Johannes Karges

Olivier Blacque

Hui Chao

Gilles Gasser

Polymeric Bis(dipyrrinato) Zinc(II) Nanoparticles as Selective Imaging Probes for Lysosomes of Cancer Cells

Keywords: Bioinorganic Chemistry, Imaging, Inorganic Chemical Biology, Nanoparticles, Targeting

Fluorescence imaging is a powerful tool in biomedical research. It has been frequently used to uncover or better understand physiological mechanisms in disease-related processes such as cancer. The majority of chromophores used for imaging are based on a 4,4-difluoro-4-bora-3a,4a-diaza-s-indacene (BODIPY) scaffold. However, their applications are limited due to their poor water solubility as well as poor cancer cell selectivity. To circumvent these drawbacks, we present herein the use of bis(dipyrrinato)zinc(II) complexes. As this class of compounds is associated with a quenching effect of the excited state in water, the lead compound of this study (3) was encapsulated in a polymer matrix with biotin as a targeting moiety (3-NP). This encapsulation improved the water solubility, overcame the quenching effects in water as well as allowed selective accumulation in the lysosomes with a bright fluorescence signal in monolayer cells as well as 3D multicellular tumour spheroids (MCTS). As benefit from the biotin targeting moiety, the nanoparticles were majorly taken up by the sodium dependent multivitamin transporter (SMVT) which is overexpressed in various cancers cells and selectively accumulated in cancerous cells over non-cancerous cells.

