Trung-Kien Pham

Ana Cameirao

Jean-Michel Herri

Experimental flowloop study on methane hydrate formation and agglomeration in high water cut emulsion systems

Keywords: Emulsion, Gas bubble, Gas hydrate formation, Agglomeration, Anti-agglomerants, Hydrate slurry transport, Suspension, Flow assurance, and Multiphase flow

As an oil or a gas field gradually matures, the amount of water produced (water cut) increases, and the hydrate risk also increases. Especially in the offshore systems, operating at low temperature and high pressure, conditions are favourable to the formation of gas hydrate, from the combination of liquid water and gas molecules, under the form of a solid phase. It is a serious issue in the flow assurance; it may cause many troubles, up to plugging.

This work brings new understanding on hydrate kinetics of crystallization, nucleation and growth, agglomeration, breakage and deposition under flowing. We focused on high water cut emulsion systems under flowing and we studied the role of commercial anti-agglomerants or low dosage hydrate inhibitors (AA-LDHIs). These additives can disperse hydrate particles to prevent plugging. More specifically, the topic of this study concerns methane hydrate crystallization, agglomeration and hydrate slurry transportation in a flowloop at high water cut systems, with and without AA-LDHI. We have been used up-to-date measurements and calculations among FBRM (Focused Beam Reflectance Measurement), PVM (Particle Video Microscope), pressure, pressure drop, flowrate/density, hydrate conversion and hydrate volume fraction. In this study, we performed the experiments at very high water volume fractions (80-90-100%) in laminar regime (150L/h) with and without anti-agglomerants. The results showed that using gas-lift system (riser), dramatic events were witnessed with a high rate of methane hydrate formation, followed with a quick agglomeration and leading to plugs, even with additive. But we also observed a dramatic foaming event, even at low AA-LDHI concentration of 0.01%.

We concluded our work with a discussion of the agglomeration mechanisms, based on a quick hydrate formation and agglomeration rates.

Introduction

Gas hydrates (or clathrate hydrates) are ice-like crystalline structures composed of molecular water cages stabilized by light hydrocarbon gases (methane, ethane, propane, etc) at low temperature and high pressure conditions (Sloan and Koh, 2008). Gas hydrate formation is a great concern in flow assurance which faces a potential hazard (plug) to oil and natural gas transportation once they form in deep-sea pipelines. There are several strategies to manage this risk including: addition of thermodynamic inhibitors (THIs), kinetic hydrate inhibitors (KHIs) and commercial anti-agglomerants or low dosage hydrate inhibitors (AA-LDHIs). The most traditional method to prevent gas hydrates is thermodynamic inhibition which shifts the hydrate equilibrium conditions to higher pressure and lower temperature. However, the severe conditions (higher pressures, lower temperatures, higher acid gas contents, higher water cuts) in oil and gas transportation make thermodynamic prevention less acceptable because of high amount of THIs used. To reduce the cost, KHIs and/or AA-LDHIs are used at a very small amount compared to THIs. In fact, KHIs may delay the nucleation or growth of hydrate formation while AA-LDHIs allow the hydrate formation but help to maintain hydrate particles dispersed in flowlines.

Low water cut systems have been studied previously. Camargo and Palermo et al., (2002) developed a rheological model for suspension and agglomeration of hydrate particles in an asphaltenic crude oil based on the data obtained from laboratory and flow loop scales. In their work, water in oil emulsion was prepared at low water cut (15-50%) and without anti-agglomerants. Turner et al., (2005) also studied gas hydrate formation in emulsion, both autoclave and flowloop, and water cut of (5%-35%) and without anti-agglomerants. Other publications concerning the understanding of kinetics and mechanisms of methane gas hydrate formation and agglomeration, as well as flow characteristics and rheological properties of natural gas hydrate slurry, in low water fraction, with and without anti-agglomerants are: Colombel et al., 2008;[START_REF] Lachance | Effect of hydrate formation/dissociation on emulsion stability using DSC and Visual techniques[END_REF]Didier et al., 2009;Bo-Hui Shi et al., 2011;Liang Mu et al., 2013;Ke-Le Yan et al., 2013. In high water cut systems, insights into gas hydrate formation and dissociation in emulsion were developed by Greaves et al., 2008. This contribution mainly focused on the effects of hydrate formation and dissociation on the stability of high water content emulsion. Afterwards, the hydrate formation and agglomeration and rheology of hydrate slurry were investigated in high water fraction with and without antiagglomerants by the following authors : [START_REF] Gao | Hydrate risk management at high watercuts with anti-agglomerant hydrate inhibitors[END_REF]Shiva Talatori et al., 2011;Hossein Moradpour et al., 2011;Prasad et al., 2011;[START_REF] Joshi | Experimental flowloop investigations of gas hydrate formation in high water cut systems[END_REF]Jun Chen et al., 2014, 2015. The study concerning gas hydrate formation and agglomeration in the Archimede flowloop (this apparatus simulates the actual pipeline conditions) at Ecole des Mine de Saint-Etienne was firstly conducted by Fidel-Dufour et al.,(2006). This work showed the rheological properties of hydrate slurry of water in dodecane emulsion at low water cuts (5-30%) and in the presence of anti-agglomerants (0-0.5%). A model was also developed to describe crystallization and rheology based on the flow loop experimental data. Leba et al., (2010) studied the formation and agglomeration of gas hydrate particles in water in oil (Kerdane ® ) emulsion with low water cuts (5-30%) and with the presence of anti-agglomerants (0.5-1%) in flow loop using the Focused Beam Reflectance Measurement (FBRM probe). Recently, Mendes- Melchuna et al., (2016) investigated gas hydrate formation and agglomeration in emulsion with different water cuts (from low to high water cuts), flowrates and different amount of additives (AA-LDHIs) by using FBRM and PVM measurements.

In general, most of the previous studies on gas hydrate formation and agglomeration and rheological characteristics of hydrate slurry merely focused on the low water cut, in small scale (autoclave) and without anti-agglomerants. In contrast, at high water cut, the gas hydrate formation and agglomeration mechanisms in the presence of AA-LDHI in flowlines are poorly understood. The aim of this work is to go deeper in this understanding of hydrate crystallization and agglomeration in pipelines at high water cut, with emulsion systems, and effects of AA-LDHIs.

Experimental Methodology

Flowloop description

The experiments were performed in the Archimede flow loop at Ecole des Mines de Saint Etienne. A schematic of the flow loop is shown in the Fig. 1. The flow loop is constructed with the following conditions: maximum pressure is 100 bar, the temperature is controlled between 0°C and ambient temperature, the flow rate is between 0 and 500 L/h. The flow loop is designed with 30 m of horizontal pipe with 1 cm diameter, a riser and a descending pipe (both of them have a height of 12m and a diameter of 1.5 cm), a gas injection system, and one sapphire window. The total liquid volume is 11.5 L and gas volume is 15L. In this experimental apparatus, some probes are equipped to monitor the gas hydrate formation and agglomeration, including: the sensors of temperature (thermocouples) and pressure; the probes of pressure drop (differential pressure transducer) for evaluating the viscosity afterwards, and identifying the plugging troubles; the Focused Beam Reflectance Measurement (FBRM) probe for detecting particle size distribution of emulsion droplets and hydrates (via their chord lengths); the Particle Video Microscope (PVM) probe to observe the emulsion, the gas hydrate formation and agglomeration phenomena; the Attenuated Total Reflection (ATR) probe for determining the concentration of methane and phase inversion of emulsion in fluid flow. Also, the mass flow meter is installed for measuring the flow rate and density. A pump Moineau and/or a gas-lift system are used to circulate the fluids in the flow loop. In this experimental set-up, we used a gas-lift or ballast system which is described in Fidel-Dufour et al., 2006. The cooling systems control the temperature in the flow loop at (4-5 o C) by using the water/ethanol circulation jacket attached to the flowloop. The gas compensation system (pressure controller) keeps the total pressure at a constant value (75 bar).

Materials and experimental procedure

Emulsions are formed by mixing water (ultrapure water, provided from Millipore filter) and oil (Kerdane ® , provided by TOTAL FLUID) with or without a small amount of commercial anti-agglomerants (0.01%). The composition of Kerdane and anti-agglomerants is shown in Table A.1, Appendix A. Because of foam formed in the mixture at 80%WC-0.01%AA-LDHI, 5 ppm of silicone oil (M.40.165.10 provided by Huber) was added only for this experiment. The mixture was charged into the flow loop and circulated at the flow rate of 150L/h to induce emulsification or dispersion. This circulation in the loop is carried out by a gas lift by injecting the compressed air (at 4-5 o C and atmospheric pressure) at the bottom of the riser. The flowloop tests are shown in the Table 1. Firstly, the mixture in the flow loop system was circulated at a desired flowrate (150L/h) and cooled down to 4-5 o C. Then it was pressurized up to 75 bar by the injection of methane (99.99 % purity supplied by AIR LIQUID) and ballast system was started. During the gas solubilization, we measured the total pressure decrease versus time to evaluate the gas transfer rate. Then, we performed a second injection up to 75 bar, and gas was consumed again by solubilization but in smaller quantity. Then, pressure stabilized and we waited for the gas hydrate formation (and agglomeration) which will again consume gas. During the hydrate formation, we monitored the gas consumption rate, the pressure drop in the horizontal section and in the separator, the temperature, the FBRM and the PVM signals. At the end of experiment (when crystallization is finished or/and pipe is plugged), the liquid and gas were depressurized to the ambient pressure and the cooling system was stopped. Finally, the remaining mixture was taken out of flow loop, which was cleaned two times with tap water and finally with pure water for the next experiment. All experimental data were online recorded by LABVIEW, V819 PVM software from METTLER-TOLEDO, LASENTEC® products and LASENTEC® FBRM data acquisition software. The experimental data were treated and analyzed on a computer.

Results and discussion

Emulsification

Fig. 2 shows the pressure drop and average chord length during emulsification for the experiments without AA-LDHI. It can be concluded from Fig. 2 that the emulsion is quite homogeneous regarding its entire volume. In fact, we observed a constant pressure drop, so the emulsion was not partitioned under sections of different water cut. However, the droplet size of emulsion was fairly stable even if the signal had little fluctuations at the beginning but it was steady at the end of experiment. It can be explained by an un-stationary dispersion/agglomeration process. The size was controlled mechanically from the shear stress only. Chord length distribution appeared in the mixture of 100%WC due to some oil still may stick in flowloop during cleaning or/and the presence of gas bubble in the flow. By addition of AA-LDHI (Fig. 3), although pressure drop measurements still remained stable, the average chord length decreased significantly with the mixture at 100%WC. In contrast, a slight fluctuation of mean droplet size was witnessed for mixture at 80%WC and 90%WC. This can be explained because oil attracts AA-LDHI more than water and the formation of oil in water emulsion may reduce the AA-LDHI aggregation in water. As a result, AA-LDHI dispersed more uniformly in water once oil is added and this helps emulsion more stable (Minwei Sun et al., 2014). From Fidel-Dufour's work, it can be concluded that concentration of 0.01%AA-LDHI in all mixtures is higher than critical micelle concentration (CMC). Furthermore, the concentration of additives is very low, and it is not enough to stabilize the emulsion chemically.

