domain: sdv chim

text:
We formulated this hypothesis more than a decade ago, stepping on years of research on Gs, G4s and G4-ligands. Our approach led to the design, synthesis and used of a broad family of synthetic G-quartets, also referred to as TASQs for template-assembled synthetic G-quartets (John Sherman). This quest led us across various chemical lands (organic and supramolecular chemistry, chemical biology and genetics), along a route on which every new generation of TASQ was a milestone in the growing portfolio of ever smarter molecular tools to decipher G4 biology. As discussed in this Account, we detail how and why we successively develop the very first prototypes of ibiomimetic ligands, which interact with G4s according to a bioinspired, like-likes-like interaction between two G-quartets, one from the ligand, the other form the G4;

iismart ligands, which adopts their active conformation only in the presence of their G4 targets; iiitwice-as-smart ligands, which act as both smart ligands and smart fluorescent probes, whose fluorescence is triggered (turned on) upon interaction with their G4 targets; and ivmultivalent ligands, which display additional functionalities enabling the detection, isolation and identification of G4s both in vitro and in vivo. As discussed in this Account, this quest thus led us to gather a panel of 14 molecular tools which were used to investigate the biology of G4s at a cellular level, from basic optical imaging to multi-omics studies.

Conspectus

Biomimetics is defined as a "practice of making technological design that copies natural processes", with the idea that "nature has already solved the challenges we are trying to solve" (Cambridge Dictionary). The challenge we decided to address several years ago was the selective targeting of G-quadruplexes (G4s) by small molecules (G4-ligands). Why? Because G4s, which are four-stranded DNA and RNA structures that fold from guanine (G)-rich sequences, are suspected to play key biological roles in human cells and diseases. Selective G4-ligands can thus be used as small-molecule modulators to gain a deep understanding of cell circuitry where G4s are involved, thus complying with the very definition of chemical biology applied here to G4 biology (Stuart Schreiber). How? Following a biomimetic approach that hinges on the observation that G4s are stable secondary structures owing to the ability of Gs to self-associate to form G-quartets, and then of G-quartets to self-stack to form the columnar core of G4s. Therefore, using a synthetic G-quartet as a G4-ligand represents a unique example of biomimetic recognition of G4s, relying on a like-likes-like approach.

Introduction

In 2004, Jeffery T. Davis brightly summarized in his review entitled 'G-quartets 40 years later' [START_REF] Davis | G-quartets 40 years later: From 5 '-GMP to molecular biology and supramolecular chemistry[END_REF] the state of the knowledge on the G-quartet, defined as "an hydrogen-bonded macrocycle formed by cation-templated assembly of guanosine". Guanosine, and more generally any guanine (G) derivative, is indeed a quite unique nucleobase: beyond its natural involvement in the double helix of DNA, in which it is associated with a cytosine, it can also create a network of different hydrogen bonds (H-bonds) via its two faces (Figure 1A): the Watson-Crick face 6 is embedded in three H-bonds with a cytosine, and the Hoogsteen face 7 can bind to various nucleobases including G itself to create either linear (G-ribbons) or cyclic suprastructures (Gquartet). [START_REF] Stefan | Applications of guanine quartets in nanotechnology and chemical biology[END_REF] I had the chance to meet Jeff Davis in Louisville, KY (USA) in 2007, during the first international meeting on quadruplex-DNA. [START_REF] Wilson | First International Meeting on Quadruplex DNA[END_REF] We notably discussed about his recently published article in which he used cross-linkable Gs to assemble a synthetic G-quadruplex (G4), [START_REF] Kaucher | A unimolecular G-quadruplex that functions as a synthetic transmembrane Na+ transporter[END_REF] defined as a "structure built from the vertical stacking of multiple G-quartets". After the cation (K + )promoted formation of several contiguous G-quartets, the lipophilic G derivative 5ʹ- (3,5-bis(allyloxy)benzoyl)-2',3'-isopropylideneguanosine 1 (Figure 1B) reacted with its neighbors via olefin metathesis to provide a covalently linked G4 comprising 4 stacked G-quartets (on average). We then discussed about the possibility of controlling the metathesis step not to obtain a synthetic G4 but a synthetic G-quartet: this would allow for obtaining an ideal, bioinspired G4-ligand, fully in line with my main interest at that time (the ligand we developed with Marie-Paule Teulade-Fichou, PhenDC3, was just published [START_REF] De Cian | Highly efficient G-quadruplex recognition by bisquinolinium compounds[END_REF] and was precisely the topic of the poster in front of which this discussion occurred). We came to the conclusion that this approach was not controllable enough for obtaining a synthetic G-quartet, the discussion stopped there but an idea was born. While challenging intermolecularly, the assembly of an intramolecular synthetic G-quartet was demonstrated to be possible one year later: John C. Sherman reported on the synthesis of a template-assembled synthetic G-quartet (TASQ), [START_REF] Nikan | Template-assembled synthetic G-quartets (TASQs)[END_REF] in which four lipophilic guanosines 5'azido-2',3'-O-isopropylideneguanosines 2 (Figure 1C) were clicked on a cavitand template. This approach provided a TASQ aimed at showing that the formation of an intramolecular Gquartet can occur in a cation-independent manner. This TASQ was used as a prototype, being conveniently soluble in organic media in which in-depth NMR studies were possible. We reasoned that a water-soluble template would lead to TASQs usable in biologically-relevant media, the design of the very first biomimetic G4-ligand was on its track.

From this very origin, the design of water-soluble TASQs has progressively evolved to provide ever smarter biomimetic G4-ligands that are now used as multivalent molecular tools to gain deep insights into G4 biology. This Account describes this evolution, after briefly reviewing what is known about G4s (their prevalence and functions) and why G4-ligands in general, and TASQs in particular, have been pivotal tools in this thrilling chemical biology quest. Native G-quartets, G-quadruplexes and ligands. While supramolecular chemists soon exploited the versatile H-bonding properties of Gs to create diverse nanoobjects, [START_REF] Davis | Supramolecular architectures generated by self-assembly of guanosine derivatives[END_REF] the recognition that naturally occurring Gs might behave alike in cells has taken a long time.

Indeed, the formation of a higher-order DNA structure that might have a functional relevance 14 somewhat countered the central dogma of biology, which placed the duplex-DNA at the very heart of every cellular biology process. [START_REF] Crick | Central dogma of molecular biology[END_REF] Massive chemical biology efforts were thus needed to demonstrate that nature also tamed the unique supramolecular property of Gs by providing G-containing nucleic acid sequences with the ability to fold into G4s. [START_REF] Varshney | The regulation and functions of DNA and RNA G-quadruplexes[END_REF] To make this possible, G-rich DNA sequences (e.g., those found at the telomeres [START_REF] Lansdorp | Telomere Length Regulation[END_REF] and in gene promoter regions) [START_REF] Chen | DNA G-Quadruplex in Human Telomeres and Oncogene Promoters: Structures, Functions, and Small Molecule Targeting[END_REF] must be freed from their duplex constraint: this occurs when DNA is at work, i.e., when being replicated or transcribed (see the schematic representation of a Gloop [START_REF] Duquette | Intracellular transcription of G-rich DNAs induces formation of G-loops, novel structures containing G4 DNA[END_REF] formed during the transcription of a G-rich region in Figure 2) or being repaired. This makes G4 formation not only tightly patrolled by the protein machinery in charge of DNA transactions but also transient in nature, explaining why their detection in fixed [START_REF] Rodriguez | Small-moleculeinduced DNA damage identifies alternative DNA structures in human genes[END_REF] then living cells was challenging. [START_REF] Antonio | Single-molecule visualization of DNA G-quadruplex formation in live cells[END_REF] DNA transactions thus represent a unique window of opportunity for G4s to fold, which is also recognized as vulnerable events that could jeopardize genomic stability. [START_REF] Zell | DNA folds threaten genetic stability and can be leveraged for chemotherapy[END_REF] This underpins the strategy based on the use of G4-targeting agents (or G4-ligands) [START_REF] Neidle | Quadruplex Nucleic Acids as Novel Therapeutic Targets[END_REF] as therapeutic agents: by stabilizing transiently folded G4s that arise during DNA transactions, G4-ligands can indeed prevent these cellular events from being properly performed. G4s can thus stall-or even trigger the collapse of-the enzymatic machinery in charge of these transactions, which is recognized as a situation of crisis, dealt with as a DNA damage, and consequently coped with by the DNA damage response (DDR) machinery. Most cancer cells being DDR-impaired, [START_REF] Jackson | The DNA-damage response in human biology and disease[END_REF] they are thus more sensitive to G4-mediated DNA damage; G4-ligands can therefore inflict severe DNA damage to cancer cells, which is at the very basis of their use as antiproliferative agents. [START_REF] Kosiol | G-quadruplexes: A promising target for cancer therapy[END_REF] Classical G4-ligands such as BRACO-19, 26 PhenDC3 11 and pyridostatin (PDS), [START_REF] Rodriguez | A Novel Small Molecule That Alters Shelterin Integrity and Triggers a DNA-Damage Response at Telomeres[END_REF] are flat aromatic molecules ideally suited to interact with G4s by stacking atop its accessible, external G-quartet (Figure 2). Their small size and moderate charge allow them to enter cells rather readily to reach their G4 targets, where they trigger their G4-mediated cellular effects. [START_REF] Olivieri | A Genetic Map of the Response to DNA Damage in Human Cells[END_REF] However, given that they are built on DNA-intercalator motifs (pyridine, quinoline, phenanthroline and acridine), off-target effects cannot be ruled out, even if an exquisite G4-selectivity has been achieved for some of them (e.g., PhenDC3, vide infra). This is where water-soluble TASQs come into play: owing to an intrinsic structural dynamism, TASQs exist under two conformations, an 'open' one in which the Gs are independent from each other, and a 'closed' one in which the intramolecular G-quartet is folded (see the schematic representation of DOTASQ (vide infra), Figure 2). TASQs thus adopt their closed, G4-affinis conformation only in the presence of G4s, which makes them uniquely actively selective for their G4 targets. This good selectivity, driven by a bioinspired, like-likes-like interaction between two G-quartets (one native from the G4, one synthetic from the TASQ), makes these ligands ideal molecular tools to investigate G4 biology. As further discussed hereafter, we spent the last decade developing different watersoluble TASQs (Figure 3) to make them evolve from biomimetic G4-ligands to multivalent molecular tools. Our goal was not to use these compounds as therapeutic agents but to design, synthesize and use ever smarter TASQ tools aimed at being used in optical imaging and large-scale, multi-omics studies in the aim of portraying G4 biology as accurately as possible. Biomimetic & smart G4-ligands. The very first prototype of water-soluble TASQ used as a biomimetic ligand was the DOTA-templated synthetic G-quartet, or DOTASQ, [START_REF] Stefan | DOTASQ as a prototype of nature-inspired G-quadruplex ligand[END_REF] comprising four Gs connected to a central DOTA (Figure 4A). This TASQ was published concomitantly (in 2011) with another water-soluble TASQ built on a macrocyclic peptide as regioselectively addressable functionalized template (RAFT). [START_REF] Murat | The Use of a Peptidic Scaffold for the Formation of Stable Guanine Tetrads: Control of a H-bonded Pattern in Water[END_REF] This 'RAFT-G4' [START_REF] Stefan | Closer to nature: an ATPdriven bioinspired catalytic oxidation process[END_REF] was not used as a G4 ligand but rather to demonstrate by nuclear magnetic resonance (NMR) that a synthetic G-quartet can fold in water. Sherman also reported on a water-soluble version of his cavitand-based TASQs 33 two years later (in 2013) but again, this TASQ was not used as a G4 ligand but rather as a platform to study the interaction between an isolated G-quartet (TASQ) and a G4 ligand (e.g., BRACO-19). DOTASQ was thus the very TASQ to be investigated as a biomimetic G4 ligand but the first results were disappointing: no G4-stabilization was obtained when subjected to FRET-melting evaluations (Figure 4B) performed with a doubly labelled G4forming oligonucleotide mimicking the human telomeric sequence (F21T). [START_REF] De Rache | Assessment of selectivity of G-quadruplex ligands via an optimised FRET melting assay[END_REF] We attributed this lack of interaction to a limited solubility in water (the poorly soluble G units, alkyl arms and amide connectors strongly counterbalanced the water-solubility of DOTA) and a high flexibility of both G arms and template (which might preclude the formation of a stable intramolecular G-quartet).

