Yong Jian Wang

Feng Li

Nicolas Rodriguez

Xavier Lafosse

Christine Gourier

Eric Perez

Frederic Pincet
email: pincet@lps.ens.fr

Snapshot of sequential SNARE assembling states between membranes shows that N-terminal transient assembly initializes fusion

Keywords: SNAREpin, Transient assembly intermediate, Surface Forces Apparatus, Förster Resonance Energy Transfer, Energy

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

Introduction

In biology, many critical protein-protein interactions occur between membrane surfaces. These include cell-to-cell adhesion [START_REF] Berrier | Cell-matrix adhesion[END_REF] to form tissues (e.g. cadherins [START_REF] Cavallaro | Cell adhesion and signalling by cadherins and Ig-CAMs in cancer[END_REF][START_REF] Thiery | Cell adhesion in development: a complex signaling network[END_REF]), infection of cells by enveloped viruses (4, 5) (e.g. viral envelope fusion proteins), and secretion when a storage vesicle containing hormones or transmitters [START_REF] Martens | Mechanisms of membrane fusion: disparate players and common principles[END_REF][START_REF] Rizo | Synaptic vesicle fusion[END_REF] fuses with the plasma membrane (this fusion is achieved by soluble N-ethylmaleimide-sensitive factor attachment protein receptors (SNAREs) [START_REF] Weber | SNAREpins: minimal machinery for membrane fusion[END_REF]). In each of these examples, the biological process is thermodynamically-coupled and energetically-driven by protein folding/assembly. This means that understanding the molecular mechanism requires a complete characterization of the various intermediate transmembrane protein complexes that appear at different intermembrane distances (Fig. 1A). Two challenges render this characterization difficult. First, because of the energy landscape, the intermediates are transient. Second, observing in the nanometer-scale gap between two membranes remains a challenge. Several existing experimental approaches provide partial information. X-ray crystallography gives atomic level images of the proteins after folding [START_REF] Poirier | The synaptic SNARE complex is a parallel four-stranded helical bundle[END_REF][START_REF] Sutton | Crystal structure of a SNARE complex involved in synaptic exocytosis at 2.4 A resolution[END_REF], providing a framework but not the pathway. With optical tweezers, continuous unfolding of single molecular complexes can be monitored [START_REF] Gao | Single reconstituted neuronal SNARE complexes zipper in three distinct stages[END_REF][START_REF] Zorman | Common intermediates and kinetics, but different energetics, in the assembly of SNARE proteins[END_REF] by applying separation forces. The Surface Forces Apparatus [START_REF] Israelachvili | Measurement of forces between two mica surfaces in aqueous electrolyte solutions in the range 0-100 nm[END_REF][START_REF] Israelachvili | Direct methods for measuring conformational water forces (hydration forces) between membrane and other surfaces[END_REF] (SFA) provides the only possible way to rigidly control intermembrane separation, the main reaction co-ordinate of this class of folding processes, by fixing the separation between two apposed membranes with subnanometer-level precision, and allows direct measurements of forces (required to unfold the protein) [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF][START_REF] Prakasam | Similarities between heterophilic and homophilic cadherin adhesion[END_REF][START_REF] Li | Complexin activates and clamps SNAREpins by a common mechanism involving an intermediate energetic state[END_REF] (Figs. 1 and2). With the SFA, a continuous series of folding intermediates can be frozen and their energetics (the energy landscape) determined [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF] (Fig. 1A). However, none of these techniques provide the assembling landscape, i.e. a complete view of the various intermediates that naturally occur during protein folding/assembly between membranes. SNARE protein assembly, that provides the energy for vesicle fusion in intracellular trafficking, is an archetype transmembrane assembly process in which the molecular landscape is strongly correlated with the membrane separation. In the classical case of synaptic fusion, there are two types of SNARE proteins which include the synaptic vesicle SNARE, Synaptobrevin VAMP2 (v-SNARE), and a heterodimer (t-SNARE) made of the presynaptic plasma membrane SNARE proteins, Syntaxin-1 and SNAP-25. v-SNARE and t-SNARE zipper to form a SNAREpin between the membranes. The efficiency of the fusion process is controlled by subtle molecular details that ensure the right timing and appropriate release of energy at each binding stage. The crystal structure of the SNARE complex [START_REF] Poirier | The synaptic SNARE complex is a parallel four-stranded helical bundle[END_REF][START_REF] Sutton | Crystal structure of a SNARE complex involved in synaptic exocytosis at 2.4 A resolution[END_REF] shows that SNAREs assemble to form a highly stable coiled-coil made of four parallel helix bundles. Zippering of the coiled-coil proceeds in a stepwise manner with a half-zippered SNARE complex being one of the intermediate state [START_REF] Gao | Single reconstituted neuronal SNARE complexes zipper in three distinct stages[END_REF][START_REF] Li | A half-zippered SNARE complex represents a functional intermediate in membrane fusion[END_REF], but the sub-molecular details of the zippering process between membranes remain unknown. Notably, it is unclear whether the initial intermediate of SNARE assembly requires the binding of their membrane distal N-terminal regions.

In this article, we demonstrate that, using an SFA equipped with a visualizing setup to observe Förster Resonance Energy Transfer (FRET), we are able to access the intermediates of protein assembly between membranes. We exemplify this by the study of the assembly of SNAREpin and show that the first step towards complete SNARE zippering is the N-terminal assembling, and we determine the exact threshold distance, 8 nm, at which this occurs.

Results

FRET/SFA

The SFA is a technique that is widely used to measure the force-distance profile between two surfaces that are often functionalized in order to mimic the surface of a colloidal particle or a biological membrane. The distance between the surfaces is determined with sub-nanometer accuracy in a spectrometer through the position of Fringes of Equal Chromatic Order (FECOs) produced from multiple-beam interferometry with back-silvered mica sheets. The force is measured via the deflection of a spring with calibrated spring constant. This powerful technique has so far been used to address a large number of unresolved scientific questions in a wide range of fields, e.g. fundamental physics [START_REF] Tabor | Surface forces: direct measurement of normal and retarded van der Waals forces[END_REF], soft-matter physics [START_REF] Luckham | Interactions between Smooth Solid-Surfaces in Solutions of Adsorbing and Nonadsorbing Polymers in Good Solvent Conditions[END_REF] or biology [START_REF] Helm | MEASUREMENT OF LIGAND RECEPTOR INTERACTIONS[END_REF]. It has also been coupled to numerous other techniques such as friction force measurements [START_REF] Israelachvili | Relationship between Adhesion and Friction Forces[END_REF], electrochemical measurements [START_REF] Valtiner | The Electrochemical Surface Forces Apparatus: The Effect of Surface Roughness, Electrostatic Surface Potentials, and Anodic Oxide Growth on Interaction Forces, and Friction between Dissimilar Surfaces in Aqueous Solutions[END_REF], fluorescence [START_REF] Wong | Fluorescence Recovery after Photobleaching Measurements of Polymers in a Surface Forces Apparatus[END_REF][START_REF] Alig | Properties of confined and sheared rhodamine B films studied by SFA-FECO spectroscopy[END_REF] or x-ray diffraction [START_REF] Golan | The x-ray surface forces apparatus for simultaneous x-ray diffraction and direct normal and lateral force measurements[END_REF], etc. When cognate proteins are anchored on the mica surfaces in order to study their assembling, the subnanometer-level resolution distance of the SFA gives access to geometrical parameters and to the energy landscape of the assembling [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF][START_REF] Perez | The Surface Force Apparatus to Reveal the Energetics of Biomolecules Assembly. Application to DNA Bases Pairing and SNARE Fusion Proteins Folding[END_REF]. However, because of the flexibility of these molecules, the force-distance profile does not allow to follow the submolecular details that could provide comprehensive information on the intermediate structures: the assembling landscape. To track the formation of molecular complexes, a distance sensitive phenomenon is required. FRET [START_REF] Stryer | Energy transfer: a spectroscopic ruler[END_REF] with its nanometric resolution on the distance between two fluorophores, is an appropriate technique to provide such information. FRET occurs between two fluorophores [START_REF] Helms | Principles of Computational Cell Biology[END_REF], one of which (the donor) emission spectrum overlaps the absorption spectrum of the other (the acceptor), and there is energy transfer between the two fluorophores when their separation distance is less than the FRET distance, R 0 , of the order of a few nanometers. FRET has so far not been incorporated in an SFA. Combining the two techniques into the FRET/SFA is uniquely suited to freeze the absolute distance between two flat membranes, measure the forces and observe the formation of molecular complexes between the surfaces at the nanometer level (almost the amino-acid level for a protein). So when two molecules (proteins) are labeled, one with a donor and the other one with an acceptor, the proximity of the labeled sites (amino-acids) on two molecules (proteins) can be estimated in one molecular (protein) complex and it becomes possible to establish whether molecules (proteins) are assembled at the level of these labeled sites (amino-acids).