Some foam was also formed in the mixtures with AA-LDHI. It is assumed that the low interfacial tension between oil, water and compressed air leads to foaming. In general, addition of AA-LDHI caused foam and instable signals in terms of average chord length and pressure drop but the average chord length is more stable at 80%WC and 90%WC thanks to oil. We assume that it is due to lower interfacial tension between oil, water and compressed air. Mean droplet size of emulsion is approximately 10 µm from the FBRM measurements in all experiments.

Crystallization 3.2.1. Onset of gas hydrate formation and plugging

The beginning of gas hydrate formation can be identified by an increase of temperature (crystallization is exothermic), an increase of pressure drop in the horizontal pipeline and in the separator, a sharp decrease of static pressure, and also from the pattern of the PVM images (see Fig. 5, 6, 7 and 9).

PLUG

Results from Table 2 show that, despite the small hydrate volume in the flowloop, in each test the plug happened very quickly (some tens of seconds) after the beginning of the gas hydrate formation. This time elapsed between the nucleation and the plugging is named "flowing time" in this study. We observed a general trend that the higher the amount of oil the longer the flowing time, both with and without AA-LDHI. Plug time is defined in this work when the flow rate falls to zero (Fig. 7). At the same time, we observed a dis-functioning of gas-lift system which began working abnormally. When hydrates form, they may agglomerate and it is evidenced by an increase of pressure drop, and the increase of the average chord length. The increases of both are observed on Fig. 6 and 7 and are also reported by (Leba et al., 2010;Jun Chen et al., 2014). In all tests, there is a slight increase of pressure drop in horizontal sections (Fig. 7) once hydrate is formed. The hydrate volume (shown in Table 2) is not important enough to yield to a big increase in pressure drop in the horizontal parts of flowloop. Also, we know from previous work (Young hoon Sohn et al., 2015 and[START_REF] Joshi | Experimental flowloop investigations of gas hydrate formation in high water cut systems[END_REF]) that agglomeration may not produce sharp increase in pressure drop but high pressure drops were observed only once hydrate deposited on the wall of pipe.

Effect of water cut on gas hydrate formation and agglomeration

Fig. 5 and 6 showed the experimental results of mixture with 90%WC, without AA-LDHI. It is a representative experimental test, showing a quick increase of pressure drop in separator, and also an important decline of static pressure once hydrate formation starts. Fig. 8 shows the hydrate conversion at different high water cuts (80-90-100%) with and without AA-LDHI. On each curve, the end of plots corresponds to the plug formation time. The longer the plot the longer the flowing time. The results illustrated that without AA-LDHI, higher water cut mixtures exhibited lower capability to prevent plugs or higher risk of hydrate blockage. And it is also observed for the mixtures in the presence of AA-LDHI. At the beginning, the rate of gas consumption is almost equal in all experiments. But after approximately one minute, the conversion rate of mixture at 90%WC-0.01%AA-LDHI increased sharply and remained nearly unchanged. On the contrary, hydrate conversion of the mixture at 100%WC-0.01%AA-LDHI and 80%WC was stable and it had slight increment afterwards. From PVM results (Fig. 9), we observed clearly hydrate particles with hydrate film covering droplets. In some cases, we can observe some agglomeration. In the experiments with AA-LDHI, PVM images had a bad quality and therefore, it was impossible to follow the crystallization processes.

Effect of additives on the gas hydrate formation and agglomeration

The results of mixture with AA-LDHI compared to mixtures without AA-LDHI, at the same water cut (Table 2), demonstrated that the addition of low quantity of AA-LDHI is not sufficient to prevent plugging. The addition of AA-LDHI also seemed to inhibit hydrate nucleation in the experiments with 100%WC and 90%WC. Interestingly, mixture with 80%WC-0.01%AA showed very short nucleation time compared to others with AA-LDHI. It is reported that the amount of oil enhanced effectively the performance of AA-LDHI (Minwei Sun et al., 2015).

Role of AA-LDHI and water continuous phase for hydrate formation and agglomeration

All experiments at high water cut systems showed higher rate of hydrate growth, and faster agglomeration and plug events, compared to previous studies at low and high water cut (Fidel-Dufour et al., PLUG [START_REF] Fidel-Dufour | Rheology of methane hydrate slurries during their crystallization in a water in dodecane emulsion under flowing[END_REF]Leba et al., 2010;Mendes-Melchuna et al., 2015). Our experiments differ due to the presence of a gas lift section. It can be hypothesized that gas hydrate is formed in the riser at a higher rate due to the good quality of the gas dispersion. We also suspect an accumulation of the particles in the separator (once a significant surge in the pressure drop of the separator was observed). Moreover, all of the experiments have been performed at high water cut. Due to the gas-lift system, it can promote a different and quicker growth and agglomeration rates when compared to literature at low water cut where the systems are mainly oil dominated. At high water cut, our system becomes water continuous, and it modifies the activity of bubbles, and interaction between continuous phase and the surface of pipelines. Furthermore, the formation of a water-gas bubble bridge between hydrate particles may be more dynamic. Firstly, it can be attributed to gas hydrate formation at the surface of gas bubbles which has been previously observed by Hideo Tajima et al., 2009 andY.-T.Luo et al., 2006. Moreover, we have to take into account an un-expected formation of foam. For the experiment at 80%WC with AA-LHDI, we observed more foam in comparison with other experiments with AA-LDHI, and also a very short time of nucleation. Both could be attributed to the high contact surface area between oil, water and gas, because a dispersed and fine population of gas bubbles and foam is formed.

Secondly, hydrate bridges between hydrate particles also contributes as one of principal reasons for a rapid agglomeration. Here, hydrate film formed on the gas and/or foam surfaces perhaps favors links between hydrate particles. In this explanation, we assumed that thanks to a lot of bubbles in the riser, hydrates can readily form on the surface of bubbles, and these bubbles, surrounded by water, may play a vital role as the water-gas bubble bridge, which later are transformed into hydrate-gas bubble easily. This created a higher rate of agglomeration resulting in a blockage.

Thirdly, at high water cut, water is the continuous phase and it also means that water covers almost the pipeline surface. As a result, when hydrate formed, in the presence of water film between the pipe surface and hydrate particles, it modified the interaction between the hydrate particle and the pipe wall. It is a different case when oil is the continuous phase at low water fraction. G. Aspenes et al., 2009 considered that the adhesion force of hydrate particles and pipe surface with water droplets is much higher than hydrate-hydrate adhesion force or hydrate-pipe wall interaction without water layer. It can explain that hydrates deposit on the pipe wall once they formed leading to plug quickly.

In addition, low flow rate and small diameter of pipe may contribute to the gas hydrate deposition. Moreover, from the average chord length results, using gas lift systems, mean droplet size of emulsion is quite small (~10µm) compared to that of previous work (Mendes-Menchuna et al., 2016) using protocol with pump Moineau. It is perhaps another reason which causes higher gas hydrate formation due to larger contact surface between oil and water. Because of lower velocity of bubbles at higher height in the riser, hydrate covering bubbles catch up at the top of riser and they agglomerate below the bulk gas-liquid interface in separator which leads to a sudden increase of pressure drop in separator and causes plug (Y.-T. Luo et al., 2006). Beside, bubbles in the riser not only enhance gas transfer to emulsion to reach faster gas saturation in emulsion but also promote gas transfer once hydrate formed causing higher rate of gas hydrate formation.

The role of oil to prevent plugging in the presence of AA-LDHI has been enlightened by (Minwei Sun et al., 2014, 2015). They supposed that surfactants absorbed on the hydrate surface, and they created hydrophobic layers which attract oil to cover on the hydrate surface, thus it led to prevent plugging. The similar explanation to the role of oil on emulsion stabilizing in the presence of AA-LDHI (discussed in section 3.1) may be applied to anti-agglomeration capacity of AA-LDHI with the support of oil. By this way, more AA-LDHI can absorb on hydrate particle surface and prevent agglomeration. In our case, the plug still happened because of low dosage of AA-LDHI. In cases without AA-LDHI, perhaps, oil dispersed into water phase slightly hindered adherence forces between hydrate particles.

Proposed mechanism

According to (Fidel-Dufour et al., 2006), flow in the riser of our flowloop shows a slug flow pattern (ρ L U L 2 =15 kg/m.s 2 ; ρ G U G 2 =3 kg/m.s 2 with U G /(U G +U L )=0.64), which is encountered in small diameter riser (<0.05 m). From proposed mechanism (Fig. 10), it is considered that once hydrate formed on the gas bubble surface in the riser, it can merge with other hydrate covering bubbles, creating a bigger one. Also, hydrate film covering gas bubbles can be broken into free hydrate particles and the free gas and it can be recovered itself by a new hydrate film, and/or can agglomerate by dint of adherence forces between hydrate particles (Hideo Tajima et al., 2009;Yang et al., 2004;Y.-T.Luo et al., 2006).

From experimental results, we witnessed very high rate of hydrate growth (gas consumption) at different water cut experiments. It can be explained by the dominance of gas bubbles in the riser comparing to oil droplets. The oil droplets may contain some dissolved gases which lead to the formation of a hydrate layer covering their surface but maybe fewer and smaller than hydrate covering bubbles. The main reason explaining the quick formation of plugs immediately after the nucleation may be the high rate of gas hydrate formation around the gas bubble, cracks from hydrate shell (this helps promoting mass transfer) and their rapid agglomeration due to water-gas bubble bridge between hydrate particles and hydrate deposition as described above (section 3.3).

Conclusions

In high water systems using gas-lift, we are confronted to a high risk of hydrate blockade because of high rate of hydrate formation (high dispersion of gas bubbles) and high risk of agglomeration, leading to a quick plug formation, even with AA-LDHI. Main reason for this phenomenon is attributed to the hydrate film formation around the surface of gas bubbles in the riser. Very low dosage of AA-LDHI is not sufficient to prevent their agglomeration, and plugging occurs rapidly. A mechanism based on a quick agglomeration rate was proposed to explain the rapid plugging. In all experiments with and without AA-LDHI, oil partially made contribution to maintain flow after gas hydrate formation.

Nomenclature & abbreviations

Fig. 1 .

 1 Fig. 1. Schematic of Archimede experimental flow loop: simplified with ballast system (above) and details with pump Moineau (below).

Fig. 2 .

 2 Fig. 2. Pressure drop (solid line) and average chord length (dashed line) during emulsification for the experiments without AA-LDHI.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Pressure drop (solid line) and average chord length (dashed line) during emulsification for the experiments with 0.01% AA-LDHI.PVM images (Fig.4) show numerous droplets in the emulsion during emulsification. In this case, in very high water cut emulsion systems, it is supposed that they are oil droplets in a water continuous phase.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Typical temperature and pressure drop (in separator) profile during crystallization experiment (90%WC without AA-LDHI).

Fig. 7 .

 7 Fig. 7. Typical flowrate and pressure drop (horizontal line) profiles during crystallization experiments (80%WC without AA-LDHI).

Fig. 8 .

 8 Fig. 8. Conversion of gas hydrate for experiments at different water cut with and without AA-LDHI.

  PVM images of gas hydrate formation for different mixtures: (a) 90%WC without AA-LDHI; (b) 80%WC without AA-LDHI; (c) 90%WC with 0.01% AA-LDHI; (d) 80%WC with 0.01% AA-LDHI.

