Alison G Tebo

Frederico M Pimenta

Yu Zhang

Arnaud Gautier

Improved chemical-genetic fluorescent markers for live cell microscopy

Inducible chemical-genetic fluorescent markers are promising tools for live cell imaging requiring high spatiotemporal resolution and low background fluorescence. The Fluorescence-Activating and absorption Shifting Tag (FAST) was recently developed to form fluorescent molecular complexes with a family of small, synthetic fluorogenic chromophores (so-called fluorogens). Here, we use rational design to modify the binding pocket of the protein and screen for improved fluorescence performances with four different fluorogens. The introduction of a single mutation results in improvements in both quantum yield and dissociation constant with nearly all fluorogens tested. Our improved FAST (iFAST) allowed the generation of a tandem td-iFAST that forms green and red fluorescent reporters 1.6-fold and 2-fold brighter than EGFP and mCherry, respectively, while having comparable size.

The ubiquitous implementation of fluorescence microscopy to address biological questions has led to the development of a series of markers based on small molecules, proteins, and hybrid techniques. The discovery of the green fluorescent protein (GFP) and the subsequent expansion of the color palette of fluorescent proteins have revolutionized biological imaging due to being genetically encoded and their absolute labeling specificity. 1 In parallel, a suite of small molecule probes with excellent brightness and photostability complements the toolbox of fluorescent markers. 2 However, both approaches suffer from drawbacks -fluorescent proteins are large and have slow maturation times, while small molecule probes suffer from off-target labeling. Thus, hybrid techniques have sought to combine the advantages of the two approaches for live cell imaging, while circumventing their respective drawbacks. [START_REF] Dean | Advances in fluorescence labeling strategies for dynamic cellular imaging[END_REF] A particularly promising hybrid approach is the use of inducible chemical-genetic fluorescent markers, which are comprised of a small protein tag that interacts specifically with a small molecule ligand, termed fluorogen, whose fluorescence is increased upon interaction with its protein tag. [START_REF] Bruchez | Dark dyes-bright complexes: fluorogenic protein labeling[END_REF][START_REF] Jullien | Fluorogen-based reporters for fluorescence imaging: a review[END_REF][START_REF] Li | Fluorogenic Labeling Strategies for Biological Imaging[END_REF] A recently developed system named the Fluorescence-Activating and absorption Shifting Tag (FAST) [START_REF] Plamont | Small fluorescenceactivating and absorption-shifting tag for tunable protein imaging in vivo[END_REF] is comprised of a small, monomeric, 14 kDa protein that was engineered from the photoactive yellow protein (PYP) to specifically and reversibly bind a series of hydroxybenzylidene rhodanine (HBR) derivatives (Figure 1A,B). Upon interaction with FAST, the fluorogen is locked into an emissive conformation. The resulting fluorescent complex provides many of the advantages of traditional autofluorescent proteins for live cell imaging while circumventing slow maturation times and oxygen dependence. The bipartite nature of this type of system allows one to optimize the fluorogen and the protein tag independently to improve or fine-tune the properties of the system. Previously, optimization of the fluorogen generated a suite of brighter and redder fluorescent markers, while still retaining the same protein tag. [START_REF] Li | Dynamic multicolor protein labeling in living cells[END_REF] To further improve the brightness, we optimized the protein tag through rational design of the binding pocket. The resulting variants were found to have increased quantum yields, particularly with the recently developed fluorogens HBR-3OM and HBR-3,5DOM. [START_REF] Li | Dynamic multicolor protein labeling in living cells[END_REF] We compared our rationally designed variants to new FAST variants that were recently identified through a new image-based, in-cell library screen to be brighter than FAST, but were not characterized as purified proteins. [START_REF] Emanuel | Highthroughput, image-based screening of pooled genetic-variant libraries[END_REF] Finally, we present tandem constructs of FAST variants as an effective strategy to increase the overall brightness of the system.

The parent protein of FAST, PYP, is a bacterial photoreceptor with a covalently attached 4-hydroxycinnamoyl (HC) chromophore, which participates in hydrogen bonding with Y42 and E46 (Figure 1C). [START_REF] Baca | Complete chemical structure of photoactive yellow protein: novel thioester-linked 4-hydroxycinnamyl chromophore and photocycle chemistry[END_REF][START_REF] Borgstahl | 1.4. Ang. structure of photoactive yellow protein, a cytosolic photoreceptor: Unusual fold, active site, and chromophore[END_REF][START_REF] Imamoto | Structure and Photoreaction of Photoactive Yellow Protein, a Structural Prototype of the PAS Domain Superfamily[END_REF] FAST was engineered through directed evolution of PYP by saturation mutagenesis of the loops near the chromophore binding pocket, resulting in replacement of the loop 94-101 in conjunction with the mutation of C69, which tethers HC, to glycine. However, the rest of the chromophore pocket of PYP remains intact, which can inform our understanding of how HBR derivatives bind to FAST. The observed The absorption red-shift is in agreement with the fluorogen being bound in its deprotonated form and stabilized by the same amino acids that interact with the chromophore of PYP, Y42 and E46. Furthermore, binding restricts the conformational freedom of the chromophore, leading to fluorescence activation. In seeking to improve the fluorescence quantum yield of FAST, we sought to further increase the conformational locking of the chromophore in the binding pocket. We identified the residues V107 and V122 as being potential targets for rational design because they are about 8 and 6 Å from the benzyl ring of 4-hydroxycinnamic acid in the PYP crystal structure (PDB: 1NWZ). [START_REF] Getzoff | Anticipatory active-site motions and chromophore distortion prime photoreceptor PYP for light activation[END_REF] Since the most successful fluorogens yet developed for FAST include modifications to the benzyl moiety to increase bulk, [START_REF] Li | Dynamic multicolor protein labeling in living cells[END_REF] we hypothesized that further introduction of bulky residues at these positions in the binding pocket of the protein would further improve the physicochemical properties of the complex (Figure 1D).

The proteins FAST V107I , FAST V122I , and double mutant FAST V107I,V122I were generated by site-directed mutagenesis and expressed in E. coli. The mutants reported by Emanuel et al, 9 mutant1, mutant2, and mutant3, were also expressed and purified. The thermodynamic dissociation constant KD, fluorescence quantum yield (Φ), and molar absorption coefficient were measured for the 24 pairs made of the six proteins with the four fluorogens: HMBR, HBR-3,5DM, HBR-3OM, and HBR-3,5DOM (Table 1, Figures 2 andS1). The mutation V107I was observed to slightly increase the affinity of the protein for the fluorogens. Furthermore, for certain fluorogens, particularly HBR-3OM and HBR-3,5DOM, which have bulky methoxy substituents on the benzyl ring, the quantum yield was dramatically increased, while the quantum yield for HBR-3,5DM exhibited a more modest increase in quantum yield. On the other hand, the mutation V122I decreased the binding affinity for the fluorogens by a factor of 3 to 8, depending on the fluorogen. This loss of affinity might result from V122I being closer to the fluorogen binding site relative to V107I. Nonetheless, this mutation also increased the quantum yield of several of the complexes. The effect of the two mutations was not additive, as seen from the characterization of the double mutant FAST V107I,V122I , which displayed lower binding affinities as might be expected from the introduction of V122I and similar increases in quantum yield as previously observed in both single mutants. The overall brightest complex is FAST V107I :HBR-3,5DM, while the largest improvement in quantum yield was achieved with FAST V107I :HBR-3OM, thus we termed this protein iFAST for improved-FAST. Interestingly, the mutant1-3 that were reported previously as brighter than FAST:HMBR by in-cell measurement in E. coli showed no significant improvement in fluorescence quantum yield relative to FAST, although the binding affinity for HMBR was increased by an order of magnitude. When tested with the three other fluorogens, mutants1-3 showed poorer performance, exhibiting lower quantum yields overall. The reported improvement in mutant1-3 brightness with HMBR in E. coli cells is probably due to the increased binding affinity favoring reporter formation in bacteria.