The FRET/SFA we have designed involves the basic version of the SFA with some modifications (Fig. 2). To allow simultaneous measurement of forces and observation of the FRET signal, the detection of a FRET signal coming from the samples must be made compatible with the interferometric distance measurements. This requires that the mica of the upper lens has high transmittance at the wavelengths of the donor excitation and the acceptor emission and good reflectance ~90-95% to observe the FECOs in the other wavelengths. This was made possible by designing an adequate 15 layers multidielectric Ti 3 O 5 /SiO 2 coating. The preparation of this special surface is detailed in the Materials and Methods. The resulting reflectivity spectrum is displayed as the black curve in Fig. S1A; the transmission in the SFA setup configuration is displayed in Fig. S1B. Secondly, a band pass filter is placed at the output of the white light source to transmit only the wavelengths used for the FECOs without interfering with the observation of the FRET signal.

Here, we used the Alexa 488 (donor) / Alexa 647 (acceptor) FRET pair . To excite the donor, a 488nm green laser (Coherent, Sapphire 488-200 mW) is sent to the mica surfaces by means of a long pass dichroic mirror that reflects the 488 nm light to the sample and that also transmits the donor and acceptor emission wavelengths. If FRET occurs, the fluorescence of the acceptor is emitted from the surfaces while the fluorescence of the donor is decreased. At the optical exit of the SFA, a dichroic band-pass beam-splitter reflects the fluorescence emission towards the CCD camera and transmits the FECOs towards the spectrometer. A band pass filter is placed at the entrance of the CCD camera (Princeton Instruments, RTE/CCD-782-V/HS) to transmit only the wavelengths related to the acceptor or the donor. The power of the laser beam is carefully tuned to prevent photobleaching while capturing the fluorescent image; in the experiments presented here, it was set at 20 mW. The CCD camera is used to capture the fluorescent image. A scheme of the optical setup is presented in Fig. 2A and the resulting image of the FECOs in the spectrometer is shown in Fig. 2B.

This sensitive system we have developed allows FRET measurements between probes placed at known locations in the proteins folding/assembling between two bilayers that are brought together (and can be withdrawn) with separation rigidly controlled (Figs. 1B), making it possible to read-out the intra-molecular separations within assembling complexes at each inter-membrane distance. The FRET/SFA can, in parallel, measure forces as in a standard SFA, so that when it is desirable the assembling and energy landscapes can be directly compared.

SNAREpin formation observed with the FRET/SFA

We have used this experimental setup with SNAREs that are labeled at their N-terminal region, t-SNARE with Alexa 488 and v-SNARE with Alexa 647 (Fig. 1B). Labeling is obtained by binding the dyes to a single cysteine at the N-terminal of each protein (see supporting information for labeling procedure). The necessary mutations of the SNAREs do not affect their ability to fuse membranes (Fig. S2). Our SNAREs also possess a 12xHis tag at their C terminal that enables their incorporation in the apposing bilayers containing 10% lipids with Nickel in their polar heads (see Material and Methods). The final SNARE density in the membrane is about 1 protein per 200 lipids. It has been previously shown that lipid anchoring the SNARE is sufficient to keep their fusion activity [START_REF] Mcnew | Close is not enough: SNARE-dependent membrane fusion requires an active mechanism that transduces force to membrane anchors[END_REF]. When the two SNARE-decorated membranes are brought together and subsequently separated, adhesion is observed (Fig. 3A). This adhesion vanishes in the control experiments in which t-SNARE is blocked to prevent SNAREpin formation. Simultaneous measurement of the FRET signal when the membranes are contacting shows that SNAREpin form in the contact area. The FRET signal disappears in the case of the control experiment. This indicates that the presence of SNARE complexes is responsible for the observed adhesion (Fig. 3B). We have previously shown that the long range repulsion in the energy profiles presented in Fig. 3A is directly correlated with the local SNARE density [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF]. Briefly, when the surfaces are between 10 and 20 nm during the approach, the SNARE are not bound yet and they behave like polymers. The repulsion is exponentially decaying with the distance and the prefactor (extrapolated repulsion when the surfaces are in contact) is proportional to the density. Dividing the adhesion energy ~0.2 k B T/nm² by the density resulting from this fit directly provides the energy per SNAREpin between the membranes which is 29 ± 4 k B T. This value is consistent with that previously measured [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF]. This confirms that SNAREpins form in a specific and normal manner between the membranes and shows that the FRET/SFA is suitable to study the arrangements and interactions of molecular complexes confined between two surfaces.

The FRET images can be further analyzed based on the fact that a continuous series of folding intermediates of SNAREpin are frozen in the SFA. The geometry of the membranes resembles that of a sphere of radius R facing a flat plane (Fig. 1A). Placing the membranes in close proximity (minimum distance h 0 ) is equivalent to observing simultaneously all separation distances, h, while moving at a distance r away from the contact location (Fig 1A

): h ≈ r 2 /(2R) + h 0
(1) Hence, a picture of the FRET signal over the whole area of the membranes directly provides a snapshot of the N-terminal assembly at all distances. The thickness of the cylindric lens (a few mm), imposes that the objective be far from the mica surface. Hence, the FRET signal remains weak. To increase the signal/noise ratio, we spin-averaged each image by successive rotations around the point of closest distance (see Materials and Methods). The main difficulty is to precisely localize this center of rotation. By testing several positions from the raw FRET signal (Fig. S3C), we estimate that our accuracy, , is better than 10 pixels, i.e. better than 3 µm. The resulting spin-averaged picture is presented in Fig. 4A. The FRET signal profile with r is displayed in Fig. 4B. It does not change when the membranes are kept in contact for 30 minutes (Fig. 4B and Fig. S4A) and it is reproducible in successive approach-separation cycles (Fig. S4B-E). As expected, the FRET profile is maximum in the center (r~0). In the case of the control, there is no FRET signal; the intensity is even minimum in the center. Given the high distance sensitivity of FRET, the intensity normalized by that at the plateau presented in Fig. 4B actually represents the ratio between the fractions of bound N-terminal regions at r and at the FRET plateau, F FRET . Using Eq. 1, it is then possible to determine the variation of F FRET with the membrane separation h from the single image obtained in Fig. 4A (Fig. 4C). Note that the error on h varies as r/R, i.e. close to 20% in the relevant range (h between 3 nm and 30 nm). F FRET (h) is maximum for h < 8 nm and is close to zero for h > 20 nm. Between 8 nm and 20 nm, the fraction of bound SNAREs decreases almost linearly. This variation is analyzed in the discussion below in view of what is already known on SNARE assembly.  Another direct conclusion that can be drawn from these observations is that SNAREpin formation is reversible. This can be probed by imaging the FRET intensity after the membranes have been separated. Indeed, a non-reversible assembling would keep the SNARE proteins bound upon membrane separation, with the FRET signal unchanged. Conversely, if the SNAREpins have been unraveled, the acceptor dye will be too far from the donor dye (typically 200 nm) to emit any FRET fluorescence. To determine whether the FRET signal disappeared, we computed the difference between the images after separation and the image at contact, just as what was done to determine the FRET signal during the approach phase. The FRET intensity profile from the spin-averaged picture of FRET signal during the separation phase is almost completely superimposable with the one obtained during the approach phase (Fig. S4F), which shows that SNAREpins can be disassembled upon separation.