Fig. 10 .

 10 Fig. 10. (a) Hydrate formation on the surface of bubble, growth, agglomeration (by water-gas bubble bridge) and breaking in the riser; (b) slug flow regime and typical hydrate agglomeration in the riser.

  of water converted to hydrate [mol] number of mol of methane consumption to hydrate [mol] initial number of mol of water in mixture [mol] conversion [%] number of cells of hydrate structure I [-] volume of one cell [m 3 /cell] volume of hydrate [m 3 ] total volume of mixture [m 3 ] hydrate volume fraction [%] water cut or water fraction [%] molecular weight of water [g/mol] initial mass of water in mixture [g] mass density of the flowing phase [kg/m 3 liquid [m/s] mass density of the gas phase [kg/m 3 ] superficial velocity of gas [m/s] repeated test anti-agglomerant -low dosage hydrate inhibitor kinetic hydrate inhibitor thermodynamic inhibitor pressure drop [bar] average chord length [µm] critical micelle concentration [%] Focused Beam Reflectance Measurement Particle Video Microscope Attenuated Total Reflection

Table 1 .

 1 Experimental conditions for flowloop tests.

	No.	WC (%)	Temp. ( o C)	Press. (bar)	Flowrate (L/h)	AA-LDHI (%wt)
	1	100	4-5	75	150	0
	2.rp	100	4-5	75	150	0
	3	100	4-5	75	150	0.01
	4	90	4-5	75	150	0
	5	90	4-5	75	150	0.01
	6	80	4-5	75	150	0
	7	80	4-5	75	150	0.01

Table 2 .

 2 Summary of the experimental results for high water cut flowloop tests.

	No.	WC	AA-LDHI	Conversion	Hydrate Volume	Nucleation time	Flowing time
		(%)	(%)	(%)	(%)	(min)	(min)
	1	100	0	1.27	1.59	1.30	1.60
	2.rp	100	0	0.99	1.24	10.50	1.60
	3	100	0.01	0.50	0.64	12.90	1.10
	4	90	0	2.17	2.44	11.50	2.10
	5	90	0.01	2.06	2.33	24.60	2.00
	6	80	0	1.07	1.07	7.10	3.50
	7	80	0.01	3.60	3.60	1.50	8.80

Acknowledgements

The authors are grateful to TOTAL E&P for the financial support in the Archimede-3 project and their approval for publishing this paper. We also would like to acknowledge the contribution on technical support and reviewing this manuscript from the members of TOTAL and the Gas Hydrate Dynamics Centre team at Ecole des Mine de Saint Etienne.

Appendix A

The composition of Kerdane and anti-agglomerants is shown in


Roqiya Nouar

Gilles Breuzard
email: gilles.breuzard@univ-amu.fr

Sonia Bastonero

Svetlana Gorokhova

Pascale Barbier

François Devred

Hervé Kovacic

Vincent Peyrot
email: vincent.peyrot@univ-amu.fr

Direct evidence for the interaction of stathmin along the length and the plus end of microtubules in cells

Keywords: fluorescein isothiocyanate, FRET: Förster Resonance Energy Transfer, Immuno-FRET: immunofluorescence resonance energy transfer, MT: microtubule, NFRET: normalized fluorescence resonance energy transfer, FRAP: Fluorescence Recovery After Photobleaching, ROI: region of interest, pSer25-, pSer38-, pSer16-, pSer63-stathmin: stathmin phosphorylated on serine 25, 38, 16, 63, respectively, TRITC: tetramethylrodhamin isothiocyanate, stathmin/oncoprotein 18, tubulin, binding, immuno-FRET, FRAP

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

In eukaryotic cells, stathmin or oncoprotein 18 is one of the most prominent destabilizers of microtubule (MT) dynamics [START_REF] Belmont | Identification of a protein that interacts with tubulin dimers and increases the catastrophe rate of microtubules[END_REF][START_REF] Curmi | The stathmin/tubulin interaction in vitro[END_REF][START_REF] Devred | Stathmin/Op18 is a novel mediator of vinblastine activity[END_REF]. Recent studies support a role for stathmin in the regulation of cell growth and motility, and show its involvement in human malignancies [START_REF] Belletti | Stathmin: a protein with many tasks. New biomarker and potential target in cancer[END_REF]. The MT-destabilizing action of stathmin remains unclear: the best accepted mechanism is one whereby stathmin reduces MT polymer by sequestering two soluble α/βtubulin dimers to form a curved complex (T2S) [START_REF] Curmi | The stathmin/tubulin interaction in vitro[END_REF][START_REF] Gigant | The 4 A X-ray structure of a tubulin:stathmin-like domain complex[END_REF]. This ternary complex is stable and unable to assemble into MT. Another model holds that stathmin may induce MT shrinkage (called 'catastrophe') directly by acting on its tips. Gupta et al. have recently shown that stathmin binds tightly to dolastatin-10 tubulin rings, which mimic curved tubulin protofilaments at MT plus-ends, and that stathmin depolymerizes stabilized protofilamentrich polymers [START_REF] Gupta | Mechanism for the catastrophe-promoting activity of the microtubule destabilizer Op18/stathmin[END_REF]. Using computer simulation, they found strong evidence for the promotion of catastrophe by stathmin through binding to the tips of MTs.

Several in vitro studies have sought to identify the stathmin domains that participate in tubulin binding [START_REF] Gupta | Mechanism for the catastrophe-promoting activity of the microtubule destabilizer Op18/stathmin[END_REF][START_REF] Howell | Dissociation of the tubulin-sequestering and microtubule catastrophe-promoting activities of oncoprotein 18/stathmin[END_REF][START_REF] Cassimeris | The oncoprotein 18/stathmin family of microtubule destabilizers[END_REF]. Using structural and biochemical approaches, Steinmetz et al. demonstrated with different truncated forms of stathmin that (i) under MT polymerizing conditions, the α-helical domain of stathmin was sufficient to stabilize tubulin heterodimers, (ii) the 40 N-terminal amino acid residues of stathmin were necessary to prevent further longitudinal stathmin-tubulin complex aggregation, and (iii) a precise length of the C-terminal domain of stathmin was necessary to form a stable ternary complex with tubulin heterodimer [START_REF] Steinmetz | Op18/stathmin caps a kinked protofilament-like tubulin tetramer[END_REF].

Stathmin activity is mainly regulated by phosphorylation on four serines (Ser16, Ser25, Ser38 and Ser63) [START_REF] Sobel | Stathmin: a relay phosphoprotein for multiple signal transduction?[END_REF][START_REF] Marklund | Serine 25 of oncoprotein 18 is a major cytosolic target for the mitogen-activated protein kinase[END_REF][START_REF] Larsson | G2/M transition requires multisite phosphorylation of oncoprotein 18 by two distinct protein kinase systems[END_REF]. It has been shown in cells that Ser25 and Ser38 are phosphorylated first, followed by phosphorylation on Ser16 and Ser63 to produce a completely inactive tetra-phosphorylated form [START_REF] Marklund | Serine 25 of oncoprotein 18 is a major cytosolic target for the mitogen-activated protein kinase[END_REF][START_REF] Larsson | Control of microtubule dynamics by oncoprotein 18: dissection of the regulatory role of multisite phosphorylation during mitosis[END_REF][START_REF] Marklund | Oncoprotein 18 is a phosphorylation-responsive regulator of microtubule dynamics[END_REF].

The impact of individual phosphorylation has been thoroughly studied in vitro using purified stathmin modified by directed mutagenesis. Stathmin phosphorylated at either Ser16 or Ser63 shows a drastic loss of affinity for tubulin dimers, whereas di-phosphorylated stathmin (on Ser25 and Ser38) still binds to tubulin, though with a reduced affinity [START_REF] Honnappa | Control of intrinsically disordered stathmin by multisite phosphorylation[END_REF][START_REF] Manna | Regulation of microtubule dynamic instability in vitro by differentially phosphorylated stathmin[END_REF]. Immunofluorescence studies of stathmin and MTs in solution have shown that both unphosphorylated and double Ser25/Ser38-phosphorylated stathmin can bind along the entire MT as well as on free tubulin [START_REF] Manna | Regulation of microtubule dynamic instability in vitro by differentially phosphorylated stathmin[END_REF][START_REF] Manna | Stathmin strongly increases the minus end catastrophe frequency and induces rapid treadmilling of bovine brain microtubules at steady state in vitro[END_REF]. Despite these extensive in vitro studies, the molecular mechanisms of stathmin binding to tubulin and/or MTs, and the effect of phosphorylation on this process, remain poorly understood, especially in the cell microenvironment.

Förster resonance energy transfer (FRET) can be used to detect the proximity between two fluorophores separated by distances of 1-10 nm [START_REF] Stryer | Energy transfer: a spectroscopic ruler[END_REF]. Classically measured by fluorescence spectroscopy, FRET can also be measured by fluorescence microscopy. Since FRET occurs over distances similar to the size of proteins, it can be used to extend the resolution of the fluorescence microscope (typically > 250 nm) to detect protein-protein interactions. FRET microscopy is thus a powerful technique to determine whether proteins that are co-localized at light resolution microscopy interact with one another in cells. In this way, Niethammer et al. developed an interesting FRET-based sensor with the expression of a double fluorescent stathmin fused to CFP on its Nterminal end and YFP on its C-terminus in Xenopus A6 cells [START_REF] Niethammer | Stathmin-tubulin interaction gradients in motile and mitotic cells[END_REF]. The FRET signal is scattered for free stathmin, owing to its high flexibility, but decreases when stathmin interacts with tubulin. In their model, they observed an anterograde gradient of the FRET signal from the perinucleus to the lamellipodia of cells, presumed to reflect differential phosphorylation of stathmin. However, direct labeling of proteins by fusing to fluorescent tags alters the protein itself and/or its endogenous level in the cell.

By examining several methodological aspects, numerous recent studies demonstrate that indirect doublelabeling immunofluorescence of proteins combined with fluorescence microscopy is a valid method to identify association of proteins by FRET in cells [START_REF] Kenworthy | Imaging protein-protein interactions using fluorescence resonance energy transfer microscopy[END_REF][START_REF] König | FRET-CLSM and double-labeling indirect immunofluorescence to detect close association of proteins in tissue sections[END_REF][START_REF] Adebiyi | Isoform-selective physical coupling of TRPC3 channels to IP3 receptors in smooth muscle cells regulates arterial contractility[END_REF][START_REF] Leo | Dynamic regulation of β1 subunit trafficking controls vascular contractility[END_REF]. In our study, we mapped and characterized the stathmintubulin/MT interactions directly in cells using immunofluorescence resonance energy transfer (immuno-FRET) and fluorescence recovery after photobleaching (FRAP) microscopy. Consistent with the in vitro studies, we detected stathmin-tubulin interactions in the cytosol and at the plus-ends of the MTs. Interestingly, we observed stathmin puncta along the MT length. We also found that this interaction occurred mostly through the Cterminal domain of stathmin. Focusing on the phosphorylation state of stathmin, we observed that Ser25-and/or Ser38-phosphorylated stathmin were bound to the MT wall. Based on our results, we propose a novel dynamic model for the role played by stathmin in MT disassembly.

