Haitao Chen

Kefeng Li

Yao Cai

Pingping Wang

Weimin Gong

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

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

Light regulation of resistance to oxidative damage and magnetic crystal biogenesis in Magnetospirillum magneticum mediated by a Cys-less LOV-like protein

Keywords: magnetotactic bacteria, Amb2291, oxidative stress, reactive oxygen species

Light-oxygen-voltage (LOV) proteins are ubiquitous photoreceptors that can interact with other regulatory proteins and then mediate their activities, which results in cellular adaptation and subsequent physiological changes. Upon blue-light irradiation, a conserved cysteine (Cys) residue in LOV covalently binds to flavin to form a flavin-Cys adduct, which triggers a subsequent cascade of signal transduction and reactions. We found a group of natural Cys-less LOV-like proteins in magnetotactic bacteria (MTB) and investigated its physiological functions by conducting research on one of these unusual LOV-like proteins, Amb2291, in Magnetospirillum magneticum. In-frame deletion of amb2291 or site-directive substitution of alanine-399 for Cys mutants impaired the protective responses against hydrogen peroxide, thereby causing stress and growth impairment. Consequently, gene expression and magnetosome formation were affected, which led to high sensitivity to oxidative damage and defective phototactic behaviour. The purified wild-type and A399C-mutated LOV-like proteins had similar LOV blue-light response spectra, but Amb2291 A399C exhibited a faster reaction to blue light. We especially showed that LOV-like protein Amb2291 plays a role in magnetosome synthesis and resistance to oxidative stress of AMB-1 when this bacterium was exposed to red light and hydrogen peroxide. This finding expands our knowledge of the physiological function of this widely distributed group of photoreceptors and deepens our understanding of the photoresponse of MTB.

Key words: LOV,

Introduction

Most bacteria must cope with frequent changes in their environment; hence, they developed various mechanisms to sense and respond to external stimuli for growth and survival. Bacteria can use light as a source of energy when exposed to it, but they can also be damaged by ultraviolet (UV) radiation or light-generated reactive oxygen species (ROS) [START_REF] Metz | Role of a short light, oxygen, voltage (LOV) domain protein in blue light and singlet oxygen-dependent gene regulation in Rhodobacter sphaeroides[END_REF]. Different types of photoreceptors, such as light-oxygen-voltage (LOV) sensing protein, photoactive yellow protein, phytochrome, sensor of blue light using flavin adenine dinucleotide, cryptochromes, and sensory rhodopsin, are involved in bacterial photosensing [START_REF] Braatsch | Blue light perception in bacteria[END_REF][START_REF] Losi | Flavin-based blue-light photosensors: a photobiophysics update[END_REF][START_REF] Purcell | Photoregulation in prokaryotes[END_REF][START_REF] Spudich | The multitalented microbial sensory rhodopsins[END_REF][START_REF] Wilde | Light-controlled motility in prokaryotes and the problem of directional light perception[END_REF].

Photoreceptor LOV belongs to a subset of the Per-Arnt-Sim (PAS) family that utilizes internally bound flavin chromophores to sense changes in blue light or redox state [START_REF] Pudasaini | Kinetics of the LOV domain of ZEITLUPE determine its circadian function in Arabidopsis[END_REF][START_REF] Taylor | PAS domains: internal sensors of oxygen, redox potential, and light[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 flavin chromophore binds non-covalently to LOV in a dark environment. Flavin photon absorption results in the formation of a flavin-C4(a)-cysteinyl adduct with a conserved cysteine (Cys) residue on the PAS-motif, thereby initiating a cascade of structural rearrangements within the LOV core, which are propagated to the domain boundaries [START_REF] Pudasaini | LOV-based optogenetic devices: light-driven modules to impart photoregulated control of cellular signaling[END_REF][START_REF] Salomon | Photochemical and mutational analysis of the FMN-binding domains of the plant blue light receptor, phototropin[END_REF]. LOV proteins are widely distributed in bacteria. The first bacterial LOV protein from Bacillus subtilis, YtvA [START_REF] Akbar | New family of regulators in the environmental signaling pathway which activates the general stress transcription factor σ B of Bacillus subtilis[END_REF], was initially reported as a positive regulator of general stress response (GSR). It binds to flavin mononucleotide (FMN) and undergoes a photocycle similar to that of the LOV domains of the plant phototropins [START_REF] Losi | First evidence for phototropin-related blue-light receptors in prokaryotes[END_REF]. In Dinoroseobacter shibae, a blue-light photoreceptor of the LOV superfamily (DsLOV) regulates photopigment synthesis [START_REF] Endres | Structure and function of a short LOV protein from the marine phototrophic bacterium Dinoroseobacter shibae[END_REF]. Purcell et al. suggested that bacterial LOV domain proteins may function as redox sensors via flavin chromophore itself [START_REF] Purcell | An analysis of the solution structure and signaling mechanism of LovK, a sensor histidine kinase integrating light and redox signals[END_REF]. Furthermore, the data hint at a link between RsLOV in Rhodobacter sphaeroides, carbohydrate metabolism and chemotaxis, and the response of cells to photooxidative stress [START_REF] Metz | Role of a short light, oxygen, voltage (LOV) domain protein in blue light and singlet oxygen-dependent gene regulation in Rhodobacter sphaeroides[END_REF]. The blue-light-modulating activity of RsLOV and RsAppA is intended to maximize photosynthesis under favourable conditions and to minimize photooxidative damage [START_REF] Losi | The evolution and functional role of flavin-based prokaryotic photoreceptors[END_REF][START_REF] Metz | Role of a short light, oxygen, voltage (LOV) domain protein in blue light and singlet oxygen-dependent gene regulation in Rhodobacter sphaeroides[END_REF]. In Brucella abortus, LOVhK is a cognate sensor histidine kinase of PhyR that functions as a stress sensor (and GSR pathway activator). GSR can be activated by red and blue lights and many other stimuli. Deletion of LOVhK severely impairs cell survival under defined conditions of oxidative and acid stresses [START_REF] Kim | The Brucella abortus virulence regulator, LovhK, is a sensor kinase in the general stress response signalling pathway[END_REF]). In addition, blue-light inhibition of Listeria monocytogenes growth is mediated by ROS and is influenced by σ B and the blue-light sensor Lmo0799 [START_REF] O'donoghue | Blue-light inhibition of listeria monocytogenes growth is mediated by reactive oxygen Species and is influenced by σB and the blue-light sensor Lmo0799[END_REF].

Previous research has found that replacement of the conserved adduct-forming Cys with alanine (Ala) abolishes the photochemical reaction of LOV [START_REF] Salomon | Photochemical and mutational analysis of the FMN-binding domains of the plant blue light receptor, phototropin[END_REF]. Recently, Yee et al. found that by removing the conserved Cys, the LOV protein still undergoes light-induced photoreaction and signal transduction through the signalling capability of neutral semiquinone (NSQ), which is photoreduced from flavin [START_REF] Yee | Signal transduction in light-oxygen-voltage receptors lacking the adduct-forming cysteine residue[END_REF]. Interestingly, naturally occurring LOV proteins with proline (Pro) at the position of the adduct-forming Cys residue can still respond to light. Additional studies are needed to determine whether the light-induced naturally Cys-less LOV undergoing signal transduction is widespread. Notably, we found a group of magnetotactic bacteria (MTB) that possess Cys-less LOV-like proteins in which the Ala occupied the position of the adduct-forming Cys. Moreover, these LOV-like proteins contain an N-terminal putative transmembrane domain. MTB can synthesize intracellular membrane-enclosed, nanosized, and chain-arranged magnetite and/or greigite crystals called magnetosomes [START_REF] Komeili | Molecular mechanisms of magnetosome formation[END_REF]Spring and Bazylinski 2002). Magnetosome chains are the magnetic sensors in MTB that act as an internal compass needle and help in the rapid orientation along the geomagnetic field, a behaviour known as magnetotaxis [START_REF] Faivre | Magnetotactic bacteria and magnetosomes[END_REF][START_REF] Frankel | Magnetite and magnetotaxis in microorganisms[END_REF][START_REF] Komeili | Molecular mechanisms of compartmentalization and biomineralization in magnetotactic bacteria[END_REF]. In addition to the magnetotaxis function, magnetosomes in Magnetospirillum gryphiswaldense MSR-1 exhibit peroxidase-like activity that reduces intracellular levels of ROS [START_REF] Guo | Magnetosomes eliminate intracellular reactive oxygen species in Magnetospirillum gryphiswaldense MSR-1[END_REF]. Moreover, light irradiation can help M. magneticum AMB-1 cells eliminate intracellular levels of ROS [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF]. Other studies also revealed the photoresponses of magnetotaxis in some MTB (de Melo and Acosta-Avalos 2017; [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[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 | Occurrence and microscopic analyses of multicellular magnetotactic prokaryotes from coastal sediments in the Yellow Sea[END_REF]. It is unknown whether and what kind of photoreceptor is involved in this process. Analysis of AMB-1 genome reveals an unusual LOV-like photoreceptor, Amb2291, of which the expected conserved adduct-forming Cys at the location of 399 was actually replaced with Ala. To assess the function of the LOV-like protein, we mutated amb2291 and analysed the effects of irradiation and hydrogen peroxide (H 2 O 2 )-induced oxidative stress (photo-oxidative stress) on cell growth and magnetosome biosynthesis in AMB-1 wild-type (WT) cells and amb2291 mutants.

Quantitative reverse transcription polymerase chain reaction (qRT-PCR) and dichlorofluorescin diacetate (DCFH-DA) methods were used to analyse the requirement of amb2291 in regulating the expression of antioxidant genes as well as magnetosome formation genes, which scavenge intracellular ROS under photooxidative stress conditions. In addition, deletion of amb2291 or A399C substitution affects the phototactic behaviour of AMB-1, which further confirms the role of Amb2291 in antioxidant damage. Moreover, we purified the Amb2291 and A399C mutant proteins and analysed their photochemical properties. This study contributes to the elucidation of the photoreaction mechanism and deepens the understanding of the photoresponse of MTB.

