Martin Gillard

Hugues Bonnet

Rémy Lartia

Hiba Yacoub

Jérôme Dejeu

Eric Defrancq

Benjamin Elias

Luminescent ruthenium(II) complexes used for the detection of 8oxoguanine in the human telomeric sequence

Detecting cancer at the early stage of the disease is crucial to keep the best chance for successful treatment. The recent development of genomic screening, a methodology that is addressed to asymptomatic patients presumably at risk of carcinogenesis, has stimulated the quest for new tools able to signal the level of risk. Carcinogenesis has been associated to chronic oxidative stress exceeding the antioxidant defenses and leading to critical genome alteration levels. The telomeric regions are presumably the most exposed to oxidative stress due to their high concentration of guanine (i.e. the easiest oxidizable nucleic base). Accumulation of 8-oxoguanine in telomeres, thus oxidative lesions, was reportedly associated with telomeric crisis and carcinogenesis. In this study, we report on the capacity of Ru(II) polyazaaromatic complexes to photoprobe 8-oxoguanine into the human telomeric sequence with the view of developing new tools for cancer risk screening.

Introduction

Diagnosis of cancer currently focuses on detecting early carcinogenesis stage in patients to give them the best chance for successful treatment. [START_REF] Crosby | A roadmap for the early detection and diagnosis of cancer[END_REF] Unfortunately, in the case of aggressive tumours, the cancer has already reached an advanced stage when the first symptoms are detected. This has stimulated the development of screening methods based on different strategies than early diagnosis. Screening is more specifically addressed to apparently healthy and asymptomatic patients that are presumably having a high risk of carcinogenesis. [START_REF] Shieh | Population-based screening for cancer: hope and hype[END_REF] Efficient screening requires the development of new tests or examinations that can be used rapidly and easily. In this context, the detection of cellular oxidative stress signals is of major interest. [START_REF] Katerji | Approaches and Methods to Measure Oxidative Stress in Clinical Samples: Research Applications in the Cancer Field[END_REF] Indeed, chronic acute oxidative stress is a great contributor to the development of numerous diseases including cancer. It consists in the presence of a too high level of reactive oxygen species (ROS) exceeding the antioxidant defences of the organism. This can be attributed to diverse environmental and intrinsic factors. [START_REF] Møller | Air pollution, oxidative damage to DNA, and carcinogenesis[END_REF][START_REF] Ames | Oxidants, antioxidants, and the degenerative diseases of aging[END_REF][START_REF] Nakabeppu | Mutagenesis and carcinogenesis caused by the oxidation of nucleic acids[END_REF][START_REF] Pizzino | Oxidative Stress: Harms and Benefits for Human Health[END_REF] Excess of ROS can cause irreversible oxidative damages to DNA, which is particularly critical at the level of telomeres as they play a pivotal role for genome stability and integrity. Studies in mouse models, human tissues and cell culture provided evidence that oxidative stress promote and accelerate telomere dysfunction. [START_REF] Ludlow | Telomeres shorten in response to oxidative stress in mouse skeletal muscle fibers. The journals of gerontology[END_REF][START_REF] Barnes | The impact of oxidative DNA damage and stress on telomere homeostasis[END_REF][START_REF] Von Zglinicki | Oxidative stress shortens telomeres[END_REF] The loss of telomeres maintenance was reported to contribute to ageing-related diseases, telomere crisis and carcinogenesis. [START_REF] Ahmed | Impact of oxidative stress on telomere biology[END_REF][START_REF] Dewhurst | Chromothripsis and telomere crisis: engines of genome instability[END_REF] This can be explained by the high sensitivity of telomeres to ROS resulting from their high concentration in guanine, the easiest oxidizable nucleic base (E G .

/G = + 1.29 V vs NHE). [START_REF] Steenken | How Easily Oxidizable Is DNA? One-Electron Reduction Potentials of Adenosine and Guanosine Radicals in Aqueous Solution[END_REF] The guanine oxidation mainly leads to the formation of 8-oxoguanine (8-oxoG), the most studied DNA damage by far, which once persistent in telomeres promotes the above mentioned telomere loss and crisis. [START_REF] Fouquerel | Targeted and Persistent 8-Oxoguanine Base Damage at Telomeres Promotes Telomere Loss and Crisis[END_REF] The detection of telomeric 8-oxoG may lead to the development of new screening tools able to sense a risk of carcinogenesis at the earliest stage. [START_REF] Nakabeppu | Cellular Levels of 8-Oxoguanine in either DNA or the Nucleotide Pool Play Pivotal Roles in Carcinogenesis and Survival of Cancer Cells[END_REF][START_REF] Manavalan | Determination of 8-hydroxy-2′-deoxyguanosine oxidative stress biomarker using dysprosium oxide nanoparticles@reduced graphene oxide[END_REF][START_REF] Kang | Detection of 8-oxoguanine and apurinic/apyrimidinic sites using a fluorophore-labeled probe with cellpenetrating ability[END_REF][START_REF] Chiorcea-Paquim | 8-oxoguanine and 8-oxodeoxyguanosine Biomarkers of Oxidative DNA Damage: A Review on HPLC–ECD Determination[END_REF] The human telomeric DNA sequence (hTel) varies from 2-50 kilobases of approximately 300-8,000 repeats of the sequence (-TTAGGG-). As a result of their high concentration in guanines, telomeres can adopt a quadruple helix structure, namely Gquadruplex (G4) consisting in the assembly of four guanines forming a guanine quartet through Hoogsteen bonds. [START_REF] Bochman | DNA secondary structures: stability and function of G-quadruplex structures[END_REF][START_REF] Burge | Quadruplex DNA: sequence, topology and structure[END_REF] Studies reporting on the impact of oxidative lesions on the human telomeric G4 DNA structure stability were recently reported. [START_REF] Miclot | Forever Young: Structural Stability of Telomeric Guanine Quadruplexes in the Presence of Oxidative DNA Lesions*[END_REF][START_REF] Bielskutė | Oxidative lesions modulate G-quadruplex stability and structure in the human BCL2 promoter[END_REF][START_REF] Singh | Oxidative Stress: Role and Response of Short Guanine Tracts at Genomic Locations[END_REF] The destabilizing effect on the G4 structure due to reduced hydrogen-bonding ability caused by the replacement of a guanine residue by 8-oxoG was evaluated. It appeared that the 8-oxoG position into the G4 structure has a major impact on its stability. [START_REF] Bielskutė | Impact of Oxidative Lesions on the Human Telomeric G-Quadruplex[END_REF] During the last decades, many studies demonstrated the abilities of ruthenium (II) complexes as "DNA light switch" or as DNA specific sites photoprobes in vitro or in cellulo. [START_REF] Friedman | A molecular light switch for DNA: Ru(bpy)2(dppz)2+[END_REF][START_REF] Smith | Groove-Binding Ruthenium(II) Complexes as Probes of DNA Recognition[END_REF][START_REF] Mcconnell | Luminescent Properties of Ruthenium(II) Complexes with Sterically Expansive Ligands Bound to DNA Defects[END_REF][START_REF] Wachter | Ruthenium Complex "Light Switches" that are Selective for Different G-Quadruplex Structures[END_REF][START_REF] Piraux | New Ruthenium-Based Probes for Selective G-Quadruplex Targeting[END_REF][START_REF] Boynton | A Ruthenium(II) Complex as a Luminescent Probe for DNA Mismatches and Abasic Sites[END_REF][START_REF] Gill | Ruthenium(ii) polypyridyl complexes and DNA-from structural probes to cellular imaging and therapeutics[END_REF][START_REF] Deraedt | Towards mismatched DNA photoprobes and photoreagents: "elbow-shaped" Ru(ii) complexes[END_REF][START_REF] Gill | A ruthenium(II) polypyridyl complex for direct imaging of DNA structure in living cells[END_REF][START_REF] Berrones Reyes | Metal complexes as optical probes for DNA sensing and imaging[END_REF][START_REF] Jiang | Ruthenium(II) Polypyridyl Complexes and Their Use as Probes and Photoreactive Agents for G-quadruplexes Labelling[END_REF][START_REF] Gillard | Photooxidizing Ruthenium(II) Complexes with Enhanced Visible Light Absorption and G-quadruplex DNA Binding Abilities[END_REF][START_REF] Weynand | Photo-induced telomeric DNA damage in human cancer cells[END_REF] In the present study, we aim at investigating the potential of Ru(II) polyazaaromatic complexes as 8-oxoG photoprobes in telomeric DNA and in duplex DNA. Complexes [Ru(bpy) 2 napp] 2+ 1, [Ru(phen) 2 cpip] 2+ 2 and [Ru(phen) 2 salphen] 2+ 3 (Figure 1) that are present as a mixture of Λ and ∆ isomers were chosen for their reported ability to discriminate DNA topologies (such as G4 DNA and DNA mismatches) thanks to their high luminescence sensitivity. [START_REF] Weynand | Towards the Development of Photo-Reactive Ruthenium(II) Complexes Targeting Telomeric G-Quadruplex DNA[END_REF][START_REF] Gillard | Flexible RuII Schiff Base Complexes: G-Quadruplex DNA Binding and Photo-Induced Cancer Cell Death[END_REF][START_REF] Gillard | Photodetection of DNA mismatches by dissymmetric Ru(ii) acridine based complexes[END_REF] We report on a brief description of the synthesis and the photophysical properties of compounds 1-3 along with a more detailed study of their 8-oxoG photoprobing capacities. This was achieved thanks to luminescence titrations using five different oligodeoxynucleotides (ODN's) that incorporate (or not) an 8-oxoG residue. Three selected ODN's hTel, hTel-oxoG21 and hTel-oxoG10 are based on the human telomeric sequence. They fold into G4 architectures as reported by Podbevšek and co-workers, [START_REF] Bielskutė | Impact of Oxidative Lesions on the Human Telomeric G-Quadruplex[END_REF] with hTel-oxoG21 and hTel-oxoG10 including an 8-oxoG at the outer or at the central quartet of the G4, respectively (Figure 2). The two others selected ODN's HP and HP-oxoG are G-rich sequences that fold into a duplex hairpin shape with HP-oxoG including an oxo-G residue in the middle of the helix (see the structures in the supporting information).

