Stéphanie Ramboarina

James A Garnett

Meixian Zhou

Yuebin Li

Zhixiang Peng

Jonathan D Taylor

Wei-Chao Lee

Andrew Bodey

James W Murray

Yilmaz Alguel

Julien Bergeron

Benjamin Bardiaux

Elizabeth Sawyer

Rivka Isaacson

Camille Tagliaferri

Ernesto Cota

Michael Nilges

Peter Simpson

Teresa Ruiz

Hui Wu

Stephen Matthews
email: s.j.matthews@imperial.ac.uk

STRUCTURAL INSIGHTS INTO SERINE-RICH FIMBRIAE FROM GRAM-POSITIVE BACTERIA

The serine-rich repeat (SRR) family of fimbriae play important roles in the pathogenesis of streptococci and staphylococci. Despite recent attention, their finer structural details and precise adhesion mechanisms have yet to be determined. Fap1 (Fimbriae associated protein 1) is the major structural subunit of SRR fimbriae from Streptococcus parasanguinis, and plays an essential role in fimbrial biogenesis, adhesion and the early stages of dental plaque formation. Combining multidisciplinary, high resolution structural studies with biological assays we provide new structural insight into adhesion by Fap1. We propose a model in which the serinerich repeats of Fap1 subunits form an extended structure that projects the Nterminal globular domains away from the bacterial surface for adhesion to the salivary pellicle. We also uncover a novel pH-dependent conformational change that modulates adhesion and likely plays a role in survival in acidic environments.

Biofilms offer protection to bacteria from hostile environments, such as adverse temperatures, pH and attack by other organisms including their hosts (1). Significant progress has been made toward understanding which gene products are involved in biofilm formation. Fimbriae, which are long, thin filamentous structures that decorate bacterial surfaces, are often employed in the reversible adhesion to surfaces at the early stages of biofilm formation (2). A large number of chronic bacterial infections involve the formation of a biofilm, with the development of dental plaque being one of the most familiar and diverse [START_REF] Costerton | Bacterial Biofilms: A Common Cause of Persistent Infections[END_REF].

The commensal streptococci of the oral cavity play a major role in the foundation of dental plaque (4). These include early colonizers of the tooth surface, viridans Streptococcus sanguinis and Streptococcus parasanguinis, which serve as substratum for the subsequent adhesion of other microflora, such as cariogenic Streptococcus mutans, and periodontal pathogens Porphyromonas gingivalis and

Aggregatibacter actinomycetcomitans (5). In addition, both S. sanguinis and S. parasanguinis have been described as etiological agents in infective endocarditis (6)(7)(8)(9)(10) with the formation of streptococcal biofilms being central to the development of disease. Fap1 (Fimbriae associated protein 1) has been identified as the major structural subunit of S. parasanguinis fimbriae and is essential for fimbrial biogenesis, adhesion and biofilm formation (11)(12)(13)(14)(15). Fap1-like subunits assemble into a new family of fimbriae in Gram-positive bacteria (16), which are characterised by large polypeptides (>200 kDa) comprising an extensive region of glycosylated, serine-rich repeats (SRR) flanked by a unique Nterminal region and a C-terminal cell wallanchoring domain. The SRR family of fimbriae play direct roles in several bacterial diseases; including the pathogenesis of streptococcal (17) and staphylococcal infective endocarditis (18), streptococcal meningitis (19), invasive pneumococcal disease (20) and neonatal disease (21)(22).

Fap1 from S. parasanguinis provides a model system to study bacterial adhesion and biofilm formation by this poorly understood family of proteins. Furthermore, these adhesins play a major role in early stages of dental plaque formation. Despite recent attention, the finer structural details of SRR fimbriae and their adhesion mechanisms have yet to be determined. Combining multidisciplinary, high resolution structural studies with biological assays we provide new insights into adhesion by Fap1. We propose a model in which the serine-rich repeats of Fap1 subunits form an extended structure that projects the N-terminal globular domains away from the bacterial surface for adhesion to the salivary pellicle. We also uncover a novel pH-dependent conformational change that modulates adhesion and likely plays a role in survival in acidic environments.

Experimental procedures

Protein preparation-Recombinant Fap1-NR (residues 106-437), Fap1-NR α (116-231) and Fap1-NR β (231-437) were expressed using pRSETA plasmid (Promega) in the BL21(DE3) E. coli strain (Novagen). Unlabelled proteins were produced in LB media whilst isotopically 15 N and 13 Clabeled proteins were produced in M9 minimal medium supplemented with 15 NH 4 Cl and 13 C-glucose. A 2 H/ 15 N/ 13 C triple labelled Fap1-NR sample was produced from Silantes media in E. coli CDN OD1. Se-Met labelled Fap1-NR protein was expressed in methionine auxotroph B834 grown in minimal media supplemented with selenomethionine. All samples were purified using nickel affinity chromatography followed by size exclusion chromatography on a Superdex-75 column (GE healthcare).

NMR samples were concentrated to 0.1-0.2 mM in 50 mM PBS, pH 8, 50 mM NaCl. X-ray diffraction samples were concentrated to ~8 mg/ml and stored at -72°C in 20 mM Tris-HCl pH 7.5, 100 mM KCl. SAXS samples were gel filtered in 50 mM PBS pH 5 or pH 8, 50 mM NaCl immediately prior to data collection.

NMR spectroscopy-NMR measurements were performed at 303K on [ 15 N, 13 C] labelled Fap1-NR  and Fap1-NR  domain samples in 50 mM PBS buffer, pH 8.0, 50 mM NaCl. NMR experiments for backbone and side-chain assignment on Fap1-NR α were performed on two different Bruker spectrometers, a DRX500 and an Avance II 800 equipped with cryoprobe. Assignments were completed using standard tripleresonance assignment methodology (23). NOE data for structure calculations of Fap1-NR α were obtained from an 800 MHz edited 1 H- 15 N nuclear Overhauser effect spectroscopy-heteronuclear single quantum coherence (NOESY-HSQC) and a 950 MHz edited 1 H-13 C NOESY-HSQC experiments with a mixing time of 100 ms recorded on a 950 MHz Oxford superconducting magnet. 15 N T 1 and T 2 relaxation times on 15 Nlabelled Fap1-NR α sample at pH 8 were recorded at 800 MHz and T 1 /T 2 ratio measured for each residue.

1 H- 15 N and 1 H α -C α residual dipolar coupling constants were measured using 2D in-phase anti-phase (IPAP)-[ 15 N-1 H] HSQC (24) and 3D HN(CO)CA experiments respectively on [ 15 N, 13 C] labelled Fap1-NR  sample at pH 8 partially aligned in Pf1 filamentous phage media. 15 N-1 H TROSY-HSQC spectra of [ 15 N-2 H] Fap1-NR were recorded at pH 5, 6 and 8 at 800 MHz. A series of 15 N-1 H HSQC on Fap1-NR α at different pHs from pH 5 to pH 8 were performed at 500 MHz and 950 MHz. 15 N-1 H HSQC on Fap1-NR  domain was performed at 800 MHz.

1 H- 15 N TROSY NMR experiments ( 25) were performed at 303K on an uniformly [ 15 N, 13 C, 2 H] labelled Fap1-NR on a Bruker Avance II 800 equipped with a cryoprobe. TROSY versions of HNCACB, HN(CO)CACB experiments for backbone assignment allowed to assign ~50% assignment of Fap1-NR spectra. All NMR data were processed using NMRpipe (26) and analyzed with NMRView (27).

NMR structure determination-The ARIA protocol for automated NOESY assignment interfaced with the CNS program was used for structure calculation (28)(29)(30). Secondary helical structure in Fap1-NR  was first identified using the chemical shift-based dihedral angle prediction software TALOS (31). For residues located in helices, hydrogen bonds,  and  backbone dihedral angles derived from TALOS and RDCs were introduced as restraints in the ARIA structure calculation. 38 D NH and 54 D CH RDCs spread evenly among the three helices were used for structure calculation. Axial Da and rhombic R components were initially estimated using the histogram method from the normalized distribution of these 92 RDCs (32). The axial (Da = -3.0 Hz) and rhombic (R = 0.55) components of the alignment tensor were further optimized with the program MODULE (33) based on the precalculated structures. For NOE derived and hydrogen bonds distances restraints, a log-harmonic potential was used for the second Cartesian cooling phase of the simulated annealing and the weight on the restraints was iteratively updated (34). The final average weights were 6.8 kcal/mol/Å 2 for NOE restraints and 25 kcal/mol/Å 2 for hydrogen bonds restraints. To avoid distortion of the peptide group during Cartesian cooling, high force constants were used to maintain planarity, while low force constants were employed for RDC restraints. Furthermore, the final structures were carefully inspected for local distortions. 100 conformers were calculated in the final iteration and 10 lowest energy structures were refined in a shell of water molecules (35). A summary of NMR-derived restraints and statistics on the ensemble of NMR structure is reported in Table 2. The solution structure of the Fap1-NR  domain was deposited in the Protein Data Bank with PDB ID code 2kub.

Crystallisation, data collection and structure determination-Crystals of both native and Se-Met labelled Fap1-NR were grown at 297K using the hanging drop vapour diffusion method in 0.1 M Hepes over a range of pH from pH 6.5 to 8.0, 5% MPD, 10-12.5% PEG 6K and 9-12.5 mM spermine-tetra HCl. Crystals were briefly soaked in the mother liquor supplemented with 20% PEG 6K before cryocooling. SeMet labelled Fap1-NR data were collected at the selenium peak energy, determined by a fluorescence scan, at beamline ID29 at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. Data were processed using MOSFLM and scaled with SCALA (36). Two selenium sites were located using SHELXD (37), and then phases were calculated using AUTO-SHARP. Two monomers were located in the electron density, and a model was built into the averaged solvent flattened experimental map using COOT (38), with rounds of refinement in REFMAC5 (39). The final model was refined with NCS restraints and TLS groups, with 5% of the reflections omitted for cross-validation. Statistics of data collection and the final model are shown in Table 1. The coordinates and structure factors have been deposited in the PDB under accession code 2x12.

Small angle X-ray scattering (SAXS)-SAXS data were recorded at the X33 beamline of EMBL (DESY, Hamburg). Samples of Fap1-NR buffered at either pH 5 or pH 8 and at 1.3, 2.7 and 4.5 mg/ml were measured using 120 s exposures to synchrotron radiation. The range of q values measured was from 0.07 to 0.48 Å -1 . SAXS data were processed in the standard manner using the program PRIMUS (40). Data points across all three concentrations were merged (points 1-300, 250-600, and 550-2000 were derived from the low, medium and high concentration samples, respectively). The radius of gyration, R g , and scattering at zero angle, I(0), were calculated from the analysis of the Guinier region by AUTORG. The distance distribution function, P(r), was subsequently obtained using GNOM, yielding the maximum particle dimension, D max . An ensemble of 20 low-resolution ab initio structures were calculated using the program GASBOR utilising default parameters (41). Models were then aligned and averaged by DAMAVER. The known atomic resolution structures of the α and β domains were docked into the low-resolution bead model of Fap1-NR at pH 8 by the program SITUS (42). This was achieved by splitting the bead model into two halves and independently minimising the fit between the high-and low-resolution structures.