INTRODUCTION

The ability to use imaging techniques either on a microscopic cellular level or even on humans has strongly influenced modern medical research. Among others, fluorescence imaging has received increasing attention over the last decades as this method was found to efficiently monitor physiological processes. Of special interest is the monitoring of disease-related organisms or cells. [START_REF] Rao | Fluorescence imaging in vivo: recent advances[END_REF][START_REF] Leblond | Pre-clinical whole-body fluorescence imaging: Review of instruments, methods and applications[END_REF] Among the different classes of imaging compounds investigated, chromophores based on a 4,4-difluoro-4-bora-3a,4a-diaza-s-indacene (BODIPY) scaffold are of special interest. These compounds are associated with a strong UV/Vis absorbance, chemical and photostability as well as sharp and strong fluorescence. For these reasons, these dyes are currently been used for numerous cellular applications including chemosensors, fluorescent switches as well as in the labelling of protein or DNA. [START_REF] Loudet | BODIPY Dyes and Their Derivatives: Syntheses and Spectroscopic Properties[END_REF][START_REF] Benniston | Lighting the way ahead with boron dipyrromethene (Bodipy) dyes[END_REF][START_REF] Wagner | Boron-dipyrromethene dyes for incorporation in synthetic multi-pigment light-harvesting arrays[END_REF][START_REF] Yee | A cell-permeable, activity-based probe for protein and lipid kinases[END_REF][START_REF] Karolin | Fluorescence and Absorption Spectroscopic Properties of Dipyrrometheneboron Difluoride (BODIPY) Derivatives in Liquids, Lipid Membranes, and Proteins[END_REF][START_REF] Boens | Fluorescent indicators based on BODIPY[END_REF][START_REF] Rezende | A Review of the Synthetic Strategies for the Development of BODIPY Dyes for Conjugation with Proteins[END_REF] However, the applications of these dyes are generally limited by their poor water solubility and cancer cell selectivity. To overcome these limitations, different dyes have been linked to a targeting moiety or formulations have been investigated to promote selective drug delivery. [START_REF] Villemin | Polymer encapsulation of ruthenium complexes for biological and medicinal applications[END_REF][START_REF] Larson | Polymeric Conjugates for Drug Delivery[END_REF][START_REF] Steichen | A review of current nanoparticle and targeting moieties for the delivery of cancer therapeutics[END_REF] Beside boron complexes, dipyrrin ligands are also able to coordinate other metal ions such as Ni(II), Zn(II), Cu(II), Fe(III), Rh(III) and Co(III). [START_REF] Sakamoto | New aspects in bis and tris (dipyrrinato) metal complexes: bright luminescence, self-assembled nanoarchitectures, and materials applications[END_REF][START_REF] Baudron | Luminescent dipyrrin based metal complexes[END_REF][START_REF] Wood | Advances in the Chemistry of Dipyrrins and Their Complexes[END_REF] During the last decade, bis(dipyrrinato)zinc(II) complexes, as an underrepresented class of compounds, have received increasing attention as fluorophores (see Figure 1 for an overview of investigated bis(dipyrrinato)zinc(II) complexes). [START_REF] Sakamoto | New aspects in bis and tris (dipyrrinato) metal complexes: bright luminescence, self-assembled nanoarchitectures, and materials applications[END_REF][START_REF] Sazanovich | Structural Control of the Excited-State Dynamics of Bis(dipyrrinato)zinc Complexes: Self-Assembling Chromophores for Light-Harvesting Architectures[END_REF][START_REF] Trinh | Symmetry-Breaking Charge Transfer of Visible Light Absorbing Systems: Zinc Dipyrrins[END_REF][START_REF] Kusaka | An extremely bright heteroleptic bis (dipyrrinato) zinc (II) complex[END_REF][START_REF] Sakamoto | Fluorescent azadipyrrinato zinc (ii) complex: hybridisation with a dipyrrinato ligand[END_REF][START_REF] Lee | Enforced Effects of Side Group Substitution Position on Luminescence Properties; Synthesis of Bis(dipyrrinato)zinc Complex Derivatives[END_REF][START_REF] Sakamoto | Dissymmetric Bis(dipyrrinato)zinc(II) Complexes: Rich Variety and Bright Red to Near-Infrared Luminescence with a Large Pseudo-Stokes Shift[END_REF][22] Unfortunately, the fluorescence quantum yields (ΦF) of these compounds reported so far remain inferior to the structurally related BODIPY derivatives. Therefore, current recent efforts are made towards improving the luminescent properties of bis(dipyrrinato)zinc(II) complexes. It was previously demonstrated that the steric hindrance of the peripheral aryl group of the dipyrrinato scaffold with methyl groups transformed the compounds from a weak emitter to a strong fluorescent chromophore. [START_REF] Sazanovich | Structural Control of the Excited-State Dynamics of Bis(dipyrrinato)zinc Complexes: Self-Assembling Chromophores for Light-Harvesting Architectures[END_REF] Further studies confirmed this observation with the introduction of peripheral 1-and 2-naphthyl groups. [START_REF] Lee | Enforced Effects of Side Group Substitution Position on Luminescence Properties; Synthesis of Bis(dipyrrinato)zinc Complex Derivatives[END_REF] Despite current research efforts, the quenching of the excited state of this class of compounds in polar solvents remains a serious problem, limiting their biological applications. This quenching effect was proposed to be caused by the formation of a non-emissive symmetry-breaking charge transfer state from the emissive excited state, which is not formed in non-polar solvents (Figure 2). [START_REF] Sakamoto | New aspects in bis and tris (dipyrrinato) metal complexes: bright luminescence, self-assembled nanoarchitectures, and materials applications[END_REF][START_REF] Trinh | Symmetry-Breaking Charge Transfer of Visible Light Absorbing Systems: Zinc Dipyrrins[END_REF] Details on the mechanism and the influence of the solvent polarity have been previously published. 17 Overview of reported bis(dipyrrinato)zinc(II) complexes. [START_REF] Sakamoto | New aspects in bis and tris (dipyrrinato) metal complexes: bright luminescence, self-assembled nanoarchitectures, and materials applications[END_REF][START_REF] Sazanovich | Structural Control of the Excited-State Dynamics of Bis(dipyrrinato)zinc Complexes: Self-Assembling Chromophores for Light-Harvesting Architectures[END_REF][START_REF] Trinh | Symmetry-Breaking Charge Transfer of Visible Light Absorbing Systems: Zinc Dipyrrins[END_REF][START_REF] Kusaka | An extremely bright heteroleptic bis (dipyrrinato) zinc (II) complex[END_REF][START_REF] Sakamoto | Fluorescent azadipyrrinato zinc (ii) complex: hybridisation with a dipyrrinato ligand[END_REF][START_REF] Lee | Enforced Effects of Side Group Substitution Position on Luminescence Properties; Synthesis of Bis(dipyrrinato)zinc Complex Derivatives[END_REF][START_REF] Sakamoto | Dissymmetric Bis(dipyrrinato)zinc(II) Complexes: Rich Variety and Bright Red to Near-Infrared Luminescence with a Large Pseudo-Stokes Shift[END_REF][22] Figure 2. Simplified Jablonski diagram demonstrating the solvent dependency of bis(dipyrrinato)zinc(II) complexes. [START_REF] Sakamoto | New aspects in bis and tris (dipyrrinato) metal complexes: bright luminescence, self-assembled nanoarchitectures, and materials applications[END_REF][START_REF] Trinh | Symmetry-Breaking Charge Transfer of Visible Light Absorbing Systems: Zinc Dipyrrins[END_REF] Keeping this in mind, we have designed homoleptic bis(dipyrrinato)zinc(II) complexes with different aryl groups in position 8 of the dipyrrin scaffold. Herein, we present the synthesis, in-depth photophysical and biological investigation of a series of homoleptic bis(dipyrrinato)zinc(II) complexes 1-4 (Figure 3) as imaging agents. Complexes 3 and 4