Results and Discussion

Synthesis

Complexes 1-3 were prepared by the chelation of napp, cpip or salphen ligands onto [Ru(bpy) 2 Cl 2 ] or [Ru(phen) 2 Cl 2 ] using previously published methods. [START_REF] Weynand | Towards the Development of Photo-Reactive Ruthenium(II) Complexes Targeting Telomeric G-Quadruplex DNA[END_REF][START_REF] Gillard | Flexible RuII Schiff Base Complexes: G-Quadruplex DNA Binding and Photo-Induced Cancer Cell Death[END_REF][START_REF] Gillard | Photodetection of DNA mismatches by dissymmetric Ru(ii) acridine based complexes[END_REF] Complexes 1-3 were isolated as orange powders after purification on silica and were characterized by 1 H-NMR spectroscopy and HRMS. Compound 1 was also characterised by X-ray crystallography. [START_REF] Gillard | Photodetection of DNA mismatches by dissymmetric Ru(ii) acridine based complexes[END_REF] The [Ru(bpy) 3 ] 2+ and [Ru(phen) 3 ] 2+ references were synthesized and purified according to reported synthetic routes from the hydrated RuCl 3 salt (see the synthesis and characterization of the complexes in the supporting information).

Electrochemistry

In terms of electrochemical properties, compounds 1-3 display oxidation potentials that are very close to those of the reference compounds [Ru(bpy) 3 ] 2+ and [Ru(phen) 3 ] 2+ (1.34 V and 1.32V vs Ag/AgCl). This corresponds to the Ru 2+ /Ru 3+ oxidation, which suggests that the HOMO of the compounds is metal centred (Table 1). Conversely, the potentials of first oxidation appear to be very different ranging from -0.77 V for complex 2 to -1.24 V for complex 1. The strong influence of the extended ligand structure on the potential of first reduction of compounds 1-3 indicates that their LUMO orbitals lay on their extended ligands. It appears that the cpip ligand in complex 2 is responsible for a strong stabilization of the LUMO orbital compared to napp and salphen ligands in compounds 1 and 3, which can be attributed to the strong electron stabilizing effect induced by the conjugated imidazole moiety. The next two reduction waves of compounds 1-3 all occur at very similar potentials to those of the reference complexes [Ru(bpy) 3 ] 2+ and [Ru(phen) 3 ] 2+ and correspond to the subsequent reductions of each bipyridine/phenanthroline ancillary ligands (-1.47 V and -1.71 V vs Ag/AgCl, for [Ru(bpy) 3 ] 2+ ).

Light absorption

The light absorption data of compounds 1-3 were measured at room temperature in acetonitrile (Figure 3 and Table 2).

The recorded spectra are typical of polyazaaromatic Ru(II) complexes and are very close to the one of [Ru(bpy) 3 ] 2+ . [START_REF] Amouyal | Synthesis and study of a mixed-ligand ruthenium(II) complex in its ground and excited states: bis(2,2[prime or minute]-bipyridine)(dipyrido[3,2-a : 2[prime or minute],3[prime or minute]-c]phenazine-N4N5)ruthenium(II)[END_REF] Analysis of the data obtained for 1-3 allows to ascribe the UV absorption bands (ε ≈ 10 5 M -1 cm -1 ) to ligand centred transitions. Particularly, the band arising at ca. 285 nm in compound 1 and [Ru(bpy) 3 ] 2+ is attributed to the absorption of the bpy ligands while the band at ca. 265 nm in compounds 2 and 3 and in [Ru(phen) 3 ] 2+ is attributed to the phen ligands. The observed strong absorption bands in the visible region at ca. 450 nm (ε ≈ 10 4 M -1 cm -1 ) arise from metal-to-ligand charge-transfer (MLCT). Complex 1 displays the strongest visible light absorptivity (ε = 2.48 x 10 4 M -1 cm -1 at 456 nm) among the reported compounds. [a] Measurements were carried out at room temperature. sh = shouldering peaks. The absorption data for [Ru(bpy) 3 ] 2+ are from references. [START_REF] Deraedt | Towards mismatched DNA photoprobes and photoreagents: "elbow-shaped" Ru(ii) complexes[END_REF][START_REF] Amouyal | Synthesis and study of a mixed-ligand ruthenium(II) complex in its ground and excited states: bis(2,2[prime or minute]-bipyridine)(dipyrido[3,2-a : 2[prime or minute],3[prime or minute]-c]phenazine-N4N5)ruthenium(II)[END_REF][START_REF] Deraedt | Design and Photophysical Studies of Acridine-Based RuII Complexes for Applications as DNA Photoprobes[END_REF] Table 1. Oxidation (E 1/2 ox ) and reduction (E 1/2 red ) potentials of complexes 1-3 and reference complexes.

Complex E 1/2ox [a] [V] [b] E 1/2 red [a] [V] [b]
[Ru(bpy)

Light emission

The light emission properties of complexes 1-3 including their luminescence wavelengths, quantum yields, and lifetimes were recorded in water and in acetonitrile and are gathered in Table 3 along with those of [Ru(bpy) 3 ] 2+ and [Ru(phen) 3 ] 2+ . Complexes 1-3 exhibit luminescence properties that are very close to those of the reference complexes [Ru(bpy) 3 ] 2+ and [Ru(phen) 3 ] 2+ with a broad unstructured emission at about 600 nm. The emission is also strongly dependent on the solvent polarity and relatively long (≈ microsecond) lifetimes of the excited states typical of [START_REF] Katerji | Approaches and Methods to Measure Oxidative Stress in Clinical Samples: Research Applications in the Cancer Field[END_REF] MLCT-emissive transition were measured. In addition, the bathochromic shift of the compounds' luminescence on going from acetonitrile to the more polar water and their large k r values (> 10 4 s -1 ) confirm the charge-separated excited states as reported for [Ru(bpy) 3 ] 2+ and [Ru(phen) 3 ] 2+ .

Luminescence titrations

As mentioned in the introduction, ruthenium (II) complexes 1-3 were mainly selected for their reported abilities to photoprobe specific DNA topologies. While compound 1 has been previously reported to photodetect DNA mismatches, compounds 2 and 3 have been shown to act as G4 DNA photoprobes. This originates from the combination of (i) the high sensitivity of the polyazaaromatic Ru(II) complexes luminescence to the microenvironment with (ii) an appropriate structural design of the napp, cpip and salphen interacting ligands. In the present work, the ability of compounds 1-3 to photoprobe 8-oxoG lesions in G-quadruplex and in duplex DNA was assessed using five short ODN's. The hTel-oxoG21 and hTel-oxoG10 sequences were recently exploited by Podbevšek and co-workers and allowed to highlight the major impact of the 8-oxoG lesion localization on the conformation and stability of the resulting G4 structures. [START_REF] Bielskutė | Impact of Oxidative Lesions on the Human Telomeric G-Quadruplex[END_REF] They showed that the hTel-oxoG10 G4 containing the 8-oxoG in the central G-quartet leads to structural rearrangement while the hTel-oxoG21 G4 containing the 8-oxoG in the outer G-quartet retains the parent G4 fold. This may lead to differences in terms of interaction and luminescence properties in the presence of photoprobes 1-3. The steady state luminescence titration curves of compound 1 in the presence of the five DNA models are displayed in showing the strongest luminescence increase when interacting with G4 DNA (+224 and +247% in the presence of hTel and hTel-oxoG21, respectively). Conversely, the increase of luminescence intensity of complexes 1-3 appeared to be drastically reduced in the presence of the central 8-oxoG containing G4 structure hTel-oxoG10. Complex 2 revealed to be the best candidate to discriminate the central 8-oxoG containing G4 from the two other sequences (+72% vs +152-166%). A similar behaviour was observed for the duplex hairpin ODN's, as the increases of luminescence intensities of the three complexes 1-3 revealed to be significantly attenuated in the presence of the HP-oxoG against HP. Complex 1 revealed to be the most discriminant compound showing a fall of the luminescence increase from +494% vs HP to a weaker +177% vs HP-oxoG. Measurements were done on air equilibrated solutions. b λ exc = 450 nm. c Measurements made with a 5 µmol L -1 concentration in complex under argon. d Measurements relative to [Ru(bpy) 3 ] 2+ in argon purged solution (Φ em = 0.063) or in nitrogen purged acetonitrile (Φ em = 0.094). [START_REF] Brouwer | Standards for photoluminescence quantum yield measurements in solution (IUPAC Technical Report)[END_REF] The photophysical data for [Ru(bpy) 3 ] 2+ and [Ru(phen) 3 ] 2+ are from reference. 20 The dissociation equilibrium constants (K D ) values of the interaction between complexes 1-3 for the five DNA models were estimated using a modified McGhee-von Hippel model that fits the titration curves (Table 4). [START_REF] Mcghee | Theoretical aspects of DNAprotein interactions -Cooperative and non-cooperative binding of large ligands to a one-dimensional homogeneous lattice[END_REF] The K D values of the interaction of the compounds with the G4 DNA models hTel, hTel-oxoG21 and hTel-oxoG10 appear to be in the tenth of micromolar range. Each of the three complexes displays very close K D values for the different studied sequences, ranging from 11 µM for compound 1 vs hTel to 43 µM for compound 3 vs hTel-oxoG10. The three compounds also display the same trend in terms of affinity for the three sequences with the affinity decreasing in this order: hTel; hTel-oxoG21; hTel-oxoG10 (complex 1: 11; 19; 22 µM, complex 2: 14; 17; 31 µM and complex 3 23, 28, 43 µM). The K D values of the interaction of the complexes with the duplex DNA models HP and HP-oxoG are in the micromolar range for compound 1 and is one to two orders of magnitude higher for compounds 2 and 3 as previously reported. Complex 1 appears to be more affine for HP vs HP-oxoG (12 vs 47 µM) while compounds 2 (205 vs 62 µM) and 3 (176 vs 44 µM) are more affine for the 8-oxoG containing hairpin. The luminescence titration data demonstrate the very low impact of the presence of the 8-oxoG lesion on the interaction strength both for G4 and duplex HP type DNA. They also highlight the high sensitivity of the compounds to their local environment, especially when interacting with DNA. The ability of compounds 1-3 to photoprobe 8-oxo-G containing ODN's in hTel-oxoG10 and HP-oxoG over their parent ODN's hTel and HP presumably arises from a poorer protection of the excited complexes in the 8-oxo-G containing ODN's towards non-radiative

