Haitao Chen

Dandan Li

Yao Cai

Dr Long-Fei Wu
email: wu@imm.cnrs.fr

Tao Song
email: songtao@mail.iee.ac.cn

Bacteriophytochrome from Magnetospirillum magneticum affects phototactic behaviour in response to light

Keywords: Bacteriophytochrome, magnetotactic bacteria, biliverdin, phototactic behaviour

Phytochromes are a class of photoreceptors found in plants and in some fungi, cyanobacteria, photoautotrophic, and heterotrophic bacteria. Although phytochromes have been structurally characterized in some bacteria, its biological and ecological roles in magnetotactic bacteria remain unexplored. Here, we describe the biochemical characterization of recombinant bacteriophytochrome (BphP) from magnetotactic bacteria Magnetospirillum magneticum AMB-1 (MmBphP). The recombinant MmBphP displays all the characteristic features, including the property of binding to biliverdin (BV), of a genuine phytochrome. Site-directed mutagenesis identified that cysteine-14 is important for chromophore covalent binding and photoreversibility.

Arginine-240 and histidine-246 play key roles in binding to BV. The N-terminal photosensory core domain of MmBphP lacking the C-terminus found in other phytochromes is sufficient to exhibit the characteristic red/far-red-light-induced fast photoreversibility of phytochromes. Moreover, our results showed MmBphP is involved in the phototactic response, suggesting its conservative role as stressprotective. This finding provided us a better understanding of the physiological function of this group of photoreceptors and photoresponse of magnetotactic bacteria.

Introduction

Light is an important environmental signal and provides sensory information for adaptation and energy for growth and metabolism. Protein-based photoreceptors have evolved to sense light quality changes in the environment and trigger critical lifestyle adaptations [START_REF] Gomelsky | Light helps bacteria make important lifestyle decisions[END_REF][START_REF] Gourinchas | Bacteriophytochromes-from informative model systems of phytochrome function to powerful tools in cell biology[END_REF][START_REF] Maresca | Light modulates the physiology of Nonphototrophic Actinobacteria[END_REF].

Phytochromes, which sense ambient light quality and quantity by responding to red and far-red light, are photoreceptor proteins used by bacteria, fungi, algae, and plants [START_REF] Duanmu | Marine algae and land plants share conserved phytochrome signaling systems[END_REF][START_REF] Nagano | From photon to signal in phytochromes: similarities and differences between prokaryotic and plant phytochromes[END_REF][START_REF] Rodriguez-Romero | Fungi, Hidden in soil or up in the air: Light makes a difference in[END_REF]. Phylogenetic analysis indicates that the phytochrome superfamily consists of five distinct clades, namely, plant phytochromes, cyanobacterial phytochromes, bacteriophytochromes (BphPs), fungal phytochromes, and a cluster of Phy-like sequences [START_REF] Blumenstein | The aspergillus nidulans phytochrome FphA represses sexual development in red light[END_REF][START_REF] Hughes | A prokaryotic phytochrome[END_REF][START_REF] Jaubert | Control of peripheral light-harvesting complex synthesis by a bacteriophytochrome in the aerobic photosynthetic bacterium Bradyrhizobium strain BTAi1[END_REF][START_REF] Jiang | Bacterial photoreceptor with similarity to photoactive yellow protein and plant phytochromes[END_REF][START_REF] Karniol | Phylogenetic analysis of the phytochrome superfamily reveals distinct microbial subfamilies of photoreceptors[END_REF].

Most of bacteriophytochromes (BphPs) use a linear tetrapyrrole chromophore (biliverdin, BV) and sense red and far-red light via a reversible shift from a redabsorbing form (Pr) to a far-red-absorbing form (Pfr) [START_REF] Rottwinkel | Bathy phytochromes in rhizobial soil bacteria[END_REF][START_REF] Vuillet | Evolution of a bacteriophytochrome from light to redox sensor[END_REF]. BphPs are composed of an N-terminal photosensory core domain (PCD) and a C-terminal output-transducing domain (OTD). The PCD comprises the period/aryl hydrocarbon receptor nuclear translocator/single-minded (PAS) that covalently binds to the open-chain BV chromophore via a thioether linkage with a conserved cysteine (Cys) residue, a cGMP phosphodiesterase/adenylyl cyclase/FhlA (GAF) domain stabilizing chromophore via polar and hydrophobic interactions, and a phytochromespecific (PHY) domain stabilizing the photoactivated Pfr state [START_REF] Burgie | Crystal Structure of Deinococcus phytochrome in the photoactivated state reveals a cascade of structural rearrangements during photoconversion[END_REF][START_REF] Gourinchas | Bacteriophytochromes-from informative model systems of phytochrome function to powerful tools in cell biology[END_REF][START_REF] Lamparter | The biliverdin chromophore binds covalently to a conserved cysteine residue in the N-terminus of Agrobacterium phytochrome Agp1[END_REF]). Upon light illumination, the PHYtongue element undergoes partial refolding into an α-helix after chromophore photoisomerization [START_REF] Burgie | Crystal Structure of Deinococcus phytochrome in the photoactivated state reveals a cascade of structural rearrangements during photoconversion[END_REF][START_REF] Gourinchas | Bacteriophytochromes-from informative model systems of phytochrome function to powerful tools in cell biology[END_REF]). These structural rearrangements around a chromophore binding site transduce the molecular signal to a downstream effector module via the PHY domain and its C-terminal OTD [START_REF] Bjorling | Structural photoactivation of a full-length bacterial phytochrome[END_REF][START_REF] Gourinchas | Bacteriophytochromes-from informative model systems of phytochrome function to powerful tools in cell biology[END_REF].

The C-terminal OTD of most BphPs possesses a two-component histidine kinase motif that transfers phosphate to a response regulator (RR) [START_REF] Gourinchas | Bacteriophytochromes-from informative model systems of phytochrome function to powerful tools in cell biology[END_REF][START_REF] Jaubert | A singular bacteriophytochrome acquired by lateral gene transfer[END_REF][START_REF] Karniol | The HWE histidine kinases, a new family of bacterial twocomponent sensor kinases with potentially diverse roles in environmental signaling[END_REF][START_REF] Rottwinkel | Bathy phytochromes in rhizobial soil bacteria[END_REF]. Other output modules, such as the PAS, GGDEF and EAL domains, which are involved in bacterial second messenger metabolism, have also been described [START_REF] Rockwell | Phytochrome structure and signaling mechanisms[END_REF]. BphP in Bradyrhizobium ORS278 (BrBphP3) is formed from only N-terminal PCD without the C-terminal OTD. BrBphP3's feauture lies in its binding to phycocyano-bilin rather than BV, and it has been acquired by lateral gene transfer from a cyanobacterial species [START_REF] Jaubert | A singular bacteriophytochrome acquired by lateral gene transfer[END_REF]). The functional roles of this BphPs that lack the C-terminal OTD have yet to be found. The analysis of magnetotactic bacteria (MTB) genomes reveals some BphP photoreceptors [START_REF] Brutesco | Bacterial host and reporter gene optimization for genetically encoded whole cell biosensors[END_REF][START_REF] Wang | Genomic evidence of the illumination response mechanism and evolutionary history of magnetotactic bacteria within the Rhodospirillaceae family[END_REF]), but only one BphP protein without a recognizable C-terminal OTD is found in the AMB-1 model strain of Magnetospirillum. MTB can synthesize intracellular magnetic crystals (magnetosomes) that allow cells to swim along geomagnetic field lines [START_REF] Faivre | Magnetotactic bacteria and magnetosomes[END_REF][START_REF] Frankel | Magnetite and magnetotaxis in microorganisms[END_REF]). This magnetotaxis of MTB can be dated back in the archean era, suggesting that MTB coevolved with the geomagneticfield over geological time [START_REF] Lin | Origin of microbial biomineralization and magnetotaxis during the Archean[END_REF]. Interestingly, these bacteria are rich in light perception systems. [START_REF] Frankel | Magneto-aerotaxis in marine coccoid bacteria[END_REF] found that the Magnetococcus strain MC-1 has a negative photoresponse behaviour to short-wavelength light. Multicellular magnetotactic prokaryotes present a photophobic response and a photokinesis behaviour [START_REF] Chen | A novel species of ellipsoidal multicellular magnetotactic prokaryotes from Lake Yuehu in China[END_REF][START_REF] De Melo | Light effects on the multicellular magnetotactic prokaryote 'Candidatus Magnetoglobus multicellularis' are cancelled by radiofrequency fields: the involvement of radical pair mechanisms[END_REF][START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF][START_REF] Shapiro | Multicellular photo-magnetotactic bacteria[END_REF][START_REF] Zhou | A novel genus of multicellular magnetotactic prokaryotes from the Yellow Sea[END_REF]. Illumination helps cell to eliminate intracellular reactive oxygen species (ROS) and triggers swimming toward a light source in Magnetospirillum magneticum AMB-1 [START_REF] Chen | Phototaxis in the magnetotactic bacterium Magnetospirillum magneticum strain AMB-1 is independent of magnetic fields[END_REF][START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF]. Although the biological roles of BphPs have been addressed in some literatures [START_REF] Bai | Bacteriophytochromes control conjugation in Agrobacterium fabrum[END_REF][START_REF] Bonomi | Xanthomonas campestris attenuates virulence by sensing light through a bacteriophytochrome photoreceptor[END_REF][START_REF] Moyano | Bacteriophytochromes from Pseudomonas syringae pv. tomato DC3000 modulate the early stages of plant colonization during bacterial speck disease[END_REF][START_REF] Mukherjee | Photosensing and quorum sensing are integrated to control Pseudomonas aeruginosa collective behaviors[END_REF][START_REF] Ricci | Visualizing the relevance of bacterial blueand redlight receptors during plant-pathogen interaction[END_REF][START_REF] Wu | Light regulation of swarming motility in Pseudomonas syringae integrates signaling pathways mediated by a bacteriophytochrome and a LOV protein[END_REF], the mechanism and biological function by which MTB respond to illumination remains unknown.

