Virginie Stoppin-Mellet

Jérémie Gaillard

Ton Timmers

Ton Timmers b Emmanuelle Neumann

James Conway

Marylin Vantard
email: marylin.vantard@cea.fr

Arabidopsis katanin binds microtubules using a multimeric microtubule-binding domain

Keywords: katanin, microtubule-binding domain, oligomerization, severing of microtubules AtKSS, Arabidopsis thaliana katanin small subunit, GST, Glutathione

teaching and research institutions in France or abroad, or from public or private research centers.

Arabidopsis katanin binds microtubules using a multimeric microtubule-binding domain.

Introduction

Plant microtubules are located in the cell cortex (cortical microtubules during interphase, preprophase band at the entry of mitosis) and in the cytoplasm (nucleusassociated microtubules, spindle and phragmoplast microtubules in mitosis), where they are arranged in highly dynamic and ordered arrays. Upon internal and external stimuli, these arrays reorganize, while acquiring specific functions including the equal segregation of chromosomes, the formation of the cell plate at the end of mitosis, the transport of cargo and the positioning of organelles [START_REF] Kost | Cytoskeleton in plant development[END_REF][START_REF] Lloyd | Microtubules and the shape of plants to come[END_REF]. The architectural changes of these microtubule arrays are dramatic manifestations of the flexibility and dynamic turnover of the microtubules in plant cells. The precise mechanisms underlying the remodeling of microtubule arrays remain difficult to elucidate, mainly because of the complexity of the plant microtubular cytoskeleton and because of the absence in plant cells of a discrete microtubule-organizing center. When compared with animal microtubules, plant microtubules possess very peculiar properties in vivo [START_REF] Chan | EB1 reveals mobile microtubule nuc leation sites in Arabidopsis[END_REF][START_REF] Dhonukshe | Alteration of microtubule dynamic instability during preprophase band formation revealed by yellow fluorescent protein-CLIP170 microtubule plus-end labeling[END_REF][START_REF] Shaw | Sustained microtubule treadmilling in Arabidopsis cortical arrays[END_REF][START_REF] Vos | Microtubules become more dynamic but not shorter during preprophase band formation: a possible "search-and-capture" mechanism for microtubule translocation[END_REF],

including the fact that cortical microtubules exhibit polymerization-biased dynamic instability at one end and slow depolymerization at the other end [START_REF] Shaw | Sustained microtubule treadmilling in Arabidopsis cortical arrays[END_REF].

In vivo, the control of microtubule assembly and disassembly requires a multitude of adaptors and regulators. These proteins associate with the polymer lattice, with the microtubule ends or with the tubulin dimer. They regulate microtubule stabilization or destabilization in a way to generate the different microtubule arrays observed in vivo [START_REF] Hashimoto | Dynamics and regulation of plant interphase microtubules: a comparative view[END_REF][START_REF] Sedbrook | MAPs in plant cells: delineating microtubule growth dyna mics and organization[END_REF]. Until recently the identification of plant microtubule regulators has been rather elusive because of their low abundance and because most of them show low sequence homology with animal counterparts. Several plant microtubulestabilizing effectors, also called Microtubule Associated Proteins, have now been identified and their functional characterization is under way [START_REF] Hussey | The plant cytoskeleton: recent advances in the study of the plant microtubule-associated proteins MAP-65, MAP-190 and the Xenopus MAP215-like protein, MOR1[END_REF][START_REF] Sasabe | MAP65: a bridge linking a MAP kinase to microtubule turnover[END_REF]. So far, AtKSS (Arabidopsis thaliana Katanin Small Subunit) is the only plant microtubuledestabilizing protein that has been functionally characterized.

Katanin is a heterodimer composed of a catalytic subunit of 60 kDa (p60) that hydrolyses ATP to sever microtubules, and a regulatory subunit of 80 kDa (p80) involved in subcellular targeting of katanin [START_REF] Mcnally | Identification of katanin, an ATPase that severs and disassembles stable microtubules[END_REF][START_REF] Hartman | Katanin, a microtubule-severing protein is a novel AAA ATPase that targets to the centrosome using a WD40-containing subunit[END_REF]. p60 oligomerizes into a ringshaped hexamer in an ATP-dependent process which is coupled to the disassembly of microtubules [START_REF] Hartman | Microtubule disassembly by ATP-dependent oligomerization of the AAA enzyme katanin[END_REF]. In animal cells, katanin is located at the centrosomes and at spindle poles where it controls the length of the mitotic spindle [START_REF] Mcnally | Katanin controls mitotic and meiotic spindle length[END_REF][START_REF] Mcnally | Katanin, the microtubulesevering ATPase, is concentrated at centrosomes[END_REF]. In plants, the Arabidopsis homologue of katanin catalytic subunit, named AtKSS, was identified on the basis of its sequence similarity with human (homo sapiens) and sea urchin katanins [START_REF] Mcclinton | cDNA isolation, characterization, and protein intracellular localization of a katanin-like p60 subunit from Arabidopsis thaliana[END_REF]. In vitro, AtKSS has a microtubule-stimula ted ATPase activity and severs microtubules in an ATP-dependent [START_REF] Stoppin-Mellet | Functional evidence for in vitro microtubule severing by the plant katanin homologue[END_REF]. In vivo, properties and functions of plant katanin are not well understood. Over the last few years, five AtKSS mutants have been described. botero1 (bot1) is a null mutant [START_REF] Bichet | BOTERO1 is required for normal orientation of cortical microtubules and anisotropic cell expansion in Arabidopsis[END_REF], while the four other mutants, fragile fiber (fra2; [START_REF] Burk | A katanin-like protein regulates normal cell wall biosynthesis and cell elongation[END_REF][START_REF] Burk | Alteration of oriented deposition of cellulose microfibrils by mutation of a katanin-like microtubule-severing protein[END_REF]), ectopic root hair 3 (erh3, [START_REF] Webb | Cell specification in the Arabidopsis root epidermis requires the activity of ECTOPIC ROOT HAIR 3 -a katanin-p60 protein[END_REF]), LUC-superexpressor 1 (lue1, [START_REF] Bouquin | The Arabidopsis lue1 mutant defines a katanin p60 ortholog involved in hormonal control of microtubule orientation during cell growth[END_REF]), dwarf and gladius leaf1 (dgl1, [START_REF] Komorisono | Analysis of the rice mutant dwarf and gladius leaf 1. Aberrant katanin-mediated microtubule organization causes up-regulation of gibberellin biosynthetic genes independently of gibberellin signaling[END_REF]) are mutated in the Cterminal ATPase domain of AtKSS. All these mutants show a common dwarf phenotype associated with an alteration of cellular anisotropic growth. But a closer look to the plants revealed differences between the mutants. For example, both fra2 and lue1 show altered composition and organization of cellulose microfibrils, as well as two-branched trichomes, whereas none of these two defects was observed in bot1. Similarly, some but not all cells of bot1, dgl1, fra2 and lue1 have disorganized cortical microtubules. Because of the complexity and the diversity of the phenotypes of AtKSS mutants, it has been difficult to get a clear view of the functions of katanin in vivo. By analyzing the induced expression of AtKSS in Arabidopsis, we recently showed that katanin favors bundling of microtubules in the cell cortex [START_REF] Stoppin-Mellet | Katanin's severing activity favors bundling of cortical microtubules in plants[END_REF]. Obviously, AtKSS is required for a proper organization of cortical microtubules in expanding plant cells, but whether the disorganization of cortical microtubule arrays is a direct or indirect consequence of AtKSS's defect is debated.

To further understand the biological roles of AtKSS, we studied the properties of full-length and truncated AtKSS both in vitro and in vivo. W e report here that AtKSS interacts with microtubules through a poorly conserved basic domain, both in vitro and in vivo. We observed that AtKSS organized into ring-shape structures, and that the microtubule-binding domain of AtKSS formed trimers. These results suggest that the microtubule-binding domain of AtKSS participate in the oligomerization of plant katanin.

Results

Identification of the AtKSS microtubule-binding domain

We previously showed that AtKSS binds microtubules in vitro [START_REF] Stoppin-Mellet | Functional evidence for in vitro microtubule severing by the plant katanin homologue[END_REF]. In a first step to understand the molecular mechanisms by which the plant katanin interacts with microtubules, we analysed the functions of the domains of AtKSS. On Caenorhabditis, Drosophila, human, and Xenopus sequences). The proteins corresponding to these domains were produced in fusion with Glutathione S-Transferase GST (GST-AtKSS1, GST-AtKSS2, GST-AtKSS3), as well as AtKSS deleted of one domain (GST-AtKSS12, GST-AtKSS13, GST-AtKSS23; figure 1B and1C). It has to be noted that whatever expression and purification conditions we used, samples of GST-AtKSS23 contained a C-terminus truncated GST-AtKSS23 (figure 1C, lane 6, lower band). As these two proteins can form hetero-oligomers (see results on katanin oligomerization), we could not separate them in native conditions.

When purified in denaturing conditions and further refolded, the full-length GST-AtKSS23 loses some of its properties. For these reasons, GST-AtKSS23 we used in this study corresponds to a mixture of the two proteins.

In order to identify the microtubule-binding domain of AtKSS, GST-AtKSS constructs were added to taxol-stabilized microtubules. After centrifugation, analysis of the supernatants and pellets showed that both GST-AtKSS12 and GST-AtKSS2 co-sedimented with microtubules, whereas neither the short N-terminus domain (GST-AtKSS1) nor the ATPase domain (GST-AtKSS3, GST-AtKSS13) interacted with microtubules (figure 2). These results indicate that amino acids 35 to 210 of AtKSS are involved in the interaction of AtKSS with microtubules in vitro.

Affinity of the microtubule-binding domain of AtKSS for microtubules in vitro and in vitro

We performed co-sedimentation assays with microtubules to evaluate the dissociation equilibrium constant of the GST-AtKSS2 and GST-AtKSS12 (figure 3).

