Brieuc Chauvin
email: brieuc.chauvin@curie.fr

Koyomi Nakazawa
email: koyomi.nakazawa@curie.fr

Alexandre Beber
email: alexandre.beber@ibt.cas.cz

Aurélie Di Cicco
email: aurelie.dicicco@curie.fr

Bassam Hajj
email: bassam.hajj@curie.fr

Manos Mavrakis
email: manos.mavrakis@fresnel.fr

Gijsje H Koenderink

João T Cabral
email: j.cabral@imperial.ac.uk

Michaël Trichet
email: michael.trichet@sorbonne-universite.fr

Stéphanie Mangenot
email: stephanie.mangenot@parisdescartes.fr

Aurélie Bertin
email: aurelie.bertin@curie.fr

SUMMARY:

Septins are cytoskeletal proteins. They interact with lipid membranes and are able to sense but also generate membrane curvature at the micron scale. We describe in this protocol bottom-up in vitro methodologies meant to analyze membrane deformations, curvature-sensitive septin binding, and septin filament ultrastructure.

ABSTRACT:

Membrane remodeling occurs constantly at the plasma membrane and within cellular organelles. To fully dissect the role of the environment (ionic conditions, protein and lipid compositions, membrane curvature) and of the different partners implicated in specific membrane reshaping processes, we undertake in vitro bottom-up approaches. In the recent years, we have been particularly interested in revealing the role of septin proteins, implicated in major diseases. Septins are essential and ubiquitous cytoskeletal proteins that interact with the plasma membrane. They are implicated in cell division, cell motility, neuro-morphogenesis and spermiogenesis among other functions. In that context, we want to understand how septins interact and are organized at membranes, subsequently induce membrane deformations and can be sensitive to specific membrane curvatures. We aim at deciphering the interplay between the ultra-structure of septins at a molecular level and the membrane remodeling occurring at a micron scale. To this end, budding yeast and mammalian septin complexes are recombinantly expressed and purified. We use a combination of in vitro assays to analyze the self-assembly of septins at the membrane. SLBs (Supported Lipid Bilayers), GUVs (Giant Unilamellar Vesicles), LUVs (Large Unilamellar Vesicles) and wavy substrates are used to study the interplay between septin self-assembly, membrane reshaping and membrane curvature.

INTRODUCTION:

Septins are cytoskeletal filament-forming proteins interacting with lipid membranes. Septins are ubiquitous in eukaryotes and essential to numerous cellular functions. They were identified as main regulators of cell division in budding yeast and in mammals [START_REF] Finger | Reining in cytokinesis with a septin corral[END_REF][START_REF] Barral | Structural insights shed light onto septin assemblies and function[END_REF] . They are involved in membrane reshaping events, ciliogenesis [START_REF] Hu | A septin diffusion barrier at the base of the primary cilium maintains ciliary membrane protein distribution[END_REF] , and spermiogenesis [START_REF] Lin | The role of the septin family in spermiogenesis[END_REF] . Within mammalian cells, septins can also interact with actin and microtubules [START_REF] Addi | Actin, microtubule, septin and ESCRT filament remodeling during late steps of cytokinesis[END_REF][START_REF] Spiliotis | Spatial regulation of microtubule-dependent transport by septin GTPases[END_REF][START_REF] Spiliotis | Cellular functions of actin-and microtubule-associated septins[END_REF] in a BORG dependent manner [START_REF] Salameh | Cdc42 and its BORG2 and BORG3 effectors control the subcellular localization of septins between actin stress fibers and microtubules[END_REF] . In various tissues (neurons 9 , cilia 3 , spermatozoa [START_REF] Myles | A localized surface protein of guinea pig sperm exhibits free diffusion in its domain[END_REF] ), septins have been identified as regulators of diffusion barriers for membrane-bound components [START_REF] Luedeke | Septin-dependent compartmentalization of the endoplasmic reticulum during yeast polarized growth[END_REF] . Septins have also been shown to regulate membrane blebbing and protrusion formation [START_REF] Gilden | The septin cytoskeleton facilitates membrane retraction during motility and blebbing[END_REF] . Septins, being multi-tasking proteins, are thus implicated in the emergence of various prevalent diseases [START_REF] Dolat | Septin functions in organ system physiology and pathology[END_REF] . Their mis-regulation is associated with the emergence of cancers [START_REF] Angelis | Septin Mutations in Human Cancers[END_REF] and neurodegenerative diseases [START_REF] Takehashi | Septin 3 gene polymorphism in Alzheimer's disease[END_REF] . Depending on the organism, several septin subunits (from two (Caenorhabditis elegans) to thirteen (humans)) assemble to form complexes whose organization varies in a tissue-dependent fashion [START_REF] Shuman | Septins From Protists to People[END_REF] . The basic septin building block gathers two to four subunits, present in two copies, self-assembled in a rod-like palindromic manner. In budding yeast, septins are octameric [START_REF] Bertin | Saccharomyces cerevisiae septins: supramolecular organization of heterooligomers and the mechanism of filament assembly[END_REF][START_REF] Iv | Insights into animal septins using recombinant human septin octamers with distinct SEPT9 isoforms[END_REF] . In situ, septins are often localized at sites with micrometer curvature: they are found at division constriction sites, at the base of cilia and dendrites, and at the annulus of spermatozoa [START_REF] Beber | reshaping by micrometric curvature sensitive septin filaments[END_REF][START_REF] Bridges | Micron-scale plasma membrane curvature is recognized by the septin cytoskeleton[END_REF] . At the membrane, the role of septins seems to be dual: they are implicated in reshaping the lipid bilayer and in maintaining membrane integrity [START_REF] Patzig | Septin/anillin filaments scaffold central nervous system myelin to accelerate nerve conduction[END_REF] . Hence, investigating the biophysical properties of septin filament-forming proteins and/or subunits at the membrane is crucial to understand their role. To dissect specific properties of septins, in a well-controlled environment, bottom-up in vitro approaches are appropriate. So far, only a limited number of labs have described the biophysical properties of septins in vitro [START_REF] Bridges | Micron-scale plasma membrane curvature is recognized by the septin cytoskeleton[END_REF][START_REF] Szuba | Membrane binding controls ordered self-assembly of animal septins[END_REF][START_REF] Tanaka-Takiguchi | Septin-mediated uniform bracing of phospholipid membranes[END_REF] . Hence, as compared with other cytoskeletal filaments, our current knowledge on the behavior of septins in vitro remains limited. We describe in this protocol how the organization of septin filaments, membrane reshaping and curvature sensitivity can be analyzed [START_REF] Beber | reshaping by micrometric curvature sensitive septin filaments[END_REF] . To this end, we use a combination of optical and electron microscopy methods (fluorescence microscopy, cryo-electron microscopy (cryo-EM) and Scanning Electron Microscopy (SEM)). The membrane reshaping of micrometric-sized Giant Unilamellar Vesicles is visualized using fluorescence optical microscopy. The analysis of the arrangement and ultrastructure of septin filaments bound to lipid vesicles is performed using cryo-EM. Septin curvature sensitivity is carried out using SEM, by studying the behavior of septin filaments bound to solid supported lipid bilayers deposited on wavy substrates of variable curvatures, which enables the analysis of curvature sensitivity for both positive and negative curvatures. As compared with previous analysis [START_REF] Bridges | Micron-scale plasma membrane curvature is recognized by the septin cytoskeleton[END_REF][START_REF] Bertin | Phosphatidylinositol-4,5-bisphosphate promotes budding yeast septin filament assembly and organization[END_REF] we propose here to use a combination of methods to thoroughly analyze how septins can self-assemble, synergistically deform membrane and be curvature sensitive. We believe our protocol will be useful and adaptable to any filamentous protein which displays an affinity for membranes.

PROTOCOL:

1. Determination of membrane reshaping using Giant Unilamellar Vesicles (GUVs) Giant Unilamellar Vesicles are generated to mimic the membrane deformations possibly induced by septins in a cellular context. Indeed, in cells, septins are frequently found at sites with micrometer curvatures. GUVs have sizes ranging from a few to tens of micrometers and can be deformed. They are thus appropriate to assay any micrometric-scale septin-induced deformations. Fluorescent lipids as well as fluorescently labelled septins (Green Fluorescent Protein (GFP)) are used to follow the behavior of both the lipids and the proteins by fluorescence microscopy.

1.1. Buffers and solution preparation 1.1.1. Prepare the 'Growth Buffer' (50mM NaCl, 50mM Sucrose and 10 mM Tris (pH = 7.8)) and the 'Observation Buffer' (75mM NaCl and 10 mM Tris (pH=7.8)).

1.1.2. Measure and adjust the osmolarity of the Observation and Growth Buffers (of 170 mOsmol.L -1 , in theory) by adding small amounts of NaCl using a commercial osmometer until their respective osmolarity are equal. Filter the buffers using a 0.2 µm filter. Aliquot the Growth buffer and store it at -20°C for further use. Store the Observation Buffer at 4°C. Note: The difference of osmolarity between both buffers should not exceed 5%. 1.1.3. Prepare a 5 mg.mL -1 β-casein solution in the Observation Buffer. Complete dissolution must be achieved (after several hours at 4°C and stirring with a magnetic stirrer). Filter the solution with a 0.2 µm filter, aliquot and store the solution at -20°C.