Accordingly, we determined whether the BphP of MTB is involved in the regulation of photoresponse of this bacterium. By combining genetics, biochemical and biophysical approaches, we found that BphP in AMB-1 (MmBphP) possesses red/far-red-light-induced photochemical properties and participates in phototactic behaviour.

Materials and methods

Strain and culture conditions

M. magneticum AMB-1 strains (ATCC 700264) were grown under microaerobic condition on Magnetospirillum growth medium at 30°C [START_REF] Chen | Efficient genome editing of Magnetospirillum magneticum AMB-1 by CRISPR-Case9 system for analyzing magnetotactic behavior[END_REF][START_REF] Matsunaga | Magnetite formation by a magnetic bacterium capable of growing aerobically[END_REF][START_REF] Yang | Effects of growth medium composition, iron sources and atmospheric oxygen concentrations on production of luciferasebacterial magnetic particle complex by a recombinant Magnetospirillum magneticum AMB-1[END_REF]. Escherichia coli strains BL21 (DE3) (Tiangen, China) and WM3064 (Philippe and Wu 2010: 309-22) were grown in a LB medium at 37°C. When appropriate, AMB-1 was grown with 15 μg/mL apramycin or kanamycin. E. coli cultures were supplemented with apramycin (50 μg/mL) or kanamycin (50 μg/mL).

Identification of MmBphP sequence

BphP was identified in the AMB-1 genomic database BLAST (protein accession number: WP_043744962.1). Phytochrome homologs were found with PDB 6FHT as a template [START_REF] Jaubert | A singular bacteriophytochrome acquired by lateral gene transfer[END_REF]). The domain was analyzed using SMART (http://smart.embl-heidelberg.de). For structural prediction, the protein sequence was examined using the Phyre2 program (http://www.sbg.bio.ic.ac.uk/phyre2/html/) [START_REF] Kelley | The Phyre2 web portal for protein modeling, prediction and analysis[END_REF]. Images were generated using AutoDock and PyMOL.

Cloning, expression, and purification of MmBphP

The sequence of MmBphP was PCR-amplified by using primers designed to introduce BamHI and XhoI sites before ATG and designated stop codons, respectively. The fragment was then cloned into the pET28a plasmid with N-terminal His6-SUMOtagged fusion proteins. C14A, D197A, C211A, R240A, H246A, Y249A, S260A, C275A, H276A, C291A, C14&R240A, C14&H246A, and C14&R240A&H246A mutants were cloned using a NEBuilder HiFi DNA assembly reaction kit in accordance with the manufacturer's instructions. These proteins were expressed in E. coli BL21 (DE3) cells and purified. After nickel chelate affinity chromatography, the proteins were further treated with SUMO protease and purified through size exclusion chromatography on a Superdex 200 column in Tris buffer (10 mM Tris/Cl, 200 mM NaCl, pH8.0). For BphPs assembled in vitro, apoproteins were incubated in darkness in a 10-fold molar excess of BV (Sigma, USA) at room temperature for 1 h. Subsequently, proteins were separated from the free BV by using a DP-10 desalting column (GE Healthcare, USA). The covalent attachment of BV to MmBphP and mutant proteins was monitored through the zinc-induced fluorescence of the chromoproteins subjected to SDS-PAGE. Blue native PAGE was performed on a natural PAGE gel consisting of 3.5% stack and 10% gradient separation gel. The cathode buffer and the anode were cooled at 4°C before it was used. Electrophoresis was started at 80 V for 30 min, increased to 120 V for 1 h, and adjusted to 200 V for 1 h [START_REF] Li | Structural and evolutionary characteristics of dynaminrelated GTPase OPA1[END_REF].

Covalent attachment of bilins to MmBphP and mutants were monitored by zinc-induced fluorescence of the chromoproteins separated by SDS-PAGE. When purified MmBphP was resolved on SDS-PAGE gels and incubated with 1.5 M zinc acetate, it was visible under ultraviolet light as an orange fluorescent band [START_REF] Berkelman | Visualization of bilin-linked peptides and proteins in polyacrylamide gels[END_REF]).

The fluorescent biliprotein-containing bands were visualized by transilluminating the gel on an AlphaImager HP UV light (365 nm) box.

Genetic manipulation of AMB-1

The in-frame deletion of bphP was created with a CRISPR-Cas9 system by using the pCRISPomyces-2 plasmid as previously described [START_REF] Cobb | High-efficiency multiplex genome editing of Streptomyces species using an engineered CRISPR/Cas system[END_REF]. A sgRNA cassette was designed and inserted into the pCRISPomyces-2 by BbsI site. PCR was performed using PrimerSTAR HS DNA polymerase (Takara, Japan) to amplify 1.0 kb upstream of the start codon in the coding region, and the 1.0 kb sequence downstream of the stop codon and to delete bphP. These fragments were then incorporated into an XbaI -digested pCRISP-sgRNA vector by using the NEBuilder HiFi DNA assembly reaction system (NEB, USA).

The pCRISP-sgRNAbphP-mutant plasmid was constructed as follows.

bphP C14A&R240A&H246A , 1.0 kb upstream sequence, and 1.0 kb downstream sequence were PCR amplified using PrimerSTAR HS DNA polymerase (Takara, Japan). The resulting 3.5 kb fragment was incorporated into the pET28a plasmid. The sgRNA sequence in the bphP C14A&R240A&H246A fragment was subjected to nonsense mutation to prevent the sgRNA of the CRISPR plasmid from editing the inserted bphP C14A&R240A&H246A . The resulting plasmid was subjected to site-directed mutagenesis in accordance with the NEBuilder HiFi DNA assembly reaction protocol to introduce the sgRNA sequence of nonsense mutation (sgRNA sequence: GGACTCGCAGCGTTGTACTA to GGACTCGgcGCGTTGTACTA) into the bphP sequence. The site-directed 3.5 kb fragment was inserted into pCRISPR-sgRNAbphP plasmid with the NEBuilder HiFi DNA assembly reaction system.

The complementing vector ComBphP was constructed through PCR amplification with

PrimerSTAR HS DNA polymerase (Takara, Japan). The ComBphP coding sequence was amplified and ligated into EcoRI-and BamHI-digested pAK0994, a pBBR1MCSbased plasmid carrying a kanamycin resistance gene and expressing the amb0994-gfp fusion from a tac promoter. This method was performed as described previously [START_REF] Chen | Efficient genome editing of Magnetospirillum magneticum AMB-1 by CRISPR-Case9 system for analyzing magnetotactic behavior[END_REF].

All the plasmids were mobilized into AMB-1 through conjugation with the E. coli strain WM3064, and CRISPR-based double-crossover events for deletions or allelic exchange were employed using a selection and screening strategy as described previously [START_REF] Chen | Efficient genome editing of Magnetospirillum magneticum AMB-1 by CRISPR-Case9 system for analyzing magnetotactic behavior[END_REF]. All the deletions were verified through PCR and sequencing. All the plasmids and primers used are listed in Tables S1 andS2.