Cell culture, cDNA cloning

The human non-small lung carcinoma cell line (ATCC, clone A549; CCL2, MD, USA) was routinely grown at 37 °C in a humidified atmosphere of 5% CO2, and maintained by regular passage in a complete medium composed of RPMI 1640 (Lonza, France) supplemented with 10% heat-inactivated fetal bovine serum (Invitrogen, France). Cells were free of mycoplasma as evidenced by mycoalert tests (Lonza). For treatment with taxol (Mr = 853.91 g/mol, in DMSO; Seripharm, France), cells were incubated with 1-50 nM taxol for 4 h at 37 °C before observations. For plasmids, full stathmin was initially subcloned into pEGFP-C1 vector (high expression pCMV promotor, ClonTech, CA, USA) between the XhoI site (forward primer: 5'-CCGCTCGAGCGGGTGGCTTCTTCTGATATCCAGG-3') and the PstI site (reverse primer: 5'-GCTGACGAGACTGAAGCTGACTAAGCTGCAGC-3'). Two mutants of stathmin were constructed: the '∆Cter-stathmin' insert encoding for stathmin deleted from Lys100 to Glu147 and mutated on Ser16, Ser25, Ser38 and Ser63 to Ala [START_REF] Howell | Dissociation of the tubulin-sequestering and microtubule catastrophe-promoting activities of oncoprotein 18/stathmin[END_REF]; and the '∆Nter-stathmin' insert encoding for stathmin deleted from Met1 to Ser40 plus from Glu141 to Asp149 and mutated on Ser63 to Ala [START_REF] Steinmetz | Op18/stathmin caps a kinked protofilament-like tubulin tetramer[END_REF]. The two genes were subcloned into a pEGFP-C1 vector as for the full stathmin, between the BglII (forward primer, respectively: 5'-TTAGATCTCCACCATGGCTTCTTCTGATATCC-3' and 5'-TTAGATCTCCACCATGAAGAAGAAGGATCTTTCCCTG-3') and BamHI site (reverse primer, respectively: 5'-TTTGGATCCCTAGTCGGCCTCTTCTGCCATTTTACTG-3' and 5'-TTTGGATCCTTATTATTTGGATTCTTTGTTCTTCCGC-3'). The pEGFP-tubulin plasmid was purchased from Invitrogen (Cergy-Pontoise, France). The pmCherry-tubulin plasmid encoding for wild-type α1B isotype was a gift from Dr Saudou [START_REF] Dompierre | Histone deacetylase 6 inhibition compensates for the transport deficit in Huntington's disease by increasing tubulin acetylation[END_REF].

Antibodies and immuno-blotting

We used, according to supplier's instructions, mouse monoclonal anti-α-tubulin antibody (Sigma-Aldrich, 1 mg/mL, clone DM1A recognizing epitope between amino acids 426 to 430), mouse monoclonal antivinculin antibody (Sigma-Aldrich, 1 mg/ml, clone hVIN-1), rabbit polyclonal anti-stathmin antibody (Sigma-Aldrich, 1 mg/mL) targeting the sequence of stathmin between amino acids 132 and 149, and recognizing all phosphorylated and unphosphorylated forms of stathmin, and rabbit monoclonal anti-pSer63 stathmin (Abcam, 1 mg/mL). Antisera containing rabbit polyclonal anti-pSer16, anti-pSer25 and anti-pSer38 antibodies were kindly given by Prof. Sobel (Institut du Fer à Moulin, UPMC, Paris, France) [START_REF] Gavet | The stathmin phosphoprotein family: intracellular localization and effects on the microtubule network[END_REF].

Phosphorylation of stathmin was analyzed by Western blotting in denaturing conditions (Fig. S3A): cells were lyzed by re-suspension in RIPA buffer (150 mM NaCl, 0.1% Triton X-100, 0.5% sodium deoxycholate, 0.1% SDS, 50 mM Tris-HCl pH 8.0), and lysates were centrifuged at 14,000 rpm for 20 min at 4 °C; equivalent quantities of proteins from the supernatant fraction underwent 15% denaturing in SDS-PAGE; proteins were blotted with anti-stathmin antibody (dilution 1:2000), anti-pSer25/pSer38/pSer16-stathmin antibodies (dilution 1:800 each) or anti-pSer63 antibody (dilution 1:1000). The anti-vinculin antibody (dilution 1:1000) was the reference. For analysis of phospho-stathmin by non-denaturing Western blotting (Fig. S2B), cells were lyzed by re-suspension in extraction buffer (50 mM Hepes pH 7.6, 150 mM NaCl, 5 mM EDTA, 10% glycerol, 0.1% Triton X-100, 10 mM 2-mercaptoethanol) and lysates were centrifuged at 12,000 rpm for 10 min at 4 °C.

Equivalent quantities of proteins from the supernatant fraction underwent 15% non-denaturing PAGE. All blots were visualized using chemiluminescent HRP substrate Immobilon Western kit (Millipore) with G:Box driven by GeneSys software (Syngene, UK), and quantified by densitometry with ImageJ.

Immunofluorescence and transfection of cells

Cells were fixed with 4% paraformaldehyde in PBS pH 7.4 for 20 min at room temperature, and permeabilized with Triton X-100 0.5% (Sigma-Aldrich, France) in PBS for 10 min at room temperature.

Immunostaining was carried out overnight with primary anti-α-tubulin (dilution 1:1000) and anti-total-stathmin antibodies (dilution 1:2000), anti-pSer16, pSer25, anti-pSr38 antibodies (dilution 1:1000 each). Next, cells were incubated for 1 h at room temperature in darkness with secondary FITC-and TRITC-conjugated antibodies (dilution 1:200 each from 1.5 mg/mL; Jackson Immunoresearch, USA). For direct immunofluorescence, primary anti-total-stathmin antibody was chemically coupled to the Atto532 fluorophore using a Lightening-Link TM Atto532 kit as recommended by Innova Bioscience, UK. Coverslips were mounted with a drop of ProLong® anti-fade solution (Invitrogen). For FRET and FRAP performed on living cells, the transient overexpression of the full stathmin, the ∆Cter-stathmin and the ∆Nter-stathmin coupled to EGFP protein was performed using the lipofection of cells with lipofectamine 2000 according to the Invitrogen protocol. Cells were also transfected to overexpress mCherry-tubulin (as acceptor). Here, 0.4 µg plasmid (0.2 µg + 0.2 µg coding for the donor and acceptor, respectively) was used.

Instrumentation and image acquisition

Imaging was performed on a Leica SP5 confocal laser scanning microscope (CLSM) with a Leica inverted microscope, equipped with a Plan-Apochromat 63× oil immersion objective (NA = 1.4). For all FRET experiments (immuno-FRET on fixed cells and FRET on living cells), images were recorded with the CLSM spectral mode selecting specific domains of the emission spectrum [START_REF] Breuzard | Molecular mechanisms of Tau binding to microtubules and its role in microtubule dynamics in live cells[END_REF]. The FRAP experiments were divided into three sequences as described elsewhere [START_REF] Breuzard | Molecular mechanisms of Tau binding to microtubules and its role in microtubule dynamics in live cells[END_REF]. The photobleaching of stathmin was carried out on a 2 µm radius circular area of the lamellipodium regions containing MTs. This step used the 488 nm wavelength laser with 10 iterations of 2 µs/pixel. To determine MT dynamics, materials, acquisition of time-lapse series and analysis of the MT dynamic instability are described elsewhere [START_REF] Breuzard | Molecular mechanisms of Tau binding to microtubules and its role in microtubule dynamics in live cells[END_REF].

The surface area of the microtubule network in the 200 µm² regions of interest (ROI) was measured by systematically executing Otsu's method via the plugin 'Otsu threshold' of ImageJ. This algorithm is an implementation of the Otsu thresholding technique (27). The histogram of pixel intensities is divided into two classes and the inter-class variance is minimized. This plugin outputs a binary image of MT and the ratios of MT on ROI surfaces are then calculated as percentages. With untreated cells, we found that 31 ± 4 % of the cell interior was occupied by the fluorescent microtubule network. As expected there was no impact of 1 nM taxol on the ratio of MT to ROI surface areas in treated cells. All these values come close to the range of 34-41% of tubulin in microtubules in tissue cultured cells previously determined by Ostlund et al. [START_REF] Ostlund | Biochemical determination of tubulin-microtubule equilibrium in cultured cells[END_REF].

FRET calculation

FRET images were corrected from both background (collected outside cells on images) and cross-talk between donor and acceptor channels using Youvan's method ( 29):

(1)

Fc = IFRET -A × ID -B × IA
where IFRET, ID, and IA were intensities (after background subtraction) in ROI of the FRET, donor (FITC or EGFP) and acceptor (TRITC, Atto532 or mCherry) channels, respectively. Parameters A and B were respectively the fraction of donor and acceptor leak-through into the FRET channel, and were calculated by quantifying the intensity ratios between images from cells labeled or expressing only the donor or the acceptor.

In our study, the values of A and B were 0.15 and 0.02 on average, respectively. No leak-through signal from the donor into the acceptor channel or vice versa was observed.

Fc was therefore normalized to the direct acceptor signal using Wouters' method (30):

(2) NFRET = Fc / IA Calculations were performed from the variation in pixel response with the PixFRET plug-in of ImageJ in ROI [START_REF] Feige | PixFRET, an ImageJ plug-in for FRET calculation that can accommodate variations in spectral bleed-throughs[END_REF]. All positive pixels positive were pixels with NFRET intensity greater than 0. The NFRET intensities of 8bit images were initially spread from 0 to 12 and from 0 to 30 on the 256-grayscale for respectively the FITC-TRITC/Atto532 pairs and the EGFP-mCherry couple, the maximum of which corresponds to 5% and 12% of NFRET in cells [START_REF] Gordon | Quantitative fluorescence resonance energy transfer measurements using fluorescence microscopy[END_REF]. FRET images were filtered with a 1-pixel-range median filter to reduce background noise.

For FRET quantifications, images showing the NFRET distributions were merged with the tubulin/MT-labeling images, and ROIs of 200 µm 2 were outlined in the cell periplasm. NFRET hotspots co-localizing with MTs and outside MTs ('in the cytosol') were then counted and expressed per unit surface area (µm²) of MT and cytosol (e.g with untreated cells, ROIs of 200 µm² were divided into ∼62 µm² and ∼138 µm² for MT and cytosol, respectively). We considered a 'NFRET hotspot' every pixel or cluster of pixels with NFRET intensities > 0%

and surrounded by pixels of with nil intensity.

Fitting of FRAP data

Fluorescence recovery was extracted from images recorded in the bleached area and corrected for experimental fluctuations during acquisition. In all, 7-15 separate FRAP measurements were made in three independent experiments, and data were independently single-normalized as ( 33):

(1)

Ffrap-norm(t) = [Ffrap(t) -Fbg(t)] / Ffrap-pre
where Ffrap(t) corresponds to the fluorescence recovery in the bleached ROI at time t; Fbg(t), the fluorescence intensity in a background ROI outside the cells, and Ffrap-pre, the mean fluorescence intensity of bleached ROIs before the bleach after background subtraction:

(2)

Ffrap-pre = {∑(t=0; t bleach-1) {[Ffrap(t) -Fbg(t)] / fprebleach}
with fprebleach corresponding to five frames during the pre-bleaching period. In FRAP, the rate of fluorescence recovery reflects diffusion and binding dynamics. The next step in FRAP analysis is to ascertain the respective contributions of diffusion and binding to the fluorescence recovery curves.