Bio-Layer interferometry studies

Bio-layer interferometry analysis (BLI) was performed to further determine the parameters of the interaction of compounds 1-3 with the five ODN's. The association and dissociation constants between the complexes and the targets were measured, which give access to the dissociation equilibrium constant (K D ). BLI was first implemented to study interactions between large biomolecules, such as protein-membrane interactions and more recently for the interaction of small compounds with G4 DNA. [START_REF] Weynand | Towards the Development of Photo-Reactive Ruthenium(II) Complexes Targeting Telomeric G-Quadruplex DNA[END_REF][START_REF] Wallner | Application of Bio-Layer Interferometry for the analysis of protein/liposome interactions[END_REF][START_REF] Picault | Characterization of the interaction of multivalent glycosylated ligands with bacterial lectins by biolayer interferometry[END_REF] The studies were done in the same saline conditions than luminescence titrations (10 mM HEPES pH 7.4, 100 mM NaCl, 50 mM KCl). BLI sensorgrams recorded for the interaction of complex 1 with the hTel and hTel-oxoG10 are displayed in Figure 6A and in Figure 6B, respectively (see the supporting information for the other sensorgrams).

As revealed by the fitting of the luminescence titration curves, compounds 1-3 exhibit K D values in the tenth of micromolar range for their interaction with the different G4 folding ODN's hTel, hTel-oxoG21 and hTel-oxoG10 (Table 5).

Conclusions

A series of three Ru(II) polyazaaromatic complexes were prepared. The compounds were selected as they were reported for their ability to photoprobe DNA specific sites thanks to their highly sensitive luminescence properties. A series of five ODN's that contain (or not) an 8-oxoG moiety were chosen and tested. This includes three G4 folding oligomers based on the human telomeric sequence, namely hTel, hTel-oxoG21 and hTel-oxoG10 and two duplex hairpin folding oligomers, namely HP and HP-oxoG. Steady state luminescence titrations revealed the ability of the three compounds to discriminate the hTel-oxoG10 sequence that imbedded an 8-oxoG lesion into the G4's central quartet over the sequences hTel and hTel-oxoG21 with respective luminescence increase of +72% vs +224-247% for the most discriminating complex. A similar trend was observed for the duplex hairpin HP that gave higher luminescence increases than for HP-oxoG (+494% vs HP to a weaker +177% vs HP-oxoG). The ability of the compounds 1-3 to photoprobe 8-oxo-G containing ODN's in hTel-oxoG10 and HP-oxoG over their parent ODN's presumably arises from their lower ability to protect the compounds against non-radiative deexcitation sources (e.g. solvent collisions and oxygen sensitization). This probably results from the lower stability and consequent looser structure of hTel-oxoG10 and HP-oxoG. Fitting of the luminescence titrations curves using a modified McGhee-von Hippel equation associated with results obtained from bio-layer interferometry studies allowed to make an in-depth study of the interaction between the complexes and the five ODN's. The results obtained showed that the compounds interact with the different ODN's within the tenth micromolar range for K D values. They also highlighted that the presence of 8-oxoG had non-significant impact on the parameters of the interaction (k on , k off or K D ) of the complexes with both duplex hairpin and G4 folding sequences. Altogether, the results from luminescence titrations presented herein show promises for the development of sensitive photoprobes for the rapid detection and quantification of 8-oxoG in human telomeric DNA.

Experimental section

Materials and instrumentation

[Ru(bpy) 2 Cl 2 ] and [Ru(phen) 2 Cl 2 ] precursors, napp, cpip and salphen ligands were prepared based on methods reported in the literature. [START_REF] Weynand | Towards the Development of Photo-Reactive Ruthenium(II) Complexes Targeting Telomeric G-Quadruplex DNA[END_REF][START_REF] Gillard | Flexible RuII Schiff Base Complexes: G-Quadruplex DNA Binding and Photo-Induced Cancer Cell Death[END_REF][START_REF] Gillard | Photodetection of DNA mismatches by dissymmetric Ru(ii) acridine based complexes[END_REF] Reagents and solvents used for synthesis were of reagent grade and were used without any further purification. The solvents for electrochemical and spectroscopic studies were of spectroscopic grade. Water was purified with a Millipore Milli-Q system. Hairpin ODNs 29-mer HP ( 3' CCGT(C) 3 TACCG(T) 5 CGGTA(G) 3 ACGG 5' ) and HP-oxoG ( 3' CCGT(C) 3 TACCG(T) 5 CGGTA/ oxo G/GGACGG 5' ) were purchased from Eurogentec. G4 ODNs 24-mer hTel ( 3' A (GGGATT) 3 GGGTT 5' ), hTel-oxoG21 ( 3' A GG/ oxo G/ATT(GGGATT) 2 GGGTT 5' ), hTel-oxoG10 ( 3' A(GGGATT) 2 G/ oxo G/GATTGGGTT 5' ) were synthesied by standard automated solid phase ODN synthesis on a 3400 DNA synthesizer. After purification by RP-HPLC, they were thoroughly desalted by size-exclusion chromatography (SEC). DNA and ODN concentrations were determined spectroscopically. Biotinylated ODN's were prepared on a Controlled Pore Glass solid support by using the phosphoramidite approach with an Applied Biosystems 3400 DNA/RNA Synthesizer (1 µmol scale). Before each experiment the oligonucleotides were annealed by heating at 90 °C for 5 minutes prior to a slow cooling to room temperature over one hour to ensure their folding into G-quadruplex or hairpin duplex structure. NMR experiments were done in d 6 -DMSO, CD 3 OD or CD 3 CN on a Bruker AC-300 Avance II (300 MHz) or on a Bruker AM-500 (500 MHz) at 20 °C. The NMR shifts were measured vs the residual peak of the solvent as the internal standard. High-resolution mass spectrometry (HRMS) spectra were recorded on a Q-Exactive orbitrap from ThermoFisher using reserpine as the internal standard. Samples were ionized by electrospray ionization (ESI; capillary temperature = 320 °C, vaporizer temperature = 320 °C, sheath gas flow rate = 5 mL min -1 ).

Cyclic voltammetry was performed using a platinum wire counter electrode, a glassy carbon disk working electrode with an area of ca. 0.03 cm 2 and a reference electrode of Ag/AgCl. The working electrode potential was monitored thanks to an Autolab PGSTAT 100 potentiostat and via a computer interface. The cyclic voltammograms were measured in dried acetonitrile using a sweep rate of 300 mV s -1 . The complex's concentration was 0.8 mmol L -1 , and tetrabutylammonium hexafluorophosphate was used as the supporting electrolyte (C =0.1 mol L -1) . The samples were purged by nitrogen before each use.