To tackle these issues, we exploited the property for which DOTA is thoroughly used, i.e., the metal chelation. We selected terbium, with an eye towards possibly benefiting from its luminescent properties (vide infra) if the presence of a metal within the DOTA cycle proved to be an effective way to tame DOTASQ structural flexibility. The coordination of a terbium ion within the DOTA cavity resulted in the tricationic complex [Tb.DOTASQ] 3+ (Figure 4A) that was consequently more geometrically constrained, water-soluble and poised to interact electrostatically with DNA. Satisfyingly, this complex stabilized G4s, although modestly, as it increased the mid-transition temperature (DT1/2) of the G4 by ca. 10°C, which is routinely used as a proxy for G4 affinity (of note: PhenDC3 imparts a >30°C-stabilization in similar conditions). This fair apparent affinity, which was confirmed with other G4-forming sequences mimicking the promoter regions of MYC and KIT genes, provided the proof of concept we were eagerly awaiting. These experiments were also performed in presence of an excess of unlabelled, 26nt long duplex-DNA competitor (ds26): the G4-stabilization imparted by the ligand was marginally affected only (-15% in the presence of 50 molar equivalents (mol. equiv.) of ds26, versus -10% for PhenDC3 in similar conditions), [START_REF] De Cian | Highly efficient G-quadruplex recognition by bisquinolinium compounds[END_REF] demonstrating a very good G4-selectivity of [Tb.DOTASQ] 3+ over duplex-DNA, which confirmed the relevance of our bioinspired design.

However, this complex turned out to be chemically unstable, likely because the DOTA was not the most suited chelator for terbium. We nevertheless tried to exploit its luminescence properties by inserting a sensitizing antenna on the DOTA scaffold (cf. [Tb.Pyro-DOTASQ] 3+ , Figure 4A), [START_REF] Laguerre | Synthetic G-Quartets as Versatile Nanotools for the Luminescent Detection of G-Quadruplexes[END_REF] but with a moderate success only. We thus envisioned two strategies to address these issues. First, we increased the stability of the intramolecular G-quartet by using a template known to strongly interact with G-quartets. To this end, we used a porphyrin template, as this scaffold is found in one of the most used G4-ligand TMPyP4 (which should nevertheless be avoided because of its indiscriminate interaction with DNA, whatever its structure). [START_REF] Stefan | Assessing the Differential Affinity of Small Molecules for Noncanonical DNA Structures[END_REF] The resulting porphyrin-templated synthetic G-quartet, or PorphySQ, [START_REF] Xu | Porphyrin-templated synthetic G-quartet (PorphySQ): a second prototype of G-quartet-based G-quadruplex ligand[END_REF] showed a very weak G4-interaction (although with an excellent G4-selectivity). These results were further compounded by a very poor water-solubility. We thus replaced the neutral G units by protonable Gs. Indeed, the Boc-PNA G-OH monomer is another G unit that harbors an ethylamine sidechain, readily protonated at pH 7.2, which makes it both water-soluble and prone to good electrostatic interactions with DNA. We thus assembled four PNA Gs around either a DOTA or a porphyrin template: the resulting TASQ, PNA DOTASQ 1 and PNA PorphySQ [START_REF] Laguerre | Porphyrin-Based Design of Bioinspired Multitarget Quadruplex Ligands[END_REF] (Figure 4A) thus displayed a tetracationic synthetic G-quartet (once folded) suited to strongly interact with a tetraanionic native G-quartet. PNA PorphySQ displayed better performance than PorphySQ (DT1/2 ca. 7 °C for both telomeric and KIT G4s); however, it suffered from a modest water-solubility and a self-immolating nature (likely via internal porphyrin-mediated photooxidation of G units). By contrast, PNA DOTASQ provided excellent results: its affinity for G4s was good (DT1/2 > 12°C for telomeric, MYC and KIT G4s, Figure 4B), its selectivity excellent, (though slightly lower than the parent compound DOTASQ, -26% in the presence of 50 mol. equiv. of ds26) and the chelation of terbium was no longer required.

We next wondered whether PNA DOTASQ interacts with both DNA and RNA G4s: competitive FRET-melting experiments showed a good affinity for RNA G4s (DT1/2 > 10°C for the sequences found in the telomeric transcript TERRA and the untranslated regions (5'-UTR) of TRF2 and VEGF mRNA), which was only marginally affected (>-13%) in the presence of 50 mol. equiv. of an unlabeled, highly stable DNA G4 competitor (TG5T). [START_REF] Haudecoeur | Multitasking Water-Soluble Synthetic G-Quartets: From Preferential RNA-Quadruplex Interaction to Biocatalytic Activity[END_REF] These results did not indicate a selectivity for G4-RNA over G4-DNA but a preferential interaction with the former, likely due to a better G-quartet accessibility. Globally speaking, PNA DOTASQ was considered as a pan-G4 interacting agent as it did interact with G-quartet whatever the G4 to which it belongs. These G4-interacting properties were satisfying enough to go further than simple G4-affinity measurement and we used it to gain insights into the way it interacts with G4s: the 1:1 stoichiometry of the PNA DOTASQ/G4 association was shown by electrospray ionization mass spectrometry (ESI-MS), [START_REF] Gabelica | Native Mass Spectrometry and Nucleic Acid G-Quadruplex Biophysics: Advancing Hand in Hand[END_REF] and no 2:1 TASQ/G4 complex was detected. The demonstration that this TASQ adopts its closed conformation upon interaction with G4s was provided by NMR (Figure 4C). [START_REF] Adrian | NMR spectroscopy of G-quadruplexes[END_REF] As duplex-DNA failed in promoting such an internal conformational switch, these results showed that only G4s can drive PNA DOTASQ into its closed, G4-affinic conformation, making it the very first prototype of a smart G4-ligand. We next aimed at endowing TASQs with additional skills in order to use them as molecular tools, notably to demonstrate the existence of G4s in human cells. We first focused on optical imaging and devised new TASQs implementable as smart G4 fluorescence probes.

Twice-as-smart G4-ligands. The TASQ design was further modified to convert them into valuable fluorescent probes. Specifically, we sought for a dye whose fluorescence properties could be modulated as a function of the TASQ conformation: a different response depending on the TASQ's open or closed status would lead to a turn-on fluorescence probe that would represent the fine art of G4 detection. [START_REF] Umar | G-Quadruplex-Based Fluorescent Turn-On Ligands and Aptamers: From Development to Applications[END_REF] This template should thus be fluorescent and sensitive to the presence of Gs in its close proximity, have a pseudo-C4 symmetry and allow for an easy introduction of four G arms.

We selected a pyrene template as its fluorescence is known to be hypersensitive to proximal G quenchers. [START_REF] Østergaard | Pyrene-functionalized oligonucleotides and locked nucleic acids (LNAs): Tools for fundamental research, diagnostics, and nanotechnology[END_REF] We used unsaturated connectors (alkyne linkers) to extend the aromaticity of the template in order to displace its excitation wavelength (lex) towards an optical window suited to optical cell imaging. This novel TASQ, named pyrene-templated synthetic G-quartet, or PyroTASQ, [START_REF] Laguerre | A Twice-As-Smart Synthetic G-Quartet: PyroTASQ Is Both a Smart Quadruplex Ligand and a Smart Fluorescent Probe[END_REF] was readily assembled through a quadruple Heck-Cassar-Sonogashira coupling (Figure 5A). [START_REF] Doucet | Palladium-based catalytic systems for the synthesis of conjugated enynes by Sonogashira reactions and related alkynylations[END_REF] Its affinity for G4s was good (DT1/2 > 13°C for telomeric, MYC and KIT G4s), its selectivity excellent (-16% in the presence of 50 mol. equiv. of ds26) and its ability to interact with G4s adopting its closed conformation demonstrated by NMR. Its photophysical properties were enticing: its fluorescence (lex = 420 nm; lem = 440 nm) was satisfyingly quenched when free in solution, strongly enhanced (up to 90-fold) upon addition of G4s and insensitive to the presence of a large excess of genomic DNA once turned-on by G4s.

The mechanism behind the spectroscopic properties of PyroTASQ was further investigated.

Free Gs (open conformation) were known to quench the pyrene fluorescence by photoinduced electron transfer (PET). We assumed that their self-association into the Gquartet (closed conformation) would trigger a redistribution of their electron density over the whole macrocycle via either resonance-assisted H-bonding (RAHB) [START_REF] Gilli | Evidence for resonance-assisted hydrogen bonding from crystal-structure correlations on the enol form of the. beta.-diketone fragment[END_REF] or charge separation, [START_REF] Fonseca Guerra | Telomere structure and stability: covalency in hydrogen bonds, not resonance assistance, causes cooperativity in guanine quartets[END_REF] or both. This, combined with the p-stacking interaction with the accessible G-quartet of a G4, would contribute to decrease the electron density of each G and thus, alleviate the PETmediated quench of the pyrene fluorescence. This explanation provided a satisfying rationale for the turn-on properties of PyroTASQ but was too simple. A combination of geometrical optimizations and density functional theory (DFT) calculations (Figure 5B) [START_REF] Zhou | Computational understanding and experimental characterization of twice-as-smart quadruplex ligands as chemical sensors of bacterial nucleotide second messengers[END_REF] showed that the open conformation actually corresponds to a conformation in which at least one G is stacked atop the template, and the highest occupied molecular orbital (HOMO) is localized on the whole TASQ structure. Upon irradiation, the energy is dissipated in a non-radiative manner by the freely vibrating G and an internal energy transfer from the pyrene to the G allows to channel the energy accumulated by the template to Gs for dissipation. In the closed conformation, the HOMO is exclusively localized on the template, precluding charge transfer with Gs, which are anyway embedded in the G-quartet and thus, no longer free to dissipate this energy through vibrational motion. The energy accumulated by the template is thus dissipated through fluorescence emission. PyroTASQ was thus the first prototype of a twiceas-smart G4-ligand, being both a smart G4-ligand and a smart fluorescent probe. These properties were used to detect G4s in vitro: polyacrylamide gel electrophoresis (PAGE, Figure 5C) was performed with both telomeric and MYC G4s along with a duplex-DNA as control. PyroTASQ provided a specific visualization of G4s using a classical gel imaging system.