Materials and Methods

Strains and culture conditions

Escherichia coli DH5α was used for general cloning. The WM3064 of E. coli was used for intergeneric conjugation between E. coli and M. magneticum. E. coli strains were grown in LB medium supplemented with apramycin (50 µg/mL) or kanamycin (50 µg/mL). M. magneticum AMB-1 strains (ATCC 700264) were grown on modified enriched Magnetospirillum growth medium comprising 5 mL of 0.01 M ferric quinate, 0.68 g of potassium phosphate, 0.12 g of sodium nitrate, 0.74 g of succinic acid, 0.05 g of L-Cys-HCl, 0.2 g of polypeptone, and 0.1 g of yeast extract. In addition, 15 µg /mL apramycin or kanamycin was used for plasmid selection [START_REF] Matsunaga | Complete genome sequence of the facultative anaerobic magnetotactic bacterium Magnetospirillum sp. strain AMB-1[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 luciferase-bacterial magnetic particle complex by a recombinant Magnetospirillum magneticum AMB-1[END_REF]. M. magneticum AMB-1 strains were cultured using a 100 mL Schott flask bottle or a 15 mL cell culture flask and incubated at 30°C.

Identification of the Amb2291 sequence

Amb2291 was identified in the M. magneticum AMB-1 genomic database BLAST [START_REF] Matsunaga | Complete genome sequence of the facultative anaerobic magnetotactic bacterium Magnetospirillum sp. strain AMB-1[END_REF]. Amino acid sequences were aligned using DNAMAN. Domain was analysed by SMART (http://smart.embl-heidelberg.de). The protein accession numbers were as follows: M. magneticum AMB-1 (WP_070108660.1, Amb2291); M. magnetotacticum MS-1 (KIL98983.1); M. gryphiswaldense Dinoroseobacter shibae (WP_044027762.1).

Purification of protein

The conserved soluble fragment (residues 351-714) of amb2291 or amb2291 A399C was cloned into the pET28a plasmid with N-terminal His6-SUMO-tagged fusion proteins. These proteins were expressed in E. coli BL21 (DE3) cells and purified as previously described [START_REF] Li | Structural and evolutionary characteristics of dynamin-related GTPase OPA1[END_REF]. The expression levels of Amb2291 variants were induced with 10 µM isopropyl-β-D-1-thiogalactopyranoside for 16 h at 16°C in the dark. For cell lysis, the soluble extract was filtered and incubated with 5 mM FMN on ice for 20 min. After nickel chelate affinity chromatography, the proteins were further purified by size exclusion chromatography on Superdex 200 column in Tris buffer (10 mM Tris/Cl, 200 mM NaCl, pH 8.0). All experiments were carried out in the dark.

Genetic manipulation of AMB-1

All plasmids and primers used in the study are listed in Tables S1 andS2. Deletion of 2045 nucleotides of Amb2291 (81-2125) was conducted by a CRISPR-Cas9 system using plasmid pCRISPomyces-2 [START_REF] Cobb | High-efficiency multiplex genome editing of Streptomyces species[END_REF]. The first step in constructing plasmid pCRISPR-sgRNAamb2291 was to select the proper sgRNA sequence. sgRNA carrying the 20 nt specific targeting sequence was synthesized (Sangon Biotech, China) and inserted into the pCRISPomyces-2 through a BbsI site [START_REF] Chen | Efficient genome editing of Magnetospirillum magneticum AMB-1 by CRISPR-Case9 system for analyzing magnetotactic behavior[END_REF][START_REF] Xie | sgRNAcas9: a software package for designing CRISPR sgRNA and evaluating potential off-target cleavage sites[END_REF]. A 2.0 kb HDR DNA fragment of the editing template was amplified from the purified genomic DNA of AMB-1 and inserted into the pCRISPR-sgRNAamb2291 with the XbaI restriction enzyme.

The vector used for the allelic exchange of WT amb2291 for amb2291 A399C was constructed as follows.

PCR amplification of amb2291 with 1.0 kb upstream and downstream sequences was conducted using Q5 high-fidelity polymerase (NEB, United States). The resulting 4.2 kb fragment was incorporated into pBBRI-mcs2. To prevent the sgRNA of the CRISPR plasmid from cutting the replaced amb2291 A399C in AMB-1, we inserted a nonsense mutation into the sequence of PAM and the sgRNA sequence in the amb2291 A399C fragment, so that sgRNA will not recognize and Cas9 will not cut amb2291 A399C in AMB-1, guaranteeing the construction of the amb2291 A399C strain. Site-directed mutagenesis of the resulting plasmid by using the NEBuilder HiFi DNA assembly reaction protocol was conducted to introduce the A399C, PAM, and sgRNA sequences of the nonsense mutation (PAM: CCT to aCT and sgRNA sequence: GTTCACCCTGCCCATGACCC to aTTCACCCTGCCCATGACCC). The site-directed 4.2 kb fragment was inserted into the plasmid pCRISPR-sgRNAamb2291 by using the same method as above.

All pCRISPR plasmids were transformed into the E. coli strain WM3064 by chemical transformation and transferred into AMB-1 through conjugation [START_REF] Chen | Efficient genome editing of Magnetospirillum magneticum AMB-1 by CRISPR-Case9 system for analyzing magnetotactic behavior[END_REF][START_REF] Philippe | An MCP-like protein interacts with the MamK cytoskeleton and is involved in magnetotaxis in Magnetospirillum magneticum AMB-1[END_REF]. All deletions were verified by PCR and sequencing.

Irradiation experiment and H 2 O 2 -induced oxidative stress in AMB-1

We investigated the effect of photooxidative stress on the growth and magnetosome biomineralization of AMB-1 cells. All cells were grown with the exposure to 20 µmol photons/m 2 /s from 625 nm or 450 nm LED light (Fig. S1a, Ouyingzhaoming, China) and were cultured in medium containing H 2 O 2 at 0, 5, 7.5, 10, and 15 µM. In general, light exerts a certain decomposition effect on H 2 O 2 . Therefore, by using the H 2 O 2 Quantitative Assay Kit (Sangon Biotech, China), medium with 15 µM H 2 O 2 was used to analyse the H 2 O 2 concentration changes versus the illumination time. The medium contains no iron ions to prevent the detection of H 2 O 2 (Fig. S1b). We also detected the change in the H 2 O 2 concentration in the dark, and the value remained unchanged within 20 h. The optical density (OD) and cellular magnetisms (Cmag) of exponentially growing cultures were measured at 600 nm under a 4.5 mT homogeneous magnetic field with a UV-visible spectrophotometer (Unico UV-2800, United States) and an electromagnetic system [START_REF] Schüler | A simple light-scattering method to assay magnetism in Magnetospirillum gryphiswaldense[END_REF][START_REF] Zhao | A simple and accurate method for quantification of magnetosomes in magnetotactic bacteria by common spectrophotometer[END_REF].

Quantitative RT-PCR assay

The procedure for qRT-PCR was based on a previously described general protocol [START_REF] Chen | Efficient genome editing of Magnetospirillum magneticum AMB-1 by CRISPR-Case9 system for analyzing magnetotactic behavior[END_REF].

Analysis was performed with cells in the plateau phase. Total RNA was extracted using an RNAeasy Plus Minikit (Qiagen, Germany). Single-strand cDNA was synthesized with a reverse transcription kit (Takara, Japan). qRT-PCR assay was performed with the SYBR Green Real-time PCR Mix (Thermo Fisher Scientific, United States) on an ABI QuantStudio 6 Flex Real-time PCR System (Applied Biosystems, United States). The relative expression level was calculated by the 2 -ΔΔC T method using the threshold cycle of 16S rRNA for normalization [START_REF] Livak | Analysis of relative gene expression data using real-time quantitative PCR and the 2 -ΔΔCT method[END_REF]. The experiment was repeated at least thrice, with each sample run in triplicate. All primers used for probing each gene in this study are listed in Table S2.

Detection of ROS generation

Intracellular ROS levels were analysed using the DCFH-DA method [START_REF] He | Involvement of reactive oxygen species in the UV-B damage to the cyanobacterium Anabaena sp[END_REF][START_REF] Wang | Changes of cell growth and nnagnetosonne biomineralization in Magnetospirillum magneticum AMB-1 after ultraviolet-B irradiation[END_REF]). After irradiation and H 2 O 2 induction, the cells grew to the late stage of exponential growth.

Then, 2 mL of bacterial culture was immediately added to 10 µL of 10 mM DCFH-DA and incubated at 37°C for 30 min in the dark. The cells were centrifuged at 12,000 rpm for 6 min and washed thrice with a PBS buffer. The fluorescence of all samples was measured using a fluorescence spectrophotometer (Hitachi F-4500, Japan) at an excitation wavelength of 485 nm and emission wavelengths of 500-600 nm.

Motility and phototactic behaviour analyses of AMB-1

AMB-1 motility behaviour was analysed at a frame rate of 33 fps by using a long working distance 40×

Olympus objective. Cells (10 µL) were observed on a glass slide with an Olympus microscope. Single cells were tracked, and the average swimming velocities were calculated using the MTrackJ plugin for ImageJ [START_REF] Meijering | Imaging and spectroscopic analysis of living cells: Optical and spectroscopic techniques[END_REF][START_REF] Murat | Opposite and coordinated rotation of amphitrichous flagella governs oriented swimming and reversals in a magnetotactic spirillum[END_REF][START_REF] Zhang | Measurement of free-swimming Mmotility and magnetotactic behavior of Magnetococcus massalia strain MO-1[END_REF]. Phototactic behaviour was quantitatively analysed using a modified mini MTB collection vessel as previously described [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF]. The cultures (80 mL) with and without H 2 O 2 induction were pooled into the reservoir and into the collecting tube. A microaerobic band formed after 30 min of growth in the dark prior to conducting the phototactic experiment. The upper interface of the microaerobic zone is parallel to the collection tube. Then, the device was placed in a cassette, where the collection tube was parallelly exposed to the light beam. Control experiments were conducted in the dark. After 30 min of collection, we measured the OD in the collecting tube and reservoir (OD collecting tube and OD reservoir , respectively). The ratio of (OD collecting tube -OD reservoir )/OD reservoir was used to determine the relative numbers of phototactic cells.

Spectrophotometry analyses

The absorbance spectra of purified Amb2291 WT and mutant proteins were determined from 700 nm to 250 nm at room temperature using a UV-visible spectrophotometer. Assays were recorded either in the dark or after exposure to a blue LED lamp (450 nm) or red LED lamp (625 nm) (Ouyingzhaoming, China). Fluorescence excitation and emission spectra were recorded at room temperature with a fluorescence spectrophotometer (Hitachi F-4500, Japan). The emission wavelength was 530 nm, and the excitation was at 450 nm.