Site-directed mutagenesis of Fap1 of S.

Parasanguinis-Surface regions containing an abundance of exposed hydrophobic amino acid side chains were selected for mutation.

The hydrophobic residues together with selected hydrophilic side chains were mutated individually to alanine using the Strategene quick change mutagenesis system as previously described (43). This included I134, E138, D142, L163 and V164 in Fap1-NR α together with L385, I291, L292, L300, N403, Q405 and I411 in Fap1-NR β. In brief, the full-length fap1 gene cloned into an E. coli and Streptococcal shuttle vector pVA838 was used as a template. Site-directed mutagenesis PCR was carried out to construct a series of alanine replacement mutants of fap1 using the mutagenic oligonucleotide primers that corresponded to the changes of selected amino acid residues to alanine residues. The amplified mutant plasmids were transformed into competent E. coli following the instructions by a QuickChange XL Mutagenesis kit (Strategene, La Jolla, CA). The plasmids with designed mutations were purified; the mutated region was sequenced to ensure the accuracy of the site-directed mutagenesis. The wild type and mutant plasmids were then transformed into a fap1 mutant of S. parasanguinis. Fap1 expression was monitored by Western blot analysis using an antibody against mature Fap1 (F51).

Adhesion of S. parasanguinis to SHA: an in vitro tooth model-Adhesion of S. parasanguinis

to saliva-coated hydroxylapatite (SHA) was performed as described previously (11). In brief, cells of wild type S. parasanguinis and site-directed mutants were labeled with 2 µCi of [methyl-3 H]thymidine/ml. The radiolabeled bacteria were incubated with SHA beads at 37°C for 1 hr to allow bacterial binding to occur and then washed. Supernatants and washed beads were transferred to scintillation vials, and their respective radioactivities were determined and used to calculate adhesion percentage as previously described. In all assays a wild-type strain FW213 and the fap1-minus mutant were used as positive and negative controls, respectively.

Adhesion blocking assays-Wild type S. parasanguinis cells were labeled with 2 µCi of [ 3 H]thymidine/ml. The radiolabeled cells were treated and reconstituted as previously described. Recombinant Fap1-NR, Fap1-NR α and Fap1-NR β were expressed and purified as described. The purified recombinant proteins were dialyzed overnight against the adhesion buffer and added into SHA beads which were at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from equilibrated with adhesion buffer and then incubated at 37°C for 1 h on a rotating shaker. The treated SHAs were incubated with labeled S. parasanguinis and subjected to the adhesion assay described above.

Electron microscopy-S. parasanguinis (FW213) was grown from frozen stocks preserved with 5% dimethylsulfoxide (44). Frozen cells were streaked onto blood agar plates, and incubated aerobically for 12-15 h in 5% CO2 at 37°C. Broth cultures were prepared by inoculation of single colonies from the blood agar plates into flasks containing Todd-Hewitt Broth (THB) and grown statically under aerobic conditions. 5 ml cell cultures in early exponential phase (A470~0.4) were centrifuged at 4000 g to remove the media. Cell pellets were resuspended in 100 µl PBS (100 mM Phosphate buffer pH 7.4, 150 mM NaCl) at 4°C.

Samples were diluted in PBS to the appropriate concentration for electron microscopy studies. A small aliquot (5µl) was applied to 400 mesh copper grids coated with a thin carbon film. The grids were washed by running several drops of PBS buffer over them and negatively stained by subsequently running a few drops of the staining solution (1% uranyl acetate (UA; Ted Pella, USA), 2% phosphotungstic acid, pH 7 with NaOH (PTA; Ted Pella, USA) and 2% methyl amine tungstate (MAT, Nanoprobes, USA) over the grids. The last drop of stain was left on the grid for 30 s; the excess liquid was wicked off, and the grids were fast air dried. The grids were observed using a dual-axis tomography holder (Fischione, USA) on a Tecnai 12 electron microscope (FEI, USA) equipped with an LaB6 cathode (Kimball, USA) operated in point mode, and a 14 μm 2048*2048 CCD camera (TVIPS, Germany). The microscope was run under identical conditions as have been used in the past to obtain images that show Thon rings beyond 0.9 nm resolution in vitreous ice preparations (45). Negatives were recorded at an accelerating voltage of 100 kV and nominal magnification of 52000X under low dose conditions on S0-163 Kodak film (~ 1000e-/nm2). Negatives were scanned with 7µm raster size on a Zeiss flat bed scanner (Intergraph, USA). The images were converted for processing with the SPIDER image processing system (46) and reduced two times to a final pixel size of 0.27 nm at the specimen level. A contrast transfer function correction was performed on the images by calculating the averaged periodogram of each image and fitting to it the theoretical transfer function calculated by varying the defocus value and astigmatism parameters (angle and strength) (47).

Each image was corrected by multiplying it with a modified version of its transfer function (47).

Straight filament regions, which showed no overlap were boxed with the program 'helixboxer' in the EMAN image processing package (48). Images containing 55 nm filament sections were windowed using the SPIDER image processing system (46). The unevenness of the background was removed using the 'ramp' command and the intensities were normalized to zero average (5 pixels strips along the longitudinal edge of the images were used to calculate the average intensity value of the background). A power spectrum was calculated from each image and an averaged power spectrum was calculated for each stain condition. The averaged power spectra were visualized using the WEB display program (46) and the height of the layer lines was extracted.

RESULTS

Major non-repeat region of Fap1 blocks S. parasanguinis adhesion to SHA. Recently, SRR fimbrial proteins known to interact with salivary components and human platelets have also been identified in other medicallyrelevant streptococcal and staphylococcal species (49). Interestingly, the sequences of regions outside the serine-repeats are implicated in adhesion and poorly conserved amongst the family (50)(51)(52)(53). To define the minimal region responsible for Fap1 adhesion to the salivary pellicle, the non-repeat portion from the N-terminus of Fap1 (Fap1-NR, residues 106-437; Fig. 1A) was recombinantly produced in E. coli, purified using metal affinity chromatography and tested for its ability to inhibit S. parasanguinis adhesion to an in vitro tooth model, saliva-coated hydroxyapatite (SHA). Binding assays revealed a dose-dependent decrease in binding in the presence of recombinant Fap1-NR indicating that this region of Fap1 is able to competitively block adhesion (Fig. 1B).

Limited trypsin proteolysis of Fap1-NR released two stable domains corresponding to a C-terminal 25 kDa polypeptide and a smaller 10 kDa N-terminal fragment, confirmed by mass spectrometry (data not shown). To investigate the contribution of these sub-domains to adhesion, we produced Fap1-NR α and Fap1-NR β separately in E. coli and tested their ability to block S. parasanguinis binding to SHA (Fig. 1B). Each domain shows a dose dependent reduction in S. parasanguinis adhesion suggesting that they both play important roles in adhesion.

Structure of the major non-repeat region

Fap1-NR. To explore the potential cooperative roles of Fap1-NR α and Fap1-NR β , and provide a better understanding of adhesion by Fap1, we embarked on a highresolution structural investigation. We initiated our structural studies using the intact 36 kDa Fap1-NR. Crystals of Fap1-NR were obtained after one day, between pH 6.5 and pH 8.5, with the highest resolution diffraction obtained for crystals grown at pH 8. Structure determination was undertaken using Se-SAD and the electron density map was refined to 2.9 Å (Table 1 and Supplementary Fig. S1). The -domain was built into the experimental electron density map from residue T236 to D435, while the electron density corresponding to all preceding residues was not observable. Crystals were washed and analysed by SDS-PAGE, which showed no sign of degradation (data not shown), therefore it was postulated that the lack of electron density for this portion of Fap1-NR results from significant disorder in the N-terminal region, which encompasses Fap1-NR α . The final model contains two identical molecules of the domain in a head to tail arrangement within the asymmetric unit. The dimer interface is small, comprising only the short loop connecting 10 and 11 in the monomer structure (Fig. 2A), which implies that dimer formation is most likely an artefact of crystal packing. Gel filtration experiments also indicate that, under these conditions Fap1-NR is largely monomeric in solution (Supplementary Fig. S2). The -domain is composed of 12 -strands arranged in two extensive -sheets built from four antiparallel -strands (Figs. 2A,B). Four additional, short -strands elaborate one face of the structure, creating an extended twisted -platform. A striking feature is the presence of large loops protruding at the top (L1-L2-L5) and the bottom (L3-L6-L9) of the sandwich, and obscuring the face of the extended -sheet (L4-L7-L8; Figs. 2A,B). These loops are well-ordered and are stabilized through hydrophobic interactions with exposed aromatic residues from the sheet core structure. A series of hydrophobic residues within L1-L2-L5 form an exposed ledge on the top of the structure (Supplementary Fig. S3).

The absence of electron density for the N-terminus of Fap1-NR α may be due to lack of structural integrity or the presence of multiple, but ordered, conformations, arising from flexibility within the 27 amino acid linker. To elucidate the structure of this domain we used solution-state nuclear magnetic resonance (NMR) spectroscopy. 2D 1 H- 15 N HSQC spectra displays excellent dispersion in backbone amide chemical shifts at pH 8, consistent with a folded polypeptide (Supplementary Fig. S4). The 15 N relaxation rates at pH 8 for backbone amides between E128 and L208 reveal a well ordered structure with no flexible loops or residues in chemical exchange, while the termini (residues 116-127 and 209-231) are highly flexible. Furthermore, NMR spectra indicate that Fap1-NR α is monomeric at at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from concentrations up to 300 μM, but selfassociates at higher concentrations (Supplementary Fig. S5).

Using a combination of manual and ARIA NMR assignment methods for Fap1-NR α (28-29), a total of 1586 nuclear Overhauser effects (NOEs) were assigned in Fap1-NR α 15 N/ 13 C-edited NOESY spectra at pH 8. The structure determination was supplemented with  and  dihedral angles and 92 residual dipolar coupling restraints (Table 2). All areas of secondary structure are well defined (Fig. 2C); the average pairwise root mean squared deviation (RMSD) for the water-refined final structures is 0.39 ± 0.06 Å for the backbone atoms and 0.81 ± 0.07 Å for the heavy atoms of residues within secondary structure. The final structure of the Fap1-NR α exhibits an αhelical bundle topology (Fig. 2C), in which the three α-helices pack against each other in an up-down-up arrangement (1 residues 129 to 150, 2 residues 153 to 178 and 3 residues 184 to 206). Hydrophobic interactions involving the side-chains of leucine, alanine and isoleucine residues stabilize the inter-helix interactions (Fig. 2D). The comparison of the 15 N-1 H TROSY HSQC of Fap1-NR with spectra of both isolated Fap1-NR  and Fap1-NR  indicates that while the majority of peaks for each isolated domain superimpose well with the corresponding peaks in Fap1-NR and a limited number of peaks exhibit chemical shifts differences, indicating the presence of a small interface between the domains (Supplementary Fig. S4). Furthermore, differences in 15 N relaxation times for α and β domains within Fap1-NR indicate some interdomain flexibility (Average T 1 /T 2 ratios are 21 and 27 for α and β domains respectively; Supplementary Figure S5).

pH-dependent conformation change within

Fap1. The oral cavity experiences a wide pH range, from neutrality at the normal saliva buffering conditions to values below 5.0, induced by the fermentation of ingested sugars by resident bacteria. To investigate the effect of pH on S. parasanguinis adhesion, SHA-binding assays were repeated over a range of pH values. A progressive increase in adhesion was observed as the pH was lowered from alkaline (pH 8) to acidic (pH 5) conditions (Fig. 3A), whereas no change in basal adhesion for the fap1 mutant strain was observed (Supplementary Fig. S6). To characterise this further 2D 1 H- 15 N TROSY-HSQC spectra of Fap1-NR were recorded over the same pH range (Supplementary Fig. S7).