1.2. Lipid preparation Prepare the lipid mixtures at a total lipid concentration of 3 mg.mL -1 in chloroform. A molar ratio of 56.8 % EggPC (Egg L-α-phosphatidylcholine), 15 % cholesterol, 10 % DOPE (1,2-dioleoyl-snglycero-3-phosphoethanolamine), 10 % DOPS (1,2-dioleoyl-sn-glycero-3-phospho-L-serine), 8 % brain PI(4,5)P2 (L-α-phosphatidylinositol-4,5-bisphosphate), and 0.2 % Bodipy-TR-Ceramide is used to enhance the protein-lipid interactions and favor the incorporation of PI(4,5)P2 [START_REF] Beber | Septin-based readout of PI(4,5)P2 incorporation into membranes of giant unilamellar vesicles[END_REF] . NOTE: Chloroform must be handled under a fume hood using nitrile gloves and safety glasses. Plastic should be avoided given that chloroform can dissolve plastic. Glass syringes must be used to pipette chloroform solutions. Before and after use, the syringes must be rinsed by pipetting chloroform 5 to 10 times. Separate syringes should be specifically used to pipet specific fluorescent lipids to prevent any contamination from one experiment to another. Lipids can be stored in chloroform at -20°C in an amber glass vial capped with Teflon. The vials must be filled with argon before closing and sealed with parafilm to prevent any lipid oxidation. 1.3. GUVs electro-formation using a platinum wires setup. A scheme of the experimental steps (1.A) and picture of the chamber at each step of the preparation procedure (1.B) is presented in Figure 1. 1.3.1. Chamber cleaning. Clean the platinum wires and the chamber first thoroughly to remove any lipids. Hence, rub the wires thoroughly as detailed below. Plunge the wires and the chamber in acetone and sonicate for 10 min. Wipe everything carefully with a paper tissue using acetone. Assemble the chamber by inserting the wires, plunge again in acetone, and sonicate for 10 min. Wipe the chamber once more with acetone, making sure the wires are fully cleaned. Plunge the chamber in ethanol, sonicate for 10 min, and wipe with ethanol. Finally, plunge the chamber in deionized water, sonicate for 10 min, and dry with a stream of nitrogen or air. Note: The Teflon chambers (Figure 1.B) were custom-made from our in-house workshop. They accommodate several chambers that can be sealed on both sides using glass coverslips. Platinum wires can be inserted all along the chamber through specific holes (1.3 mm in diameter).

1.3.2. Deposition of lipids: Deposit about 3 to 4 drops (less than 1 μL) of the lipid mixture at 3 mg. mL -1 on each of the platinum wire. The wires are rotated by 180° and deposit 3 to 4 to lipid droplets on the opposite side. Note: The drops should not contact one another. About 5 μL of the lipid mixture are required per chamber.

1.3.3. Insert the growth chamber within a vacuum chamber for 30 min to remove any trace of chloroform. NOTE: When dried, the lipids are vulnerable to oxidation. Deep vacuum (0.1 mbars) is best. The dried lipids must not be left in air for over a few min. 1.3.4. Chamber bottom sealing: deposit high vacuum grease at the bottom of the chamber (side closest to the wires) at the periphery of the three wells using a syringe. Press a clean (22*40 mm) coverslip against the grease to ensure a perfect sealing. Both extremities of the chamber are sealed (at the entry/exit sites of the wires) using sealing paste (wax plates). Place vacuum grease in a similar fashion on the other side of the chamber.

1.3.5. Electro-formation. Fill the chambers with growth buffer (about 1 mL is required per chamber). Do not stir too rapidly and strongly the solution to prevent any detachment of the lipid film from the wires. Seal the top of the chamber hermetically using a 22*40 mm coverslip by pressing it against the grease. Discard air bubbles from the chamber. Place the chamber at 4°C, in a fridge, and connect the wires to a wave function generator with the sine function at 500 Hz. For short growths (6 h), set an effective voltage of 350 mV while for longer growths (12 to 16 h) apply a voltage of 250 mV as already presented and optimized in Beber et al [START_REF] Beber | Septin-based readout of PI(4,5)P2 incorporation into membranes of giant unilamellar vesicles[END_REF] .

1.3.6. Chamber cleaning after use. Remove the windows. Wipe off all sealant and grease.

Remove the wires, and wash and scrub the chamber with a tissue using water and ethanol (≥70%) alternately. NOTE: The optimal voltage and time scale for GUVs growth depend on many parameters, from the buffer salt concentration to the chamber geometry (distance between the wires and size). The same chamber should be used every time the experiment is repeated to ensure reproducibility. NOTE: The wires are in proximity with the bottom of the chamber, so that the lipids can be imaged using fluorescent microscopy. Imaging the lipids at each step (See Figure 1.B) is advised to make sure the electro-formation process is successful.

[Place Figure 1 here] 1.4. Septin incubation with the vesicles 1.4.1. GUVs gentle collection. Collect GUVs from the wires by using pre-cut pipette tips that are brought in close vicinity of the wires. Then, pipet the solution all along the wire. This procedure prevents the generation of strong lamellar flows that could disrupt GUVs. Following this step, cutting pipette tips is no longer required. Indeed, lamellar flows do not damage GUVs in solution. NOTE: Because of the presence of PI(4,5)P2 in the lipid mix, GUVs that have been collected must be stored for no more than 2-3 hours prior to the experiment. Indeed, PI(4,5)P2 gets solubilized rapidly and septins no longer bind to the membranes a few hours after their formation. Once septins are bound, however, they remain bound to the membrane for a few days.

1.4.2. Dilute the septins stock solution in Tris 10 mM (pH 8) exclusively to reach an osmolarity equal to the one of the Growth Buffer. The septin solution can, if needed, be further diluted in the Observation Buffer. Add the intended volume of collected GUVs. The incubation can be performed directly in the observation chamber after passivation with β-casein (see below). 20 to 30 min waiting time is required to reach equilibrium. NOTE: The expression and purification of septin octameric complexes (human or budding yeasts) are extensively described in other articles [START_REF] Bertin | Saccharomyces cerevisiae septins: Supramolecular organization of heterooligomers and the mechanism of filament assembly[END_REF][START_REF] Iv | Insights into animal septins using recombinant human septin octamers with distinct SEPT9 isoforms[END_REF] . But briefly, septins are expressed in E. coli and purified following the dedicated protocol, in the lab. Affinity, size exclusion and ion exchange purification steps are performed. After production, septins are kept at -80° in a Tris-HCl 50mM, pH 8, KCl 300 mM, MgCl2 5 mM solution at about 1mg.mL -1 (3 μM). Septin complexes should not be concentrated through a filter centrifugation device, which induces aggregation and thus yield reduction.

1.5. Imaging with confocal and/or spinning disk microscope. 1.5.1. Passivation: To prevent the GUVs from sticking to the surface and/or explode, passivate the observation chamber with a 5mg.mL -1 β-casein solution which is left for a 30 min incubation.

1.5.2. Remove the β-casein solution and insert the collected GUVs in the observation chamber, in the observation buffer, using a pipette. The composition discrepancy between the interior of the GUV and the external buffer creates both a density and refractive index mismatch. Let the GUVs sediment at the bottom of the chamber for 10 to 15 min. 1.5.3. Observation: Thanks to the refractive index mismatch, GUVs are visible with transmission light optical microscopy. Using confocal microscopy, visualize the fluorescent signal of the lipids to check for the GUV quality and membrane lamellarity state. Assess the density of septins bound to GUVs by recording the septins fluorescent signal, after performing a proper calibration [START_REF] Beber | Septin-based readout of PI(4,5)P2 incorporation into membranes of giant unilamellar vesicles[END_REF] . Perform Z-stack acquisitions at 0.4 μm spatial interval to analyze and visualize the 3D deformations of the vesicles induced by the interaction between the septins and the membrane. NOTE: Oil immersion objectives with 60-or 100-fold magnifications were used. Nikon confocal or spinning disk microscopes with pixel sizes of 250 nm and 110 nm, respectively were used. To collect images, a EZ-C1 Nikon software was employed. Those microscopes are standard microscopes that can be found in any imaging facility. Any given lab would have to adapt their imaging to their own piece of equipment. No specific anti-photo bleaching agent were added in solution.

Analysis of septin filaments ultrastructural organization by cryo-electron microscopy

Vesicles are not suitable for imaging with standard electron microscopy methods. Indeed, using standard negative stain methods, samples are dried. Upon dehydration, vesicles are likely to undergo unspecific deformations, frequently resulting in lipid protrusions. Cryo-electron microscopy is thus a much better strategy to observe specific deformations of vesicles. Indeed, using cryo-EM, the samples are embedded within a thin (about 100-200 nm) layer of vitrified ice, which preserves the samples in a close to native state. GUVs are too large (several tens of micrometers) to be embedded within thin ice and thus imaged by transmission electron microscopy. Hence, Large Unilamellar Vesicles (LUVs), whose diameters range from about 50 to 500 nm, are generated to determine both how septins can deform vesicles and how they arrange on vesicles.