We calculated an apparent K d of 5.10 µM for GST-AtKSS2 and 6.87 µM for GST-AtKSS12. The apparent K d of GST-AtKSS12 and unfused AtKSS12 had similar values (6.87 µM for GST-AtKSS12 versus 18.03 µM for AtKSS12), showing that in our experimental conditions GST did not significantly interfere with the binding of the proteins to microtubules. These results show that AtKSS binds microtubules with a low affinity, and that the N-terminus part of AtKSS (domain 1) is not involved in the binding of the protein to microtubules in vitro.

We wondered whether the microtubule-binding domain of AtKSS interacts with microtubules in vivo, where microtubules are covered with microtubule-associated proteins. We transiently expressed AtKSS12 or AtKSS2 fused to DsRed in onion epidermal cells. Sixteen hours after transformation, the AtKSS12-DsRed (figure 4A)

and AtKSS2-DsRed were detected along filaments. We did not detect any difference between the localization of AtKSS12 and AtKSS2 in vivo. Transformed cells were treated with 20 µM oryzalin, a potent plant microtubule destabilizer. This concentration of oryzalin is sufficient to depolymerize all microtubules detectable at the light microscope level in vivo [START_REF] Binet | Disruption of microtubular cytoskeleton induced by cryptogein, an elicitor of hypersensitive response in tobacco cells[END_REF]. After 20 min of incubation, almost all the stained filaments had disappeared (figure 4B-D). These results show that the domain of AtKSS delimited by the amino acids 35 to 210 binds microtubules in living cells.

Oligomerization of AtKSS

In order to determine whether AtKSS is able to form oligomers, we examined the structure of AtKSS molecules at the electron microscopy level. AtKSS appeared as irregular rings of 25-30 nm diameter composed of globular subunits (figures 5A-C). Rings of AtKSS often contained a crack radiating outward. These rings are very similar to hexamers of animal p60 [START_REF] Hartman | Katanin, a microtubule-severing protein is a novel AAA ATPase that targets to the centrosome using a WD40-containing subunit[END_REF].

To identify domains of AtKSS that may be involved in the o ligomerization, we studied the oligomeric state of the different AtKSS constructs. First, the molecular mass of the different katanin constructs was estimated by gel filtration. The Nterminus part of katanin (GST-AtKSS1), the ATPase domain of katanin (GST-AtKSS3), and the chimeric protein composed of these two domains (GST-AtKSS13)

were eluted from the column as dimers (table I). Because GST exists as a dimer in its native state [START_REF] Ji | The threedimensional structure of a glutathione S-transferase from the mu gene class. Structural analysis of the binary complex of isoenzyme 3-3 and glutathione at 2.2-A resolution[END_REF], and because in our experimental conditions GST was eluted as a dimer (table I), we concluded that dimers of GST-AtKSS1, GST-AtKSS3 and GST-AtKSS13 resulted from the dimerization of GST. In contrast the microtubule-binding domain of katanin, either alone (GST-AtKSS2) or combined to the N-terminus domain (GST-AtKSS12) was eluted as trimers. We also performed cross-linking experiments with the different GST constructs using the zero-length cross-linker EDC together with NHS. These experiments could not be done with GST-AtKSS2, as the protein is not stable at room temperature. Incubation of GST-AtKSS12 with the crosslinker yielded to a band of an apparent molecular mass of 160 kDa (figure 5D). This result suggests that AtKSS12 forms trimers. None of the chimeric proteins lacking the microtubule-binding domain (GST alone, GST-AtKSS1, GST-AtKSS3 (data not shown) or GST-AtKSS13 (figure 5E)) formed oligomers. Trimers of AtKSS12 without GST were also observed after cross-linking (figure 5F). All together these data suggest that AtKSS oligomerizes into a ring-shaped oligomers, and that the microtubule-binding domain of AtKSS is invo lved in the oligomerization process.

ATPase and microtubule-severing activity of katanin.

We studied the ATPase activity and severing activity of the different domains of katanin. From all the chimeric AtKSS we studied, only those containing domain 3

(GST-AtKSS13, GST-AtKSS23 and GST-AtKSS3) hydrolyzed ATP (figure 6A). GST-AtKSS2 and GST-AtKSS12 had no detectable ATPase activity (data not shown).

GST-AtKSS23 had the highest ATPase activity, and its activity was stimulated by microtubules, as observed for baculovirus-expressed full-length AtKSS [START_REF] Stoppin-Mellet | Functional evidence for in vitro microtubule severing by the plant katanin homologue[END_REF].

Hydrolysis of ATP by GST-AtKSS13 and GST-AtKSS3 was not stimulated by microtubules. These data are in agreement with our results showing that the ATPase domain of katanin is not able to interact with microtubules (figure 2).

We also analysed the microtubule-severing activities of different katanin constructs using turbidity experiments and video-microscopy assays. We observed that the chimeric katanin containing the microtubule-binding domain and the ATPase domain (GST-AtKSS23) destabilized microtubules in vitro (figure 6B

Discussion

Up to now, data obtained from the analysis of four AtKSS mutants [18-20, 22, 23] and from the induced expression of AtKSS in arabidopsis plants [START_REF] Stoppin-Mellet | Katanin's severing activity favors bundling of cortical microtubules in plants[END_REF] clearly indicate that AtKSS is required for a proper organization of cortical microtubules in expanding plant cells. But how katanin achieves this function is still unknown. In this study, we examined some of the molecular properties of katanin, including its interaction with microtubules and its ability to form oligomers.

Our data show that amino acids 35 to 210 of AtKSS are sufficient for AtKSS to bind microtubules both in vitro and in vivo. Noticeably plant and animal catalytic subunits of katanin bind microtubules via the same domain (domain 2, see figure 1A)

and with a comparable K d (this study, [START_REF] Hartman | Microtubule disassembly by ATP-dependent oligomerization of the AAA enzyme katanin[END_REF][START_REF] Mcnally | Two domains of p80 katanin regulate microtubule severing and spindle pole targeting by p60 katanin[END_REF]). This may be due to the fact that although the amino-acid sequences of plant and animal microtubule-binding domains of katanin are poorly conserved, they are enriched in basic amino acids that can interact with the acidic tubulin surface.

We found that besides mediating the interaction of katanin with microtubules, the microtubule-binding domain of AtKSS allo ws intermolecular interactions between katanin subunits. Indeed, the microtubule-binding domain of AtKSS oligomerizes into trimers and AtKSS forms rings similar in size and in shape to sea urchin katanin [START_REF] Hartman | Katanin, a microtubule-severing protein is a novel AAA ATPase that targets to the centrosome using a WD40-containing subunit[END_REF].

These results suggest that the microtubule-binding domain is involved in the oligomerisation of katanin. It has been shown that the microtubule-severing activity of animal katanin is correlated with its ability to oligomerize into a ring-shaped hexamer [START_REF] Hartman | Microtubule disassembly by ATP-dependent oligomerization of the AAA enzyme katanin[END_REF]. Assuming that the active AtKSS is also a hexamer, we propose that the oligomerization of the catalytic subunit of katanins takes place in at least two steps:

the formation of a trimer and then the addition of two trimers to form a hexamer. More detailed structural data are now required to understand how katanin subunits interact to form a hexamer, and to evaluate the role of the catalytic state of the ATPase domain in this process.

Our data did not allow us to specify a function to the N-terminus of AtKSS (AtKSS1). This domain does not bind microtubules by itself, neither does it interfere with the interaction of the microtubule-binding domain with microtubules as K d values of GST-AtKSS2 and GST-AtKSS12 were similar. It is probably not essential for the oligomerization of AtKSS as GST-AtKSS23 fragments microtubules. It is possible that this domain, although not necessary for severing activity of AtKSS, favors one or several steps of the microtubule-binding/oligomerization/ATP hydrolysis cycle of AtKSS, and that the procedures we used were not sensitive enough to detect such a fine regulation. The N-terminus part of AtKSS may also be involved in the recruitment of AtKSS-regulatory proteins, as it has been shown for animal katanin [START_REF] Mcnally | Two domains of p80 katanin regulate microtubule severing and spindle pole targeting by p60 katanin[END_REF]. AtKSSinteracting proteins have recently been identified by two-hybrid screen [START_REF] Bouquin | The Arabidopsis lue1 mutant defines a katanin p60 ortholog involved in hormonal control of microtubule orientation during cell growth[END_REF].

Characterization of the ability of these proteins to regulate katanin in vitro and in vivo will be of great help in the understanding of the roles of microtubule severing in higher plants.

Non-ectopic fragmentation of microtubules in plant cells implies that the activity of katanin is regulated in vivo. One way to control katanin's activity is to modulate its binding to microtubules. Several lines of evidence suggest that katanin transiently interacts with microtubules. First, anti-katanin antibodies did not detect AtKSS along microtubules [START_REF] Mcclinton | cDNA isolation, characterization, and protein intracellular localization of a katanin-like p60 subunit from Arabidopsis thaliana[END_REF]. Second, in our hands, co-localization of katanin with microtubules in vivo was only observed in transiently transgenic cells in which AtKSS was overexpressed after biolistic transformation (our unpublished data). Neither in transgenic Arabidopsis cells expressing AtKSS in an inducible manner [START_REF] Stoppin-Mellet | Katanin's severing activity favors bundling of cortical microtubules in plants[END_REF], nor in tobacco cells expressing AtKSS after agro-infiltration (our unpublished results), could we detect an interaction between katanin and microtubules. Indeed, there is only one study reporting the co-localization of AtKSS with microtubules in vivo [START_REF] Bouquin | The Arabidopsis lue1 mutant defines a katanin p60 ortholog involved in hormonal control of microtubule orientation during cell growth[END_REF]. In this work, stably overexpressed AtKSS-GFP-GUS decorates microtubules in arabidopsis plants. However the functionality of the tripartite chimeric protein used in the study has been questioned by the authors. Third, the K d value we calculated for the microtubule-binding domain of AtKSS is at least ten times higher than the K d we estimated for plant MAP65-1b [START_REF] Icker-Planquart | Interactions of tobacco microtubule-associated protein MAP65-1b with microtubules[END_REF]. All these observations are consistent with a transient interaction of plant katanin with microtubules. The absence of colocalization of a microtubule-interacting protein with microtubules in vivo is not unique to katanin. Recent studies showed that spastin, an AAA katanin-related protein that destabilizes microtubules does not co-localized with microtubules in animal cells, whereas mutated spastin with altered ATPase activity does [START_REF] Evans | Linking axonal degeneration to microtubule remodeling by spastin-mediated microtubule severing[END_REF][START_REF] Roll-Mecak | The drosophila homo logue of the hereditary spastic paraplegia protein, spastin, severs and disassembles microtubules[END_REF]. These results

show that ATP hydrolysis is involved in the interaction of spastin with microtubules.