Discussion

The F FRET (h) variation is intriguing because we previously reported in independent SFA measurements that there is no adhesion between two SNARE-decorated membranes when their separation is larger than 8 nm. This conclusion was based on the sharp adhesion drop in Fig. 4D and Ref. [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF]. These two results may seem contradictory: how can SNARE complex start assembling without producing any significant adhesion between 8 and 20 nm? A closer look at the data shows that there is indeed a small adhesion above 8 nm (1.3 k B T ± 2 k B T per SNAREpin). This value, which is much smaller than the 35 k B T observed below 8 nm, may be slightly underestimated (at most 2 k B T) because of the polymer repulsion of the unstructured SNAREs. Hence, it corresponds to a weak state with an energy lower than 5 k B T per SNAREpin. The weakness of the bond between the N-terminals of SNAREs is therefore compatible with the detection of FRET and with the relative lack of adhesion we previously reported. We also observed that it was necessary to wait for tens of minutes in close contact before maximum adhesion was achieved in the SFA [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF]. The adhesion increased with a characteristic time of ~20 min (see Fig. 2b in ref [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF]). The reason for this long wait was unknown. We suspected it may come from the confinement and density between the relatively flat surfaces. Here, we also waited for at least 30 minutes before separation and took FRET images at various times during this wait (Fig. 4B and Fig. S4A). We observed that the FRET signal is fully established as soon as the surfaces are in contact. This means that the N-terminal domains of the SNAREs assemble immediately but that the complex is not yet zippered with enough energy to generate adhesion. Hence, the fast setting-up of the FRET also proves the existence of low lowenergy state in which a small fraction of the N-terminal residues of the SNARE domains are bound. This weak state is the first one that occurs during SNAREpin formation. Previous optical tweezers measurements showed that, upon separation, SNAREpins oscillate between several intermediate states depending on the force applied to disassemble it (11) but would not have been able to observe this initial state. However, the final state in the SFA is relatively well characterized [START_REF] Gao | Single reconstituted neuronal SNARE complexes zipper in three distinct stages[END_REF][START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF] and corresponds to a half-zippered state (or possibly zippered beyond). This allows us to separate the 2 situations: upon contact and after 30 minutes in contact. First, upon contact, N-terminal regions of the SNARE immediately bind in the low energy state. Below 8 nm, virtually all SNAREpins are assembled in their N-terminal region. Between 8 and 20 nm, only a fraction of them are bound; this fraction decreases as h increases. Beyond 20 nm, no SNAREpins are assembled. Second, after 30 minutes, the SNAREpins have transited towards the half-zippered state for h < 8 nm. Above 8 nm, and because there is no easily detectable adhesion, the SNAREpins are only in the weak energy state. These results are summarized in Fig. 4E. The long delay for the transition towards the half zippered state indicates a high activation energy barrier. Assuming a standard density of the transition time, the activation energy can be written E a =k B Tln( 0 ) where  0 is the frequency of escape attempt,  the characteristic time and k B T the thermal energy. In water, for this type of molecules,  0 is between 10 10 s -1 and 10 8 s -1 and, here,  ~ 1000s. This leads to E a ~25 -30 k B T. These results imply that, in vivo, synaptic vesicles must be brought as close as 20 nm, and preferably 8 nm, form the presynaptic plasma membrane to start SNAREpin formation. The long transition time, , between the N-terminal weakly bound state and the high energy half-zippered state suggests that an active mechanism exists to accelerate the transition process, possibly through the use of regulatory factors. This observation of the first transient state during SNARE zippering between membranes shows the sensitivity and efficiency of the FRET/SFA to detect and characterize intermediate structures between closely apposed membranes. It opens up the way to monitor other transmembrane transient states that cannot be observed otherwise.

Materials and Methods

Chemicals

The lipids used in this study are, 1,2-Dimyristoyl-sn-Glycero-3-Phosphoethanolamine(DMPE) (850745X), 1,2-Dioleoyl-sn-Glycero-3-Phosphocholine (DOPC) (850375C), 1,2-Dioleoyl-sn-Glycero-3-[Phospho-L-Serine] (Sodium Salt) (DOPS) (840035C), and 1,2-dioleoyl-sn-glycero-3-[(N-(5-amino-1-carboxypentyl)iminodiacetic acid)succinyl] (nickel salt) (NTA-Ni) (79404) which are purchased from Avanti Polar Lipid. More details on buffer and other chemicals are provided in the supporting information.

SNARE Proteins (Fig. S5) Cytosolic v-SNARE with end-of-sequence 12x Histidine (v-SNARE). v-SNARE for the

FRET/SFA study is made of the cytoplasmic domain of mouse VAMP2 (residues 1-96 with a single Cysteine, S28C, and C-terminal 12x His).

Cytosolic t-SNARE with end-of-sequence 12x Histidine (t-SNARE). t-SNARE for the FRET/SFA

study is made of the cytoplasmic domain of rat Syntaxin1A (residues 1-265 with a single Cysteine, S193C, and C-terminal 12x His) and mouse His6-SNAP25 (residues 1-206, Cysteinefree). Details on protein constructs, expression and purification are given in the supporting information.

Reflective coating on Mica

The lower mica has the backside surface coated with a 67 nm silver layer prepared by a thermal evaporator in a clean room of Class1000. The backside of the upper mica has a custom coating made of 15 or 13 alternating Ti 3 O 5 and Si0 2 layers starting with Ti 3 O 5 , with a total thickness of 1.1µm. More precisely, the technique used is an IAD Ion assisted deposition i.e E-beam evaporation (Plassys MEB 800) with ion assistance provided by an ion gun (KRI EH 1000 source). The design, synthesis and refinement of the custom coating have been done theoretically with Essential MacLeod software. Targets of reflectivity, R, have to be set. In our case, we wanted low R (~5%) for both 488nm-520nm and above 660 nm spectral regions and high R (~95%) between 570 nm and 630 nm. The optimization is using Si0 2 and Ti 3 O 5 mean refractive indices for the entire considered spectral range. From given initial thicknesses to the theoretical stack at an arbitrary wavelength of this range, the algorithm finds solutions with different figures of merit. As an example, for 15 layers, we chose one of these solutions for which the thickness of each layer is (starting from the mica surface): 65.69 nm (Ti 3 O 5 ) / 129.59 nm (Si0 2 ) / 72.65 nm (Ti 3 O 5 ) / 35.48 nm (Si0 2 ) / 29.32 nm (Ti 3 O 5 ) / 182.25 nm (Si0 2 ) / 66.43 nm (Ti 3 O 5 ) / 89.79 nm (Si0 2 ) / 79.51 nm (Ti 3 O 5 ) / 111.51 nm (Si0 2 ) / 25.16 nm (Ti 3 O 5 ) / 24.02 nm (Si0 2 ) / 136.10 nm (Ti 3 O 5 ) / 67.05 nm (Si0 2 ) / 94.86 nm (Ti 3 O 5 ). The deposition is made with the following protocol. Both Ti 3 O 5 and Si0 2 are deposited with a rate of 0.25nm/s. Starting materials are Ti 3 O 5 and SiO 2 of 1-3 mm pieces put in Molybdenum liners. To avoid stress issues on the mica that would curve it, we set the ion gun discharge voltage at 100V and discharge Current at 1A. The background pressure before adding gases was 1.0×10 -7 mbars, the working pressure was ~3.0×10 -4 mbars with flows of 5sccm for Ar and 5 sccm for the ion source and 10 sccm of Ar for the Keeper (plasma bridge). The layer thicknesses were followed in real time using both a quartz microbalance and an ellipsometer (here, we used 60.5° angle of incidence and 580 nm wavelength). The reflectivity of the coating is provided in Fig. S1.