The fluorescence response was surprisingly smeared and two-color, being indigo for the nicely defined bands and green for the higher molecular weight smear. We attributed these observations to a clean PyroTASQ/G4 interaction (indigo band) and a higher-order PyroTASQ/G4 complex, likely driven by the well-known tendency of pyrene to self-associate to form excimer (green smear). This tendency was confirmed during preliminary cell-based investigations, in which aggregates were found to accumulate around the cells, making the resulting images uninterpretable. A modification of the TASQ chemical scaffold was thus required, sufficient to prevent self-aggregation but not too much to keep the finely tuned G4interacting and photophysical properties of PyroTASQ.

Our options were limited. Hinging on a recent report describing the straightforward synthesis of the naphthalene precursor 3,6-dibromo-2,7-bis(trifluoromethanesulfonyloxy)naphthalene [START_REF] Shinamura | Linear-and Angular-Shaped Naphthodithiophenes: Selective Synthesis, Properties, and Application to Organic Field-Effect Transistors[END_REF] suited to be engaged in a quadruple Heck-Cassar-Sonogashira coupling, we synthesized a new probe that we named naphthalene-templated synthetic G-quartet, or NaphthoTASQ (N-TASQ for short, Figure 5A). [START_REF] Laguerre | Visualization of RNA-quadruplexes in live cells[END_REF] We hoped that halving the aromatic surface would decrease self-association while maintaining the peculiar turn-on properties of the reference compound PyroTASQ. The affinity of N-TASQ for G4s was good (DT1/2 > 12°C for telomeric, MYC and KIT G4s) and its selectivity excellent (-11% in the presence of 50 mol. equiv. of ds26). The connection of four alkyne linkers weakly displaced the excitation wavelength (to 320 nm), leading us to exploit an indirect sensitization mode, based on an energy transfer from the G4 to the bound ligand. This energy transfer, known to be particularly efficient within G4s as mediated by cations, [START_REF] Dumas | Cation-Mediated Energy Transfer in G-Quadruplexes Revealed by an Internal Fluorescent Probe[END_REF] was fully suited to the bioinspired TASQ/G4 interaction as it implies the presence of a cation in between the synthetic and native Gquartets. In doing so, we provided an interesting DNA structure-specific sensitization of N-TASQ, as only the addition of G4s lighted its fluorescence up (ca. 20-fold with MYC G4).

Satisfyingly, this molecule did not aggregate and was both bioavailable (readily entering cells without permeabilization) and non-toxic (IC50 >100 µM after 72 h-treatment).

To circumvent the low absorbance maxima of N-TASQ, poorly convenient for optical imaging (leading to an elevated cell autofluorescence background), the first series of optical images using N-TASQ were collected with a two-photon microscope adjusted at lex = 720 nm. Thanks to its intrinsic cell-permeability, N-TASQ was usable in living cells: MCF7 and osteosarcoma (U2OS) cells were live-incubated with N-TASQ, prior to be imaged without further manipulations (no fixation, permeabilization, mounting steps). The collected images, which confirmed the perinuclear accumulation of RNA G4/N-TASQ complexes, represented the first visualization of G4s in living cells (Figure 5D). We next used N-TASQ on fixed and permeabilized breast cancer cells (MCF7); surprisingly (further discussed below), high-quality images were collected using a confocal microscope through the different emission filters (blue, <495 nm; green, 495 -590 nm; and red channels, >590 nm). The actual nature of the observed foci was confirmed via by different treatments (DNase, RNAse) and colocalization with the G4-specific BG4 antibody. [START_REF] Biffi | Quantitative visualization of DNA G-quadruplex structures in human cells[END_REF] The quality of the N-TASQ labelling was found to be strongly dependent on the fixation method (MeOH should be privileged); in these conditions, N-TASQ labelled preferentially RNA G4s (in both cytoplasmic and nucleolar sites), likely due to the higher accessibility of these targets.

These good results nevertheless raised two questions: why was a sensitization of N-TASQ at 720 nm efficient (it is more than double its maximal lex) using a two-photon microscope? And why was the fluorescence observable through the blue/red/green channels using a confocal microscope? Deep spectroscopic investigations revealed that N-TASQ did not behave as a classical fluorescence dye: rather counterintuitively, this molecule can be sensitized at wavelengths at which it does not absorb light. This behavior, known as the red-edge effect (REE), is studied for more than 50 years notably in polymer science [START_REF] Demchenko | The red-edge effects: 30 years of exploration[END_REF] but rarely, if ever, reported in optical cell imaging investigations. REE was timely described for a series of PNA oligomers during the course of our investigations, [START_REF] Berger | Light-emitting self-assembled peptide nucleic acids exhibit both stacking interactions and Watson-Crick base pairing[END_REF] which provided a rationale for the peculiar behavior of N-TASQ. Briefly, when two aromatic partners (here a G-quartet and a naphthalene template) strongly interact with each other, the water molecules trapped in between them (belonging to their hydration shell) are dissociated from bulk water. Their motion is restricted and their dielectric relaxation participates to the deexcitation of the probe: therefore, the relaxation of each water molecule provides a virtual excited state of lower energy (Figure 5E), which can be thus excited with photons of lower energy, that is, of higher wavelengths (i.e., on the red edge of the absorption spectrum of the dye). This effect, possible only in media with very slow dynamics (here, in fixed cells), made the sensitization of N-TASQ possible at wavelengths >320 nm.

We used this properties to show that high-quality images could be collected using a classical confocal microscope, with lasers adjusted at 408, 488 and 555 nm (Figure 5F). On this basis, we devised an experimental setup to use N-TASQ to label either RNA G4s (live-cell N-TASQ incubation before fixation and imaging) or DNA G4s (cell fixation before N-TASQ incubation and imaging) (Figure 5G). [START_REF] Amor | Cellular Detection of G-Quadruplexes by Optical Imaging Methods[END_REF] We also devised a quantitative use of N-TASQ fluorescence to assess the modulation of the G4 landscape in cancer cells (e.g., HeLa) upon incubation with G4-interacting compounds, e.g., the G4-stabilizer BRACO-19 was found to increase it, [START_REF] Yang | Real-time and quantitative fluorescent live-cell imaging with quadruplex-specific red-edge probe (G4-REP)[END_REF] or the G4-destabilizer PhpC 4 to decrease it (unpublished). N-TASQ was also found to be compatible with live-cell imaging systems which enables to characterize real-time at which pace it enters cells (ca. 12 h) and its residence time in cells (ca. 100 h). [START_REF] Yang | Real-time and quantitative fluorescent live-cell imaging with quadruplex-specific red-edge probe (G4-REP)[END_REF] The use of N-TASQ was extended to other cancer cells, notably to uncover the origin of their replicative immortality (telomerase-dependent (TERT + ) vs. telomerase-independent (ALT + ) mechanisms). 56 N-TASQ was also used for the detection and quantification of G4 landscapes in a series of nervous cells including neurons, 57 astrocytes 58 and microglia [START_REF] Tabor | Differential responses of neurons, astrocytes, and microglia to G-quadruplex stabilization[END_REF] (without or with treatment with PDS). These different applications clearly highlighted the versatility of this probe, being usable in either fixed or live cells, with different imaging apparatus, in different cell types. To go a step further, we needed to devise new molecular tools in order to identify the nucleic acid sequences (and eventually the associated proteins) the TASQs interact with in cells.

Multivalent G4-ligands. The chemistry of polycyclic aromatic hydrocarbons (PAHs, such as pyrene or naphthalene) being rather challenging, [START_REF] Harvey | Advances in the synthesis of polycyclic aromatic compounds[END_REF] we did not try to synthesize PAH derivatives that would have allowed for endowing the corresponding TASQ (PyroTASQ, N-TASQ) with additional functionalities. Our experience with Pyro-DOTASQ (vide supra) convinced us about the convenient use a the DOTA derivative named AMC (aminomethylcyclen, Figure 4A) to introduce new functionalities. We thus used the aminomethyl appendage of AMC to introduce various functional moieties on the PNA DOTASQ scaffold. The first one was a biotin: the corresponding biotinylated TASQ, named BioTASQ (Figure 6A), [START_REF] Yang | Transcriptome-wide identification of transient RNA G-quadruplexes in human cells[END_REF] was intended to be used as a molecular bait for fishing G4-forming sequences thanks to the well-established affinity precipitation using streptavidin-coated beads. [START_REF] Laitinen | Brave new (strept) avidins in biotechnology[END_REF] Unexpectedly, the introduction of a biotin handle almost abolished the G4-affinity of the resulting TASQ, as only very weak stabilizations (DT1/2 < 2°C) were obtained during FRETmelting investigations. We ascribed this lack of affinity to an internal poisoning of one of the G arm by H-bonding with biotin (Figure 6B). To discard simple steric reasons, we synthesized a derivative with a polyethylene glycol (PEG) linker in between the AMC template and the biotin: the resulting BioTASQ v.2 displayed an even worse G4-stabilization, [START_REF] Renard | Small-molecule affinity capture of DNA/RNA quadruplexes and their identification in vitro and in vivo through the G4RP protocol[END_REF] confirming that the ability of biotin to create intramolecular H-bonds was indeed responsible for the loss of G4 interaction.