Whether a comparable mechanism also affects the interaction of plant katanin with microtubules remains to be shown.

The present study provides new insights into the molecular properties of plant katanin. However many questions remain to be answered before we can fully understand the molecular mechanisms underlying microtubule severing by katanin in plants. Does katanin preferentially associate with a subset of microtubules in vivo? If so, how is this association regulated? Does it depend on the dynamic properties of microtubules, and/or on the presence of microtubule-associated proteins such as MAP65 proteins? W hat is the role of the homologue of the katanin-accessory protein p80? The ongoing characterization of plant microtubule-interacting proteins should soon provide insights into these challenging issues.

Methods

Chemicals and antibodies

Unless specified all chemicals were purchased from Sigma (St Louis, MO, USA).

GST antibodies were purchased from Amersham Pharmacia Biotech (Little Chalfont, England). Anti-AtKSS antibodies raised against the N-terminus peptide of the protein were previously described [START_REF] Stoppin-Mellet | Functional evidence for in vitro microtubule severing by the plant katanin homologue[END_REF].

AtKSS cloning and expression

Three domains of AtKSS were defined: the 34 first amino-acids (domain 1), amino acids 35 to 210 (domain 2) and the ATPase domain (aa 211 to 523; domain 3).

These domains are further referred to as AtKSS1, AtKSS2 and AtKSS3 respectively.

AtKSS1, AtKSS2, AtKSS3, as well as combinations of the domains, AtKSS12 (aa 1 to 210), and AtKSS23 (aa 211 to 523), were cloned into the pGEX-4T-1 plasmid (Amersham Pharmacia Biotech) in frame with the Glutathione S-Transferase GST (Amersham Pharmacia Biotech) using the following primers: for AtKSS1, primers Biosciences, Darmstadt, Germany) except GST-AtKSS23 that was expressed in Rosetta-gami (DE3)pLysS bacteria (Novagen). Proteins were purified on a glutathione column (Amersham Pharmacia Biotech) according to the manufacturer procedures. Purified proteins were frozen in liquid nitrogen in the presence of 10% (v/v) glycerol and stored at -80°C.

Expression of AtKSS in vivo

For transient expression in onion epidermal cells, full-length AtKSS and truncated AtKSS were cloned into the pOL-LT plasmid [START_REF] Mollier | The gene encoding Arabidopsis thaliana mitochondrial ribosomal protein S13 is a recent duplication of the gene encoding plastid S13[END_REF], under the control of a double strong 35S promoter and upstream the DsRed2 fluorophore (Clontech, BD Biosciences, NJ, USA). Cells were transformed by bombardment with DNA-coated gold beads as described [START_REF] Chiu | Engineered GFP as a vital reporter in plants[END_REF]. Cells were observed 12 to 25 hours after transformation using a Leica TCS-SP2 confocal microscope (Leica Microsystems, Wetzlar, Germany). DsRed2 fluorophore was visualized under the follo wing illumination conditions: 543 nm excitation, 580 to 620 nm emission.

Microtubule cosedimentation assays

Tubulin was purified from beef brain according to [START_REF] Hyman | Preparation of modified tubulin[END_REF]. Taxol-stabilized microtubules were prepared using a procedure adapted from [START_REF] Belmont | Identification of a protein that interacts with tubulin dimers and increases the catastrophe rate of microtubules[END_REF] according to the recommendations described at the web site of Tim Mitchison's lab (http://mitchison.med.harvard.edu/protocols/poly.html). For quantification, 10 µl of taxol-stabilized microtubules were depolymerized for 10 min at 4°C in 90 µl of cold depolymerizing buffer (50 mM KCl, 5 mM CaCl 2 ,1 mM DTT in cold assembly buffer).

The concentration of tubulin dimers was calculated using the absorbance of the sample at 280 nm and an extinction coefficient of 115,000 M -1 cm -1 . Considering the quantification procedure, the amount of taxol-stabilized microtubules used in the assays is indicated as concentrations of tubulin dimers (figures 2 and 3). To limit adsorption of AtKSS on plastic surfaces, all eppendorf tubes and tips were siliconized [START_REF] Sambrook | Molecular cloning: a laboratory manua l[END_REF]. To identify the domains of AtKSS involved in microtubule binding, taxolstabilized microtubules (0 to 3 µM) were incubated for 20 min at 25°C with 0.

[ ]= ( K d + MT tot [ ]+[ K tot ]-( K d +[MT tot ]+[ K tot ]) 2 -4[MT tot ][ K tot ] 2
where [KMT] is the concentration of AtKSS bound to microtubules, [MT tot ] and [K tot ] are the total concentrations of tubulin dimers and AtKSS respectively. For K d calculation, data were analysed using Kaleidagraph (Synergy Software, PA, USA).

Electron microscopy

AtKSS was purified from insect cells as previously described [START_REF] Stoppin-Mellet | Functional evidence for in vitro microtubule severing by the plant katanin homologue[END_REF]. Specimens at a concentration of 0,1 mg/ml were imaged by negative stain electron microscopy using a JEOL 1200 EXII microscope operating at 100 kV and a nominal magnification of 40,000x. The stain used was sodium silico-tungstate.

Gel filtration chromatography

Gel filtration of katanin constructs was performed on a fast protein liquid chromatography system with a Superdex TM 200 HR 10/30 column (Amersham Pharmacia Biotech). Column equilibration and chromatography were performed in column buffer (20 mM Tris pH 8, 0.1 M KCl, 2 mM MgCl 2 , 1 mM DTT). The column was calibrated with the following molecular mass standards: ribonuclease A (13.7 kDa), ovalbumin (43 kDa), albumine (67 kDa), aldolase (158 kDa), catalase (232 kDa), ferritin (440 kDa), Dextran Blue 2000 (Amersham Pharmacia Biotech). After 15 min in the presence of GST-AtKSS23, most microtubules were severed (F).

Chemical cross-linking

Bar= 10 µm

  and 6E-F). In contrast, neither GST-AtKSS1, nor the microtubule binding domain alone (GST-AtKSS2) or fused to the domain 1 (GST-AtKSS12), nor the ATPase domain alone (GST-AtKSS3) severed microtubules in vitro (data not shown). Our data did not allo w us to quantitatively compare the severing activities of the GST-AtKSS23 and the fulllength katanin and thus to determine if the presence of the short NH 2 -terminal domain (AtKSS1) modulates katanin's activities in vitro. Our results show that the microtubule-binding domain and the ATPase domain are sufficient for katanin to sever microtubules in vitro.

  3'; for AtKSS12, primers 5HAtKSS60 and R210; for AtKSS23, primers F34 and 3HAtKSS. The chimeric AtKSS truncated of the domain 2, AtKSS13 (aa 1 to 34 and 211 to 523) was obtained by PCR using the QuickChange TM site-directed mutagenesis kit (Stratagene, CA, USA), the full length AtKSS in pGEX-4T-1 as substrate and the following primers: Fsequences were verified by sequencing. All GST-AtKSS proteins were expressed overnight at 15°C in Rosetta (DE3)pLysS bacteria strain (Novagen-Merck

  EDC), along with N-hydroxyccinimide (NHS) were obtained from Pierce (Rockford, IL, USA). Stock solutions of 100 mM EDC and NHS were made fresh in dry DMSO immediately before use. Purified AtKSS proteins (20 µg) were incubated with 1 mM EDC and 1 mM NHS at room temperature. After 10 min, 1 mM of EDC and NHS were again added to the samples. After 20 min the reaction was stopped by the addition of 9 mM glycine. The cross-linked AtKSS constructs were observed by Coomassie-blue stained gels and western-blots.

Fig. 1 .

 1 Fig. 1. AtKSS constructs used in this study. (A) Alignment of amino acid sequences of most known katanin p60 including human (Hs), Xenopus laevis (Xl), Strongylocentrotus purpuratus (Sp), Drosophila melanogaster (Dm), Arabidopsis thaliana AtKSS (At), Chlamydomonas reinhardtii (Cr) and Caenorhabditis elegans (Ce). Identical and similar amino acids are highlighted in black and gray respectively. Dashes represent gaps. (B) Diagram of the different AtKSS constructs corresponding to combination of domain 1 (aa 1 to 34, black box), domain 2 (aa 35 to 210, white box) and domain 3 (aa 211 to 523, gray box). C, Coomassie blue-stained SDS-PAGE gel of the purified AtKSS constructs. Lane 1, GST. Lane 2, GST-AtKSS1. Lane 3, GST-AtKSS2. Lane 4, GST-AtKSS12. Lane 5, GST-AtKSS3. Lane 6, GST-AtKSS23. Lane 7, GST-AtKSS13. Masses of molecular markers are indicated in kDa.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Binding of AtKSS constructs to microtubules in vitro. 0.4 µM of GST-AtKSS constructs were incubated with various amount of microtubules. Samples were centrifuged to pellet microtubules and the amount of GST-AtKSS proteins bound to microtubules was evaluated by densitometry. A, Western-blot of supernatant (S) and pellet (P) of GST-AtKSS proteins probed with GST antibodies. Concentrations of microtubules used in each assay (indicated as the concentration of tubulin dimers) are indicated. B, Results are expressed as percentage of total GST-AtKSS constructs that pelleted with microtubules versus tubulin dimers concentration.

Fig. 4 .