Next, we hypothesized that dimerizing FAST could generate significantly brighter markers for imaging through increase of the apparent absorption coefficient, as has previously been done with some autofluorescent proteins such as tdTomato. [START_REF] Shaner | Improved monomeric red, orange and yellow fluorescent proteins derived from Discosoma sp. red fluorescent protein[END_REF] A tandem of FAST (td-FAST) results in a probe of about 28 kDa, which is about the same size as fluorescent proteins, with a brightness 1.4fold higher than EGFP [START_REF] Day | The fluorescent protein palette: tools for cellular imaging[END_REF] when used with HBR-3,5DM. Furthermore, to capitalize on the mutations that improve the quantum yield of FAST, we generated a tandem of iFAST (td-iFAST) as well, which provides an additional improvement in overall brightness depending on the fluorogen chosen. The brightness of td-FAST and td-iFAST along with FAST and iFAST was tested in live cell confocal microscopy with HBR-3OM, which exhibits the largest change in quantum yield between FAST and iFAST, to evaluate their relative brightness in the context of cell imaging (Figure 3). We observed that tandems of FAST are expressed homogenously in the cell, confirming that neither the size nor the construction of the tandem impedes diffusion into the nucleus (Figure 3A). Relative to FAST, iFAST was found to be 1.7-fold brighter, which is slightly higher than predicted based on the brightness measured in vitro (Figure 3B,C and Figures S2-S3). Furthermore, td-FAST and td-iFAST were found to be 2.8fold and 3.8-fold brighter than FAST, respectively (Figure 3B,C and Figures S2-S3).

Next we examined the photobleaching of FAST, iFAST, FAST V122I , and mutant1 with HMBR in living cells. It has previously been reported that FAST exhibits a biphasic photobleaching behavior composed of a rapid phase and a slow phase. [START_REF] Emanuel | Highthroughput, image-based screening of pooled genetic-variant libraries[END_REF][START_REF] Pimenta | Chromophore Renewal and Fluorogen-Binding Tags: A Match Made to Last[END_REF] Long acquisitions by confocal microscopy allowed us to compare the photobleaching behavior of the four proteins with 10 µM HMBR (Figure S4A). In the conditions used, iFAST, FAST V122I , and mutant1 show a reduction of the amplitude of the rapid bleaching phase compared to FAST. After this rapid initial fluorescence decrease, FAST, iFAST, FAST V122I showed almost steady fluorescence, while mutant1 seemed to be slightly more prone to photodamage (Figure S4A). Mutant1 contains a cysteine residue, which is prone to photooxidation, in its sequence that is absent in the three other variants, which might explain the observed difference of behavior. The photobleaching of FAST and iFAST were also examined in the presence of 10 µM HBR-3,5DM, HBR-3OM, and HBR-3,5DOM (Figure S4B-D). FAST and iFAST showed similar photobleaching curves for all three fluorogens. The reporters with HBR-3,5DM and HBR-3OM, while being brighter, appear to be slightly more sensitive to photobleaching than with HMBR, particularly in the case of HBR-3OM. Imaging with 10 µM HBR-3,5DOM revealed that both FAST and iFAST are as photostable as mCherry.

One major advantage of fluorogenic and hybrid systems comprised of a small molecule and a protein tag is that further development of the system can be done by optimizing either the protein or the small molecule. Here, we have presented work to improve the fluorescence performance of the FAST system by manipulation of the protein tag by rational design to introduce a single mutation that increases steric hindrance around the chromophore. The use of rational design was complementary to the directed evolution approach used originally for developing FAST, as it has facilitated the finetuning of properties difficult to further improve by high-throughput approaches. Not only has rational design provided us with improved tags but also supports our hypothesis that HBR derivatives bind to FAST in a similar fashion as 4-hy-droxycinnamic acid to PYP. We also showed that dimerizing FAST is an effective way to increase the overall brightness, particularly when the size of the tag is less of an issue. Optimized td-iFAST is 1.6-fold brighter than EGFP, while having comparable size. Since the color of FAST-based systems can be changed at-will, td-iFAST presents a particularly attractive alternative to red fluorescent proteins. In particular, td-iFAST with HBR-3,5DOM is 2-fold brighter than mCherry. These new variants of FAST exhibit quantum yields on par with fluorescent proteins while retaining the advantages of the FAST system -namely, no maturation time, on-demand imaging, small tag size, and the ability to change the color at-will.

Furthermore, the introduction of these mutations demonstrates that FAST could be further modified in efforts to expand the FAST system, such as to further improve the fluorescence performance and physicochemical properties, but also to expand the color palette or engineer new fluorogen specificity.

ASSOCIATED CONTENT

Supporting Information. Supporting information contains Figures S1-S3 and Materials & Methods. The Supporting Information is available free of charge on the ACS Publications website. a total number of cells analyzed of 373, 324, 374, and 329 for FAST, iFAST, td-iFAST, and td-FAST, respectively.

Photobleaching measurements for HMBR, HBR-3,5DM, and HBR-3OM were carried out at 10 µM fluorogen at 488 nm excitation (4.6 kW / cm 2 , 1.27 µsec pixel dwell); EGFP was used as a control. HBR-3,5DOM and mCherry were excited at 541 nm (2.7 kW / cm 2 , 1.27 µsec pixel dwell). Samples were imaged continuously for 500 images.

Western Blots Samples used for microscopy were washed twice with Dulbecco's Phosphate-Buffered Saline (DPBS) and lysed in 100 uL CellLytic (Sigma). The samples were then centrifuged for 5 minutes at 6,000 x g at 4 °C and the supernatant was used for further analysis. Total protein concentration was determined by BCA assay (Sigma) in parallel with purified protein samples. Samples were fixed to have the same amount of total protein and separated by SDS-PAGE (4-12% Bis-Tris NuPAGE). Proteins were transferred by semi-dry blotting to a nitrocellulose membrane (Amersham Protran 0.2 µm). Membranes were imaged with Ponceau-S (Sigma) to ensure homogenous transfer and loading and then incubated with blocking buffer (Tris Buffer Saline with 0.1% Tween-20 [TBS-T] + 5% dry milk) for 1 hr. Membranes were incubated overnight at 4 °C with antimyc (Cell Signaling Technology) at 1:2000 in blocking buffer and then washed with TBS-T for 1 hour. The membranes were then stained with diluted (1:2500) secondary antibody, anti-rabbit (Cell Signaling Technology), in blocking buffer for one hour. Membranes were then washed for 1 hour in TBS-T. Antibody was visualized with BioRad ClarityMax reagent after 5 min incubation using a BioRad ChemiDoc MP. Blots were analyzed in ImageJ using the gel analyzer tool without background correction. Standard curves using purified FAST-myc and td-FAST-myc were used to interpolate the quantity of protein in the sample.