To tackle this, we pre-mixed BioTASQs and streptavidin-coated magnetic beads, in order to hijack biotin from internal poisoning, and used this assembly to isolate G4s from solution. This was validated using fluorescently labelled G4-forming sequences (FAM-G4s) in vitro, along with a FAM-duplex as control. The capture efficiency of TASQs was quantified by measuring the fluorescence resulting from the dissociation of the FAM-G4/TASQ/bead complex (thermal treatment). Both BioTASQs pulled down RNA G4s (11-to 44-fold fluorescence enhancement, normalized to TASQ-less control) more readily than DNA G4s (2-to 26-fold enhancement). This efficiency was slightly altered (-25%) in the presence on an excess of duplex-DNA, which was further confirmed by the lack of FAM-duplex capture (0.7-fold enhancement). This opened the way towards cell-based investigations: the ability of BioTASQs to fish G4s out from cell lysates was first validated by a RT-qPCR analysis targeting well-established RNA G4forming sequences. [START_REF] Lyu | RNA G-quadruplexes (rG4s): genomics and biological functions[END_REF] We selected RNA targets and developed the corresponding protocol G4RP (G4-RNA precipitation, Figures 6C andD) to disentangle a paradoxical situation resulting from two conflicting reports: the rG4-seq 64 method concluded that RNA G4s were pervasive structures in the human transcriptome, while a combination of DMS-seq and RT-stop profiling [START_REF] Guo | RNA G-quadruplexes are globally unfolded in eukaryotic cells and depleted in bacteria[END_REF] concluded that RNA G4s were mostly unfolded in vivo. We hypothesized that the formation of G4s might be transient only in cells; we thus devised the G4RP protocol introducing a fixation step to capture transiently folded G4s in vivo prior to cell lysis and BioTASQ-mediated G4-precipitation. This approach was validated by RT-qPCR against a series of well-known G4-prone transcripts including the sequence found in the 5'-UTR of NRAS and VEGF mRNAs. [START_REF] Yang | Transcriptome-wide identification of transient RNA G-quadruplexes in human cells[END_REF][START_REF] Renard | Small-molecule affinity capture of DNA/RNA quadruplexes and their identification in vitro and in vivo through the G4RP protocol[END_REF] Both BioTASQs efficiently pulled down RNA G4s from cell lysates (16-to 59fold enhancement, normalized to biotin control), with a slightly better performance of BioTASQ v.2. We artificially forced G4-folding in cells live-incubating them with BRACO-19: this treatment increased the overall efficiency of the capture, more markedly with BioTASQ (up to 9-fold enrichment, normalized to untreated control) than BioTASQ v.2 (4-fold enrichment only). These results confirmed the ability of G4-ligands to modulate G4 landscapes in cells, in line with N-TASQ results described above. We next performed a transcriptomewide analysis (G4RP-seq, Figure 6E) 3 of the same cancer cells, without or with cell incubation with BRACO-19: the results obtained first confirmed the prevalence of RNA G4s in vivo, with >6,000 G4-containing transcripts isolated. They also uncovered the prevalence of G4s in long non-coding RNAs (lncRNAs). And very importantly, they helped redefine both the targets and the cellular modes of action of G4-ligands: our results showed that cells evolved a mechanism (likely helicases) [START_REF] Brosh | History of DNA Helicases[END_REF][START_REF] Lejault | How to untie G-quadruplex knots and why?[END_REF] to avoid G4-formation in lncRNAs that is counteracted by G4-ligands, which thus affect their functions by stabilizing a folding that is not compatible with their regular cellular activity. We then wanted to address the overall low G4-affinity of biotinylated TASQs. We thus revisited their synthetic schemes to circumvent the biotin internal poisoning issue and make their access more straightforward. We changed the nature of the G arms, removing one amide connector to alleviate H-bonding with biotin. In doing so, the central template was not a DOTA anymore but a cyclene, which made the resulting molecule cyclene-templated synthetic Gquartets, or CyTASQs. We assessed the validity of this approach with the naked CyTASQ (i.e., without biotin handle) to verify that using such an alkyl arm did not modify the overall G4interacting properties of the parent PNA DOTASQ compound. CyTASQ stabilized G4s modestly (DT1/2 = 5 and 9 °C for telomeric and MYC G4s, respectively), lower than PNA DOTASQ (DT1/2 > 12°C), [START_REF] Sperti | Biomimetic, Smart, and Multivalent Ligands for G-Quadruplex Isolation and Bioorthogonal Imaging[END_REF] but its G4-selectivity was excellent (-8% in the presence of 50 mol. equiv. of ds26), better than PNA DOTASQ. We next synthesized the biotinylated CyTASQ, or BioCyTASQ (Figure 6A), and evaluated its G4-interacting properties: quite satisfyingly, its G4-affinity was unaffected by the presence of the biotin handle (DT1/2 = 5 and 10 °C for telomeric and MYC G4s, respectively) and its selectivity even better (-2% in the presence of 50 mol. equiv. of ds26), making BioCyTASQ a far better G4-ligand than BioTASQ (DT1/2 < 2°C). Its affinity was also assessed by a fluorescence quenching assay (FQA) [START_REF] Le | G-quadruplex ligands exhibit differential G-tetrad selectivity[END_REF] that confirmed its good G4 affinity (Cy5-MYC, app KD ca. 1 µM; of note, no 2:1 TASQ:G4 complex was detected, thus confirming aforementioned ESI-MS results), while being 1 order of magnitude lower than PhenDC3 ( app KD ca. 0.1 µM). We next confirmed that BioCyTASQ did capture RNA G4s as did BioTASQ by performing a G4RP-RT-qPCR analysis of MCF7 cell lysate against NRAS and VEGF (Figure 6D). With these results in hand, we wanted to meet another challenge: using biotinylated TASQs to capture and identify DNA G4s in vivo. Inspired by the G4RP-seq protocol, we developed the G4DP-seq (G4-DNA precipitation and sequencing) using a model genome, the rice genome.

This model benefitted from a straightforward access to high-quality genomic DNA, ideally suited to perform comparative studies in which G4s were captured by either BioTASQ, BioCyTASQ or BG4. This approach thus represented an unprecedented comparison of the efficiency of affinity-based G4 purification protocols, i.e., chemoprecipitation (G4DP-seq) [START_REF] Feng | Chemo-versus immuno-precipitation of G-quadruplex-DNA (G4DNA): a direct comparison of the efficiency of the antibody BG4 versus the small-molecule ligands TASQs for G4 affinity capture[END_REF] vs. immunoprecipitation (BG4-IP-seq), [START_REF] Feng | Epigenomic features of DNA G-quadruplexes and their roles in regulating rice gene transcription[END_REF] in line with our comparison of chemodetection (N-TASQ) vs. immunodetection (BG4) of G4s. [START_REF] Yang | G-quadruplexes mark alternative lengthening of telomeres[END_REF] The collected results (Figure 6E) confirmed that small molecules competed with antibodies for the isolation of G4s: while BG4-IP-seq identified ca. 30,000 G4 peaks, G4DP-seq identified ca. 70,000 G4s with BioTASQ and 50,000 G4s with BioCyTASQ. [START_REF] Feng | Chemo-versus immuno-precipitation of G-quadruplex-DNA (G4DNA): a direct comparison of the efficiency of the antibody BG4 versus the small-molecule ligands TASQs for G4 affinity capture[END_REF] These numbers were not commensurate with that of G4s detected in human cells (>500,000 G4 peaks), [START_REF] Chambers | High-throughput sequencing of DNA G-quadruplex structures in the human genome[END_REF] likely originating in a more suppressive effect on higher-order DNA structure formation in plant genomes, as a similar trend was observed with Cquadruplexes (or i-motifs), 73 ca. 25,000 peaks in plants [START_REF] Ma | Genome-wide characterization of i-motifs and their potential roles in the stability and evolution of transposable elements in rice[END_REF] vs. >600,000 peaks in human cells. [START_REF] Martinez | Human genomic DNA is widely interspersed with i-motif structures[END_REF] G4DP-seq and BG4-IP-seq captured G4s in in vitro conditions, that is, with G4-forming sequences from fragmented DNA properly folded prior to be precipitated by affinity capture.

A fine analysis of the difference between BioTASQ and BioCyTASQ revealed a majority of common G4 peaks (ca. 82%) but a longer average length of G4 peaks for BioCyTASQ. We then compared the efficiency of TASQs in in vitro vs. in vivo conditions, where naturally folded G4s were cross-linked live prior to cell lysis, chromatin fragmentation and affinity capture steps. A similar amount of G4 peaks was detected (ca. 60,000 peaks), with >70% common peaks with in vitro conditions, highlighting the suitability of TASQ probes to the two approaches.

We then modified again the synthetic access to biotinylated TASQs to assemble the G arms by click chemistry, a guarantee of efficiency and rapidity. The resulting TASQs, which harbored a triazole linker within their G arms, were consequently termed TriazoTASQ and BioTriazoTASQ (Figure 6A). [START_REF] Sperti | Click-Chemistry-Based Biomimetic Ligands Efficiently Capture G-Quadruplexes In Vitro and Help Localize Them at DNA Damage Sites in Human Cells[END_REF] Both compounds displayed high affinity and selectivity for G4s: while BioTriazoTASQ has not yet been used for G4RP/G4DP investigations, it was used in a pretargeted G4 imaging protocol to detect G4s in human cells thanks to the labelling of BioTriazoTASQ once in its cellular binding sites by a fluorescently labelled streptavidin (Figure 6G). This strategy was validated using BioTASQ and BioCyTASQ, the live-incubation of which revealed an accumulation of TASQs in both the cytoplasm and nucleoplasm, therefore testifying for a high bioavailability. [START_REF] Sperti | Biomimetic, Smart, and Multivalent Ligands for G-Quadruplex Isolation and Bioorthogonal Imaging[END_REF] Though indirect, it allowed for a wise selection of excitation/emission wavelengths, which opened the way towards colocalization studies: it was used to colocalize nuclear G4 and DNA damage sites, which provided unique insights into the mechanism of action of G4 targeting agents as DNA damage increased upon cell incubation with the G4-ligand PDS, while the G4-destabilizer PhpC reversed G4-mediated DNA damage (unpublished).

Finally, the most recent TASQ prototypes, named MultiTASQ, photoMultiTASQ and azidoMultiTASQ (Figure 6A), combined the skills of the previous TASQ generations with an additional level of modularity. [START_REF] Sperti | The multivalent G-quadruplex (G4)-ligands MultiTASQs allow for versatile click chemistrybased investigations[END_REF] Their clickable handle (an alkyne for MultiTASQ and photoMultiTASQ, an azide for azidoMultiTASQ) allowed them for being implemented in a series of bioorthogonal investigations: they can indeed be clicked (the copper-catalyzed azidealkyne cycloaddition (CuAAC) [START_REF] Tornøe | Peptidotriazoles on solid phase:[1, 2, 3]triazoles by regiospecific copper (I)-catalyzed 1, 3-dipolar cycloadditions of terminal alkynes to azides[END_REF][START_REF] Rostovtsev | A Stepwise Huisgen Cycloaddition Process: Copper(I)-Catalyzed Regioselective "Ligation" of Azides and Terminal Alkynes[END_REF] for MultiTASQ and photoMultiTASQ, the strain-promoted azide-alkyne cycloaddition (SPAAC) [START_REF] Agard | A Strain-Promoted [3 + 2] Azide-Alkyne Cycloaddition for Covalent Modification of Biomolecules in Living Systems[END_REF] for azidoMultiTASQ) once in interaction with their cellular partners with, for instance, a biotin for affinity capture (e.g., click-seq) [START_REF] Tyler | Click chemistry enables preclinical evaluation of targeted epigenetic therapies[END_REF] or a fluorophore for visualization purposes (e.g., in situ click imaging, Figure 6H). [START_REF] Cañeque | Visualizing biologically active small molecules in cells using click chemistry[END_REF] The photoactivatable diazirine handle of photoMultiTASQ was specifically designed to capture and profile G4-interacting proteins (i.e., the co-binding-mediated protein profiling (CMPP) protocol), [START_REF] Zhang | Chemical profiling of DNA G-quadruplex-interacting proteins in live cells[END_REF] which add a proteomics component to the already well-advance genomics use of TASQ. This patented technology [START_REF] Monchaud | Biomimetic G-quartet compounds[END_REF] thus offers new and wide-ranging opportunities to go increasingly further in the study of cellular G4s at a multi-omics scale.

Outlook

The logic behind the development of the 14 different TASQs described herein was to provide the G4 community with ever smarter and user friendly molecular tools to decipher the biology of DNA and RNA G4s. Our philosophy was to address different but complementary questions regarding G4s with tools form the same family, in order to make these sets of results comparable. We believe this is being achieved, even if TASQs are course not devoid of limitations including their cumbersome synthesis, their moderate G4-affinity, their inability to discriminate between G4s of different nature (DNA, RNA) and topologies, and a possible but not-yet-observed random association with C-rich sequences. Another limitation is the scope of application of TASQs, which remains, for now, limited to cell-based investigations. An extension towards small animal studies could be envisaged but will require to both rethink the design of TASQs (to make them, for instance, detectable via whole-body imaging) and address key points (such as their physiological stability, metabolization, drug-induced organ toxicity, etc.). This is an open canvas where much remains to be done and uncovered.