Spectroscopic characterization

The absorbance spectra of the purified MmBphP wild-type (WT) and mutant proteins were obtained at room temperature from 850 nm to 250 nm by using a UV-visible spectrophotometer (Unico UV-2800, USA). The assay was recorded either in the dark or after 15 min of illumination with 625 nm, 680 nm, 710 nm, or 750 nm LED light with a half-width of 30 nm at 10 μmol photons/m 2 /s (Hasunopto, China). Fig. S1 shows the spectra of these light sources. Dark recovery kinetic was followed at 708 nm and 753 nm after 15 min under 710 nm light. For dark conversion, the half-life time was estimated as the time in which 50% of the Pfr-to-Pr conversion took place considering the saturation value as 100%. The two phase association exponential fit was used to derive the half-life parameter. Room-temperature fluorescence emission and excitation spectra were recorded using a fluorescence spectrophotometer (Hitachi F-4500, Japan).

Fluorescence spectra were obtained at 680 nm excitation or 750 nm emission.

Phototactic behaviour analyses of AMB-1

Phototactic behaviour was quantitatively analyzed as previously described [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF]. Using a modified mini MTB collection vessel, the cultures with or without H2O2 induction were pooled into the reservoir that is connected to the collecting tube at one side. And then the device was placed in a cassette, where the collection tube was exposed parallelly to the light beam. Before conducting the phototactic experiment, a microaerobic band formed after growing in the dark for 30 min. The upper interface of the microaerobic zone is parallel to the collection tube. After 30 min of 710 nm or 750 nm light irradiation, the OD in the collecting tube and the reservoir was measured (ODcollecting tube and ODreservoir, respectively). The relative numbers of phototactic cells were analyzed by the ratio of (ODcollecting tube -ODreservoir)/ODreservoir. Control experiments were performed in the dark. In this study, we analyzed the effect of H2O2 on the photoresponse between WT and mutants. To prevent the difference in photoresponse caused by H2O2 on the state of bacterial movement for WT and mutants, we observed that at 40 µM concentrations H2O2 affected nonsignificantly the velocities of these strains (Fig. S2, Supporting Information). Thus, we studied the difference in phototactic behaviour between the WT and mutants under the conditions of without or with 40 µM H2O2. The statistical analyse of phototactic behaviour was conducted through Mann-Whitney U-test using SPSS 22.0 (SPSS, IBM).

Results

Identification of BphP

According to the composition of the C-terminal OTD, the domain organization of BphPs can be classified into nine types (Fig. 1A). Three of them (e, f and h) are found in freshwater AMB-1, MSR-1, MS-1, XM-1, MV-1 or marine QH-2 MTB.

Interestingly, BphP proteins in some of these MTB are found to be type (h) BphP which lack the C-terminal OTD (Fig. 1B). The representative MmBphP in the model strain AMB-1 of Magnetospirillum sp. consists of a 496 aa PAS-GAF-PHY domain (Fig. 1B). Fig. 1C shows the arrangement of the genes located downstream or upstream of MmBphP in AMB-1, revealing a heme oxygenase gene close to MmBphP. The heme oxygenase opens the heme ring to form the linear tetrapyrole BV which is the first intermediate in phytochrome chromophore synthesis [START_REF] Muramoto | The Arabidopsis photomorphogenic mutant hy1 is deficient in phytochrome chromophore biosynthesis as a result of a mutation in a plastid heme oxygenase[END_REF], suggesting MmBphP may be combined with BV in its natural state to perform its functions. Accordingly, by taking AMB-1 as the research object, we studied the possible involvement of BphP with no recognizable OTDs in the photoresponse regulation in MTB. To further enrich the function assessment of BphPs lacking OTDs, we purified the MmBphP protein and constructed the MmBphP deletion mutant in AMB-1 strain.

Characteristics and photochemical properties of recombinant MmBphP protein

Recombinant MmBphP was detected through SDS-PAGE electrophoresis, and a single band migrating at ~54 kDa was observed (Fig. S3 B, Supporting Information).

According to the peak position of the elution profiles of a Superdex 200 size-exclusion column, MmBphP displayed a molecular weight of approximately 108 kDa, suggesting the dimer property of these proteins in the solution (Fig. S3 A, Supporting Information).

Blue native electrophoresis results revealed that the protein could form stable homodimers (Fig. S3 B, Supporting Information).

In Fig. 2, the purified MmBphP binding to the BV chromophore appears blue green (insert in Fig. 2A). The absorption spectrum of the dark-adapted MmBphP sample is typical of the Pr ground state with a main band centered at 708 nm (Fig. S4, Supporting Information). Illumination of the Pr state with 625 nm, 680 nm or 710 nm light induces only a small absorption increase in the near infra-red region around 753 nm (Fig. 2A; [START_REF] Gourinchas | Bacteriophytochromes-from informative model systems of phytochrome function to powerful tools in cell biology[END_REF]. In this study, the 710 nm light was used as red light illumination, and a 750 nm light was used as far-red illumination source.

To assess whether MmBphP has similar structural characteristics as other BphPs, we constructed a predictive structural model of MmBphP using a homologue protein (PBD code 6FHT) as template through Phyre2 search. As shown in Fig. 2B. Like other BphPs [START_REF] Yang | Crystal structure of the chromophore binding domain of an unusual bacteriophytochrome, RpBphP3, reveals residues that modulate photoconversion[END_REF][START_REF] Gourinchas | Bacteriophytochromes-from informative model systems of phytochrome function to powerful tools in cell biology[END_REF], MmBphP is composed of PAS, GAF, and PHY domains. The enlarged dotted line on the right of the predictive structure, by using AutoDock and PyMOL, shows that MmBphP is associated with a bounded BV.

The occurrence of a fast and near complete reversion to the dark-adapted state in Fig. 2C shows that the absorption kinetics changes at 708 nm or 753 nm after 710 nm illumination treatment. The half-time of the dark recovery is 7.8 s at 25°C (Table 1) and several orders of magnitude faster than those reported in phytochromes or BphPs in which dark recoveries occur in periods ranging from minutes to days [START_REF] Karniol | The pair of bacteriophytochromes from Agrobacterium tumefaciens are histidine kinases with opposing photobiological properties[END_REF]. However, a faster dark recovery than MmBphP has been reported for BrBphP3 in Rhizobium ORS278 that shows a half-time of 460 ms in dark recovery after illumination [START_REF] Jaubert | A singular bacteriophytochrome acquired by lateral gene transfer[END_REF]).

To further investigate the photochemical characteristics of MmBphP, we analyzed the fluorescent properties of this protein. Fluorescence spectra were obtained at 680 nm excitation or 750 nm emission. Fig. 2D shows the excitation (black line) and emission (gray line) spectra for a suspension of MmBphP recorded at room temperature. The fluorescence excitation spectra of MmBphP closely match the absorption spectra at 708 nm. The fluorescence emission spectra are centered at 723 nm, which indicates that the fluorescent originates from the BphP-bound BV chromophore [START_REF] Toh | Fluorescence quantum yield and photochemistry of bacteriophytochrome constructs[END_REF]. Similar to the emission spectra of MmBphP, DrBphP emits fluorescence in the near-infrared region at 720 nm, which is substantially longer than the wavelength emitted by GFPderived fluorescent proteins [START_REF] Shu | Mammalian expression of infrared fluorescent proteins engineered from a bacterial phytochrome[END_REF].

Identification of MmBphP variants unable to bind to BV

In general, a conserved Cys is in the PAS domain in most bacteria, or in GAF domain in cyanobacterial and plant phytochromes forms the covalent linkage to the bilin chromophore [START_REF] Kreslavski | The impact of the phytochromes on photosynthetic processes[END_REF]. To determine the Cys residues involved in covalent attachment to the bilin chromophore in AMB-1, we carried out site-directed mutant experiments (C14A, C211A, C275A, and C291A). In addition AMB-1 BphP residues D197, R240, H245, Y249, S260, and H276, which are conserved to those interacting with the propionate side chains of BV in Rhodopseudomonas palustris [START_REF] Yang | Crystal structure of the chromophore binding domain of an unusual bacteriophytochrome, RpBphP3, reveals residues that modulate photoconversion[END_REF], were also substituted for alanine to analyze their potential binding to BV (Fig. S5 A, Supporting Information).