Light absorption data were recorded on a Shimadzu UV-1700 spectrophotometer. Luminescence data were recorded on a Varian Cary Eclipse instrument. Quantum yields were obtained using [Ru(bpy) 3 ] 2+ as a reference. [START_REF] Brouwer | Standards for photoluminescence quantum yield measurements in solution (IUPAC Technical Report)[END_REF] The luminescence lifetime measurements were performed at the second harmonic of a titanium crystal at λ=450 nm with a sapphire laser (picosecond Tsunami laser Spectra at a repetition rate of 80 kHz). A Fluotime 200 instrument from AMS Technologies was used for the decay acquisition. It consists of a GaAs microchannel plate photomultiplier tube (Hamamatsu model R3809U-50) followed by a time-correlated single-photon counting system from Picoquant (PicoHarp300). Luminescence decays were analyzed with FLUOFIT software. The time resolution of the system is of ca.4 ps. Luminescence titrations were done in the following buffer: 10 mM HEPES pH 7.4, 50 mM NaCl and 100 mM KCl. The complex concentration was kept at 5 µM. Bio-layer interferometry experiments were performed using sensors coated with streptavidin (SA sensors) purchased from Forte Bio (SARTORIUS). Prior to use, they were immersed for 10 minutes in a buffer before functionalization to dissolve the sucrose layer. Then the sensors were dipped for 15 minutes in DNA containing solutions (biotinylated hairpin ODN's) at 100 nM and rinsed in the buffer solution (10 mM HEPES, 50 mM NaCl, 100 mM KCl (pH 7.4), 0.5% v/v surfactant P20) for 10 minutes. The functionalized sensors were next dipped in the ruthenium complex containing solution at different concentrations (see supporting information) for 4 minutes interspersed by a rinsing step in the buffer solution for 4 minutes. Reference sensors without DNA immobilization were used to subtract the non-specific adsorption on the SA layer. The sensorgrams were fit using a 1:1 interaction model. The reported values are the means of representative independent experiments, and the errors provided are standard deviations from the mean. Each experiment was repeated at least two times using a concentration range from 0.5 μM to 150 μM.

Synthetic procedures and characterization

[Ru(bpy) 2 napp] 2+ 1. [Ru(bpy) 2 Cl 2 ] (20 mg, 0.041 mmol, 1.0 eq.) and the napp ligand (20 mg, 0.061 mmol, 1.5 eq) were dissolved in an EtOH/H 2 O mixture (5/5 -v/v, 5 mL). The reaction medium was heated at 80 °C under stirring to the entire consumption of the ligand while monitored by TLC (3 h). EtOH was then removed under vacuum and NH 4 PF 6 was added yielding precipitation of the complex. The so-formed orange solid was washed with water, EtOH and dried in vacuo. The crude product was finally purified by chromatography on silica (10:1:0. [Ru(phen) 2 cpip] 2+ 2. The same method used for complex 1 with the precursor [Ru(phen) 2 Cl 2 ] (20 mg, 0.038 mmol) and the cpip ligand (15 mg, 0.045 mmol) gave the crude product that was purified by chromatography on silica (CH 3 CN/H 2 O/ KNO 3sat 10:1:0.25) to provide the title compound 2 as an orange powder (30 mg, 0.025 mmol, 68%). R f 0.30 (CH 3 CN/H 2 O/ KNO 3sat 10:1:0.25); [START_REF] Crosby | A roadmap for the early detection and diagnosis of cancer[END_REF] [Ru(phen) 2 salphen] 2+ 3. The same method used for complex 1 with the precursor [Ru(phen) 2 Cl 2 ] (20 mg, 0.038 mmol) and the salphen ligand (16 mg, 0.038 mmol) and gave the crude product that was purified by chromatography on silica (CH 3 CN/H 2 O/ KNO 3sat 10:1:0.25) to provide the title compound 3 as an orange powder (33 mg, 0.028 mmol, 75%). R f 0.

Figure 1 .

 1 Figure 1. Structures of [Ru(bpy) 2 napp] 2+ 1, [Ru(phen) 2 cpip] 2+ 2 and [Ru(phen) 2 salphen] 2+ 3.

Figure 2 .

 2 Figure 2. Schematic representation of the hTel, hTel-oxoG21 and hTel-oxoG10 oligonucleotides used in this study. The sequences are: hTel ( 3' A (GGGATT) 3 GGGTT 5' ), hTel-oxoG21 ( 3' A GG/ oxo G/ATT(GGGATT) 2 GGGTT 5' ), hTel-oxoG10 ( 3' A(GGGATT) 2 G/ oxo G/GATTGGGTT 5' ).

Figure 3 .

 3 Figure 3. Absorption spectra of complexes 1-3 and [Ru(bpy) 3 ] 2+ in acetonitrile under air.

  Figure 4A and 4B (see the supporting information for compounds 2 and 3). The great increases of luminescence intensity in the presence of hTel, hTel-oxoG21 and hTel-oxoG10 are reported in Figure 5A and allow to assess the photoprobing ability of compounds 1-3 for the different G4 folding sequences. As displayed in Figure 5B, complexes 1-3 show very close increases of luminescence intensity in the presence of hTel and hTel-oxoG21, with complex 3

Figure 4 .

 4 Figure 4. Relative luminescence intensity (I/I 0 ) of complex 1 as a function of the relative (A) G-quartet or (B) base pair concentration. I 0 is the luminescence of the complex without DNA. Measurements were carried out in solutions of 2.5 μM of complex in 10 mM HEPES buffer, with 100 mM NaCl and 50 mM KCl at pH 7.4. Excitation at 450 nm. The McGhee-von Hippel equation was implemented to fit the curves. equation.

A

  McGhee-von Hippel type equation was used to assess the K D values; the binding site was fixed to three base pairs or to one G4 per complex (best fit). Errors are estimated to 5%.

Figure 5 .

 5 Figure 5. Increase of luminescence intensities (in %) for complexes 1-3 in the presence of (A) hTel, hTel-oxoG21 and hTel-oxoG10 and (B) HP and HP-oxoG.The results are reported as increase percentage compared to the luminescence of the compound without DNA. Measurements were done on 2.5 μM solutions of complex in 10 mM HEPES buffer, with 100 mM NaCl and 50 mM KCl at pH 7.4.

Figure 6 .

 6 Figure 6. BLI response curves for the interaction of complex 1 with hTel (A) and hTel-oxoG10 (B). Measurements are performed using 0.15 to 150 µM of complex in 10 mM HEPES pH 7.4, 100 mM NaCl, 50 mM KCl.

Table 2 .

 2 Light

absorption bands and absorption coefficients in acetonitrile for complexes 1-3 and references.

Complex Absorption λ max [nm] (ε [10 4 L.mol -1 .cm -1 ]) [a]

[Ru(bpy) 3 ] 2+ 250, 285 (8.71), 345 (sh), 452 (1.45) [Ru(phen) 3 ] 2+ 265, 287 (sh), 418 (sh), 447 (1.84) [Ru(bpy) 2 napp] 2+ 1 246, 289 (13.6), 343 (sh), 456 (2.48) [Ru(phen) 2 cpip] 2+ 2 265, 287 (sh), 335 (sh), 460 (1.61) [Ru(phen) 2 salphen] 2+ 3 265, 288 (sh), 338 (sh), 458 (1.04)

Table 3 .

 3 Emission data in CH 3 CN and H 2 O at 298 K for complexes 1-3 and reference complexes.

		Emission λ max a,b [nm]	Φ em c,d		τ em [ns] c		k r [10 3 s -1 ]
	Complex
		CH 3 CN	H 2 O	CH 3 CN	H 2 O	CH 3 CN	H 2 O	CH 3 CN	H 2 O
	[Ru(bpy) 3 ] 2+	604	604	0.094	0.063	855	630	77	69
	[Ru(phen) 3 ] 2+	604	606	0.028	0.072	460	920	61	75
	[Ru(bpy) 2 napp] 2+ 1	601	606	0.072	0.12	812	841	89	143
	[Ru(phen) 2 cpip] 2+ 2	597	603	0.058	0.14	380	1315	152	106
	[Ru(phen) 2 salphen] 2+ 3	599	600	0.018	0.075	688	1222	26	61

Table 4 .

 4 Dissociation equilibrium constants (K

	ODN	Constan	1	2	3
		ts
	hTel	I/I 0 max	3.0	2.5	3.2
		K D (µM)	11	14	23
	hTel-oxoG21	I/I 0 max	2.9	2.6	3.4
		K D (µM)	19	17	28
	hTel-oxoG10	I/I 0 max	1.95	1.6	1.7

D ) estimated for 1-3 with the five DNA models.

Table 5 .

 5 BLI analysis data for the interactions of complexes 1-3 with the studied ODN's.Equilibrium dissociation constants were deduced from the kinetic rate constants. a The weak binding of the complex the oligonucleotides did not allow to determine the kinetic parameters of the interactions (n.d.) in the studied concentration range. Running buffer: 10 mM HEPES pH 7.4, 100 mM NaCl, 50 mM KCl. Measurements are performed using 0.15 to 150 µM of complex in 10 mM HEPES pH 7.4, 100 mM NaCl, 50 mM KCl.