2.1. Generation of Large Unilamellar Vesicles 2.1.1. Prepare 50 µg of a mix of lipid solubilized in chloroform with a compositions (56.8% EggPC, 15% cholesterol, 10% DOPE, 10% DOPS, 8% brain PI(4,5)P2; molar ratios) optimized to enhance septin interaction with the membrane in a glass vial.

2.1.2. Dry the solution under a flow of argon to generate a dried lipid film in the vial. Put the vial under vacuum for 30 min to achieve full lipid drying.

2.1.3. Resolubilize the lipid film in an aqueous solution (50 mM Tris-HCl (pH 8); KCl 50 mM; MgCl2 2 mM) at 1 mg.mL -1 (50 μL) by vortexing for 10 s and transfer the solution into an Eppendorf tube. Note: LUVs must be used at once for incubation with septins. Otherwise, the protein-lipid interaction will be weak. This crude re-solubilization process generates a heterogeneous population of vesicles with diameters ranging from 50 nm to 500 nm. Hence, a whole range of diameters, and thus curvatures, are simultaneously assayed.

Incubation with proteins

Incubate the septins with the solubilized lipids at a final septin and lipid concentrations of 0.1 mg. mL -1 (about 300 nM) and 20 nM respectively in a high salt buffer (50 mM Tris-HCl (pH 8), 300 mM KCl, 2 mM MgCl2). Incubate the sample for one hour at room temperature.

2.3. Plunge freezing to vitrify the sample 2.3.1. Glow-discharge holey carbon grids (300 mesh) on the carbon side for 30 s at 5 mA using a plasma generator equipment and insert the grid inside a plunge freezing machine in a humid environment.

2.3.2. Adsorb 4 μL of the sample on the grid, on the carbon glow-discharged side. Immediately before sample adsorption, add 5 to 10 nm gold beads in solution, in case the samples are used to generate tilted series by cryo-tomography. Note: The density of gold beads must be screened and adjusted empirically and depends on the provider. Optimized density should be 10 to 15 gold bead in the field of view.

2.3.3. Blot dry the samples from the bare side to aspirate the sample drop on the opposite side. Note: The blotting time is typically 4 s, and the position of the filter paper and blotting force are adjusted empirically by testing and screening alternative positions of the filter paper to optimize the ice quality (thickness) and the material density.

2.3.4.

Transfer the grids into a microscope or store them in a liquid nitrogen container.

NOTE: Using holey carbon grids is essential to accommodate the polydispersity in sizes of the vesicles. Blotting the sample from the opposite side favors an enhanced adsorption of the biological material on the grid.

2.4. Cryo-electron microscopy imaging 2.4.1. Insert the grids into an electron microscope equipped for cryo-EM observation and cooled down with liquid nitrogen. Screen the whole grid by generating a map of the whole sample at low magnification (120 magnification, typically), to select areas displaying better ice (thin and well vitrified).

2.4.2. Cryo-electron microscopy 2D data collection. Collect images at a pixel size of about 2 Å per pixels to check for the quality of the sample. Note: Both the septin filaments and the lipid bilayer should be visible.

Cryo-electron tomography data collection.

Select areas of interest displaying deformed vesicles. Note: Enough gold beads (at least 10) should be present in the field of view. According to the software used for tilted series data collection, select focus and tracking positions to be far enough from the area of interest. Collect tilted series by varying the tilt angle from -60 to +60 degrees, collecting an image every two or three degrees. Note: The total dose should be about, or less than, 100 electrons/Å 2 . The pixel size varies from 1.3 to 2.1 Å, depending on the microscope used for data collection. Ideally, an angular symmetric scheme for data collection is preferred as follows: 0, -3, +3, -6, +6, -9, +9[...]-60, +60 degrees. However, goniometers of side-entry microscopes do not provide enough mechanical stability to achieve those symmetric schemes. Alternatively, the acquisition can be started at 0 degrees to 34 degrees, followed by a second angular sequence from -2 degrees to -60 degrees, and ended by a final sequence from 36 to 60 degrees. The purpose is to collect the first images (with the lowest radiation damages) at the lowest angles. NOTE: To visualize the ultrastructure of vesicles and bound septin filaments, a standard cryo-EM microscope can be used (200kV, LaB6 (Lanthanum hexaboride) filament and sample side entry). However, if one aims at pursuing further image processing (sub-tomogram averaging, for instance), it is best to use last generation FEG (Field Emission Gun) microscopes equipped with direct detectors. In this protocol, we will restrict our description to the acquisition of 3D reconstruction and leave out sub-tomogram averaging.

2.5. Three-dimensional reconstruction generation from cryo-tomography and segmentation 2.5.1. Image alignment and 3D reconstruction. Use the IMOD software suite for tilted series image alignment and 3D reconstruction [START_REF] Mastronarde | Automated tilt series alignment and tomographic reconstruction in IMOD[END_REF][START_REF] Kremer | Computer visualization of three-dimensional image data using IMOD[END_REF] (https://bio3d.colorado.edu/imod/). Within IMOD, perform tilt series alignment based on fiducials (gold beads) positioning. In addition, if needed, carry out contrast transfer function (CTF) determination and correction within IMOD [START_REF] Mastronarde | Automated tilt series alignment and tomographic reconstruction in IMOD[END_REF] . Finally, achieve 3D reconstruction with IMOD following each step rigorously.

2.5.2. Segmentation. Segment manually the lipid bilayers and septin filaments using 3Dmod [START_REF] Kremer | Computer visualization of three-dimensional image data using IMOD[END_REF] from the IMOD software suite, for display.

Analysis of septin curvature sensitivity using scanning electron microscopy

To understand how septins can be sensitive to micrometer curvatures, we have used an in vitro approach and incubated septin filaments complexes with solid supported lipid bilayers deposited on micrometer-scale wavy undulated patterns.

3.1. Design of wavy NOA (Norland Optical Adhesive) replica from wavy PDMS (polydimethylsiloxane) patterns. 3.1.1. Typically, use patterns of 250 nm amplitude and 2 μm lateral periodicity or other dimensions to assay the curvatures suitable to your protein of interest. Note: PDMS (polydimethylsiloxane) undulated patterns are designed and generated as described in Nania et al. [START_REF] Nania | Sub-100 nm wrinkling of polydimethylsiloxane by double frontal oxidation[END_REF][START_REF] Nania | Frontal vitrification of PDMS using air plasma and consequences for surface wrinkling[END_REF] . Briefly, centimeter large PDMS coupons were subjected to a plasma irradiation while mechanically stretched by a linear strain stage. A thin (~ 10 nm) layer of oxidized, glassy material is formed on the surface of the PDMS. After the mechanical tension is released, the PDMS surface spontaneously buckles, adopting a wavy periodic pattern. The periodicity and amplitude of the undulations can be controlled by tuning the mechanical strain and the power and duration of the plasma treatment. Hence, wavy undulated patterns of prescribed curvature could be produced.

3.1.2. NOA resin replica. From the PDMS substrates generate NOA replicas. In a clean room environment, deposit 5 μL of liquid NOA on a circular glass coverslip of 1 cm in diameter. Appose the PDMS wavy template on the drop. Treat with a UV light treatment at 320 nm for 5 min to photo-polymerize the liquid NOA into a thin polymer film. The slide with the freshly polymerized NOA is then gently peeled off from the PDMS piece. NOTE: NOA is a common optically transparent glue and is thus suitable for optical microscopy imaging. In addition, NOA is resistant to the chemical fixation and staining processes carried out prior to SEM imaging. Both NOA 71 and NOA 81 resins are used with similar outcomes. The PDMS initial pattern could be used several times to produce NOA replicas. After their production NOA replicas can be kept at room temperature, in a box for months.

3.2. Fusion of a lipid supported bilayer on the NOA pattern and protein incubation 3.2.1. Treat the NOA films using a plasma cleaner for 5 min to make the surface of the pattern hydrophilic.

3.2.2. Prepare a solution of SUVS (Small Unilamellar Vesicles) with a molar composition of 56.8% EggPC, 15% cholesterol, 10% DOPE, 10% DOPS, 8% brain PI(4,5)P2 and total lipid of 1 mg.mL -1 . Prepare the SUVs by resuspending a dried lipid film in the Observation Buffer, like the one prepared in section 2.1.3. Sonicate gently the solution using a bath sonicator until the solution is transparent. Sonicate the solution for 5 to 10 min. Note: The solution of SUVs can be stored frozen for several weeks at -20°C.

Insert the slides supporting the NOA patterns within cell culture boxes enclosing wells

whose dimensions accommodate the 1 cm circular coverslips. Deposit 100 μL of the SUV solution at 1 mg.mL -1 on the freshly glow-discharged NOA patterns and leave for incubation for 30 min, at room temperature. Following this incubation, induce the fusion of SUVs with the surface of the pattern to generate a supported lipid bilayer. Note: The volume is large enough so that evaporation is not an issue.