Statistical analysis

All statistical analyses were performed using SPSS 22.0 (SPSS, IBM, United States). Hierarchical clustering analysis (HCA) was performed to analyse the values of OD 600 and Cmag. ROS values and the transcription-level results were analysed using two-tailed Student's t-test. Mann-Whitney U-test was used to analyse phototactic behaviour. One-way analysis of variance was used to investigate the difference in velocities under H 2 O 2 -induced oxidative stress. Each experiment was repeated three times or more. All data were expressed as the mean±standard deviation. In all cases, statistical tests were considered significant at p<0.05.

Results

Genomes of Magnetospirillum sp. encode a group of natural Cys-less LOV-like proteins

Genomic analysis allowed us to identify the amb2291 as a CDS coding a LOV-like protein in AMB-1.

This 713-residue protein contains the LOV-like domain encompassing PAS and PAC domains, histidine kinase, and histidine kinase-like ATPase (Fig. 1a). Notably, this LOV domain includes the key residues of the canonical LOVs, such as the signalling relevant glutamine (Gln), the asparagines that interact with the polar side of the isoalloxazine ring, and the arginines that interact with the phosphate moiety of the flavin [START_REF] Fedorov | Crystal structures and molecular mechanism of a light-induced signaling switch: The Phot-LOV1 domain from Chlamydomonas reinhardtii[END_REF]), but it does not contain the Cys in the highly conserved motif GXNCRFLQ (Fig. 1b). The conserved Cys residue is involved in the photochemical generation of a flavin-Cys adduct upon blue-light absorption by the flavin in LOV photoreceptors [START_REF] Briggs | The LOV domain: a chromophore module servicing multiple photoreceptors[END_REF][START_REF] Christie | LOV (light, oxygen, or voltage) domains of the blue-light photoreceptor phototropin (nph1): binding sites for the chromophore flavin mononucleotide[END_REF]. At the Cys position, an Ala substitutes for the strictly conserved Cys in the LOV domain of Amb2291 and the homologues from other Magnetospirium sp. strains MS-1, MSR-1, ME-1, and XM-1 (Fig. 1b). In other naturally occurring Cys-less LOV, this position can be Pro, glycine (Gly), or lysine (Lys). This observation would challenge the photoresponse and physiological function of this group of natural Cys-less, LOV-like proteins in magnetospirilla. Moreover, all Magnetospirillum sp.

LOV-like full-length proteins contain transmembrane domains with 1 to 3 predicted transmembrane segments. To assess the function of this group of LOV-like proteins, we constructed an amb2291 deletion mutant and a site-directed A399C mutant in AMB-1 (substitution of Cys for Ala 399) (as described in "Materials and Methods").

Characteristics of recombinant Amb2291 and Amb2291 A399C proteins

To verify whether the Cys-less Amb2291 has the characteristics of LOV and to investigate the effect of light illumination of Amb2291 or variant, the soluble LOV-like sensory domain of Amb2291 and Amb2291 A399C from residues 351 to 714 (Fig. S2) were expressed in E. coli BL21 (DE3) as N-terminal His6-SUMO-tagged protein and purified via nickel chelate affinity, SUMO protease treatment and size exclusion chromatography. Recombinant proteins on denaturing gels of electrophoresis have a yellow colour and an apparent size of ~40 kDa (Fig. S3). Based on the peak position of the elution profiles of a Superdex 200-size exclusion column, Amb2291 and Amb2291 A399C displayed a molecular weight of approximately 80 kDa (Fig. S3), thereby suggesting the dimer property of these proteins in solution.

Amb2291 and Amb2291 A399C exhibited a broad absorption band in the blue region that might be composed of sharp peaks near 449 and 475 nm and a shoulder near 420 nm, revealing a mixture of oxidized (450 nm) states of FMN. In addition, the absorption spectrum of Amb2291 had a double-peak structure at 361 and 375 nm, whereas the component at 361 nm was reduced to a minor shoulder in the absorption spectrum of Amb2291 A399C . The major UV-A absorption maximum occurred at 378 nm (Fig. 2a). Fig. 2b andc show the fluorescence emission and excitation spectra of the Amb2291 and Amb2291 A399C proteins, respectively. Compared with Amb2291, the fluorescence emission spectrum maximum of A399C mutant shows a redshift of 4 nm. The highest peaks of Amb2291 and Amb2291 A399C proteins are at 529 and 533 nm, respectively, and those in the excitation spectrum are at 467 nm. Thus, both Amb2291 and Amb2291 A399C demonstrated the characteristic properties for the LOV core, but a slight difference was obvious between the two chromophore binding domains.

Fig. 3 shows that dark-state LOV proteins contain an oxidized flavin cofactor that maximally absorbs blue light at 449 nm in the ground state. Blue-light irradiation results in the bleaching of the ground state and in the formation of a photoproduct, and absorption increased at 390 nm (Fig. 3a,b). The results show a faster reaction of the A399C mutant to blue-light illumination compared with the WT strain (Fig. 3c), demonstrating the efficiency of adduct formation in the presence of Cys. Moreover, similar to the BAT P188C mutant of H. hochstenium [START_REF] Yee | Signal transduction in light-oxygen-voltage receptors lacking the adduct-forming cysteine residue[END_REF], Amb2291 and Amb2291 A399C appear to form the hydroquinone (HQ) directly with little NSQ intermediate observed on our observed time scale.

Effects of amb2291 deletion on cell growth under conditions with photooxidative stress

To investigate the role of Amb2291, we performed a CRISPR-Cas9-based gene knockout experiment to construct in-frame deletion of amb2291 as described in the "Materials and Methods" section. The effects of photooxidative stress on cell growth in AMB-1 WT cells and Δamb2291 mutant were analysed. AMB-1 cells were treated with different concentrations of H 2 O 2 and immediately incubated for 20 h under multiple light ranges (red light 625 nm or blue light 450 nm, 20 µmol photons/m 2 /s) or dark conditions. OD at 600 nm was measured to evaluate the oxidative stress effect on growth.

As shown in Fig. 4, the OD values can be grouped into clusters a, b, and c by using the HCA method when the squared Euclidean distance is equal to 4. Without irradiation in the dark, the OD of both AMB-1 WT cells and Δamb2291 mutant decreased with the increase in the H 2 O 2 concentration in the growth media starting from 5 µM (Fig. 4, black curves). The OD of AMB-1 WT cells drastically decreased from 10 µM H 2 O 2 and continued to decrease with increasing H 2 O 2 concentration under blue-light irradiation, while the OD of Δamb2291 mutant decreased from 7.5 µM (group c, blue curves in Fig. 4). The reason may be that blue light induces the formation of ROS that adds to the stress induced by H 2 O 2 addition. However, red-light irradiation abolished the deleterious effect of H 2 O 2 on the WT cells at all analysed concentrations (Fig. 4, red curve, group a). Interestingly, the sensitivity of the Δamb2291 mutant to H 2 O 2 under the red-light irradiation showed no significant change compared with darkness treatment as well as WT cells under darkness (Fig. 4, red dashed-line, group b).

Collectively, these results revealed that red light protects the AMB-1 WT cells from the deleterious effect of H 2 O 2 -generated oxidative stress, which was dependent on the function of Amb2291. For better understanding the response of AMB-1 to photooxidative stress, red light was the main light source in the following studies.

Expression of the amb2291 gene in AMB-1 cells

To investigate the functional differences between Amb2291 and Amb2291 A399C , we constructed an allelic exchange of WT amb2291 for the amb2291 A399C mutant. By qRT-PCR analysis, we observed that red-light irradiation increased the expression of the amb2291 gene in WT cells, regardless of the H 2 O 2 treatment (Fig. 5). The deletion of the amb2291 gene from 81 to 2125 nucleotides substantially enhanced amb2291 expression, especially under the conditions of red-light irradiation and the presence of H 2 O 2 (p<0.01), since the first 80 nucleotides of amb2291 were still kept in Δamb2291, but the transcription level is higher than that of WT and amb2291 A399C . By contrast, site-specific substitution for the conserved Cys, amb2291 A399C mutation displayed the same phenotype of the WT strain. These results unveil the autoregulation of amb2291 to a certain extent of the transcription level compared with the higher transcription level in Δamb2291 caused by the incomplete function of amb2291. In addition, the conservation of Ala 399 is not required for the autoregulation, although it is indispensable for the attenuation of photooxidative stress.

Effect of H 2 O 2 and light illumination on magnetosome formation in AMB-1 cells

We analysed the effect of photooxidative stress on magnetosome formation by measuring Cmag in AMB-1 WT, Δamb2291, and amb2291 A399C mutant cells [START_REF] Schüler | A simple light-scattering method to assay magnetism in Magnetospirillum gryphiswaldense[END_REF][START_REF] Zhao | A simple and accurate method for quantification of magnetosomes in magnetotactic bacteria by common spectrophotometer[END_REF]. By the HCA method, the results showed that Cmag can be divided into clusters (a) and (b) when the squared Euclidean distance is equal to 4 (Fig. 6a). The Cmag value in WT cells under red-light irradiation and with H 2 O 2 -induced oxidative stress formed cluster a, which was higher than all those under the other conditions. This result suggested that Amb2291, red light, and oxidative stress contributed to the maximum synthesis of magnetosomes.

We previously reported that magnetosome-formation-associated genes mms13, mms6, and mmsF significantly responded to visible light [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF]. To investigate the mechanism by which photooxidative stress influences magnetosome formation in AMB-1 WT and mutant cells, we also performed qRT-PCR to detect the expression levels of mms13, mms6, and mmsF. After 20 h of illumination, the expression levels of mms13 and mmsF were significantly upregulated in WT cells and amb2291 A399C mutants compared with their dark controls (Fig. 6b,d We further examined the involvement of the amb2291 gene in controlling intracellular ROS levels by the DCFH-DA method (Fig. 8). The ROS levels of WT cells under photooxidative stress conditions were significantly lower than those of mutant cells (p<0.01), whereas the ROS level significantly increased (p<0.01) in mutant cells under light irradiation compared with those in the dark. The amb2291 gene helps AMB-1 eliminate intracellular ROS under oxidative stress through the light regulation system.

amb2291 affects the phototactic behaviour under photooxidative stress

We analysed the difference in the phototactic behaviour in AMB-1 WT cells and mutants. The phototactic behaviour was quantitatively analysed by a modified mini MTB collection vessel as previously described [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF] concentration, the velocities of all cells showed a consistent downward trend (Fig. S4). Thus, we studied the difference in phototactic behaviour between the WT and mutant strains under 0, 10, and 20 µM H 2 O 2 treatments.

Phototrophic behaviour was observed only with red light for all three strains analysed. Phototactic behaviour of the mutant strains all decreased with the increasing concentration of H 2 O 2 (Fig. 9, p<0.05).