Figure 1 .

 1 Figure 1. FAST principle and improvement. (A) FAST variants bind and activate the fluorescence of HBR analogs. (B) The structures of the HBR analogs used in this study. (C) Detailed view of the chromophore binding pocket in PYP including the relative positions of V107 and V122 (PDB: 1nwz). (D) Proposed fluorogen binding pocket in iFAST. characteristics of FAST, namely the absorption red-shift and fluorescence activation upon fluorogen binding are due to the binding mode of the fluorogen to the protein.The absorption red-shift is in agreement with the fluorogen being bound in its deprotonated form and stabilized by the same amino acids that interact with the chromophore of PYP, Y42 and E46. Furthermore, binding restricts the conformational freedom of the chromophore, leading to fluorescence activation. In seeking to improve the fluorescence quantum yield of FAST, we sought to further increase the conformational locking of the chromophore in the binding pocket. We identified the residues V107 and V122 as being potential targets for rational design because they are about 8 and 6 Å from the benzyl ring of 4-hydroxycinnamic acid in the PYP crystal structure (PDB: 1NWZ).[START_REF] Getzoff | Anticipatory active-site motions and chromophore distortion prime photoreceptor PYP for light activation[END_REF] Since the most successful fluorogens yet developed for FAST include modifications to the benzyl moiety to increase bulk,[START_REF] Li | Dynamic multicolor protein labeling in living cells[END_REF] we hypothesized that further introduction of bulky residues at these positions in the binding pocket of the protein would further improve the physicochemical properties of the complex (Figure1D).

Figure 2 .

 2 Figure 2. Spectral characteristics of FAST and iFAST. Absorption spectra of FAST (A) and iFAST (C) and fluorescence spectra of FAST (B) and iFAST (D) with HMBR, HBR-3,5DM, HBR-3OM, and HBR-3,5DOM. Fluorogen concentration was 3 µM and FAST mutant concentration was 40 µM. Spectra were recorded in pH 7.4 PBS at 25°C.

Figure 3 .

 3 Figure 3. Relative in-cell brightness of FAST, iFAST, td-FAST, and td-iFAST. (A) Representative confocal micrographs of HEK 293T cells expressing FAST, iFAST, td-FAST, and td-iFAST. The images were all acquired at 16-bit with the same settings, each pixel defined by 65,536 grey levels. Given the difference of brightness between FAST, iFAST, td-FAST and td-iFAST, the intensity scale used for the images of tdFAST and td-iFAST did not allow the proper visualization of FAST and iFAST. Therefore we provided one panel (top) that is optimized for visualizing td-FAST and a second (bottom) that is optimized for visualizing FAST. Intensity (grey level) scales are given on the right of each panel. Cells were labeled with 20 µM HBR-3OM. Scale bars are 20 µm. (B) Relative brightness in cells (mean ± sem of 4 experiments, each with an average of 80-90 cells quantified). Fluorescence levels were normalized by the protein expression level determined by quantitative western blotting as shown on Figure S2. Figure S3 presents the individual measurements. (C) Comparison of the brightness of FAST, iFAST, td-FAST and td-iFAST labeled with HBR-3OM in mammalian cells (in vivo brightness) and solution (in vitro brightness).

Table 1 . Physico-chemical properties of mutant:fluorogen complexes in PBS pH 7.4.

 1 Abbreviations are as follows: labs, wavelength of maximal absorption; lem, wavelength of maximal emission; e, molar absorption coefficient at labs (standard error is typically 10%); Φ, fluorescence quantum yield; KD thermodynamic dissociation constant.

	Mutant	Fluorogen	λabs (nm)	λem (nm)	ε (mM -1 cm -1 ) Φ	KD
							(µM)
	FAST	HMBR	481	540	45	0.23 ± 0.03	0.13
		HBR-3,5DM	499	562	48	0.49 ± 0.05	0.08
		HBR-3OM	494	561	40	0.36 ± 0.04	0.31
		HBR-3,5-DOM	518	600	39	0.31 ± 0.03	0.97
	iFAST	HMBR	480	541	41	0.22 ± 0.03	0.07
		HBR-3,5DM	499	558	46	0.57 ± 0.08	0.06
		HBR-3OM	495	560	39	0.49 ± 0.07	0.2
		HBR-3,5-DOM	516	600	38	0.40 ± 0.04	0.41
	FAST V122I	HMBR	480	542	41	0.35 ± 0.05	0.35
		HBR-3,5DM	499	559	43	0.52 ± 0.07	0.57
		HBR-3OM	495	560	38	0.47 ± 0.07	1.4
		HBR-3,5-DOM	516	596	~30	0.32 ± 0.03	8.3
	FAST V107I,V122I	HMBR	480	541	35	0.24 ± 0.03	0.3
		HBR-3,5DM	499	558	50	0.48 ± 0.07	0.29
		HBR-3OM	495	560	33	0.49 ± 0.07	2.0
		HBR-3,5-DOM	516	598	~30	0.33 ± 0.04	6.4
	mutant1	HMBR	486	545	56	0.23 ± 0.03	0.024
		HBR-3,5DM	507	565	48	0.25 ± 0.03	0.58
		HBR-3OM	501	555	51	0.26 ± 0.03	0.17
		HBR-3,5-DOM	515	595	36	0.26 ± 0.03	2.0
	mutant2	HMBR	486	544	59	0.23 ± 0.03	0.024
		HBR-3,5DM	504	565	47	0.34 ± 0.04	0.49
		HBR-3OM	501	555	48	0.26 ± 0.03	0.2
		HBR-3,5-DOM	516	595	41	0.27 ± 0.03	2.5
	mutant3	HMBR	488	547	39	0.27 ± 0.04	0.012
		HBR-3,5DM	504	565	47	0.34 ± 0.04	0.21
		HBR-3OM	502	555	50	0.24 ± 0.03	0.2
		HBR-3,5-DOM	515	596	38	0.25 ± 0.03	3.0

ACKNOWLEDGMENT (This work has been supported by the European Research Council (ERC-2016-CoG-724705 FLUOSWITCH), the Agence National de la Recherche (ANR-14-CE09-0002-01), France BioImaging (ANR-10-INBS-04), and the Equipex Morphoscope 2 (ANR-11-EQPX-0029).

AUTHOR INFORMATION

Corresponding Author *Arnaud Gautier arnaud.gautier@ens.fr

Notes

The authors declare the following competing financial interest: A.G. is co-founder and holds equity in Twinkle Bioscience, a company commercializing the FAST technology. scanning confocal microscope equipped with a 488 nm laser (with a power of 4.6 kW / cm 2 at the specimen plane) and a 541 nm laser (with a power of 2.7 kW / cm 2 at the specimen plane) at a frame rate of 1 image / 750 ms (pixel dwell time of 1.27 µs), n = 3 cells per curve.

Supporting Information

MATERIALS AND METHODS

General FAST is a variant of the photoactive yellow protein (PYP) containing the mutations C69G, Y94W, T95M, F96I, D97P, Y98T, Q99S, M100R, T101G. HMBR, HBR-3,5DM, HBR-3OM, and HBR-3,5-DOM were prepared as previously described 1,2 .