SNARE layer reconstitution

To prepare the sample for the deposition of the lipid bilayer, the backside-coated mica is glued onto the lens of SFA by thermal epoxy with the coated surface contacting the glue. The mutidielectric coated mica is glued onto the upper lens, and the silver coated mica on the lower lens. Then the lenses are transferred into water and held vertically in a home-built Langmuir trough. The DMPE chloroform solution is directly used as purchased for the first lipid leaflet on both mica surfaces. For the second leaflet, a mixture of the chloroform solutions of DOPC, DOPS and NTA-Ni with the lipid molar ratio of 80%, 10% and 10% was used. The mixture has undergone three freeze (by liquid nitrogen) and thaw cycles to homogenize the solution before use. The rest of the procedure is the same as the one we previously published [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF]. Details are provided in the supporting information In the control experiment that follows a fluorescently labeled t-and v-SNARE experiment, the lower lens is taken off from the SFA chamber, kept immersed in buffer in a 5mL beaker and further incubated with non-labeled His-tag free v-SNARE (final concentration of 0.2 µM) at 4 o C overnight. It undergoes the same procedure to rinse off un-bound protein and is remounted into the apparatus.

SFA force measurement

A homemade SFA similar to the original design ( 13) is used, except for the modification on the top mount, as shown in Figs. 1B and2A. It can adapt a large optical window with the SFA lens at the center. This modification increases the angle through which the emission of the acceptor is collected and therefore improves the optical sensitivity for the FRET signal. The upper lens is initially glued on an optically smooth polished glass slide, provided by Optique Fichou, France. This special upper lens is used in the deposition of the lipid bilayer. One band pass filter (from 570nm to 630nm) is added in front of the white light source, which limits the wavelength range of the fringes for the FECOs but does not disturb the force measurement. The on/off switching of the white light does not give any difference on the intensity of the dark image captured by the CCD camera at the same exposure time for the fluorescent imaging, which proves that the filtered white light source does not interfere with the fluorescent signal. The characteristic wavelengths of the mercury lamp are used for the calibration of the spectrometer. The procedure of the force measurement is the same as described before [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF]. The spring constant of the cantilever is pre-calibrated and its value is 109N/m.

Fluorescence Detection

Genesis MX 488-1000 STM laser purchased from Coherent is used to excite Alexa 488. The laser beam is reflected to the sample by a long pass dichroic mirror with a cut off at 500nm, purchased from Edmunds Optics. In between the laser source and the long pass dichroic mirror, an engineered 20 o diffuser from Thorlabs is used to enlarge the illuminated area on the sample. The total optical power illuminating the sample is only about 400μW, as measured by a photo detector. A band pass dichroic beam splitter (transparent from 565nm to 655nm) from Chroma Technology Corp is used to reflect the fluorescent light to the CCD camera while letting the 570 -630nm light pass to the spectrometer. In front of the CCD camera (Princeton Instruments, RTE/CCD-782-V/HS), there is a multi-filters unit with two band pass filters, both from Edmunds Optics: the transparent wavelength ranges are from 515nm to 560nm and from 650nm to 700nm for the emissions of Alexa 488 and Alexa 647 respectively. The setup of the fluorescent detection is displayed in Fig. 2.

When the separation distance between the samples, controlled by monitoring the fringes, is less than 500nm, one 30% neutral filter is added in front of the laser before turning on the laser to adjust the focus of the fluorescent image. The focus is adjusted with the observation of the fluorescent image of t-SNARE. The 30% neutral filter is also present to capture fluorescent images of t-SNARE, but not for fluorescent images of v-SNARE. The longest exposure time for each fluorescence image is 5s to avoid any bleaching effect or saturation of the detector.

Spin-Averaged Image and Intensity Profile

To obtain the spin-averaged image, the initial FRET image (Fig. S4) is rotated 119 times by 3 degrees. This process produces 120 images representing the initial image turned by 0°, 3°, 6°, …, 357°. Then these 120 images are averaged. The whole process is done by a macro programed in Image-J. Intensity profiles such as the one presented in Fig. 4B are plots of the intensity values along a straight line across the center of the spin-averaged image. Because the region close to the spin center does not contain many pixels to perform the average large fluctuations can be observed. In our system, the data I FRET (r) of r less than 0.7μm (2 pixels from the center) are not representative of a real average. They should be disregarded for the analysis.

FIGURE LEGENDS

Fig. 1 Transient transmembrane molecular intermediates -how to visualize them (A) Top: Two molecules assemble to form a molecular complex between a spherical surface and a flat surface with a fixed closest separation distance, h 0 , at the center. This geometry is the same as that in the FRET/SFA. As the surfaces come closer, the level of assembly of the molecules increases. This is represented in the lower panel cross section in which the complex formed by two imaginary molecules A and B exhibits 4 distinct states: disassembled (left), two intermediate states (middle) and fully assembled (right). In the case of the geometry presented here, when the surfaces are covered with molecules A and B respectively, the level of molecular assembly varies with the distance r to the closest approach on the right hand side of the diagram because the separation distance increases gradually with r. Hence, at a given r, because the local separation distance is fixed, assembling may only proceed until a certain level, and therefore the two molecules are trapped in an intermediate state. Then, a single snapshot is sufficient to observe all the intermediate states at various r and therefore at all separation distances. In the presented situation, when r is larger than r 1 , molecules A and B are not bound. Between r 1 and r 2 they are in intermediate state 1, between r 2 and r 3 intermediate state 2 is reached and below r 3 , the complex is fully assembled. Considering three regions in molecules A and B, a membrane distal region (blue), an middle region (orange) and a membrane proximal region (green), each state can be fully identified. When none of the regions are bound, the molecules are disassembled. If the blue regions are the only ones bound, the molecular complex is in the transient intermediate state 1. If regions blue and orange are assembled but not the green ones, it is in intermediate state 2. When all regions are bound, the complex is fully assembled. (B) SNARE proteins are used to represent molecules A and B. They zipper into a four-helix coiled coil between the membranes. In vivo, the progression of the zippering brings the two membranes together and drives them to fuse. The cytosolic domains are His-tagged to be bound to NTA(Ni)lipids, mimicking the natural transmembrane anchorage. The membranes are deposited on a mica sheet the other side of which is coated with silver (bottom) or a custom made multi-dielectric layer. This setup allows the simultaneous measurement of the intermembrane separation distance by interferometry and the visualization of FRET signal between SNAREs that are labeled with dyes in their N-terminal regions (excitation at 488 nm). The intermembrane distance can be frozen, just as in (A), in which case, all intermembrane distances can be observed along r.