 4 Fig. 4. Localization of AtKSS12 in vivo. (A) Onion epidermal cell transiently expressing AtKSS12-DsRed and observed 16 hours after bombardment. A dense and disorganized network of filaments is detected in the cortex of the cell. (B to D) Time-lapse observation of an onion cell expressing AtKSS-DsRed and treated with 20 µM oryzalin, a microtubule-depolymerizing drug, for 0 min (B), 10 min (C) and 20 min (D). After 20 min of treatment, no microtubules were observed.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Structure and oligomerization of AtKSS. (A to C) Electron micrographs of AtKSS purified from baculovirus-infected insect cells. Rings are 25-30 nm diameter. (D to F) Cross-linking of GST-AtKSS12, GST-AtKSS13 and AtKSS12. GST-AtKSS12 (D), GST-AtKSS13 (E) and AtKSS12 (F) were incubated in the absence (lanes -) or in the presence (lanes +) of cross-linker molecules. Samples were analysed by coomassie blue-stained SDS-PAGE (gel) and by western-blot an anti-GST antibody (GST-AtKSS12 and GST-AtKSS13) or anti-AtKSS antibodies (AtKSS12). Masses of molecular markers are indicated in kDa. Bar= 20 µm

  4 µM of the different GST-tagged truncated AtKSS in Binding Buffer (20 mM Hepes pH 7.2, 2 mM MgCl 2 , 0.1 mM EGTA, 20 µM taxol, 0.5 mg/ml casein). Reactions (50 µl) were spun through a glycerol cushion (30 % (v/v) glycerol in binding buffer) for 5 min at 100.000 g as described[START_REF] Mcnally | Two domains of p80 katanin regulate microtubule severing and spindle pole targeting by p60 katanin[END_REF]. Supernatant and the resuspended microtubule pellet were analysed by western blot using an anti-GST antibody and the ECL kit (Amersham Pharmacia Biotech). For K d determination, 1 µM of taxol-

	stabilized microtubules and various concentrations of AtKSS constructs (from 0 to 5
	µM) were used in the assays. The amount of AtKSS constructs free and bound to
	microtubules were determined by scanning Coomassie blue stained gels and
	comparing the intensities with standards that were loaded on the same gel using
	NIH-image software (http://rsb.info.nih.go/nih-image/). The amount of AtKSS bound
	to microtubules is described by
	KMT

TABLE LEGEND Table I .

 LEGENDI Number of oligomers of the different AtKSS constructs.

Acknowledgements

We would like to thank Herman Höfte (INRA, Versailles) for sharing data on botero mutants, and Guy Schoehn (EMBL, Grenoble) for access to the electron microscope. This work was supported by the French Ministry for Research and the CNRS (ACI Dynamique et Réactivité des Assemblages Biologiques), and by Rhône-Alpes County (Plan Etat-Région Rhône-Alpes "Nouvelles approches des Sciences du vivant").


Maud Marchal
email: maud.marchal@imag.fr

Jocelyne Troccaz

A one-DOF freehand haptic device for robotic tele-echography

Tele-echography may be an interesting alternative to conventional care when the medical expert is not close to the patient to be examined. In a system for tele-echography, medical expert proprioception and gesture feelings must be preserved to synchronize the ultrasound images with the motion made by the physician. It is why we develop a master-slave system for robotic tele-echography integrating force feedback. Because existing haptic devices are not fully adapted to this application -their workspace is often small and their basis is fixed, we have developed a one-dimensional free-hand force feedback device. In this paper, we describe the design and a first evaluation of this new haptic device for robotic teleechography.

Introduction

Echography is a difficult examination based on specialized skills. Performing an ultrasound examination involves good eye-hand coordination and the ability to integrate the acquired information over time and space. The physician has to be able to build mentally 3D information from both the 2D images and the gesture information and to put a diagnosis from this information. Some of these specialized skills may lack in some healthcare centers or for emergency situations. Several systems are developed worldwide for tele-echography: some of them include a robot controlled remotely by the expert. The TER1 project [START_REF] Gonzales | TER: a system for robotic tele-echography[END_REF] developed by the TIMC laboratory in cooperation with clinical and industrial partners belongs to this category. TER is a master-slave system (see figure 1). The motion of a master manipulator carrying a virtual probe is controlled by the expert and remotely reproduced at the patient site by a slave robot carrying the echographic probe. The contact force between the probe and the patient is fed back to the operator allowing him to have a haptic virtual environment. The objective of this haptic control is to preserve medical expert proprioception and gesture feeling which are necessary to synchronize the ultrasound images with the motion made by the physician.

In the first two versions of TER, the virtual probe is attached to the PHANToM haptic device (Sensable Technologies Inc.) in order to facilitate the expert hand-eye coordination. It renders force information from local models and measurements from the slave site. Experiments have been successfully performed on phantoms and humans (see [START_REF] Vilchis | A New Robot Architecture for Teleechography[END_REF] for more details). However, this haptic system has several drawbacks for this application: it has a rather limited workspace and its basis is fixed whilst the natural gesture is free and takes place in a potentially large workspace. Moreover, during echography the crucial information for the control of the probe is the pressure exerted on the body of the patient. Very few haptic systems correspond to such needs; [Sato] is probably the closest one. Thus, we decided to develop a one degree of freedom (DOF) free-hand haptic device for the remote robot-based echographic examination. We name it "haptic probe".

Description of the system

As already introduced, the aim of the haptic probe is to provide the interface to a robot which carries an echographic probe on a patient located afar. The probe offers the user two main functions:

• To transmit position and orientation parameters to the slave site. In particular, the two DOF corresponding to the position of the probe on the surface on which it moves, the DOF corresponding to the probe axial translation (in other words: the probe pressure onto the body), the 3 orientation DOF. • To receive the information of the slave robot in order to realize the force feedback. The prototype has a cylindrical form and is composed of two parts (see Figure 2). The external part is held by the operator. Its position and orientation are tracked in real time using a localizer and positional information is sent to the robot. In the current version, we use a passive optical localizer (Polaris from Northern Digital Inc.). A rigid body attached to the haptic probe is equipped with reflecting markers. Its design is such that it remains visible when the haptic probe is rotated by the operator during the examination. The rigid body is located at the top of the probe. This first packaging element is rigidly attached to a second part integrating a DC motor and a transmission. This second part, which contacts the working surface (a table or a manikin), moves relatively to the rest of the haptic probe to render force information to the operator. The motor transmits a torque to a ball-screw mechanism which transforms rotational motion of the motor into linear motion of the mobile part. A force sensor is placed at the extremity of the haptic probe to measure the applied force and to control the motor. The probe is connected to a PC which acquires and treats the data and realizes the force feedback.

Results and discussion

Current work deals with the fine tuning of the motor control parameters in order to provide a good haptic perception for the remote control of the robot.

The haptic probe has still to be integrated into the TER system but preliminary experiments are promising. Size and weight issues were not considered for this first prototype but the gesture can be very natural with the haptic probe which does not constrain the workspace and renders a relevant whilst limited (1D) force information. There are no equivalent devices dedicated to tele-echography. In similar projects of robotic teleechography, few include force feedback while the gesture is very important for echographic image interpretation. Systems which include force feedback integrate complex devices have the same drawbacks than the Phantom.

Conclusion

In this paper, the design and first elements of evaluation of a new haptic probe dedicated to robotic tele-echography were presented. This new probe is really adapted to remote echographic examination. The probe enlarges the workspace of the physician, which can also accomplish actions with more precision and has an environment close to the one of a conventional echography. The force feedback adds the tactile perception to the visual perception and also improves the quality of the ultrasound examination.

Figure 1 :

 1 Figure 1: TER: (left) Master site: the expert interprets the echographic image whilst remotely controlling the echographic probe using the Phantom -(right) Slave site: the robot moves the echographic probe on the patient body.

Figure 2 :

 2 Figure 2: The haptic probe

TER is a french acronym for robotic tele-echography


Martin Victor Johansson

Fabrice Testa

Imen Zaier

Pierre Perrier

Jean Philippe Bonnet

Philippe Moulin

Irina Graur

Mass flow rate and permeability measurements in microporous media

Introduction

The determination of the permeability of porous media like the micro and nanoporous membranes or ultra-tight shale-gas reservoirs is still a challenge up to now. The low porous membranes find a broad application in medicine [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF] and biotechnology for separation and filtration [START_REF] Abedini | Application of membrane in gas separation processes: its suitability and mechanisms[END_REF]. The recent development of low porous ceramic media with high thermal, chemical and structural stability and the ability to have catalytic properties has opened up new horizons for this kind of membrane applications, for example, in high-temperature gas separation and catalytic reactions [START_REF] Lin | Experimental studies on pore size change of porous ceramic membranes after modification[END_REF]. Another type of porous media, the ultra-tight shale-gas reservoirs of tiny pores (in nanoscale) play a significant role in securing hydrocarbon energy because of their potential to offset declines in conventional gas production [START_REF] Naraghi | A stechastic permeability model for the shal-gas system[END_REF]. In all of these type of applications, the porous media permeability has to be known.

Permeability is a measure of how readily a fluid can flow through a porous material [START_REF] Sander | Laboratory measuremant of low permeability unconventional gas reservoir rocks: A review of experimental methods[END_REF]. Gas permeability is an important parameter to understand the transport characteristics of a fluid through a porous medium, which can be obtained from the mass or volume flow rate. For the determination of low permeability, either the steady-state or the transient methods can be used. The steady-state method needs a precise flow meter to measure very slow flow, so when the permeability is very low the conventional gas flowmeters may be inappropriate [START_REF] Jannot | A new quasi-steady method to measure gas permeability of weakly permeable perouse media[END_REF]. Therefore, the transient "pulse-decay" or "draw-down" techniques, [START_REF] Bruce | Calculations of unsteady-state gas flow through porous media[END_REF], [START_REF] Brace | A test of the low effective stress for cristalline rocks of low porosity[END_REF], [START_REF] Dana | Gas relative permeability and pore structure of sanstones[END_REF], [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF], [START_REF] Jannot | A new quasi-steady method to measure gas permeability of weakly permeable perouse media[END_REF], are also used to determine the low permeabilities.

By using these techniques, the permeability can be calculated directly from the pressure variations in time, without going first through the mass flow rate measurements [START_REF] Brace | A test of the low effective stress for cristalline rocks of low porosity[END_REF], [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF].

