The WizardCam provides a physical prototyping area for paper prototyping. The Canvas provides a digital prototyping area for the creation of digital sketches. The UserCam provides an enactive [START_REF] Bruner | Toward a theory of instruction[END_REF] prototyping area where a user-actor can perform the interactions in context. The system is flexible enough to accommodate a variable number of designers. For example, the same person can act as paper prototyper and digital sketcher (Figure 1).

TOWARDS COLLABORATIVE INTERACTIVE PROTOTYPING

Video is a great medium to explore interactive behaviors within a given context of use, and systems such as M enable the collaborative co-creation of video artifacts. However, the final design artifact is a video prototype with no interactive capabilities for the viewer: the presenter can play, pause, rewind and fast-forward the video but the audience cannot interact with the prototype.

In order to create interactive prototypes of non-standard interactions, paper and video representations need to evolve into soware-based representations. Research has produced a number of novel tools for creating interactive prototypes beyond coding, but these tools are targeted at a single community of practice rather than the collaborative work of designers and developers. Approaches such as programming by demonstration [START_REF] Myers | Programming by example: intelligence in demonstrational interfaces[END_REF], state machines [START_REF] Appert | SwingStates: adding state machines to the swing toolkit[END_REF] or inference engines, for example, are interesting but have not been studied in a collaborative context. Aer studying current designer-developer collaboration practices and challenges, we have identified three main collaborative design breakdowns [START_REF] Maudet | Design Breakdowns: Designer-Developer Gaps in Representing and Interpreting Interactive Systems[END_REF]: missing information, when designers do not communicate specific details; edge cases, when designers do not think about particular cases; and technical constraints, when designers are not aware of technical limitations. These studies also showed that current workflows and tools induce unnecessary rework: Designers create a multitude of redundant design documents and developers must recreate them with their own tools, introducing mismatches with the original design. These findings show that designers still struggle to have a "conversation" with the soware material [START_REF] Kursat Ozenc | How to support designers in geing hold of the immaterial material of soware[END_REF].

We created E (Figure 3), a tool that reduces these design breakdowns during the collaborative prototyping of touch-based mobile interaction. Through multiple interconnected representations of the interaction under construction, E reduces reworking, redundancies and design breakdowns 4 . 4 See hps://www.youtube.com/watch?v= uMJ9uNbPvwE E features a linear storyboard that propagates changes from past to future screens, thus reducing redundancies within the storyboard. To facilitate navigation between visual and symbolic representations, the code editor is aware of the design elements in the storyboard: Visual elements can be dragged from the storyboard into the editor or directly accessed by their symbolic names, using intelligent code completion. Thanks to this tight integration, E supports visual representations for elements that are usually only available symbolically, such as measures that represent, e.g., the distance between two elements (Figure 4), or touch events representing the position of the finger and the radius of the contact area. Intermediate representations, such as the interactive state machine diagram, let designers and developers develop a common vocabulary, quickly explore the interaction under construction and detect edge cases. The most important aspect of E is the use of the target device, in this case a mobile phone, not only as a testing device but also as a design device. The system relies on a similar approach to programming with examples [START_REF] Kato | Programming with Examples to Develop Data-Intensive User Interfaces[END_REF] that we call design by enaction : Designers or developers can perform the desired user inputs directly on the target device as if they were the final user, in the same way as the user-actor in M; While acting out the interaction, E highlights the corresponding state in the state machine diagram and records every user input event. Designers can then define which screen of the storyboard is associated with which input event in the recorded sequence of user inputs. Developers can use this concrete example as a test to guide development [START_REF] Beck | Test-driven development: by example[END_REF]. E also provides an assisted testing feature that automatically replays the recorded example and highlights the mismatches between the implementation and the desired design. Initiatives such as the Hour of Code [8] try to spread programming to a broader audience. Undoubtedly, more designers will know how to program in the future than today. However, code is not the only and most adequate representation for every aspect of interaction design. We need to provide multiple representations, e.g. symbolic, visual or enactive, to create collaborative spaces in which professionals with dierent skills, mindsets and values can work together. Digital tools need to provide ways of navigating these representations in a fluid way, thus reducing reworking and increasing reuse.

CONCLUSION

We outlined two collaborative prototyping tools for interaction design. Both projects emphasize sharing a common context, not only to prototype the interaction but also to prototype the user experience [START_REF] Buchenau | Experience prototyping[END_REF]. M encourages designers to create scenarios that can be explored with dierent alternatives of the system, or to assess a system in multiple scenarios. E supports design by enaction, where the target device is not only used for testing but also as a design medium to explore user interactions. While we focused on early stage prototyping, the ideas behind these tools can be applied to other collaborative tools. The ultimate goal is to beer integrate such tools with the final digital product in order to create truly evolutionary prototypes [START_REF] Beaudouin-Lafon | Prototyping tools and techniques[END_REF].

Figure 2 :

 2 Figure 2: The Canvas sketching interface: The final composition (le) and the interface (right) show the "user overlay". Both sides have a list of sketches and animation controls at the boom.

Figure 3 :

 3 Figure 3: E uses a target mobile device and a desktop interface with five areas: a storyboard with consecutive screens, an event timeline with a handle for each screen, a state machine, a code editor and a device mirror

Figure 4 :

 4 Figure 4: The designer created two measures, one between the touches and the other between the two rectangles. These measures are invisible on the target device (le) but they are revealed in the device mirror (right) and updated in real time to facilitate exploration.

Figure 5 :

 5 Figure 5: On the le, the developer performs an o-device mimicking gesture with his le hand to understand the proposed design. On the right, the designer performs an on-device gesture with both hands to communicate the design.

See hps://www.youtube.com/watch?v=7z_ y8AX9gu8

ACKNOWLEDGMENTS

This work was partially supported by European Research Council (ERC) grants № 321135 "CREATIV: Creating Co-Adaptive Human-Computer Partnerships" and № 695464 "ONE: Unified Principles of Interaction".