Intriguingly, strong pH dependences were observed for many 1 H and 15 N chemical shifts of residues located within the α-domain, while the majority of the residues from the -domain were not affected (Supplementary Data S7). CD experiments performed between pH 5 and pH 8 indicate that the α-helical content remains unchanged over the pH range (data not shown). This was confirmed by repeating the pH titration and monitoring 1 H and 15 N chemical shifts of amides within the isolated Fap1-NR  domain (Fig. 3B). After reassignment of the NMR spectra at pH 5, an analysis of backbone secondary shifts indicates that the three α-helices remain largely intact, consistent with CD measurements, and that no unfolding occurs between the pH 8 and pH 5. Amide chemical shift changes are localised to one end of the α-helical bundle (Fig. 3C,D). Although no histadine residues are present, the Fap1-NR  domain possesses eighteen charged residues (twelve acidic and six basic; Supplementary Data S8). Interestingly, the amide resonances for three of these residues (R146, D152, E154 and K207) titrate with pH.

Furthermore, chemical shift changes of several hydrophobic residues (L148, L151, L157, A165, I206 and L208) proximal to charged groups are also observed. (Supplementary Data S8). Decreasing the pH may cause a rearrangement of the relative orientation and/or dynamics of the α-helices.

Small angle X-ray scattering (SAXS) provides useful low resolution information on the global structural features of proteins in solution. This technique is sensitive to changes in relative position of domains or positions of secondary structure and is therefore highly suitable for determining the overall structure of Fap1-NR under acidic and alkaline conditions. After exhaustive gel filtration chromatography of recombinant Fap1-NR at pH 8 and pH 5, SAXS measurements were recorded (Figs. 4A and4B; Table 3). The resulting SAXS density indicates that Fap1-NR exists as a curved, extended structure comprising two domains (Fig. 4B). The relative sizes of Fap1-NR α and Fap1-NR β domains are consistent with two halves of the density. Comparison of the data at pH 8 and pH 5 reveals a subtle difference in conformation, as judged by a change in the shape of the pair-distance distribution function (Fig. 4A). Upon reducing the pH the pair-distance distribution shifts to longer distances, for which a plausible interpretation would be an opening of the two domain structure (Fig. 4C). It is also likely that the extent of interdomain mobility is affected by pH and would contribute to an altered SAXS profile. It is interesting to note that the residues affected by the change in pH within the αdomain would be situated adjacent to the 27 residue linker between Fap1-NR α and Fap1-NR β (Fig. 4D).

Analysis of the atomic resolution details of Fap1-NR reveals two patches on the surfaces of the α-and β-domains that comprise a number of exposed hydrophobic side chains (Fig. 4D).

The exposed hydrophobic residues together with selected adjacent residues within these regions were selected for mutation; including 5 amino acids in Fap1-NR α (I134, E138, D142, L163 and V164) and 7 amino acids located within the hydrophobic patch within L1-L2 and L5 in Fap1-NR β (L385, I291, L292, L300, N403, Q405 and I411) (Fig. 2A,B; Fig 4D and Fig. S3). We hypothesized that these residues may form part of a binding site for the salivary pellicle and created a series of site-specific Fap1 mutants in this region within S. parasanguinis and measured their ability to adhere to SHA (Fig. 4E). As a control a charge mutation at Fap1-E204A on the opposite face of the α-domain was also constructed and tested. Western blot analysis revealed that all the mutants expressed Fap1 (data not shown), however all exhibited deficiencies in bacterial adhesion with the exception E204A.

Overall structure of Fap1. We analyzed the feasibility of imaging the fimbriae on the bacterial surface of the wild-type strain (FW213) for structure determination by single particle electron microscopy using several negatively stained preparations. Good results were obtained with preparations in 1% uranyl acetate (UA), 2% phosphotunstic acid, pH 7 (PTA) and 2% methyl amine tungstate (MAT) (Fig. 5A). The fimbriae are very flexible structures that bend easily in all the preparations. A close look at individual fimbriae reveals a diameter of ~5.5 nm and an internal substructure with a periodicity of ~6.5 nm. At neutral pH, fimbriae are close together forming rafts and the internal substructure is more apparent (Fig. 5A, open white arrows). At low pH (UA staining), the tip of the fimbriae come together to form aggregates (Fig. 5A, white arrows), in which the adhesion points are located uniquely at the tip of the fimbriae. Strikingly, despite the fimbriae being more abundant on the cell surface at pH 7.0 using staining reagents PTA and MAT, the tips do not come together. Even though an additional contribution from the electrostatic charges of the different stains at the experimental pH cannot be ruled out at the present time, these data support the notion of a pH-dependent conformational change for Fap1.

Once electron micrographs of wholemount bacterial preparations in different negative stains had been collected, digitized, and corrected for the contrast transfer function (CTF) of the microscope, fimbrial regions were extracted from the images following strict criteria: the fimbriae should be straight; they should not overlap other fimbriae and they should be separated by at least two fimbrial diameters from its nearest neighbor (Fig. 5B). As many images as possible, with a length of ~55 nm, were at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from extracted from the boxed fimbrial regions. After centering the images, an average image (Fig. 5B) and an average power spectrum (Fig. 5C) were calculated for each staining condition. The power spectra, in all stain preparations, show predominant layerlines at 1/(6.5 nm) and 1/(33 nm), which relate to structural repeats in the fimbriae. At the present time, an unambiguous assignment of the Bessel orders is not possible due to the weak scattering power of the fimbrial regions and the uncertainty in the diameter of fimbriae.

DISCUSSION

Recent studies have suggested that Fap1 belongs to a wider family of cell wallanchored, serine-rich repeat proteins (49) that exhibit features distinct from the pili of other Gram-positive organisms, such as Corynebacteria diphteria, group A and group B Streptococcus (GAS and GBS) and Gram-negative bacteria expressing chaperone usher (CU) and Type IV secretion systems.

Typically, these pili are characterised by small subunits noncovalently or covalently linked to each other (54)(55)(56), whereas SRR fimbriae are characterised by the presence of extensive glycosylated serine-rich repeats within a large protein subunit (>200 kDa). SSR fimbriae are found on several pathogenic streptococci and staphlococci, which include GspB from Streptococcus gordonii (17), SSR-1 and SSR-2 from Streptococcus agalactiae (21)(22)57), SrpA from S. sanguinis (50) and SraP from Staphylococcus aureus (18). Many of these proteins mediate specific interactions with human platelets through their N-terminal regions, which share little sequence homology to each other and are implicated in binding to a diverse array of host receptors (20,57).

We have demonstrated that a 36 kDa fragment from the non-repeat region of Fap1 (Fap1-NR) harbours the binding properties for colonisation of the oral cavity (58). Fap1-NR comprises two domains with distinct secondary structure and topology, namely an -helical domain (residues 116-231, Fap1-NR α ; Fig. 1A) and another composed predominantly of β-strands (residues 231-437, Fap1-NR β ; Fig. 1A). The protein sequence outside the non-repeat region is composed of ~2000 amino acid residues of S(V/I/E) dipeptide repeats. Perfect sequence repeats of less than four amino acids tend to form fibrous extended structures that are often super-helical (59); for example the collagen triple helix (60) and the zigzag beta structure of the silk I (61-62) and the β-turn helix of the Plasmodium falciparum circumsprozoite protein (63)(64). The extensive repetition of the serine dipeptides together with our electron microscopy data suggests a model for the fimbriae in which Fap1 strands form a super-helical extended structure with a principal repeat at 6.5 nm and serine sidechains exposed for O-glycosylation (Fig. 6). The larger 33 nm periodicity may be attributed to an independent helical stripe of surface glycosylation, the exact pattern of which has yet to be determined. Furthermore, since there is an ambiguity in layer line indexing, fimbriae comprising a coiled coil arrangement of two or more Fap1 helical strands cannot be excluded. Fap1 fimbriae are variable in length (in 300-600 µm range), similar to other SRR fibrils e.g. SrpA on the surface of S. cristatus (65). These length variations could be attributed to more than one subunit present per fimbriae, linked in a head-to-tail fashion. Although the classic pilin motif for head-to-tail crosslinking is absent (56,66); lysines residues present at the N-terminus of Fap1 could act as substrates for this crosslinkage.

In this model the N-terminal nonrepeat domain, Fap1-NR, comprises two globular domains that are located at the distal end of the structure for adhesion. Although no significant primary sequence similarity can be detected with proteins of known structure (no E value < 1 PHYRE hits (67)), both domains display structural similarity to two separate families of surface proteins from Gram positive bacteria (68).

The structure of the β-domain resembles the ligand-binding A-region from the Microbial Surface Components Recognizing Adhesive Matrix Molecules (MSCRAMM) family of bacterial adhesins (Supplementary Fig. S8A). Specifically, a high degree of similarity is observed with the fibrinogenbinding protein SdrG from S. epidermidis (pdb:1r17; Supplementary Fig. S8A) (55,(69)(70), the collagen binding ACE from Enterococcus faecalis (pdb:2z1p; Supplementary Fig. S8B) (71)(72) and the clumping factor A from S. aureus (pdb:1n67; Supplementary Fig. S8C) (73)(74)(75); typically with a backbone RMSD of 3.5 Å over ~120 residues. The MSCRAMM ligand-binding region encompasses two similar β-domains, in which a ligand peptide docks into an inter-domain groove of the open structure and a C-terminal extension closes the complex resulting in a change in relative domain orientation (74). Although the α-domain is unrelated to the flanking βrich domain from the MSCRAMMs, it displays a striking similarity with the imperfect helical repeats of EBH, the giant, extracellular matrix-binding protein from staphylococci (pdb:2dgj; Supplementary Fig. S8D) (76). The lack of electron density for the α-domain within the crystal structure of Fap1-NR, coupled with its well-defined structure in solution also implies that some flexibility exists at the interdomain boundary. It is tempting to speculate that the dual similarity with EBH and MSCRAMM surface proteins implies that host extracellular matrix proteins (ECM) are putative targets for Fap1 and supports the wider role for SRR fimbriae. The interaction between Fap1 and the salivary pellicle also suggests a role for calcium binding, however, no direct interaction was observable in our NMR assays (data not shown).