Sample preparation for SEM analysis

Various protocols have been developed in electron microscopy to analyze protein structures and their assembly. Our protocol preserves the organization of septin filaments, using a fixation protocol derived from Svitkina et al. [START_REF] Svitkina | Correlative light and electron microscopy of the cytoskeleton of cultured cells[END_REF] that is easy to implement. Besides, this protocol optimizes high-resolution SEM observations.

and preparation of stock solutions

protocol requires several reagents and solutions that can be prepared in advance, or just prior to incubation. Follow our instructions to avoid any artifacts or lack of chemical reactivity.

Sodium cacodylate 0.2M stock solution: prepare this double-strength stock solution of cacodylate buffer by dissolving powder into distilled water with a magnetic stirrer. After complete dissolution, adjust the pH to 7.4 by gently adding 0.1 M HCl (almost 1 mL is required in case of a 50 mL solution). Add distilled water to adjust to the final volume. This solution can be stored 24 -48 h at 4°C. 2% glutaraldehyde (GA) in 0.1 M sodium cacodylate: Prepare this solution right before the fixation. It is important for glutaraldehyde to have a high level of purity (EM Grade). Prepare the fixative solution to obtain a final concentration of 2% GA in 0.1 M sodium cacodylate. Use 25 to 50 % commercial stock solution of GA because of its optimized storage capacity at -20°C. Dilute this GA solution with the double strength stock solution of sodium cacodylate (see above). 1% Osmium tetroxide (OsO4) in 0.1 M sodium cacodylate: Prepare this solution right before use. Use commercial 4% OsO4 solution in cleavable glass bulbs because of their storage and handling properties. It is important to note that OsO4 is highly reactive. Make sure that its color is slightly yellow and not dark. Prepare this second fixative solution to obtain a final concentration of 1% OsO4 in 0.1 M sodium cacodylate. This OsO4 solution is diluted with the strength stock solution of sodium cacodylate described above. 1% Tannic acid (TA) in water: Prepare this solution right before use. Prepare the TA solution to obtain a final concentration of 1% TA in distilled water at room temperature. Dissolve for a few min by vortexing. TA solution cannot be stored and must be filtered with a 0.2 μm filter before being used. 1% Uranyl acetate (UA) solution in water: prepare UA solution to obtain a final concentration of 1% UA in distilled water. Dissolve for at least 30 min -1h by vortexing or shaking at room temperature. This solution can be stored for a month at 4°C but can easily precipitate and must be filtered with 0.2 μm filters before being used. Graded concentration of ethanol solutions: Prepare 50% -70% -95% -100% ethanol solutions in water. As ethanol is highly hydroscopic, use a final bath in an anhydrous solution to remove all traces of water that might interfere with drying and coating. This last solution must be prepared from newly opened bottles of 100% ethanol, or from 100% ethanol that has been dehydrated for at least 24h with molecular sieves (nominal pore diameter: 4 nm). Be careful not to shake this solution as it might cause a resuspension of silicate particles. Note: Be careful when handling the chemical reagents used in this protocol. Glutaraldehyde, osmium tetroxide, sodium cacodylate and uranyl acetate are all highly toxic, the last of this list being radioactive as well. All manipulation of the reagents and their waste must be done by using individual (gloves, lab coats, safety glasses) and collective protection (fume hoods and plexiglass shields), according to the procedures of your laboratory. Note: Be careful, as the evaporation of ethanol between baths is fast. The sample handling must also be fast to avoid any air-drying.

3.3.3.2.

Transfer the glass slides inside the pre-ethanol-filled critical point dryer and follow manufacturer's instructions. Note: In our protocol, we use an automatic apparatus, but any system can be used. Protocols may differ for each apparatus but must be optimized to completely remove ethanol (25 baths in our protocol) and reduce speeds for both the different solvent exchanges (CO2 inlets and ethanol/CO2 outlets) and for the final depressurization (almost 1h in our protocol).

3.3.3.3. At the end of the critical point drying, store samples immediately in a desiccator mounting and coating. As dried samples are highly hygroscopic, we recommend coating them as soon as possible.

Alternately, HMDS can offer a cheaper and faster method. Note that we have never used or tested it on our samples, but this approach produced great results for the observation of proteins at the inner face of cellular membranes [START_REF] Franck | Clathrin plaques and associated actin anchor intermediate filaments in skeletal muscle[END_REF][START_REF] Elkhatib | Tubular clathrin/AP-2 lattices pinch collagen fibers to support 3D cell migration[END_REF] .

Sample mounting and coating

biological samples present poor electrical conductivity properties, they need to be coated with a conductive metal film before SEM observation. Plasma-magnetron sputtering is thus used. 3.3.4.1. Attach the glass slides to stubs that will be used for future SEM observations. Use silver paint because of its improved electrical conductivity, as compared with carbon disks. Add a strip of silver paint to the upper face of the coverslip and make sure that the connection with the stub is satisfactory. Avoid any paint agglomerates. Note: The paint strip must be thin, to avoid any contact between the sample and the objective lens of the SEM when working at high resolutions (i.e., low working distance).

3.3.4.2. Wait for the solvent to evaporate completely. Note: The duration of this step may vary depending on the amount and the thickness of the silver paint deposite. This step can be shortened by using a bell jar or the coater and a primary vacuum for 10-30 min.

3.3.4.3. Use an apparatus equipped with a plasma-magnetron sputtering head and a rotary-planetary stage. Use the standard protocols that are provided by the manufacturer. The apparatus is vacuumed to 2.5 x 10 -5 mbar, purged once with high quality Argon before adjusting the pressure to 8.0 x 10 -3 mbar for a pre-sputtering phase (120 mA for 60 s, to the oxide layer at the surface). Deposit 1.5 nm of tungsten (90 mA, working distance 50 mm) controlled by the film thickness monitor. Note: The coater must be perfectly cleaned to ensure the reproducibility of the film evaporation. The coating must be stopped once the targeted thickness (indicated by the user) has been reached. The final film thickness is calculated and corrected afterward. Films of about 1.5 nm on average, a post-correction of 0.7 nm using our apparatus. Because those correction values are close to the target, a series of coatings are performed to assess and then subtract this correction from the target value.

3.3.4.4. the samples under vacuum to protect them from the ambient air until, and throughout all the SEM analyses. Note: the nature of the metal used for sputtering matters. Platinum, despite being a common material, results in a coating of poor quality with a Pt film thickness adapted to septin filaments (1.5 nm). At high resolutions, a 1.5 nm platinum film exhibits a lack of cohesivity. The size of the platinum clusters and the septin filaments thereby become similar, leading to misinterpretations during the filament segmentation process [START_REF] Stokroos | A comparative study of thin coatings of Au/Pd, Pt and Cr produced by magnetron sputtering for FE-SEM[END_REF] (see Figure 2.A, insert). Tungsten is a good alternative to platinum because it displays a smaller grain size, barely visible at high resolution SEM (see Figure 2.B, insert). Nonetheless, pure tungsten is easily oxidized, leading to strong charging effect artifacts during SEM observation if the procedure detailed in section 3.3.4. is not strictly followed.

[place Figure 2 here] 3.4 Sample observation by SEM Acquire the images using a Field-Emission gun SEM (FESEM) microscope. Note: the technology was recently upgraded with a novel electron optic design to improve resolution. This gain in resolution (accessible with other manufacturers), especially at low accelerating voltage (0.9 nm at 1 kV), is necessary to resolve nanometric structures similar to septin networks. Achieve high resolution imaging using the detection of primary secondary electrons (SE1) with the "in-lens" detector and with the settings below:

-accelerating voltage: 3 kV, -the beam current is fixed with the 20 mm (34 pA) aperture or decreased with the 15 mm (18,5 pA) aperture if the suppression of charging effects is needed, -observations are done with resolutions ranging from 21.25 to 1.224 nm/pixel, and at about 5.58 nm/pixel for data analysis (20 k magnification according to the Polaroid 545 reference), -working Distance: between 1 and 2 mm for high resolution observation, around 3 mm if increased depth of field is required, -scan speed and line integration are continuously adjusted to ensure a constant signal to noise ratio with an acquisition time around 30 to 45 s per image.

REPRESENTATIVE RESULTS:

1. GUVs deformations.

Typical confocal fluorescence images of GUVs reshaped after being incubated with septins are displayed in Figure 3, in conditions where septins polymerize. Bare GUVs (Figure 3.A) are perfectly spherical. Upon their incubation with more than 50 nM budding yeast septins filaments, the vesicles appear deformed. Up to a concentration of 100 nM budding yeast septin octamers, the vesicles appear facetted, and the deformations remain static and thus do not fluctuate (Figure 3.B). Above 200 nM budding yeast septins, with the membrane being saturated with septins, periodic deformations are observed (Figure 3.C). Using budding yeast septins, spikes are visualized with a periodicity varying from 2 to 6 micrometers and with an amplitude of about 1 micrometer. Hence septins strongly reshape membranes in a specific fashion. These amplitudes and associated curvatures of the observed deformations strikingly reflect the affinity of septins for micrometer curvatures observed in cells. GUVs, being deformable, are thus appropriate to assay how proteins reshape membranes.