The binding of BV to WT or mutated AMB-1 MmBphP was analyzed by a zinc-induced fluorescence assay. As demonstrated by zinc-induced fluorescence, apo-MmBphP and all the variants containing C14A show negative results in zinc-induced fluorescence test, confirming the importance of Cys14 for covalently binding to BV (Fig. 3A). To further compare kinetic properties among variants and WT phytochromes, we analyzed the absorption spectroscopy and calculated the half-life time obtained from the experiment that the absorption kinetics changes at 708 nm after 710 nm illumination treatment (Fig. 3B; Table 1). Fig. 3B shows that the absorbance of C14A, D197A or H246A at 708 nm is greatly reduced upon 710 nm light illumination compared with WT. Moreover, the illuminated states possess different life times. The half-life times are 387.9 s, 44.2 s and 5790.1 s for C14A, D197A and H246A, respectively (Table 1).

The final photoproduct of these three variants is much longer stable than that of WT. C211A, R240A, C275A, and C291A exhibit minimal photoconversion and a fast reversion to the dark-adapted state, which is similar to WT. The H276A mutant adopts the Pr dark state but shows no detectable formation of the Pfr state upon illumination at 710 nm. This may result either from completely blocked photoconversion to the Pfr state or from formation of a Pfr state that is too short-lived to be detected in our experiments [START_REF] Yang | Conformational differences between the Pfr and Pr states in Pseudomonas aeruginosa bacteriophytochrome[END_REF]). Interestingly, Y249A and S260A resulted in a characteristic of the Pfr state upon 710 nm light illumination and exhibited slower dark recovery compared with WT.

Red light/darkness induced spectral analysis showed that compared with WT protein, C14A could form a photoconvertible holoform with blue-shifted extrema and had detectable absorption at 702 nm in the dark, as well as R240A and H246A (Fig. 3; Table 1). The crystal structures of the chromophore-binding pocket of RpBphP2 and RpBphP3 showed that the Arg240 and His246 residues interact with the propionate side chains of BV [START_REF] Bellini | Dimerization properties of the RpBphP2 chromophorebinding domain crystallized by homologue-directed mutagenesis[END_REF][START_REF] Yang | Crystal structure of the chromophore binding domain of an unusual bacteriophytochrome, RpBphP3, reveals residues that modulate photoconversion[END_REF]. Experimental data confirmed that the H247Q mutation played a role in the stabilization and coordination of the chromophore [START_REF] Tasler | Biochemical and spectroscopic characterization of the bacterial phytochrome of Pseudomonas aeruginosa[END_REF]. Fig. 3 shows that the single substitution of these residuces affects zinc-induced fluorescence assay or photochemical spectra at different levels. To assess the accumulation effect of these mutations, we further constructed the double mutants C14&R240 and C14&H246, and the triple mutant C14A&R240A&H246. We observed additional effect of the substitutions, i.e. they almost completely abolished the absorption spectra at 600-800 nm (Fig. 3B).

The Pr and Pfr states of BphPs typically display maximum absorption peaks at approximately 700 nm and 750 nm, respectively, with some overlap in the region between the two peaks. To avoid cross excitation, the red-light sources used to illuminate Pr state are shifted towards lower wavelengths from the maximum absorption peak to 625 nm or 680 nm in order to avoid excitation of the Pfr state.

However, for some BphPs that the Pr state displays an absorbance maximum at 700-710 nm, some researchers used 700 nm or 705 nm LED light for red light illumination of Pr, and 750 nm for Pfr [START_REF] Giraud | A new type of bacteriophytochrome acts in tandem with a classical bacteriophytochrome to control the antennae synthesis in Rhodopseudomonas palustris[END_REF][START_REF] Fixen | Apo-bacteriophytochromes modulate bacterial photosynthesis in response to low light[END_REF]. We compared spectra of WT MmBphP or mutants illuminated at 710 nm with those at 625 nm, 680 nm or 750 nm. The spectra results by using 625 nm, 680 nm, and 710 nm light sources were similar (Fig. S4, Supporting Information; Fig. S6, Supporting Information). Therefore, we reported the results obtained using LEDs at 710 nm as red light illumination, and 750 nm as far-red illumination source in most case except otherwise specified in this study.

Analysis of phototactic behaviour in AMB-1

Previously we reported that illumination triggers cells to swim toward a light source and helps cells to eliminate intracellular ROS in AMB-1 [START_REF] Chen | Phototaxis in the magnetotactic bacterium Magnetospirillum magneticum strain AMB-1 is independent of magnetic fields[END_REF][START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF]). In addition, research reveals that RpBphP4 is redox sensitive [START_REF] Vuillet | Evolution of a bacteriophytochrome from light to redox sensor[END_REF].

To investigate the effect of MmBphP on phototactic response, we constructed mutant strains and analyzed the phototactic behaviour with or without H2O2 in AMB-1 cells by using a modified MTB collection vessel [START_REF] Chen | Light regulation of resistance to oxidative damage and magnetic crystal biogenesis in Magnetospirillum magneticum mediated by a Cys-less LOV-like protein[END_REF][START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF]). The phototactic behaviour was analyzed by illuminating with 710 nm and 750 nm light sources. For AMB-1 WT and mutant strains, the relative number of cells swimming toward 710 nm or 750 nm light was significantly higher (p<0.01) than that under the dark condition (Fig. 4). Moreover, for the 710 nm illumination and H2O2 treatment groups, the photoresponse of ΔBphP or BphP C14A&R240A&H246A mutant cells was significantly weaker than that of WT cells (Fig. 4, p<0.01). In contrast, for 750 nm illumination and H2O2 treatment groups, there was no significant difference in the phototactic behaviour between WT and mutant cells (p>0.05). For strains without H2O2 treatment, the number of WT cells swimming toward light was not significantly different from that of the mutant strain (p>0.05). In addition, there was no significant difference in the phototactic response between WT and ComBphP cells (p>0.05).

Discussion

Photosensing proteins play key roles in the regulation of physiological activities in response to light, a critical element in the acclimatization to the environment. In this study, we described the properties of photosensing protein MmBphP, a new type of BphPs, presenting in the MTB model strain AMB-1. MmBphP has red/far-red-lightinduced photoreversibility features among all members of the phytochrome superfamily. The characteristic absorbance spectra and photoconversion of MmBphP reflect its association with BV chromophore, which are covalently attached at the conserved Cys14 in the PAS domain as most BphPs and Fugal phytochromes do, while plant and cyanobacterial phytochromes are covalently attached at a conserved Cys in the GAF domain. BV adopts the special conformation in the Pr state through the vinyl group of ring A linking to the cysteine covalently and goes through an isomerization at its C15-C16 double bond upon light activation, which results in rotation of ring D [START_REF] Gourinchas | Bacteriophytochromes-from informative model systems of phytochrome function to powerful tools in cell biology[END_REF]. The MmBphP protein lacking the C-terminal part showed a fast recovery to the stable state after illumination. Our result is consistent with previous findings that the dark recovery is fast in BphPs lacking C-terminal region. BrBphP3 lacking natural C-terminal region in Rhizobium ORS278 showed a fast dark recovery with half-life time of 460 ms [START_REF] Jaubert | A singular bacteriophytochrome acquired by lateral gene transfer[END_REF]). In Agrobacterium tumefaciens and Xanthomonas campestris, the Agp2 variant lacking the output module and XccBphP lacking the PAS9 domain showed much faster kinetics, pointing to the modulation of the dark reversion of the chromophore by the output module [START_REF] Velazquez | A protonation-coupled feedback mechanism controls the signalling process in bathy phytochromes[END_REF][START_REF] Otero | Structure of the full-length bacteriophytochrome from the plant pathogen Xanthomonas campestris provides clues to its longrange signaling mechanism[END_REF]).

In addition, our results showed that light induced photoconversion had a small absorption increase in the near infra-red region around 753 nm, and long-term irradiation would not affect these absorption spectra (data not shown). An obvious Pfr state could not be fully detected in various conditions assayed in this study. We speculate MmBphP may experience a 'fast' Pfr reaction, which evolved on a picosecond or microsecond time scale [START_REF] Heyne | Ultrafast dynamics of phytochrome from the Cyanobacterium Synechocystis, reconstituted with phycocyanobilin and phycoerythrobilin[END_REF][START_REF] Jaubert | A singular bacteriophytochrome acquired by lateral gene transfer[END_REF][START_REF] Singer | Spectroscopic investigation on the primary photoreaction of bathy phytochrome Agp2-Pr of Agrobacterium fabrum: isomerization in a pH-dependent H-bond network[END_REF]. This may result from formation of an obvious Pfr state that is too short-lived to be detected in our experiments. Another possibility is that a supplementary protein or component in AMB-1 cells is involved in controlling Pfr to Pr conversion and they are absent in hetero-expressed and purified recombinant MmBphP protein complex.