The primary objective of the present work is to develop the transient method to measure the pressure evolution in time in high and low-pressure tanks generated by the gas flow through a homogeneous porous medium. This experimental methodology, based on the constant volume technique, was initially developed for the isothermal and non-isothermal measurements of the mass flow rate through the microchannels [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF], [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF], [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF]. We provide here the physical justifications of the exponential fitting of the pressure variations with time as well as the physical conditions of its implementation. From the measured pressure variations in time, the mass flow rate through the porous medium is deduced.

It is shown that the gas permeability can be easily obtained directly from the pressure variations with time without going first through the mass flow rate calculations. The main advantages of the proposed approach are: its simplicity, the possibility of further extraction of the Klinkenberg coefficient and average pore size, and, finally, its further generalization for the case of temperature gradient driven flows.

The paper is organized as follows. After a brief introduction, the experimental apparatus and methodology are presented in Section 2. Then, in Section 3, the relation between the pressure variations in the tanks and a porous medium permeability is established by introducing the pressure relaxation time, which properties are analyzed in detail in Section 4. The behaviors of the measured mass flow rate and permeability are discussed in next two Sections. The paper is closed with final comments and conclusions.

Experimental methodology

Experimental apparatus

The experimental setup is a high vacuum system capable of measuring up to 5 decades of pressure. In the presented experiment the mean pressure is varied from 75 Pa up to 131 kPa. This large pressure measurement range is achieved by using three pairings of four capacitance diaphragm manometers (CDM) with full-scales: 133 kPa -133 kPa, 133 kPa -13.3 kPa and 13.3 kPa -1.33 kPa. Four high purity gas bottles with test gases, Helium, Neon, Nitrogen, Argon (Air Liquide, France) are used. The pumping is performed by a two-stage diaphragm vacuum pump (DVP) and a turbomolecular pump (TMP), see the schematic of the experimental setup in Fig. 1. Each side of the porous medium is connected to two reservoirs of volumes V 1 and V 2 for the high and low-pressure, respectively. Both tanks' volumes, including the volumes of the valves, connecting tubes and pressure sensors, are measured accurately, and these volumes are equal to V 1 = 255.8 ± 5.5 cm 3 and V 2 = 238.8 ± 5.1 cm 3 , for the high and low-pressure tanks, respectively. Several leakage tests were performed for this setup showing the absence of the detectable increase in pressure, measured with the lowest F.S. pressure sensor of 1.33 kPa over a period of 30 minutes. In addition, for all realized measurements we did not detected any linear increase of the mean pressure.

Two microporous samples, used in the experiments and mentioned in the following as the first and second discs, have a cylindrical shape (disc) with the same diameter and thickness (in main flow direction) equal to D =9.5 ± 0.01mm and L = 2.3 ± 0.01mm, respectively. The structure of these microporous discs is the same as that used to support the active layers of micro-to-ultra filtration ceramic membranes.

For such ceramic microporous medium, depending on manufacturer, the porosity is in the range 15 -30% with pore diameter ranging from 1 to 10 µm. The total volume of each porous disc is 0.14 cm 3 , so by taking 30% of porosity a gas volume inside the medium is approximately 0.047 cm 3 , which is much smaller than the volume of each tank.

The experiments are performed within a narrow temperature range, excluding any heat source in an environment. The temperature is measured using the thermocouple with the accuracy of 0.6 K.

Mass flow rate measurements

The constant volume technique, used previously for the measurements of the mass flow rate through the microchannels [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF], [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF], [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF], [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF], was implemented here to measure the mass flow rate through different samples of a microporous ceramic medium. This technique allows deducing the mass flow rate from the pressure variation in time. The mass flow rate through a microporous medium is The high pressure and low-pressure side are separated by a porous ceramic media which is fixated with vacuum epoxy glue. Each side of the porous media has a reservoir with the highpressure tank volume of V 1 = 255.8 ± 5.5 cm 3 , low-pressure tank volume V 2 = 238.8 ± 5.1 cm 3 and capacitance diaphragm manometers (CDM) measuring pressure p 1 and p 2 . The room temperature is measured with a thermocouple. The vacuum system is connected to a gas delivery system with four gas bottles, Helium, Neon, Nitrogen, and Argon.

p 2 V 1 L R p 1 T ambient > p 2 p 1
generated by setting an initial pressure drop between the reservoirs, see Fig. 1.

This method requires very large tank volumes relative to the volume occupied by a gas inside a microporous medium: in our experimental setup this ratio is larger than 10 3 . The applied method is similar to the Brace method [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF] (pulse decay method), usually used to analyze the permeability of the porous samples.

In addition, in this work we take into consideration the effects of rarefaction.

Even if the permeability can be deduced directly from the pressure (or pressure difference) variation in time, we prefer to start by providing the expressions of the mass flow rate through a microporous medium and the conditions of its derivation. The mass flow rate could be a useful quantity to characterize a porous sample; it can be used to derive the characteristic pore dimensions and the gas-surface interaction characteristics.

Under the quasi-steady conditions, i.e. when the flow through a porous medium is established, we assume that the gas temperatures in each tank, denoted T 1 and T 2 , are in thermal equilibrium with the walls of the tanks and that both tanks are in thermal equilibrium with the environment. Therefore, we assume that the ambient temperature, denoted T ambient , determines the gas temperatures T 1 and T 2 in each tank such that:

T 1 ≈ T 2 ≈ T ambient = T. (1)
A possible variation in temperature of the gas leaving and entering the tanks could directly perturb the significance of the measurement. To make this clearer, let us write the ideal gas law in each tank in the following form:

p 1 V 1 = M 1 RT, p 2 V 2 = M 2 RT, (2)
where R, p i , and M i , i = 1, 2, are, respectively, the specific gas constant, the pressure and the mass of the gas in tank i. In the present study the maximum considered pressure is of the order of atmospheric pressure; therefore we do not consider here the real gas effects. However, the proposed approach can be generalized to take into account the real gas effects by using, for example, the van der Waals equation instead of the ideal gas law.

Under our experimental conditions, the volume of each reservoir is constant during an experiment, so it is possible to differentiate each expression in Eqs.

(2), as it has been done in Ref [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF]:

dM i = V i RT dp i 1 - dT /T dp i /p i , i = 1, 2. (3)
If the relative temperature variation in a tank is negligible in relation to the relative pressure variation in time, then the mass flow can be considered to be isothermal. Therefore, by defining a specific small time interval, dt, it is then possible to obtain from Eqs. (3) the isothermal mass flow rates dM 1 /dt and dM 2 /dt as:

dM i dt = V i RT dp i dt , if i = dT /T dp i /p i 1, i = 1, 2. (4)
Following the differentiation technique, we consider here the variation of any thermodynamic parameter, dM and dp, sufficiently small to approximate dM/dt and dp/dt as the time derivative of the mass (i.e. mass flow rate Ṁ ) and the time derivative of the pressure, respectively. If the relative temperature variation is small compared to the pressure variation, the values of i are small in Eqs. (4), so by adjusting the sign, dM i /dt can be considered as the mass flow rate Ṁi through the microporous media

Ṁ1 = - dM 1 dt = - V 1 RT dp 1 dt , Ṁ2 = dM 2 dt = V 2 RT dp 2 dt . (5)
In the frame of the quasi-stationary flow assumption, i.e. when the flow through a porous medium is established, it is clear that the mass flow rate leaving the first tank is necessarily equal to the mass flow rate entering the second tank and also to the mass flow rate at any point inside the porous medium

Ṁ1 = Ṁ2 = Ṁ . (6)
Along with this study, we will continue to admit relation ( 6) at any time. However, the assumption of the mass conservation, Eq. ( 6), neglects possible storage of gas in the porous medium. For the small ceramic porous sample used in present experiment, this hypothesis of the absence of the gas accumulation inside a porous medium is justified. In the same time, for the porous media such as the coal and shales, the compressible fluid storage needs to be accounted [START_REF] Sander | Laboratory measuremant of low permeability unconventional gas reservoir rocks: A review of experimental methods[END_REF].

The experimental verification of mass conservation for the porous samples used in present experiments is discussed in Section 5.2.

Sometimes it is convenient to express the mass flow rate in function of the pressure difference between two tanks. From the mass conservation law, the mass of a gas leaving the first tank is necessarily equal to the mass of a gas entering the second tank, so Eq. ( 6) is valid. From Eqs. ( 5) and ( 6) we can obviously deduce:

Ṁ (t) = - V 0 RT d(∆p(t)) dt , V 0 = V 1 V 2 V 1 + V 2 . (7)
From the previous reasoning, it is clear that the mass flow rate can be calculated using expressions ( 5) and [START_REF] Bruce | Calculations of unsteady-state gas flow through porous media[END_REF], when the pressure variation in each tank or the pressure difference between them in time is known. During the experiments, the pressure variations in time in each tank are measured and then fitted by using the exponential fitting function.

Exponential pressure fitting

The experimental procedure starts by setting the initial pressure difference ∆p 0 between the tanks at time t 0 as:

∆p 0 = p 1 (t 0 ) -p 2 (t 0 ) = p 01 -p 02 , (8)
where p 01 and p 02 are the initial pressures in the high and low-pressure tanks, respectively. This first step is done by opening the valve v pump , see Fig. 1, for a short time, and then closing it. Further, we have a closed system with a pressure difference. Afterward, the gas begins to flow through a microporous medium from the high-pressure tank to the low-pressure tank up to the same final equilibrium pressure, p f , is reached in both tanks, see Fig. 2. The timedependent pressure difference between the tanks is noted as:

∆p(t) = p 1 (t) -p 2 (t). (9)
One example of the pressure evolution in each tank as well as of the pressure difference between two tanks is shown in Fig. 3.

Usually, when using the "pressure pulse technique", [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF], the small pressure "pulses" (pressure differences between the tanks) are applied to the system, i.e.

∆p 0 p m , where p m = 0.5(p 1 +p 2 ) is the mean pressure between two tanks. We will discuss below that finally we do not really need to respect this restrictive condition and the an arbitrary pressure difference between the tanks can be used in the system if both tanks volumes are equal. However, the restriction for p 1 /p 2 ratio exists when the tanks volumes are different, see Section 3.1.