were previously reported in a study describing the photophysical properties of bis(dipyrrinato)zinc(II) complexes. [START_REF] Trinh | Symmetry-Breaking Charge Transfer of Visible Light Absorbing Systems: Zinc Dipyrrins[END_REF]22 In this work, the ability of these compounds to be used as imaging probes is investigated. The compounds were found to selectively accumulate in the cytoplasm, while being taken up through passive diffusion. These complexes are highly fluorescent in toluene but undergo quenching in a polar environment. To overcome this limitation as well as their poor water solubility, the quenching effects of the excited state in water and to make the compounds selective for cancerous cells, the lead compound of this study, namely complex 3, was encapsulated inside a polymeric matrix with a biotin end group to form nanoparticles (3-NP). Biotin is a compound of the vitamin B family, which is majorly internalised by the sodium dependent multivitamin transporter (SMVT). Since this receptor is overexpressed in various cancers and the demand for biotin in rapidly growing tumours is higher than in normal tissue, the conjugation of an active molecule to biotin offers the possibility for specific cancer cell targeting. [START_REF] Maiti | Biotin conjugated organic molecules and proteins for cancer therapy: A review[END_REF][24][START_REF] Roth | Biotin in clinical medicine-a review[END_REF] The nanoformulated complex was found to be highly water soluble and fluorescent. The particles selectively accumulated in the lysosomes as shown by the bright fluorescence signal observed in this organelle as well as being highly luminescent in 3D multicellular tumour spheroids (MCTS). Importantly, the particles benefit from the biotin targeting as demonstrated by a selective accumulation in transfected cancerous cells in comparison to non-transfected non-cancerous human lung cells. To the best of our knowledge, this is the first report of bis(dipyrrinato)zinc(II) complexes, which are highly fluorescent in an aqueous environment and which are selective for cancer cells.