Fap1 exhibits an unusual pHdependent conformational change that affects key residues located at the interdomain junction of the two subdomains of Fap1-NR, with concomitant effects on the relative arrangement of these domains that allow Fap1 to adhere more tightly to SHA. The mean resting saliva pH observed in human oral cavities lies in the range 6.5 to 7.1. However, the buffering capacity of saliva is unable to cope with the rapid drop in pH induced by microbial acid production after the ingestion of fermentable carbohydrates. Plaque pH can reach values below 5.0 and remain low for some time before returning to neutrality. Some streptococci, particularly S. mutans, alter their physiology to survive in acidic environments. This is known as the acid tolerance response (ATR) and includes the production of stress-responsive proteins, increased glycolytic activity and the regulation of intracellular pH (77)(78). S. parasanguinis and many other oral streptococci are primary colonisers of the oral cavity and will be frequently exposed to acidic stress; however, they are unable to thrive at low pH and in response shut down their metabolic functions (79). A plausible mechanism for survival would be for S. parasanguinis to modulate adhesion to outcompete acid-tolerant species, a mechanism that has been proposed for other species of bacteria (80)(81)(82)(83)(84). Enhanced Fap1mediated adhesion at low pH would provide a more tenacious attachment to the salivary pellicle and increase the likelihood of S. parasanguinis survival during periods of acidic conditions. Moreover, its role as substrate for biofilm formation would also be enhanced thereby increasing bacterial cell density and providing respite.

Using a combination of low and high resolution methods we provide new insights into the architecture of Fap1 fimbriae and adhesion by a primary coloniser of the human oral cavity.

Furthermore, our conclusions likely extend to other SRR fimbriae from pathogenic organisms and suggest new therapeutic strategies. SM) and the National Institutes of Health (grant R01DE017474 awarded to TR, and grants R01DE011000 and R01DE017954 to HW) for financial support. We would like to thank the beamline scientists at ID29 of the European Synchrotron Radiation Facility (ESRF) and X33 at the DORIS-III ring of the Deutsches Elektronen-Synchrotron (DESY). The authors are also indebted to Christina Redfield at Oxford University for the provision of 950 MHz instrument time. The 950 MHz NMR facility was funded by the Wellcome Trust Joint Infrastructure Fund (JIF) and the E.P. Abraham Fund. b) Fap1-NR, Fap1-NR α and Fap1-NR β block bacterial adhesion in a dose dependent manner. 0, 10 and 100 µg of recombinant Fap1 proteins were preincubated with SHA before adding 3 H thymidine labelled wild-type S. parasanguinis. Wild-type binding is normalised to 100% and the fap1 deletion mutant was used as a negative control. The protomer is colored as in Fig. 1a with the cell wall anchor colored cyan, the major and minor SRR regions colored dark (protein) and light (glycan) orange, whilst the major and minor NR regions are colored green. The model on the left represents a super-coiled Fap1 monomer with repeating units of 6.5 nm along the fibre axis of the SRR region and protomer length of approximately 250-300 nm. Extra subunits may be incorporated within a coiled-coil to produce higher oligomeric states and the model on the right represents multiple

Figure Captions

Fig. 1 :

 1 Fig. 1: Domain organisation of S. parasanguinis Fap1 and adhesion to SHA a) Schematic representation of the domain structure of Fap1. The amino acid positions at which the domains begin and end are indicated. Mature Fap1 is numbered 1 to 2502.

Fig. 2 :

 2 Fig. 2: Three-dimensional structures of the two sub-domains Fap1-NR α and Fap1-NR β from the major non-repeat region from S. parasanguinis Fap1 fimbriae. a) Ribbon representation of the X-ray structure of Fap1-NR β The β-strands are colored in yellow and orange, short helices in red. L1 to L9 refer to loops 1 to 9 in the structure. b) Topology diagram of the X-ray structure of Fap1-NR β with the same colored code as in (a). c) Superimposition of the ten best NMR structures of Fap1-NR α with the three helices α1, α2 and α3 shown in blue. d) Ribbon representation of the NMR structure of Fap1-NR α in two different orientations with the three helices α1, α2 and α3 in blue. Leucine, alanine and isoleucine sidechains involved in inter-helix interactions are colored in dark blue, cyan and magenta respectively.

Fig. 3 :

 3 Fig. 3: pH-dependent structural change in Fap1-NR α a) S. parasanguinis adhesion to SHA is pH dependent. Adhesion of 3 H labelled wildtype bacteria to SHA was carried out at pH 5.0, 6.0, 7.0, 8.0 and 9.0. b) Superimposition of regions from the 15 N-1 H HSQC spectra of Fap1-NR α measured at 500 MHz at pH 8.0 (blue), pH 6.2 (black), pH 5.7 (green) and pH 5.2 (red). Full TROSY-HSQC spectra of Fap1-NR are shown in Supplementary fig. S7. c) Weighted differences in the observed 1 H and 15 N chemical shifts of Fap1-NR α between pH 8.0 and pH 5.2, where Σδppm = (4 x δ 1 H) + δ 15 N. Residues significantly affected by pH (Σδppm>0.95) are highlighted in red, whilst those affected to a lesser extent (0.95<Σδppm>0.8) are highlighted in yellow. The secondary structure of the helical bundle is shown on the top. More details are also given in Supplementary Figs. S7 & S8. d) Localisation of the chemical shift changes between pH 8 and 5.2 on the NMR structure of Fap1-NR α at pH 8. The three helices and the surface not affected by pH are in blue, whilst residues displaying chemical shift changes due to pH are colored in red (large) and yellow (small). The orientation is the same as Fig 2C. The chemical shift changes are localised towards the C-terminal pole.

Fig. 4 :

 4 Fig. 4: Overall structure of Fap1-NR and mutagenesis a) SAXS scattering profile of Fap1-NR at pH 5 (red) and pH 8 (blue). b) SAXS derived electron density for Fap1-NR at pH 8 with the crystal structure of Fap1-NR β and the solution structure of Fap1-NR α rigid body fitted into the envelope. The 27 residue linker is shown as a blue dashed line and the N-and C-termini of Fap1-NR are annotated. c) Effects of pH change on the electron density of Fap1-NR. SAXS derived electron density at pH 8 is blue and at pH 5 is red. d) Potential receptor binding surface of Fap1. Two exposed hydrophobic regions were mutated Fap1-NR. Residues including I134, E138, D142, L163 and V164 in Fap1-NR α together with L385, I291, L292, L300, N403, Q405 and I411 in Fap1-NR β were mutated to alanine. Mutants affecting adhesion of Fap1-NR are shown as green spheres and line the concave face of the SAXS envelope. The position of the negative control E204A is shown as yellow spheres and is on the non-concave face. Residues in Fap1-NR α affected by changes in pH are colored red and congregate at the interdomain boundary. e) Effect of site-directed mutagenesis of amino acid residues located at the putative binding interfaces in Fap1-NR on bacterial adhesion to SHA. Adhesion of S. parasanguinis FW213 (1), fap1 mutant (2) and site-directed fap1 mutants: I134A (3), E138A(4), I134AE138A (5), D142A (6), L163A(7), V164A(8) and E204A (9) in the α-domain; and L385A(10), I411A(11), L300A(12), I291AL292A(13), N403A(14), Q405A(15) and N403AQ405A(16) in the β-domain.

Fig. 5 :

 5 Fig. 5: Electron microscopy of Fap1 fimbriae a) Micrographs of long fimbriae from wild type S. parasanguinis stained with UA (left), PTA (centre) and MAT (right). Scale bar = 100 nm; Black arrows indicate fimbriae; White arrows indicate tip aggregates; Open white arrows indicate fimbrial bundles. b) Averaging of narrow fimbrial sections; micrograph of S. parasanguinis stained with PTA (left), three representative fimbrial sections (centre) and average of all fimbrial regions (right). Scale bar = 10 nm; Fimbriae diameter 5.5 nm. c) Average power spectra from fimbrial regions stained with UA (left), PTA (middle) and MAT (right) lines show the principal 1/6.5 nm layer line and a weaker one at 1/33 nm, present in all staining conditions.

Fig. 6 :

 6 Fig. 6: Model for the super-coiled strand of Fap1 fimbriae.

at

  Fig. 1

Table 1 : Crystal data collection and refinement statistics for Fap1-NR β . Crystal parameters

 1 at Institut Pasteur -CeRIS onFebruary 13, 2017 http://www.jbc.org/ Downloaded from polymerized subunits with just two displayed here. The Fap1 model was created from a lefthanded helix in which single atoms (representing individual amino acids) were arranged on a helical axis, with additional atoms (representing glycan residues) associated with alternate amino acid residues. This helix was in turn rotated about a central super-helical fibre axis. The SAXS envelope was scaled to the SRR model and placed at the C-termini, followed by a short stretch of the helical SRR model representing the minor SRR. An alignment of SRR sequences (middle) from S. parasanguinis Fap1, S. aureus SrpA, S. gordonii M99 GspB and L. johnsonii NCC533 LJ1711 is shown. The consensus SRR dipeptide sequence (SRRc) is SX, where X can be any amino acid. Numbers in parentheses refer to the outermost resolution shell. R meas = hn h / (n h -1))l |I hl -<I h >| / (h)(l) <I h > where I is the integrated intensity of a given reflection, <I> is the mean intensity of multiple corresponding symmetryrelated reflections and n h is the number of observations of reflection h.R work =  ||F o | -|F c || /  F owhere F o and F c are the observed and calculated structure factors respectively. R free = R work calculated using ~5% random data excluded from the refinement. rmsd stereochemistry is the deviation from ideal values. Ramachandran analysis was carried out using Molprobity(85) at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ By normalisation against data for BSA, calculated using the formula [I(0) FAP ÷ I(0) BSA *66 kDa] where I(0) BSA was 188.3.

	Space group

b. For the definition of normalised spatial discrepancy (NSD), see

Downloaded from at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from

at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from

Kozin & Svergun, 2001) at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from

ACKNOWLEDGEMENTS

The authors would like to thank the Wellcome Trust (programme grant number 079819; equipment grant number 085464 to at Institut Pasteur -CeRIS on February 13, 2017 http://www.jbc.org/ Downloaded from

Supplementary Figure S6:

Adhesion of a fap1 mutant of S. parasanguinis to SHA Adhesion of 3 H labeled fap1 mutant to SHA was carried out at pH 5.0, 6.0, 7.0, 8.0 and 9.0 as described in materials and methods. The fap1 mutant exhibits only background level of adhesion at different conditions. This article cites 0 references, 0 of which can be accessed free at


Pierre-André Jouvet
email: pjouvet@u-paris10.fr

Elodie Le Cadre
email: elodie.lecadre@agroparistech.fr

Caroline Orset
email: caroline.orset@agroparistech.fr

Irreversible investment, uncertainty, and ambiguity: the case of bioenergy sector $

Keywords: Ambiguity, Bioenergy, Irreversible investment, Uncertainty. JEL Classification: D21, D81, Q42, Q23

We analyze production and investment decisions of an agent in industrial activities that are characterized by two forms of uncertainty: demand uncertainty (in terms of number of buyers) and competitive effect uncertainty (in terms of other energy resource). We apply our model on the bioenergy industries. We compare the case of an ambiguity neutral agent with that of an ambiguity averse agent. We show that the investment decision of an agent depends on the effects of both the capital investment and the level of production on the cost and the uncertainty the agent is confronted with. Moreover, we find that ambiguity aversion tends to decrease the agent's optimal levels of production and investment. Our numerical analysis of the French case illustrates the different effects associated with demand uncertainty and competitive effect uncertainty.