[Place Figure 3 here] 2. Self-assembly of septin filaments bound to LUVs imaged by cryo-EM.

As compared with fluorescence microscopy, cryo-EM provides better spatial resolution. Individual septin filaments and lipid bilayers can thus be discerned unambiguously in images. Figure 4.A displays an image of LUVs decorated with budding yeast septin filaments (some of them being highlighted in blue) at 50 nM. Parallel sets of filaments, about 10 nm apart, are observed on the LUVs. This typical image is a two-dimensional projection. Hence, to decipher any 3D deformation and to know whether septin filaments directly interact with the membrane, cryo-electron tomography is carried out (Figure 4.B. and 4.C). Figure 4.B displays a slice in a three-dimensional reconstruction, while Figure 4.C. presents a segmentation of the same area highlighting the membrane in yellow and septin filaments in blue. As seen on the side view, septin filaments remain essentially straight bound to the liposome and the vesicle has been considerably flattened as compared with naked spherical vesicles.

[Place Figure 4 here] 3. Curvature dependent arrangement of septin filaments visualized by Scanning Electron Microscopy.

Septins localize, in cells, at locations displaying micrometer curvatures (e.g., at the cell division furrow, the base of cilia, etc.). In addition, in vitro studies show that septins, as seen above, can reshape membranes so they eventually display micrometer curvatures. To assay the curvature sensitivity of septins for both positive (convex) and negative (concave curvatures), we have incubated septins with a supported lipid bilayer fused with the wavy patterns described above. The outcome is displayed in Figure 5. Figure 5.A recapitulates the different steps required to obtain the samples. Figure 5.B presents a wavy pattern at low magnification, where the periodicity of negative and positive curvatures is clearly visible. Two successive waves are pointed by orange solid lines. Besides defects, on average orthogonal to the waves are pointed by white arrows. On Figures 5.C and 5.D, at higher magnification, the ultrastructural organization of septin filaments is displayed. Septins assemble into sets of parallel filaments, whose orientation depends on the curvature (see filaments highlighted in blue). Indeed, septin filaments are not oriented randomly. Instead, septins can bend when interacting with negative (concave curvatures) to follow the imposed curvature. Conversely, on positive curvatures, septin filaments remain straight, unbent, and align along the convex waves. Hence, septins can interact with both positive and negative curvatures but adopt specific organizations on given curvatures. This methodology thus appears particularly relevant to highlight the curvature sensitivity of binding and self-assembly of filamentous proteins.

[Place Figure 5 here] Effect of the material deposited on septin filaments on wavy PDMS patterns. SEM of filaments which are coated by sputtering with either A) 1.5 nm of platinum, exhibiting the "arid cracked soil" pattern typical of a lack of cohesivity between the clusters of platinum nuclei, or B) 1.5 nm of tungsten covered with a smooth and cohesive layer. Scale bar 200 nm, white-boxed squares represent magnified views on lower right. The spherical globules are small lipid vesicles interacting with the septins. As stated above, we have used a lipid mixture which enhances PI(4,5)P2 incorporation within the lipid bilayer and thus facilitates septin-membrane interactions. Indeed, we have shown elsewhere [START_REF] Beber | Septin-based readout of PI(4,5)P2 incorporation into membranes of giant unilamellar vesicles[END_REF] that budding yeast septins interact with vesicles in a PI(4,5)P2 specific fashion. This composition was adjusted empirically from screening multiple compositions and is now widely used in our lab. PI(4,5)P2 lipids have to be handled carefully. Stock solutions must be aliquoted in small volumes so that a specific vial is not opened more than twice for pipetting. Besides, lipid aliquots should be stored under argon to prevent lipid oxidation. Finally, once the lipid mixes are solubilized in an aqueous solution, the mixture must be used at once and cannot be stored for further experiments. We have noticed that after GUVs reconstitution or supported lipid bilayer fusion, the interaction between septins and lipids is greatly reduced after two hours. We have chosen to present some of the results obtained with octameric budding yeast septin complexes (enclosing Cdc10, Cdc11, Cdc12 and Cdc3). However, the experimental procedures followed to analyze the behavior of septin complexes from other species should be identical. To generate GUVs, we prefer the electro-formation method using platinum wires over electroformation on ITO plates or PVA swelling. As demonstrated earlier [START_REF] Beber | Septin-based readout of PI(4,5)P2 incorporation into membranes of giant unilamellar vesicles[END_REF] , this method reproducibly produces unilamellar GUVs of good quality without any defects. The wavy substrates can be designed with a wide range of curvatures. After screening various designs, we have consistently used patterns with a periodicity of 2 μm and an amplitude of 250 nm which generate curvatures ranging from -3.5 μm -1 to 3.5 μm -1 (± 0.5 um -1 ). Using patterns with a short periodicity and higher amplitude, the lipid bilayers did not nicely conform to the surface. Often, in the case of higher curvature, the bilayer would be suspended in between two convex waves instead of being fully supported on the NOA substrate. With lower amplitudes and larger periodicity, septins organized with random nematic-like orientations, indicating that the patterns were too flat. Besides, using wavy patterns instead of cylindrical tubes [START_REF] Bridges | Micron-scale plasma membrane curvature is recognized by the septin cytoskeleton[END_REF] or spheres [START_REF] Bridges | Micron-scale plasma membrane curvature is recognized by the septin cytoskeleton[END_REF] has the benefit that both positive and negative curvatures can be tested simultaneously. the future, examining the behavior of filaments on gaussian "saddle-like" curvatures would be relevant to mimic the curvature of more realistic cellular contexts. In solution, mixing LUVs with septins induced the generation of septin-lipid aggregates even at low concentrations of proteins and lipids. The samples thus were heterogeneous, and it is wise to look for thinner ice domains where smaller and more defined septin-vesicle objects are found. This is particularly crucial when choosing areas of interest where cryo-tomography will be performed. Often, even at low magnifications, vesicles displaying protrusions of strong deformations as compared with spherical vesicles are more likely to have a high density of septin filaments bound. This protocol focuses on obtaining a description of the global ultrastructure of filaments on biomimetic membranes. Currently higher resolutions are reachable. The resolution will result from the limited number of views available by imaging. However, this remains beyond the scope of this protocol. We have shown through this protocol that a combination of complementary methods is essential to understand and dissect how the specific arrangement of cytoskeletal septin filaments can sense and reshape membranes in a curvature dependent manner.

FIGURE AND TABLE LEGENDS:

Figure 1 .

 1 Figure 1. A. Schematic representation of the electro-formation process using platinum wires. B. Picture of the teflon home-made device used to generate GUVs by platinum wire electron formation. Wires are 0.5 mm in diameter and 3 mm apart. C. GUVs (spherical objects) observed by transmission microscopy during the growth process. The opaque zone at the bottom of the image is the platinum wire. D. GUVs (round-shaped fluorescent objects) observed with fluorescent microscopy during the growth on the platinum wire. Scale bars are 100 m.

Figure 2 .

 2 Figure 2.Effect of the material deposited on septin filaments on wavy PDMS patterns. SEM of filaments which are coated by sputtering with either A) 1.5 nm of platinum, exhibiting the "arid cracked soil" pattern typical of a lack of cohesivity between the clusters of platinum nuclei, or B) 1.5 nm of tungsten covered with a smooth and cohesive layer. Scale bar 200 nm, white-boxed squares represent magnified views on lower right. The spherical globules are small lipid vesicles interacting with the septins.

Figure 3 .

 3 Figure 3. A. Equatorial plane of a Control GUVs labelled with 0.5% rhodamine PE, images by confocal microscopy. Scale bar 3 μm. B.GFP-septins (100 nM) bound to a deformed GUVs and imaged by fluorescence confocal microscopy. Scale bar 1μm. C. GFP-septins (600 nM) bound to a deformed GUVs and imaged by fluorescence spinning disk microscopy. Scale bar 10μm. The GUVs are made of a molar ratio of 56.8 % EggPC, 15 % cholesterol, 10 % DOPE, 10 % DOPS, 8 % brain PI(4,5)P2 , and 0.2 % Bodipy-TR-Ceramide. The observation buffer includes 75mM NaCl and 10 mM Tris (pH=7.8).

Figure 4 .

 4 Figure 4.

Figure 5 .

 5 Figure 5. A.Schematic representation of the steps required to obtain the "wavy" samples meant to analyze septin curvature sensitivity imaged by SEM. B. Low magnification SEM image of a wavy pattern with septin filaments bound but not resolved at this resolution. Two consecutive waves are highlighted in orange. Orthogonal defects are pointed by white arrows. Scale bar 10 μm. C. SEM Image at 10 k magnification where septin filaments are visible. Scale bar 1 μm. D. SEM Image at 20k magnification. Some of the filaments are highlighted in blue for visibility. Scale bar 200 nm. The spherical objects present in the images are small vesicles interacting with the septins and substrate. The septin concentration was set at 200 nM for B, C and D.