Molecular Biology

The point mutations were inserted in FAST via site-directed mutagenesis using complementary primers and Gibson assembly using the previously reported pAG87. The tandem construct was synthesized by Eurofins and subcloned into plasmids for bacterial expression using Nhe I and Xho I restriction sites. Mutations were subsequently introduced sequentially in the same manner as for FAST. The genes for mutants (1-3) were synthesized by Eurofins and subcloned into plasmids for bacterial expression by Gibson assembly. All proteins were then subcloned into plasmids for mammalian expression by Gibson assembly.

Protein expression and purification

Expression vectors were transformed in Rosetta (DE3) pLysS E. coli (New England Biolabs). Cells were grown at 37°C in LB medium complemented with 50 μg/ml kanamycin and 34 μg/ml chloramphenicol to OD600nm 0.6.

Expression was induced for 4 h by adding isopropyl β-D-1-thiogalactopyranoside (IPTG)

to a final concentration of 1 mM. Cells were harvested by centrifugation (4,000 × g for 20 min at 4°C) and frozen. The cell pellet was resuspended in lysis buffer (phosphate buffer 50 mM, NaCl 150 mM, MgCl2 2.5 mM, protease inhibitor, DNase, pH 7.4) and sonicated

(5 min at 20 % of amplitude, 3 sec on, 1 sec off). The lysate was incubated for 2 h at 4 °C

where S and R are the sample and reference, respectively, I is the integrated intensity of fluorescence emission, Abs is the absorbance at the excitation wavelength and n is the refractive index of each solution. FAST:HMBR was used as a reference.

Mammalian cell culture HEK 293T cells were cultured in Dulbecco's Modified Eagle Medium (DMEM) supplemented with phenol red, Glutamax I, and 10% (vol/vol) fetal calf serum (FCS), at 37 °C in a 5% CO2 atmosphere. For imaging, cells were seeded in μDish IBIDI (Biovalley) coated with poly-L-lysine. Cells were transiently transfected using Genejuice (Merck) according to the manufacturer's protocol for 24 hours prior to imaging.

Fluorescence microscopy Confocal micrographs were acquired on a Zeiss LSM 710

Laser Scanning Microscope equipped with a Plan Apochromat 63×/1.4 NA oil immersion objective. Cells were imaged in DMEM supplemented with 20 µM HBR-3OM for quantification. Images were collected as a series of z-stacks by defining the first and last image whereby positive intensity is detected so as to capture all in and out of focus light.

The number of images varied on the thickness of the cells but the imaging interval between stacks was kept constant for all stacks. Stacks were analyzed by generating a sum intensity projection across the stack and the integrated density was measured for each cell for about 80-90 cells per sample. The experiment was repeated four times, with


Robin Scheibler

Diego Di

Antoine Deleforge

Ivan Dokmanić

SEPARAKE: SOURCE SEPARATION WITH A LITTLE HELP FROM ECHOES

Keywords: Source separation, echoes, room geometry, NMF, multi-channel

It is commonly believed that multipath hurts various audio processing algorithms. At odds with this belief, we show that multipath in fact helps sound source separation, even with very simple propagation models. Unlike most existing methods, we neither ignore the room impulse responses, nor we attempt to estimate them fully. We rather assume to know the positions of a few virtual microphones generated by echoes and we show how this gives us enough spatial diversity to get a performance boost over the anechoic case. We show improvements for two standard algorithms-one that uses only magnitudes of the transfer functions, and one that also uses the phases. Concretely, we show that multi-channel non-negative matrix factorization aided with a small number of echoes beats the vanilla variant of the same algorithm, and that with magnitude information only, echoes enable separation where it was previously impossible.

INTRODUCTION

Source separation algorithms can be grouped according to how they deal with sound propagation: those that ignore it [START_REF] Le Roux | Deep NMF for speech separation[END_REF], those that assume a single anechoic path [START_REF] Rickard | The duet blind source separation algorithm[END_REF], those that model the room transfer functions (TFs) entirely [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF][START_REF] Nugraha | Multichannel audio source separation with deep neural networks[END_REF], and those that attempt to separately estimate the contribution of the early echoes and the contribution of the late tail [START_REF] Leglaive | Multichannel audio source separation with probabilistic reverberation modeling[END_REF]. In this paper we propose yet another route: we assume knowing the locations of a few walls relative to the microphone array, which enables us to exploit the associated virtual microphones. This assumption is easy to satisfy in living rooms and conference rooms, but the corresponding model incurs a significant mismatch with respect to the complete reverberation. We show that it nonetheless gives sizable performance boosts while being simple to estimate. The approach we propose is reminiscent of acoustic rake receivers [START_REF] Dokmanić | Raking the cocktail party[END_REF]; we thus call it separake.

A typical setup is illustrated in Figure 1. We consider J sources emitting from J distinct directions of arrival (DOAs) {θj} J j=1 , and an array of M microphones. The array is placed close to a wall or a corner. There are two reasons why this is useful: first, it makes echoes from the nearby walls significantly stronger than all other echoes; second, it ensures that the resulting virtual array (real and virtual microphones) is compact. The latter justifies the far field assumption which in turn simplifies exposition.

Real and virtual microphones form dipoles with diverse frequencydependent directivity patterns. Our goal is to design algorithms

Virtual microphones

Fig. 1: Typical setup with two speakers recorded by two microphones. The illustration shows the virtual microphone model (grey microphones) with direct sound path (dotted lines) and resulting first-order echoes (colored lines) which benefit from this known spatial diversity.

Echoes have been used previously to enhance various audio processing tasks. It was shown that they improve indoor beamforming [START_REF] Dokmanić | Raking the cocktail party[END_REF][START_REF] Scheibler | Raking echoes in the time domain[END_REF][START_REF] Scheibler | Rake, Peel, Sketch: The signal processing pipeline revisited[END_REF], aid in sound source localization [START_REF] Ribeiro | Turning enemies into friends: Using reflections to improve sound source localization[END_REF], and enable lowresource microphone array self-localization [START_REF] Dokmanić | From acoustic room reconstruction to SLAM[END_REF]. They have, however, rarely been analyzed in the context of source separation with non-negative source models.

Our Goal and Main Findings

Our emphasis here is different than that in [START_REF] Leglaive | Multichannel audio source separation with probabilistic reverberation modeling[END_REF]. Rather than fitting the echo model, we aim to show that separation in the presence of echoes is in fact better than separation without echoes. We ask the following questions:

1. Is speech separation with echoes fundamentally easier than speech separation without echoes? Are there specific settings where this is true or false?

2. Is it necessary to fully model the reverberation or can we get away with a geometric perspective where we know the locations of a few virtual microphones?

To answer these questions we set up several simple experiments. We take two standard, well-understood multi-channel source separation algorithms which estimate the channel (the TFs), and instead of updating the channel estimate we simply fix it to the TFs of real and a few virtual microphones. The first algorithm-non-negative matrix factorization (NMF) via multiplicative updates (MU)-only uses the magnitudes of the transfer functions, while the second oneexpectation maximization (EM)-also uses the phases. In this initial investigation we look at the (over)determined case (J ≤ M ), leaving the analysis of the underdetermined case to future work. Our findings can be summarized as follows:

• (MU) With magnitudes only, multi-channel anechoic separation is hardly any better than single-channel separation: as the magnitude of the transfer functions is the same at all microphones, channel modeling offers no diversity. The situation is different in rooms where the direction-dependent magnitude of TFs varies significantly from microphone to microphone. We show that replacing the transfer functions with a few echoes (even just one) gives significant performance gains compared to not modeling the TFs at all, but also that it does better than learning the TFs through multiplicative updates.