Fig. 2 Diagram of the FRET/SFA setup (A)

The optical pathways for the SFA fringes, fluorescence excitation and fluorescence emission are denoted by orange, blue and red lines respectively. For the SFA, the modified top mount can adapt the optical window with the large diameter. The 30% neutral density filter (ND filter) is presented in the optical pathway only during the time period of either adjusting the focus or imaging the fluorescently labeled t-SNARE. A magnification of the interacting surfaces is shown in detail in Fig. 1B. (B) Resulting image and spectrum of the FECOs in the spectrometer by this setup. The range of wavelengths used for force measurements, i.e. for observation of the FECOs, is 570 -630 nm. A single-line home-made camera with 6000 pixels captures the spectrum along the white dashed line. The resulting intensities are presented below. Out of this range of wavelengths, only fluorescence (or FRET) can pass through the upper multi-dielectric layer presented in Fig. 1B.

Fig. 3. Correlation between adhesion and SNARE N-terminal binding

FRET/SFA can measure interaction energies while the surfaces are approached and subsequently separated. Simultaneously, the FRET signal (i.e. the assembly of the molecular complex) can be monitored at any separation distance. (A) SNARE induced adhesion between membranes. In black, the energy per SNAREpin is presented during an approach/separation cycle. During the approach (open disks) only positive, unfavorable energy is observed. Upon separation (closed disks), negative, favorable energy is measured, indicating the SNAREs have assembled. The maximum energy, corresponding to the SNARE binding energy, is the maximum one, i.e. e m ~ 30 k B T here. The red disks represent an approach/separation cycle in the control experiment in which Alexa488 labeled t-SNARE is blocked by pre-incubation with non-labeled and His-tag free v-SNARE. No adhesion is observed in the control experiment, showing unambiguously that the adhesion observed in the fluorescently labeled t-and v-SNARE experiments is SNARE assembling induced. (B) FRET signal is correlated with the adhesion of the membranes. The measurement of the FRET signal at the closest separation distance during the approach separation cycle is strongly correlated with the presence of adhesion: a positive FRET signal is observed only when there is adhesion, i.e. in the fluorescently labeled t-and v-SNARE experiment. Control experiments do not display any FRET signal. The relative intensity of the FRET signal, I FRET is equal to the mean value of the fluorescent intensity before it decays minus that of the control experiment (plateau in Fig. 4B). The FRET intensities and the corresponding adhesion energies in the FRET/SFA and control experiments were measured repeatedly and at different positions on each sample. The mean values are presented in the plot.

Fig. 4. Stable N-terminal assembly is required for initial binding of the SNAREs (A)

Left: spin-averaged picture (see Materials and Methods and Fig. S3) of the FRET signal between two membranes decorated with fluorescently labeled t-and v-SNARE respectively. Right: spin-averaged picture of the FRET signal in the control experiment. The scale bar represents 20μm. The intensities taken along the dashed line through the center to form the intensity profiles are displayed in (B): I a (black and red lines) represents the intensity of the fluorescently labeled t-and v-SNARE experiment while I c (blue) is the control. r equals 0 at the center. The black line shows the intensity right after the contact, and the red line was obtained after the surfaces were kept in contact for 30 min. The two lines are overlapping, revealing no time-dependence of the FRET signal. The variation in I c are due to optical interferences. Hence, the actual FRET signal is the difference between I a and I c . (C) Fraction of the N-terminal assembly in the total population of SNARE pairs with the intermembrane separation distance h (obtained from r through the equation provided in the main text). The N-terminal domains are only 100% assembled when h < 8 nm. The error bars are standard deviation obtained by averaging 20 to 40 data points on 4 separate experiments. (D) Adhesion of t-and v-SNARE decorated membranes in the SFA with h 0 , the minimal intermembrane separation reached during the approach/separation cycle, is normalized by the strongest adhesion (same data as in [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF]). The membranes were kept at distance h 0 for at least 20 min. (E) Various SNARE complex states in the SFA, their respective energies and the distribution of these states after 0 and 30 min in contact at h=8 nm, 13 nm and 20 nm.

  Fig. 4. Stable N-terminal assembly is required for initial binding of the SNAREs (A)Left: spin-averaged picture (see Materials and Methods and Fig.S3) of the FRET signal between two membranes decorated with fluorescently labeled t-and v-SNARE respectively. Right: spin-averaged picture of the FRET signal in the control experiment. The scale bar represents 20μm. The intensities taken along the dashed line through the center to form the intensity profiles are displayed in (B): I a (black and red lines) represents the intensity of the fluorescently labeled t-and v-SNARE experiment while I c (blue) is the control. r equals 0 at the center. The black line shows the intensity right after the contact, and the red line was obtained after the surfaces were kept in contact for 30 min. The two lines are overlapping, revealing no time-dependence of the FRET signal. The variation in I c are due to optical interferences. Hence, the actual FRET signal is the difference between I a and I c . (C) Fraction of the N-terminal assembly in the total population of SNARE pairs with the intermembrane separation distance h (obtained from r through the equation provided in the main text). The N-terminal domains are only 100% assembled when h < 8 nm. The error bars are standard deviation obtained by averaging 20 to 40 data points on 4 separate experiments. (D) Adhesion of t-and v-SNARE decorated membranes in the SFA with h 0 , the minimal intermembrane separation reached during the approach/separation cycle, is normalized by the strongest adhesion (same data as in[START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF]). The membranes were kept at distance h 0 for at least 20 min. (E) Various SNARE complex states in the SFA, their respective energies and the distribution of these states after 0 and 30 min in contact at h=8 nm, 13 nm and 20 nm. Solid bars represent 0 min and open columns represent 30 min.

  Figure 2

Figure 4

 4 Figure 4

Acknowledgments: We thank Prof. Jacob Israelachvili, Department of Chemical Engineering, University of California, Santa Barbara, Prof. James Rothman and Prof. E. Karatekin, School of Medicine, Yale University for valuable discussions. This work was supported by Agence Nationale de la Recherche grant ANR-12-BSV5-0002 to F.P. and a Fondation Pierre-Gilles de Gennes fellowship to Y. J. W.


F Decremps

Guillaume Morard

G Garbarino

M Casula

Polyamorphism of a Ce-based bulk metallic glass by high-pressure and high-temperature density measurements

Keywords: numbers: 61.43.Fs, 64.70.kj, 81.30.Hd

Polyamorphism of a Ce-based bulk metallic glass by

high-pressure and high-temperature density measurements

I. INTRODUCTION

Bulk metallic glasses (BMG) are among the most interesting amorphous materials which combine the electric conductivity of metals with glassy structural disorder, thus promising to be industrial products with appealing performances [1][2][3]. Recently, rare earth based metallic glasses including cerium based ones (Ce-MGs) have been found to display thermoplastic behavior with very low glass transition temperature (T g < 373 K) and low Young's modulus in comparison with classical amorphous matter [4][5][6][7]. At room temperature, these alloys are strong (elastic strain of 2 % [4]) and brittle. In their supercooled-liquid state, these materials can however be repeatedly shaped into very fine structures down to the nanometer scale, of great interest for micro electro-mechanical systems or nanotechnology applications such as high-density data storage [8].