These shortcomings notwithstanding, we believe that these tools have proven efficient and versatile enough for a broad series of in vitro investigations to be successful, which address most of the questions relevant today on G4 biology. We also believe that the time is ripe to benefit from this outstanding wealth of knowledge to meet new challenges. Indeed, G4s can boast of being at the very heart of a special attention for 2 decades now. However, accumulating evidence now shows that other higher-order nucleic acid structures deserves to be granted such special treatment: [START_REF] Wang | Dynamic alternative DNA structures in biology and disease[END_REF] this includes i-motifs, 73 R-loops, [START_REF] Crossley | R-Loops as Cellular Regulators and Genomic Threats[END_REF] triplex-DNA [START_REF] Dalla Pozza | Three's a crowdstabilisation, structure, and applications of DNA triplexes[END_REF] and DNA junctions, [START_REF] Mcquaid | Interactions of small molecules with DNA junctions[END_REF] for which ligands and protocols have been yet devised, but not in a finely orchestrated manner (as G4s did), which has somewhat dampened the advance of these investigations. Let us wager that the coming years will deliver fascinating advances in these areas, which will undoubtedly find multiple applications in the field of genetics and genetic diseases.

On a more personal note, the results presented herein provide a strong message that unformal discussions occurring during conferences (and poster sessions) might lead to an entirely new area of research, even if I think that neither Jeff and I would have envisioned at that time that this open discussion would take us from simple supramolecular chemistry considerations to deep, multi-omics investigations.

  UMR6302, UBFC, 21078 Dijon, France david.monchaud@cnrs.fr

Figure 1 .

 1 Figure 1. A. Chemical structure of guanosine and its possible supramolecular assemblies (GC base pair, G-ribbon and G-quartet). B. J. T. Davis' approach to synthesize a unimolecular, covalent G4 via olefin metathesis. C. J. C. Sherman's approach to synthesize an intramolecular G-quartet referred to as template-assembled synthetic Gquartet (TASQ) via click chemistry.

Figure 2 .

 2 Figure 2. Schematic representation of a G-loop, which results from the cotranscriptionally formation of a R-loop (DNA:RNA hybrid, green and yellow strands) and a G4 (red strand); details about the G4 structure and the G4ligand preferred binding site (external G-quartet); chemical structure of classical G4-ligands (BRACO-19, PhenDC3 and PDS), which stack atop the accessible G-quartet of a G4, and of a biomimetic G4-ligand (DOTASQ), with a schematic representation of its conformational plurality (open and closed conformations) at the origin of its bioinspired, 'like-likes-like' quartet/quartet interaction with a G4.

Figure 3 .

 3 Figure 3. Evolution of water-soluble TASQs used as G4-ligands, from biomimetic to multivalent TASQs (for patented and commercially available TASQs, see 29 ).

Figure 4 .

 4 Figure 4. A. Chemical structure of biomimetic (Tb.DOTASQ, PorphySQ and Pyro-DOTASQ) and smart ligands ( PNA PorphySQ and PNA DOTASQ). B. Schematic representation of the FRET-melting assay (left) and results collected with a doubly labelled G4 in absence (center, whose stability is quantified by the mid-transition temperature T1/2)or presence of a ligand (right), here PNA DOTASQ (with the DT1/2 value used as a proxy for G4 apparent affinity), without or with an excess of the duplex competitor ds26. C. NMR results collected with a G4 (left) or a duplex (right) upon addition of increasing concentrations of PNA DOTASQ, which demonstrate the conformational switch of the ligand triggered by its G4 target.

Figure 5 .

 5 Figure 5. A. Chemical structure of the twice-as-smart G4 ligands PyroTASQ and NaphthoTASQ (N-TASQ) and schematic representation of their interaction with G4s, which triggers their fluorescence (turn-on probe). B.Schematic representation of the mechanism behind the turn-on fluorescence properties of TASQs (molecular orbital analyses performed at BLYP-D3/TZP level of DFT gas phase calculations). C. Fluorescence analysis of a polyacrylamide gel performed with DNA (telomeric and MYC G4s or the duplex ds26) and increasing amounts of PyroTASQ. D. Live-cell imaging of U2OS cells treated with N-TASQ (5 µM for 48 h) collected with a two-photon microscope (lex = 720 nm). E. Simplified Jablonski diagram depicting the ground (S0) and excited states (S1) of a TASQ in dilute solution (left) or ordered environment (right) where the solvent relaxation is slowed down, which give rise to the red edge effect (a continuous model of solvent relaxation with intermediate ground (S0', S0", etc.) and excited states (S1', S1", etc.) leading to red-shifted excitation (lem', lem " , etc.) and emission (lex', lex " , etc.) wavelengths). F. Dependence of the emission maxima (lex max ) on the excitation wavelength (lex) (left), typical of the red edge effect, which was confirmed by confocal microscopy (right) with images of MCF7 cells collected through the blue, green and red channels. G. Use of N-TASQ in either live cells (left) or fixed cells (right) which allow for a direct visualization of either RNA (left) or DNA (right) G4 foci.

Figure 6 .

 6 Figure 6. A. Chemical structure of the multivalent TASQs including the biotinylated BioTASQ, BioCyTASQ and BioTriazoTASQ and the MultiTASQs (MultiTASQ, azidoMultiTASQ ( az MultiTASQ) and photoMultiTASQ). B. Schematic representation of the possible internal structural poisoning of BioTASQ. C-E. Schematic representation of the G4RP (G4-RNA precipitation) protocol (C) and examples of results obtained by RT-qPCR (D) or sequencing (E). F. Number of G4 peaks obtained by the G4DP-seq protocol (G4-DNA precipitation and sequencing) in the rice genome using BioTASQ or BioCyTASQ in comparison with that of BG4-ChIP-seq in similar conditions (using the antibody BG4); left: example of motifs enriched by the two TASQs. G,H. Schematic representation and results of pre-targeted G4 imaging (G) using BioTriazoTASQ (to colocalize G4 and DNA damage sites) and in situ click imaging (H) using MultiTASQ (being used as a live-cell or post-fixation agent).

Acknowledgments

This work would not have been possible without the support of different funding agencies (the CNRS, ANR, ISITE BFC, ERDF, Plan Cancer and European Union) and the contribution of talented students, including Loïc Stefan and Romain Haudecoeur (DOTASQs), Aurélien Laguerre (PyroTASQ and N-TASQ), Pauline Lejault (BioTASQ) and Francesco Rota Sperti (CyTASQs, TriazoTASQs and MultiTASQs); of enthusiastic coworkers, either locally (chiefly, Ibai E. Valverde and Marc Pirrotta) or abroad (chiefly, Sunny Y. Yang and Judy M. Wong (CA), Andrey S. Tsvetkov (USA) and Wenli Zhang (China)); of dedicated collaborators I have the chance to work with on different (more or less G4-free) projects; and of all scientists worldwide who contribute every day to make the G4 field a lively and thrilling research area. France, Paris) under the supervision of Dr. Marie-Paule Teulade-Fichou, before moving to Institut Curie (Orsay) in 2007 with Dr. Teulade-Fichou. He then joined the Institut de Chimie Moléculaire de l'Université de Bourgogne (ICMUB, Dijon, France) in 2009, where he became CNRS Research Director in 2017, to develop chemical genetics programs on DNA/RNA secondary structures, smart ligands and probes.

Conflict of interest

The CNRS has licensed BioCyTASQ and MultiTASQs to Merck KGaA for commercialization.

Biography


Giorgio Palma
email: giorgio.palma@uniroma3.it

Stefano Meloni
email: stefano.meloni@unitus.it

Roberto Camussi
email: roberto.camussi@uniroma3.it

Umberto Iemma
email: umberto.iemma@uniroma3.it

Christophe Bogey
email: christophe.bogey@ec-lyon.fr

A data-driven multi-objective optimization of wave-packets for near-field subsonic jet noise

Aeroacoustics

of innovative aircraft cannot disregard the development of low-order models for the jet noise source, which are essential to assess the propulsion-airframe interactions from the conceptual design stage. The main scope of this work is to provide a noise source model that can be coupled with relatively low computational cost methods for aeroacoustic scattering. To this end, this paper presents for the first time a multi-objective optimization of the 0 𝑡 ℎ mode wave-packet in the jet near-field. The importance of calibrating the model with near-field pressure data stems from the fact that in new aircraft the engine nacelles are typically positioned at a few diameters from the wing or fuselage. In this work, the near-field of a high-subsonic jet at a Mach number of 0.9 is represented as a cylindrical surface radiating the pressure disturbances of a wave-packet source, optimized using large-eddy simulation data from three lines at different radial distances. The optimized model has been tested at Strouhal numbers up to 1, and the optimized solutions have been chosen using a Pareto-ranking criterion which considers the wave-packet prediction over an extra line. A good agreement is achieved between the reference data and model predictions, for multiple near-field radial distances.

I. Introduction

Since the beginning of aeroacoustics, jet noise has been considered a hot topic in aviation noise because of its dominant role in community exposure. The incoming of increasingly strict noise regulations makes essential the development of modern strategies to reduce the noise emitted by jets and aircraft engines. The design of a modern commercial aircraft must consider the acoustic emissions from the early conceptual phase, hence reliable and fast prediction methods are of fundamental importance. Tremendous efforts have been and are being deployed in the development of more effective acoustic treatments and devices for quieter aviation, such as quieter high lift devices [START_REF] Burghignoli | Evaluation of the noise impact of flap-tip fences installed on laminar wings[END_REF], chevrons for jet exhaust [START_REF] Kamliya Jawahar | Experimental Investigation on the Jet Noise Sources for Chevron Nozzles in Under-expanded Condition[END_REF][START_REF] Meloni | A Wavelet-Based Time-Frequency Analysis on the Supersonic Jet Noise Features with Chevrons[END_REF], acoustic liners, and/or innovative treatments [START_REF] Palma | On the integration of acoustic phase-gradient metasurfaces in aeronautics[END_REF][START_REF] Palma | Innovative Acoustic Treatments of Nacelle Intakes Based on Optimised Metamaterials[END_REF][START_REF] Jones | Evaluation of Variable-Depth Liners with Slotted Cores[END_REF][START_REF] Dodge | An Acoustic Liner with a Multilayered Active Facesheet[END_REF][START_REF] Palani | Experimental and numerical assessment of novel acoustic liners for aero-engine applications[END_REF] for turbofans ducts. Innovative configurations are being studied to avoid technological saturation in the race for aviation noise abatement. Blended and Hybrid Wing Body (BWB) aircraft is probably the most promising alternative to the currently dominating tube-and-wing configuration, both in terms of efficiency and community noise reduction [START_REF] Liebeck | Design of the Blended Wing Body Subsonic Transport[END_REF][START_REF] Centracchio | Noise shielding models for the conceptual design of unconventional aircraft[END_REF][START_REF] Burghignoli | Multi-objective optimization of BWB aircraft for noise shielding improvement[END_REF][START_REF] Xie | Integrated Sizing and Optimization of Hybrid Wing Body Aircraft in Conceptual Design[END_REF]. The main characteristic of a BWB is the non-net distinction between the fuselage occupying the center body and wings, with the former being wider and shaped like an airfoil to provide a non-negligible contribution to the overall lift, resulting in a sensible efficiency increase. The large center body surface offers the possibility of the upper installation of the propulsion system, which can be exploited for the acoustic shielding of the engine noise [START_REF] Papamoschou | Jet noise shielding for advanced hybrid wing-body configuration[END_REF][START_REF] Denisov | Investigation of noise-shielding efficiency with the method of sequences of maximum length in application to the problems of aviation acoustics[END_REF][START_REF] Palma | Optimized metamaterials for enhanced noise shielding of innovative aircraft configurations[END_REF]. The simulation of the shielding effect in the audible frequency range involves the solution of the scattering problem of the whole aircraft up to extremely high Helmholtz number 𝐻𝑒 𝑙 = 𝑘𝑙, with 𝑘 the wave-number and 𝑙 a characteristic length of the aircraft, e.g. the center body length, which makes it computationally very expensive in particular when introduced in an optimization process.