• (EM) With both phases and magnitudes, anechoic separation will be near-perfect since it corresponds to a determined linear system. Therefore, any uncertainty from imperfections in channel modeling will make things worse. Surprisingly, approximating the TFs with one echo matches learning them through EM updates and using more outperforms it.

For a sneak peak at the gains, fast forward to Figure 3.

MODELING

Suppose J sources emit inside the room and we have M microphones. Each microphone receives

ym(t) = J j=1 cjm(t),
with cjm being the spatial image of the jth source at the mth microphone. Spatial images are given as

cmj(t) = (xj * hjm)(t),
where hjm is the room impulse response between the source j and microphone m. The room impulse response is a central object in this paper. We model it as

hjm(t) = K k=0 α k jm δ(t -t k jm ) + jm(t),
where the sum comprises the line-of-sight propagation and the earliest K echoes we want to account for (at most 6 in this paper), while the error term jm(t) collects later echoes and the tail of the reverberation. We do not assume ejm(t) to be known. We assume that the sources are in the far field of real and virtual microphones so the times t k jm depend only on the source DOAs which we assume are known. Assuming K echoes per source are known, we can form an approximate TF from source j to microphone m,

Hj,m(e jω ) = K k=0 α k jm e -iω tk jm . (1)
We only assume relative arrival times and amplitudes to be known, that is tjm = t k jm -t 0 jm and α k jm = α k jm /α 0 jm , respectively. In practice, these parameters can be estimated [START_REF] Remaggi | Acoustic reflector localisation for blind source separation and spatial audio[END_REF]. In addition, we assume all walls to be spectrally flat in the frequency range of interest.

As usual, we process by frames. In the short-time Fourier transform (STFT) domain the mth microphone signal reads

Ym[f, n] = J j=1 Hjm[f ]Xj[f, n] + Bm[f, n] (2)
with f and n being the frequency and frame index, Xj[f, n] the STFT of the jth source signal, and Bm[f, n] a term including noise and model mismatch. It is convenient to group the microphone observations in vector form,

Y[f, n] = H[f ] X[f, n] + B[f, n]. (3)
where

Y[f, n] = Ym[f, n] m , H[f ] = Hjm[f, n] m,j , X[f, n] = Xj[f, n] j , and B[f, n] = Bm[f, n] m . Let the squared magni- tude of the spectrogram of the jth source be Pj = |Xj[f, n]| 2 f n .
We postulate a non-negative factor model for Pj:

Pj = DjZj, ( 4
)
where Dj is the non-negative dictionary, and the latent variables Zj are called activations. Source separation can then be cast as an inference problem in which we maximize the likelihood of the observed Y over all possible non-negative factorizations (4). This normally involves learning the channel (frequency-domain mixing matrices). Instead of learning, we fix the channel to the earliest few echoes.

SOURCE SEPARATION BY NMF

To evaluate the usefulness of echoes in source separation, we modify the multi-channel NMF framework of Ozerov and Févotte [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF] as follows. First, we introduce a dictionary learned from available training data. We explore both speaker-specific and universal dictionaries [START_REF] Sun | Universal speech models for speaker independent single channel source separation[END_REF]. Speaker-specific dictionaries can be beneficial when speakers are known in advance. Universal dictionary is more versatile but gives a weaker regularization prior. Second, rather than learning the TF from the data, we use the approximate model of (1).

In the following we briefly describe the two algorithms used.

NMF using Multiplicative Updates (MU-NMF)

Multiplicative updates for NMF only involve the magnitudes and are simpler than the EM updates. The updates are guaranteed nonnegative as long as the intitialization is. They have been originally proposed by Lee and Seung [START_REF] Lee | Algorithms for non-negative matrix factorization[END_REF]. We use the Itakura-Saito divergence [START_REF] Févotte | Algorithms for nonnegative matrix factorization with the β-divergence[END_REF] between the observed multi-channel squared magnitude spectra

Vm = [|Ym[f, n]| 2 ]
f n and their non-negative factorizations,

Vm = J j=1 diag(Qjm)DjZj, m = 1, . . . , M (5)
where

Qjm = | Hjm[f ]| 2
f is the vector of squared magnitudes of the approximate TF between microphone m and source j. We add an 1-penalty term to promote sparsity in the activations due to the potentially large size of the universal dictionary [START_REF] Sun | Universal speech models for speaker independent single channel source separation[END_REF]. The cost function is thus [START_REF] Dokmanić | Raking the cocktail party[END_REF] where

C MU (Zj) = mf n d IS (Vm[f, n]| Vm[f, n]) + γ j Zj 1,
d IS (v|v) = v v -log v v -1.
By adapting the original MU rule derivations from Ozerov and Févotte, we obtain the following regularized MU update rule:

Zj ← Zj m (diag(Qij)Dj) Vj V -2 j m (diag(Qij)Dj) V -1 j + γ , (7)
where multiplication , power, and division are element-wise. Importantly, neglecting the reverberation (or working in the anechoic regime) leads to a constant Qjm for all j and m. A consequence is that the MU-NMF framework breaks down with a universal dictionary. Indeed, (5) becomes the same for all m, Vm = j DZj = D j Zj, so even with the correct atoms identified, we can assign them to any source without changing the value of the cost function. Therefore, anechoic multi-channel separation with a universal dictionary cannot work well. This intuitive reasoning is corroborated by numerical experiments in Section 4.3. Of course, in line with the message of this paper, this problem is overcome by using echoes.

NMF using Expectation Maximization (EM-NMF)

Unlike the MU algorithm that independently maximizes the loglikelihood of TF magnitudes, EM-NMF maximizes the joint loglikelihood over all complex-valued channels [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF]. Hence, it takes into account observed phases. Each source j is modeled as the sum of components with complex Gaussian priors of the form

c k [f, n] ∼ CN 0, d f k z kn such that Xj[f, n] ∼ CN (0, (DjZj) f n ) , (8)
and the magnitude spectrum Pj of (4) can be understood as the variance of source j. Under this model, and assuming uncorrelated noise, the microphone signals also follow a complex Gaussian distribution with covariance matrix

Σ Y [f, n] = H[f ] Σ X [f, n] H H [f ] + Σ B [f, n], (9)
and the negative log-likelihood of the observed signal is

CEM(Zj) = f n trace Y[f, n]Y[f, n] H Σ -1 Y [f, n] + log det ΣY[f, n].
This quantity can be efficiently minimized using the EM algorithm proposed in [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF]. We modify the original algorithm by fixing the source dictionaries Dj and the early-echo channel model H[f ] throughout the iterations. Since adding sparsity priors is not straightforward in the EM framework, the universal dictionary was left for future work.

NUMERICAL EXPERIMENTS

We test our hypotheses through computer simulations. In the following, we describe the simulation setup, dictionary learning protocols, and we discuss the results.