In this context, the structural [9][10][11][12], electronic [9,13], elastic [7,[14][15][16] and thermal [17] properties of Ce-MGs have been extensively studied. Under pressure, an amorphousamorphous phase transition occurs from a low-density amorphous (LDA) phase to a highdensity amorphous (HDA) one. Indeed, previous x-ray diffraction measurements on Ce-MGs [9,15,18] investigated such properties on polyamorphic transition in Ce-MGs upon high pressure. Such works have shown an important structural change under pressure which is reversible and exhibits hysteresis under decompression. Sheng's group [9,10] has proposed a structural model for the LDA and HDA phases based on ab initio molecular dynamics simulations in which a splitting of the Ce-Ce nearest-neighbors distance is observed at the transition. The local structure at high pressure exhibits a double-shell feature, with an inner shell corresponding to a significant shorter bond compared to the low-pressure structure. In these calculations, based on the density functional theory with explicit local Coulomb repulsion (DFT+U), the LDA and HDA phases have been characterized by a localized and itinerant 4f electron, respectively. Also using x-ray techniques, Duarte et al [15] reported a low-and high-density amorphous phases at different pressure ranges, with the occurrence of a hysteresis at the transition, and of an intermediate density region expected to coincide with a mixture of both phases, or with a new metastable phase. This last hypothesis has been recently reported by Luo et al [19], through an exploration of the LDA-HDA transformation path at the temperature of 390 K.

In order to elucidate the nature of a polyamorphic transition and the corresponding phase diagram, one needs to extract the density of the compound as a function of pressure and temperature. However, all previous works have analyzed the experimental data using theoretical calculations or approximations in which the momentum transfer values for the first sharp diffraction peak (FSDP) can be roughly related to volume variation. In this work, we present a comprehensive experimental study of Ce 69 Al 10 Cu 20 Co 1 bulk MG by means of angle dispersive x-ray diffraction performed up to 16 GPa along two distinct isotherms (300 and 340 K). The whole diffuse signals have been here processed following the procedure as detailed in the Appendix, devised by [20][START_REF] Egami | Underneath the Bragg Peaks: Structural Analysis of Complex Materials[END_REF][START_REF] White | [END_REF] and later by Morard et al [23]. In this way, we have been able to extract directly the structure factor S(Q), the pair distribution g(r), the atomic density ρ and the compressibility as a function of pressure and temperature. These

II. EXPERIMENTAL TECHNIQUE AND ANALYSIS

Ce 69 Al 10 Cu 20 Co 1 bulk metallic glass was prepared in the form of thin ribbons (thickness ≈ 30 µm) using the single-roller melt-spinning technique. Its composition was confirmed by electron microprobe, scanning electron microscopy images showing uniform contrast. The glassy nature of the sample was verified by x-ray diffraction (no detectable Bragg peaks) and differential scanning calorimetry (presence of exothermic peak indicative of a glass transition temperature). This sample has been cut as few microns of dimensions and loaded in a rhenium gasket hole of 100 µm diameter placed in a membrane diamond anvil cell (DAC), resistively heated and equipped with 300 µm culet size diamonds. Neon was used as pressure transmitting medium. We used DACs able to independently control pressure and temperature without bringing the cell back to ambient conditions. The temperature was regulated using an electronic module and measured with an accuracy better than 1 K by two K-type thermocouples glued one on each anvil. Pressures were determined from the ruby and SrB 4 O 7 :Sm 2+ fluorescence lines before and after collection of each diffraction pattern.

These two readings always provided both pressure and temperature values (corresponding uncertainties given by symbols sizes in all the following figures).

Angle dispersive x-ray diffraction experiments in diamond anvil cell were performed at ID27 beam-line at the European Synchrotron Radiation Facility (ESRF). Incident energy of 33 keV was focused on the sample by using a focalized micrometric-sized x-ray beam. At each (P,T) point, a diffraction pattern of the background I bk (Q) was recorded through the pressure transmitting medium (but out of the Ce 69 Al 10 Cu 20 Co 1 glass sample), in order to be subtracted from the measured signals I meas (Q) (see Fig. 1).

The scattering signal from the Ce 69 Al 10 Cu 20 Co 1 bulk metallic glass sample only I S (Q) has been extracted from the measured signal after removing the background signal (using the classical Krogh-Moe Norman method [24]). Then, an iterative procedure has been applied

to calculate the pair distribution function g(r) as well as the density of the glassy alloy based on the Fourier treatment:

g(r) = 1 + 1 2π 2 rρ 0 ∞ 0 Q(S(Q) -1) sin(Qr)dQ = ρ(r) ρ 0 . (1)
The main steps of the corresponding numerical procedure are detailed in the Appendix, together with the basic assumptions and sources of uncertainties.

III. POLYAMORPHISM OF CERIUM-MG AT HIGH PRESSURE

Using the previous experimental setup and analysis, the pressure evolution of the density of a prototypical Ce-based amorphous sample (Ce 69 Al 10 Cu 20 Co 1 ) has been studied up to 16

GPa at ambient temperature. The occurrence of a pressure-induced phase transition from two glassy states can be markedly observed by a comparison of diffraction patterns I S (Q) at, for example, 0.5 GPa and 11 GPa (see Fig. 2). At low pressure, the FSDP is observed at 22.7 nm -1 , followed by two other wider bumps around 38.7 and 55.2 nm -1 (respectively labeled a and b). Such patterns evolved at higher pressure, where the FSDP is shifted to 25.2 nm -1 with roughly the same intensity, whereas the 38.7 nm -1 bump intensity mainly weakens while shifting to higher Q-values. In parallel, a new structure at around 50 nm -1 is observed in the high pressure (11 GPa) S(Q) function. All these observations can also be found on the corresponding pressure evolution of the g(r), represented in the right-hand side of Fig. 2.

The emergence of this new peak (labeled c) first demonstrates that a LDA-HDA transition occurs with local structure transformation probably related to the critical change of electronic properties of Ce atoms [25,26]. Secondly, the shift of the FSDP without intensity change shows that such transition does not significantly alter the mean distance order (taken, here, as the total coordination number). It suggests a transition related to a collapse of the main architecture of the first atomic shells, with shorter average atomic distance, apparent at high pressure. Note that this is a quite unusual behavior since in the majority of glassy oxides where polyamorphism has been demonstrated, the LDA-to-HDA transition goes along with a critical change of the mean distance order [11,19].

In order to go further, an integration of the first g(r) peak is known to give an information proportional to the total coordination number N c of the first atomic shell following the equation:

N c = rcut 0 4πr 2 ρ 0 g(r)dr, (2)
where r cut is set as the location of the minimum at the right side of the first peak in the pair distribution function g(r).

In the present work, and taken into account the experimental uncertainty, the evolution of N c is found to be pressure independent (inset of Fig. 3), with a typical value for a metallic alloy (here, 10±1). In the same Figure, the pressure evolution of the relative density ρ = ρ/ρ 0 up to 16 GPa is also represented. Moreover, note here that ρ has been defined as the number of atoms per volume units in proportion to the ambient conditions sample density (in this case ρ 0 has taken the value of ρ 0 LDA = 38.2 ± 2 at/nm 3 , as the sample is in the LDA state at ambient pressure).

A smooth but clear change in the pressure evolution of ρ is observed between 6 and 9

GPa, a finding which reinforces the previous arguments in favor of a LDA-HDA transition in Ce 69 Al 10 Cu 20 Co 1 . It must be emphasized that the iterative procedure to extract the density from the experimental signal did not converge in this pressure range. This could be due either to the presence of an intermediate local structure between the LDA and HDA phases [19], or to the coexistence between the two phases, as expected in a first-order phase transition.

In experiments [25], the LDA-to-HDA transformation in cerium-based bulk metallic glasses can be tightly linked to the behavior of the single 4f electron of Ce, as for the γ ⇄ α first-order transition in pure cerium. In the latter transition, studies under high pressure and high temperature have revealed the existence of a critical point (CP) at high temperature in the solid phase [28,29], driven by entropy difference, and probably also in the liquid state [30].