Introduction

Investments into renewable technologies will have to develop in order to reach the renewable energy target of 20% fixed by the European Union (EU) for 2020. 1 To reach the future targets set out by the EU, significant amounts of biomass and investments into biomass based technologies will be necessary. 2 Biomass is key to the development of renewable energies, but it must undergo a pretreatment and densification process before it can be transported and stored. Indeed, biomass is a resource that is heterogeneous in quality and is not homogeneously distributed across space. Therefore, the large range of biomass types is not directly usable in some feeding systems and conversion processes. Investment in new pre-treatment facilities is a necessary step in the total biomass supply chain in order to save transport, material, handling costs for users and to reduce investments in transformation facilities.

These pre-treatment processes are still in progress and the biomass market is emerging. Although a potential investor has information about the demand and the competitive effect on the supply market, this information still remains imperfect.

Indeed, due to the novelty of this market, the agent cannot get a perfect knowledge on the number of buyers before starting the production. He will either have to supply a few potential buyers such as heat and electricity producers, needing to replace coal, or a larger number of potential buyers including producers of second generation biofuel and heat and electricity producers. This uncertainty then affects the agent's perception of the average price. Here and hereafter, we define this uncertainty as the demand uncertainty.

Moreover, the competition effect from other energy resource on the price of pretreated biomass is also not well-known by the agent. In fact, the biomass may be sold either to heating or power units as a substitute for coal (the selling price could then be indexed with coal prices) or to Biomass to Liquid (BtL) units as a substitute for fossil fuel and prices could then be indexed with oil prices, which fluctuate even more sharply than coal prices. So, uncertainty about competition affects the agent's perception of the average price and mostly the variance price. We define this uncertainty as the competitive effect uncertainty.

Considering these two kinds of uncertainty and their impact on the selling price, a biomass agent has to decide how much capital investment and produced units he will make in biomass activities. Capital investment, also called in the literature the cost of entry, in bioenergy production represents a quasi-sunk cost due to the fact that biomass torrefaction is a specific, and relatively expensive, process. This naturally raises the issue of the effect of both types of uncertainty and of the irreversibility on the investment level and production.

Furthermore, in the energy market, the instability of the economy may lead the agent to have uncertainties about his evaluation of the variance of the output price. We use the term 'ambiguity' to indicate situations in which the odds of an uncertain event are not precisely known. In other words, a situation in which there is an 'uncertainty about uncertainty'. 3An agent who has doubts about the odds is considered as an ambiguity-averse agent. So a question arises: how an ambiguity-averse agent behaves when he makes his decisions concerning investment and production?

To understand the impact of uncertainty on investment and production in biomass activities, we propose a two-period model in which there is incomplete information about the number of buyers and the competitive effect. Under these uncertainties, an agent has to choose his capital investment for the production of pre-treated biomass units at the following period. We study the cases of an ambiguity-neutral agent and of an ambiguity-averse agent. Following [START_REF] Klibanoff | A smooth model of decision making under ambiguity[END_REF], we extend our work by presenting ambiguity as a second order prior probability distribution over the set of plausible distributions of the competitive effect. This approach allows us to analyze the impact of ambiguity on the investment and production choices.

The standard theory of irreversible investments or quasi sunk cost [START_REF] Henry | Investment decisions under uncertainty: The "irreversibility effect[END_REF][START_REF] Sutton | Sunk costs and Market structure[END_REF] and options values suggests a negative relation between investment and uncertainty [START_REF] Dixit | Investment under uncertainty[END_REF]. Empirical studies also confirm this negative relation [START_REF] Carruth | What do we know about investment under uncertainty[END_REF][START_REF] Bond | Microeconometric Evidence on Uncertainty and Investment[END_REF][START_REF] Fan | A real options based model and its application to china's overseas oil investment decisions[END_REF]. However, [START_REF] Kulatilaka | Strategic growth options[END_REF][START_REF] Sarkar | On the investment-uncertainty relationship in a real options model[END_REF] point out that an increase in uncertainty could increase the probability of investing, and thereby has a positive impact on investment. Moreover, [START_REF] Mohn | Investment and uncertainty in the international oil and gas industry[END_REF] argue that any positive impact on investment arising from the fact that greater uncertainty, under certain circumstances, increases the marginal profitability of capital. In all these papers the effect of price uncertainty has been analyzed as the effect of demand uncertainty on capacity choice [START_REF] Trigeorgis | Real Options[END_REF][START_REF] Dangl | Investment and capacity choice under uncertain demand[END_REF][START_REF] Isik | A model of entry-exit decisions and capacity choice under demand uncertainty[END_REF][START_REF] Elder | Oil price uncertainty in canada[END_REF], 2010). Considering real options approach, [START_REF] Murto | Timing of investments in oligopoly under uncertainty: A framework for numerical analysis[END_REF] are interested with in the timing of investment projects under demand uncertainty and oligopolistic competition. The important characteristic is that the output price is influenced by both exogenous uncertainty and new capacity investments. This paper is closed to our approach with demand uncertainty and competitive effect even if there is no real uncertainty on the competition. [START_REF] Murto | Timing of investment under technological and revenue-related uncertainties[END_REF] introduces two types of uncertainty by combining effect of technological uncertainty and uncertainty in output price with real options approach. However, no work has been done on the two types of uncertainty (demand uncertainty and competitive effect uncertainty) that affect prices in different ways: the perception of the average and the variance of the price. Concerning ambiguity, we refer to the basic literature on ambiguity with [START_REF] Ellsberg | Risk, ambiguity, and the savage axioms[END_REF] and [START_REF] Fellner | Two propositions in the theory of induced innovations[END_REF][START_REF] Fellner | Probability and Profit[END_REF], the empirical investigations by [START_REF] Slovic | Who accepts savage's axiom?[END_REF] and the recent literature with [START_REF] Klibanoff | A smooth model of decision making under ambiguity[END_REF] and [START_REF] Gollier | Does ambiguity aversion reinforce risk aversion? applications to portfolio choices and asset prices[END_REF] to indicate situations for which the odds of an uncertain event are not precisely known. Determining how an ambiguity-averse agent decides to invest and produce in emerging technologies is an important line of research in entrepreneurial decision-making in BtL.

Using an analytical approach and numerical analysis, we first note that whatever the certainty or uncertainty context, the agent never invests or produces when he thinks that an increase in capital increases the cost of one more unit. Moreover, we show that the agent's capital investment decision depends on the effects of the amount of capital invested, of the level of production on the cost and on the uncertainty to which the agent is confronted. Then, we observe asymmetric effects of demand uncertainty and competitive effect uncertainty on the optimal amount of investment and optimal production. Finally, we find that ambiguity aversion tends to decrease the agent's level of capital investment and production.

The French biomass pre-treatment industry (torrefaction) is taken as an example, and the empirical results show that the model developed here can provide useful advice for pretreatment biomass investment programs.

The remainder of the paper is organized as follows. Section 2 consists of a description of the model. Section 3 analyzes and compares the optimal investment and production decisions of both an ambiguity neutral agent and an ambiguity averse agent. Section 4 presents a numerical analysis. Finally, section 5 concludes.

Model description

We consider a two period model with a risk-neutral agent. The agent faces two types of uncertainty: demand uncertainty, in terms of number of buyers, and a competitive effect uncertainty. Indeed, whereas the agent knows that he is competing on the market of fuel providers,4 i.e. he is a price taker,5 he only has a subjective perception of his potential customers and of the severity of the market competition. Both types of uncertainty affect prices in different ways: the demand uncertainty pertains to the perception of the price average while the competition effect uncertainty pertains to the average and mostly the price variance.

We define four possible states of the world: a Low number of buyers and a Weak competition effect (LW ), a High number of buyers and a Weak competition effect (HW ), a Low number of buyers and a Strong competition effect (LS) and High number of buyer and a Strong competition effect (HS). We propose to divide the agent's subjective probabilities on these states in two kinds of beliefs: first, the agent's subjective probabilities are ψ on the low number of buyers, and (1 -ψ) on the high number of buyers; second, the agent's subjective probabilities are θ on the strong competition, and (1 -θ) on the weak competition. In addition, we consider that the 'right' value of the probability associated with the competitive effect uncertainty θ may be unknown. In this case, θ is a random variable, and it is called θ. The agent associates a probability distribution F (θ) on [θ, θ] which measures the subjective relevance of a particular θ probability. The competitive effect is then ambiguous in the sense that his beliefs depend on a probability distribution. Instability in the energy market can cause the agent to become uncertain about the true value of probability θ, which pertains to the variance of the output price. So there may be a great deal of ambiguity associated with the competition based on the output selling price.6 Following [START_REF] Klibanoff | A smooth model of decision making under ambiguity[END_REF], we describe the agent's behaviour towards ambiguity by a function φ. An increasing and concave φ means that the agent is ambiguity averse. Similarly, ambiguity neutrality is characterized by the linear function φ.

We associate a selling price P i with each state i ∈ {LW, HW, LS, HS}. A larger number of buyers is likely to be able to support a higher price, so we get that P HW > P LW and P HS > P LS . Moreover, competition between fuel suppliers leads to a lower price, P HW > P HS and P LW > P LS .

In period 0, the agent has the opportunity to invest in plant in order to produce pretreated biomass. Let be K ≥ 0, the stock of capital and the investment costs I(K). I is an increasing and convex function such that I(0) = 0. As in [START_REF] Cairns | Sunk Cost and Cost Functions[END_REF], we assume a sunk capital, i.e., a capital amount that is specific to the firm.

In period 1, if the agent has invested I(K) in period 0, he has to choose his production q which represents the units of torrefied biomass. This yields a pay-off equals to P i q in state i ∈ {LW, HW, LS, HS}. From this pay-off the cost of production c(q, K) which is an increasing and convex function in q and a decreasing and convex function in K must be subtracted. Moreover, we assume that if q > 0 then c(q, 0) = ∞, c(0, K) = 0 for all K > 0.

So, with a discount factor β < 1, the agent's expected pay-off V (K, q; ψ, θ) is expressed as follows:

V (K, q; ψ, θ) = -I(K) + βψ [θ (P LS q -c(q, K)) + (1 -θ) (P LW q -c(q, K))] +β(1 -ψ) [θ (P HS q -c(q, K))) + (1 -θ) (P HW q -c(q, K))] .
Likewise, considering the KMM ambiguity approach, the agent's expected pay-offs is given by:7

W (K, q, ψ, θ) = φ -1 θ θ φ (V (K, q, ψ, θ)) dF (θ) .
(1)

Optimal decision making

Neutrality to ambiguity

In this section, we consider that the agent is aware of the true value of θ . In other words, there is uncertainty about the price variance and he believes that the probability associated with this uncertainty is relevant. In this case, we consider W (K, q; ψ, θ) = V (K, q; ψ, θ).8

In period 0, the agent has to determine his optimal stock of capital K * for producing pretreated biomass. Then, in period 1, he could decide which quantity q * to produce. By consequence, we propose to solve this model through backward induction.

We define the expected price under demand uncertainty, the expected price under competitive effect uncertainty, and the expected price under both demand and competitive effect uncertainties, respectively, as follows:

E ψ P m = ψP Lm +(1-ψ)P Hm , E θ P j = θP jS +(1-θ)P jW , and E ψθ P = ψE θ P L +(1-ψ)E θ P H .
with j ∈ {L, H} and m ∈ {W, S}.