First of all, we examined whether diffusion could be ignored or not (see Fig. S3). The results indicate that diffusion is so fast relative to binding that it can be ignored. Hence, the fluorescence recovery curve reflects the binding interactions. Later, recovery curves were fitted using a binding-dominant model given by [START_REF] Sprague | Analysis of binding reactions by fluorescence recovery after photobleaching[END_REF][START_REF] Hallen | Fluorescence recovery kinetic analysis of gamma-tubulin binding to the mitotic spindle[END_REF]:

(3)

F(t) = F∞ -Ceq.exp -k off .t
The kinetic curves were analyzed for single exponential by nonlinear least-squares fitting based on the Levenberg-Marquardt algorithm to adjust F∞, the fluorescence intensity at infinite time, Ceq, the fraction of fluorescence at equilibrium due to binding, and the dissociation rate constant koff. This model was applied with no foreknowledge of either the geometry of the bleaching or the process of fluorescence recovery.

Using koff from the curve fit, the pseudo first-order binding constant, k*on, was calculated by:

(4)

k * on = Ceq.koff / (1 -Ceq)
And the turnover time was calculated by:

(

RESULTS

Immuno-FRET reveals that stathmin interacts with microtubules and cytosolic tubulin.

The spatial distribution of the interaction between tubulin and MT with endogenous stathmin was determined in A549 cells fixed with paraformaldehyde. Here, we performed immuno-FRET of α-tubulin using a secondary antibody coupled to FITC as FRET donor. Stathmin was detected using a primary antibody specific to unphosphorylated and phosphorylated stathmin ('total-stathmin' on figures), and then with a secondary antibody coupled to TRITC as FRET acceptor (Fig. 1A). The immunolocalization of stathmin yielded punctuated staining in the cytoplasm as previously described [START_REF] Gavet | The stathmin phosphoprotein family: intracellular localization and effects on the microtubule network[END_REF], which sometimes co-localized with the MTs NFRET images were obtained using Wouters normalization, for which possible artifacts can occur, as previously well described [START_REF] Wang | Noise-induced systematic errors in ratio imaging: serious artefacts and correction with multi-resolution denoising[END_REF]. To validate the significance of the measured NFRET signals in spots we compared the quenched and unquenched FITC donor emission after specific TRITC acceptor photobleaching (Fig. 1B, panels in false colors). Prior to TRITC photobleaching ('pre-bleaching -recovery' images), fluorescence emission of MTs was partially quenched when there were co-localized spots of stathmin (arrows).

After the TRITC photobleaching ('post-bleaching -recovery' images), a recovery of the fluorescence of MT was observed where spots of stathmin occurred before the acceptor bleaching. At these bleached areas (n = 5 cells), we calculated a significant NFRET efficiency [START_REF] Siegel | Measurement of molecular interactions in living cells by fluorescence resonance energy transfer between variants of the green fluorescent protein[END_REF], on average %E = 4.5 ± 1.3. No relevant fluorescence intensity fluctuation in the channels of donor (%E = 1.1 ± 0.9) and acceptor (%E = 0.5 ± 0.9) was measured inside the 'control' region. Our observations confirm the interactions between stathmin and tubulin/MT as NFRET hotspots in cells.

Two experimental controls were performed to validate the range of NFRET efficiencies (Fig. S1A-B).

As a negative control for NFRET (Fig. S1A), cells expressing the free EGFP (as donor) were fixed, followed by indirect immunofluorescent labeling of α-tubulin (secondary antibody coupled to TRITC, as acceptor). As expected, we observed no FRET signal, despite the apparent co-localization of EGFP and tubulin. As a positive control for NFRET (Fig. S1B), we performed an indirect immunofluorescent labeling of α-tubulin using two secondary antibodies, one coupled to FITC and the other to TRITC. On the images, MTs are yellow because of the superimposition of co-labeled tubulin. The NFRET signal is continuous over the entire length of MTs, with an efficiency value between 1% and 5%. This double labeling of tubulin is ideal for obtaining the highest NFRET efficiency: a closely similar NFRET efficiency is observed for the interaction of stathmin with tubulin/MTs. Lastly, in order to rule out the possibility of artifacts due to large size or misorientation of secondary antibody complexes, we used direct immunofluorescence of α-tubulin (antibody coupled to FITC, as donor) and total-stathmin (coupled to Atto532, as acceptor) (Fig. S1C). The NFRET signals were still observed in hotspots within the same range of NFRET efficiency, indicating that the two immunofluorescence approaches were equivalent in our cell model. All these experiments confirm that immuno-FRET is well-suited to explore the interaction of stathmin with tubulin and/or MTs in cells.

Using FRET, we clearly demonstrate in cells that stathmin interacts with tubulin in the cytosol, as well as throughout the lengths and at the plus-ends of MTs. The interaction at the plus-ends of MTs is consistent with prior in vitro observations [START_REF] Gupta | Mechanism for the catastrophe-promoting activity of the microtubule destabilizer Op18/stathmin[END_REF]. Our data are consistent with a model where stathmin acts at MT ends. More surprising is the interaction of stathmin with the MT wall. Gupta and co-workers suggest that stathmin does not bind significantly to the MT lattice. By contrast, phosphoforms of stathmin, i.e. double phosphorylation of stathmin on Ser25 and Ser38, bind to MTs [START_REF] Manna | Regulation of microtubule dynamic instability in vitro by differentially phosphorylated stathmin[END_REF][START_REF] Manna | Stathmin strongly increases the minus end catastrophe frequency and induces rapid treadmilling of bovine brain microtubules at steady state in vitro[END_REF]. We therefore went on to explore, by immuno-FRET, the binding of phosphorylated forms of stathmin with tubulin and/or MTs and their localization.

Stathmin phosphorylated on serine 38 and/or 25 interacts with microtubules.

The destabilizing activity of stathmin is attenuated by phosphorylation of four serine residues: Ser16, Ser25 and Ser38, located in the N-terminus region, and Ser63 at the beginning of the C-terminal domain of stathmin. To analyze the amounts of all phosphoforms of stathmin, we examined protein content of cells by Western-blot in denaturing conditions using antibodies against anti-stathmin phosphorylated on serine 16

('pSer16'), serine 25 ('pSer25'), serine 38 ('pSer38') and serine 63 ('pSer63') (Fig. S2A). We observed that pSer38-and pSer25-stathmin were the main phosphoforms in our cell model. The higher phosphorylation levels of stathmin were weakly represented (for pSer16-stathmin) or undetectable (for pSer63-stathmin). Our results are consistent with expected ratios of phosphoforms of stathmin [START_REF] Marklund | Serine 25 of oncoprotein 18 is a major cytosolic target for the mitogen-activated protein kinase[END_REF][START_REF] Larsson | Control of microtubule dynamics by oncoprotein 18: dissection of the regulatory role of multisite phosphorylation during mitosis[END_REF][START_REF] Marklund | Oncoprotein 18 is a phosphorylation-responsive regulator of microtubule dynamics[END_REF]. We then performed indirect immunofluorescence of α-tubulin and pSer38-, pSer25-and pSer16-stathmin in fixed cells. Since no phosphorylation of Ser63 was observed by Western blot, pSer63-stathmin was not included in this assay. The quantification and distribution of NFRET hotspots between MTs and the cytosol are reported in Fig. 2. For FRET analysis, we selected two quantitative parameters: the number of NFRET hotspots belonging to MTs or the cytosol, and the percentage of MT plus-ends presenting NFRET hotspots. Images showing the NFRET distributions in cells were merged with the tubulin/MT-labeling images, and ROIs of 200 µm 2 were outlined in the cell periplasm. NFRET hotspots co-localizing with MTs and outside MTs ('in the cytosol') were counted and expressed per unit surface area (µm²) of MT or cytosol (see 'Materials and Methods for details). From the same ROIs, a 0.5 µm long segment was drawn on the plus-end of MTs, and a NFRET hotspot was counted each time that one occurred in this calibrated area. For the immunofluorescence of total-stathmin, the density of NFRET hotspots in the cytosol was 1.0 ± 0.1 hotspots/µm 2 (Fig. 2A, white bars). The density of NFRET hotspots on MTs was significantly higher (1.5 ± 0.2 hotspots/µm 2 , Fig. 2B) than in the cytosol. Our data indicate an appreciable fraction of stathmin interacting with the MT wall. Moreover, up to 82% of MT plusends showed NFRET hotspots (Fig. 2C), which is consistent with the presence of highly dynamic MTs in the cell periphery [START_REF] Sammak | Direct observation of microtubule dynamics in living cells[END_REF][START_REF] Walker | Dynamic instability of individual microtubules analyzed by video light microscopy: rate constants and transition frequencies[END_REF].

For pSer38-and pSer25-stathmin (gray and black bars, respectively in Fig. 2A-C), we counted 1.0 ± 0.1 and 0.7 ± 0.1 NFRET hotspots/µm 2 on MTs respectively, and 0.1 ± 0.1 hotspots/µm 2 in the cytosol for both phospho-stathmins. Also, 37% and 17% of MT plus-ends with NFRET hotspots were measured for pSer38-and pSer25-stathmin, respectively. In cells labeled for pSer38-(left panel) and pSer25-stathmin (middle panel),

NFRET images show a punctuated distribution of NFRET signals mainly on MTs (Fig. 2D, white arrows).

Thus, pSer25-and/or pSer38-stathmin could bound to the wall in addition to the plus-ends of MTs. By contrast, for anti-pSer16 stathmin, the number of NFRET hotspots was very low (Fig. 2A-B), and no hotspots were colocalized with the MT plus-ends (Fig. 2C). This is consistent with a loss of interaction between tubulin/MT and pSer16-stathmin [START_REF] Manna | Stathmin strongly increases the minus end catastrophe frequency and induces rapid treadmilling of bovine brain microtubules at steady state in vitro[END_REF], suggesting that phosphorylation of Ser16 abolished the ability of stathmin to bind to tubulin and/or MTs. By contrast, the phosphorylation of Ser25 and/or Ser38 did not affect the binding of stathmin to tubulin or MTs (wall and tips).