This CP plays an important role in the two-scale theory of polyamorphism despite it has never been observed [31]. Hence, studying the high-temperature behavior of solid Ce-BMG in order to search for the occurrence of such a CP in their phase diagram is an exciting area of research that should finally unravel the origin and mechanisms of polyamorphism.

Experimentally, the critical barrier is obviously related to the low kinetic stability of glasses which limits the temperature range of study. We performed a differential scanning calorimetry study of the present Ce 69 Al 10 Cu 20 Co 1 glass alloy, which has revealed the presence of an exothermic peak (indicative of a glass transition temperature T g ) at 365 K, and a crystallization temperature (at ambient conditions) of 416 K. Given these constraints, we also determined the equation of state of the sample at 340 K (well below the glass transition temperature), similarly to the previous angle dispersive x-ray diffraction experiments at 300 K.

Typical structure factors S(Q) and pair correlation function g(r) at different pressure along the 340 K isotherms are shown in Fig. 4. Qualitatively, the same behavior as the ambient temperature one is here observed, with an obvious structural modification related to the appearance at about 3 GPa of a peak at 55 nm -1 while the width and the intensity of the FSDP remain unchanged. Similarly, the evolution of total coordination number is found to be pressure independent, with N c = 11 ± 1, while the relative density ρ = ρ/ρ 0 up to 14 GPa needs to be fitted using two distinct Murnaghan equations of state (see Fig. 5) giving B 0 LDA = 17.4 ± 0.8 GPa (ρ 0 LDA = 37.9 ± 2 at/nm 3 ) and B 0 HDA = 53.2 ± 13 GPa (ρ 0 HDA = 42.4 ± 2 at/nm 3 ). The density variation in the transition region (between 4 and 6 GPa) is here about 10 %, without in Ref [19]. The discontinuous volume change measured in our work between 4 and 6

GPa at 340 K, as it occurs for a first-order transformation, can actually be hidden by the coexistence of the low-and high-pressure phases between 4 and 6 GPa, or by the presence of an intermediate medium-density amorphous state [19].

Finally, the present data support the conclusion that the hypothetical existence of a critical point in Ce-MGs still has to be demonstrated, the temperature effect previously observed [25] on the electronic properties of such a type of sample being probably related to a classical kinetic effect.

V. CONCLUSION

The present angle dispersive x-ray diffraction study at high pressure and temperature on Ce 69 Al 10 Cu 20 Co 1 provides a direct experimental evidence that the LDA-to-HDA transformation occurs at high pressure and ambient temperature. Along an isotherm at higher temperature, a similar behavior is observed without any evidence of a clear first-order transition [19] nor of a critical point existence. In both cases (300 and 340 K), the density pressure evolution is shown to be compatible with the general picture of structural rearrangement previously proposed by numerical simulations [9]. We observed in particular that the density change at the transition, roughly 10 %, occurs without variation of the total coordination number. As this polyamorphism in Ce-rich MG is known to be driven by a modification of Ce 4f states, our findings are in good agreement with the expected Ce-Ce bond shortening at the transition [9,11]. We emphasize that angle dispersive x-ray diffraction experiments only provide global information on the structure of the sample, and the role played by each type of atoms, particularly the cerium ones, cannot be discriminated here. However, it is interesting to highlight the connection between the LDA-to-HDA transformation in the Ce-MGs and the volume collapse transition in pristine Ce. In both cases, the density change is not accompanied by a rearrangement of the local geometry, nor by a modification of the coordination number, at variance with standard structural phase transitions. In a previous theoretical study of pure Ce [26] based on quantum Monte Carlo simulations, some of us pointed out that the Ce-Ce chemical bond variation, driven by strong local electron correlation and forbital hybridization, is key to understand the volume collapse. The corresponding phase transition is found already at zero temperature, even before the entropy effects dictated by the Kondo energy scale kick in, whose magnitude exponentially varies as a function of the f-electron hybridization [32]. In this paper we presented a compelling experimental evidence that also Ce-MGs, although very different from pure Ce, show a transition related to a significant volume variation. Volume collapse effects found in Ce-bearing compounds, such as Ce-MGs, are fascinating signatures of a general phenomenon, not specific to pristine Ce or Ce-alloys crystals, which calls for a further theoretical and experimental understanding.

VI. APPENDIX: ANALYSIS OF DIFFUSE SIGNAL FROM BULK METALLIC GLASSES A. Theoretical background

Within the general frame of a polyatomic compound, the spherically averaged coherent

x-ray scattering from disordered atoms can be written as:

I coh (Q) = N p f 2 p (Q) + m n =m f m (Q)f n (Q) sin Qr mn Qr mn , (3)
where N is the total number of compositional (or "elementary") units, m and n are the atoms in the entire volume of the x-ray-illuminated sample (labeled as Ω) whereas p defines atoms within an elementary unit volume dV, f n the atomic form factor of the n-th atom, and Q = 4π sin θ/λ the scattering momentum.

In the following, the average atomic distribution ρ pq (r) will be defined in such a way that the quantity ρ pq (r)dV ≡ < ρ p (r pq )dV q > Ω is the number of q atoms in an elementary volume dV at a distance r of a p-type atom taken as the center of the chemical unit. Then, Eq. 3 can be rearranged as an integral over the x-ray-illuminated sample volume Ω:

I coh (Q) = N p f 2 p (Q) + N p q Ω f p (Q)f q (Q)ρ pq (r) sin Qr Qr dV. (4)
Following the Faber-Ziman method [33], the previous I coh (Q) can also be expressed as:

I coh (Q) = f 2 + f 2 ∞ 0 4πr 2 (ρ(r) -ρ 0 ) sin Qr Qr dr, (5)
where ρ 0 is the atomic density of the sample and ρ(r) is the effective density function defined as:

ρ(r) = m n ρ mn (r)X m f m (Q) Q X n f n (Q) Q m n X m f m (Q)X n f n (Q) Q (6)
where X i is the occurrence of i atoms within the sample volume, and . . . Q denotes the average over the Q range at which the x-ray scattering experiment is performed.

The intermediate function f 2 and f 2 are respectively defined as:

f 2 = N p q f p (Q)f q (Q) = m n X m f m (Q)X n f n (Q), (7)
and

f 2 = N p f 2 p (Q) = n X n f 2 n (Q), (8)
with p and q in the above equations running on the elementary unit, while n and m run over the whole sample Ω.

The structure factor normalized to unity (here called Faber-Ziman structure factor S F Z ) can then be expressed as:

S F Z (Q) = I coh (Q) -( f 2 -f 2 ) f 2 = 1 + ∞ 0 4πr 2 (ρ(r) -ρ 0 ) sin Qr Qr dr. (9)
A Fourier transform of the previous equation thus leads to the experimental distribution function F (r):

F (r) = 4πr(ρ(r) -ρ 0 ) = 2 π Qmax 0 Q(S F Z (Q) -1) sin(Qr)dQ, (10)
from where the pair distribution function g(r) can be extracted as:

g(r) = ρ(r) ρ 0 . (11)

B. Krogh-Moe Norman normalization

The intensity of the signal scattered from the sample I S (Q) is calculated from the measured intensity I mes (Q) as the following:

I S (Q) = I mes (Q) -bI bk (Q), (12)
with I bk the intensity of the signal coming from the sample environment (anvils, pressure transmitting medium, etc.). This signal, here acquired under the same pressure and temperature conditions but without scattering from the Ce-MG, is adjusted by a scale factor b to account for the slight difference in experimental conditions (coming from, as example, the thickness of pressure transmitting medium or the pressure gradient within the experimental volume).