So the first order condition on quantity is:

9 ∂c(q, K) ∂q = c q = E ψθ P. (2)
If for all q > 0 we get V (K, q; ψ, θ) ≤ 0, i.e., if the project is never profitable, then the agent does not produce. On the other hand, if there exists q > 0 such that V (K, q; ψ, θ) > 0, the project is profitable for a certain level of production. This relation implies that q is an implicit function of K, q ≡ q(K).

Now, we study the optimal stock of capital K which maximizes the expected pay-off V (K, q(K); ψ, θ). The condition is given by the solution of the following program:

- dI(K) dK + β dq(K) dK E ψθ P -β c q dq(K) dK + c K = 0.
Using equation (2), we obtain,

βc K = - dI(K) dK = -I (K). (3)
If for all K > 0 we get V (K, q(K); ψ, θ) ≤ 0, i.e., if the project is never profitable, then the agent does not invest in it. On the other hand, if there exists K > 0 such that V (K, q(K); ψ, θ) > 0, the project is profitable for certain level of investment.

As already mentioned in [START_REF] Mcdonald | The value of waiting to invest[END_REF] as in [START_REF] Gollier | Comment intégrer le risque dans le calcul économique ?[END_REF], it is optimal for an agent to invest only if investment value exceeds his cost. Finally, the optimal decisions (q * , K * ) are defined by equations ( 2) and (3). We can summarize some static comparative results in the following lemma Lemma 1 (i) A higher price, P LW , P HW , P LS , and/or P HS , always increases the level of production, q * . That is not the case for the investment K * , which increases when ∂ 2 c(q * , K * )/∂q∂K < 0, decreases when ∂ 2 c(q * , K * )/∂q∂K > 0 and does not vary when 9 Here and hereafter, we will equally use the following notations: ∂c(q,K)

∂K = c K , ∂ 2 c(q,K) ∂q∂K = c qK , ∂ 2 c(q,K) ∂K 2 = c KK and ∂ 2 c(q,K) ∂q∂q = c qq .
∂ 2 c(q * , K * )/∂q∂K = 0.

(ii) A higher subjective probability on the realization of a low number of buyers, ψ, and/or a higher subjective probability on the realization of a strong competition effect, θ, always decreases the level of production, q * . That is not the case for the investment K * which decreases when ∂ 2 c(q * , K * )/∂q∂K < 0, increases when ∂ 2 c(q * , K * )/∂q∂K > 0 and does not vary when ∂ 2 c(q * , K * )/∂q∂K = 0.

Proof. Part (i)

Increasing any price induces an increase of E ψθ P . Conditions ( 2) and (3) imply q * ≡ q * (E ψθ P ) and K * ≡ K * (E ψθ P ). Denoting P , the price expected value and differentiate ( 2) and ( 3), we obtain respectively, if ∂ 2 c(q * ( P ), K * ( P ))/∂q∂K = 0, dq * ( P )

d P = 1 -c qK ∂K * ( P ) ∂ P c qq and dK * (P ) dP = - c qK βc qq c KK -β[c qK ] 2 + c qq I (K)
With cost convexity assumption and the convexity of I(.), we assume βc qq c KK -β[c qK ] 2 + c qq I (K) > 0. Then, if c qK < 0, ∂K * ( P )/∂ P > 0 and ∂q * ( P )/∂ P > 0.

Part(ii) Conditions ( 2) and (3) imply q * ≡ q * (ψ, θ) and K * ≡ K * (ψ, θ). We differentiate (2) and (3) with respect to ψ and θ, we obtain respectively, if ∂ 2 c(q * (θ, ψ), K * (θ, ψ))/∂q∂K = 0,

∂q * (θ, ψ) ∂θ = ψ (P LS -P LW ) + (1 -ψ) (P HS -P HW ) -c qK ∂K * (θ,ψ) ∂θ c qq ∂q * (θ, ψ) ∂ψ = θ (P LS -P HS ) + (1 -θ) (P LW -P HW ) -c qK ∂K * (θ,ψ) ∂ψ c qq and ∂K * (θ, ψ) ∂θ = - c qK [ψ (P LS -P LW ) + (1 -ψ) (P HS -P HW )] c qq c KK -[c qK ] 2 + c qq I KK ∂K * (θ, ψ) ∂ψ = - c qK [θ (P LS -P HS ) + (1 -θ) (P LW -P HW )] c qq c KK -[c qK ] 2 + c qq I KK
where ψ (P LS -P LW )+(1-ψ) (P HS -P HW ) < 0 and θ (P LS -P HS ) + (1 -θ) (P LW -P HW ) < 0. Then, ∂q * (θ, ψ)/∂θ < 0, ∂K * (θ, ψ)/∂θ < 0, ∂q * (θ, ψ)/∂ψ < 0 and ∂K * (θ, ψ)/∂ψ < 0 if

∂ 2 c(q * (θ, ψ), K * (θ, ψ))/∂q∂K < 0.
So the opportunity to sell each unit at a higher price, and then getting a higher pay-off, prompts the agent to produce more. This opportunity may come from an increase in the possible selling prices, a lower belief in the realization of a low number of buyers, and/or a lower belief in the realization of a strong competition effect.

Moreover, if the marginal production cost decreases in the capital investment, a higher price, a lower belief in the realization of a low number of buyers, and/or a lower belief in the realization of a strong competition effect increases the optimal level of investment in capital. Besides, if the capital investment has no impact on the marginal production costs then the prices and the two beliefs do not affect the agent's decision concerning the level of investment in capital. For simplicity in the following, we consider c qK > 0 and we can get that K * = 0 and q * = 0.

In addition, if the marginal production cost decreases with the capital investment, a higher price, a lower belief in the realization of a low number of buyers, and/or a lower belief on the realization of a strong competition effect prompts the agent to invest and produce.

So the effect of the capital investment on the marginal production cost plays a major role in the agent's decision concerning his investment in capital and production.

It is natural now to compare the situation of certainty with the situations in which there is one type of uncertainty (either demand uncertainty, or competitive effect uncertainty), and with the situation in which there are both types of uncertainty. To do so, we define the marginal rate of substitution associated with the cost as follows:

T M SC(q, K) = - ∂c(q,K) ∂q ∂c(q,K) ∂K
, which represents the increase in K for which the cost is maintained when the agent produces one more unit. Using relations (2) and (3), we have

T M SC(q, K) = - c q c K = E ψθ P I (K) . ( 4
)
The situation of certainty, i.e. that in which the agent knows the number of buyers on the market and the severity of the market competition effect, corresponds to E ψθ P = P . In this case, we denote by q * C and K * C the optimal quantity and investment . The cases in which there is only one type of uncertainty: first, demand uncertainty, i.e., the agent has perfect knowledge of the level of the effect of market competition , correspond to E ψθ P = E ψ P m with θ = 0 or θ = 1. In the case of Demand Uncertainty, we denote by q * DU and K * DU the optimal quantity and investment; secondly, the competition effect uncertainty (i.e., the agent does not initially know the effect of competition but he knows the number of buyers of the future market) corresponds to E ψθ P = E θ P j with ψ = 0 or ψ = 1. In the case of Competitive effect Uncertainty , we denote by q * CU and K * CU the optimal quantity and investment.

We first note that regardless of the certainty or uncertainty level, the agent never invests nor produces when he thinks that an increase in capital increases the cost of one more unit.

The lack of information on the true level of the price implies that the agent expects a lower value of the price than the realized one when it is high and expects an upper value of the price than the realized one when it is low. This directly impacts on the level of production, which decreases when the expected price is lower than the realized one, and increases when the expected price is higher than the realized one. Even though the level of capital investment is affected by this erroneous evaluation, the agent's decision also takes into account the effect on the cost of both the level of production and the level of capital invested. Actually, producing more leads the agent to choose a level of investment in capital that reduces his unit production cost. Then under uncertainty, the agent makes a lower (higher) investment when the expected price is lower (higher) than the realized one and the increase in capital decreases the cost of one more unit. [START_REF] Elder | Oil price uncertainty in canada[END_REF], 2010) find empirical evidence that uncertainty about oil prices has tended to depress investment in Canada and United States. Our model could explain their result with considering that the investors expect at once a price that is lower than the true one and that an increase in capital reduces the marginal cost of production.

Moreover, under both uncertainties, the agent produces less when he does not know that the realized price is the highest, i.e., the number of buyers is high and there is little market competition. In this context, he invests less capital when he thinks that an increase in capital increases the cost of one more unit. On the contrary, the agent produces more when he does not know that the realized price is the lowest, i.e., the number of buyers is low and there is a strong market competition effect. Then, he invests more in capital when he thinks that an increase in capital decreases the cost of one more unit.

Aversion to ambiguity

In this section, we seek to understand how choices concerning capital and capacity investment are affected by ambiguity aversion. We then propose to compare the optimal production and capital investment decisions of an agent who is averse to ambiguity with those of an ambiguity neutral agent. To formalize the aversion to ambiguity, we consider that the 'right' value of the probability associated with the competition severity uncertainty θ may be unknown. The agent's belief, θ, is then represented not as a single probability measure on the set of states but as a set of probability measures. Such a framework is relevant to the decision concerning investment and production; indeed, as quoted in [START_REF] Heath | Preference and belief: Ambiguity and competence in choice under uncertainty[END_REF]: the ambiguity aversion is particularly strong in cases in which people feel that their competence in assessing the relevant probabilities is low.

We then extend the model by considering that θ is a random variable. The agent now associates a probability distribution F (θ) on [θ, θ] which measures the subjective relevance of a particular θ probability. Following [START_REF] Klibanoff | A smooth model of decision making under ambiguity[END_REF], we assume that the preferences of the agent indicate smooth ambiguity aversion. So, the agent considers that his expected pay-off is defined by equation (1):

W (K, q, ψ, θ) = φ -1 θ θ φ (V (K, q, ψ, θ)) dF (θ)
with φ(.) defined by an increasing and concave function when the agent is ambiguity averse.

As mentioned previously, we consider the problem in two steps. First, we focus on the impact of ambiguity aversion on the optimal production, q * and second on the optimal capital investment, K * .

For a given stock of investment, the first order condition for production is given by: θ θ φ (V (K, q; ψ, θ))

∂V (K, q; ψ, θ) ∂q dF (θ) = 0 (5) where ∂V (K, q; ψ, θ) ∂q = (E ψθ P -c q ). ( 6)

Proposition 1. For a given initial stock of capital investment, ambiguity aversion tends to decrease the agent's optimal level of production, q * < q * .

Proof. We use the following notations:

∆(q, θ) = φ (V (K, q; ψ, θ)) Λ(q, θ) = E ψθ P -c q .
By definition the covariance is: cov(∆(q, θ), Λ(q, θ)) = E(∆(q, θ)Λ(q, θ)) -E(∆(q, θ))E(Λ(q, θ)).

Then, with condition (5), we have

E(∆(q * , θ)Λ(q * , θ)) = J(q * ) = 0
Comparison to neutrality ambiguity case.