To address the question of how stathmin interacts with the MT wall, we over-expressed in living cells two forms of stathmin truncated in either the C-terminus (∆Cter-stathmin coupled to EGFP, as donor) or the Nterminus (∆Nter-stathmin also coupled to EGFP) of the protein. Tubulin tagged with mCherry was co-expressed as an acceptor. To avoid any effect of phosphorylation, the ∆Cter-stathmin was mutated on Ser16, 25, 38 and 63, and the ∆Nter-stathmin on Ser63 to alanine. All the NFRET quantifications were performed directly in living cells. When cells expressed ∆Cter-stathmin, a similar density of NFRET hotspots was measured on MTs (2.2 ± 0.8 hotspots/µm 2 ) (Fig. 3A) and in the cytosol (1.9 ± 0.9 hotspots/µm 2 ) (Fig. 3C, right panel) where many NFRET hotspots were observed in the cytosol (arrows in ROI1) and along the MTs (arrows in ROI2). These results suggest that the ∆Cter-stathmin protein interacts similarly with cytosolic tubulin and MTs. On the other hand, the expression of ∆Nter-stathmin led to 2.5 ± 0.7 hotspots/µm 2 on MTs (Fig. 3B), and 0.5 ± 0.2 hotspots/µm 2 in the cytosol (Fig. 3B). These data suggest that the deletion of the N-terminus domain of stathmin facilitates its binding to the MT wall. In the T2S complex, the N-terminal region of stathmin is known to bind to the exposed surface of α-tubulin. The steric hindrance between dimers of tubulin in a protofilament does not permit correct positioning of the N-terminus domain, and so the ∆Nter-stathmin has an ability to bind preferentially to the surfaces and/or structures of tubulin exposed on the MT wall. Dynamic molecular interactions are fundamental to all cellular processes. In cells, analyses of these interactions are frequently carried out using fluorescence recovery after photobleaching (FRAP). The binding exchange of stathmin with tubulin/MT was next investigated by FRAP.

A binding-dominant model describes stathmin-tubulin/microtubule interactions.

In FRAP imaging, the rate of fluorescence recovery indicates how fast neighboring fluorescent molecules fill a bleached zone. This mobility of proteins depends on both diffusion and potential binding interactions. In our study, FRAP experiments were carried out on full stathmin (coupled to EGFP) expressed by living cells that also expressed mCherry-tubulin in order to locate the bleaching spots on MTs more clearly.

We first evaluated the contribution of diffusion in the FRAP curve. In cells over-expressing EGFPstathmin, we used two photobleaching spot sizes (radius 2 µm and 4 µm) and compared fluorescence recovery curves (Fig. S3). We observed that the two curves were closely similar for the two spot sizes. Data were fitted using a single exponential equation. They gave comparable fractions of fluorescence at equilibrium (small ROI: Ceq = 0.13 ± 0.01; large ROI: Ceq = 0.15 ± 0.02), and similar rate constants (small ROI: k = 0.57 ± 0.07 s -1 ; large ROI: k = 0.49 ± 0.06 s -1 ). These data indicate that the diffusion of free stathmin is so fast relative to its binding to MTs that it can be ignored. The time curve of fluorescence recovery is dominated by the binding reaction of stathmin with tubulin and/or MTs and other protein partners (kinases and/or phosphatases).

We then used the binding-dominant model to determine how EGFP-stathmin interacts with MTs. FRAP experiments were carried out in the presence of MTs vs. in their nocodazole-induced absence (Fig. 4 and Table 1). We note that EGFP-stathmin can be phosphorylated by the enzyme machinery of cells. With MTs ('stathmin + microtubules' in Table 1), k*on < koff for the binding phase (k*on = 0.07 ± 0.02 s -1 and koff = 0.36 ± 0.06 s -1 ) together with Ceq = 0.13±0.01 (the fraction of stathmin bound at equilibrium) are evidence that stathmin binds to tubulin/MTs. With nocodazole ('stathmin -microtubules' in Table 1), k*on = 0.14 ± 0.03 s -1

<< koff = 1.29 ± 0.12 s -1 , together with the decreased Ceq = 0.10 ± 0.01, indicate a faster binding exchange of the stathmin/tubulin complexes and other protein partners than with MTs. The longer t1/2 (1.91 ± 0.30 s) obtained with MTs than without them (t1/2 = 0.54 ± 0.05 s) also shows that stathmin binds more strongly to tubulin when MTs are assembled.

When the MT polymerization is prevented by nocodazole, disappearance of the hyperphosphorylation forms and a progressive decrease in stathmin phosphorylation on Ser16 are observed [START_REF] Küntziger | Stathmin/Op18 phosphorylation is regulated by microtubule assembly[END_REF].

Here, the decrease in hyperphosphorylation forms of stathmin did not offset the lack of MT: the overall binding process of stathmin was reduced. Under regular conditions (no drug) a large proportion of stathmin forms are bound to MTs, and a smaller share to soluble tubulin dimers.

(Table 1). For ∆Cter-stathmin with and without MTs, similar short t1/2 (0.63 ± 0.05 s and 0.40 ± 0.02 s, respectively), k*on << koff, and Ceq values (0.05 ± 0.01 and 0.04 ± 0.01, respectively) were measured. With or without MTs, FRAP parameters were similar, suggesting that the N-terminal domain of stathmin binds preferentially to the soluble tubulin dimers. In addition, FRAP analysis of ∆Nter-stathmin in the presence of MTs gave a 2.6-fold lower koff (0.61 ± 0.06 s -1 ) and 2.6-fold longer t1/2 (1.13 ± 0.11 s) than without MT (koff = 1.59 ± 0.08 s -1 and t1/2 = 0.44 ± 0.1 s). The value of Ceq = 0.14 ± 0.01 was significantly higher than that without MT (Ceq = 0.04 ± 0.01). These data are comparable to those obtained with EGFP-stathmin obtained in the presence of MTs, indicating an appreciable fraction of stathmin interacting with the MT wall.

FRAP analysis showed that exchange at the binding site of stathmin is significantly modified by the presence of MTs, and that the C-terminal domain of stathmin is involved in this reaction. The above experiments yield compelling evidence that stathmin can bind to MTs in cells. We then sought to elucidate how the MT dynamics and their structure were involved in this interaction. For this purpose we used taxol, a wellknown ligand able to stabilize MTs and change dynamics.

The interaction of pSer25-and/or pSer38-stathmin with microtubules is promoted by a low concentration of taxol.

It is well established that the structure and stability of MTs are modified by the binding of taxol [START_REF] Green | The effects of taxol on cytoskeletal components in cultured fibroblasts and epithelial cells[END_REF].

Taxol also induces an increase in stathmin phosphorylation [START_REF] Küntziger | Stathmin/Op18 phosphorylation is regulated by microtubule assembly[END_REF][START_REF] Martello | Elevated levels of microtubule destabilizing factors in a Taxol-resistant/dependent A549 cell line with an alpha-tubulin mutation[END_REF]. To further explore the molecular mechanism of the binding of stathmin to MTs, cells were exposed to 1-50 nM taxol (4 h at 37 °C) to determine a concentration of taxol that disturbed the MT dynamics without modifying the phosphorylation of stathmin.

Non-denaturing Western blots revealed that in cells incubated with 1 nM and 3.5 nM of taxol, the phosphorylation levels of stathmin did not change compared with untreated cells (Fig. S2B). Beyond 3.5 nM taxol, tri-and tetra-phosphorylated stathmin increased significantly, indicating large amounts of pSer16-and pSer63-stathmin. In addition, parameters of the MT dynamics were measured (Table S1). Compared with untreated cells, 1 nM taxol did not modify the rate of shrinkage or the time spent in shortening phases, whereas longer pause and shorter growing time (32%) were recorded. Higher concentrations of taxol caused lower rates of shrinkage (up to -52%, at 20 nM taxol).

For the immuno-FRET experiment, we analysed the ability of stathmin (using 'total-stathmin' labelling) and pSer25-/pSer38-stathmin (using specific phosphoform labelling) to interact with tubulin and MTs in the presence of 1nM taxol (4h, 37°C) (Fig. 5). With 1 nM taxol, we found a 1.6-fold higher density of hotspots on MTs (2.4 ± 0.4 NFRET hotspots/µm 2 , Fig. 5B black bar) compared with untreated cells (1.5 ± 0.2 hotspots/µm 2 , Fig. 5D, white bar), showing a large number of NFRET hotspots on MTs (lower panel). No significant change in the number of NFRET hospots in the cytosol (1.0 ± 0.1 hotspots/µm 2 vs. 0.8 ± 0.2 hotspots/µm 2 with taxol, Fig. 5A) was found. The percentages of NFRET hotspots at MT plus-ends were comparable between untreated and 1 nM taxol-treated cells (82% and 95%, respectively) (Fig. 5C). These observations indicate that 1 nM taxol increases the binding of stathmin to the MT wall. Furthermore, we detected more NFRET hotspots for pSer38stathmin (1.9 ± 0.2 hotspots/µm 2 ) and pSer25-stathmin (1.7 ± 0.1 hotspots/µm 2 ) in taxol-treated than in untreated cells (1.0 ± 0.1 NFRET hotspots/µm 2 for pSer38-stathmin, and 0.7 ± 0.1 NFRET hotspots/µm 2 for pSer25-stathmin). No significant change in the counted NFRET hotspots was measured in the cytosol (0.10 ± 0.10 hotspots/µm 2 vs. 0.15 ± 0.10 hotspots/µm 2 with 1 nM taxol, for the two phosphorylated stathmins). Thus, with 1 nM of taxol, the forms of stathmin interacting with MTs were mainly pSer25-/pSer38-stathmin. In addition, for pSer38-and pSer25-stathmin, 37% and 25% of MT plus-ends showed NFRET hotspots respectively, which was similar to untreated cells. Altogether, 1 nM taxol drastically increased the number of interactions of phosphorylated stathmin with the tubulin protofilaments and the plus-ends of MTs.

Our data reveal that in the presence of 1 nM taxol, pSer25-and/or pSer38-stathmin interacted largely with the MT wall and at the plus-ends. The increase in the fraction of stathmin bound to MTs could be related to significant changes in surfaces and/or structures of tubulin exposed on the MT wall and/or to a decrease in the MT dynamic instability mediated by taxol. Thus, our results suggest strongly that low concentration of taxol potentiated the interaction of stathmin by its C-terminal domain with the MT wall.

DISCUSSION

The binding of stathmin to tubulin/MT has been extensively studied using a broad variety of in vitro methods [START_REF] Gupta | Mechanism for the catastrophe-promoting activity of the microtubule destabilizer Op18/stathmin[END_REF][START_REF] Wilson | Microtubules, in vitro[END_REF][START_REF] Barbier | Stathmin and interfacial microtubule inhibitors recognize a naturally curved conformation of tubulin dimers[END_REF][START_REF] Malesinski | Stathmin potentiates vinflunine and inhibits Paclitaxel activity[END_REF]. Stathmin binds to two tubulin heterodimers to form the "curved" T2S tubulin sequestering complex. To date, clear proof of stathmin binding to tubulin/MTs is lacking, especially in cells where interactions depend on the microenvironment. Here, in A549 cells, we demonstrate by immuno-FRET and FRAP imaging that the interaction of stathmin with tubulin can occur not only in the cytosol, but also along the length and at the plus-end of MTs. This pattern had not been observed in cells before.

In a step toward understanding stathmin-tubulin/MT interactions in cells, we focused on the impact of the phosphorylation of stathmin. Recent in vitro data obtained from purified proteins indicate that di-pSer25-/pSer38-stathmin binds to pre-assembled MTs, unlike mono-pSer16-or pSer63-stathmin cannot [START_REF] Manna | Regulation of microtubule dynamic instability in vitro by differentially phosphorylated stathmin[END_REF][START_REF] Manna | Stathmin strongly increases the minus end catastrophe frequency and induces rapid treadmilling of bovine brain microtubules at steady state in vitro[END_REF].