There is, hence, a strong need for fast models for predicting the effects of the propulsion system installation on the acoustic emissions at the aircraft level, enabling its assessment from the first design stages and also for disruptive configurations. Adaptive metamodeling techniques have been recently applied to this class of problems [START_REF] Burghignoli | Adaptive RBF with hyperparameter optimisation for aeroacoustic applications[END_REF][START_REF] Centracchio | Noise shielding surrogate models using dynamic artificial neural networks[END_REF] to reduce the computational effort required in determining the optimal position of the propulsion system that minimises the noise directed towards the ground and community. Boundary Elements Method (BEM) simulations involving the monopole as a noise source are often used to feed the model creation. However, such a simple source has been demonstrated not to be able to satisfactory model the jet noise [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF]. In some low-order models, the noise sources of a single circular jet were represented by a set of uncorrelated quadrupoles [START_REF] Kopiev | Simulation of azimuthal characteristics of turbulent jet noise by correlation model of quadrupole noise sources[END_REF]. However, it was shown that the directivity at shallow angles was not well reproduced [START_REF] Denisov | Using the Correlation Model of Random Quadrupoles of Sources to Calculate the Efficiency of Turbulent Jet Noise Screening with Geometric Diffraction Theory[END_REF].

In this framework, the discovery of coherent structures in jets changed the perspective of jet noise and provided a basis for introducing the wave-packet approach [START_REF] Jordan | Wave Packets and Turbulent Jet Noise[END_REF]. As suggested by Papamoschou [START_REF] Papamoschou | Wavepacket modeling of the jet noise source[END_REF], the wave-packet is an amplitude-modulated travelling pressure wave. Several authors have widely used this approach to predict and model the jet noise source from far-field measurements having parameters such as envelope amplitude, wavelength, position, and convection velocity (see e.g. references [START_REF] Papamoschou | Wavepacket Modeling of the Jet Noise Source[END_REF][START_REF] Koenig | Farfield filtering and source imaging of subsonic jet noise[END_REF]). For example, Cavalieri et al. [START_REF] Cavalieri | Axisymmetric superdirectivity in subsonic jets[END_REF] used azimuthally decomposed far-field measurements to determine envelope parameters for higher-order azimuthal modes. However, for the mentioned BWB architecture, the jet is typically in a very closely coupled configuration with the center body scattering surface. A large body of literature demonstrated that the low-frequency amplification of the jet noise in the classical jet-wing architecture can be ascribed to the scattering of the jet hydrodynamic field [START_REF] Meloni | An experimental investigation into model-scale installed jet-pylon-wing noise[END_REF][START_REF] Meloni | Wall-pressure fluctuations induced by a compressible jet in installed configuration[END_REF][START_REF] Jordan | Jet-flap interaction tones[END_REF]. The prediction of the acoustic shielding, hence, needs source models able to capture the near field characteristics of the emitted noise in order to accurately address the effects of the aircraft scattering. Applications of the wave-packet procedure using near-field measurements have been carried out in various research works. Mollo-Christensen [START_REF] Mollo-Christensen | Measurements of near field pressure of subsonic jets[END_REF][START_REF] Mollo-Christensen | Jet Noise and Shear Flow Instability Seen From an Experimenter's Viewpoint[END_REF] provided the first observations of the wave-packet features from the point of view of hydrodynamic instability and aeroacoustics, whereas Crighton and Huerre [START_REF] Crighton | Shear-layer pressure fluctuations and superdirective acoustic sources[END_REF] suggested various simple models to predict near-field structures. Following the literature, the near-acoustic field is characterized by acoustic fluctuations that are small enough to permit linearization but close to the jet so that they could be contaminated by the irrotational hydrodynamic pressure field. In this region, most of the kinetic energy of fluctuations is related to azimuthally-coherent structures that lead to nonlinear effects on the wave-packet evolution. Thus, for the complexity mentioned above, the high subsonic jet noise source prediction is still a challenging task, which is crucial because it represents the operative condition of modern turbofan engines. In this framework, the principal focus of the present work is to find out an optimized amplitude-modulated wave-packet able to model the jet near-field noise that can be integrated in future into the BEM formulation to obtain a reliable prediction of the jet-surface aeroacoustic scattering also for innovative aircraft configurations. To tune the present model, we used a numerical database obtained using a LES simulation of an isothermal round free jet at a Mach number of 𝑀 = 0.9 and a diameter-based Reynolds number of 𝑅𝑒 𝐷 = 10 5 with the nozzle exhaust turbulence level fixed at 𝑇 𝐼 = 9%. The database contains pressure data at different axial and radial locations, that we considered in a subset ranging from x/D=0 up to x/D=20 in the axial direction and from 𝑟/𝐷 = 0.5 up to 𝑟/𝐷 = 3 in the radial direction, allowing us to optimize the wave-packet jet near-field domain. The near-field domain depicted in this database is representative of all the jet zones that could be influenced by solid boundaries in reality (i.e. wing or fuselage) see, [START_REF] Meloni | An experimental investigation into model-scale installed jet-pylon-wing noise[END_REF][START_REF] Meloni | Wall-pressure fluctuations induced by a compressible jet in installed configuration[END_REF].

A preliminary version of this study has been presented at the 28 th AIAA/CEAS Aeroacoustics 2022 Conference [START_REF] Palma | A Multi-Objective Optimization of a Wave-Packet Model Using Near-Field Subsonic Jet Data[END_REF], in which the optimization of the wavepacket involved a single Strouhal number. This paper reports an extensive analysis performed at multiple Strouhal numbers, namely 𝑆𝑡 𝐷 = 𝑓 𝐷/𝑈 𝑗 = 0.25, 0.5, 0.75 and 1 where f and 𝑈 𝑗 are the frequency and the nozzle exhaust jet velocity respectively, and for the 0 𝑡 ℎ azimuthal mode which has a relevant intermittency around the frequencies associated with the lower Strouhal numbers [START_REF] Camussi | Intermittent statistics of the 0-mode pressure fluctuations in the near field of Mach 0.9 circular jets at low and high Reynolds numbers[END_REF][START_REF] Micci | Intermittency and Stochastic Modeling of Low-and High-Reynolds-Number Compressible Jets[END_REF]. This has been confirmed by Cavalieri [START_REF] Cavalieri | Axisymmetric superdirectivity in subsonic jets[END_REF], who found a superdirective wavepacket to be consistent with the polar structure of the sound field for azimuthal Fourier modes 𝑚 = 0, 1 and 2 and 0.2 < 𝑆𝑡 𝐷 < 0.8 [START_REF] Albuquerque Maia | Two-point wavepacket modelling of jet noise[END_REF]. As suggested by Rodriguez et al. [START_REF] Rodríguez | Wavepacket models for subsonic twin jets using 3d parabolized stability equations[END_REF], at low and moderate Strouhal numbers, the 0 𝑡 ℎ mode is the most relevant one for the noise radiated by single jets. This is due to the more elongated structure of the corresponding wave-packet, the higher peak amplitudes, and slower radial decay.

For this reason, this mode can be considered representative of the whole signal in the analyzed domain, and being this the first work that considers near-field data in the optimization of the wavepacket, we take into account only the axisymmetric mode 𝑚 = 0. The optimization has been performed using a Multi-Objective Particle Swarm Optimization (PSO) algorithm [START_REF] Sierra | Improving PSO-Based Multi-objective Optimization Using Crowding, Mutation and ∈-Dominance[END_REF], originally introduced by Kennedy and Eberhart [START_REF] Kennedy | Particle swarm optimization (PSO)[END_REF], which is based on the social-behavior metaphor of a flock of birds or a swarm of bees searching for food, and belongs to the class of heuristic algorithms for evolutionary derivative-free global optimization.

The paper is structured as follows. Key information about the numerical simulation used to generate the database are presented in section II. In section III details about the wave-packet model are reported and the optimization algorithm.

Results are shown in section IV. Some concluding remarks are proposed in section V.

II. Numerical setup

The near-field of the isothermal round free jet at a Reynolds number 𝑅𝑒 𝐷 = 10 5 used for this paper has been computed by (LES). The nozzle exhaust jet Mach number has been fixed at 𝑀 = 0.9, with the nozzle-exhaust boundary-layer thickness set at 𝛿 𝑏𝑙 = 0.15𝑟 0 and the nozzle exit turbulence intensity at 9% (see [START_REF] Bogey | Acoustic tones in the near-nozzle region of jets: characteristics and variations between Mach numbers 0.5 and 2[END_REF] for details). The LES has been carried out using an in-house solver of the three-dimensional filtered compressible Navier-Stokes equations in cylindrical coordinates (r, 𝜃, x) based on low-dissipation and low-dispersion explicit schemes. The quality of the grid for the present jet LES has been assessed in previous papers [START_REF] Bogey | Grid sensitivity of flow field and noise of high-Reynolds-number jets computed by large-eddy simulation[END_REF]. Specifically, the grid contains approximately one billion points. Pressure has been recorded at several locations spanning a large near-field domain and gaining time-resolved signals, see references [START_REF] Bogey | Grid sensitivity of flow field and noise of high-Reynolds-number jets computed by large-eddy simulation[END_REF], and [START_REF] Bogey | Effects of nozzle-exit boundary-layer profile on the initial shear-layer instability, flow field and noise of subsonic jets[END_REF] for a description of the available data. In addition, the near-pressure field of this jet has been also investigated in [START_REF] Adam | Imprint of Vortical Structures on the Near-Field Pressure of a Turbulent Jet[END_REF]. Being the present study limited to the near-field domain, we consider arrays of virtual microphones parallel to the nozzle exhaust, positioned at r/D=1, 2, 2.5, and 3. Each array contains 1024 probes that cover a domain that spans between 𝑥=0 up to 𝑥/D=20. The data have been stored at a sampling frequency corresponding to 𝑆𝑡 𝐷 = 12.8, with a total of 3221 time snapshots. A representative one is shown in Fig. 1.

The LES dataset selected for the present work is over-resolved for the purpose, and less resolved simulations or, also, experimental data might be, in principle, used successfully. Nevertheless, we preferred a very well-assessed simulation with a very high spatial resolution for this first study on optimizing a wavepacket with near-field data. The original pressure signals are represented in terms of their azimuthal components through the azimuthal decomposition [START_REF] Michalke | On turbulence and noise of an axisymmetric shear flow[END_REF]. The Fourier coefficients are stored for the first four azimuthal modes that dominate the sound field for low polar angles. As aforementioned the wavepacket model presented in this paper has been carried out for the 0 𝑡 ℎ azimuthal mode, which is dominant for the noise generation at Strouhal numbers lower than 1 [START_REF] Cavalieri | Axisymmetric superdirectivity in subsonic jets[END_REF].