  .3.2.3. Incubate samples in the OsO4 second fixative solution for 10 min with maximum protection from light. This second step allows for fixation of membranous structures and increases the electrical conductivity of the samples. Remove the second fixative solution and wash samples thrice in distilled water, 5 min per washing with gentle shaking. Incubate washed samples in a freshly filtered UA fixative solution for 10 min with maximum protection from light. Remove UA solution and wash samples thrice in distilled water, 5 min per washing with gentle shaking.3.3.3. Sample dehydration and critical point dryingAir-drying is not permitted, to avoid damaging samples through unwanted surface tensions when changing from the liquid to gaseous state. Two methods have been established in EM: the physical method of reaching the supercritical state of CO2 and its critical point (31°C,

	3.3.2.4. Incubate washed samples in filtered TA fixative solution for 10 min maximum.
	Remove TA solution and wash samples thrice in distilled water, 5 min per washing with
	gentle shaking.
	3.3.2.5.
	3.3.2. Sample fixation
	3.3.2.1. Wash samples (NOA slides with fused supported lipid bilayers and incubated
	protein) with PBS then replace it with the fixative solution prewarmed at 37°C. Let the
	reaction proceed for 15 min. Samples can then be stored at 4°C.

3.3.2.2. Remove the fixative solution and wash fixed samples thrice with 0.1 sodium cacodylate, 5 min per washing with gentle shaking.

374 bar), or the chemical method of evaporating hexamethyldisilazane (HMDS), a drying agent with a reduced surface tension. CO2 and HMDS are poorly miscible with water. Consequently, all traces of water must be substituted to a transitional solvent (Ethanol) to avoid any damage during later drying processes.

3.3.3.1. Incubate samples 2-3 min in each ethanol solution starting at 50% to reach the anhydrous ethanol bath.

ACKNOWLEDGMENTS: (Instructions)

We thank Patricia Bassereau and Daniel Lévy for useful advice and discussions. This work benefited from the support of the ANR (Agence Nationale de la Recherche) for funding the project "SEPTIME", ANR-13-JSV8-0002-01 and the project "SEPTSCORT", ANR-20-CE11-0014-01. B. Chauvin is funded by the Ecole Doctorale "ED564: Physique en Ile de France" and Fondation pour lea Recherche Médicale. K. Nakazawa was supported by Sorbonne Université (AAP Emergence). We thank the Labex Cell(n)Scale (ANR-11-LABX0038) and to Paris Sciences et Lettres (ANR-10-IDEX-0001-02). We thank the Cell and Tissue Imaging (PICT-IBiSA), Institut Curie, member of the French National Research Infrastructure France-BioImaging (ANR10-INBS-04).

DISCLOSURES: (Instructions)

The authors declare no conflicts of interest to be disclosed.

REFERENCES: (Instructions)


H Dhaouadi
email: hassen.dhaouadi@ipeit.rnu.tn

N Bitri

S Essid

T Soltani

A Gharbi

J P Marcerou
email: marcerou@crpp-bordeaux.cnrs.fr

Extension of the Hamaneh -Taylor model using the macroscopic polarization for the description of chiral smectic liquid crystals

Keywords: numbers: 61, 30, Cz

Chiral smectic liquid crystals exhibit a series of phases, including ferroelectric, antiferroelectric and ferrielectric commensurate structures as well as an incommensurate Sm -C * α phase. We carried out an extension of the phenomenological model, recently presented by M. B. Hamaneh and P. L. Taylor, based upon the distorted clock model. The salient feature of this model is that it links the appearance of new phases to a spontaneous microscopic twist : i.e. an increment α of the azimuthal angle from layer to layer. The balance between this twist and an orientational order parameter J gives the effective phase. We introduce a second orientational order parameter I which physical meaning comes from the macroscopic polarization, the effect of an applied electric is also studied. We derive new phase diagrams and correlate them to our experimental results under field showing the sequence of phases versus temperature and electric field in some compounds.

I. INTRODUCTION

Chiral smectics are allowed to become ferroelectric and present a helical precession of the optical axes around the layer normal when a tilt ofthe molecules appears in the layers [1]. In the order of decreasing temperature and increasing tilt angle θ, one can observe a subset of the following full sequence [2,3,4,5] : the smectic A (Sm-A) without tilt angle (θ = 0) ; the smectic-C * α (Sm -C * α ) with a tilt angle θ and an azimuthal angle Φ precessing with a short incommensurate period along the layer normal ; the smectic-C * (Sm -C * ) with Φ precessing with a long period and an helicity sign depending on chirality, it is locally ferroelectric (P S = 0) ; the smectic-C * Ferri2 (Sm -C * F i2 ) where Φ is periodic over four layers and has a non regular increment (∆Φ = 2π/4) within the unit cell, the whole structure shows a long pitch helix with the same sign as the Sm -C * , it has no macroscopic polarization (P S = 0) ; the smectic-C * Ferri1 (Sm -C * F i1 ) where Φ has a non regular increment (∆Φ = 2π/3) periodic over three layers, a long pitch helix with the opposite sign as in SmC * , it is truly ferrielectric (P S = 0) ; the smectic-C * A (Sm -C * A ) with Φ periodic over two layers, a regular increment (∆Φ = π), a long pitch helix with the opposite sign to Sm -C * , it referred to as antiferroelectric (P S = 0) or anticlinic. Some of these phases may be missing when varying the chemical formula (tail length) [6] but the order of appearance is con-served. Two of them present a macroscopic polarization, four of them a long pitch helical precession with a sign change in the middle of the sequence [7,8,9]. Although most of these phases present a biaxiality of the unit cell, the global structure is uniaxial because of the helical precession around the layer normal and an optical activity that can be huge results from the rotation of the biaxial structure [10]. Other nomenclatures are also adopted : [2,11]. To characterize the different phases several experimental methods can be used : optical observations, calorimetric measurements and resonant X-ray scattering [3,4]. Other subphases have been proposed [12,13,14] but are linked to assumptions which are not accepted unanimously.

Sm -C * → Sm -C * β , Sm -C * F i1 → Sm -C * γ and Sm -C * F i2 → AF
Let us briefly mention some theoretical models which have been proposed to describe the structures and behavior of chiral smectic liquid crystals. The devil's staircase model, also called Ising model because one direction only is allowed for the azimuth, has been proposed soon after the discovery of the tilted smectic subphases by Chandani et al. [2,12,15]. It is based on the assumption that the competition between the synclinic sequence (where one layer and the following one have the same azimuth) and the anticlinic one (opposite azimuths) is at the origin of subphases which present a periodic succession of such sequences. An infinite number of phases with various ratios of synclinic versus anticlinic sequences are predicted making the so-called devil's staircase [12]. One can define the q T index as the fraction S/(S+A) of synclinic ordering versus the total number. Unfortunately the same index applies to phases with different symmetries so it is simply irrelevant. This model is still supported [13] although it has been ruled out by the results of X-ray resonant scattering experiments [16].

The nearest-neighbors models are based on the defini-tion of a quantity often called ξ αz which measures the tilt inside a layer by the coordinates of the c-director [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF]:

ξ xz ξ yz = sin θ cos Φ sin θ sin Φ (1)
When ξ αz is considered as the macroscopic order parameter, it can be chosen to describe the Sm-A to Sm-C phase transition [START_REF] Carlsson | [END_REF]19] as its modulus is nearly proportional to the tilt angle θ. If one considers it at the layer level, defining a different ξ j αz for each layer j, it can be used to build up a local free energy taking into account the interactions between layers. Theories have been proposed which deal with these interactions by means of nearest-neighbors couplings ξ j ξ j+1 or nextnearest-neighbors ξ j ξ j+2 . The form of the free energy is discrete as one has to sum up the interaction terms over all layers in the integration domain. One can find theories by Sun et al. [20], Roy et al. [21,22], Vaupotic [23]. In all cases a suitable combination of coupling terms could lead to phase diagrams compatible with existing phases. Lorman has introduced linear combinations of this parameter ξ αz over up to four layers [24] and his treatment leads to the prediction of the well known tilted subphases except for the Sm-C * α phase. ¿From an experimentalist's point of view these models are of little usefulness and it is better to stay in the frame of the distorted-clock model as it gives the best description of the currently encountered phases.

A. The distorted-clock model

This is a purely experimental model describing the observations without ab initio theory, also called XY model because all the azimuthal directions in the layer plane are allowed. It has been derived by modeling the orientation of the molecules in the layers and then fitting the resonant scattering experiments with success. With consecutive iterations of the initial regular model [4,16], the authors have introduced some asymmetries in the azimuthal angle distribution as it is reported in figure 1. The model is still in evolution concerning the Sm -C * α phase [5,25] but it is coherent and is at the base of the calculations reported here.