Moreover, the photoresponse of WT cells was significantly stronger than that of Δamb2291 or amb2291 A399C mutants after adding H 2 O 2 (Fig. 9, p<0.05). When H 2 O 2 was not added, all three strains exhibited a photoresponse. Amb2291 is likely involved in the reduction of the toxicity of H 2 O 2 to cells and helped WT cells to actively swim towards light.

Discussion

Flavin-based photoreceptor proteins of the LOV superfamily are ubiquitous in plants, algae, fungi, and prokaryotes [START_REF] Losi | The evolution and functional role of flavin-based prokaryotic photoreceptors[END_REF]. During photoreaction, the non-covalently bound flavin chromophore forms a covalent adduct with a conserved Cys residue in LOV under blue light to initiate signal transmission to various output domains [START_REF] Losi | Flavin-based blue-light photosensors: a photobiophysics update[END_REF]. Current studies have reported that some LOV photoreceptors apparently retained biological activity even after the substitution of the Cys thiol for non-reactive side chains [START_REF] Aihara | Mutations in N-terminal flanking region of blue light-sensing light-oxygen and voltage 2 (LOV2) domain disrupt its repressive activity on kinase domain in the Chlamydomonas phototropin[END_REF][START_REF] Chen | Physical interaction between VIVID and white collar complex regulates photoadaptation in Neurospora[END_REF][START_REF] Okajima | Photosensitivity of kinase activation by blue light involves the lifetime of a cysteinyl-flavin adduct intermediate, S390, in the photoreaction cycle of the LOV2 domain in phototropin, a plant blue light receptor[END_REF][START_REF] Vaidya | Structure of a light-activated LOV protein dimer that regulates transcription[END_REF][START_REF] Yee | Signal transduction in light-oxygen-voltage receptors lacking the adduct-forming cysteine residue[END_REF]. In GenBank, 70 LOV-like proteins contain the strictly conserved LOV domain, except that the adduct-forming Cys is replaced with other residues, including Ala, Pro, Gly and Lys [START_REF] Yee | Signal transduction in light-oxygen-voltage receptors lacking the adduct-forming cysteine residue[END_REF]. In the present study, a group of LOV-like proteins from Magnetospirillum sp. in which the conserved Cys residue is substituted with Ala were reported. The Amb2291 and Amb2291 A399C mutant proteins in M. magneticum AMB-1 were purified. Accordingly, the pale yellow Amb2291 and Amb2291 A399C purified from overexpression in E.coli indicate the binding of FMN to the chromophore domains and potential partial reduction of FMN, regardless of the presence or absence of the conserved Cys. Combined with Fig. 2, these results demonstrate the characteristics of LOV-like proteins. Then, we examined the spectral change of Amb2291 and Amb2291 A399C under blue-light illumination to analyse the photoreaction. Consequently, a faster reaction of Amb2291 A399C to blue light was observed compared with the WT strain, which shows that Cys is an effective electron donor to the photoexcited flavin. Despite of the change in a prolonged illumination studied here cannot be well compared with those in a few minutes, FMN-bound Amb2291 indeed goes through photoreaction as a response to blue light by some mechanism. As the spectrum does not show any change under the same dose of red light (Fig. S5), it demonstrates that the effect of photodamage is negligible. In a conventional LOV photocycle, blue-light absorption drives the formation of a covalent adduct between Cys and flavin, promoting rotation of a conserved glutarnine residue to initiate a cascade of structural rearrangements within the LOV core that are propagated to the domain boundaries [START_REF] Zoltowski | Tripping the light fantastic: blue-light photoreceptors as examples of environmentally modulated protein-protein interactions[END_REF] and then generating downstream signalling responses. Signal transduction through flavin photoreduction to the NSQ was recently found to be a general mechanism in natural Cys-less, LOV-like regulators. The WT BAT of H. hochstenium shows properties similar to LOV proteins, reducing slowly to the NSQ upon blue-light illumination. However, when the WT BAT is converted to a traditional LOV mechanism by mutating the active-site Pro residue to Cys, the BAT P188C mutant shows a faster reaction to blue-light illumination, appears to the fully reduced HQ with little NSQ intermediate observed in their designed time scale [START_REF] Yee | Signal transduction in light-oxygen-voltage receptors lacking the adduct-forming cysteine residue[END_REF]. Interestingly, the absorption changes measured for Amb2291 and Amb2291 A399C on exposure to blue light are similar to those of the BAT P188C variant. Although photoreduction of the Cys-less Amb2291 is inefficient, the FMN-bound protein readily undergoes complete chemical reduction to the fully reduced HQ, with little NSQ intermediate observed on the carried-out time scale (Fig. 3a). Moreover, the aromatic residues of phenylalanine and tyrosine close to the FMN-binding domain motif, as revealed in the sequence analysis in Fig. 1, may generate reductive quenching, resulting in a short excited-state life time of Amb2291, which exhibits inefficient photoreduction [START_REF] Yee | Signal transduction in light-oxygen-voltage receptors lacking the adduct-forming cysteine residue[END_REF].

To analyse the biological functions of amb2291 under photooxidative stress, we used the CRISPR technique to construct Δamb2291 and amb2291 A399C mutants. Growth and magnetosome biosynthesis of AMB-1 WT cells were better than that of Δamb2291 or amb2291 A399C mutants. Interestingly, we found that red light protects the WT cells from the deleterious effect of H 2 O 2 -generated oxidative stress, which depends on the function of Amb2291. Dual sensing of blue light and oxidative stress has been observed in R. sphaeroides, wherein the blue-light-modulating activity of RsLOV and RsAppA is intended to affect photosynthesis and minimize photooxidative damage [START_REF] Losi | The evolution and functional role of flavin-based prokaryotic photoreceptors[END_REF][START_REF] Metz | Role of a short light, oxygen, voltage (LOV) domain protein in blue light and singlet oxygen-dependent gene regulation in Rhodobacter sphaeroides[END_REF]. The structural biochemistry of a fungal LOV photoreceptor reveals an evolutionarily conserved pathway that integrates light and oxidative stress [START_REF] Lokhandwala | Structural biochemistry of a fungal LOV domain photoreceptor reveals an evolutionarily conserved pathway integrating light and oxidative stress[END_REF]. To some extent, the light modulating activity of amb2291 affected magnetosome synthesis and minimized photooxidative damage. Red-light irradiation and H 2 O 2 treatment upregulated the expression of the amb2291 gene. In addition, the expression results in amb2291 deletion strains revealed the autoregulation of the amb2291 gene. Thus, Amb2291 played an important role in photooxidative stress.

Many environmental stimuli, including light, H 2 O 2 , and hyperthermia, generate high levels of ROS that can damage cellular DNA and proteins, perturb the normal redox balance, and cause oxidative stress [START_REF] Lushchak | Oxidative stress and mechanisms of protection against it in bacteria[END_REF][START_REF] Touati | Iron and oxidative stress in bacteria[END_REF]. Therefore, organisms have developed methods to protect themselves from ROS, enabling cells to survive oxidative stress. For example, oxyR and sod are stress-related genes that regulate the transcription of defence genes via a rapid kinetic reaction in response to oxidative stress [START_REF] Jo | Structural details of the OxyR peroxide-sensing mechanism[END_REF][START_REF] Lu | Endogenous superoxide is a key effector of the oxygen sensitivity of a model obligate anaerobe[END_REF]). In the present study, the expression levels of oxyR and sod genes were compared between dark and light conditions with or without H 2 O 2 treatment.

Compared with the mutant controls, simultaneous red-light illumination and H 2 O 2 treatment increased the oxyR and sod expression levels only in the WT strains. This result is inconsistent with the expression of oxyR and sod genes in the WT strains reported by [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF], which may be due to the differences among the concentrations of H 2 O 2 and light treatment. Amb2291 is involved in the regulation of oxidative stress genes and is beneficial for resisting the damage caused by photooxidative stress. A previous study showed that singlet oxygen induces carotenoid biosynthesis in Phaffia rhodozyma and Myxococcus xanthus and is presumed to be light dependent [START_REF] Hodgson | Genetics of regulation and pathway of synthesis of carotenoids[END_REF][START_REF] Schroeder | Singlet oxygen and peroxyl radicals regulate carotenoid biosynthesis in Phaffia rhodozyma[END_REF]. Further studies showed that carotenoids can protect bacteria from photooxidative stress damage [START_REF] Glaeser | Photo-oxidative stress in Rhodobacter sphaeroides: protective role of carotenoids and expression of selected genes[END_REF]. Zeaxanthin accumulates only under photooxidative stress and plays an essential role in protecting the photosynthetic apparatus via the xanthophyll cycle [START_REF] Demmig-Adams | Ecophysiology of the xanthophyll cycle[END_REF]). The Amb2291 protein may also serve the function of both pigments to protect cells from photooxidative stress damage. In addition, the ROS level of WT cells was lower than that of mutant strains under the addion of H 2 O 2 , which is consistent with the notion that light helps AMB-1 cells eliminate intracellular ROS [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF]). These findings further provide new evidence that Amb2291 reduces the increase in ROS through light regulation.

AMB-1 exhibited a photoresponse to red light (Fig. 9), which is consistent with our previous study [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]. After H 2 O 2 treatment, the number of amb2291 deletion or amb2291 A399C mutant cells swimming towards light was significantly less than that of the WT cells.

This finding might imply that red light under H 2 O 2 conditions reduced the phototaxis of the mutant, which may be related to the increased cytotoxicity of H 2 O 2 or ROS in the absence of functional Amb2219. Amb2291 is involved in light regulation to help eliminate ROS under oxidative stress, thereby creating the optimum niche for bacterial growth. It has been shown that Fe 3 O 4 magnetosomes in MTB exhibit peroxidase-like activity, which can catalyse the photoFenton reaction according to the following equation to scavenge the intracellular levels of ROS [START_REF] Guo | Magnetosomes eliminate intracellular reactive oxygen species in Magnetospirillum gryphiswaldense MSR-1[END_REF]): Fe 2+ + H 2 O 2 = Fe 3+ + HO• + OH - [START_REF] Goldstein | The fenton reagents[END_REF][START_REF] Touati | Iron and oxidative stress in bacteria[END_REF]. Moreover, light irradiation enhances this activity by promoting the synthesis of magnetosome and helps MTB eliminate intracellular ROS [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF].

amb2291 expression was significantly increased with red-light irradiation. Moreover, red-light illumination and H 2 O 2 treatment significantly increased the expression of magnetosome formation genes (mms13, mms6, and mmsF) and antioxidative genes (oxyR and sod) in WT cells compared with those in Δamb2291 and amb2291 A399C strains. Therefore, Amb2291 participates in the light-regulated signalling pathway and further enhances the synthesis of magnetosomes and resistance to oxidative damage, which helped clear intracellular ROS, thereby improving phototactic behaviour and bacterial growth.