III. Wave-packet model

The noise source model used, i.e. the wave-packet model for the jet noise, has been introduced by Papamoschou in references [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF][START_REF] Papamoschou | Wavepacket modeling of the jet noise source[END_REF][START_REF] Papamoschou | Wavepacket Modeling of the Jet Noise Source[END_REF][START_REF] Huang | Numerical Study of Noise Shielding by Airframe Structures[END_REF], who in turn developed it from the works by Morris [START_REF] Morris | Jet noise prediction: Past, present and future[END_REF][START_REF] Morris | A Note on Noise Generation by Large Scale Turbulent Structures in Subsonic and Supersonic Jets[END_REF] and the previous ones by Tam

and Burton [START_REF] Tam | Sound generated by instability waves of supersonic flows. Part 2. Axisymmetric jets[END_REF], Crighton and Huerre [START_REF] Crighton | Shear-layer pressure fluctuations and superdirective acoustic sources[END_REF], and Avital et al. [START_REF] Avital | Mach Wave Radiation by Mixing Layers. Part I: Analysis of the Sound Field[END_REF]. The fundamental assumption at the basis of the model is that the peak noise radiation from the jet in the aft region is related to the large-scale coherent structures in the jet flow which can be modeled as instability waves at its boundary, growing and then decaying along the axial distance [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF]. In the model, the jet is substituted with a cylindrical surface, surrounding the original jet, radiating the pressure perturbation imposed on it. Applying the wave-packet ansatz, the pressure on the cylindrical surface at 𝑟 0 surrounding the jet is prescribed as

𝑝 𝑤 (𝑚, 𝑟 0 , 𝑥, 𝜃, 𝑡) = 𝑝 0 (𝑥)𝑒 -𝑖 𝜔𝑡+𝑖𝑚𝜃 (1)
where m is the azimuthal mode number, x denotes the axial coordinate, 𝜃 is the azimuthal angle, 𝜔 = 2𝜋 𝑓 is the pulsation. In the present study, the reference surface is taken at 𝑟 0 = 𝐷 and the wave-packet axial shape 𝑝 0 (𝑥) is given in the form [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF] 𝑝

0 (𝑥) = tanh (𝑥 -𝑥 0 ) 𝑝 1 𝑏 𝑝 1 1 1 -tanh (𝑥 -𝑥 0 ) 𝑝 2 𝑏 𝑝 2 2 𝑒 𝑖 𝛼( 𝑥-𝑥 0 ) (2)
The coordinate 𝑥 0 is used to locate the relative position between the origin of the wave-packet function and the nozzle exit. The signal growth is controlled by the parameters 𝑏 1 and 𝑝 1 , while 𝑏 2 and 𝑝 2 define its decaying rate. Following

Morris [START_REF] Morris | A Note on Noise Generation by Large Scale Turbulent Structures in Subsonic and Supersonic Jets[END_REF] and Papamoschou [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF], the solution in the linear regime (i.e., solution for the 3D wave equation in cylindrical polar coordinates) for an arbitrary radial distance 𝑟 ≥ 𝑟 0 can be evaluated as

𝑝 𝑤 (𝑚, 𝑟, 𝑥, 𝜃, 𝑡) = 1 2𝜋 𝑒 -𝑖 𝜔𝑡+𝑖𝑚𝜙 ∫ ∞ -∞ p0 (𝑘)
𝐻 (1) 𝑚 (𝜆𝑟) 𝐻 (1) 𝑚 (𝜆𝑟 0 )

𝑒 𝑖𝑘 𝑥 𝑑𝑘 with 𝜆 = 𝜔 𝑐 ∞ 2 -𝑘 2 1/2 , - 𝜋 2 < 𝑎𝑟𝑔(𝜆) < 𝜋 2 (3)
where p0 (𝑘) is the Fourier transform of 𝑝 0 (𝑥), 𝑐 ∞ is the speed of sound of the unperturbed flow, and 𝐻 (1) 𝑚 is the Hankel function of the first kind and order 𝑚. The model adopted is derived from the 3D wave equation in cylindrical coordinates, which solution is radially decaying as a combination of a Bessel of the first and second kind (i.e. a Hankel function of the first kind [START_REF] Morris | A Note on Noise Generation by Large Scale Turbulent Structures in Subsonic and Supersonic Jets[END_REF]). The pressure field generated by the wave-packet can be easily separated in its radiative and decaying components looking at the supersonic (|𝜔/𝑘 | ≥ 𝑐 ∞ ) and subsonic (|𝜔/𝑘 | < 𝑐 ∞ ) values of the phase speed, respectively.

In Papamoschou [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF], the parameters of the deterministic wave-packet were obtained through a numerical optimization aimed at matching the experimentally measured far field directivity of the jet, hence involving only the radiative part of the wave-packet. The tuned wave-packet was then employed as an equivalent noise source for the jet in BEM scattering calculations to predict the shielding effect from a thin plate. However, the interactions between the pressure perturbation generated by the wave-packet and the obstacle typically happen in the jet near-field. The wave-packet axial location was adjusted to match the near-field peak emission obtained from phased-array measurements to include some information on the acoustic near-field.

In this work, the same deterministic wave-packet model is used, but its parameters are defined from near-field data on co-axial lines at several radial distances from the jet axis, namely 𝑟/𝐷 = 1, 2, and 2.5. A multi-objective optimization procedure aims at matching the complete pressure fluctuation envelope from the model with the one from the numerical simulations for each of the considered lines. A generic unconstrained optimization problem consists of the research of the set of variables v that yields to a minimum of the 𝑁 𝐽 objective functions 𝐽 𝑛 (v, q) minimize/maximize [𝐽 𝑛 (v, q)] , 𝑛 = 1, ..., 𝑁 𝐽 and v ∈ D v with bounds 𝑣 𝐿 𝑚 ≤ 𝑣 𝑚 ≤ 𝑣 𝑈 𝑚 , 𝑚 = 1, ..., 𝑁 𝑣

where q is the vector of the parameters, v is the vector of the 𝑁 𝑣 design variables bounded by 𝑣 𝐿 𝑛 and 𝑣 𝑈 𝑛 in the design space D v , In the present application, v represents the vector collecting the wave-packet parameters v = 𝑝 1 , 𝑏 1 , 𝑝 2 , 𝑏 2 , 𝜔/(𝛼𝑈 𝑗 ), 𝑥 0 and 𝑁 𝐽 = 3 defines the three objective functions to be minimized, one for each 𝑟 𝑛 considered

𝐽 𝑛 (x, y) = √ ∫ | 𝑝 𝑛 (𝑟 𝑛 ) -𝑝 𝐿𝐸𝑆 (𝑟 𝑛 )| max (| 𝑝 𝐿𝐸𝑆 (𝑟 𝑛 )|) 2 𝑑𝑥 (5)
The objective functions represent the L2-norm of the difference between the pressure predicted by the wavepacket source model and the reference pressure from the LES over the axial extension of the considered lines, divided by the peak value from the reference curve for each radial distance to normalize the objective function values. Multiple

Strouhal numbers have been considered, optimizing the wavepacket source model separately for each value in the set 𝑆𝑡 𝐷 = 0.25, 0.5, 0.75 and 1, using pressure data from the numerical database for the dominant axisymmetric azimuthal mode. With the values of the three objective functions tending to zero simultaneously, the wave-packet model would perfectly trace the simulations for the considered radial distances. A solution reaching this goal would occupy the so-called utopia point in the codomain, which, as its name suggests, is not reachable for non-trivial multi-objectives problems in which the involved functions are even just partially conflicting. In these cases, including the one considered in this paper, the optimization process identifies a set of compromise solutions that identify an (approximated) Pareto front in the codomain. The defining property of solutions lying on a Pareto frontier is that moving from one to the other none of the objective functions can be improved in value without degrading some of the other. Hence, the Pareto front identifies all the, possibly infinite, non-dominated solutions of the multi-objective optimization problem, which is considered optimal in a Paretian sense, and equally good. Any choice of the preferred solution within this set is subjective, and to be made depending on the problem under analysis with a specific grade of arbitrariness [START_REF] Bragge | Bibliometric Analysis of Multiple Criteria Decision Making/Multiattribute Utility Theory[END_REF][START_REF] Martins | Engineering Design Optimization[END_REF].

IV. Results

Since the interest is primarily on matching the shape of the modelled pressure at several radial distances and the relative amplitudes among them rather than the absolute ones, the data from the simulations have been normalized

with respect to the maximum value at 𝑟 0 . For each 𝑆𝑡 𝐷 , a Multi-Objective Particle Swarm Optimization algorithm is employed to find the solutions minimizing all the objective functions together. This heuristic optimization algorithm introduced originally by Kennedy and Eberhart [START_REF] Kennedy | Particle swarm optimization (PSO)[END_REF] was extended to handle multiple objective functions by Coello et. al. [START_REF] Sierra | Improving PSO-Based Multi-objective Optimization Using Crowding, Mutation and ∈-Dominance[END_REF][START_REF] Coello | Handling multiple objectives with particle swarm optimization[END_REF]. The optimization has been performed with a Matlab implementation using a fixed budget of 500 iterations, with a swarm composed of 140 individuals, whose initial positions were randomly defined in the domain with uniform distribution,for a total time of about 200s per optimization on a single core. . The total computation time on a workstation with an Intel Xeon(R) CPU E5-2620 v4 @ 2.10GHz took about 200s per optimization on a single core. The subroutines might be better optimized to further reduce the computation time. At the present moment, the method is applicable to one azimuthal mode at a time, and when more than one is expected to give a significant contribution, the total noise source can be obtained with a linear combination. A possible future extension of the method might simultaneously optimise more than one wavepacket, each one with a different azimuthal mode. The drawback of this approach is that a good solution is expected to be more difficult to be found as the dimension of the domain increases (there will be four variables per wavepacket); this problem is typically referred to as "the course of dimensionality".

As anticipated in the previous section, when the objectives conflict, the solutions resulting from the minimization are optimal in a Paretian sense. The set of Pareto-optimal solutions lying on the Pareto front plotted in Fig. 2, has equal dignity in terms of minimization of the objective functions. Since it represents the set of the non-dominated solutions, moving along the front is not possible to improve one of the objective values without worsening at least one of the others. One of the techniques that may be employed to identify the preferred solution among the optima is to identify a ranking criterion [START_REF] Palma | Optimized metamaterials for enhanced noise shielding of innovative aircraft configurations[END_REF][START_REF] Bragge | Bibliometric Analysis of Multiple Criteria Decision Making/Multiattribute Utility Theory[END_REF][START_REF] Zakeri | Ranking based on optimal points multi-criteria decision-making method[END_REF][START_REF] Iemma | Aircraft sound quality as Pareto ranking criterion in multi-objective MDO[END_REF][START_REF] Iemma | Sound-Quality-Based Decision Making in Multiobjective Optimisation of Operations for Sustainable Airport Scenarios[END_REF] (also called Decision Maker algorithm), to be used as an added objective, evaluating the solutions performance on it and then selecting the solution resulting in the most suitable. Any selection criterion is valid in principle and may be used reasonably, from simple subjective preferences to more complex analyses of the results. In this study, a Pareto ranking criterion is formulated by combining the analysis of the performance of the optimized source model when predicting the pressure fluctuations over the 𝑛 + 1 line at 𝑟 𝑛+1 /𝐷 = 3 (a farther radial distance from the ones used during the optimization), and the distance of the solutions from the utopia point in the codomain of the problem. In particular, solutions are ordered on the basis of their 𝐽 𝑛+1 value, and all the solutions with a relative difference in their value under 100% with respect to the best one are included in a subset of optimal solutions; the one closest to the utopia point is taken as the preferred one. The only exception is the 𝑆𝑡 𝐷 = 0.5 case, in which the solution minimizing the error on the closest line 𝑟/𝐷 = 1 is selected. This kind of decision-maker (DM) algorithm can be classified in the class of the a posteriori articulation of preferences [START_REF] Zakeri | Ranking based on optimal points multi-criteria decision-making method[END_REF], implying that the DM's involvement starts posterior to the explicit revelation of "interesting" solutions.