Setup

An array of three microphones arranged on the corners of an equilateral triangle with edge length 0.3 m is placed in the corner of a 3D room with 7 walls. We select 40 sources at random locations at a distance ranging from 2.5 m to 4 m from the microphone array. Pairs of sources are chosen so that they are at least 1 m apart. The floor plan and the locations of microphones are depicted in Figure 2.

The scenario is repeated for every two active sources out of the 780 possible pairs. The sound propagation between sources and microphones is simulated using the image source model implemented in pyroomacoustics Python package [START_REF] Scheibler | Pyroomacoustics: A Python package for audio room simulations and array processing algorithms[END_REF]. The wall absorption factor is set to 0.4, leading to a T60 of approximately 100 ms. An example RIR is shown in Figure 2. The sampling frequency is set to 16 kHz, STFT frame size to 2048 samples with 50% overlap between frames, and we use a cosine window for analysis and synthesis. Partial TFs are then built from the K nearest image microphones. The global delay is discarded.

With this setup, we perform three different experiments. In the first one, we evaluate MU-NMF with a universal dictionary. In the other two, we evaluate the performance of MU-NMF and EM-NMF with Speaker-specific dictionaries. We vary K from 1 to 6 and use three baseline scenarios:

1. anechoic: Anechoic conditions, no model mismatch.

learn:

The TFs are learned from the data along the activations as originally proposed [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF].

no echoes:

Reverberation is present but ignored (i.e. K = 0).

With the universal dictionary, the large number of latent variables warrants the introduction of sparsity-inducing regularization. The value of the regularization parameter γ was chosen by a grid search on a holdout set with the signal-to-distortion ratio (SDR) as the figure of merit [START_REF] Vincent | First stereo audio source separation evaluation campaign: data, algorithms and results[END_REF] (Table 1).

Dictionary Training, Test Set, and Implementation

Universal Dictionary: Following the methodology of [START_REF] Sun | Universal speech models for speaker independent single channel source separation[END_REF] we select 25 male and 25 female speakers and use all available training sentences to form the universal dictionary

D = [D M 1 • • • D M 25 D F 1 • • • D F 25 ]
. The test signals were selected from speakers and utterances outside the training set. The number of latent variables per speaker is 10 so that with STFT frame size of 2048 we have D ∈ R 1025×500 .

Speaker-Specific Dictionary: Two dictionaries were trained on one male and one female speaker. One utterance per speaker was excluded to be used for testing. The number of latent variables per speaker was set to 20. All dictionaries were trained on samples from the TIMIT corpus [START_REF] Garofolo | TIMIT acousticphonetic continuous speech corpus[END_REF] using the NMF solver in scikit-learn Python package [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF].

Implementation: Authors of [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF] provide a Matlab implementation of MU-NMF and EM-NMF methods for stereo separation. We ported their code to Python and extended it to arbitrary number of input channels. 1 The number of iterations for MU-NMF (EM-NMF) was set to 200 (300) and simulated annealing in EM-NMF implementation was disabled.

Results

We evaluate the performance in terms of signal-to-distortion ratio (SDR) and source-to-interference ratio (SIR) as defined in [START_REF] Vincent | First stereo audio source separation evaluation campaign: data, algorithms and results[END_REF]. We compute these metrics using the mir eval toolbox [START_REF] Raffel | mir eval: A transparent implementation of common MIR metrics[END_REF].

The distributions of SDR and SIR for separation using MU-NMF and a universal dictionary are shown in Figure 3a, with a summary in Figure 4. We use the median performance to compare the results from different algorithms. First, we confirm that separation fails for flat TFs (anechoic and K = 0) with SIR at around 0 dB. Learning the TFs performs somewhat better in terms of SIR than in terms of SDR, though both are low. Introducing approximate TFs dramatically improves performance: the proposed approach outper- 1 Our implementation and all experimental code are publicly available in line with the philosophy of reproducible research.

forms the learned approach even with a single echo. With up to six echoes, gains are +2 dB SDR and +5 dB SIR. Interestingly, with more than one echo, 1 regularization becomes unnecessary; nonnegativity and echo priors are sufficient for separation.

Separation with speaker-dependent dictionaries is less challenging since we have a stronger prior. Accordingly, as shown in Figures 3b and 4, MU-NMF now achieves a certain degree of separation even without the channel information. The gains from using echoes are smaller, though one echo is still sufficient to match the median performance of learned TFs. Using an echo, however, results in a smaller variance. Adding more echoes further improves SDR (SIR) by up to +2 dB (+3 dB).

In the same scenario, EM-NMF (Figure 3c) has near-perfect performance on anechoic signals which is expected as the problem is overdetermined. For MU, a single echo suffices to reach the performance of learned TFs and further improve it. Moreover, echoes significantly improve separation quality as illustrated by up to 3 dB improvement over learn. It is interesting to note that in all experiments the first three echoes near-saturate the metrics. This is good news since higher order echoes are hard to estimate.

CONCLUSION

In this paper we began studying the role of early echoes in sound source separation-a challenging task in computational auditory scene analysis. We found that a simple echo model not only improves performance, but it also enables separation in conditions where it is otherwise not possible. One such example is separation with non-negative speaker-independent models. Echoes seem to play an essential role in magnitude-only algorithms like nonnegative matrix factorization via multiplicative updates. They improve separation as measured by the standard metrics even when compared to approaches that learn the transfer functions. We believe these results are only a first step in understanding the potential of echoes in computational auditory scene analysis. They suggest that simple models used in this paper could be used as regularizers in other common audio processing tasks. Ongoing work includes running real experiments, studying the underdetermined case, and blindly estimating the wall parameters.

Fig. 2 :

 2 Fig. 2: Left, a typical simulated RIR. Right, the simulated scenario.

Fig. 3 :Fig. 4 :

 34 Fig.3: Distribution of SDR and SIR for male and female speakers as a function of the number of echoes included in modeling, and comparison with the three baselines.

Table 1 :

 1 Value of the regularization parameter γ used with the universal dictionary.


Determining Ti-17 β-phase Single-Crystal Elasticity Constants through X-Ray Diffraction and inverse scale transition model

The scope of this work is the determination of single-crystals elastic constants (SEC) from X-ray diffraction lattice strains measurements performed on multi-phase polycrystals submitted to mechanical load through a bending device. An explicit three scales inverse selfconsistent model is developed in order to express the SEC of a cubic phase, embedded in a multiphase polycrystal, as a function of its X-ray Elasticity Constants. Finally, it is applied to a twophases (α+β) titanium based alloy (Ti-17), in order to estimate Ti-17 β-phase unknown SEC. The purpose of the present work is to account the proper microstructure of the material. In particular, the morphologic texture of Ti-17 α-phase, i.e. the relative disorientation of the needle-shaped grains constituting this phase, is considered owing to the so-called Generalized Self-Consistent model.