Collectively, Amb2291 showed a similar LOV spectra based on blue-light irradiation but was unable to undergo the canonical photochemistry of LOV domains with the formation of a flavin-Cys covalent adduct. Upon red-light irradiation, Amb2291 improved the resistance to oxidative stress by some mechanism. Our understanding of the downstream physiological role of LOV protein is far less than that of its biochemical properties, because only several of LOV proteins' roles had been identified [START_REF] Wu | Light regulation of swarming motility in Pseudomonas syringae integrates signaling pathways mediated by a bacteriophytochrome and a LOV protein[END_REF]). Whether Amb2291 is directly involved in the red-light response needs further confirmation.

We propose the following assumptions. First, structural prediction analysis found that Amb2291 has a specific extra membranous domain, while most LOV domains have no transmembrane structure in general, except the one that is found in the cytoplasm and membrane components in R. sphaeroides [START_REF] Hendrischk | Characterization of an unusual LOV domain protein in the α-Proteobacterium Rhodobacter sphaeroides[END_REF]). We are not sure whether the specificity of this protein structure plays a role.

Second, studies have shown that the red/far-red-light-sensing bacteriophytochrome (BphP) can form an integrated signalling network with LOV in response to red, far-red, and blue light [START_REF] Wu | Light regulation of swarming motility in Pseudomonas syringae integrates signaling pathways mediated by a bacteriophytochrome and a LOV protein[END_REF].

Strikingly, genomic analysis revealed the presence of BphP protein in AMB-1, named MagBphP, which is a red/far-red photoreceptor that mediates photosensory responses [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]. We speculate that Amb2291 and the red/far-red-light-induced MagBphP may form an integrated signalling network; one of the biological significances of light-mediated behaviours in AMB-1 is to avoid environmental stress, such as oxidative stress. Fig. 10 shows the functional model of LOV-like Amb2291. The Cys-less LOV-like Amb2291 protein participates in the light-regulated signalling pathway and improves resistance to oxidative damage and magnetic crystal biogenesis in M.

magneticum. However, the present work does not provide enough data to allow for a conclusion about the mechanism between Amb2291 and MagBphP signalling pathways. In addition, MTB usually live in aquatic environments and navigate towards sediment. They may move to find better living environments that contain trace oxygen through migration towards light [START_REF] Wang | Genomic evidence of the illumination response mechanism and evolutionary history of magnetotactic bacteria within the Rhodospirillaceae family[END_REF]). One of the major challenges that ancient life may have adapted to is ROS. The ancestral role of intracellular magnetosomes may have helped early life cope with oxidative stress through photoresponse [START_REF] Lin | On the origin of microbial magnetoreception[END_REF]. In short, Amb2291 acts as an LOV-like protein with a complex sensory role when cellular redox state and/or light were functional input signals.

  , p<0.05). Importantly, red-light illumination led to significant upregulation of mms13, mms6, and mmsF in WT cells compared with Δamb2291 and amb2291 A399C strains (Fig. 6b, c and d, p<0.05). H 2 O 2 treatment also led to significant upregulation of mms13 in WT cells compared with Δamb2291 or amb2291 A399C mutant strains (Fig. 6b, p<0.05). mms6 expression was significantly downregulated in the H 2 O 2 treatment of Δamb2291 strains compared with dark controls (Fig. 6c, p<0.05). Red light improved the magnetosome synthesis of WT AMB-1 cells, which was related to the function of Amb2291. amb2291 regulates the expression of antioxidant genes and helps AMB-1 scavenge intracellular ROS At the end of exponential growth, after 20 h of incubation, we analysed the difference in the expression of the stress-related genes (oxyR, sod) between the dark and light groups with or without H 2 O 2 by qRT-PCR. For WT and mutant cells, regardless of the H 2 O 2 treatment, red-light illumination led to significant upregulation of the oxyR gene compared with their dark controls (Fig. 7a, p<0.01). By contrast, red-light illumination significantly downregulated the expression of sod compared with the dark controls in both the WT cells and the amb2291 mutants (Fig. 7b, p<0.05). The H 2 O 2 treatment without red-light illumination had no obvious effect on oxyR and sod expression. However, simultaneous red-light illumination and H 2 O 2 treatment increased the expression of oxyR and sod only in the WT strain (Fig. 7, p<0.05).

FiguresFig. 1

 1 Figures

Fig. 2

 2 Fig. 2Spectral properties of purified Amb2291 and Amb2291 A399C proteins. a Absorption spectra of Amb2291 (solid line) and Amb2291 A399C (dashed line) in the dark. b Fluorescence emission spectra for Amb2291 and Amb2291 A399C . Fluorescence excitation was monitored at 450 nm. c Fluorescence excitation spectra for Amb2291 and Amb2291 A399C . Emission was at 530 nm.

Fig. 3

 3 Fig. 3 Photoreduction of purified Amb2291 and Amb2291 A399C proteins by blue-light irradiation. Absorbance changes measured for Amb2291 (a) and Amb2291 A399C (b) on exposure to high-intensity blue light (1700 µmol photons/m 2 /s). c The kinetic data were obtained by following the absorbance changes at 449 nm for Amb2291 and Amb2291 A399C proteins. A 0 represents the amount of unreacted flavoprotein at time zero. A x represents the corresponding amount of unreacted flavoprotein at time x.

Fig. 4

 4 Fig. 4 Evaluation of the H 2 O 2 -generated oxidative stress effect on AMB-1 cells. The OD values of AMB-1 WT cells (solid lines) and Δamb2291 mutant (dashed lines) under irradiation or in the dark at different concentrations of H 2 O 2 were the average of four independent experiments with standard deviations (error bars). The OD values are grouped into a, b, and c using the HCA method.

Fig. 5

 5 Fig. 5 Relative expression levels of the residual amb2291 gene determined via qRT-PCR under red light and/or by adding H 2 O 2 to the medium (n=3). # represents the statistical analyses performed between red light and dark. & shows the statistical analyses between the H 2 O 2 group and their respective normal controls. * describes the statistical analyses between the mutants and WT control. Δ shows the statistical analyses between Δamb2291 and amb2291 A399C mutant cells.

Fig. 6

 6 Fig. 6 Effects of amb2291 deletion or mutant on magnetosome formation under photooxidative stress conditions. a Value of Cmag under red-light irradiation or dark at different concentrations of H 2 O 2 in AMB-1 cells (n=4). The Cmag of each group is shown and analysed using HCA. Relative expression levels of mms13 (b), mms6 (c), and mmsF (d) determined via qRT-PCR under red light and/or by adding H 2 O 2 to the medium (n=3). 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 one or two asterisks ( * p<0.05, * * p<0.01).

Fig. 7

 7 Fig. 7 Relative expression levels of oxyR (a) and sod (b) genes determined under red light and/or by adding H 2 O 2 in the medium (n=3). The 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

Fig. 8

 8 Fig. 8 Effects of Amb2291 on intracellular ROS level in AMB-1 cells (n=6). The level of significance of the differences observed between the 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 one or two asterisks ( * p<0.05, * * p<0.01).

Fig. 9

 9 Fig.9Effects of amb2291 deletion or site-directed mutagenesis on the phototactic behaviour of AMB-1 cells. Relative phototactic behaviour (%) of the WT and mutant strains (n=4). Level of the significance of the differences observed between light and dark control is expressed as a hashtag sign (#p<0.05). The significant differences between the two groups depicted by the lines in the graph are

  . To prevent the difference in phototactic behaviour caused by H 2 O 2 on the state of bacterial movement, we first examined the speed of bacteria movement under different H 2 O 2 concentrations. Low H 2 O 2 concentrations (10, 20, 30, and 40 µM) exhibited insignificant effects on the velocities of AMB-1 WT cells and mutants. However, with increasing H 2 O 2

Acknowledgements

We thank Dandan Li (National Institute of Biological Sciences) for the plasmid of pET28a-His6-SUMO. We greatly appreciate Changyou Chen with the help of useful discussion and suggestions.

Funding information

This work was supported by the State Key Program of National Natural Science of China (Grant No.

51937011) and the Research Project Funded by the Institute of Electrical Engineering, Chinese

Academy of Sciences (grant number Y650141CSA).

Compliance with ethical standards

Conflict of interests

The authors declare that they have no conflict of interest. The arrows show the red-light signal transduction pathway; Amb2291 may interact with red/far-red-light-induced bacteriophytochrome (BphP) by some mechanisms to form the signalling network response to red light. The signal transduction intermediates can be anti-oxidative genes, such as oxyR and sod, and magnetosome, the magnetite crystal in MTB.

Ethical approval


Baparou Danhane
email: baparou.danhane@univ-lorraine.fr

Jérôme Lohéac
email: jerome.loheac@univ-lorraine.fr

Marc Jungers
email: marc.jungers@univ-lorraine.fr

Contributions to output controllability for Linear Time Varying systems

Keywords:

The purpose of this paper is to provide some contributions to one notion of Output controllability for Linear Time Varying (LTV) systems which is the Complete State to Output Controllability (CSOC), notion introduced in the 60s by P. E. Sarachik and G. M. Kranc. More precisely, we consider LTV systems with direct transmission of the input to the output and establish criteria to ensure the CSOC in finite time of these systems. We also give, under the assumption of CSOC in finite time, an explicit expression of a continuous control built by means of a Gramian matrix.

Introduction

The Complete State to Output Controllability (CSOC) for Linear Time Varying (LTV) systems was introduced in [START_REF] Sarachik | On optimal control of systems with multinorm constraints[END_REF] by P. E. Sarachik and G. M. Kranc in 1963. They considered LTV systems without direct transmission of the input to the output and extended the Kalman state controllability Gramian criterion to the case of output. In [START_REF] Kreindler | On the concepts of controllability and observability of linear systems[END_REF], E. Kreindler and P. E. Sarachik extended this criterion to LTV systems with direct transmission from the input to the output. For this notion, the aim is to study if one can drive in finite time the output of these systems to a prescribed value from any initially given state data. Another notion of output controllability for LTV systems without direct transmission of the input to the output, which is Complete Output Controllability can be found in [START_REF] Angelo | Linear time-varying systems: Analysis and synthesis[END_REF]. For this second notion, as stated in [3, Definition 4.2], the question is whether it is possible or not to steer the output from any initial output value to a prescribed output value in finite time. Note that even if the criteria given in [START_REF] Angelo | Linear time-varying systems: Analysis and synthesis[END_REF] are the same as those given in [START_REF] Kreindler | On the concepts of controllability and observability of linear systems[END_REF], these two notions are rigorously distinct as it was discussed in [4, Section 6] for Linear Time-Invariant systems. So, the criteria is the set of real functions defined on R + that are essentially bounded on any compact set of R + . For all of these spaces, we use the same convention, as the one used for R to define matrices of such functions.