B. The Hamaneh-Taylor model

It is only recently that a new phenomenological way to describe the chiral smectic phases was proposed [26,27] that we call afterwards the H&T model. It is based on the balance between a short range twisting term trying to impose an increment α of the azimuthal angle between adjacent layers and a long range term linked to the anisotropy of curvature energy in the layer planes. They derived an order parameter J = cos2Φ l where the average is taken on the azimuthal angles inside the unit cell. It is non null in the phases enumerated above and associated to an energy η J 2 where η is a coefficient that describes the strength of the long-range interactions. The short order term reads cos(∆Φ -α) with ∆Φ = Φ l -Φ l-1 , the elastic term is η J 2 and the free energy of the sample is :

F F 0 = cos (∆Φ -α) + ηJ 2 (2)
The order of magnitude of F 0 is the electrostatic energy (-P S • E c ) necessary to drive at the field E c the phase transition to a ferroelectric phase with a polarization P S [26,27] while η is of order unity.

This leads to a phase diagram in the (η, α) plane showing the sequences of subphases which can be observed in a given liquid crystal. This model presents some limitations. First, it introduces a phase with six layers which was never observed experimentally. Second, the extent of the three layers phase is very small.

After this review we introduce in the next sections a new orientational order parameter I that will describe the contribution of the macroscopic polarization P S to the ordering. We present then the phases diagrams obtained from a numerical calculation. Eventually we compare the theoretical results with our experimental data obtained on several compounds

II. THE ORIENTATIONAL ORDER PARAMETER

In H&T model it is shown clearly that the average J = cos2Φ l is non zero in all the phases described by the distorted clock model, by analogy we state that another average I = cosΦ l is also non null in the phases like Sm -C * and Sm -C * F i1 which possess a macroscopic polarization. The origin Φ 0 is such that the averages over sine functions are zero [26,27], it will be the azimuth Φ 1 of the first layer except in the Sm -C * F i2 phase where it is equal to Φ 1 + υ/2. We get the following table where µ and υ stand for the characteristic angles of asymmetry in the Ferri phases : The symmetry argument of R. Meyer [1] stating that there exists a polarization P as soon as the the layer normal -→ N and the director -→ n make an angle θ can be translated by introducing in the free energy the mixed product of -→ P , -→ N and -→ n . Taking into account the table I, the angle between -→ N and -→ n reads Iθ, so with the addition of the self energy of the polarization, one gets :

I = cosΦ l J = cos2Φ l origin Φ0 Sm -C * α 0 0 X Sm -C * 1 1 Φ1 Sm -C * A 0 1 Φ1 Sm -C * F i1 [1 + 2cosµ]/3 [1 + 2cos2µ]/3 Φ1 Sm -C * F i2 0 -cosυ Φ1 + υ/2
∆F P = P 2 2ε 0 χ -CP Iθ (3)
by minimizing over P one finds

P S = ε 0 χCIθ (4) ∆ FP = - P 2 S 2ε 0 χ = - ε 0 χC 2 θ 2 2 I 2
We have thus demonstrated that the term due to the macroscopic polarization P S , present only when I is non zero, can be written γθ 2 I 2 and we can add it to H&T free energy after a little bit of algebra on the orientational order parameter.

B. relationship between I 2 and J 2 terms.

Let us start from the de Gennes orientational order parameter in the Sm-A phase, in what follows one considers the z axis to be perpendicular to the smectic layers.

Q 0 ij = n i n j - 1 3 δ ij =   -1/3 0 0 0 -1/3 0 0 0 +2/3   (5)
One can build up an orientational order parameter Q ij for all the tilted phases of the distorted clock model by first computing for each layer, in an axis frame where z is the layer normal, the expression of Q 0 ij after a tilt of the director n by an angle θ. This rotation is made around an axis inside the layer such that Φ is the angle between the c-director (projection of the director n in the layer plane) and the x axis.

Q θ Φ ij = 1 - 3 2 sin 2 θ   -1/3 0 0 0 -1/3 0 0 0 +2/3   + 1 2 sin 2 θ   cos 2Φ sin 2Φ 0 sin 2Φ -cos 2Φ 0 0 0 0   (6)
sin θ cos θ

  0 0 cos Φ 0 0 sin Φ cos Φ sin Φ 0  
Finally Q ij is computed by averaging over the unit cell of each phase. For the Sm-C * α the average over Φ is null for the second and the third matrix and there remains :

Q α ij = 1 - 3 2 sin 2 θ   -1/3 0 0 0 -1/3 0 0 0 +2/3   (7)
For all other phases, one can write a general formula for Q ij which is a function of Φ 0 defined in table I as the angle between the origin of azimuthal angles in the unit cell and the x axis. The resulting order parameter Q ij is unique ; it is only its expression in a given frame which depends on Φ 0 . One readily finds that the order parameters J = cos2Φ l and I = cosΦ l can be factorized in the last two matrices. For example in the Sm -C * F i1 phase where Φ 1 = Φ 0 , Φ 2 = Φ 0 + µ, Φ 3 = Φ 0 -µ, the averages in the unit cell give (cos Φ 1 + cos Φ 2 + cos Φ 3 ) = 3I cos Φ 0 , (cos 2Φ 1 + cos 2Φ 2 + cos 2Φ 3 ) = 3J cos 2Φ 0 , (sin Φ 1 + sin Φ 2 + sin Φ 3 ) = 3I sin Φ 0 , (sin 2Φ 1 + sin 2Φ 2 + sin 2Φ 3 ) = 3J sin 2Φ 0 and so on for all the phases :

Q ij = 1 - 3 2 sin 2 θ   -1/3 0 0 0 -1/3 0 0 0 +2/3   + J 2 sin 2 θ   cos 2Φ 0 sin 2Φ 0 0 sin 2Φ 0 -cos 2Φ 0 0 0 0 0   (8)
-

I sin θ cos θ   0 0 cos Φ 0 0 0 sin Φ 0 cos Φ 0 sin Φ 0 0  
It is straightforward to remark that the only parameters that should be retained for building the free energy are the factorized matrix coefficients J sin 2 θ ∼ θ 2 J and I sin θ cos θ ∼ θI. So the H&T term should read ηθ 4 J 2 and the polarization term as demonstrated above γθ 2 I 2 .

We took advantage of this by considering that we could write the modified H&T free energy under the form :

F F 0 = cos (∆Φ -α) + ηθ 4 J 2 + γθ 2 I 2 (9)
= cos (∆Φ -α)

+ ηJ 2 + γ √ ηI 2
We then assume that the temperature dependence of the η coefficient is due to θ 4 ∼ (T c -T ) 2 , where T c is the temperature of appearance of the tilt angle and that the coefficient γ depends only on the compound and not on the temperature. We eventually build phase diagrams in the (η, α) plane, each one for a different value of γ taking it to be of order unity, from 0 to 1 (see e. g. figure 2). These results show that the Sm -C * and Sm -C * F i1 domains grow with γ i.e. with the permanent polarization, let us consider now what happens in the presence of an external electric which is known to induce phase transitions to polar phases [28].

C. effect of electric field in the layer plane

An electric field applied to the sample always creates a small dielectric polarization proportional to it which is the same in all studied phases to first approximation. But when there is already a spontaneous polarization P S it will displace the energy by a term which reads roughly -P S • E. The free energy can be written by slightly modifying equation (3):

∆F E = P 2 2ε 0 χ -CP Iθ -P • E - ε ⊥ E 2 2 (10)
by minimizing over P, one finds:

P = ε 0 χ (CθI + E) = P S + ε 0 χE ∆ FE = - P 2 S 2ε 0 χ -P S • E - ε ⊥ E 2 2 ( 11
)
For a given value of the electric field, the third term -ε ⊥ E 2 /2 is the same for all the phases and does not depend on the orientational order parameters I and J so we simply forget about it. The first two terms have the same order of magnitude F 0 ∼ -P S • E c and are respectively quadratic and linear versus θ, so by taking η as the main parameter we can write down :

F F 0 = cos (∆Φ -α) + ηJ 2 + γ √ ηI 2 + δ 4 √ ηI (12)
We use this expression to compute the phase diagrams in the (η, α) plane, each one for a different value of γ and δ, see e.g. figure 3.

III. NUMERICAL STUDY AND PHASE DIAGRAMS

In the different phases of the distorted clock model represented on the figure 1, we establish at first the ex- pression of the free energy for a given (η,α) by computing the order parameters I and J as well as the quantity cos(∆Φ -α) for each structure. The H&T model predicts the existence of a phase with six layers, we are going to disregard this phase on the one hand by the fact that it was never observed on the other hand because it disappears of the diagram once the term due to the polarization P S is added. Other structures not observed experimentally have been briefly tested like a four layers asymmetric phase which has a less favorable energy than the Ferri2 phase. Let us point out that as F 0 is negative we look for an absolute maximum of F/F 0 to get the best phase.