Introduction

The knowledge of Single-crystal Elasticity Constants (SEC) is important in the field of engineering applications of materials. Actually, it is necessary to predict the micro-mechanical behaviour of polycrystalline materials, such as their X-Ray Elasticity Constants (XEC). SEC characterization through the classical methods, such as analyzing the ultrasonic waves velocities [1], usually requires millimetre sized single-crystals. However, many materials are difficult to obtain in an appropriate single-crystal form. It is the case of numerous multi-phases alloys such as (α+β) titanium alloys. In these circumstances, the traditional SEC determination methods are no more suitable. Owing to X-ray diffraction (XRD) techniques, SEC can be evaluated from the mechanical macroscopic stiffness and XEC measured on polycrystals. The required procedure was historically introduced for single-phase polycrystals [2]: experimental XEC are fitted using a multiscale approach, such as Kröner's model [3]. The method was firstly applied to cubic polycrystals [4] and then, extended to hexagonal single phases [5]. The last reference features a geometric sets average mathematical framework instead of the traditional arithmetic mean, classically used for achieving multiscale modelling. Discussions about the consequences of such a choice can be read in [6]. In the case that a multiphase polycrystal is characterized, the interphase elastic interactions which occur should properly be taken into account for an accurate estimation of the SEC from the experimental study of such materials [7,8]. A method has recently been proposed to deal with the question of SEC determination from measurements performed on multiphase materials [9]. In the same reference, an application to the determination of Ti-17 β-phase SEC from lattice strains measurements performed on both phases of an (α+β) Ti-17 specimen submitted to controlled applied mechanical loads, in-situ in a diffractometer, was provided. The present work is focused on the question of accounting a more realistic morphologic microstructure. Actually, Ti-17 β-phase is constituted by needle-shaped grains, randomly oriented in the reference frame associated to the sample. The historical numerical treatment of the experimental data provided in [9] was achieved owing to the traditional Eshelby [10] and Kröner [3] Self-Consistent (EKSC) elastic model (chosen due to the satisfactory results obtained during previous investigations [11][12][13][14][15]). When EKSC model can handle a single grain-morphology, shared by every inclusion constituting the Representative Equivalent Volume (REV) of the material, this theoretical approach fails to represent a morphologic texture featuring various grain shapes and/or a relative disorientation of the morphologies coexisting among the same REV. It was actually numerically shown in [16], that in such a situation, the classical EKSC model does not simultaneously fulfil both the so-called "Hill's average relations over the mechanical states", historically established in [17], anymore. The first aim of the paper is thus to develop an inverse scale-transition model enabling to by-pass this problem. In a second part, this new approach will be applied to the practical determination of Ti-17 β-phase SEC from lattice strains measurements performed on both phases of a specimen submitted to controlled applied mechanical loads in-situ in a diffractometer. The results obtained through either the new or the traditional approaches, respectively enabling to account the morphological texture of the material, or neglecting this very microstructural feature, will be compared in this second section of the paper.

On the identification of materials properties through scale-transition models

Introduction. Scale transition models have a long history and rich literature. Among their applications, their ability to predict the effective behavior of macroscopically homogeneous but microscopically heterogeneous materials is often valued. Numerous papers have actually dealt with the question of estimating the effective stiffness of single-phase [18] and two-phases [8] metallic bulk polycrystals, thin solid films [19], or even metal / organic matrix composites [20], for instance. Scale transition models are also often used for calculating the XEC [8,21] required for achieving internal stress determination from XRD peak-shifting measurements. Another centre of interest of scale transition models is their ability to provide links between local fields (such as the mechanical states, either strains or stresses, as an example), and their macroscopic counterparts. This information is valuable within multi-physical phenomena such as mechanical states enhanced diffusion [12,22]. The third driving force behind the extensive use of scale-transition models is for identifying materials properties. The question of determining the multi-scale properties of some constituents of heterogeneous materials has been extensively addressed in the field of materials science, especially for studying polycrystals (like titanium alloys [9,11,15], metal matrix composites [13,14] and iron oxides from the inner core of the Earth [5], for instance).

Accounting a morphologic texture through "self-consistent" scale transition models. It was shown by many authors, that the application of traditional scale transition models such as Mori-Tanaka estimates or EKSC model to the case of multi-phase materials containing heterogeneities of different shapes or alignments was questionable, since several fundamental relations are not satisfied anymore [23][24][25][26]. In order to overcome the theoretical difficulties related to accounting various grain morphologies within the same RVE, Li proposed to add an intermediate level (denoted by the superscript a ) between the traditional mesoscopic ( i ) and macroscopic ( I ) scales, as displayed on figure 1 [27]. The resulting scale-transition relations respectively write:

a i i T ε ε ε ε ε ε ε ε : =
, and

I 1 i a T ε ε ε ε ε ε ε ε : - = , (1)
where ε ε ε ε stands for the strains, whereas T denotes the intermediate to mesoscopic strain localization tensor. The square brackets < > represent the volume average. Within the context of Eshelby-Kröner model, T satisfies

( ) [ ] 1 I i Ii i I C c E T - + - = :
, where c i and C I are respectively the mesoscopic and the macroscopic stiffness tensors. I represents the fourth order identity tensor, and Ii E is the socalled Morris tensor, which expresses the interaction between an inclusion with a given morphology and the Homogeneous Equivalent Medium (HEM) [28]. Furthermore, specific attention must be paid to the fact that the calculation of the Morris tensor must be made with the stiffness tensor C I dropped in the coordinate system Ri, attached to the main axes of the ellipsoidal inclusion. According to Eq. 1, the mesoscopic to macroscopic scale transition relation becomes:

I 1 i i i T T ε ε ε ε ε ε ε ε : : - = (2)
As a consequence, the strain localization tensor A i , traditionally enabling to express the mesoscopic strains from their macroscopic counterparts, is defined by

1 i i i T T A - = :
. The homogenization relation enabling to find the macroscopic stiffness C I from the mesoscopic stiffnesses c i satisfies:

1 i i i I T T c C - = : : (3)
Eq. 1 to Eq. ) and iii) XEC of the analyzed phase for a single {hkl} reflection.

In need of the GSC model: Ti-17 morphologic texture. Ti-17 is an (α+β) titanium alloy containing volume fractions f α and f β of the phases, being equal to f β = 30 % and f α = 70 %, respectively. According to the micrographs published in [9], the structure of the studied Ti-17 polycrystal consists of acicular α (needle-shaped crystallites) mixed to slightly equiaxed prior β grains, as shown on figure 2. According to figure 2, the morphologic microstructure exhibited by Ti- 17 α-phase cannot be properly taken into account through the traditional EKSC model. The GSC model, however, was specifically built in order to model such a material. Macroscopic elasticity constants of Τ Τ Τ Τi-17. In order to determine the XEC required for SEC calculation according to [9] Ti-17 macroscopic and pseudo-macroscopic elastic constants are necessary. The macroscopic characterization has been described in [11] (see Table 1), whereas determining Ti-17 β-phase average properties requires the use of an inverse multiscale model. Identification of Ti-17 average (pseudo-macroscopic properties) through inverse GSC model. Ti-17 β-phase crystallites can be represented by spherical grains. Thus, only the relative disorientation between α-Ti needles should be represented at the mesoscopic scale. The average pseudo-macroscopic behaviour C β of Ti-17 β-phase results from an inversion of Eq. 3:

Macroscopic scale I

( ) ( )

1 i 1 i I T T c T T C C - - - = β β β β α α α α α α α α α α α α β β β β β β β β α α α α β β β β , m c m c β α , β : θ , θ : θ , θ f f : : f 1 ( 4
)
Where

{ }

the disorientation of the single-crystal crystalline structure / morphologic misalignment of the needle-shaped grains, with respect to the sample reference frame. The lengths of the principal axis of the ellipsoid representing the needles were taken equal to a 1 = 100, a 2 = 1 and a 3 = 1 (arbitrary units). 20,000 α-Ti17 inclusions with random values of c θ and m θ were considered for constituting the REV of the material. The resulting average properties of β-Ti17, deduced from Eq. 4 are displayed in Table 1. According to Table 1, accounting the morphologic texture of Ti-17 α-phase results in a slight softening of the predicted elastic behavior of the second phase.