Preliminaries and problem formulation

Consider, for t > 0, the LTV system ẋ(t) = A(t)x(t) + B(t)u(t), (1a)

y(t) = C(t)x(t) + D(t)u(t), (1b)
where x(t) ∈ R n , u(t) ∈ R m and y(t) ∈ R q are respectively the state, the input (or the control) and the output of the system at time t where n, m and q are positive integers. These three integers will always refer to the dimensions of the state, input and output. We also consider

A ∈ L ∞ loc (R + ) n n , B ∈ L ∞ loc (R + ) n m , C ∈ C 0 (R + ) q
n and D ∈ C 0 (R + ) q m . Unless otherwise stated, these regularities will be assumed all along this paper.

Some reminders on differential equations

Since in this paper, we will investigate the state to output controllability in finite time of system (1), we consider a generic time t f > 0 and focus on the restrictions of solutions of (1) on the interval [0, t f ]. It is well-known, see e.g. [5, Lemma 2], that for u ∈ L ∞ (0, t f ) m , equation (1a) coupled with the initial condition x(0) = x 0 ∈ R n admits a unique solution on C 0 ([0, t f ]) n defined, for every t ∈ [0, t f ], by

x u (x 0 , t) = R(t, 0)x 0 + t 0 R(t, s)B(s)u(s) ds. ( 2
)
This solution is expressed in terms of the resolvent matrix R associated to the system ẋ(t) = A(t)x(t). Definition and properties of this matrix are recalled below.

Definition 1

The resolvent of the system

ẋ(t) = A(t)x(t) is the map R : (t, τ ) ∈ R 2 + → R(t, τ ) ∈ R n n such that for every (t, τ ) ∈ R 2 + , M (t) = R(t, τ ) is the unique solution of the Cauchy problem Ṁ (t) = A(t)M (t) with M (τ ) = I n .
The following properties of this map can be found for instance in [START_REF] Agarwal | An introduction to ordinary differential equations[END_REF]Lecture 18].

Proposition 1 We have R ∈ C 0 (R 2 + ) n n , and for every t 0 , t 1 , t 2 ∈ R + , R(t 0 , t 0 ) = I n and R(t 0 , t 1 )R(t 1 , t 2 ) = R(t 0 , t 2 ). In addition, if A ∈ C k (R + ) n n , with k ∈ N, then R ∈ C k+1 (R 2 + ) n n and for every t ∈ R + , ∂R(t, t 2 ) ∂t = A(t)R(t, t 2 ), ∂R(t 1 , t) ∂t = -R(t 1 , t)A(t).

Definition and problem statement

In the framework of state controllability, it is well-known, and can be seen through [START_REF] Coron | Control and nonlinearity[END_REF], that even if we consider the set of essentially bounded controls as a set of admissible inputs, the state trajectories are continuous. But, when one wants to control the output of the system (1) with D = 0 q m , one can readily see that allowing jumps on the input induces discontinuities on the output trajectories. This is often not desirable in practice. For this practical purpose, it is more natural to consider the set of continuous functions as the set of admissible controls. Therefore, the following definition will be considered in the sequel.

Definition 2

The system (1) is said to be state to output controllable (SOC) in time t f , if for every (x 0 , y 1 ) ∈ R n × R q , there exists a control u ∈ C 0 ([0, t f ]) m such that the output solution y u (x 0 , t) of system (1) with x 0 ∈ R n as initial state data and u as input satisfies at time t f , y u (x 0 , t f ) = y 1 .

This definition gives rise to the following problems.

Problem 1 Determine efficient and practical characterizations of the SOC in finite time of system (1).

Problem 2 For any (x 0 , y 1 ) ∈ R n × R q , find, when system (1) is SOC in a time t f > 0, an explicit expression of a continuous control steering x 0 to y 1 in time t f in the sense of Definition 2.

From (1b), we observe that in any case, y u (x 0 , t f ) ∈ Im(C(t f )|D(t f )). Hence, a necessary condition for system (1) to be SOC in time t f is rk C(t f )|D(t f ) = q. This condition also tells us that if system (1) is SOC in some time t f > 0, it is not necessarily SOC in another time tf = t f .

Main results

The proofs of the results given in this section are postponed in Section 5. All along this paper, the following notations will be used: for all t ∈ [0, t f ] and

u ∈ C 0 ([0, t f ]) m , M t f (t) := C(t f )R(t f , t)B(t) ∈ R q m , H t f (t) := t f t M t f (s) ds + D(t f ) ∈ R q m , E t f (u) := t f 0 M t f (t)u(t) dt + D(t f )u(t f ) ∈ R q , K t f := t f 0 M t f (t)M t f (t) dt + D(t f )D(t f ) ∈ R q q , G t f := t f 0 H t f (t)H t f (t) dt ∈ R q q .
The following theorem links the notion of SOC in time t f to the surjectivity of the Endpoint map E t f , and to the positive definiteness of the Gramian matrices K t f and G t f .

Theorem 2

The following assertions are equivalent.

(i) The system (1) is SOC in time t f .

(ii) The map E t f defined on C 0 ([0, t f ]) m is surjective.

(iii) The symmetric matrix K t f is positive definite.

(iv) The symmetric matrix G t f is positive definite.

We will see in Theorem 8 that the Gramian matrix G t f , involved in condition (iv), will also lead to an expression of a continuous control, while the Gramian matrix K t f proposed in [START_REF] Kreindler | On the concepts of controllability and observability of linear systems[END_REF] will only lead to a piecewise continuous control. The above SOC criteria (iii) and (iv) might be difficult to check in practice. Indeed, to use them, we need to compute the resolvent matrix, and to check the invertibility of a matrix. This is difficult in general, even in low dimensions. For that reason, we propose in Theorem 3 a sufficient condition which does not require the knowledge of the resolvent matrix. Furthermore, this condition becomes necessary under analytical assumption.

Theorem 3 Let k ∈ N ∪ {∞}, and assume that A ∈ C k-1 (R + ) n n (or A ∈ L ∞ loc (R + ) n n for k = 0), B ∈ C k (R + ) n m . If there exists a time t f > 0 such that rk (D(t f )|C(t f )B i (t f ), i ∈ N k ) = q, (3)
where B 0 (t) = -B(t) and B i+1 (t) = -Ḃi (t) + A(t)B i (t), then system (1) is SOC in time t f . Moreover, if A and B are analytic matrices, then (3), with k = ∞, is necessary.

Remark 1 Theorem 3 is also valid if the matrices A(t) and B(t) only have the required regularity at time t = t f .

Remark 2 Since the rank of a matrix is invariant under a certain threshold of perturbation (see e.g. [8, Chapter 5]), if (3) holds in time t f , then (3) also holds in a neighborhood of t f . Indeed, one can see that the matrices D and CB i (for i ∈ N k ) are all continuous at time t f . Furthermore, if A(t), B(t), C(t) and D(t) are analytic matrices with respect to t, and (3) holds for some time t f > 0 and some k ∈ N, then (3) holds for almost every t f ∈ R + .

Remark 3 Bearing in mind Cayley-Hamilton theorem, one can readily see that if matrices A, B, C and D are time independent, condition (3), with k n, is exactly the Kalman extended rank criterion given in [4, Theorem 3.1

(soc 2 )], that is rk(D|CA 0 B| • • • |CA n-1 B) = q.
If A and B are not constant, we can no longer use Cayley-Hamilton in (3) due to the presence of the time derivatives of these matrices.

We insist on the fact that condition (3) is sufficient, but not necessary as it is illustrated on the following example. 2 dt is positive definite, thanks to the assumptions made on f and g.

Example 1 Consider system (1) with

A = 0 3 3 , D = 0 2 1 , C = (I 2 | 0 2 1 ) and B(t) = (f (t), g(t), h(t)) , where f, g, h ∈ C k (R + ), for some k ∈ N ∪ {∞}, non identically null functions on [0, t f ] such that the support of f and g are re- spectively included in [0, t f /2] and [t f /2, t f ]. We deduce that Span{D(t f )u, u ∈ R m } + Span{C(t f )B i (t f )u, i ∈ N k , u ∈ R m } ⊂ {0} × R. Therefore, we infer that condition (3) does not hold in time t f . But one can see through condi- tion (iii) that K t f = diag t f 0 f (t) 2 dt, t f 0 g(t)
Contrary to Theorem 3, the following results use the resolvent matrix R. Thus, they are only usable in practice if one knows how to calculate it. For the sake of completeness, we recall in the following lemma the expression of the derivatives of H t f . The proof of this lemma is not given but can be obtained by induction, together with Proposition 1.

Lemma 4 Let k ∈ N∪{+∞}. If A ∈ C k-1 (R + ) n n (or A ∈ L ∞ loc (R + ) n n for k = 0), B ∈ C k (R + ) n m then H t f ∈ C k+1 (R + ) q m and H (i) t f (t) = C(t f )R(t f , t)B i-1 (t) for all i ∈ N * k+1 and t ∈ R + .
The following proposition extends [2, Proposition 1.19] to the case of the controllability of the output of system (1).

Proposition 5 Assume there exists a time t f > 0 such that (3) holds with k = ∞. Then there exists ε > 0 such that for every t

∈ (t f -ε, t f +ε)∩(R + \{t f }), rk H (i) t f (t), i ∈ N <q = q, (4)
Moreover, if A and B are analytic then (3) and (4) are equivalent.

Remark 4 Without the analyticity hypothesis, (3) and (4) are not equivalent. One can get convinced by considering system (1) with m = n = q = 1, A = D = 0, C = 1 and B(t) = (2/(t -1) 3 )e -1/(t-1) 2 for t = 1 and B(1) = 0. With this example, one can easily see that taking t f = 1, (4) holds but (3) does not.

The following theorem generalizes Theorems 3 and Proposition 5. The proof of this theorem will be sketched at the same time as the one of Theorems 3 and Proposition 5.