	ODN	Constant	1	2	3
	hTel	k on (10 3 M -1 s -1 ) 2.4 ± 0.7 1.1 ± 0.3 3.6 ± 1.1
		k off (s -1 )	50 ± 10	40 ± 20	47 ± 15
		K D (µM)	19 ± 3	10 ± 5	15 ± 8
	hTel-	k on (10 3 M -1 s -1 ) 2.9 ± 1.2 1.1 ± 0.2 4.4 ± 1.3
	oxoG21	k off (s -1 )	70 ± 20	6 ± 3	64 ± 20
		K D (µM)	34 ± 8	7 ± 3	24 ± 11
	hTel-	k on (10 3 M -1 s -1 ) 2.5 ± 0.9 0.8 ± 0.2 2.8 ± 1.2
	oxoG10	k off (s -1 )	80 ± 20	6 ± 2	78 ± 12
		K D (µM)	33 ± 7	7 ± 2	35 ± 16
	HP	k on (10 3 M -1 s -1 ) 1.3 ± 0.3 n.d. a	2.0 ± 0.9
		k off (s -1 )	90 ± 10	n.d. a	100 ± 30
		K D (µM)	82 ± 28	n.d. a	76 ± 16
	HP-oxoG	k on (10 3 M -1 s -1 ) 2.7 ± 0.9 n.d. a	1.4 ± 0.4
		k off (s -1 )	110 ± 40 n.d. a	148 ± 4
		K D (µM)	73 ± 29	n.d. a	116 ± 34

  5 CH 3 CN/H 2 O/ KNO 3sat ) which gave [Ru(bpy) 2 napp] 2+ 1 (31 mg, 0.030 mmol, 73%). The exchange from of counter anion from PF 6 -to Cl -was done by the addition of NBu 4 Cl to an acetone solution of the complex. R f 0.35 (CH 3 CN/H 2 O/ KNO 3sat 10:1:0.5); 1 H NMR (CD 3 CN, 500 MHz) δ (ppm), 10.47 (1H, s, H d ), 9.81 (1H, d, J a-b = 8.2, J a-c = 1.2 Hz, Ha), 9.59 (1H, dd, J c-b = 8.2, J c-a = 1.2 Hz, H c ), 9.24 (1H, d, J m-l = 8.2 Hz, H m ), 8.53 (4H, m, H 5 , H 5' , H 6 , H 6' ), 8.34 (1H, d, J i-j = 9.1 Hz, H i ), 8.25 (1H, d, J j-i = 9.1 Hz, H j ), 8.16 (2H, m, H 4 , H 4' ), 8.12 (3H, m, H e , H f , H g ), 8.01 (2H, m, H 7 , H 7' ), 7.95 (1H, m, H l ), 7.91-7.84 (5H, m, H b , H 2, H 2' , H m, H h ), 7.73 (1H, d, J 9-8 = 5.4 Hz, H 9 ), 7.69 (1H, d, J 9'- 8' = 5.6 Hz, H 9' ), 7.49-7.44 (2H, m, H 3 , H 3' ), 7.28-7.21 (m, 2H, H 8 , H 8' ); HRMS-ESI calculated for [C 43 H 29 N 7 F 6 PRu] + : m/z 890.11728, found: m/z 890.11700 and for [C 43 H 29 N 7 Ru] 2+ : m/z 372.57617, found: m/z 372.57640.

  H NMR (CD 3 CN, 500 MHz) δ 9.05 (d, 1H, J = 8.2 Hz, H a /H a' ), 8.89 (d, 1H, J = 8.4 Hz, H a /H a' ), 8.59 (dd, 4H, J = 8.3 Hz, J = 1.2 Hz, H 2 /H 2' and H 9 /H 9' ), 8.28 (d, 2H, J = 8.7 Hz, H e and H e' ), 8.25 (s, 4H, H 5 and H 5 , ; H 6 and H 6' ), 8.07 (dd, 2H, J = 10.5 Hz, J = 4.8 Hz, H 8 and H 8' ), 8.01 (d, 2H, J = 5.4 Hz, H 7 and H 7' ), 7.96 (d, 2H, J = 5.2 Hz, H 4 and H 4' ,), 7.67-7.59 (m, 8H, H 3 H 3',H b , H b' , H c , H c' , H d and H d' ). HRMS Calcd for [C 43 H 27 N 8 ClF 6 PRu] + : m/z 931.07595, found: m/z 931.07688 Da.

  31 (CH 3 CN/H 2 O/ KNO 3sat 10:1:0.25); 1 H NMR (CD 3 CN, 500 MHz) δ 9.03 (broad m, 4H, H d and H c ), 8.62 -8.60 (m, 2H, H 2 or H 9 ), 8.60 -8.58 (m, 2H, H 2 or H 9 ), 8.25 (s, 4H, H 5 and H 6 ), 8.12 -8.06 (m, 4H, H 7 and H e ), 8.02 (dd, J = 5.2, 1.2 Hz, 2H, H 4 ), 7.99 (dd, J = 5.3, 1.2 Hz, 2H, H a ), 7.68 (dd, J = 8.3, 5.3 Hz, 2H, H b ), 7.63 (d, J = 5.2 Hz, 2H, H 3 or H 8 ), 7.61 (d, J = 5.2 Hz, 2H, H 3 or H 8 ), 7.51 -7.44 (m, 2H, H g ), 7.17 -7.06 (m, 4H, H f and H h ). HRMS-ESI calculated for [C 43 H 28 N 8 O 96 Ru] 2+ : m/z 384.07255, found: m/z 384.07322.

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Acknowledgements

M. G. and B. E. gratefully acknowledge the Université catholique de Louvain (UCLouvain) and the F.R.S-FNRS (Grant Number U.N021.21) for financial support. This work was partially supported by the "Agence National de la Recherche", Labex ARCANE (ANR-16-CE11-0006-01), CBH-EUR-GS (ANR-17-EURE-0003), and the région Auvergne-Rhône-Alpes. The NanoBio-ICMG platforms (UAR 2607) are acknowledged for their support.

Conflicts of interest

The authors declare no competing interest.

Supporting Information Available

The supporting information describes the synthesis of the complexes and shows their NMR spectra. Analytical data such as cyclic voltamograms, luminescence titration curves and BLI sensograms are also provided along with a decscription of the models used for the fitting of BLI and luminescence titrations curves.


Guilhem Poy
email: guilhem.poy@umontpellier.fr

Slobodan

Slobodan Žumer
email: slobodan.zumer@fmf.uni-lj.si

Scattering-based microscope imaging of light beams in soft birefringent media with orientational fluctuations

Keywords: Liquid crystals, spatial optical solitons, light scattering, microscopy

est destinée au dépôt

INTRODUCTION

Liquid crystals (LCs) have led to countless technological applications in the domain of optics (displays, flat lenses, modulators...) thanks to their optical anisotropy and facile response under external fields. One fascinating development, initiated 2 decades ago in the groups of G. Assanto 1 and M. Karpierz, 2 is the generation and manipulation of spatial optical solitons called nematicons. These self-focused beams can be generated at powers as low as ∼ 1 mW, thanks to the giant Kerr-like non-linear optical response of LCs associated with the opticalfield-induced nonlocal reorientation of the director field. Since nematicons are effectively associated with a localized waveguide along their axis and can be efficiently steered with bias voltage control of the beam walkoff, [START_REF] Peccianti | Routing of anisotropic spatial solitons and modulational instability in liquid crystals[END_REF] they are good candidates to carry optical informations in photonics applications. Nematicons can adopt complex 3D trajectories due to modulated interaction potentials, [START_REF] Alberucci | Nonlinear bouncing of nonlocal spatial solitons at the boundaries[END_REF][START_REF] Poy | Chirality-Enhanced Periodic Self-Focusing of Light in Soft Birefringent Media[END_REF] and have nontrivial interactions between themselves with fascinating orbiting kinematics. [START_REF] Kwasny | Nematicon-nematicon interactions in a medium with tunable nonlinearity and fixed nonlocality[END_REF] Nematicons can also be used to assist random lasing in dye-doped LCs with high efficiency and directional control. [START_REF] Perumbilavil | Beaming random lasers with soliton control[END_REF] More generally, it was recently realized that the self-focusing properties of optical solitons in LCs can not only come from the beam-induced modulation of the effective refractive index but also from spin-orbit interactions without any effective index changes. [START_REF] Jisha | Self-Trapping of Light Using the Pancharatnam-Berry Phase[END_REF][START_REF] Assanto | Spin-optical solitons in liquid crystals[END_REF] Finally, the nonlinear optical interactions at the core of optical solitons in LCs can be exploited to efficiently transform optical modes 10 and even lead to complex optomechanical interactions with topological solitons [START_REF] Poy | Interaction and co-assembly of optical and topological solitons[END_REF] -robust and localized birefringent structures embedded in the continuous orientational field of the LC.

In experiments, nematicons are usually directly imaged in a microscope without a conventional transmission or reflection light source. Indeed, the orientational fluctuations of the LC scatter light from the soliton's laser beam, which can therefore form an image through the microscope's objective. Such images are quite hard to interpret quantitatively: for example, the apparent beam waist on the microscope image can be quite different from the physical beam waist. [START_REF] Conti | Observation of Optical Spatial Solitons in a Highly Nonlocal Medium[END_REF] We note that experimental optical setups allowing to image scattered fields are commonly used in the context of dynamic light scattering of soft matterials (e.g. with differential dynamic microscopy [START_REF] Giavazzi | Scattering information obtained by optical microscopy: Differential dynamic microscopy and beyond[END_REF] or photon correlation imaging [START_REF] Duri | Resolving Long-Range Spatial Correlations in Jammed Colloidal Systems Using Photon Correlation Imaging[END_REF] ) or visualization of nanoscale objects (e.g. with interferometric scattering microscopy [START_REF] Young | Interferometric Scattering Microscopy[END_REF] ). However, the theory behind these setups is not directly applicable to the transverse visualization of optical solitons or laser beams in scattering media since in this context there are a priori no interferences with a fundamental beam (only scattered fields are captured by the microscopic objective) and since a large number of fluctuation modes contributes to the final image.

To better understand how the intensity profile of laser beams is related to the imaged intensity of scattered fields, we introduce a theoretical framework allowing to simulate microscope images originating from the scattered fields of light propagating in a bulk scattering medium. We apply this framework to analyse microscope images of simple Gaussian beams and more complex bouncing solitons in LCs 5 -self-focused beams bouncing between the confining plates of an LC sample-and discuss specific features of scattered field imaging. Finally, we suggest how our framework could be at the basis of a novel tomography technique of optical fields based on fluctuating environments.