Nevertheless, we do have observed a rapid absorption kinetics change at 708 nm or 753 nm after 710 nm illumination treatment in Fig. 2C, which indicates the occurrence of photoconversion in MmBphP after irradiation.

In the spectral measurement (Fig. 3B; Table 1), substitution of Cys14, Asp197 or His246 exhibited a significantly weakened and broadened absorption band in the red, and detected a slower dark recovery upon 710 nm light illumination compared with WT, suggesting their possibility of being the BV binding pocket which is lined by many highly conserved residues in MmBphP. Alanine substitutions at Cys211, Cys275, and Cys291 show Pr/Pfr photoconversion efficiency similar to the WT MmBphp, demonstrating the variants could still covalently bind to BV and exhibit a little effect on BV configuration or photoconversion. Moreover, the maximum absorption peaks of C14A, R240A and H246A mutants showed a blue shift of 6 nm to 702 nm in the dark state, indicating that they may affect the binding to BV. Strikingly, illumination of red light on single mutant Y249A and S260A resulted in the appearance of an obvious absorption band (753 nm), characteristic of the Pfr state. These two mutants underwent reversible Pr/Pfr photoconversion, but its rate of reversion to the Pr state in the dark is significantly slower than that of WT, with a half-life time of about 247.8 s and 411.5 s, respectively. Thus, introduction of a single alanine residue into the location of Tyr249 or Ser260 in MmBphP was sufficient to replace the photoconversion of WT MmBphP, suggesting Tyr249 and Ser260 may locate at the binding pocket of MmBphp, which has a direct impact on the Pr/Pfr photoconversion. Photoconversion of the H276A, is barely detectable in our experiments, but it has normal Pr absorption spectra in the dark. The phenotype indicates that it is likely to be involved in engaging the GAF domain for photoconversion, as the residue in the GAF domain usually interact with the chromophore primarily [START_REF] Yang | Crystal structure of the chromophore binding domain of an unusual bacteriophytochrome, RpBphP3, reveals residues that modulate photoconversion[END_REF]. The spectral properties of the variants tested can be noncovalent associations with BV. Because photoconversion does not appear to take covalent attachment as the prerequisite, covalent attachment may provide a more stable holoprotein that make phytochrome suited to reversible photoswitching better by utilize bilin chromophores [START_REF] Rockwell | Phytochrome structure and signaling mechanisms[END_REF]). In addition to the output module affecting dark recovery [START_REF] Velazquez | A protonation-coupled feedback mechanism controls the signalling process in bathy phytochromes[END_REF][START_REF] Otero | Structure of the full-length bacteriophytochrome from the plant pathogen Xanthomonas campestris provides clues to its longrange signaling mechanism[END_REF], our results suggest that the amino acids at the BV binding pocket also affect Pr/Pfr photoconversion efficiency.

Interestingly, BphP C14A&R240A , BphP C14A&H246A or BphP C14A&R240A&H246A exhibited little red, far-red, or near-red absorption and failed to covalently bind to BV, indicating the importance of Cys14, Arg240 and His246 forming the BV binding pocket to some extent. The substitution of arginine with alanine, which abolishes two salt bridges that form with the propionate side chain of the B pyrrole ring, is sufficient to prevent the covalent and noncovalent association of BV with RpBphPs [START_REF] Fixen | Apo-bacteriophytochromes modulate bacterial photosynthesis in response to low light[END_REF]). Thus, the conserved arginine in MmBphP may be required for the autocatalytic incorporation of BV. The histidine was characterized to be of important for the chromophore attachment in Calothrix sp. PCC7601 CphB [START_REF] Quest | Chromophore selectivity in bacterial phytochromes-Dissecting the process of chromophore attachment[END_REF]. H247Q mutation in Pseudomonas aeruginosa resulted in a spectral shift of the Pr and Pfr forms. This result suggested that these mutations play a role in the stabilization and coordination of the chromophore [START_REF] Tasler | Biochemical and spectroscopic characterization of the bacterial phytochrome of Pseudomonas aeruginosa[END_REF]. In comparison with the other BphPs, these different characteristics observed in these variants of MmBphP may be related to an increase in the number of hydrogen bonds in the BV-binding pocket, which increases the rigidity of the BV environment [START_REF] Fixen | Apo-bacteriophytochromes modulate bacterial photosynthesis in response to low light[END_REF].

Currently, studies have shown that ultraviolet radiation and free-iron-generated ROS have been major environmental challenges in life on early Earth. Therefore, the initial biomineralization of iron nanoparticles in early life cells has developed into a mechanism to reduce the toxicity of ROS [START_REF] Lin | On the origin of microbial magnetoreception[END_REF]. [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF] found that light helps to eliminate intracellular ROS in AMB-1 cells. [START_REF] Fei | The role of phytochromes in Nicotiana tabacum against Chilli veinal mottle virus[END_REF] reported that phytochrome A and phytochrome B can help reduce ROS in the cell and are potent regulators of plant defense. Our result showed that under the oxidative stress of H2O2 treatment, the number of WT cells moving toward 710 nm light is significantly higher than that of MmBphP mutant strains, while the phototactic behaviour has no significant difference between WT and mutants by illuminating with 750 nm light. The result

Supplementary data

Supplementary data are available at FEMSLE online

Fig

  Fig.S4, Supporting Information), which is similar to the 'Meta' state of Agp1 BphP

indicates

  MmBphP can respond to red light to activate antioxidant enzymes such as catalase and peroxidase when transformation of Pr into Pfr occurred (Kreslavski et al. 2018), and further help fight toxic effects of oxidative stress or ROS, allowing WT cells to swim toward the light source. The lacking C-terminal region results in a 'classical' output module missing, one may, however, consider MmBphP a genuine photochrome because of the properties of photo response shown in this study. The mechanism of how the C14, R240, H246 affect the photoconversion behaviours through interaction with BV configurations (Fig. S5 B, Supporting Information) and how MmBphP are involved in the light transduction pathway need further study.

Fig 1

 1 Fig 1 Domain organization and arrangement of genes for BphPs. (A) The domain organization of BphPs in bacteria can be classified into nine types (a-i). (B) Domain organization of various BphP sequences in MTB. (C) Arrangement of the genes located downstream or upstream of MmBphP in AMB-1. The arrows indicate the direction of transcription. HmuO encodes heme oxygenase.

Fig 2

 2 Fig 2 Biochemical characterization of MmBphP. (A) Absorption spectra of purified MmBphP, MmBphP apoprotein (purple dotted line), BV (green dotted line), after photoconversion with red light (red line, 710 nm) and far red light (dark red line, 750 nm), and their difference spectrum (orange dotted line). The insert shows the images of BphP apoprotein, BV and MmBphP with BV binding. (B) The possible molecular configuration of BV used in this work that could be triggered by red or far red light (upper). A predictive structure of MmBphP with BV binding was constructed through Phyre2 search (lower). MmBphP is composed of PAS (blue), GAF (green), and PHY (orange) domains. BV is shown in light blue. (C) Dark reversion of MmBphP after 710 nm illumination. Dark recovery kinetic was followed at 708 nm (black line) and at 753 nm (dotted line). (D) The fluorescence excitation spectrum (black line) and emission spectrum (gray line) of MmBphP.

Fig 3

 3 Fig 3 Zinc-induced fluorescence and absorbance spectra of MmBphP variants. (A) Purified proteins were subjected to SDS-PAGE and assayed for assembly with BV by using a zinc-induced fluorescence assay. (B) Absorption spectra of purified WT and selected mutants of MmBphP. The spectra were obtained for protein incubated in the dark (black line) and after illumination with 710 nm light for 15 min (red dashed line).

Fig 4

 4 Fig 4 Relative phototactic behaviour (%) of AMB-1 WT cells and mutants (n=5). Bars and error bars depict the mean±SD. Level of significance of the differences observed between light and dark control is expressed as hashtag sign (#p<0.05 and ##p<0.01). The significant differences between the two groups depicted by the lines in the graph are indicated by asterisks ( * p<0.05 and * * p<0.01).