EXPERIMENTAL SECTION

Instrumentation and Methods

1 H-and [START_REF] Sakamoto | New aspects in bis and tris (dipyrrinato) metal complexes: bright luminescence, self-assembled nanoarchitectures, and materials applications[END_REF] C-NMR spectra were recorded on a 400 MHz NMR spectrometer (Bruker). Chemical shifts (δ) are reported in parts per million (ppm) referenced to tetramethylsilane (δ 0.00) ppm using the residual proton solvent peaks as internal standards. Coupling constants (J) are reported in Hertz (Hz) and the multiplicity is abbreviated, as follows: s (singlet), d (doublet), and m (multiplet). Electrospray Ionization-Mass Spectrometry (ESI-MS) experiments were carried out using a LTQ-Orbitrap XL from Thermo Scientific (Thermo Fisher Scientific) and operated in positive ionization mode, with a spray voltage at 3.6 kV. No sheath and auxiliary gas was used. Applied voltages were 40 and 100 V for the ion transfer capillary and the tube lens, respectively. The ion transfer capillary was held at 275 °C. Detection was achieved in the Orbitrap with a resolution set to 100,000 (at m/z 400) and an m/z range between 150 and 2000 in profile mode. Spectrum was analyzed using the acquisition software XCalibur 2.1 (Thermo Fisher Scientific). The automatic gain control (AGC) allowed for the accumulation of up to 2 * 10 5 ions for Fourier Transform Mass Spectrometry (FTMS) scans, maximum injection time was set to 300 ms and 1 µs scan was acquired. 10 µL was injected using a Thermo Finnigan Surveyor HPLC system (Thermo Fisher Scientific) with a continuous infusion of methanol at 100 µLꞏmin-1. Elemental microanalyses were performed on a Thermo Flash 2000 elemental analyzer.

Materials

Synthesis

General synthetic procedure for the synthesis of the dipyrrin ligand:

The aldehyde (2 mmol) and 2, 4-dimethylmethylpyrrole (5 mmol) were dissolved in a dichloromethane solution (40 mL). Trifluoroacetic acid (100 L) was added and the mixture stirred at room temperature overnight under nitrogen atmosphere. Tetrachlor-p-benzochinon (2 mmol) was added and the reaction mixture stirred another 2 h. After this time, the solution was filtered and the solvent evaporated. The crude product was purified by column chromatography on neutral alumina with a gradient of dichloromethane/hexane. The fractions containing the product were united and the solvent was removed. The solid was washed with Et2O and dried.

Yield: 50-70%.

General synthetic procedure for the complexation with Zn:

The dipyrrin ligand (1 mmol) was dissolved in a 3:2:3 butanol/ethanol/H2O (200 mL) mixture to which Zn(OAc)2 * 2 H2O (0.4 mmol) and Et3N (4 mmol) was added. The reaction mixture was heated at reflux overnight under nitrogen atmosphere. After completion of the reaction, the solvent was removed. The crude product was purified by column chromatography on silica with a gradient of dichloromethane/pentane. The fractions containing the product were united and the solvent was removed. The solid was washed with Et2O and dried. Yield: 45-63%.

Bis((Z)-2-((3,5-dimethyl-2H-pyrrol-2-ylidene)(phenyl)methyl)-3,5-dimethyl-1H- pyrrole)zinc(II) 1: 1 H-NMR (CD2Cl2,

X-ray crystallography

X-ray single-crystal data were collected at 160(1) K on Rigaku OD diffractometers with an Oxford liquid-nitrogen Cryostream cooler: a XtaLAB Synergy Dualflex (Pilatus 200K detector) diffractometer for dipyrrin_1, 2 and 4, and a Xcalibur (CCD Ruby detector) diffractometer for 3. A single wavelength X-ray source from a micro-focus sealed X-ray tube were used with the Cu Kα radiation (λ= 1.54184 Å) 26 for the analyses of dipyrrin_1 and 4, and with the Mo Kα radiation (λ= 0.71073 Å) 26 for the analysis of 2 and 3. The selected single crystals were mounted using polybutene oil on a loop fixed on a goniometer head and transferred to the diffractometer.