THEORETICAL MODEL

In this section, we derive the equations needed to simulate microscope images formed by the scattered optical fields of a laser beam propagating in a scattering medium. We first derive a general formula for any media with permittivity fluctuations, and then specialize our result to the case of oriented unixial liquid crystal layers with orientational fluctuation.

Imaging of bulk scattering sources

We start from the wave equation for a monochromatic optical field E(r) exp (ik 0 ct):

∇ × ∇ × E = k 2 0 ϵ • E (1)
with k 0 the wavevector in empty space and ϵ the relative permittivity tensor. We assume that ϵ = n 2 I + δϵ, with δϵ a fluctuation field which is small with respect to the base isotropic permittivity n 2 . Let us define E 0 as any solution of Eq. 1 without fluctuations (δϵ = 0), for a given light source. Using the Maxwell divergence equation and Eq. 1, we then find that the scattered optical fields E ′ are given at first order by:

∆ + k 2 0 n 2 E ′ = -k 2 0 δϵ • E 0 ≡ -k 2 0 Σ, (2)
with Σ ≡ δϵ • E 0 representing the bulk scattering source. Note that this equation is valid only in the quasi-static approximation, assuming that the typical time scales for the fluctuation field are much larger than 1/(k 0 c)-which is the case of the orientational fluctuations of the next subsection.

By solving Eq. 2, one can, in principle, find the total scattered field anywhere in space. Here, we want to restrict the calculation of the scattered field on the image plane of a 2f lens projection setup (with f the focal length of the lens), which will serve as a minimalistic model for the objective of a microscope (neglecting magnification since it can always be absorbed in a scaling operation [START_REF] Goodman | Introduction to Fourier optics[END_REF] ). Such a system is shown schematically in Fig. 1. We assume that the scattering volume (layer z ∈ [-h/2, h/2]) is confined between a lower and upper glass plate (layer z ∈ [h/2, h/2 + e]) of refractive index n g . The distance between the lens and upper glass plate is ∆, and we consider the general case of an immersion medium with refractive index n i . For simplicity, we assume that the mean isotropic refractive index of the scattering volume matches the one of glass.

Having defined the geometry of our system, we can now explicitly solve Eq. 2 by switching to the transverse Fourier plane: where the subscript ⊥ indicates a projection onto the xy plane and where we defined the transverse Fourier transform of any function f as:

∂ 2 z + k 2 0 n(z) 2 -|p ⊥ | 2 Ẽ⊥ (p ⊥ , z) = -k 2 0 Σ⊥ (p ⊥ , z), (3)
f (p ⊥ , z) ≡ d 2 r ⊥ f (r ⊥ , z) exp [ik 0 p ⊥ • r ⊥ ] (4)
Following the conventional approach of Fourier optics [START_REF] Goodman | Introduction to Fourier optics[END_REF] we model the lens as an ideal diffraction-limited Fourier filter t with object-side immersion medium of index n i :

t(p ⊥ ) = P * (n i , 2f, p ⊥ )P * (1, 2f, p ⊥ )Θ (NA -|p ⊥ |) , (5)
with NA the numerical aperture of the lens, Θ the Heaviside function, and P(n, l, p ⊥ ) ≡ exp ik 0 l n 2 -|p ⊥ | 2 the Helmhotz phase propagator. We solve Eq. 3 assuming only forward propagating modes and neglecting Fresnel reflections at the index discontinuity surfaces. By using the Green function of the Helmholtz equation inside the scattering layer [START_REF] Goodman | Introduction to Fourier optics[END_REF] and then composing with the appropriate Helmhotz propagators and lens filter in each subsequent layer of Fig. 1, we find the Fourier-transformed optical fields on the image plane:

Ẽ(im) ⊥ = ik 0 2n g h/2 -h/2 dz P(1, 2f, p ⊥ ) t(p ⊥ ) P(n i , ∆, p ⊥ ) P(n g , l s -z, p ⊥ ) Σ⊥ (p ⊥ , z) (6)
with l s ≡ e + h/2. Coming back to real-space and applying the Fourier convolution theorem, we finally find the optical fields on the image plane:

E (im) ⊥ (r ⊥ ) = h/2 -h/2 dz d 2 u ⊥ T (u ⊥ , z)Σ(r ⊥ -u ⊥ , z) (7)
where we introduced the amplitude point-spread-function (PSF) of the lens (up to a constant phase factor which was removed):

T (u ⊥ , z) ≡ iπe -ik0ngz λ 3 0 n g d 2 p ⊥ e ik0p ⊥ •u ⊥ +iϕ(|p ⊥ |,z) Θ(NA -|p ⊥ |) = 2iπ 2 e -ik0ngz λ 3 0 n g NA 0 dp e iϕ(p,z) J 0 (k 0 p|u ⊥ |) p, (8)
ϕ(p, z) = k 0 n 2 g -p 2 -n g (l s -z) + n 2 i -p 2 -n i (∆ -2f ) , (9)
with J 0 the zeroth-order Bessel function of the first kind. In the previous equation, ∆ can be seen as an adjustable parameter allowing us to vary the focusing of the lens in our optical setup. A perfect focus can be attained when the phase in Eq. 9 is zero for all radial frequencies p, since it implies a maximal center amplitude for the PSF in Eq. 8 (no fast phase variation which averages out the PSF). This is only possible when the glass and immersion media are index-matched (n i = n g ), in which case we find that the optical fields are perfectly focused in the plane z = z f ≡ l s + ∆ -2f . When n i ̸ = n g , the focus is always non-ideal and one must rely on an approximate criterion for finding the focusing plane. In practive, we found that solving ϕ(NA, z) = 0 works well for numerical aperture lower than 0.4, which gives a more general definition than the previous one for the z-coordinate of the focusing plane:

z f ≡ l s + n 2 i -NA 2 -n i n 2 g -NA 2 -n g (∆ -2f ) (10)
Note that Eq. 7 is very similar to the usual formula relating the optical fields at the object and image planes of a lens through a convolution with the amplitude PSF, [START_REF] Goodman | Introduction to Fourier optics[END_REF] except that here our source object is not an plane but a bulk scattering source represented by the quantity Σ. Since the latter is by nature a stochastistic quantity related to the permittivity fluctuations, the image fields are also random quantities; we focus here on the (ergodic) averaged intensity fields measured by a camera captor placed on the image plane:

I (α) (r ⊥ ) ≡ ⟨ E (im) α (r ⊥ ) 2 ⟩ = V d 3 u V d 3 u ′ [T (r ⊥ -u ⊥ , u z )E 0 (u)] ⊺ G (α) (u, u ′ ) [T (r ⊥ -u ′ ⊥ , u z )E 0 (u ′ )] * (11)
In this formula, α = x, y is the polarisation of the measured optical fields (which can be selected with a polarizer), V is the integration volume corresponding to the scattering layer z ∈ [-h/2, h/2], and G (α) is the correlation matrix of permittivity fluctuations associated with α-polarized scattered fields:

G (α) ij (u, u ′ ) ≡ ⟨δϵ αi (u) δϵ αj (u ′ )⟩ (12)
Eq. 11 will be our master equation for simulating microscope images of bulk scattering source, and fully takes into account the partially incoherent nature of scattering through the correlation function G (α) . This formula is very similar to one discussed at the beginning of Ref. [START_REF] Val'kov | Fluctuations and light scattering in liquid crystals[END_REF] in a context different than the microscopy setup discussed here. We emphasize that our derivation of the PSF T in this formula is only approximate and quite similar to the simple scalar approach described by Gibson and Lanni. [START_REF] Gibson | Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy[END_REF] However, it can be generalized without any technical difficulties to include more complex phenomena such as vectorial abberations and Fresnel reflections. [START_REF] Haeberlé | Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy[END_REF] We focus here on a simpler model of PSF in order to gain a general understanding of the image formation process of bulk scattering sources, specialized in the next section to the case of orientational fluctuations in a nematic LC layer.

Scattering induced by orientational fluctuations

We now assume that the scattering layer is formed of a nematic LC layer oriented along n 0 ≡ e z (i.e. with homeotropic boundary conditions). The director field can be written as n = n 0 + δn with δn the orientational fluctuations induced by the thermal noise. Note that since the director field is unit normed, δn is necessarily orthogonal to n 0 . At first order in the fluctuation, the relative permittivity tensor is given by ϵ = ϵ m + δϵ, with the fundamental and fluctuating contributions given by:

ϵ m = ϵ o I + ϵ a n 0 ⊗ n 0 (13) δϵ = ϵ a (n 0 ⊗ δn + δn ⊗ n 0 ) (14)
with

ϵ o ≡ n 2 o , ϵ a ≡ n 2 e -n 2 o ,
and n e (n o ) the extraordinary (ordinary) index of the LC. Note that the unperturbed permittivity tensor ϵ m is not isotropic, contrary to what was assumed in the previous section. In practice, this anisotropy can introduce vectorial aberations not taken into account by our simple PSF model. Here, we will choose to ignore these abberations by assuming that the anisotropy of relative permittivity ϵ a is sufficiently small, in which case ϵ m ≈ n 2 o I.