Furthermore, the catastrophe-promoting activity of stathmin cannot be disrupted by phosphorylation of Ser25 and Ser38. In our study, no significant NFRET signal was observed in cells when we used energy transfer between Ser16-phosphorylated stathmin and tubulin/MTs, confirming that stathmin does not bind to MT when phosphorylated on Ser16. We also found a significant quantity of pSer25-and/or pSer38-stathmin interacting with the plus-end and the wall of MT. The MT plus-end can be considered as an intermediate zone where a few curved protofilaments (interacting with unphosphorylated stathmin) coexist with laterally unbound protofilaments (interacting with pSer25-and/or pSer38-stathmin) [START_REF] Mandelkow | Microtubule dynamics and microtubule caps: a time-resolved cryo-electron microscopy study[END_REF]. Furthermore, surfaces and/or structures of tubulin exposed on the MT wall seem to be more favorable for the binding of stathmin by its C-terminus domain. This is strongly supported by the interaction of truncated ∆Nter-stathmin, particularly abundant on MTs. In this study we also examined the effect of low concentration of taxol (1nM), an anticancer drug that modulates mechanical properties of MTs, also reported elsewhere [START_REF] Nogales | Structure of tubulin at 6.5 A and location of the taxol-binding site[END_REF][START_REF] Downing | New insights into microtubule structure and function from the atomic model of tubulin[END_REF], without changing the level of stathmin phosphorylation. Our FRET data mainly evidenced a strong increase in the interaction of pSer38and/or pSer25-stathmin forms with MTs in the presence of this drug. Taxol could induce structural modifications of MTs with substantial effects on the accessibility and/or three-dimensional structure of the binding loci of pSer25-and/or pSer38-stathmin with the MT wall.

Despite the central role of stathmin in the regulation of the MT assembly-disassembly, no direct measurement of the binding exchange of stathmin with MTs in living cells has been yet reported. This is probably because of the difficulties met in detecting rapid stathmin-tubulin exchanges [START_REF] Charbaut | Stathmin family proteins display specific molecular and tubulin binding properties[END_REF][START_REF] Krouglova | Fluorescence correlation spectroscopy analysis of the dynamics of tubulin interaction with RB3, a stathmin family protein[END_REF]. Our systematic analysis of stathmin-tubulin/MT binding through a FRAP approach indicates that stathmin interaction with tubulin/MT in living cells is a very dynamic process, with a t1/2 value of the order of 2 s. Also, FRAP analysis using truncated stathmin revealed similar t1/2 values for the ∆Nter-stathmin protein and full stathmin, but significantly shorter t1/2 for the ∆Cter-stathmin form with and without MTs. The ∆Cter-stathmin should interact very weakly with MTs (k*on << koff). Our FRAP data provide compelling evidence that stathmin can bind by its C-terminal domain to MT.

From all thet findings reported here, we can propose a model for the mechanism by which stathmin and Ser25-and/or Sr38-phosphoisoforms interact with tubulin and/or MTs (Fig. 6, Assembly). Unphosphorylated stathmin binds to tubulin dimers, giving the "curved" T2S assembly-incompetent complex, which indirectly promotes "catastrophe" or MT disassembly. The phosphorylation of Ser16 and Ser63 by kinases (51) leads to an inactive stathmin (as revealed by the loss of the NFRET signal). Ser25 and Ser38 can first be phosphorylated by cyclin-dependent kinase in T2S [START_REF] Brattsand | Cell-cycle-regulated phosphorylation of oncoprotein 18 on Ser16, Ser25 and Ser38[END_REF], releasing the N-terminus domain of stathmin and diminishing a number of interactions with tubulin to induce changes in the three-dimensional structure of the curved T2S complex.

This leads to a "straight" T2S complex that can be incorporated into MTs during the assembly step, as previously suggested [START_REF] Steinmetz | Op18/stathmin caps a kinked protofilament-like tubulin tetramer[END_REF]. In addition, we cannot rule out the possibility that pSer25-and/or pSer38-stathmin bind directly to tubulin protofilaments by their C-termini (as revealed with NFRET hotspots with truncated stathmin). During the disassembly phase (Fig. 6, Disassembly), the pSer25-and/or pSer38-stathmin found next to the ruffled MT plus-end may be dephosphorylated by a protein kinase such as PP2A [START_REF] Mistry | Role for protein phosphatases in the cell-cycle-regulated phosphorylation of stathmin[END_REF]. This allows active non-phosphorylated stathmin forms to act at the tip of MT, favoring the formation of tightly curved protofilament (T2S complex) and contributing directly to the catastrophe event. Being already present on the protofilament of MT, stathmin will only need to be dephosphorylated in response to a signaling pathway to become fully active. The same molecular mechanism has been proposed for MCAK (mitotic centromereassociated kinesin, a depolymerizing protein), which rapidly targets MT plus-ends [START_REF] Helenius | The depolymerizing kinesin MCAK uses lattice diffusion to rapidly target microtubule ends[END_REF]. Taken together, our observations made in cells are consistent with the earlier in vitro studies of Manna et al. showing binding of phosphorylated stathmin throughout the length of MT, and Gupta et al., who demonstrate an interaction of unphosphorylated stathmin with the MT plus-ends [START_REF] Gupta | Mechanism for the catastrophe-promoting activity of the microtubule destabilizer Op18/stathmin[END_REF][START_REF] Manna | Stathmin strongly increases the minus end catastrophe frequency and induces rapid treadmilling of bovine brain microtubules at steady state in vitro[END_REF].

In conclusion, using FRET and FRAP measurements, we have found direct evidence for the interaction of stathmin with both the length and the plus-end of MTs in cells. Here we give a molecular basis for the understanding of the effects of stathmin on the MT cytoskeleton in eukaryotic cells. Our data show the coexistence of the two models of stathmin action, the tubulin-sequestering role and the catastrophe-promoting activities at the plus-ends of MT, and the existence of a new population of phospho-stathmin (pSer25-and/or pSer38-stathmin) capable of binding to MTs without depolymerizing them.

(

  white arrows in region of interest ROI 1, lower panel) and with weakly labeled tubulin in the cytosol (arrows in ROI2). On the right panels, the normalized fluorescence resonance energy transfer (NFRET) signal was observed in pixel clusters, which we name NFRET hotspots. Beneath NFRET image, we superimposed images of the α-tubulin staining (gray in figure) with NFRET signal (purple) and we observed that these NFRET hotspots are located in the cytosol (arrows in ROI2), probably representing stathmin-tubulin complexes, and at the ends of MTs (blue arrowhead in ROI1). Interestingly, we observed NFRET hotspots throughout the length of MTs (arrows in ROI1).

Figure 1 .

 1 Figure 1. FRET imaging detects stathmin-tubulin interactions in the cytosol, at the plus-end and along the length of MT walls. Immuno-FRET (A) and FRET by acceptor photobleaching (B) were performed following an indirect immunofluorescence of α-tubulin (FITC, as donor) and endogenous stathmin (TRITC, as acceptor), phosphorylated and unphosphorylated ('total-stathmin' on images). (A) The co-localization of αtubulin (green) and endogenous stathmin (red) are juxtaposed to NFRET images (right); below, two regions of interest (ROI 1 and 2) with white arrows pointing to NFRET hotspots on MTs and in the cytosol and a blue arrowhead in ROI 1 to a NFRET hotspot at the MT plus-ends; for cropped NFRET images, the co-localization of NFRET hotspots with MTs and cytosolic tubulin are figured; side bars for NFRET intensities: color pixels (grey and purple) score for NFRET efficiencies from 0% to 5%. (B) The recovery of FITC (α-tubulin) emission after TRITC (total-stathmin) photobleaching confirms the NFRET signal in the initial state: the comparison of fluorescence intensities of FITC-labeled α-tubulin between pre-and post-bleaching of TRITC-labeled stathmin (white arrows) reveals a fluorescence recovery of donor in spots on MTs; no relevant fluorescence intensity fluctuation is observed inside the 'control' regions. Side bars: color pixels score for fluorescence intensity in 256 colored levels. Scale bar for (A-B): 15 µm.

Figure 2 .

 2 Figure 2. Stathmin interacting with MTs is phosphorylated on serine 38 and/or 25. Immuno-FRET of αtubulin and stathmin using primary antibodies specific for serine 38 (pSer38), 25 (pSer25) and 16 (pSer16) phosphoforms of stathmin was performed. Numbers of NFRET hotspots located in the cytosol (A) and on MT (B); each reported value corresponds to the mean number ± s.e.m of NFRET hotspots counted in 40 ROIs of 200 µm 2 , a ROI per cells, and expressed per µm 2 of MT or cytosol. Bars refer to immunofluorescence of totalstathmin (white bars), pSer38-stathmin (gray bars), pSer25-stathmin (black bars), pSer16-stathmin (hatched bars). (C) Percentages of MT plus-ends displaying NFRET hotspots (means ± s.e.m of 100 MTs from 20 ROIs); significant statistical differences (*) were calculated using Student's t test with p < 0.01. (D) Images showing the superimposition of immunofluorescence of α-tubulin and phosphoforms of stathmin (upper panels), the distribution of NFRET hotspots (middle panels) and the superimposition of NFRET signal with the α-tubulin labelling (lower panels): the bound forms of pSer25-and/or pSer38-stathmin are mainly located on the MT wall (white arrows), whereas no NFRET hotspots for pSer16-stathmin are detected (white empty arrowheads) despite co-localization with tubulin/MTs; side bars for NFRET intensities: color pixels (grey and purple) score for NFRET efficiencies from 0% to 5%; scale bars: 5 µm.

Figure 3 .

 3 Figure 3. Stathmin can interact by its C-terminal domain with the MT. (A-B) Living cells cooverexpressing truncated EGFP-∆Cter-stathmin (A), EGFP-∆Nter-stathmin (B) forms (as donor) and mCherrytubulin (as acceptor); each reported value corresponds to the mean ± SD number of NFRET hotspots counted in 18 ROIs of 200 µm 2 , a ROI per cells, and expressed per µm 2 of MT (white bars) or cytosol (black bars); the significant statistical difference (**) was calculated using Student's t test with p < 0.001. (C) Data are illustrated by a representative head image with two enlarged ROIs underneath (panels on the right): white arrows show NFRET hotspots on MTs. White arrowheads point to hotspots in cytosol; side bars for NFRET intensities: color pixels (gray and purple) score for NFRET efficiencies from 0% to 12%; scale bars: 10 µm.

Figure 4 .

 4 Figure 4. FRAP curves reveal a binding-dominant model of stathmin with tubulin/MTs. After photobleaching, fluorescence intensity of EGFP-stathmin corresponding to the full stathmin was plotted against time. The fluorescence recovery of EGFP-stathmin was monitored in the presence (filled circle) or in the absence (empty circle) of the MT network. MTs were disassembled by nocodazole (1 mg/ml for 30 minutes and maintained in medium during FRAP experiments). The full line curves correspond to the average fit of 10-15 recovery courses. Compared with no MT, an apparent longer half-time t1/2 of the stathmin fluorescence measured with MTs shows a slower binding exchange of the complex stathmin/MTs than stathmin/tubulin dimers in the cytosol.

Figure 5 .