Ti-17 β

β β β-phase XEC calculation. X-Ray analysis of {310} β and {21.3} α lattice strains on a Ti-17 polycrystal (1 mm thick plate) submitted, in-situ, to applied stresses through a cylindrical bending device which radius is R=99.5 mm, was achieved in [9]. The longitudinal macroscopic stress yields biaxial pseudo-macroscopic stresses, the components of which are given in Table 2: accounting the morphologic texture of Ti-17 α-phase affects the stresses experienced by the phases. The macroscopic applied stress being uniaxial along longitudinal axis x 1 , the resulting measured strain in α-phase for a pseudo-macroscopically isotropic sample, is given in [7]:

      + +       + +       - - + = α,R α,A σ σ Tr Tr S σ σ S 2 1 ψ sin σ σ σ σ S 2 1 ε α 1 R α, 33 A α, 33 α 2 2 R α, 33 A α, 33 R α, 11 A α, 11 α 2 α ψ (5)
Where the exponents A and R respectively denote the Applied and Residual stresses in the considered α-phase (a similar equation is obtained for β-phase). The measurement without applied stress yields the strains due to residual stresses only. That condition enables XEC determination from the subtraction of the lattice strains measured under and without applied load:

( ) ( ) ∞ → - = = R α ψ ε 99.5 R α ψ ε α ψ ∆ε = s α .sin 2 ψ+i α =       +       +       - α,A σ Tr α 1 S A α, 33 σ α 2 S 2 1 ψ 2 sin A α, 33 σ A α, 11 σ α 2 S 2 1 (6)
The XEC deduced from the experimental results published in [9], through the pseudo-macroscopic stresses listed in table 2 are: ½ S 2 {21.3} α = 12.05 10 -6 /MPa; ½S 2 {310} β = 14.97 10 -6 /MPa, according to the GSC model, whereas the traditional EKSC model gives : ½ S 2 {21.3} α = 11.72 10 -6 /MPa; ½ S 2 {310} β = 14.56 10 -6 /MPa. Ti-17 β-phase SEC calculation and discussion. The results obtained for Ti-17 β-phase XEC was reported in the analytical relations already presented in [9], accounting for the macroscopic and pseudo-macroscopic elastic constants listed in Table 1. Numerical computations yield the values given in Table 3, where Ti-17 β-phase SEC obtained through either the GSC model or the EKSC model, have been compared to Ti-40Nb (100 % β) mesoscopic elastic properties. The deviation observed in the SEC of the two materials is consistent with the measurements of β-Ti macroscopic elasticity constants in various (100 % β) alloys [31,32]. Contrary to α-phase elastic properties, which remain almost constant in two-phases Titanium based alloys, due to the small concentration of alloying elements in this phase, β-Ti SEC and its related pseudo-macroscopic elasticity constants vary from one alloy to the other. This is due to the occurrence of very strong variations in the nature and concentration of β-stabilizers elements. Nevertheless, accounting the morphologic texture of Ti-17 α-phase through the GSC model yields slight discrepancies from the SEC previously determined using the classical EKSC model.

Conclusion

Inverse scale transition models, involving explicit simplified forms, were used in order to handle the question of determining the unknown single-crystal elasticity constants of Ti-17 β-phase embedded in an (α+β) Ti-17 alloy. The present work features, for the first time in the context of SEC identification from X-Ray diffraction analyses performed on polycrystals, the so-called Generalized Self-Consistent model. Contrary to the classical Eshelby-Kröner Self-Consistent model, the recently proposed generalized version of the model enables accounting a morphologic texture within the Representative Elementary Volume. A more realistic morphological texture for both Ti-17 phases was taken into account, owing to the generalized self-consistent model: i) Ti-17 β-phase was assumed to be constituted by an assembly of spherical grains, whereas ii) needle-shaped crystallites, randomly oriented in the coordinate system associated to the specimen, were considered for Ti-17 αphase. A comparison between the numerical results, obtained at each step of the experimental data treatment method, through either the classical EKSC model or the GSC model was provided. Both scale-transition models yield to very close values for either the pseudo-macroscopic properties of Ti-17 β-phase, the concentration of the macroscopic stress state, imposed to the sample, at the pseudo-macroscopic scale, the X-Ray Elasticity constants ½ S 2 {21.3} α and ; ½ S 2 {310} β ) and the researched Ti-17 β-phase SEC. This study demonstrates, at least numerically, that a random morphologic texture can be neglected without much consequence, within the context of modelling the multi-scale behaviour of heterogeneous materials made of quasi-isotropic elementary inclusions.

  3 are the fundamental relations of the Generalized Self-Consistent (GSC) model. It should be highlighted, that this model corresponds to a "normalization" of the EKSC model, where actually T EKSC =A EKSC , since the intermediate medium and the macroscopic scale are merged.Characterization of Ti-17 β-phase SEC through GSC modelIntroduction. Explicit forms for the XEC of a cubic phase embedded in a macroscopically quasiisotropic multi-phase material were demonstrated from the traditional EKSC model in[START_REF] Fréour | ! ! ! ![END_REF]. The model can be inverted in order to express the SEC of the diffracting cubic phase whose mesoscopic stiffness tensor components ( relations already presented in[9]. Application of those analytical relations requires the knowledge of: i) the macroscopic elasticity constants of the multi-phase material (

Figure 1 :

 1 Figure 1: Introducing an intermediate medium (a) between the macroscopic scale (I) and the mesoscopic scale (i), for modelling multi-phase polycrystals presenting a morphologic texture.

Figure 2 :

 2 Figure 2: The morphologic texture of Ti-17 twophase polycrystal featuring disoriented needle-shaped α-Ti crystallites and equiaxed β-Ti grains.

2 :

 2 Predicted pseudo-macroscopic stresses due to the applied macroscopic stresses in the bended Ti-17 plate.Comparison between the GSC model and the classical EKSC model.

Table 1 :

 1 macroscopic and β-phase pseudo-macroscopic elastic properties of Ti-17. Comparison between the values predicted by the GSC model and EKSC model. Y and ν respectively stand for Young's modulus and Poisson's ratio.

		C 11 [GPa]	C 12 [GPa]	Y [GPa]	ν
	Ti-17 [9]	169.4	87.4	110.0	0.340
	Ti-17 β-phase (GSC model -present work)	180.3	108.5	98,8	0.376
	Ti-17 β-phase (EKSC model -[9])	183.0	111.0	99.3	0.377

Table 3 :

 3 SEC and anisotropy elastic coefficient A c of Ti-17 β-phase. Comparison to Ti-40Nb (100 % β) SEC.

		c 11 [GPa]	c 12 [GPa]	c 44 [GPa]	Ac
	Ti-17 (β) GSC model	167	115	44	1.7
	Ti-17 (β) EKSC model [9]	174	116	41	1.4
	Ti-40Nb (β) [30]	156.5	111.6	39.6	1.76