Theorem 6 Let k ∈ N∪{∞}. Assume that A ∈ C k-1 (R + ) n n (or A ∈ L ∞ loc (R + ) n n for k = 0), B ∈ C k (R + ) n m and let ∈ N * q . The system (1) is SOC in time t f if there exist t 1 , • • • , t in [0, t f ], not necessarily distinct, such that rk H (i) t f (t j ), i ∈ N k+1 , j ∈ N * = q, (5)
Moreover, if (5) holds with k = ∞ and = 1, then there exists ε > 0 such that for every t

∈ (t 1 -ε, t 1 + ε) ∩ ([0, t f ]\{t 1 }), we have rk H (i) t f (t), i ∈ N <q = q.
Let us compare Theorem 6 with [3, Theorem 4.9]. In [START_REF] Angelo | Linear time-varying systems: Analysis and synthesis[END_REF]Theorem 4.9], it is required that the regularity of the matrices A and B are respectively n -2 and n -1, and the rank condition shall be satisfied on a nonempty interval. In Theorem 6 we do not really have regularity constraints, and the rank condition is pointwise. Needless to say that if the necessary condition of [3, Theorem 4.9] is fulfilled, then the necessary condition of Theorem 6 is also fulfilled. In addition, all the discussions that we had in the analytic case can be applied to Theorem 6.

As title of example, let us apply Theorem 6 to Example 1 with k = 0, i.e., the functions f , g and h are just continuous. In that case,

H t f ∈ C 1 (R + ) 2 , H t f (t) = ( t f t f (s)ds, t f t g(s)ds) and H (1) t f (t) = -(f (t), g(t)) for all t ∈ [0, t f ]. By assumption, there exists (t 1 , t 2 ) ∈ [0, t f /2] × [t f /2, t f ] such that f (t 1 ) = 0, f (t 2 ) = 0, g(t 1
) = 0 and g(t 2 ) = 0. From these computations, we have rk H

(1)

t f (t 1 ), H (1)
t f (t 2 ) = q = 2. Hence, Theorem 6 ensures the SOC of the system in time t f .

In the case we do not have enough regularity to apply Theorems 3 or 6, the following theorem can be considered. This theorem, while being simple, is relevant since it gives necessary and sufficient condition for SOC without requiring supplementary regularities on the matrices A and B other than being essentially bounded. Recall that in that case, H t f ∈ C 0 (R + ) q m . Theorem 7 The system (1) is SOC in time t f if and only if there exists

t 1 , • • • , t q ∈ [0, t f ], not necessarily distinct, such that rk(H t f (t 1 ), • • • , H t f (t q )) = q. (6)
Moreover, if A and B are analytic, (t 1 , • • • , t q ) can be found in any nonempty open set contained in [0, t f ] q .

Since the rank of a matrix is invariant under a certain threshold of perturbation, if there exists (t 1 , • • • , t q ) ∈ [0, t f ] q such that (6) holds, then there exists a neighborhood N of (t 1 , • • • , t q ) in [0, t f ] q such that for every ( t1 , • • • , tq ) ∈ N , rk(H t f ( t1 ), . . . , H t f ( tq )) = q. From the continuity of H t f with respect to t f if (6) holds, there exists ε > 0 such that for every tf ∈ (t f -ε, t f + ε) ∩ R + the system is SOC in time tf .

To illustrate Theorem 7, let us consider again Example 1, with f and g uniformly bounded. Due to the support assumptions made on f and g, there exist (t

1 , t 2 ) ∈ [0, t f /2]×[t f /2, t f ] such that t f t1 f (s)ds = 0, t f t2 f (
s)ds = 0 and t f t2 g(s)ds = 0. That is to say that rk(H t f (t 1 ), H t f (t 2 )) = q = 2, and hence, according to Theorem 7, this system is controllable.

We end this section by giving Theorem 8 which solves Problem 2.

Theorem 8 Assume that the system (1) is SOC in time t f . For every (x 0 , y 1 ) ∈ R n × R q and every u 0 ∈ R m , the control, given, for every t ∈ [0, t f ], by

u(t) = u 0 + t 0 H t f (τ ) (G t f ) -1 (y 1 -y u0 (x 0 , t f )) dτ, (7)
steers x 0 to y 1 in time t f , where we have set

y u0 (x 0 , t f ) = C(t f )R(t f , 0)x 0 + H t f (0)u 0 .
Furthermore, this control is the unique minimizer of

min 1 2 t f 0 | ẇ(t)| 2 dt w ∈ H 1 (0, t f ) m , w(0) = u 0 , y 1 = y w (x 0 , t f ). ( 8
)
Remark 5 The expression of the control given in (7) is based on the inverse of the Gramian matrix G t f . As already mentioned, computing this matrix and checking its invertibility might be hard in practice. From a numerical point of view, it is more interesting to consider the minimization problem (8) than trying to compute directly the expression (7).

When system (1) is SOC in time t f > 0, even if the matrix K t f does not lead to a continuous control as mentioned earlier, we observe from [START_REF] Kreindler | On the concepts of controllability and observability of linear systems[END_REF] that it leads to a piecewise continuous one. More precisely, the control given by

u(t) = M t f (t) (K t f ) -1 δ y , for t ∈ [0, t f ), u(t f ) = D(t f ) (K t f ) -1 δ y , (9)
with δ y = y 1 -C(t f )R(t f , 0)x 0 , steers x 0 to y 1 in time t f . Furthermore, this control is the unique minimizer of

min 1 2 t f 0 |u(t)| 2 dt + 1 2 |ξ| 2 u ∈ L 2 (0, t f ) m , ξ ∈ R m , y 1 -C(t f )R(t f , 0)x 0 = t f 0 M t f (t)u(t) dt + D(t f )ξ,
where ξ stands for the final control value u(t f ). One can see from the above expression of u that, this control is discontinuous unless

C(t f )B(t f ) = D(t f ).

Illustration

To illustrate our results, we consider the system (1) with n = 3, m = 1, q = 2, and matrices A, B, C and D given for t 0 by

A(t) =   0 -1 0 1 0 0 0 0 t   , B(t) =   cos(t) sin(t) 0   , C(t) = 0 1 t 2 t 0 cos(t) , D(t) = 0 sin(t) . (10)
For any time t f > 0, one can check that the state controllability Gramian in time t f , given in [2, Theorem 1.11] applied to system (1) with (10) is not invertible. It follows that this system is not state controllable in time t f .

• Criteria (iii) and (iv). Let us check if this system is SOC in some time t f > 0. For any time t f > 0, we get after some computations det(K t f ) = t f sin(t f ) 4 and det(G t f ) = t 4 f sin(t f ) 4 /12, where K t f and G t f are the matrices defined respectively in (iii) and (iv). It follows from Theorem 2 that system (1) with matrices given in (10) is SOC in time t f > 0 if and only if t f ∈ πN.

• Theorem 3. Since the matrices defined in (10) are analytic, condition (3) turns to be necessary and sufficient. From (iv), we have

H t f (t) = sin(t f )(t f - t), sin(t f ) + t f cos(t f )(t f -t) . We then have H (0) t f (t) = H t f (t), H (1)
t f (t) = -(sin(t f ), t f cos(t f )) and H (i)
t f (t) = 0 for all i 2. It follows from the previous computations that system [START_REF] Agarwal | An introduction to ordinary differential equations[END_REF] with ( 10) is SOC in time t f if and only if t f ∈ πN.

• Proposition 5. For every t > 0, we have det 2 . It follows that system (1) with ( 10) is SOC in time t f if and only if criterion (3) is fulfilled. Here again, we have used the analyticity of the matrices defined in (10) to get the necessary condition.

H (i) t f (t), i ∈ N <2 = sin(t f )
• Theorem 7. For every t 1 , t 2 ∈ [0, t f ], we have det H t f (t 1 ), H t f (t 2 ) = (t 2 - t 1 ) sin(t f ). We deduce that (t 1 , t 2 ) → det H t f (t 1 ), H t f (t 2 ) is not identically null if and only if t f ∈ πN.
• Control computation. To illustrate Theorem 8, we take t f = π/2. From the above, the system considered is SOC in this time. Now we would like to build a continuous control u, steering x 0 = (1 0 1) to y 1 = (1 2) in time t f . For this purpose, we decide to construct u such that u(0) = 1. Applying formula [START_REF] Sarachik | On optimal control of systems with multinorm constraints[END_REF], we get for every t

∈ [0, π/2] u(t) = 12 π 2 3 + πe π 2 /8 t 2 -2 π 8 + 3πe π 2 /8 t + 1.
One can also see that the piecewise control built in ( 9) is given by u(t) = -π 2 e π 2 /8 for t ∈ [0, π/2) and u( π 2 ) = 2. Solving (1) with these two controls, and x 0 as initial data, we get the time trajectories depicted on Figure 1.

Proofs of the results

Proof of Theorem 2

We do not give the proof of condition (iii) since it can be found in [START_REF] Kreindler | On the concepts of controllability and observability of linear systems[END_REF]Theorem 2]. We only prove the equivalence between (i), (ii) and (iv).

• Equivalence between (i) and (ii). First note that if system (1) is SOC in time t f then the origin can be steered to any point of R q in time t f . This implies that (ii) holds. Reciprocally, if E t f is surjective, for every (x 0 , y 1 ) ∈ R n × R q , there exists a control u * ∈ C 0 ([0, t f ]) m such that E t f (u * ) = y 1 -C(t f )R(t f , 0)x 0 , i.e., x 0 can be steered to y 1 in time t f . • Equivalence between (ii) and (iv). Let us first assume that there exists a non-zero vector η ∈ R q such that

η G t f η = 0. ( 11
) Since H t f ∈ C 0 (R + ) q m , we deduce that η H t f (t) = 0 1 m for all t ∈ [0, t f ], i.e. η t f t M t f (s) ds + η D(t f ) = 0 1 m , ∀t ∈ [0, t f ]. ( 12
)
Taking t = t f in (12) and differentiating the same equation with respect to t, one deduce that η D(t f ) = 0 1 m and η M t f (t) = 0 1 m for almost every t ∈ [0, t f ]. On the right-hand side, we have the time trajectories obtained using the control built with K t f . One can see that the output trajectories with our proposed Gramian G t f are continuous and those built by means of K t f are not.