Assuming that the director field is negligibly perturbed by the optical fields, a straightforward calculation gives us the permittivity correlation matrix as a function of the director correlation matrix g ij (u, u ′ ) = ⟨δn i (u)δn j (u ′ )⟩:

G (α) (u, u ′ ) = ϵ 2 a g αα (u, u ′ ) e z ⊗ e z (15)
Assuming that the Frank elastic constants are all equal to K, we find that the director correlation function of a homeotropic layer of thickness h calculated by Zel'dovich and Tabiryan 20 simplifies to:

g(u, u ′ ) = k b T πKh ∞ n=1 K 0 (q n |u ⊥ -u ′ ⊥ |) sin q n u z + h 2 sin q n u ′ z + h 2 [I -n 0 ⊗ n 0 ] , (16)
with q n ≡ nπ/h, k b the Boltzmann constant, and K 0 the zeroth-order modified Bessel function of the second kind. Since fluctuation modes are not confined in the xy plane, the correlation function is translationally invariant in this plane and only depends on the difference u ⊥ -u ′ ⊥ . For the z-direction, however, this translational invariance is broken because of the confining glass plates, and the correlation function must fulfill the boundary conditions g = 0 when either u z or u ′ z is equal to ±h/2, which can be readily verified on the last equation. Using Eqs. 15 and 16, we find that our master equation 11 simplifies to:

I (α) (r ⊥ ) = d 2 u ⊥ d 2 u ′ ⊥ ∞ n=1 F n (r ⊥ , u ⊥ ) g n (|u ⊥ -u ′ ⊥ |) F n (r ⊥ , u ′ ⊥ ) * , (17)
with g n (u) ≡ k b T ϵ 2 a /(πKh) K 0 (q n u) an effective in-sample-plane correlation function and F n the input optical field E 0z averaged by the PSF and the n-th mode of fluctuation along z:

F n (r ⊥ , u ⊥ ) ≡ h/2 -h/2 du z sin q n u z + h 2 T (r ⊥ -u ⊥ , u z ) E 0z (u). (18)
In the geometry considered here, the scattered intensity is fully depolarised which is why the right-hand-side of Eq. 17 does not depend on the polarization index α. This result is fully expected since our system is invariant by rotation around the axis z. In the following, we therefore omit the (α) index and denote by I(x, y, z f ) the intensity measured at point {x, y} by the camera when the focusing plane of the lens is at z = z f . In experiments, the 3D signal I can be measured using optical sectioning, i.e. by measuring images of the sample z-layer by z-layer while changing the distance between sample and microscope objective.

In the next section, we numerically integrate Eq. 17 to calculate scattered images of Gaussian beams and bouncing solitons propagating inside the LC layer, and discuss the influence of the focusing optics and possible applications.

NUMERICAL RESULTS

Numerical methods

To calculate a scattered image for a given focus z f and input field E 0 , we repeatedly apply Eq. 17 with different r ⊥ taken from a Cartesian mesh defining our image. The 4D integral in Eq. 17 cannot be easily integrated using traditional quadrature methods because of the high dimensionality. Instead, we used the stochastic Monte-Carlo integration algorithm VEGAS, which uses adaptive importance sampling to accelerate the convergence. To efficiently calculate the fields F n , we used the discrete sine transform algorithm of type I (DST-I), with typically 256 z-discretization steps (which also corresponds to the cut-off limit for the discrete sum in Eq. 17). Our numerical code implementing this calculation was written in C++, using the Cuba library [START_REF] Hahn | Concurrent cuba[END_REF] for Monte-Carlo integration and the FFTW library [START_REF] Frigo | FFTW: An adaptive software architecture for the FFT[END_REF] for the DST-I transform.

All our calculations were done with a wavelength λ = 0.5 µm for the optical fields, a thickness e = 1 mm for the upper glass plate, and n g = 1.5 for the refractive index of glass. For the extraordinary and ordinary indices, we took typical room-temperature values associated with thermotropic LCs: [START_REF] Li | Refractive Indices of Liquid Crystals for Display Applications[END_REF] n o ≈ n g and n e ≈ 1.75. For the refractive index of the lens' immersion medium, we chose either n i = 1 (dry objective) or n i = n g (oil immersion objective). In the VEGAS Monte-Carlo method, we chose to use the same Sobol quasi-random samples for all the image points, which means that the calculated intensities at different point of the image are strongly correlated. In short, the calculated images do not include very visible spatial noise, and typically corresponds to the exact image up to a random multiplicative factor which approaches 1 when a sufficiently high number of integrand evaluations was reached. We repeatedly evaluate the integrand until the L 2 -error (evaluated from the variance of intermediate integral calculations in the VEGAS algorithm) for the calculated image is smaller than 0.1. On a desktop computer with 12 threads, our code can evaluate 10 4 integrand values in typically 0.25 s. Depending on the numerical aperture of the lens, between 4 10 3 and 10 5 integrand evaluations are needed to obtain sufficient convergence, with lower numerical aperture being more difficult to integrate * . All calculated images are rescaled such that the maximal intensity is 1, which means that one can ignore the factor k b T ϵ 2 a /(πKh) in the definition of the correlation function g n and use arbitrary units for the input field E 0 .

We focus our numerical study on two examples for the optical field E 0 . The first one is a simple Gaussian beam with a z-polarization and propagating along the y axis in the mid-sample-plane of a h = 40 µm-thick homeotropic LC slab:

E (gaussian) 0 (x, y, z) = y r y -iy r exp ik y + x 2 + z 2 2(y -iy r ) -i arctan y y r e z (19)
with k = k 0 n e the wavevector in the LC slab, y r = kw 2 0 the Rayleigh length and w 0 = 2 µm the beam waist. The second one is more complex and corresponds to the bouncing optical soliton experimentally observed in a previous paper. [START_REF] Poy | Chirality-Enhanced Periodic Self-Focusing of Light in Soft Birefringent Media[END_REF] We recall that a spatial optical soliton is a beam that can preserve its transverse profile as it propagates in a medium with an optical nonlinearity. For the particular case of bouncing optical soliton in homeotropic LC layers, the beam has an extroardinary polarization, propagates in the yz plane of the sample and is bouncing between the confining glass plates of the sample thanks to total internal reflection. With a sufficiently high beam power, the nonlocal molecular reorientation of LC molecules induced by the beam polarization acts as a waveguide for the beam, which can therefore propagate without diffracting and can be addressed as an optical soliton. We refer the reader to our previous paper [START_REF] Poy | Chirality-Enhanced Periodic Self-Focusing of Light in Soft Birefringent Media[END_REF] for all details concerning the physics of these peculiar solitons, and focus here on a simulated bouncing optical soliton with an input waist w 0 = 5 µm and insertion angle of 15 • in a homeotropic LC slab of thickness h = 15 µm. The simulation was done thanks to a generalized beam propagation method and a simple director relaxation algorithm that are described in the supplementary information of Ref., [START_REF] Poy | Chirality-Enhanced Periodic Self-Focusing of Light in Soft Birefringent Media[END_REF] and gave us the optical field E (bouncing) 0

. In Fig. 2, we show 3D volumetric rendering of the optical fields intensity for the two example of beams that we study here.

Scattered images of a gaussian beam

Let us start by discussing our results with the Gaussian beam of Fig. 2a. All simulations presented in this subsection assumes an oil-immersion microscope objective with n i = n g . On Fig. 3a andb, we show xy and xz slices of the calculated microscope intensity field I(x, y, z f ), assuming a numerical aperture of 0.4. These images suggest that the transverse profile of the Gaussian beam is well-captured by the microscope objective, but that the axial resolution along z is limited. This is expected, since for NA=0.4 the typical axial resolution ∆z ≡ 2n g λ/NA 2 ≈ 9.4 µm for the associated PSF is bigger than the beam waist, whereas the transverse resolution ∆r ≡ λ 0 /(2NA) ≈ 0.63 µm is smaller than the beam waist † . To better visualize the axial and transverse resolution of scattered fields, we analyzed the x and z profiles of the calculated optical stack I(x, y, z f ). In the following, we indicate by Ĩ any profile of I rescaled by its maximal value. On Fig. 3c, we show x-profiles of Ĩ calculated for NA = 0.4 at different foci z f . These profiles shows that the transverse resolution is optimal when the beam is focused by the lens (z f = 0), but worsen when the beam is out-of-focus, similarly to conventional visualization of micron-sized objects in a transmission microscope. Finally, Fig. 3d shows x and z-profiles calculated for different NA. From these plots, we conclude that both transverse and axial resolution of the scattered fields improves with higher numerical aperture. We note that this would not be the case if the index of the immersion medium would not be matched to the one of glass, which we verified by direct simulations and can also be visible on Fig. 3 of Ref., [START_REF] Haeberlé | Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy[END_REF] where a PSF model similar to ours was studied. More specifically, we noticed that when n i = 1, the axial resolution of scattered fields in our simple PSF model is roughly the same as in index-matched objectives for numerical aperture below 0.3, but is seriously degraded above this-an observation that can be intuitively understood as a geometrical abberation induced by refraction at the glass-air interface.

To summarize, these preliminary numerical experiments shows that, in terms of transverse and axial resolution, the visualization of bulk scattering fields obeys very similar laws than classical microscope visualization of objects under Köhler illumination, despite the partially incoherent nature of light scattering and nonlocalized spatial extent of fluctuations. One could therefore hope, given an index-matched objective with a sufficiently high NA, to be able to reconstruct in 3D the shape of a laser beam propagating in a scattering volume, similar to the 3D visualization of microscopic objects using confocal microscopy or optical sectioning techniques. We numerically examine the feasability of this new tomography technique in the next section by looking at bouncing optical solitons, and also highlight other important features of scattered field visualization.