Pre-experiments, data collections, data reductions and analytical absorption corrections [START_REF] Clark | The analytical calculation of absorption in multifaceted crystals[END_REF] were performed with the program suite CrysAlisPro. 28 Using Olex2, [START_REF] Dolomanov | OLEX2: a complete structure solution, refinement and analysis program[END_REF] the structures were solved with the SHELXT [START_REF] Sheldrick | SHELXT-Integrated space-group and crystal-structure determination[END_REF]

Spectroscopic measurements

The absorption of the samples was recorded on a Lambda 850 UV/VIS spectrometer (PerkinElmer) at room temperature. Emission spectra were recorded on a LS 55 fluorescence spectrometer (PerkinElmer) at room temperature. For the determination of the luminescence quantum yield, the samples were prepared in Toluene solution with an absorbance of 0.1 at 500 nm. The luminescence quantum yields were calculated by comparison with the reference Rhodamine B in Ethanol (Φem=0.5) [START_REF] Brouwer | Standards for photoluminescence quantum yield measurements in solution (IUPAC Technical Report)[END_REF] applying the following formula: The stability of the complex in Toluene/H2O/DMEM (2% DMSO, v%) was investigated by UV/Vis spectroscopy. The complex was dissolved with about 0.5 absorption at its absorption maximum and stored at room temperature in the dark. The absorption spectrum from 210-700 nm was recorded with a SpectraMax M2 Microplate Reader (Molecular Devices) after each time interval (0, 1, 2, 4, 8, 12, 24, 48 h) and compared.

Photostability

The photostability of the complex in Toluene (2% DMSO, v%) was investigated by UV/Vis spectroscopy. The complex was dissolved with about 0.5 absorption at its absorption maximum and exposed to a constant LED irradiation 510 nm (4.2 mW/cm 2 ) with an Atlas Photonics LUMOS BIO irradiator. The absorption spectrum from 210-700 nm was recorded with a SpectraMax M2 Microplate Reader (Molecular Devices) after each time interval (0, 5, 10, 15, 20, 25, 30 min) and compared.

Distribution coefficient

The lipophilicity of a compound was determined by measuring its distribution coefficient between the PBS and Octanol phase by using the "shake-flask" method. For this technique, the used phases were previously saturated in each other. The compound was dissolved in the phase (A) with its major presence with an absorbance of about 0.5 at 500 nm. This solution was then mixed with an equal volume of the other phase (B) at 80 rpm for 8 h with an Invitrogen sample mixer and then equilibrated overnight. The phase A was then carefully separated from phase B.

The amount of the compound before and after the sample mixing was determined by UV/Vis spectroscopy at 500 nm using a SpectraMax M2 Microplate Reader (Molecular Devices). The evaluation of the complexes was repeated three times and the ratio between the organic and aqueous phase calculated.

Cell culture

Human cervical carcinoma (HeLa), adenocarcinomic human alveolar basal epithelial cells (A549) human lung fibroblasts (HLF) cells were cultured using DMEM media and retinal pigment epithelium (RPE-1) cells using DMEM/F-12 with addition of 10% FBS and 1%

penstrep. The cells were cultivated and maintained at 37 °C in a cell culture incubator at 37 °C with 5% CO2 atmosphere. Before an experiment, the cells were passaged three times.

Cellular uptake

The cellular uptake of the complex was investigated by the determination of the Zn content inside the cells. The complex with a final concentration of 25 μM (2% DMSO, v%) was incubated at 37 °C on a cell culture dish with a density of ca. 4-6 . 10 6 cells in 10 mL of media.

After this time, the media was removed and the cells washed with cell media. The cells were trypsinised, harvested, centrifuged and resuspended. The number of cells on each dish was accurately counted. Each sample was the digested using a 60% HNO3 solution for three days.

Each sample was diluted to solution of 2% HNO3 in water. The Zn content was determined using an ICP-MS apparatus and comparing the results with the Zn references. The Zn content was then associated with the number of cells.

Cellular uptake mechanism

The mechanism of the cellular uptake was investigated by systematic inhibition of different uptake pathways and afterwards determination of the amount of Zn inside the cells via ICP-MS. For the experiment, 1 . 10 6 cells were pretreated with the corresponding inhibitor.

Control: The cells were incubated with the complex (25 μM, 2% DMSO, v%) for 1 h at 37°C.

After this time, the cells were washed with PBS. The cells were detached with trypsin, harvested, centrifuged and resuspended. The number of cells on each dish was accurately counted. Each sample was the digested using a 60% HNO3 solution for three days. Each sample was diluted to solution of 2% HNO3 in water. The Zn content was determined using an ICP-MS apparatus and comparing the results with the Zn references. The Zn content was then associated with the number of cells.