The authors of Ref. [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF] proposed to use a linear fit of the natural logarithm of an exponential function for the pressure variation in time, while we suggest to use directly the exponential pressure fit to describe the pressure difference decay in time in the following form:

∆p(t) = ∆p 0 exp (-(t -t 0 )/τ ), (10)
where τ is the pressure relaxation time, which is constant during an experiment.

The same exponential representation of the pressure evolution in time in the first p 1 (t) and second p 2 (t) tanks is written in the form [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF]:

p 1 (t) = p f + (p 01 -p f ) exp (-(t -t 0 )/τ 1 ) , p 2 (t) = p f + (p 02 -p f ) exp (-(t -t 0 )/τ 2 ) , (11)
here τ 1 , τ 2 are the gas pressure relaxation times in the reservoir 1 and 2, respectively, p f is the final pressure. The pressure variations with time p i (t) in each tank 1 and 2 can be thus associated with an exponential decay. In practice the pressure relaxation times τ i are obtained from the fit of the measured pressure evolution in each tank, see Fig. 3. The properties of the pressure relaxation time are discussed in Section 4.

By using Eq. ( 7) we can now express the mass flow rate using the exponential representation of the pressure difference in time, Eq. ( 10), and then its analytical derivative, so the mass flow rate expression becomes

Ṁ (t) = - V 0 RT d(∆p(t)) dt = V 0 RT ∆p 0 τ exp - t -t 0 τ . (12)
From Eqs. [START_REF] Sander | Laboratory measuremant of low permeability unconventional gas reservoir rocks: A review of experimental methods[END_REF] we can also express the mass flow rate using the exponential representations of the pressure variation in time in each tank, Eqs. [START_REF] Cardenas | Thermal transpiration flow: a circular cross-section microtube submitted to a temperature gradient[END_REF], and then their analytical derivatives, so the mass flow rate expressions become:

Ṁ (t) = Ṁ1 (t) = - V 1 RT d(p 1 (t)) dt = V 1 RT p 01 -p f τ 1 exp - (t -t 0 ) τ 1 , Ṁ (t) = Ṁ2 (t) = V 2 RT d(p 2 (t)) dt = V 2 RT p f -p 02 τ 2 exp - (t -t 0 ) τ 2 . (13)
Equations ( 12) and ( 13) provide the time-dependent expressions for the mass flow rate. Therefore, with one experiment we can calculate the mass flow rate for various different pressure ratios between the reservoirs, see Section 4 for more details. The conditions related to the implementation of this technique, as the choice of the tank volume size, possible pressure ratio and the impact of the thermal effects are discussed in Section 3.1.

Relation with Darcy law and the permeability

The Darcy law [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF] allows us to relate the instantaneous discharge (or volumetric) flow rate through a porous medium, Q, to the pressure drop over a given distance L, which is the thickness of a porous sample:

Q = KS µ (p 1 -p 2 ) L , (14)
where S is the surface of the porous sample, µ is the fluid viscosity, K is the permeability. Initially, the Darcy law was derived for an incompressible fluid with constant viscosity and the permeability K refers to the hydrodynamic (intrinsic permeability). In this article we use the Darcy law for the gases, i.e., a compressible fluid and we do not make any preliminary assumption of the rarefaction level of a gas.

The viscosity coefficient for a gas depends on gas temperature and gas nature and it is calculated as [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF]:

µ = µ ref T T ref ω , ( 15
)
where ω is the gas viscosity index, µ ref is the gas viscosity at temperature

T ref = 273.15 K [1]
. The reference values of the viscosity, µ ref , for each gas used here as well as the viscosity index ω, are given in Table 1. For the liquid flows the volumetric flow rate is constant along a porous sample. For the gases only the mass flow rate is conserved along the porous sample. To express the permeability for a gas we can rewrite Eq. ( 14) in following form by replacing the pressure difference through a sample by the pressure gradient [START_REF] Sander | Laboratory measuremant of low permeability unconventional gas reservoir rocks: A review of experimental methods[END_REF]:

Q = - KS µ dp dx . ( 16
)
By using the relation between the volumetric and mass flow rate

Q = Ṁ ρ = Ṁ RT p , (17)
then by integrating along the porous sample and by using the mass conservation property, we obtain the expression, analogous to Eq. ( 14), which relates the mass flow rate and the permeability

Ṁ = KS µ ∆p L p m RT . (18)
Then, using Eqs. ( 5) and admitting the mass conservation again, Eq. ( 6), and using Darcy law in form [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF], we can relate the pressure variation in each tank to the gas permeability:

V 1 dp 1 dt = -p m K µ S L (p 1 -p 2 ), V 2 dp 2 dt = p m K µ S L (p 1 -p 2 ). (19)
To obtain previous relations we replaced the local pressure in Eq. ( 17) by the mean pressure, which does not vary during an experiment. This mean pressure is constant during an experimental run, when the tanks volumes are equal,

V 1 = V 2
, and it varies only slightly when these volumes are slightly different, see Section 3.1, where the conditions of the mean pressure constancy are provided.

From Eqs. [START_REF] Arkilic | Gaseous slip flow in long microchannels[END_REF] we obtain the differential equation for the pressure difference ∆p(t) between the tanks:

d(∆p(t)) ∆p(t) = - dt τ , (20)
where

τ = 1 p m µ K LV 0 S . (21)
This differential equation, subjected by the initial condition, ∆p(t = t 0 ) = ∆p 0 , is easily solved and the variation in time of the pressure difference between the tanks is obtained in the form of Eq. [START_REF] Billiotte | Experimental study on gas permeability of mudstones[END_REF].

From previous discussion it is clear that we do not use here any assumption about the smallness of the pressure "pulses" compared to the mean pressure in the tanks. Therefore, this technique can be implemented for any pressure difference between the tanks under the condition of equality of the tanks volumes.

When the tanks volumes are different some restrictions have to be respected in order to keep the mean pressure close to a constant value during an experiment duration, see Section 3.1. In addition, to integrate the differential equation [START_REF] Sharipov | Rarefied gas dynamics. Fundamentals for research and practice[END_REF] the relaxation time τ has to be time independent. By analyzing Eq. ( 21) one can see that only mean pressure can be time-dependent. The experimental conditions of the mean pressure constancy in time are discussed in Section 3.1.

The permeability depends on time only through the mean pressure, so the mean pressure constancy in time ensures that τ is constant in time and so justifies the integration of Eq. ( 20) and consequently the use of its solution, Eq. ( 10), for the experimental data (pressure) treatment. Therefore, the permeability K of a porous sample can be derived directly from the pressure measurements.

The experimental curve of pressure variation in time can be fitted by exponential expression, Eq. ( 10), with τ as a fitting parameter. Then, the permeability can be found from the analytical expression for the pressure relaxation time τ , Eq. ( 21), as:

K = 1 p m µ τ LV 0 S . (22)
Other expression for the permeability can be obtained from ( 18):

K = Ṁ p 2 1 -p 2 2 2µRT L S = Ṁ ∆pp m µRT L S . (23)
Both expressions, Eqs. ( 22) and ( 23), can be used to derive the permeability from the measurements.

Conditions of constancy of mean pressure

In this section we establish the conditions of the constancy of mean pressure during the measurement procedure, which leads to the pressure relaxation time constancy and justifies the use of expression [START_REF] Cercignani | Mathematical methods in kinetic theory[END_REF] for the permeability calculations. In addition, as it has been pointed out in Ref. [START_REF] Brace | A test of the low effective stress for cristalline rocks of low porosity[END_REF], in general case, the constancy of the mean pressure is an important point, especially for high pressure experiments, where the viscosity and compressibility factor may change as a function of pressure. In presented here experiments the implemented pressure and temperature conditions allow to us to stay under the ideal gas flow assumptions and the viscosity does not change with pressure. However, it is still important to have a constant mean pressure as for low permeable porous media the permeability can be a function of mean pressure due to rarefaction effects.

A pressure difference between the tanks is fixed initially, at t 0 , equal to ∆p 0 , Eq. ( 8), then the gas flows through the porous medium up to the final stage, when a pressure equality in both tanks is reached, see Fig. 2. A relation between the pressure variation in each tank, i.e. from the initial pressure in each tank, p i (t 0 ), i = 1, 2, to the final equilibrium pressure p f , reached in the system, can be calculated a priori as it is closely related to the tanks volumes ratio. From the ideal gas law and admitting again the mass conservation along the microporous medium at any time, we can write the following relation for two tanks if they are maintained at the same temperature

dp 1 V 1 = -dp 2 V 2 . ( 24
)
The previous relation is then integrated in time from an initial (at time t 0 ) state of a gas in each tank, p i (t 0 ), to its final state, p f . It is worth to underline that in Eq. ( 24) the expressions are exact (perfect) differential, and so their integration does not depend on the form (linear or exponential) of the pressure variation in time. Therefore, we obtain

p f -p 2 (t 0 ) p 1 (t 0 ) -p f = V 1 V 2 . ( 25
)
Without the loss of generality we can assume here that in the beginning of each experiment we have p 1 > p 2 . It is clear from Eq. ( 25) that by adjusting the tanks volumes ratio we could control the pressure variation between initial and final stages. From Eq. ( 25) we obtain the estimation of the maximal variation of the mean pressure p m with time, from its initial value p m (t 0 ) to its final value

p m (t f ) = p f : p f p m (t 0 ) = p 1 (t 0 )V 1 + p 2 (t 0 )V 2 (V 1 + V 2 )p m (t 0 ) . ( 26
)
It is clear from Eq. ( 26) that when the tanks volumes are equal, the mean pressure p m (t) does not vary in time, i.e., between its initial value, p m (t 0 ), and its final value, p f , so p m (t 0 ) = p f . When the volumes are different we can estimate the maximal amplitude of mean pressure variation between its initial state p m (t 0 ) and its final state p f , using Eq. ( 26) and the ratio of tanks volumes.

To do this expression (26) can be rewritten as:

p f p m (t 0 ) = 2(1 + k V k p ) (1 + k V )(1 + k p ) . ( 27
)
We introduced here the tanks volumes ratio k V = V 1 /V 2 and the initial pressure ratio k p = p 1 (t 0 )/p 2 (t 0 ). By using Eq. ( 27), we can calculate the variation of the mean pressure during an experiment, i.e. the ratio between its final and initial values p f /p m (t 0 ).