Table 1

 1 The spectral properties of MmBphP variants

		Absorption spectrum
	Proteins		Half-life time (s)
		Pr λmax (nm)
	MmBphP	708	7.8
	C14A	702	387.9
	D197A	708	44.2
	C211A	708	9.1
	R240A	702	14.6

Acknowledgements

We thank Changyou Chen with the help of useful discussion and suggestions.

The authors declare no conflict of interest.

Funding

This work was supported by the State Key Program of National Natural Science of China (Grant No. 51937011), the National Youth Foundation of China (Grant No. 51907192), the Research Project Funded by the Institute of Electrical Engineering, Chinese Academy of Sciences (Grant No. Y650141CSA), and CNRS-CAS for LIA-MagMC.

Author contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. HC, L-FW and TS conceived and designed the experiments. HC, DL and YC prepared samples and carried out the experiment. HC, L-FW and TS reviewed, analyzed and interpreted data. HC and TS wrote the paper and all authors discussed the results and commented on the manuscript.

Competing interests


Alexander Fedotov

Frédéric Klopp

Pointwise

Pointwise existence of the Lyapunov exponent for a quasi-periodic equation

POINTWISE EXISTENCE OF THE LYAPUNOV

where v : R → R is continuous and periodic, v(x + 1) = v(x), 0 < ω < 1 and 0 ≤ θ < 1.

When ω ∈ Q, the mapping n → v(nω + θ) is quasi-periodic.

The spectral theory of such quasi-periodic equations is very rich, and the study has generated a vast literature; among the authors are A. Avila, Y. Avron, J. Bellissard, J. Bourgain, V. Buslaev, V. Chulaevsky, D. Damanik, E. Dinaburg, H. Eliasson, A. F., B. Helffer, M. Hermann, S. Jitomirskaya, F. K., R. Krikorian, Y. Last, L.Pastur, J. Puig, M. Shubin, B. Simon, Y. Sinaï, J. Sjöstrand, S. Sorets, T. Spencer, M. Wilkinson and many others (see e.g. [START_REF] Puig | Reducibility of quasi-periodic skew products and the spectrum of Schrödinger operators[END_REF] for a recent survey). Speaking about intriguing spectral phenomena, one can mention for example that • for such equations, the spectral nature depends on the "number theoretical" properties of the frequency ω; and that one expects that

• typically such equations exhibit Cantorian spectrum;

• σ pp (H θ ), the singular continuous spectrum, is topologically typical. This has been well understood only for a few models, most prominently, for the almost Mathieu equation when v(x) = 2λ cos(x)). 0.2. Lyapunov exponent. One of the central objects of the spectral study of the quasi-periodic equations is the Lyapunov exponent. Recall its definition. Equation (1) can be rewritten as [START_REF] Cycon | Schrödinger Operators[END_REF] ψ(n + 1)

ψ(n) = M(nω + θ) ψ(n) ψ(n -1) , M(x) = E -v(x) -1 1 0 .
So the large n behavior of solutions to (1) can be characterized by the limits (when they exist) :

γ + (E, θ) = lim n→+∞ 1 n log M((n -1)ω + θ) • • • M(θ + ω) M(θ) (3) γ -(E, θ) = lim n→+∞ 1 n log M -1 (θ -nω) • • • M -1 (θ -2ω) M -1 (θ -ω) (4)
Furstenberg and Kesten have proved Theorem 1 ( [START_REF] Cycon | Schrödinger Operators[END_REF]). Fix E. For almost every θ, these limits exist, coincide and do not depend on θ.

For energies E such that the limits exist, coincide and do not depend on θ, their common value is called the Lyapunov exponent; we denote it by γ(E). We are interested in the pointwise (in both E and θ) existence of the limits γ + (E, θ) and γ -(E, θ). We call them the right and left Lyapunov exponents. Speaking about the pointwise existence of the Lyapunov exponent itself, we say that it does not exist for a pair (E, θ) when either at least one of γ ± (E, θ) does not exist or both of them exist, but at least one of them differs from γ(E). As, in general, singular continuous spectrum can be present, in this statement, one cannot replace σ s , the singular spectrum, with σ pp , the pure point spectrum. One may ask if it is possible to characterize the singular continuous spectrum in terms of the Lyapunov exponent. Consider equation (1) on the interval E ∈ I where γ(E) > 0. Almost surely, for a given θ, the Lyapunov exponents exist a priori only almost everywhere in E. Denote by I Lyapunov the subset of I where γ + (E, θ) and γ -(E, θ) both exist and are positive. For E ∈ I Lyapunov , the solutions to (1) have to increase or decrease exponentially (see, e.g., [START_REF] Cycon | Schrödinger Operators[END_REF]). This implies that the singular continuous component of the spectral measure vanishes on I Lyapunov . So, it can be positive only on I \ I Lyapunov . And, the latter must happen if the spectrum on I is singular continuous. 0.4. B. Simon's example: We now recall an example by B.Simon showing that, for quasi-periodic operators, one can find singular continuous spectrum on an interval where the Lyapunov exponent is positive. Consider the Almost Mathieu equation, i.e., equation [START_REF] Buslaev | The monodromization and Harper equation[END_REF] with v(θ) = 2λ cos θ. For this equation, by Herman's theorem ( [START_REF] Michael | Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnol'd et de Moser sur le tore de dimension 2[END_REF]), γ(E) ≥ log λ. We assume that λ > 1. Then, γ(E) is positive for all E, and the spectrum is singular. Let the frequency ω be such that, for some infinite sequence (p m , q m ) ∈ N × N * ,

ω - p m q m ≤ m -qm .
Such Liouvillean frequencies are topologically typical but of zero measure. One has

Theorem 3 ([2]
). Under the above conditions, there are no eigenvalues and the spectrum is purely singular continuous.

Note that this result is a consequence of a theorem by A. Gordon (see [START_REF] Ya | A sufficient condition for continuity of the spectrum of a discrete Schrödinger operator[END_REF][START_REF] Ja | The point spectrum of the one-dimensional Schrödinger operator[END_REF]) which roughly says that when the quasi-periodic potential can be super-exponentially well approximated by periodic potentials, the equation (1) does not admit any decreasing solutions. Note that actually, in the case of the almost Mathieu equation, Gordon's result implies that any of its solution φ satisfies the inequality

lim m→∞ max ( φ(±q m ), φ(±2q m ) ) ≥ 1 2 φ(0), φ(n) = (|ψ(n + 1)| 2 + |ψ(n)| 2 ) 1/2 .
This means that the corresponding generalized eigenfunctions have to have infinitely many humps located at some of the points ±q m , ±2q m , m ∈ N. These humps prevent the solutions from being square summable. 0.5. Non-trivial model problem. In the present note, we concentrate on the model equation

ψ(n + 1) + ψ(n -1) = λv 0 (nω + θ) ψ(n), n ∈ Z, (5)
v 0 (θ) = 2 e iπω/2 sin(πθ). [START_REF] Hardy | Some problems of Diophantine approximation[END_REF] where 0 < ω < 1 is an irrational frequency, 1 < λ is a coupling constant, and θ is the ergodic parameter. Actually, up to a shift in θ, this is an Almost Mathieu equation with the spectral parameter equal to zero.

We study this equation for the following reasons:

(1) a large part of analysis is quite simple whereas (we believe that) to carry it out one has to use a non trivial renormalization procedure; (2) the techniques developed in this study can be generalized to the case of real analytic potentials v;

(3) this model is related to various self-adjoint models via a cocycle representation (see [START_REF] Fedotov | Renormalization of exponential sums and matrix cocycles[END_REF]), e.g., it comes up naturally when studying the spectral properties of the equation

-ψ ′′ (t) + α l≥0 δ ( l(l -1)/2 + lφ 1 + φ 2 -t ) ψ(t) = Eψ(t).
For the model equation ( 5), our ultimate goal is to describe the set of θ for which the Lyapunov exponent exists or does not exist and to describe the solutions both when the Lyapunov exponent exists and does not exist.

We concentrate on the case of frequencies complementary to the frequencies occurring Simon's example. And, in the case when the Lyapunov exponent does not exist, this leads to a new scenario for the behavior of solutions of (1).

Our main tool is the the monodromization renormalization method introduced by V. Buslaev -A. Fedotov originally for the semi-classical study of the geometry of the spectrum of one dimensional finite difference almost periodic equations, see [START_REF] Buslaev | The monodromization and Harper equation[END_REF]. The idea was to construct Weyl solutions outside the spectrum but, at each step of the renormalization, closer to spectrum so as to uncover smaller and smaller gaps in the spectrum. Now, essentially, we use it to study the solutions of the model equation on the spectrum.