Low temperature: The cells were incubated at 4°C for 1 h. The cells were further incubated with the complex (25 μM, 2% DMSO, v%) for 1 h at 4°C. After this time, the cells were washed with PBS. The cells were detached with trypsin, harvested, centrifuged and resuspended. The number of cells on each dish was accurately counted. Each sample was the digested using a 60% HNO3 solution for three days. Each sample was diluted to solution of 2% HNO3 in water.

The Zn content was determined using an ICP-MS apparatus and comparing the results with the Zn references. The Zn content was then associated with the number of cells.

Metabolic Inhibition: The cells were incubated with 2-Deoxy-D-glucose (50 mM) and oligomycin (5 μM) for 1h. After the preincubation, the cells were washed with PBS and further incubated with the complex (25 μM, 2% DMSO, v%) for 1 h at 37°C. After this time, the cells were washed with PBS. The cells were detached with trypsin, harvested, centrifuged and resuspended. The number of cells on each dish was accurately counted. Each sample was the digested using a 60% HNO3 solution for three days. Each sample was diluted to solution of 2% HNO3 in water. The Zn content was determined using an ICP-MS apparatus and comparing the results with the Zn references. The Zn content was then associated with the number of cells.

Endocytic inhibition: The cells were incubated with NH4Cl (50 mM) or chloroquine (100 μM) for 1h. After the preincubation, the cells were washed with PBS and further incubated with the complex (25 μM, 2% DMSO, v%) for 1 h at 37°C. After this time, the cells were washed with PBS. The cells were detached with trypsin, harvested, centrifuged and resuspended. The number of cells on each dish was accurately counted. Each sample was the digested using a 60% HNO3 solution for three days. Each sample was diluted to solution of 2% HNO3 in water.

The Zn content was determined using an ICP-MS apparatus and comparing the results with the Zn references. The Zn content was then associated with the number of cells.

Cation transporter inhibition: The cells were incubated with tetraethylammonium chloride (1 mM) for 1h. After the preincubation, the cells were washed with PBS and further incubated with the complex (25 μM, 2% DMSO, v%) for 1 h at 37°C. After this time, the cells were washed with PBS. The cells were detached with trypsin, harvested, centrifuged and resuspended. The number of cells on each dish was accurately counted. Each sample was the digested using a 60% HNO3 solution for three days. Each sample was diluted to solution of 2% HNO3 in water. The Zn content was determined using an ICP-MS apparatus and comparing the results with the Zn references. The Zn content was then associated with the number of cells.

Sodium-dependent multivitamin transporter inhibition: The cells were incubated with pantothenic acid (50 μM) or lipoic acid (50 μM) for 1h. After the preincubation, the cells were washed with PBS and further incubated with the complex (25 μM, 2% DMSO, v%) for 1 h at 37°C. After this time, the cells were washed with PBS. The cells were detached with trypsin, harvested, centrifuged and resuspended. The number of cells on each dish was accurately counted. Each sample was the digested using a 60% HNO3 solution for three days. Each sample was diluted to solution of 2% HNO3 in water. The Zn content was determined using an ICP-MS apparatus and comparing the results with the Zn references. The Zn content was then associated with the number of cells.

Intracellular distribution by confocal luminescence imaging

The co-localisation of the complex was determined under usage of its luminescence properties.