From condition (2), we know that E(∆(q * , θ)) = 0 and then cov(∆(q * , θ), Λ(q * , θ)) = E(∆(q * , θ)Λ(q * , θ)) So if cov(∆(q * , θ), Λ(q * , θ)) < 0 that implies E(∆(q * , θ)Λ(q * , θ)) < 0. This is equivalent to J(q * ) < 0 = J(q * ). Since φ is increasing and concave, J(.) is decreasing function and q * > q * . The sign of covariance is given by differentiate ∆(q, θ) and Λ(q, θ) with respect to θ where ∂Λ(q, θ) ∂θ = ∂E ψθ ∂θ < 0 and ∂∆(q, θ) ∂θ = φ (V (K, q; ψ, θ)) ∂V (K, q; ψ, θ) ∂θ > 0 with φ (V (K, q; ψ, θ)) < 0 and ∂V (K, q; ψ, θ)/∂θ < 0. Therefore, cov(∆(q * , θ), Λ(q * , θ)) < 0 and q * > q * .

Let us now turn to the analyze of the agent's optimal investment in capital. Equation (5) implies that q * ≡ q * (K) and the first order condition is:

θ θ φ (V (K, q * (K); ψ, θ)) ∂V (K, q * (K); ψ, θ) ∂K dF (θ) = 0 (7)
where

∂V (K, q * (K); ψ, θ) ∂K = -I (K) + β dq * (K) dK E ψθ P -β(c q dq * (K) dK + c K ). ( 8
)
Proposition 2. If ∂ 2 c(q * , K * )/∂q∂K < 0 then ambiguity aversion tends to decrease the agent's optimal investment level, K * < K * .

Proof. Similar to the proof of Proposition 1 with conditions (3) and ( 7). Aversion to ambiguity concerning the competition effect leads the agent to reduce his investment in capital and his production. Actually, the agent has doubts about its own subjective beliefs on the competition effect. This adds a new uncertainty dimension for him and discourages him from investing and producing. Ambiguity aversion then restrains investment and production in the new process. This may have drastic consequences on the development of emerging processes.

Numerical analysis

The empirical analysis is based on the French biomass pre-treatment industry. The case of France is a particularly interesting subject of study, because active research studies are being conducted on second generation biofuel technologies (ADEME, 2009). One of the pilot programmes in which five French partners and one German partner participate, has launched BioTfueL, a million Euro project that uses the Fischer-Tropsch process to convert torrefied wood biomass into drop-in renewable fuel. This group will launch pilot projects in France that will commence in 2012. The domestic biomass resources available are also large.10 Prospects for the diffusion of torrefaction technology in such a dynamic and expanding market are also of particular interest if the economical profitability is to be enhanced.

To determine the profit flow the firm receives when the project is implemented, we suppose, as is frequently done, that sunk investment costs are linear: I(K) = p K K, p K > 0 and I (K) = p K with p K ∈ [0, 1], the investment coefficient [START_REF] Cairns | Sunk Cost and Cost Functions[END_REF]. The quantity of pre-treated biomass is a function of the amount of capital, K, that have to be paid for the installation of a production facility. Using (4), we can easily define different probability thresholds, θ and ψ by comparing different cases.

Determination of the cost function

Like [START_REF] Cairns | Sunk Cost and Cost Functions[END_REF] and [START_REF] Tsatsaronis | On avoidable and unavoidable exergy destructions and investment costs in thermal systems[END_REF] , we consider the avoidable cost of production c(q, K) as a function of the amount of capital, K and the output production, q. The avoidable costs are commonly calculated by subtracting the unavoidable cost from the respective total cost excluding the sunk cost I(K) such that:

c(q, K) = c T (q, K) -c U N (q, K)
where c T (q, K) is the total cost and c U N (q, K), the unavoidable costs.

The total cost function is a convex function composed of the capital costs and the production costs. We use a limited development at the order one of the translog function to represent the cost minimizing behaviour of the agent who uses the amount K of capital to produce a quantity q of output. For the torrefaction technology, the cost function is: ln(c T (q, K)) = a 1 + a 2 ln(q) + a 3 ln(K) + a 4 ln(q) ln(K)

where a 1 > 0, is a fixed cost, a 2 and a 3 are the cost elasticity of the production and the capital respectively, a 4 , is the cross elasticity between production and capital. We assume that the unit costs of production are increasing in accumulated production so a 2 > 0 and the investment costs of invested capital K are decreasing in accumulated capital so a 3 < 0. The data on operating costs of a torrefaction plant were taken from the existing literature and consists of engineering estimates. The technology exists today but it is tested at pilot scale, so we estimate our coefficients on the basis of economic data for different possible capacities of units (c.f. table (B.1) in appendix). We assume that unit runs at full capacity. The estimates are presented in table B.2 in appendix.

We then determine the unavoidable cost rate related to the production and the investment as follows [START_REF] Tsatsaronis | On avoidable and unavoidable exergy destructions and investment costs in thermal systems[END_REF]. Due to technical limitations imposed by the availability and/or costs of materials and manufacturing methods, a maximum value of the mass efficiency of the torrefaction process cannot be exceeded regardless of the amount invested. This efficiency is achieved at the point where the investment cost becomes infinite. This point determines the unavoidable destruction of raw biomass per unit of torrefied biomass . Thus, we could determine the cost rate associated with the unavoidable raw biomass destruction Z U N q . Similarly, the unavoidable investment costs per unit of torrefied biomass, Z U N K , are obtained by considering an extremely inefficient version of the technology, that is a version that would never be feasible in practice because of the very high biomass costs associated with it. 11 We assume that the percentage of the total costs that cannot theoretically be avoided, in view of today's technology and economic environment of the torrefaction, technology is between 20% and 50% [START_REF] Tsatsaronis | On avoidable and unavoidable exergy destructions and investment costs in thermal systems[END_REF]. We take an average unavoidable cost Z U N such as Z U N = Z U N q = Z U N K = 35%. Then the avoidable costs are calculated by subtracting the unavoidable cost rates from the respective cost rates:

c(q, K) = (1 -Z U N )e a 1 +a 4 ln(K) ln(q) K a 3 q a 2 .
Then we have to select an appropriate discount rate. It is an important topic in investment decision.12 Various ways of calculating discount rates adjusted for risk, have been proposed by [START_REF] Trigeorgis | Real Options[END_REF]. [START_REF] Schmit | Ethanol plant investment using net present value and real options analyses[END_REF] assume a discount rate of 8% to reflect a relatively high credit risk for the investment in ethanol plant, whereas [START_REF] Uslu | Pre-treatment technologies, and their effect on international bioenergy supply chain logistics. techno-economic evaluation of torrefaction, fast pyrolysis and pelletisation[END_REF] chooses a discount rate of 12.5% for an investment in torrefaction. The discount rate of the refinery unit is between 8 and 10% [START_REF] Dangl | Investment and capacity choice under uncertain demand[END_REF][START_REF] Felfli | Torrefied briquettes: technical and economic feasibility and perspectives in the brazilian market[END_REF]. In our analysis, we assume it is equal to 10% because torrefaction units will supply BtL and refinery units. We will vary this rate in the sensitive analysis.

We illustrate the results determined in section 3 for an ambiguity neutral agent and an ambiguity averse agent.

Agent's preferences 4.2.1. Ambiguity-neutral agent

According to operating data for torrefaction plant, the marginal cost of production decreases in K and increases in q (table (B.1)). Furthermore, the cost function does not vary with the number of uncertainty such that c(q * C , K * C ) = c(q * , K * ) and c(q * DU , K * DU ) = c(q * CU , K * CU ). To illustrate our results, we consider the following scenario: p K = 0.5, 13 θ = 0.5, ψ = 0.5. 14 We take the prices of torrefied biomass collected during a survey conducted among potential buyers of torrefied biomass in France such as P LW = 100 Euros/t, P HW = 200 Euros/t, P LS = 80 Euros/t and P HS = 148 Euros/t. 15 From ( 2) and (3), we determine the optimal level of production and investment for the situation in which there are both types of uncertainty, the situation in which there is certainty and the situation in which there is only one uncertainty. The results are summed up in table (B.3) for the different cases P i = P jm for j ∈ {L, H} and m ∈ {W, S}.

Taking into account of the operating costs, the optimal investment in capital, K * and the optimal levels of production, q * are ranked to the uncertainty which the agent faces.

We observe the asymmetric effects of uncertainty on the optimal amount of investment and optimal production. We show that the effect of number of buyers related uncertainty is stronger than that of the competition related uncertainty as the investment and the production levels are higher. This is true when, in the case of certainty, the effect of competition is weak and the number of buyers is low, or high if and only if the agent's subjective probability on the weak competition is lower than a certain threshold . Furthermore, if in the situation of certainty the agent knows the competition effect is weak and the number of buyers is high, the combination of both types of uncertainty leads him to invest less. He behaves similarly if the number of buyers is low in the certainty case if and only if his subjective probability concerning number of buyers is lower than a certain threshold (cases P LW = 100 Euros/t and P HW = 200 Euros/t of table (B.3)).

Secondly, the effect of the competition related uncertainty is stronger than the uncertainty concerning the demand uncertainty as the investment and production levels are higher. This is true when in the case of certainty the competition is strong and the number of buyers is high, or low if and only if the agent's prior belief concerning the weak competition is higher than a certain threshold . Besides, if in the certainty case the agent knows the competition is strong and the market size is low, the combination of both types of uncertainty leads him 13 We study the effect of different values of this parameter in the sensitivity analysis. 14 We take a average value between 0 and 1 for ψ and θ. 15 These values were determined during a confidential survey conducted in 2010 among the French energy companies which are potential consumers of torrefied biomass. For more details, please contact the authors.

to invest more. If the market size is high in the certainty case, he has the same behaviour if and only if his prior belief on market size is higher than a certain threshold (cases P HS = 148 Euros/t and P LS = 80 Euros/t of table (B.3)).

Finally, we study the effect of increasing and decreasing the investment coefficient. We sum up the results in table (B.4) in appendix for different value of p K ∈ [0, 1]. We notice that K * decreases when p K increases whether there is one or two types of uncertainty. As proved in the Lemma 1, a higher investment coefficient increases the cost of investment so the agent decreases his capital investment.

Ambiguity-averse agent

We now illustrate propositions 1 and 2 to examine the difference ambiguity causes in the results. Following [START_REF] Judd | Numerical methods in economics[END_REF]; [START_REF] Miranda | Applied computational economics and finance[END_REF], we use a Gaussian quadrature to produce the Legendre-Gauss weights and nodes for computing the integral of the continuous function W on interval [θ, θ] with θ = 0 and θ = 1. We use a beta distribution B(θ; η, µ) with the parameter η = 0.5 and µ = 0.5 to specify the probability distribution over the set of plausible distribution of the competitive effect. Indeed, the beta distribution is often used to describe the distribution of an unknown probability value, typically, as the prior distribution over a probability parameter. It is defined on the interval [0, 1]. η and µ give the shape of the probability density function. If η < 1 and µ < 1, the beta density function is U-shaped and symmetric about 1/2 if η = µ. Following [START_REF] Klibanoff | A smooth model of decision making under ambiguity[END_REF]; Engle-Warnick et al. ( 2008), we consider a constant absolute ambiguity aversion (CAAA) utility function such as:

φ(V (K, q; ψ, θ)) = 1-e -τ V 1-e -τ if τ > 0 V if τ = 0
where τ is the coefficient of ambiguity aversion. For the figure clarity, we take τ = 15 to compare the case of an agent who is highly averse to ambiguity with that of an ambiguity neutral agent.16 Figures (A.1) and (A.2) below show the marginal payoffs, ∂W (q, K; 0.5, θ)/∂q in function to q to determine q * and ∂W (q(k), K; 0.5, θ)/∂K in function to K to determine K * for both an ambiguity neutral and an ambiguity averse agents.