In our experimental conditions two tanks volumes are related as

k V = V 1 /V 2 = 1
.071, so from Eq. ( 27) we can find that for the initial pressure ratio between the tanks, k p , equal to 1.5, 2 and 3, the ratio p f /p m (t 0 ) is equal 1.0069, 1.0114 and 1.0171, respectively. Therefore, the initial pressure ratio equal to 2 leads to approximately 1% of deviation of the mean pressure from its initial value. Under our experimental conditions the experimentally evaluated value of p f /p m (t 0 ) was found lower than 1%.

Pressure relaxation time

By fitting the measured time variation of the pressure difference between two tanks using exponential law, Eq. ( 10), we obtain a function ∆p(t) that describes the relaxation process with the help of a single fitting parameter τ , that is the characteristic time of the experiment or pressure relaxation time.

Similar expressions for the pressure variation in each tank, Eqs. [START_REF] Cardenas | Thermal transpiration flow: a circular cross-section microtube submitted to a temperature gradient[END_REF], involve the relaxation parameters τ 1 and τ 2 , which can also be obtained by the fitting of the pressure variation in one (high or low-pressure) tank. From different expressions of the mass flow rates, Eqs. ( 12) and ( 13), and using the mass conservation property in form [START_REF] Lasseux | An improved macroscale model for gas slip dlow in porous media[END_REF], we can find that the ratios between the characteristic times are finally independent from the tanks volumes and are equal to one:

τ 1 τ 2 = τ 1 τ = τ 2 τ = 1. (28)
This analytical finding, Eq. ( 28), was confirmed experimentally. As it can be seen from Table 2 for the most cases the difference between three relaxation times, τ 1 , τ 2 and τ , is small, of the order of 1%.

Finally, to obtain the mass flow rate through the microporous medium and its permeability, see Section 6, we can use either the exponential fit of the pressure difference between the tanks (in the case of the use of the differential pressure sensors), or just pressure evolution in a tank.

Gas conduction time

Now we can compare the gas conduction time τ c to the gas relaxation time τ

to have an additional estimation of the importance of the thermal effects. If the tank represents an infinite heat sink at constant temperature to the gas, then, the time it takes for the gas to reach equilibrium with the tank can be modeled.

In Ref. [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF] the transient heat conduction equation was solved analytically and the solution was presented as the infinite series of the Bessel functions. When keeping only the first leading term of the series the characteristic conduction time can be estimated as:

τ c = ρR 2 res P r 2.4µ , ( 29
)
where ρ is the gas density, R res is the characteristic reservoir dimension, P r is the Prandtl number. The reservoir characteristic dimension (its radius) is equal to 19.6mm, the Prandtl number is equal to 2/3 and 0.71, for the monoatomic and polyatomic gases, respectively. The gas conduction time, Eq. ( 29), is proportional to the gas density and so to the gas pressure under our experimental conditions. It depends also on the gas nature through the gas viscosity. The value of the gas conduction time for some experimental conditions are provided in Table 2, last column. For all considered cases the pressure relaxation time τ is much longer than the gas conduction time τ c . Therefore, we have a new experimental confirmation that the gas temperature remains close to the constant temperature during the measurements.

Properties of the pressure relaxation time

As the mass flow rate through a porous medium, Eqs. ( 12) and ( 13), and its permeability, Eq. ( 22), depend on the pressure relaxation time it is interesting to study its properties.

Figure 4 shows the pressure relaxation time, expressed in seconds, as a function of the inverse molecular mean free path, -1 . The equivalent molecular mean free path is defined as following [START_REF] Abedini | Application of membrane in gas separation processes: its suitability and mechanisms[END_REF]:

= µv 0 p m , (30)
where v 0 is the most probable molecular speed

v 0 = √ 2RT . ( 31
)
It is clear from Eq. (30) that the inverse equivalent mean free path is a function of the mean pressure. By analyzing Fig. 4 and Table 2 we can conclude that the pressure relaxation time is proportional to the molar mass. That is, the shortest relaxation time is obtained for Helium, which has smaller molar mass, the longest relaxation time is found for Argon, which has greatest molar mass, see Table 1. All gases have similar behaviors as a function of the inverse molecular mean free path, which is a function of mean pressure. For the low mean pressure (large mean free path) the relaxation time is quasi-constant, then it decreases linearly with pressure increasing (the mean free path decreasing). This behavior is related to the number of collisions (molecule-molecule and molecule-wall collisions): when the intermolecular collisions are numerous (small mean free path) the relaxation time is short. With increasing of the molecular mean free path (decreasing of pressure) the number of intermolecular collision decreases which leads to the increase of the relaxation time, because the gas reaches its equilibrium state through the intermolecular collisions. When the molecular mean free path becomes large enough the number of the intermolecular collisions becomes negligible in comparison to the number of collisions with the wall (Knudsen diffusion regime) and the relaxation time becomes constant, see Fig. 4(b). In this case, the pressure relaxation time is determined only by the morphological parameters of a porous medium, i.e., mean pore size, porosity, tortuosity and particularities of gas-surface interaction.

It is worth to underline that the pressure relaxation time measured for two discs is different even if the same gas is considered. This fact lets us conclude that the internal structure of the microporous discs could be different. We comment on this observation in Section 6.3.

The pressure relaxation time can be normalized by the characteristic time of the flow, which is defined as follows:

t c = L v 0 . ( 32
)
This characteristic time depends on the gas nature through the most probable gas velocity v 0 , Eq. ( 31). The relaxation time, normalized by the characteristic time, is shown in Fig. 5, as previously in function of the inverse molecular mean free path. It is interesting to note that now all gases follow the same curve, so all gases have the same pressure relaxation time for the same value of the inverse molecular mean free path. Similar behavior of the relaxation time of the thermal creep flow was observed in Refs. [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF], [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF], [START_REF] Abedini | Application of membrane in gas separation processes: its suitability and mechanisms[END_REF], where the gas flow driven by only a temperature gradient through the microchannels of the circular and rectangular cross-sections was studied.

By taking into account the definition of the characteristic time, Eq. ( 32), we can rewrite expression of the permeability, Eq. ( 22), in the following form by using the equivalent molecular mean free path, Eq. ( 30),

K = V 0 S t c τ . ( 33
)
When the pressure relaxation time is used, the previous expression of the permeability allows calculating the permeability of a porous sample for different gases. This is because the normalized relaxation time is the same for all the considered here gases for a given value of molecular mean free path.

Another formula for the permeability can be derived by introducing gas relaxation time [START_REF] Abedini | Application of membrane in gas separation processes: its suitability and mechanisms[END_REF], t f , which is inversely proportional to the collision frequency of the gas molecules, and it can be calculated as

t f = µ p m = v 0 , (34)
so the expression of permeability becomes

K = V 0 L S t f τ . (35)
Previous expression shows that the gas permeability depends on sample dimensions, S and L, and on the tanks volume, V 0 , used in experiments. The increasing (or decreasing) in this volume leads to the corresponding change in the pressure relaxation time, so that the ratio V 0 /τ remains the same. Therefore, besides the geometrical characteristics, the permeability depends on the ratio between two characteristic times: gas and pressure relaxation times.

Mass flow rate

The mass flow rate was calculated from the pressure measurements in each tank by using the techniques, explained in Section 2.3. Three noble gases and one diatomic gas were used: Helium, Argon, Neon and Nitrogen.

Measurement uncertainty of the mass flow rate

The classical uncertainty calculation technique is used to estimate the measurement uncertainty of the mass flow rate, when it is calculated from the pressure or pressure difference exponential evolution, Eqs. ( 12) or (13). To ensure the upper limit of uncertainty, we use the maximum uncertainty on the mass flow rate measurements in the case of the pressure difference fit, Eq. ( 12), which reads:

δ Ṁ Ṁ = δV 0 V 0 + δ (∆p) ∆p + δT T + ε 0 , where δ (∆p) ∆p = δτ τ + δp 1 p 1 + δp 2 p 2 . (36)
In previous expression the uncertainty on the pressure difference in time involves the uncertainty on the fitting parameter τ and the uncertainty on the pressure sensors, provided by the manufacturer. The uncertainty on τ was obtained from the difference in magnitude of a 95% confidence interval for τ to represent the experimental data. The parameter ε 0 in Eq. ( 36) represents the uncertainty coming from the non-isothermal effects [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF] and it is equal to the maximum value in time of the ratio (dT /T )/(dp/p). To evaluate the value of ε 0 the standard deviation and the mean temperature is used for dT /T term calculations, while the pressure difference and mean pressure are used to evaluate the pressure differential and pressure, respectively. The maximal relative uncertainties of each term in Eq. (36), obtained for two microporous discs, are summarized in Table 3. The uncertainty of the mass flow rate lies in the range 3.6 -5.1%.

δV0 V0 δ ∆p ∆p δT T ε 0 δ Ṁ Ṁ
Uncertainty 3.0% < 1.9 % < 0.2 % < 1.2 % 3.6% -5.1 % Table 3: Measurement uncertainties of the mass flow rate, when the pressure difference between the tanks is used for the calculation. The maximal values obtained for two discs are given.

Results on the mass flow rate

Typical pressure variations over time in both reservoirs are shown in Fig. 3.

The exponential shape decay of the pressure in each tank and the pressure difference between two tanks are clearly visible on this figure. The exponential fit of the pressure (and pressure difference) variation curves during the total measurements duration with the pressure relaxation time as a single fitting parameter allow for very smooth reproduction of the experimental pressure recording.

In the current experimental setup the measured mass flow rate through the microporous media lies in the range 5 • 10 -7 -5 • 10 -12 [kg s -1 ]. This range can be extended by modifying the experimental setup configuration.

The typical mass flow rate variations in time, calculated from the pressure variation in inlet tank, -Ṁ1 , in outlet tank, Ṁ2 , are shown in Fig. 6. It is clear that all three curves practically coincide, which confirms the mass conservation property within instrumental uncertainty.