Intracellular distribution by ICP-MS

The co-localisation of the complex was determined by measuring the Zn content inside the cell via ICP-MS. 10 . 10 6 cells were incubated with the complex (25 μM, 2% DMSO, v%) for 4 h at 37°C in the dark. After this time, the cells were detached with trypsin and harvested. The number of cells was accurately counted. The amount was equally divided into four portions. In the first portion, the nucleus was extracted using a nucleus extraction kit (Sangon Biotech) following manufacturer recommended protocol; in the second portion, the mitochondria was extracted using a mitochondria extraction kit (Sangon Biotech) following the manufacturer recommended protocol and in the third portion, the lysosome was extracted using a lysosome extraction kit (GenMed Scientific) following manufacturer recommended protocol. Using the fourth portion, the cytoplasm was extracted. For this, the cells were detached with trypsin, harvested and centrifuged. The cell pellet was resuspended in lysis buffer and the cells were lysed. The cell organelles were separated using a vacuum ultracentrifuge (Optima MAX-XP ultracentrifuge, Beckman Coulter) for 150 min at 200000 g at 4 °C. The supernatant solution was separated. Each sample was the digested using a 60% HNO3 solution for three days and was diluted to a solution of 2% HNO3 in water. The Zn content was determined using an ICP-MS apparatus and comparing the results with the Zn references. The Zn content was then associated with the number of cells.

(Photo-)cytotoxicity on 2D cell monolayers

The cytotoxicity of the complexs was assessed by measuring cell viability using a fluorometric resazurin assay. The cultivated cells were seeded in triplicates in 96 well plates with a density of 4000 cells per well in 100 μL of media. After 24 h, the medium was removed and the cells were treated with increasing concentrations of the complex diluted in cell media achieving a total volume of 200 μL. The cells were incubated with the complex for 4 h. After this time, the media was removed and replaced with 200 μL of fresh medium. For the phototoxicity studies, the cells were exposed to light with an Atlas Photonics LUMOS BIO irradiator. Each well was constantly illuminated with either a 510 nm or 540 nm irradiation. During this time, the temperature was maintained constantly at 37 °C. The cells were grown in the incubator for additional 44 h. For the determination of the dark cytotoxicity, the cells were not irradiated and after the medium exchange directly incubated for 44 h. After this time, the medium was replaced with fresh medium containing resazurin with a final concentration of 0.2 mg/mL. After 4 h incubation, the amount of the fluorescent product resorufin was determined upon excitation at 540 nm and measurement its emission at 590 nm using a SpectraMax M2 Microplate Reader (Molecular Devices). The obtained data was analyzed with the GraphPad Prism software.

Generation and analysis of 3D multicellular tumor spheroids (MCTS)

A suspension of 0.75% agarose in PBS buffer was heated inside a high-pressure autoclave. The hot emulsion was transferred into wells (50 μL per well) of a 96 cell culture well plate. The plates were exposed for 3h to UV irradiation to ensure the sterility and allow the agarose solution to cool down. After this time, the agarose was overlayed with a HeLa cell suspension at a density of 3000 cells per well in 150 μL of media. The MCTS were cultivated and maintained at 37 °C in a cell culture incubator at 37 °C with 5% CO2 atmosphere. The culture media was replaced every two days. Within two-three days MCTs were formed from the cell suspension. The formation as well as integrity and diameter of the MCTs was monitored by an Axio Observer Z1 (Carl Zeiss) phase contrast microscope. The luminescence images along the z-axis were captured by (λex = 514 nm, λem = 530 -610 nm) excitation in the z-stack mode with a an LSM 880 (Carl Zeiss) laser scanning confocal microscope equipped with a GaAsP detector.

Preparation of particles

A solution of 3 mg of 3 in 0.5 mL DCM was added to a solution of 6 mg of 1,2-distearoyl-snglycero-3-phosphoethanolamine-N-[biotin(polyethylene glycol)-2000] ammonium salt (DSPE-PEG2000-biotin) in 19.5 mL H2O. The solution was further treated with ultrasonic pulses with a Scientz -II D ultrasonic homogenizer using a 2 x 10 min method (tsonication = 2 s, Power = 15%, tbreak = 1 s) while keeping the sample constantly at 25 °C. After this, the DCM was removed by evaporation at 50 °C. The mixture was filtered through a 0.2 μm syringe filter to remove large aggregates. After that a clear transparent solution in H2O was obtained. The amount of encapsulated complex was determined by ICP-MS as 0.129 mg/mL (Yield: 86%).