[Insert figures (A.1)and (A.2) ]

The marginal values are decreasing in q and K. Thus we find that an ambiguity averse agent produces less than an ambiguity-neutral agent and invests less as defined in the proposition 1 and 2. The results are the same for different value of τ . Due to ambiguity, an agent who invest in biomass torrefaction facilities chooses a lower capacity for his units than he would if he were ambiguity neutral. Ambiguity aversion leads the investor to evaluate probabilities distribution according to the least-favourable state, in this case the lowest pay-off.

This behaviour could have consequences on the development of emerging BtL process. Indeed, as mentioned before, the pre-treatment could enhance the deployment of BtL process because it can improve the economics of the overall production chain. If the producer invests less, the buyer takes the risk of not being supplied the right quantity. The buyer of the torrefied biomass perceives uncertainty about the availability of their inputs. They would be reluctant to invest in the new renewable energy process.

Finally, we study the effect of an increase and decrease in the investment coefficient p K when the agent is ambiguity averse. Figure (A.3) below shows the marginal payoffs ∂W (q(K), K; 0.5, θ)/∂K in function to K for different values of p K ∈ [0, 1].

[Insert figure (A.3)] We notice that K * decreases when p K increases whatever the coefficient of ambiguity aversion level. A higher investment coefficient increases the cost of investment so the agent decreases his capital investment when the agent is ambiguity averse. 4.3. Parameter sensitive analysis 4.3.1. Sensitivity of the optimum strategy as regards the unavoidable cost rate Z U N In the reference example, the percentage of total costs that could be not avoided in view of today's technology and economic environment, is an average of maximum and minimum percentages that are possible today. The reference unit produced torrefied biomass relatively expensively compared with units for which the unavoidable cost rate is higher but the mass efficiency of the torrefaction is also better. Increasing the unavoidable cost rates should increase the competitive advantage of a high-output capacity due to the fact the avoidable cost is lower, but regardless the amount invested, the mass efficiency is lower. This can be verified in the table (B.5) in which tests are made for minimum and maximum values in the range of possible unavoidable cost rates determined for this torrefaction process. A decrease in the proportion of the unavoidable cost reduces the profitability by increasing the avoidable cost price and therefore deters the entrepreneur for investing in high capacity. An increase in the unavoidable cost rate increases the optimal capital investment. However, whatever the unavoidable cost rate Z U N , the ranking of optimal investment in capital is unchanged.

Variation in the discount rate

In our reference simulation, we considered a discount rate of 10% per year. In this section, we examine the effect of an increase in the discount rate from 10% to 12% and a decrease from 10% to 8%. A change in the discount rate modifies the optimum investment strategy. A higher discount rate penalizes the waiting time and therefore encourages the entrepreneur to invest earlier. Indeed, the sensitive analysis (c.f. table (B.6)) shows the increase (decrease) in the discount rate leads to a high (low) investment in capital. Nevertheless, whatever the discount rate, the ranking of optimal investments in capital is unchanged.

Conclusion

In this paper, we assess the impact of two types of uncertainty and of the ambiguity aversion of the agent on his investment and production strategy. We develop a formal model for decision making in which agents are neutral to risk and averse to ambiguity about the true distribution of the competitive effect. We analyze the optimal capacity and production choices in this model. Using an analytical approach and numerical analysis, we first note that whatever the certainty or uncertainty context, the agent never invests or produces when he thinks that an increase in capital increases the cost of one more unit. Moreover, the agent's capital investment decision depends on the effects of the amount of capital invested, of the level of production on the cost and on the uncertainty to which the agent is confronted. Then, we observe asymmetric effects of demand uncertainty (in terms of number of buyers) and competitive effect uncertainty on the optimal amount of investment and optimal production. Finally, in the presence of ambiguity about the competition effect, agents will invest less in their units and their level of production is lower. The main feature of this model is that it helps to understand the behaviour of an agent who faces uncertainty about the market size and market competition if he is averse to ambiguity. This paper emphasizes the need to reduce the effects of ambiguity in the European policy framework that encourages the development of renewable energy production. The introduction of long-term contracts could contribute to reducing them. Actually, these contracts could be defined as agreements between a pretreated biomass producer (seller) and a renewable energy generator owner (buyer) for the purchase of torrefied biomass. By hedging against price volatility, these contracts would reduce the ambiguity impact of the competition effect. They could take the forms of competitive procurement process or bilateral contract negotiation (see [START_REF] Michaud | Long-term Renewable Energy Contracts, Net-Metering, and Ocean Wind Initiatives in New England[END_REF]).

An attractive feature of the model is to determine how the risk and ambiguity aversions of the buyer will affect the investment strategy of torrefied biomass producers. Finally, it will be important to check empirically, with potential agents (private forest owners, cooperatives...) the theoretical results obtained in our model and to evaluate the degree of their ambiguity aversion. The marginal payoff ∂W (K, q; 0.5, θ)/∂K in function to K for q * (K) = 7.18 and q * (K) = 7.26. Calculated with β=0.1, τ =15 and p K =0.5. K 2 Ranking of optimal levels of q and K Two Uncertainties 0.5 0.5 12.96 6.45

P i = P LW
Certainty(C) 1 1 13.91 6.84 q * DU > q * C > q * > q * CU Competitive effect (CU) 1 0.5 11.83 5.98

K * DU > K * C > K * > K *
CU Demand (DU) 0.5 1 15.39 7.44

P i = P HW
Certainty(C) 0 1 16.79 8.00 q * C > q * DU > q * CU > q * Competitive effect(CU) 0 0.5 14.03 6.88 K * C > K * DU > K * CU > K * Demand(DU) 0.5 1 15.39 7.44

P i = P LS
Certainty (C) 1 0 9.49 5.00 q * > q * CU > q * DU > q * C Competitive effect(CU) 1 0.5 11.83 5.98

K * > K * CU > K * DU > K * C
Demand (DU) 0.5 0 10.21 5.31

P i = P HS
Certainty (C) 0 0 10.90 5.59 q * CU > q * > q * C > q * DU Competitive effect (CU) 0 0.5 14.04 6.88 (1) In MEuros.

K * CU > K * > K * C >
Table B.5: Sensitive analysis as regards the unavoidable cost rate for θ = 0.5, ψ = 0.5 in P i = P HS = 148 Euros/t.

Z U N Number of uncertainties q 1 K 2 Ranking of optimal levels of q and K 20 % Two Uncertainties 11.39 6.21 q * CU > q * > q * C > q Discount rate Number of uncertainties q 1 K 2 Ranking of optimal levels of q and K 8 % Two Uncertainties 12.96 5.99 q * CU > q * > q * C > q

Figure

  Figure A.2: The marginal payoff ∂W (K, q; 0.5, θ)/∂K in function to K for q * (K) = 7.18 and q * (K) = 7.26. Calculated with β=0.1, τ =15 and p K =0.5.

  Figure A.3: The marginal payoff ∂W (K, q; 0.5, θ)/∂K in function to K for q * (K) = 7.18 for different values of p K when the agent is ambiguity averse. Calculated with β=0.1 and τ =15.

  In ton per hour; (2) In MEuros.

Table B .

 B 3: Optimal level of production and investment in function of the uncertainties for P LW = 100 Euros/t, P HW = 200 Euros/t, P LS = 80 Euros/t and P HS = 148 Euros/t

	Case	Number of uncertainties ψ	θ	q 1

Table B .

 B 4: Sensitive analysis as regards the investment coefficient, p K for θ = 0.5, ψ = 0.5 in case P i = P HS = 148 Euros/t.

	Number of uncertainties K 1		Values of p K
			0.2	0.5	0.8	1
	Certainty (C) Two Uncertainties	K * C K *	8.73 6.45 5.52 5.13 7.56 5.59 4.79 4.46
	Competitive effect (CU) K * CU Demand(DU) K * DU	9.33 6.88 5.89 5.47 7.16 5.31 4.55 4.23

Table B .

 B 6: Sensitive analysis as regards the discount rate for θ = 0.5, ψ = 0.5 in P i = P HS = 148 Euros/t.

		Certainty (C)	9.58 5.39	DU
		Competitive effect (CU) 12.33 6.63
		Demand (DU)	8.97 5.12	K * CU > K * > K * C > K * DU
		Two Uncertainties	15.26 6.77
	50 %	Certainty (C)	12.84 5.87	q * CU > q * > q * C > q * DU
		Competitive effect (CU) 16.52 7.23
		Demand (DU)	12.03 5.56	K * CU > K * > K * C > K * DU
	(1) In ton per hour; (2) In MEuros.

*

In

2007, the European Commission has fixed the renewable energy target in the EU's overall mix to 20% in the final energy consumption by 2020 regarding 1990. To reach this goal, the member states have adopted the pack energy-climate and renewable energy(European Commission, 2009) in particular which defines the operational measures to develop 20% of renewable energies by 2020.2 Currently biomass delivers around 4% of the EU's primary energy(EEA, 2008).

For more details on ambiguity approach, see[START_REF] Camerer | Ambiguity-Aversion and Non-Additive Probability: Experimental Evidence, Models and Applications[END_REF];[START_REF] Etner | Decision theory under uncertainty[END_REF].

Indeed, the agent knows that there already exists substitute of pre-treatment process which could provide the biomass consumers.

The energy price is determined by the total supply of energy and each unit considers the price as given. Then we do not consider the impact of additional capacity on price.

In a context in which there are two uncertainties affecting simultaneously the level and the variability of the price, an agent is more sensitive to the price variability than to the price average. The demand uncertainty affects the perception of the average price while the competitive effect uncertainty affects the perception of the average price and mostly the variance price. Therefore, we assume an ambiguity associated with the competition effect uncertainty.

Notice that we are interested by optimal decision and so the φ -1 has no influence on our results. We thank an anonymous referee for pointing out this property.

More precisely, a neutral agent maximizes the expected pay-off φ(V (K, q; ψ, θ)) where φ is linear function. For notation convenience, we assume φ is a scalar equal to 1 when agent is neutral to ambiguity.

The French potential of forest residues was estimated at over 30 Mt per year available for energetic use in 2015[START_REF] Meeddat | La fiscalité des biocarburants en France[END_REF].

In practical applications, this term is determined by arbitrarily selecting a set of parameters for this technology that lead to a very inefficient solution and by estimating the investment costs for this solution.

See Kumbaroglu et al. (2008) for references.

The results are robust for any other lower value of τ . In the portfolio choice examples proposed by[START_REF] Klibanoff | A smooth model of decision making under ambiguity[END_REF], the coefficient of ambiguity aversion varies between 1 and 20. For sake of clarity, we then take τ = 15.

Appendix A. Figures The marginal payoff ∂W (6.34, q; 0.5, θ)/∂q in function to q. Calculated with β=0.1, τ =15 and p K =0.5.

Appendix B. Tables