6. Permeability data

Measurement uncertainty on the permeability

The uncertainty of the permeability measurements, when using Eq. ( 22), is calculated by the classical way similar to the calculation of the uncertainty on the mass flow rate, where the maximum uncertainty is used:

δK K = δV 0 V 0 + δL L + δS S + δµ µ + δp m p m + δτ τ . (37)
The relative measurement uncertainty on the permeability is presented in Table 4, where only the maximum values (of two porous discs) for each term of Eq.

(37) are given, so the permeability uncertainty lies in the range 5.0 -6.4%. Uncertainty 3.0% 0.5% 0.1% 1.0% < 0.6 % < 1.4 % 5.0% -6.4 % Table 4: Measurement uncertainties of the microporous media permeability. The maximum of uncertainty of two discs is provided.

Permeability results

The microporous media permeability, calculated using Eq. ( 22) from the measured mean pressure and pressure relaxation time, is shown in Fig. 7 as a function of inverse gas mean pressure. The results, obtained for two microporous discs and different gases, are presented and plotted in the log-log axis.

Theoretically, for the large values of mean pressure, the permeability has to be constant, and the same for all gases tested for each disc. This part of the permeability curve corresponds to the classical Darcy law (intrinsic permeability), K ∞ , where a porous medium permeability does not depend on the nature of a fluid flowing inside. However, as it is clear from Fig. 7, only for the first disc with Helium this regime is reached in the present measurements, see "plateau" in Fig. 7(a). Larger mean pressure values, above atmospheric pressure, need to be applied to reach this regime for other gases and the second disc.

When the mean pressure decreases the permeability increases, and it becomes larger than the intrinsic permeability, the phenomenon of the apparent permeability appears, so-called Klinkenberg effect [START_REF] Abedini | Application of membrane in gas separation processes: its suitability and mechanisms[END_REF]. In this case, the permeability is usually expressed as

K = K ∞ 1 + b p m , ( 38
)
where b is a correction factor. As it is clear from Fig. 7 the apparent permeability becomes gas dependent: for a fixed mean pressure, the permeability is higher for lighter gases. However, if the same curves are plotted as a function of the mean free path, , see Fig. 8, the data for different gases are located on the same curve. A similar property was also observed for the pressure relaxation time τ , see Section 4.2 and Fig. 5 This finding leads to an interesting property:

when the characteristic time is determined for a microporous medium for one gas, then the permeability can be calculated using Eq. ( 33) for other gases.

If we compare expression derived in the present paper to calculate the permeability, Eq. ( 22), with the classical expression of apparent permeability, Eq.

(38), we find that Eq. ( 22) has the same asymptotic properties as Eq. ( 38), which can be confirmed by analyzing the measured data. When the mean pressure tends to infinity, then the product τ p m , tends to a constant value, see Fig. 9 and we find the constant intrinsic permeability. For the small values of the mean pressure p m the pressure relaxation time keeps its constant value, see Fig. 4, therefore the apparent permeability tends to infinity.

Recently several papers were published, where the dependence of the correction factor b in Eq. (38) from the pressure is discussed [START_REF] Abedini | Application of membrane in gas separation processes: its suitability and mechanisms[END_REF], [START_REF] Abedini | Application of membrane in gas separation processes: its suitability and mechanisms[END_REF]. Usually the permeability is plotted as a function of inverse pressure in the linear-linear coordinate system. If we plot the experimental permeability curves, shown on Fig. 7, but using the linear-linear instead of logarithmic-logarithmic scale for both axis we can observe the typical behavior of the permeability: it seems to be increase linearly as the mean pressure decreases demonstrating the well-known Klinkenberg effect [START_REF] Abedini | Application of membrane in gas separation processes: its suitability and mechanisms[END_REF], [START_REF] Adiga | Nanoporouse membranes for medical and biological applications[END_REF]. From this evident linear dependence of the permeability on the mean pressure (K = A+B/p m ) we could conclude that the correction factor B is pressure independent and can be used for large pressure variation range.

However, the linear-linear representation can mask some behavior, because it is difficult to present correctly the different order of magnitude of the parameter variation in linear scale. If we try to fit the whole permeability experimental curve with the same expression, K = A+B/p m , we cannot find the same pair of A and B coefficients for the whole curve, see Fig. 11b). The relative deviation between the measured and fitted experimental points, (K exp -K f it )/K f it , for helium curve is shown of Fig. 12. It is clear that this curve fit very well the low pressure range and the large deviation exists in the high pressure (Darcy regime). This result demonstrates that the permeability points cannot be fitted with unique b coefficient in the large pressure range when using the Klinkenberg expression. Of course, the more detailed study of this property is needed to find the numerical values of the correction factor b.

Difference between two discs

Two porous discs considered in the present study were fabricated to have the same expected properties, i.e. the smallest average pore size of the order of 3 µ m. However, the measured permeability has very different values for two discs, especially for the low pressure. Therefore we assume that two discs have different internal structure. The tomographic analysis confirms this experimental finding:

the averaged pore size was much larger for the first disc compared to the second one. Therefore, this technique can be used for the non-destructive analysis of the permeability of the microporous media. In addition this technique can be implemented to derive the characteristic pore size of a microporous sample.

Conclusion

The experimental procedure for the measurements of mass flow rate and permeability through the microporous media is proposed and analyzed. In the frame of this procedure the pressure evolution in each tank (or the evolution of the pressure difference between two tanks) is successfully fitted with an expo- Figure 3: Color on-line: pressure evolution as a function of time. Red squares is upstream tank pressure, p 1 , green pentagons is the downstream tank pressure, p 2 , magenta line -is the mean pressure, blue triangles is the pressure difference between tanks. Red, green and blue lines represent the exponential pressure fits, which correspond to Eqs. [START_REF] Cardenas | Thermal transpiration flow: a circular cross-section microtube submitted to a temperature gradient[END_REF] and [START_REF] Billiotte | Experimental study on gas permeability of mudstones[END_REF].

τ [s] ℓ -1 [m -1 ] NITROGEN ARGON HELIUM 10

Figure 1 :

 1 Figure1: Schematic of the experimental setup. A diaphragm vacuum pump (DVP) and a turbomolecular pump (TMP) is connected to the experimental setup separated by a valve vpump. The high pressure and low-pressure side are separated by a porous ceramic media which is fixated with vacuum epoxy glue. Each side of the porous media has a reservoir with the highpressure tank volume of V 1 = 255.8 ± 5.5 cm 3 , low-pressure tank volume V 2 = 238.8 ± 5.1 cm 3 and capacitance diaphragm manometers (CDM) measuring pressure p 1 and p 2 . The room temperature is measured with a thermocouple. The vacuum system is connected to a gas delivery system with four gas bottles, Helium, Neon, Nitrogen, and Argon.

  nential function using one fitting parameter: the pressure relaxation time. The simple expressions for the mass flow rate and the permeability, derived from the exponential fitting of the pressure relaxation in each tank, are proposed. It was found that besides of the sample dimensions the gas permeability can be characterized by the ratio between gas relaxation time (inverse of gas collision frequency) and the pressure relaxation time. With the present experimental setup we measured the mass flow rate in the range 5 • 10 -7 -5 • 10 -12 [kg/s] and the permeability in the range 10 -14 -10 -11 [m 2 ]. However, we are not restricted to these ranges of parameters with the present experimental setup (volume configuration and sample size). We estimate that we could measure at least 50 times lower mass flow rate and permeability. To go further in low permeability measurements the experimental setup has to be modified. The proposed approach is the first very promising stage to evolve towards measurements of even lower permeabilities and also the characteristic dimension (pore size) of membranes used for microfiltration (> 100 nm) and ultrafiltration (> 10 nm).

Figure 4 :Figure 5 : 2 × 10 - 8 4 × 10 - 8 6 × 10 - 8 8 × 10 - 8 1 × 10 - 7 1. 2 × 10 - 7 0 8 1 × 10 - 7 1. 5 × 10 - 7 2 × 10 - 7 2. 5 × 10 - 7 3 × 10 - 7 3. 5 × 10 Figure 6 :K [m 2 ]Figure 7 :K [m 2 ]Figure 8 :Figure 9 :Figure 10 :Figure 11 :Figure 12 :

 4521081081081081072107810751071075107107510627289101112 Figure 4: Color on-line: variation of the relaxation time as a function of inverse molecular mean free path, -1 , for different gases and two porous media: (a) first disc, (b) second disc.

Table 1 :

 1 Parameters of the gases used in present experiments

	Gas µ ref × 10 -5 [P a.s] He 1.865	ω 0.66	R [J/kgK] Molar mass M [g.mol -1 ] 2077.1 4.003
	Ne	2.976	0.66	412.02	20.18
	N 2	1.656	0.74	296.80	28.00
	Ar	2.117	0.81	208.13	39.95

Table 2 :

 2 Relaxation times τ 1 , τ 2 and τ , in seconds (s) for the second disc, measured in the high and low-pressure tanks and by using the pressure difference between two tanks, respectively.

		τ1[s]	τ2[s]	τ [s]	pm [10 5 Pa] τc[s]
	HELIUM	54.11	53.94	54.03	1.13	0.96
		66.94	66.87	66.90	0.79	0.68
		153.83 155.60 154.72 0.04	0.03
	NEON	92.38	92.48	92.43	1.16	3.11
		119.98 118.59 119.28 0.79	2.13
		333.25 331.46 332.34 0.03	0.08
	NITROGEN 62.30	61.75	62.03	1.12	7.45
		92.98	92.04	92.50	0.70	4.64
		371.56 369.32 370.44 0.03	0.23
	ARGON	79.27	78.61	78.94	1.11	8.11
		112.34 111.33 111.83 0.74	5.41
		444.07 444.22 444.15 0.04	0.26

For each of four gases the relaxation time is provided for two pressure differences between the tanks. The mean pressure of each experiment is given in fifth column. The last column provides the conduction time τc, Eq.(

29

).

Acknowledgment This work has been carried out in the framework of the Labex MEC (ANR-10-LABX-0092) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government program managed by the French National Research Agency (ANR). The authors (M.V. Johansson, P. Perrier, and I. Graur) would like to acknowledge financial support provided by the European Union network program H2020, MIGRATE project under Grant Agreement No.643095.