Scattered images of a bouncing soliton

As explained in Sec. 3.1, optical solitons in LCs are associated with a beam-induced reorientation of the director field, which should in principle be taken into account in the calculation of the correlation function and scattered fields. Here, we simplify the discussion by assuming that the reorientation angle needed to stabilize the optical soliton is very small, which is the case for the large waist w 0 = 5 µm of the simulated bouncing optical soliton of Fig. 2b (|δn| ∼ 9 10 -3 ). On Fig. 4a andb, we show xy-slices of the scattered intensity signal I at z f = 0, calculated for different NAs with either a dry objective (a) or oil-immersion objective (b). Similar to what was described in the previous subsection, the resolution increases with the numerical aperture when the immersion index is matched to the one of glass, with well-separated maxima associated with the points where the beam crosses the mid-sample-plane for the highest numerical aperture.

Interestingly, one also observe that the intensity along the soliton axis y is modulated with an alternance of bright and dark areas. To understand the physical origin of this modulation, we plot on Fig. 4c yz-slices of the signal I calculated for different NAs with n i = n g and x = 0. Therein, one observe that the dark (bright) areas correspond to parts of the beam moving away from (towards) the objective lens, and conclude that the modulation is due to the different cross-scattering sections of forward and backward scattering processes. More precisely, let k i be the (local) wavevector of the bouncing soliton, k f ≡ k 0 n g e z the mean observation wavevector associated with the lens projection setup of Fig. 1, and q ≡ k f -k i the scattering wavevector. Since the scattering cross section in nematic LCs scales like [START_REF] Oswald | Nematic and cholesteric liquid crystals: concepts and physical properties illustrated by experiments[END_REF] 1/|q| 2 , we find that the intensity ratio between dark and bright areas in Fig. 4 should typically be r ≡ q (+) 2 / q (-) 2 , with q (+) (q (-) ) the scattering wavevector associated with part of the soliton with k iz > 0 (k iz < 0). Using the theoretical expression of the extraordinary wavevector [START_REF] Poy | Ray-based optical visualisation of complex birefringent structures including energy transport[END_REF] in terms of the Poynting vector direction u s = cos θ e y + ± sin θ e z (with θ = 15 • the angle between the beam and the sample plane), we find:

r = n g n eff -n 2 o sin θ 2 + n 4 e cos 2 θ (n g n eff + n 2 o sin θ) 2 + n 4 e cos 2 θ
(20) † These formula for the axial and transverse resolutions can be obtained from the theoretical profiles of the PSF along the transverse and longitudinal directions, and corresponds to the first zeros of the PSF in the paraxial limit. Using the values of the parameters used in our simulations, we estimate r ≈ 0.67. To compare this with our numerical simulations, we plotted on Fig. 4d the y-profiles calculated for a very low numerical aperture of 0.05, as well as the experimental profile obtained from the data of Ref. [START_REF] Poy | Chirality-Enhanced Periodic Self-Focusing of Light in Soft Birefringent Media[END_REF] which was measured with a low-NA microscope objective and a similar angle θ. To ease the comparison between experiments and simulations, these signals are plotted as a function of y/Λ with Λ the signal periodicity (286 µm in the experiments, 140 µm in the simulations). From this plot, we obtain r ≈ 0.6 for the numerical simulation and r ≈ 0.7 for the experiment, in reasonable agreement with our estimation above. We emphasize that an intensity modulation can also be experimentally observed on polarized optical micrographs of the solitonic sample due to the beam-induced director field perturbation, 5 with exactly the same periodicity Λ.

We note that in the bouncing soliton experiments previously mentioned, 5 the apparent full width at half maximum FWHM ≈ 30 µm of the soliton is bigger than the one estimated from the director field perturbation visualized in direct polarized optical microscopy (∼ 15 µm), which is really surprising since with a nonlocal optical response one expect exactly the opposite. This suggests that the experimentally visualized scattered fields are associated with a wider transverse profile than the true solitonic optical field inducing the director field reorientation. Several explanations are possible. First, multiple scattering processes across the thickness of the sample-which are neglected in this paper-could play a role in spreading the apparent profile, although we think this explanation is unlikely since the mean photon free path (which can be estimated from the scatteringinduced beam power decrease along the beam axis) is typically 1 mm and therefore much bigger than the sample thickness. Second, the low NA of the objective used in these experiments and possible out-of-focus effects could also contributes to spreading the profile, as discussed in our simulations; although this effect is rather small in our simple model of PSF even with low-NA objectives, additional vectorial geometrical abberations not taken into account here could amplify the loss of transverse resolution. Third, the perturbed director field at the core of the soliton could affect the structure of fluctuation modes and the propagation of the scattered fields-two effects which are neglected in our simulations since we assumed |δn| ≪ 1. Finally, it is possible that the bouncing soliton visualized in the experiment exists on top of another wider beam which does not contribute to the molecular reorientation but still contributes to the scattered fields; this may happen if the input beam is strongly scattered at the sample entrance by topological defects typically present near the boundary of thin homeotropic LC slabs.

We think the latter two possibilities are the most likely, since other solitonic experiments done in the group of Assanto [START_REF] Conti | Observation of Optical Spatial Solitons in a Highly Nonlocal Medium[END_REF] showed two relevant results in a system with much cleaner beam coupling at the sample entrance than in the bouncing soliton experiments: 5 small solitonic waist smaller than 5-10 µm (which are associated with a strong reorientation of the director) are systematically overestimated from the scattered light images, whereas bigger waists can be reliably measured. Nevertheless, more experiments and comparison with simulations are likely needed to find which effect contributes the most, especially in the strongly nonlinear optical regime.

CONCLUSION

To conclude, we introduced a theory for modeling microscope images of scattered optical fields originating from a laser beam propagating in a bulk scattering medium. We applied this theory to the case of fluctuating nematic LCs, and demonstrated with simulations of scattered images of a Gaussian beam and a bouncing optical soliton that the axial and transverse resolution of the scattered fields is comparable to the classical resolution of physical objects visualized in a transmission microscope. We also discussed the role of the scattering wavevector in the intensity modulation of the scattered images and additional mechanisms that can affect the resolution of scattered images of localized beams in LCs.

Assuming that these mechanisms can be kept in check (for example by restricting the experiments to the weakly nonlinear regime and controlling carefully the beam coupling at the sample entrance), our results suggest that 3D tomography of optical fields propagating in scattering media is in principle possible with high-NA immersion objective-as visible in Fig. 4c. Simple optical sectioning measurements with a standard microscope could provide a first experimental test of this hypothesis, but ultimately the best axial resolution could be attained with confocal microscopy. Of course, the proposed technique has some limitations: depending on the director alignment, not all polarization of the optical fields can be visible in the scattering direction (i.e. the sample normal), and the scattered intensity is modulated by the local scattering wavevector and the shape of the correlation function. Provided that supporting simulations are used to analyze experiments, these limitations

Figure 2 .

 2 Figure 2. (a) Gaussian beam of waist w0 = 2 µm propagating in a 40 µm-thick homeotropic LC slab. (b) Simulated bouncing optical soliton of waist w0 = 5 µm propagating in a 15 µm-thick homeotropic LC slab.

6 Figure 3 .

 63 Figure 3. (a) Calculated xy-slice of the imaged scattered intensity for the Gaussian beam of Fig. 2a, with z f = 0 and NA=0.4. (b) xz-slices along the dashed lines of (a). The white bars in (a,b) represent 5 µm and apply to both horizontal and vertical axes. (c) Rescaled scattered intensity profile Ĩ(x, 0, z f ) as a function of x for different z f , using the same calculation parameters as in (a,b). (d) Profiles of Ĩ(x, 0, z f ) along x (left, y = z f = 0) or z f (right, x = y = 0) for different numerical apertures.

Figure 4 .

 4 Figure 4. (a-b) xy-slices of the imaged scattered intensity for the bouncing optical soliton in Fig. 2b, calculated with z f = 0 and either a dry objective (a) or oil-immersion objective (b) with different NAs indicated on the right. (c) xy-slice of the same signal as in (b), calculated with x = 0 and an oil-immersion objective with different NAs indicated on the right. The white bars in (a,b,c) represent 10 µm and apply to both horizontal and vertical axes. (d) Longitudinal y-profile Ĩ(0, y, 0) numerically calculated with a dry objective and NA=0.05, and comparison with the same signal experimentally observed with similar parameters than in our simulations.[START_REF] Poy | Chirality-Enhanced Periodic Self-Focusing of Light in Soft Birefringent Media[END_REF]

* Small NA are more difficult to integrate because the associated PSF covers a larger region of space, but the scale of the transverse phase variations in the integrand (typically λ0) stays the same. These fast phase variations are difficult to integrate accurately, but they are also a fundamental ingredient for correctly calculating the scattered intensity.

ACKNOWLEDGMENTS

The authors acknowledge funding from the ARSS (Javna Agencija za Raziskovalno Dejavnost RS) through Grant No. P1-0099.

should not be too much a problem and our suggested technique could be for example used to track in 3D the mode transformation of optical solitons, [START_REF] Izdebskaya | Self-Induced Mode Transformation in Nonlocal Nonlinear Media[END_REF] or the complex 3D trajectories of interacting solitons. [START_REF] Kwasny | Nematicon-nematicon interactions in a medium with tunable nonlinearity and fixed nonlocality[END_REF]