Existence of the Lyapunov exponent for the model equation

We now formulate our results on the pointwise existence of the right Lyapunov exponent γ + (θ) for the model equation ( 5); as we have set the energy parameter, to a fixed value, we omit it in the Lyapunov exponents. The right Lyapunov exponent is defined by the formula (3) with ( 7)

M(x) = λv 0 (θ) -1 1 0 ,
where v 0 is given by [START_REF] Hardy | Some problems of Diophantine approximation[END_REF]. Note that for γ -(θ), the left Lyapunov exponent, one has similar results.

1.1. Main result. Here, we formulate a sufficient condition for the existence of the Lyapunov exponent. Therefore, we need to introduce some notations. For L = 0, 1, 2 . . . , define

ω L+1 = 1 ω L , ω 0 = ω.
where {a} is the fractional part of a ∈ R, and

λ L+1 = λ 1 ω L L , λ 0 = λ Remark 1.
The numbers {ω l } ∞ l=1 are related to the continued fraction expansion of ω:

ω = 1 a 1 + ω 1 = 1 a 1 + 1 a 2 + ω 2 = 1 a 1 + 1 a 2 + 1 a 3 + ω 3 = ... = 1 a 1 + 1 a 2 + 1 a 3 + 1 a 4 + . . .
where a 1 , a 2 , a 3 . . . ∈ N are the elements of the continuous fraction for ω. It is well known that, for any l ∈ N, one has ω l ω l+1 ≤ 1/2. This implies that the numbers λ l increase super-exponentially.

Furthermore, for a given ω ∈ (0, 1) \ Q and s ∈ (0, 1), define the following sequence ( 8)

s L = s L-1 ω L-1 , s 0 = s.
One has

Lemma 1. If s = k 0 + ω 0 l 0 , where k 0 , l 0 ∈ Z, then • for all L, one has s L = k L + ω L l L with k L , l L ∈ Z; • if k 0 > 0, then the sequence (k 2L ) L≥0
is monotonically decreasing until it vanishes and then it stays constant equal to 0; • let k 0 > 0 and L be the first number for which k 2L = 0, then

k 0 ω 0 ω 1 . . . ω 2L-1 ≤ 2.
For a given L ∈ N, define K(2L, ω) being the maximal k 0 such that k 2L = 0 and set K(2L -1, ω) = K(2L, ω). Now, we are ready to discuss the Lyapunov exponent. We have Theorem 4. Pick λ > 1 and ω ∈ (0, 1) irrational. Assume that there exists a function M : N → N such that M(L) < L and that, for L → ∞,

(9) ω M (L) ω M (L)+1 . . . ω L-1 → 0, and λ M (L) ω M (L) ω M (L)+1 . . . ω L → ∞.
For a given 0 ≤ θ ≤ 1, the Lyapunov exponent γ + (θ) for equation [START_REF] Ya | A sufficient condition for continuity of the spectrum of a discrete Schrödinger operator[END_REF] exists if, for all sufficiently large L, one has:

(10) |θ -k -lω 0 | ≥ ω 0 ω 1 . . . ω M (L)-1 e - 1 ω 0 ω 1 •••ω M (L)-1 , for all k, l ∈ Z such that 0 ≤ k + lω 0 ≤ 1 and (11) K(M(L), ω) < k ≤ K( L, ω), L = L if L is even, L + 1 otherwise.
Furthermore, when γ + (θ) exits, it is equal to log λ.

One also has a similar statement on the pointwise existence of the left Lyapunov exponent γ -. Note that for γ -to exist, θ has to avoid neighborhoods of the points k + lω 0 with negative k. Now, turn to a discussion of the results given in Theorem 4.

1.2. Admissible frequencies. Denote by Ω the set of ω ∈ (0, 1) satisfying the conditions of Theorem 4 1.2.1. The measure of Ω. Khinchin's famous result (see e.g. [START_REF] Ya | Continued fractions[END_REF]) on the geometric means of the products of the elements of the continued fractions implies Lemma 2. mes Ω = 1.

Proof. Let {a l } be the elements of the continued fraction for ω. By Khinchin, for almost all ω, one has lim L→∞ (a 1 a 2 . . . a L )

1 L = C, where C = 2, 6... is a universal constant. Pick l ∈ N. One has 1 2a l < ω l-1 < 1 a l . Therefore, for almost all frequencies ω, lim L→∞ (ω 0 ω 1 . . . ω L-1 ) 1 L ≤ 1 C , lim L→∞ (ω 0 ω 1 . . . ω L-1 ) 1 L ≥ 1 2C .
Such ω belong to Ω: in (9) one can take

M(L) = [L/2 ].
1.2.2. Liouvillean numbers in Ω. Recall that an irrational number ω is called Liouvillean if, for any n ∈ N, there are infinitely many (p, q) ∈ Z × N such that

ω - p q ≤ 1 q n .
(see e.g. [START_REF] Ya | Continued fractions[END_REF]).

One has Lemma 3. The set Ω contains Liouvillean numbers satisfying

(12) ω - p q ≤ 1 qλ cq , c = c(ω) > 0,
for infinitely many (p, q) ∈ Z × N.

Proof. We construct a Liouvillean ω ∈ Ω by choosing inductively (a l ) l≥1 , the elements of its continued fraction. Therefore, we pick a 1 ≥ 1 large and, for all L ≥ 1, we choose a L+1 so that

(13) 1 2 a L+1 ≤ (a 1 a 2 . . . a L ) -1 λ a 1 a 2 ...a L ≤ a L+1 .
We now check that such an ω belongs to Ω. Therefore, we check that one has [START_REF] Puig | Reducibility of quasi-periodic skew products and the spectrum of Schrödinger operators[END_REF] for M(l) = l -1. As λ > 1, the sequence (a l ) l is quickly increasing, and so

(14) ω l-1 → 0, l → ∞.
Furthermore, as, for all l ≥ 0, one has ω l = (a l+1 + ω l+1 ) -1 , we get

ω l-1 ω l λ l-1 > 1 4a l a l+1 λ a 1 a 2 ...a l ≥ a 1 a 2 . . . a l-1 8 .
This implies that λ l-1 ω l-1 ω l → ∞, and so ω ∈ Ω. Now, let us check that ω satisfies (12) (and, thus, is a Liouville number). Consider p l q l , the sequence of the best approximates for ω. Recall that (see e.g. [START_REF] Ya | Continued fractions[END_REF]), for all l ∈ N,

ω - p l q l ≤ 1 a l+1 q 2 l , ( 15
)
q l+1 = a l+1 q l + q l-1 , q 1 = a 1 , q 0 = 1. ( 16
)
The relations (16) imply that (17) a l a l . . . a 2 a 1 < q l < P a l . . . a 2 a 1 , P =

∞ l=1 1 + 1 a l a l+1 ;
the product P converges as the sequence (a l ) l is quickly increasing. Relations ( 17) and ( 13) imply that a l+1 ≥ q -1 l λ q l /P . This and ( 15) imply (12).

1.3. The set of "bad" phases. For given λ > 1 and ω ∈ Ω, denote by Θ the set of phases θ not satisfying (10) for infinetly many L. One has

Lemma 4. The set Θ is topologically typical (countable intersection of dense open sets) and, under the condition

(18) ∞ L>0 λ M (L) ω M (L) ω M (L)+1 . . . ω L -1 < ∞
(which is stronger than ( 9)), it has zero Lebesgue measure.

Proof. For a given L > 0, denote the set of θ not verifying (10) by Θ L . Then

(19) Θ = N ≥0 L≥N Θ L .

Thus, Θ is a countable intersection of open sets.

As ω is irrational, the points θ k,l = k + ω 0 l (k, l ∈ Z, k ≥ 0) are dense in the interval (0, 1). So, to complete the proof of the first property of Θ, it suffices to show that the set L≤N Θ L contains all the points θ k,l with k sufficiently large. But, this follows from (11) and the inequality M(L) < l.

Finally, note that, by (10),

mes Θ L ≤ 1 ω K(L)ω 0 ω 1 . . . ω M (L)-1 λ M (L) -1 ≤ 2 ω λ M (L)ω M (L) . . . ω L-1 ω L -1 .
Under the condition (18), this implies that the Lebesgue measure of Θ is zero.