Following the method developed by Krogh-Moe [24] and Norman [34], I S (Q) is then converted into the normalized structure factor units:

S F Z (Q) = I coh -( f 2 -f 2 ) f 2 = (αI S (Q) -I incoh (Q)) -( f 2 -f 2 ) f 2 , (13)
where the normalization factor α is:

α = -2π 2 ρ 0 + Qmax 0 I incoh (Q)+ f 2 f 2 Q 2 dQ Qmax 0 Q 2 I S (Q) f 2 dQ . (14)
In the above equation, Q max is the maximum scattering momentum, typically around 70-80 nm.

-1 for DAC experiments, and I incoh (Q) is the sum of the incoherent scattering signals from the sample. The effect of the limited Q range, in particular on the density, will be discussed later. Note that the Q range used in the present study is suitable for the study of amorphous compounds and does not lead to spurious peaks in the g(r) [START_REF] Waseda | The structure of non-crystalline materials[END_REF].

C. Iterative procedure

To extract the sample density ρ 0 , an numerical iterative procedure with basic assumptions has to be carried out in order to minimize the error in the determination of g(r).

The first step here is to calculate the experimental distribution function F (r) using equation 10 with given ρ 0 et b values, as starting guess.

In a second step, a minimal distance r min is defined in such a way that the distance between 0 and r min represents the largest distance where no atom can be found. For r < r min , no atom can be present, i.e. ρ(r) = 0, so that no oscillation should be observed in the g(r)

function. As a consequence, we also have a particular distribution function F (r) which is

F (r < r min ) = -4πrρ 0 .
Third, one can calculate the difference between model and real data ∆F 0 (r) (where index 0 represents the value of the function before the first iteration) such as:

∆F 0 (r) = -F 0 (r) -4πrρ 0 , for 0 < r < r min . ( 15
)
∆F 0 (r) thus represents the error on the calculation of F (r) before the first iteration. Two types of systematic errors can usually take place here.

The first one is related to the determination of the normalization factor α, mainly affected by the experimental limited Q range (Q max in Eq. 14). Loss of information present at large Q obviously affects the values of density determined using the hereby procedure. This source of error is however quite small since the structure factor of glasses naturally damps quickly with increasing Q, so that the main part of the information is comprised in the first two oscillations.

Beside this truncation effect, another source of uncertainty comes from the determination of the b scale factor in the background subtraction. In the present work, the background was substantial which did not allow for a stringent fitting procedure of b. We thus fixed it manually, using exactly the same procedure for each pressure point, so that no relative error are expected to occur in the final determination of the pressure evolution of the sample density.

At this point, following the work of Kaplow et al. [START_REF] Kaplow | [END_REF], the iterative procedure can be written as:

i(Q) = S(Q) -1, (16)
i(Q) = i 0 (Q) + ∆i(Q), (17)
α = α 0 (1 + ∆α), (18) with ∆i(Q) et α 0 ∆α the error on the i et α variables, respectively. We emphasize that small variations of the normalization factor α could lead through this iterative scheme to large variations in S(Q), and so in g(r).

It goes:

Q∆i(Q) = Q∆αS 0 (Q) = ∆α(Qi 0 (Q) + Q) = ∆αQi 0 (Q) + ∆αQ, (19)
where the first term is a scaling factor for S(Q), and the second one, called ramp term, leads to large oscillation in the low r region of the g(r) [20]. The basic assumption is then to assert that the ramp term is the main source of ∆F 0 (r), which leads to the following equation, at the heart of the iterative process: ∆αQ = r min 0 ∆F 0 (r) sin(Qr)dr. (20) ∆α is directly determined by the experimental ∆F (r). Using the combination of Eq. 19 and Eq.. 20, the error on S(Q) coming from ∆α is determined, and a new structure factor is obtained through:

S 1 (Q) = S 0 (Q)(1 - 1 Q r min 0 ∆F 0 (r) sin(Qr)dr). ( 21
)
The iterative process is then built by re-injecting S 1 (Q) into Eq. 10 to obtain F 1 (r) and recalculate ∆F 1 (r).

In the present work, typically five iterative steps were necessary to converge. The corresponding fifth χ 2 is thus calculated as:

The above function exhibits one well defined minimum [20], here determined through a the simplex minimization procedure method [37] which allows to determine the unique (ρ 0 , b) couple at convergence.

  results are then discussed in the context of polyamorphism and in relation with the γ ⇆ α transition in pure cerium, as well as with the existence of a critical point in the phase diagram of Ce-MGs. The paper is organized as follows. The experimental technique, together with the method of data analysis are presented in Sec. II and in the Appendix. The resulting high-pressure studies of the equation of state and polyamorphism on Ce 69 Al 10 Cu 20 Co 1 bulk MG are given in Sec. III (300 K isotherm) and in Sec. IV (340 K isotherm). The conclusion is drawn in Sec. V.

2 FIG. 1 :

 21 FIG. 1: (color online) (Left) Measured diffraction pattern I meas (Q) of Ce 69 Al 10 Cu 20 Co 1 bulk metallic glass at 2.2 GPa and 300 K, including the background (red line, I bk (Q), mostly due to Compton signal from the diamonds here) which has been recorded at the same conditions in the experimental volume and beside the Ce-MG sample. (Right): Scattering signal of the sample I S (Q) (background removed).

FIG. 2 :

 2 FIG. 2: Structure factor S(Q) (left) and pair correlation function g(r) (right) of Ce 69 Al 10 Cu 20 Co 1 at 300 K and high pressure: 0.6 GPa (black) and 11.8 GPa (red).

FIG. 3 :

 3 FIG. 3: Relative density of Ce 69 Al 10 Cu 20 Co 1 as a function of pressure at 300 K. Inset: pressure evolution of the corresponding total coordination numbers N c .

  both low-and high-pressure pure phase domains, a Murnaghan equation of state has been used to fit the ρ(P) data, with B ′ fixed to 4. It leads to B 0 LDA = 17.4 ± 2 GPa (ρ 0 LDA = 38.2 ± 2 at/nm 3 ) and B 0 HDA = 45.2 ± 8 GPa (ρ 0 HDA = 42.4 ± 2 at/nm 3 ). Note here that the two Murnaghan equations of state give a density variation of about 12 % at 7 GPa (the average pressure along the transformation process), a discontinuity probably hidden experimentally by the kinetics of the transition between the different states. While the polyamorphism in Ce 69 Al 10 Cu 20 Co 1 has been demonstrated at ambient temperature in the previous section, the existence of such a feature along a higher-temperature isotherm stands as an open question. Firstly, because this sample belongs to the material family with long-range disordered structure, including amorphous and glasses, where the intriguing hypothesis of the two-scale theory, initially proposed by Rapoport [27], theoretically confirmed but never validated experimentally, is still debated. Secondly because this sample is cerium-based: a crucial consequence is that, as we recently observed by x-ray absorption

FIG. 4 :

 4 FIG. 4: Pressure evolution of the structure factor S(Q) (left) and pair correlation function g(r) (right) patterns of Ce 69 Al 10 Cu 20 Co 1 along the 340 K isotherm.

FIG. 5 :

 5 FIG.5: Relative density as a function of pressure at 340 K. Inset: pressure evolution of the corresponding total coordination numbers.

χ 2 (ρ 0 , b) ≡ r min 0 (

 20 ∆F 5 (r)) 2 dr.

Acknowledgments

We acknowledge G. Le Marchand and P. Munsch (IMPMC) for the DAC preparation.

We also thank G. Weck for fruitful discussions.

[1] W. Wang, Advanced Materials 21, 4524 (2009).