 5 Figure 5. The interaction of pSer38-and/or pSer25-stathmin with the MTs is promoted by taxol. Immuno-FRET of α-tubulin and stathmin (for total-stathmin, pSer38-stathmin and pSer25-stathmin) was performed on cells incubated without (white bars) and with 1 nM taxol (black bars) (4 h, 37 °C). Numbers of NFRET hotspots located in the cytosol (A) and on MT (B); each reported value corresponds to the mean ± s.e.m number of NFRET hotspots counted in 40 ROIs of 200 µm 2 , a ROI per cells, and expressed per µm 2 of MT or cytosol. (C) Percentages of MT plus-ends displaying NFRET hotspots (means ± s.e.m of 100 MTs from 20 ROIs); significant statistical differences (*) were calculated using Student's t test with p < 0.01. (D) All data are illustrated with (D) panels of images on the right; side bars for NFRET intensities: color pixels (gray and purple) score for NFRET efficiencies from 0% to 5%; scale bars: 5 µm.

Figure 6 .

 6 Figure 6. Conceptual model for the catastrophe-promoting mechanism by which stathmin depolymerizes MTs. Data in cells show that the interaction of stathmin with tubulin/MTs is modulated by its phosphorylation status (numbers in brackets refer to the figure above): [1] Binding of stathmin (green) to tubulin promotes MT catastrophe by sequestering free tubulin [1a] (NFRET hotspots) and by acting on protofilaments at MT plusends, thus destabilizing the tips and increasing the likelihood of catastrophe [1b] (6,55); [2a] Binding of pSer25-

ACKNOWLEDGEMENTS

R.N. received fellowships from Région Provence Alpes Côtes d'Azur and from ONET Technologies. We thank Prof. André Sobel for the gift of the antisera containing rabbit polyclonal anti-pSer16, anti-pSer25 and anti-pSer38 antibodies. V.P. gratefully acknowledges the help of S. Boulay. We acknowledge the financial support of INCa, INSERM and DGOS (SIRIC label). The authors declare that they have no conflict of interest.

/pSer38-stathmin (orange) to MT wall constitutes a pool of inactive stathmin (NFRET hotspots), possibly a scaffold with phosphatases [START_REF] Mistry | Role for protein phosphatases in the cell-cycle-regulated phosphorylation of stathmin[END_REF], dephosphorylated and contributing to the catastrophe; or [2b] phosphorylation of Ser25 and/or Ser38 of stathmin by kinases may change curved to "straight" T2S complex so it can be incorporated into the growing MT [START_REF] Manna | Regulation of microtubule dynamic instability in vitro by differentially phosphorylated stathmin[END_REF][START_REF] Curmi | Stathmin and its phosphoprotein family: general properties, biochemical and functional interaction with tubulin[END_REF]; [START_REF] Devred | Stathmin/Op18 is a novel mediator of vinblastine activity[END_REF] fully phosphorylated stathmin detaches from tubulin/MTs and is unable either to sequester tubulin or to promote MT disassembly [START_REF] Curmi | Stathmin and its phosphoprotein family: general properties, biochemical and functional interaction with tubulin[END_REF].


A new simple recursive algorithm for finding prime numbers using Rosser's theorem

Keywords: Prime numbers, Algorithm prime numbers, Rosser's theorem

)) we defined a new method derived from Rosser's theorem (2) and we used it in order to approximate the nth prime number. In this paper we improve our method to try to determine the next prime number if the previous is known. We use our method with five intervals and two values for n (see Methods and results). Our preliminary results show a reduced difference between the real next prime number and the number given by our algorithm.

However long-term studies are required to better estimate the next prime number and to reduce the difference when n tends to infinity. Indeed an efficient algorithm is an algorithm that could be used in practical research to find new prime numbers for instance.

Introduction

For more details, see the first study (The distribution of prime numbers: overview of n.ln(n))

We define the difference ∆ as below:

1 ∆ = N -(n.ln(n)) n ∈ N * (1)
if N is the real nth prime number and n.ln(n) the approximated nth prime number.

We define ζ as below:

ζ = (n.ln(n)) -∆ n ∈ N * (2)
We define as below:

= ζ ∆ ∆ = 0 (3)
The aim is to know ∆ to find the real nth prime number. In fact ∆ is the difference between the real nth prime number and the number given by the empirical formula. According to [START_REF] Rosser | The n th prime is greater than n.log(n)[END_REF] we have:

ζ = (n.ln(n)) -∆ n ∈ N * ζ = (n.ln(n)) -ζ = 0 ζ = (n.ln(n))-ζ = 0 ζ + ζ = (n.ln(n)) ζ( + 1) = (n.ln(n)) ζ = (n.ln(n))

+1

According to (3) we have:

2 = ζ ∆ ∆ = 0 ∆ = ζ = 0 ∆ = (n.ln(n)) 2 + = 0 (4)
Finally the real nth prime number is given by the following formula:

N = (n.ln(n)) + ∆ n ∈ N * N = (n.ln(n)) 2+ 1+ = -1 N = (n.ln(n))p with 1 < p < 2, p = 2+ 1+ and n > 2 (5)
We must know p to find ∆ and the real nth prime number. In the previous study we noticed that the value of (and consequently p) for a next prime number was very close to the value of p for a previous prime number (for the third interval of the previous study). In this work we confirm that two consecutive prime numbers seem to have values of p that are very close to each other. For this reason we try to use the value of p that is associated with the nth prime number to approximate the next prime number. However there are errors in the determination of the next prime number when we use this value to determine the next prime number. Because the value of p of the next prime number is unpredictable and is either greater than, smaller than or equal to the value of p of the previous prime number, we change the value of p that is associated with the nth prime number and we use three values of p to approximate the next prime number (see Methods and results).

Methods and results

Methods

In this work we use five intervals and two values for n as described below:

n ∈ [2, 200] n ∈ [1000, 1195] n ∈ [1800, 1999] n ∈ [2600, 2903] n ∈ [999971, 10 6 ] n = 100000 n = (2 * 10 17 ) -1
By using Microsoft Excel 2016 we calculate the value of p for each n of each interval and for each value of n (for more details see Introduction and the previous study). Then, in order to approximate the next prime number, we change the value of p as described below:

p ± ln(ln(ln(n+1))) n+1
with ln(ln(ln(n + 1))) = ln 3 (n + 1)

We use these three values of p to approximate the next prime number, as below:

p 1 = p -ln(ln(ln(n+1))) n+1
with ln(ln(ln(n + 1))) = ln 3 (n + 1)

p 2 = p p 3 = p + ln(ln(ln(n+1))) n+1
with ln(ln(ln(n + 1))) = ln 3 (n + 1)

Finally we have:

p 1 * ((n + 1).ln(n + 1)) p 2 * ((n + 1).ln(n + 1)) p 3 * ((n + 1).ln(n + 1))
Our results show that the next prime number defined as p(n+1) is close to one of these three values, with a margin of error and exceptions.

Proposed algorithm

We propose the following recursive algorithm as a way to approximate the next prime number if we know a nth prime number:

1. We define p(n) as the nth prime number. We want to approximate the next prime number p(n+1) with the value of p that is associated with n.

2. We calculate n.ln(n)

3. We calculate ∆ 4. We calculate n.ln(n) -∆ 5. We calculate 6. We calculate p 1 , p 2 and p 3 as described above 7. We approximate p(n+1) using the three values of p :

p(n + 1) ≈ p 1 * ((n + 1).ln(n + 1))
Or p(n + 1) ≈ p 2 * ((n + 1).ln(n + 1))

Or p(n + 1) ≈ p 3 * ((n + 1).ln(n + 1))

The real next prime number p(n+1) is close to one of these three values, with a margin of error and several exceptions.

Results

Discussion

We notice that the real next prime number is close to one of the three values of p(n+1).

However there is often a margin of error and there are several exceptions. Moreover it is impossible to predict the exact value of p that is associated with the next prime number.

Because the value of p of the next prime number is unpredictable and is either greater than, smaller than or equal to the value of p of the previous prime number, we must use the three values of p. Consequently the speed of the algorithm is slow and this is a problem if we want to find new prime numbers.

We fail to approximate the next prime number with p 2 or p 3 when there are twin primes (n and n+2 are both prime) but not with p 1 .

Finally our method seems to be particularly effective even if the value of n is increased (n = (2.10 17 ) -1)) because we found the exact value (-0.4) of the next prime number using p 2 , suggesting that our algorithm may be used to find new prime numbers. However we fail to determine the exact value of other next prime numbers (with n = (2.10 17 ), n = (2.10 17 )+1

and n = (2.10 17 ) + 2, data not shown here).

Conclusion

In this work we approximated the next prime number using the value of p of the previous prime number. It is clear that the value of the next prime number is close to one of the three values of p(n+1) but there is a margin of error and several exceptions. Further investigations are needed to improve this algorithm. Is it possible to predict the exact value of p of the next prime number using the value of p of the previous prime number? Is it possible to decrease the margin of error (when n tends to infinity)?

  ) refers to the number given by our method with the value of p 1 , p 2 or p 3 .

	p(n+1) (p
	n	p(n)	p1	p2	p3	p(n+1) (p1) p(n+1) (p2) p(n+1) (p3) Real next prime number
	175	1039	1.14672182 1.14954286 1.15236391	1043.5228	1046.08996	1048.65713	1049
	176	1049	1.14993182 1.15274069 1.15554956 1053.54281	1056.11624	1058.68966	1051
	1158	9349	1.14386326 1.14444114 1.14501902 9353.49293	9358.21834	9362.94375	9371
	1180	9533	1.14159283 1.14216111 1.14272938 9537.47382	9542.22146	9546.9691	9539
	1900	16363	1.14036498 1.14073525 1.14110551 16367.4388	16372.7531	16378.0675	16369
	1901	16369	1.14010367 1.14047375 1.14084384 16373.4365	16378.7515	16384.0665	16381
	2642	23719	1.13912624 1.13940047	1.1396747	23723.4062	23729.1173	23734.8283	23741
	2643	23741	1.13969691 1.13997104 1.14024518 23745.4113	23751.1228	23756.8343	23743
	100000	1299709 1.128903959 1.128912894 1.128921828 1299712.84 1299723.126 1299733.412	1299721
	999971	15485473 1.12090981 1.12091077 1.12091174 15485476.3	15485489.6	15485502.9	15485497
	999972	15485497 1.12091034 1.12091131 1.12091227 15485500.3	15485513.6	15485526.9	15485537
	999973	15485537 1.12091203	1.120913	1.12091397 15485540.3	15485553.6	15485566.9	15485539
	999974	15485539 1.12091098 1.12091194 1.12091291 15485542.3	15485555.6	15485568.9	15485543
	(2.10 17 ) -1	1	2	3	4	5	6	7	8
	1 8512677386048191019
	2 1.06843604568915229910775060
	3 1.06843604568915230562882979
	4 1.06843604568915231214990898
	5 8512677386048191010.6
	6 8512677386048191062.6
	7 8512677386048191114.5
	8 8512677386048191063

The list of prime numbers used in this study. http://compoasso.free.fr/primelistweb/page/prime/ liste online.php

The next prime numbers. The next prime numbers were found using http://www.numberempire.com/primenumbers.php