In Fig. 2 the optimal solutions are highlighted by red circles on their respective three dimensional Pareto fronts. Each dot representing a solution is coloured on the basis of its fitness over the 𝑛 + 1 line. It can be seen that the Pareto ranking criterion tends to prefer solutions that privilege the results on 𝐽 1 and 𝐽 3 more than the performance on the second radial distance 𝐽 2 .

Figure 3 shows the shapes of the optimal wave-packets identified by the Pareto ranking criterion. The envelope of the complex pressure is represented in black, with the real and imaginary parts plotted with blue and red lines, respectively. The sound emission of a wave-packet source depends on both the spatial envelope and temporal growth and decay [START_REF] Crighton | Shear-layer pressure fluctuations and superdirective acoustic sources[END_REF][START_REF] Sandham | Sound radiation from exponentially growing and decaying surface waves[END_REF]. As expected, the wave-packet shapes strongly depend on the Strouhal number, which affects in particular the peak amplitude location. It can be seen that the number of spatial oscillations inside the envelope increase with the Strouhal number, as the spatial and temporal frequencies are linearly connected by the sound speed. a good match on the propagated information at 𝑟/𝐷 = 3, the wave-packet prediction capabilities would be comparable to an interpolation model. On the contrary, the agreement on the fourth line confirms that the optimized wave-packets correctly catch some of the jet noise source characteristics and the relative importance of the hydrodynamic and acoustic part of the pressure perturbation, since the wave-packets were not informed of the LES pressure fields for r/D>2.5.

𝑆𝑡 𝐷 r/D = 1 r/D=2 r/D=2. The mean difference between the wave-packet and the LES SPL curves is representative of the model's average error. The values for all radial distances and Strouhal numbers are reported in Tab. 1. They are lower than 3 dB for 75% of the analysed data. Some grade of conflict of the objective functions is somehow expected. The ideal jet noise source model would simultaneously fit the reference pressure curves at all distances, leading to a unique optimum solution instead of a Pareto front. However, it has to be reminded that the wave-packet model used in this study is a simplified interpretation of the actual noise production and propagation phenomena related to the jet. The line at 𝑟/𝐷 = 1, moreover, is partially immersed in the jet, due to the jet expansion angle, while the wave-packet model assumes the pressure propagation to higher radial distances to happen in a quiescent fluid. The use of the information on this line introduces an approximation in the model, which, however, needs to be informed with perturbations from the very proximity of the jet axis to predict also the hydrodynamic field.

V. Conclusion

For the first time, a multi-objective optimization of a wave-packet in the jet near-field is presented to predict the behaviour of the 0 𝑡 ℎ azimuthal mode. The results of the optimizations are found to provide a good agreement between the numerical reference data and the model in the tested Strouhal range 0.25 ≤ 𝑆𝑡 𝐷 ≤ 1. The optimizations have been performed for a wide range of Strouhal numbers where the 0 𝑡 ℎ mode is known to be an essential component in the whole jet pressure field, and for radial distances relevant for the jet-surface scattering phenomena in innovative aircraft configurations.

A Pareto front has been obtained as a solution of each optimization due to the objectives being partially conflicting, i.e. the model-simulation agreement at three different radial distances from the jet axis. The preferred solution on the front is then selected using a Pareto ranking criterion method, considering the wavepacket prediction over an extra line.

The optimized noise source model prediction can reproduce the LES data from the free jet with a mean error for each radial distance of the probe arrays within 3 dB for most cases and is suitable to be integrated into the BEM solver to evaluate of jet-surface aeroacoustic scattering. Specifically, the idea, which is currently under development, is to use this optimized noise source to generate the incident field over the scattering body (such as the wing pressure side for standard aircraft, or the trailing part of a BWB center body section), and then evaluate solution for the jet installed noise including the scattering and shielding effects from the surfaces. In the presented approach, being the wavepacket related to a free jet, in order to address different engine-airframe arrangements it is sufficient to adjust the evaluation of the incident field, changing only the relative positioning between the noise source and the scattering surfaces without modifying the wavepacket parameters. The process can be easily included in an optimization loop to find the solution minimizing, for example, the noise radiated towards the ground. The computational related to the acoustic simulation of each configuration can be significant, in particular for high frequency acoustic simulations, making the whole optimization process barely or even not affordable. To mitigate the effort required, some metamodeling techniques can be applied.

Among other techniques, adaptive radial basis functions (also in their stochastic version) and/or artificial neural networks may be used, which have been recently applied to the metamodelling of similar problems, both for shielding and jet noise prediction [START_REF] Burghignoli | Adaptive RBF with hyperparameter optimisation for aeroacoustic applications[END_REF][START_REF] Centracchio | Noise shielding surrogate models using dynamic artificial neural networks[END_REF]. Such surrogate models allow for very fast evaluations of the specific metrics they are built for, e.g. insertion loss at some observation points, dramatically reducing the optimization cost. The computational effort is substantially moved to the building of the metamodel, which must be trained with some (possibly costly) acoustic simulations, however it is expected to be lower than what a direct approach would require.

The optimized wave-packet capability to predict the solution at a higher radial distance shows that it captures at least some of the features of the modelled noise source, such as the relative contributions of the radiating and non radiating parts of the pressure perturbations. It has been evidenced how the probe line closer to the jet axis is partially immersed in the flow, which is somehow a limit of the model. However, the final aim of the work is to develop a jet noise source model able to predict the scattering effects in closely coupled installed jet configurations, where the flow grazes the scattering surfaces and the hydrodynamic part of the perturbation is relevant. The method is, in principle, applicable to any noise signal produced by jets in which the contribution from coherent structures is significant. Furthermore, the input for the wavepacket optimization is not restricted to numerical simulations, as experiment data is also eligible.

In any case, if we consider turbulent jets and the underlying hypotheses of the wavepacket modelling of jet noise are satisfied, the optimization procedure is expected to be able to find a set of parameters for the wavepacket, also for a Reynolds number typical of flight conditions (i.e. about two orders of magnitudes higher of what has been considered in this study). The method can also be coupled with existing empirical scaling law to estimate the SPL for different flow conditions, including Mach number variations [START_REF] Lighthill | On sound generated aerodynamically[END_REF][START_REF] Zaman | Power spectral density of subsonic jet noise[END_REF] and partially considering the effect of the Reynolds number [START_REF] Zaman | Power spectral density of subsonic jet noise[END_REF][START_REF] Meloni | Experimental investigation of jet-induced wall pressure fluctuations over a tangential flat plate at two Reynolds numbers[END_REF]. It is worth noting that two orders of magnitude of Reynolds number can involve a variation of the shape of the SPL spectrum, yielding in non-perfect effectiveness of the empirical laws at all the 𝑆𝑡 numbers. Further investigations that involve multimodal analyses including other azimuthal contributions and different turbulence levels are ongoing.

The multimodal approach will be needed to extend the presented model towards complex geometries, like Chevrons and Scarfed nozzle, in which jet flows are no longer characterized by axial symmetry, and hence higher azimuthal order modes need to be involved in the analysis.

  angular frequency in radians 𝑘 = 𝜔/𝑐 ∞ acoustic wave-number 𝑅𝑒 𝐷 = 𝜌𝑈𝐷/𝜇 nozzle exhaust Reynolds number 𝐻𝑒 𝑙 = 𝑘𝑙 Helmholtz number with characteristic lenght 𝑙 𝑆𝑡 𝐷 = 𝑓 𝐷/𝑈 Strouhal number 𝑈 𝑗 = nozzle exhaust jet velocity 𝑀 = 𝑈 𝑗 /𝑐 ∞ jet Mach number 𝛿 𝐵𝐿 = nozzle exhaust boundary layer 𝐽

Fig. 1

 1 Fig. 1 Snapshot in the (x,r) plane of the pressure signals. The black dashed lines represent the probe arrays.

Fig. 2

 2 Fig. 2 Codomains of the optimization problems. Non-dominated solutions of the multi-objective optimization for the analysed Strouhal numbers.

( a )

 a 𝑆𝑡 𝐷 = 0.25v opt = [1.0558, 6.03, 0.7017, 16.8122, 1.0112, 0.0001] (b) 𝑆𝑡 𝐷 = 0.5v opt = [1.0664, 6.8254, 2.4476, 15.7218, 1.0625, 0.0018] (c) 𝑆𝑡 𝐷 = 0.75v opt = [0.8120, 9.2314, 2.9113, 19.6413, 1.1268, 0.0122] (d) 𝑆𝑡 𝐷 = 1.0v opt = [1.1596, 4.7583, 1.0663, 20.0931, 1.1350, 0.0169]

Fig. 3

 3 Fig. 3 Shape of the optimized wave-packets and their parameters.

3 Fig. 4 3 Fig. 5 3 Fig. 6 3 Fig. 7

 34353637 Fig. 4 Solution selected by the Pareto ranking criterion for 𝑆𝑡 𝐷 = 0.25. Results for the radial distances used in the optimization (a)-(c) and for the validation line (d).

Figures 4 -

 4 Figures 4-7shows the sound pressure levels (SPL) predicted by the LES simulations and by the optimised wave-packets at the four 𝑆𝑡 𝐷 and radial distances considered. The optimized wave-packets selected with the Pareto ranking criterion give satisfactory agreement with the LES SPL on the lines located at 𝑟/𝐷 =1, 2 and 2.5 used in the optimization for all the Strouhal analysed in Figures4-7 (a-c). A good prediction of the pressure perturbation by the wave-packet model is also observed at 𝑟/𝐷 = 3 in Figures4-7 (d). It should be noted that the propagation of the pressure perturbation considered by the wave-packet model assumes that the radiating surface is outside of the jet plume, while the first line at 𝑟/𝐷 = 1 is immersed in the flow for large part of its axial extension. It is necessary to use this kind of approximation of the model to include the near-field hydrodynamic contribution inside the optimized source, which is important to predict the scattering noise generated by a jet installed close to a solid boundary. Without

Table 1 Mean error in dB on each virtual probe line between predicted and LES levels for each Strouhal number.

 1

	5 r/D=3

Acknowledgments

C. Bogey was partially supported by the LABEX CeLyA (ANR-10-LABX-0060/ANR-16-IDEX-0005). The numerical data analyzed in this work were obtained using the HPC resources of PMCS2I (Pôle de Modélisation et de Calcul en Sciences de l'Ingénieur et de l'Information) of Ecole Centrale de Lyon, and the resources of IDRIS (Institut du Développement et des Ressources en Informatique Scientifique) under the allocation 2021-2a0204 made by GENCI (Grand Equipement National de Calcul Intensif).