• In the Sm -C * α phase the short order term reduces to 1 while the additional terms vanish. J = 0 and I = 0 thus the free energy is F = F 0 • In the Sm -C * phase J = 1 and I = 1 thus

F F 0 = η + cos α + γ √ η + δ 4 √ η (13)
• In the Sm -C * A phase J = 1 and I = 0 thus

F F 0 = η -cos α (14)
• In the Sm -C * F i2 phase J =cos υ, I = 0 and by minimizing F over υ we find for η > 0.5 that the preferred angle is such that sin υ = sin α/2η and the free energy reads :

F F 0 = η + sin 2 α 4η (15)
• In the Sm -C * F i1 phase J = (1 + 2 cos 2µ) /3, I = (1 + 2 cos µ) /3 so all the terms of equation( 12) must be explicited :

cos(∆Φ -α) = (2 cos(µ -α) + cos(2µ + α)) /3 ηJ 2 = η 4 cos 2 µ -1 2 /9 γ √ ηI 2 = γ √ η (2 cos µ + 1) 2 /9 δ 4 √ ηI = δ 4 √ η |2 cos µ + 1| /3 ( 16
)
The maximization of the energy has to be made numerically giving the preferred value of µ and F/F 0 ;

A. Computation of phase diagrams

Let us first discuss the physical meaning of these diagrams. On the x-axis one reports the values of the parameter η that we take in mean field approximation as being the fourth power of the tilt angle θ, thus the second power of the distance in temperature from the Sm-A phase -i.e. the tilt angle appears at the Sm-A to Sm-C phase transition and follows a √ T c -T law. So the xaxis represents decreasing temperatures from the Sm-A at left to the right. The y-axis shows the scale of variation of the ad-hoc angle α from 0 to π. This parameter makes the richness of the H&T model, it means physically that due to the chirality the director wants to be twisted from layer to layer by the angle α and it is the balance between this tendency and a more uniform azimuthal angle distribution measured by the I and J order parameters that gives rise to the effective structures. It is remarkable that with so few parameters one can get all the structures determined experimentally before the theory was developed by H&T. One is not as usual trying to fit the experiment to a theory imposed ab initio. What we introduce in this paper are new terms linked to the polar nature of two phases in the tilted chiral smec- tics nomenclature, the Sm -C * and the Sm -C * F i1 . We express these terms as functions of the x-axis parame- ter, one measuring the spontaneous polarization and the other its coupling with external field. Our aim is to show that the domain filled in by these two phases will be extended in the diagram. We first present in figure 2 some diagrams obtained for different values of γ that correspond to the ground states without applied field. As γ increases one notices an expansion of the domains corresponding to the Sm -C * and Sm -C * F i1 phases which are the only ones with a spontaneous polarization. It is to be noticed that for γ = 1 the domain corresponding to Sm-C * F i2 has completely disappeared.

The diagram obtained with γ = 0.2 is chosen to illustrate qualitatively (figure 3) the sequences of phases which appear for the two compounds we have studied experimentally in our group : the C10F3 and the C7F2 [6,28,29,30].

With the applied field i.e. the parameter δ one notices also an expansion of the Sm-C * and Sm-C * F i1 domains. Furthermore it appears a band corresponding to Sm -C * F i1 in full centre of the alpha domain, this band widens gradually as δ increases.

B. Comparison with experiment

In the figure 3 4). For a large enough field the SmC * domain will cover almost all the length of the path (arrow d).

The C10F3 compound presents at zero field the following phase sequence:

Sm -A → Sm -C * α → Sm -C * F i2
. For weak electric field (arrow a' in figure 5) the sequence remains almost the same. For a higher electric field (arrow b') the curve corresponding to the C10F3 begins in the Sm -C * domain then passes in the Sm -C * α before the Sm -C * F i1 grows bigger at the expense of the Sm-C * F i2 domain. One then encounters successively two triple points : the (Sm-C

* α )-(Sm-C * F i1 )-(Sm-C * F i2 ) and the (Sm -C * ) -(Sm -C * α ) -(Sm -C * F i1
). For a strong electric field the Sm -C * phase is dominant (arrow c'). structures in mixtures at two temperatures, they found a value of υ of about 166 • and that of µ of about 152 to 160 • with no discernable dependence on temperature.

Starting from our equations we made the calculation of υ and µ for different points from the two curves of figure 3. We found values of µ varying from 156 to 161 • for both compounds (under field) which is comparable to that reported by Cady, Roberts et al. [3,31]. On the other hand the values we computed for the υ angle, about 142 • , are slightly lower than the measured values [31,32] but still far from the regular clock model (υ = 90 • ).

We have shown that taking into account the macroscopic polarization in H&T theory one is led to an expansion of the Sm -C * and Sm -C * F i1 existence domains which is correlated with the (E-T) phase diagrams we have obtained experimentally.

As noticed by H&T the translation of the theory in a quantitative way requires the knowledge of the physical path α(η) or separately α(T ) and η(T ). What we add is the requirement for the new γ and δ parameters which can be derived from the measurement of P S and E (see e.g. equation 11). The parameter η should be considered to be zero at the temperature T c where the tilt angle appears and to follow a (T c -T ) 2 law below. In the H&T frame it gets remarkable values at the phase boundaries like η = 1 + cos α at the Sm -C * α to Sm -C * A phase transition. So the measurement of α in the Sm -C * α phase is required.

Several experimental methods have been reported to measure the parameter α, let us quote the resonant Xrays diffraction which allows to measure a periodicity ranging usually from eight to three layers in Sm -C * α with a decrease (i.e. an increase of α) when cooling down the sample [5,16,25,33]. However there are a few exceptions where the periodicity varies from about ten to fifty layers (α is decreasing) when lowering the temperature in the assumed Sm-C * α range [4,33]. This is not obviously an experiment accessible to everyone. Differential optical reflectivity has been also used to acquire the temperature variation of the helical pitch in the Sm -C * α phase and has also shown an increase of α when cooling down [34]. Another method used by Isaert [35] consists in measuring the spacing of the Friedel fringes which appear in the reflected light texture on the free surface of drops, this simple method could allow a fast measurement of α. Finally another candidate is the gyrotropic method given by Ortega and al [36] who by a measurement of the ellipticity of eigenmodes claim to get the pitch and thus the angle α.

IV. CONCLUSION

In this paper we report a new successful method for the description of chiral smectic liquid crystals based on the Hamaneh -Taylor model. The introduction of the polarization and the electric field contributions give results that sound in good agreement with experiment and allow explaining the appearance under field of an intermediate ferrielectric phase. The measurement of the parameter α and of the polarization P S should allow to trace the paths followed by a given compound in the (η, α) plane.

In the future we plan to introduce the helicity and its sign evolution in the phase sequence.

FIG. 1 :

 1 FIG. 1: Schematic description of the different phases in the distorted clock model. (a) side view (same tilt angle θ) period p of 1 to 4 layers, (b) top view (periodic azimuth such that Φ l+p = Φ l +2π). ∆Φ = ν or ∆Φ = π-ν in the Sm-C * F i2 and ∆Φ = µ or ∆Φ = 2(πµ) in the Sm -C * F i1 . Arrows indicate the direction Φ0 taken as the origin of azimuthal angles. Any gap or overlap between molecules is due to the hand made drawing and has no physical meaning.

1 FIG. 2 :

 12 FIG. 2: (Color online) Ground states diagram for γ = 0, 0.5, 0.75 and 1. The symbols 1 and 2 stand for ferri1 and ferri2.

1 FIG. 3 :

 13 FIG.3:(Color online) Diagrams obtained with applied field for γ = 0.2 and δ = 0, 0.2, 0.4 and 0.6. The black plain curve corresponds to an estimate of the path η(α) followed by C10F3 and the dotted line to η(α) for C7F2.

FIG. 4 :

 4 FIG.4: (E,T) phase diagram of the compound C7F2. The constant field paths a to d correspond to increasing values of the parameter δ.

FIG. 5 :

 5 FIG.5: (E,T) phase diagram of the compound C10F3. The constant field paths a' to c' correspond to increasing values of the parameter δ.

  are represented the estimated paths followed on decreasing the temperature by two compounds studied experimentally in our group : the C7F2 (dotted curve) and C10F3 (plain). The paths are unchanged in the different panels of the figure as they depend only on the temperature, the nature of the phase encountered at a given point changes as the Sm -C * F i1 and Sm -C * domains grow. The C7F2 compound shows the following ground phase sequence: SmA → Sm -C * α → Sm -C * A . The corre-sponding path (dotted line in fig. 3) has to include the (Sm -C * α ) -(Sm -C * F i1 ) -(Sm -C * A ) triple point (below arrow a in fig. 4) as a very weak field reveals the Sm -C * F i1 phase. The increase in the electric field enlarges the Sm -C * and Sm -C * F i1 domains. At a certain point the path crosses the Sm -C * domain then the Sm -C * α and the Sm -C * F i1 finishing in the Sm -C * A (b and c in fig.

  C. DiscussionAnother comparison to experiment can be made with the published values of the distortion angles µ and υ. A. Cady et al.[3] find for the Sm -C * F i2 structure a value of υ of about 164 • , Roberts et al.[31] measured the angular distortion of the Sm -C * F i2 and Sm -C * F i1

TABLE I

 I

: order parameters I and J and origin of angles in the different phases A. introduction of an I 2 term in the free energy

Acknowledgments

We wish to acknowledge the support of CMCU grant # 06/G 1311 and CNRS/DGRSRT grant # 18476.