It follows that the linear map E t f is not surjective. Conversely, assume there exists η ∈ R q \{0 q } such that η E t f (u) = 0 for every

u ∈ C 0 ([0, t f ]) m . It follows that η t f 0 M t f (t)u(t) dt + η D(t f )u(t f ) = 0, for every u ∈ C 1 ([0, t f ]) m , and therefore, for every v ∈ C 0 ([0, t f ]) m , η t f 0 M t f (t) t 0 v(s) ds dt + η D(t f ) t f 0 v(t) dt = 0, (13)
Noticing that by an integration by parts we have

t f 0 M t f (t) t 0 v(s) ds dt = t f 0 t f t M t f (s)v(t) ds dt, (14)
we deduce from (13) that

η t f 0 H t f (t)v(t) dt = 0, ∀v ∈ C 0 ([0, t f ]) m . ( 15
) Setting v(t) = H t f (t) η for all t ∈ [0, t f ], v ∈ C 0 ([0, t f ]
) m and it follows from (15) that (11) holds for some non-zero vector η, that is to say that G t f is not positive definite.

Remark 6 It can be deduced from the above proof that denoting by R t f the set of all output points that can be reached from 0 n in time t f , we have

R t f = E t f (u), u ∈ C 0 ([0, t f ]) m = t f 0 H t f (t)u(t) dt, u ∈ C k ([0, t f ]) m for every k ∈ N.
This observation shows that [4, Lemma 4.1] remains true in the case of LTV systems.

Proof of Theorem 3

Let us prove Theorem 3 by contraposition. To this end, we assume that system (1) is not SOC in time t f . Then, according to Theorem 2, there exists a non-zero vector η ∈ R q such that η G t f η = 0. Using the continuity of H t f , we infer that

η H t f (t) = 0 1 m , ∀t ∈ [0, t f ]. (16)
Taking t = t f in (16), we get η D(t f ) = 0 1 m . Using Lemma 4, we deduce by differentiating (16) with respect to t and evaluating the resulting equations at time t f that for every

i ∈ N k , η C(t f )B i (t f ) = 0 1 m .
From the above, we deduce that (3) does not hold. Note that in (16), we decided to evaluate this equality and its derivatives at time t f in order to eliminate the resolvent. But one can also evaluate this relation and its derivatives at any other time taken in [0, t f ]. This leads to the sufficient condition given in Theorem 6. Observe that if in addition, matrices A and B are analytic, then the map t → η H t f (t) is analytic in time. It follows that if its derivatives of all order vanish at some time, then that map vanishes identically on R + . This implies that G t f η = 0 q and therefore that G t f is not positive definite.

Proof of Proposition 5

The proof of this proposition is divided into two steps. The first step deals with the case m = 1. In the second step, we show how the case m > 1 can be reduced to the case m = 1. This proof is inspired by the proof of [START_REF] Coron | Control and nonlinearity[END_REF]Proposition 1.19]. Note that when A and B are analytic, the reverse implication is deduced from Theorem 3. 3) holds for t f thanks to the assumption in Proposition 5, we have from (3) and Lemma 4, rk Lemma 1.22], and the discussion following this lemma, to the function H t f , we deduce that there exist

• Case m = 1. Note that since A ∈ C ∞ (R + ) n n and B ∈ C ∞ (R + ) n , then H t f ∈ C ∞ (R + ) q . Since (
H (i) t f (t f ), i ∈ N = q. Applying [2,
q integers d 1 < • • • < d q , q functions a i ∈ C ∞ (R + ) and q vectors v i ∈ R q such that rk(v 1 , • • • , v q ) = q, a i (t f ) = 0, ∀i ∈ N * q , (17a)
H t f (t) = q i=1 a i (t)(t -t f ) di v i , ∀t 0. ( 17b
)
Hence, when t goes to t f , we have det

H (i) t f (t), i ∈ N <q = K(t -t f ) δ + O (t -t f ) 1+δ , (18)
with δ = -q(q -1)/2 + q i=1 d i and

K = 1 i<j q (d j -d i ) q i=1 a i (t f ) det(v 1 , • • • , v q ).
From (17a) and the fact that d i = d j for i = j, we deduce that K = 0. It follows from (18) that there exists ε > 0 such that for every t

∈ (t f -ε, t f + ε) ∩ (R + \{t f }), det H (i) t f (t), i ∈ N <q = 0. This ends the case m = 1. • Case m > 1. Set H t f (t) = (h 1 (t), • • • , h m (t)).
From the hypothesis of Proposition 5 and from Lemma 4, we have R q = Span h (i)

j (t f ), j ∈ N * m , i ∈ N . Hence, there exist p ∈ N * such that R q = Span h (i) j (t f ), j ∈ N * m , i ∈ N <p . We then set, for t 0, H(t) := m j=1 (t -t f ) (j-1)p h j (t) and obtain rk H(i) (t f ), i ∈ N = q. Since h j ∈ C ∞ (R + ) q , we have H ∈ C ∞ (R + ) q .
We therefore use the result of the case m = 1 to get the existence of ε > 0 such that for every t ∈

(t f -ε, t f + ε) ∩ (R + \{t f }), rk H(i) (t), i ∈ N <q = q. Since Span H(i) (t), i ∈ N <q ⊂ Span h (i) j (t), j ∈ N * m , i ∈ N <q , we infer that for every t ∈ (t f -ε, t f + ε) ∩ (R + \{t f }), rk H (i) t f (t), i ∈ N <q = q.

Proof of Theorem 7

It can be seen through (16) that if system (1) is not SOC in time t f , condition [START_REF] Kreindler | On the concepts of controllability and observability of linear systems[END_REF] does not hold for every t

1 , • • • , t q in [0, t f ]. Assume by contraposition that for every t 1 , • • • , t q ∈ [0, t f ], rk H t f (t 1 ), • • • , H t f (t q ) < q. Choose τ 1 , • • • , τ q ∈ [0, t f ] that realize max t1,••• ,tq∈[0,t f ] rk H t f (t 1 ), • • • , H t f (t q ) . It follows that for every u ∈ R m and every t ∈ [0, t f ], H t f (t)u ∈ Span H t f (τ i )u, i ∈ N * q , u ∈ R m R q .
Hence, we deduce that there exists a vector η ∈ R q \{0 q } such that η H t f (t) = 0 1 m for every t ∈ [0, t f ]. It follows that η is in the kernel of G t f , and therefore that system (1) is not SOC in time t f . Now, assume that matrices A and B are analytic, and that the system (1) is SOC in time t f . From what precedes, there exist t

1 , • • • , t q ∈ [0, t f ] such that rk H t f (t 1 ), • • • , H t f (t q ) = q. This implies the existence of τ 1 , • • • , τ q ∈ {t 1 , • • • , t q } and u 1 , • • • , u q ∈ R m such that det H t f (τ 1 )u 1 , • • • , H t f (τ q )u q = 0. Since A and B are analytic, the application (s 1 , • • • , s q ) ∈ [0, t f ] q → det H t f (s 1 )u 1 , • • • , H t f (s q )u q ∈ R
is also analytic and non identically null. Hence, its zeros set is of null measure. This implies that for every nonempty open set O, in [0, t f ] q , there exist ( t1 , t2 ,

• • • , tq ) ∈ O such that det H t f ( t1 )u 1 , • • • , H t f ( tq )u q = 0.

Proof of Theorem 8

Let x 0 ∈ R n , y 1 ∈ R q and u 0 ∈ R m . Since the system (1) is SOC in time t f , there exists a control u ∈ C 0 ([0, t f ]) m steering x 0 to y 1 in time t f . There even exists an infinite number of controls doing the same transfer since, by the rank Theorem, the kernel of the linear map E t f defined in Theorem 2 is of infinite dimension. We can even build a continuous control such that u(0) = u 0 as it will be done in the sequel of this proof. To prove Theorem 8, we will look for the control solution of the minimization problem [START_REF] Stewart | Matrix perturbation theory[END_REF]. We prove this theorem by analysis and synthesis. First suppose that there exists a function w ∈ H 1 (0, t f ) m such that w(0) = u 0 that steers x 0 to y 1 in time t f . Since w ∈ H 1 (0, t f ) m and w(0) = u 0 , there exists v ∈ L 2 (0, t f ) m such that w(t) = u 0 + t 0 v(τ )dτ . Replacing w by this expression in y w (x 0 , t f ) and using (14), we deduce that y w (x 0 , t f ) = y u0 (x 0 , t f ) + Φ(v), with Φ(v) = t f 0 H t f (t)v(t) dt. The minimization problem (8) can then be rewritten as min 1 2

t f 0 |v(t)| 2 dt v ∈ L 2 (0, t f ) m , y 1 -y u0 (x 0 , t f ) = Φ(v). ( 19
)
Note that problem (19) is the minimization of a strictly convex functional under non-empty affine constraints. Hence, it admits a unique solution in L 2 (0, t f ) m . Taking ψ as the Lagrange multiplier associated to the equality constraint, the Lagrangian of this optimal control problem is given by

L(v, ψ) = 1 2 t f 0 |v(t)| 2 dt + ψ, y 1 -y u0 (x 0 , t f ) -Φ(v) .
Deriving the first order optimality conditions, we get v(t) = H t f (t) ψ, where ψ is solution of G t f ψ = y 1 -y u0 (t f , x 0 ). Since G t f is invertible, thanks to the SOC in time t f of system (1), we deduce that ψ = (G t f ) -1 (y 1 -y u0 (x 0 , t f )).

Of the foregoing, we have v(t) = H t f (t) (G t f ) -1 (y 1 -y u0 (x 0 , t f )), and w(t) = u 0 + t 0 H t f (τ ) (G t f ) -1 (y 1 -y u0 (x 0 , t f )) dτ . Finally, setting u = w, we have u ∈ C 0 ([0, t f ]) m with u(0) = u 0 and by construction, this control steers x 0 to y 1 in time t f .

Conclusion

In this paper, we provided criteria for state to output controllability of linear time varying control systems. When the system is state to output controllable, we gave an explicit expression of a continuous control realizing the desired transfer, this even if there is a direct transmission between the input and the output. In this paper, we have assumed that the initial state is known. However, combining the arguments used in this paper and the strategy used in [4, Section 6], one can also derive similar conditions for the output to output notions introduced in [START_REF] Danhane | Characterizations of output controllability for LTI systems[END_REF].

Figure 1 :

 1 Figure1: On the left, we have the continuous control (Theorem 8), state and output trajectories built by means of the matrix G t f taking the output of system (10) to y 1 = (1 2) in time t f = π/2 with x 0 = (1 0 1) as initial state data. On the right-hand side, we have the time trajectories obtained using the control built with K t f . One can see that the output trajectories with our proposed Gramian G t f are continuous and those built by means of K t f are not.