1.4. Heuristics and the statement of Theorem 4. Let us now describe some heuristics "explaining" Theorem 4.

Consider a continuous version of equation ( 5) :

(20) φ(s + ω) + φ(s -ω) = λv 0 (s) φ(s), s ∈ R.
If φ satisfies this equation, then the formula ψ(n) = φ(nω + θ), n ∈ Z, defines a solution to [START_REF] Ya | A sufficient condition for continuity of the spectrum of a discrete Schrödinger operator[END_REF].

If λ >> 1, then one can expect that, on a fixed compact interval, equation ( 20) has an exponentially increasing solution φ + with the leading term φ + 0 satisfying the equation

(21) φ + 0 (s + ω) = λv 0 (s) φ + 0 (s), s ∈ R.
For the last equation, one can easily construct a solution φ + 0 that is analytic and has no zeros in the band 0 < re s < 1+ω. One can extend this solution analytically to the left of this band using equation (21). As v 0 vanishes at integers, φ + 0 has zeros at all the points of the form s k,l = k + lω where k, l > 0 are integers. If there is a true solution to (20) with the leading term φ + 0 , then (5) has a solution ψ + with the leading term φ + 0 (nω + θ). Furthermore, if θ ∈ (0, 1) admits the representation θ = k 0l 0 ω with some positive integers k 0 and l 0 , then, at least for sufficiently large λ, the leading term of ψ + increases exponentially on the "interval" where -k 0 ω + l 0 < n < l 0 + 1 and then vanishes at the points n = l 0 + 1, l 0 + 2, . . . . The equality θ = k 0l 0 ω can be interpreted as a quantization condition: when this condition is satisfied, the solution ψ + that is exponentially growing up to the point n = l 0 , at this point, changes to the exponential decay. So, it is natural to expect that the solution ψ + keeps growing up to the infinity if θ is "far enough" from all the points of the form k 0l 0 ω with positive integers k 0 and l 0 . Hence, the right Lyapunov exponent should exist.

Non-existence of the Lyapunov exponent

Theorem 4 is rather rough in the sense that the sizes of the "secure intervals" that θ has to avoid for the Lyapunov exponent to exist (see (10)) are too big. This is actually due to the fact that, under the conditions of Theorem 4, one has much more than the existence of Lyapunov exponent. Roughly, under these conditions, for each L large enough, equation ( 5) has solutions that, locally, on intervals of length of order (ω 0 ω 1 . . . ω M (L)-1 ) -1 , can have complicated behavior whereas globally, on the interval 0 < k < K of length of order (ω 0 ω 1 . . . ω L ) -1 , they are nicely exponentially increasing.

Our method also allows a precise description of the set of θ where the Lyapunov exponent does not exist. The structure of this set is quite complicated; in the present note, we only describe it for frequencies in Ω 1 ⊂ Ω, the set of ω satisfying the conditions (22) ω L → 0, and λ L-

1 ω L-1 ω L → ∞.
instead of ( 9). One has the following two statements:

Theorem 5. Pick λ > 1. Let ω ∈ Ω 1 . For a 0 < θ < 1, define the sequence {s L } by [START_REF] Ya | Continued fractions[END_REF] with s 0 = θ. Assume that there is a positive constant c such that for infinitely many even positive integers L one has

dist ( s L-1 , ω L-1 • N ) ≤ ω L-1 λ -c L and s L-1 ≥ c.
Then, the right Lyapunov exponent γ + (θ) does not exist. and Theorem 6. Pick λ > 1. Let ω ∈ Ω 1 . For a 0 < θ < 1, define the sequence {s L } by ( 8) with s 0 = θ. Assume that there is a positive constants c and N such that for infinitely many odd positive integers L one has

dist ( s L , ω L • N ) ≤ λ -c L , s L-1 ≤ 1 -c, and s L ≤ ω L N.
Then, the right Lyapunov exponent γ + (θ) does not exist.

The above two theorems are sharp: in the case of ω ∈ Ω 1 , if the Lyapunov exponent does not exist, then θ satisfies the conditions of one of the them.

As for the behavior of the solutions, in both cases, roughly, we find that, for infinitely many L, even if we forget of the complicated local behavior of the solutions on the intervals of the length of order (ω 0 ω 1 . . . ω M (L)-1 ) -1 , one can see that globally, on the interval 0 < k < K of the length of order (ω 0 ω 1 . . . ω L ) -1 , the solutions change from exponential growth to the exponential decay. For example, in the case of Theorem 6, there exists solutions that, at first, are globally exponentially increasing then are globally exponentially decaying, the length of the interval of increase and the interval of decrease being of the same order. Here we use the word globally to refer to the fact that this exponential growth or decay happens at a large scale.

The main ideas of the proof

As we have mentioned in the introduction, the main tool of the proof is the monodromization renormalization method. The new idea is that one can consider the infinite sequence of the almost periodic equations arising in the course of the monodromization as a sequence of equations describing a given solution of the input equation on larger and larger intervals, the ratio of their length being determined by the continued fraction of the frequency. Now, the renormalization formulas can be written in the form

M(θ + (k -1)ω) . . . M(θ + ω) M(θ) ∼ Ψ({kω + θ}) [ M 1 (θ 1 -ω 1 ) M 1 (θ -2ω 1 ) . . . M 1 (θ -k 1 ω 1 ) ] t Ψ -1 (θ). (23)
Here, "∼" means "equal up to a sign",

• M(θ) = 2λ sin(πθ) -e -iπθ e iπθ 0 ; the second order difference equation (20), the continuous analog of ( 5), is equivalent to the first order matrix difference equation

(24) Ψ(s + ω) = M(s)Ψ(s), s ∈ R;
• Ψ is a fundamental solution to (24), i.e., such that Ψ(s) ∈ SL(2, C) for all s; • t denotes the transposition; • M 1 is a monodromy matrix corresponding to this solution, i.e., the matrix defined by Ψ(s + 1) = Ψ(s)M t 1 (s/ω). The new constants ω 1 , θ 1 and the number k 1 are defined by

ω 1 = { 1/ω }, θ 1 = { θ/ω }, k 1 = [ θ + kω ].
And, as usual {a} and [a] denote the fractional and the integer part of a ∈ R. Formula (23) relates the study of the matrix product M(θ + (k -1)ω) . . . M(θ + ω) M(θ) to that of a similar product: the monodromy matrix M 1 is unimodular and, as the matrix M, it is 1 anti-periodic. One can apply the same renormalization formula for the new matrix product and so on. It is easy to check that after a finite number of renormalizations, one gets a matrix product containing at most ... one matrix. This feature recalls the renormalization of the quadratic exponential sums carried out by Hardy and Littlewood (see [START_REF] Hardy | Some problems of Diophantine approximation[END_REF][START_REF] Fedotov | Renormalization of exponential sums and matrix cocycles[END_REF]). At each step of the monodromization, one has to study similar difference equations Ψ L (s + ω k ) = M L (s)Ψ L (s), L = 0, 1, 2, . . . . One needs to have a good enough control of their solutions but only on one fixed compact interval namely [0, 1]. For our model, one can choose the fundamental solutions so that all the matrices M L have the same functional structure, and the numbers (λ L ) are the successive coupling constants in these equations. For λ = λ 0 > 1, the sequence (λ L ) L tends to infinity very rapidly; this enables an effective asymptotic analysis of the successive equations. For general almost periodic equations, one finds an analogous effect at least when the coupling constant in the input equation is large enough.

  EXPONENT FOR A QUASI-PERIODIC EQUATION ALEXANDER FEDOTOV AND FR ÉD ÉRIC KLOPP 0. Introduction 0.1. Quasi-periodic finite difference equations. Consider the finite difference Schrödinger equation (1) (H θ ψ)(n) = ψ(n + 1) + ψ(n -1) + v(nω + θ)ψ(n) = Eψ(n)

0. 3 .

 3 Lyapunov exponents and the spectrum. For ω ∈ Q, one has the following theorem by Ishii -Pastur -Kotani Theorem 2 ([2]). The absolutely continuous spectrum, σ ac (H θ ), is the essential closure of the set of energies where the Lyapunov exponent vanishes. This theorem immediately implies Corollary 1 ([2]). If γ(E) is positive on I, an interval, then the spectrum in I (if any) is singular, σ ∩ I ⊂ σ s .

Both authors acknowledge support of the CNRS and RFBR through the PICS grant n • 07-01-92169. 1