Release kinetics of the particles

The release kinetics of the particles were investigated by dialysis against 1x PBS at physiological pH (7.4) and low pH (5.0) at 37 °C. The particles (200 μL) were added to into a dialysis bag and placed in the PBS bath with gentle shaking. PBS was changed every 12 h. At various time points (1h, 2h, 4h, 12h, 24h, 48h) the dialysis units were removed and the amount of compound determined by ICP-MS.

Selectivity assay between cancerous and non-cancerous cells

Adenocarcinomic human alveolar basal epithelial cells (A549) were transfected with the fluorescence protein mCherry using a transfection kit (Beyotime) and following the manufacturer recommended protocol. Not successfully transfected cells were removed using the antibiotic G-418. The cells were mixed with an equal amout of non-cancerous human lung fibroblasts (HLF) cells and allowed to adhere overnight. The cells were incubated with the complex (10 μM) for 4 h at 37°C in the dark. After this time, the cells were washed with PBS.

Confocal images were taken with a LSM 880 (Carl Zeiss) laser scanning confocal microscope equipped with a GaAsP detector. The investigated Zn complexes (λex = 514 nm, λem = 530 -610 nm) and the fluorescence protein mCherry (λex = 633 nm, λem = 650 -720 nm) were detected under usage of their luminescence properties. these structural parameters, we can assume that there is no significant influence from the substituent of the dipyrrinato ligand (dimethylphenyl, trimethylphenyl and anthracene) on the coordination geometry of the metal center.

RESULTS AND DISCUSSION

Photophysical Characterisation of 1-4

The photophysical properties (Table S3) of 1-4 were investigated to evaluate their potential as imaging probes. As expected from previous works on bis(dipyrrinato)zinc(II) complexes, [START_REF] Sakamoto | New aspects in bis and tris (dipyrrinato) metal complexes: bright luminescence, self-assembled nanoarchitectures, and materials applications[END_REF][START_REF] Sazanovich | Structural Control of the Excited-State Dynamics of Bis(dipyrrinato)zinc Complexes: Self-Assembling Chromophores for Light-Harvesting Architectures[END_REF][START_REF] Trinh | Symmetry-Breaking Charge Transfer of Visible Light Absorbing Systems: Zinc Dipyrrins[END_REF][START_REF] Kusaka | An extremely bright heteroleptic bis (dipyrrinato) zinc (II) complex[END_REF][START_REF] Sakamoto | Fluorescent azadipyrrinato zinc (ii) complex: hybridisation with a dipyrrinato ligand[END_REF] these compounds have a strong and sharp absorption at 493-495 nm. The different aryl groups in position 8 of the complex did neither cause a significant blue or red shift. The maximum of the emission signal was measured to be between 505-520 nm, resulting in an overlay between absorption and emission spectra (Figure 4a) and a small Stokes shift of 12-25 nm. As documented for various BODIPY complexes and other bis(dipyrrinato)metal complexes, [START_REF] Loudet | BODIPY Dyes and Their Derivatives: Syntheses and Spectroscopic Properties[END_REF][START_REF] Sakamoto | New aspects in bis and tris (dipyrrinato) metal complexes: bright luminescence, self-assembled nanoarchitectures, and materials applications[END_REF][START_REF] Sazanovich | Structural Control of the Excited-State Dynamics of Bis(dipyrrinato)zinc Complexes: Self-Assembling Chromophores for Light-Harvesting Architectures[END_REF] the complexes 2-4, which were designed with a steric hinderance of the rotation of the aryl groups, were found to have highly increased fluorescence quantum yields (ΦF) in comparison to the unsubstituted, model compound 1. As the compound with the strongest fluorescence signal, 3 was identified as the lead compound of the series.

Interestingly, the ΦF values of the compounds 1-4 vary highly based on the solvent employed during the measurements (Figure 4b). Fluorescence was found to be very intense in non-polar solvents while it was quenched in polar solvents. Recent investigations have demonstrated that this effect was caused by the formation of a nonemissive symmetry-breaking charge transfer state, which is only formed in polar solvents. [START_REF] Trinh | Symmetry-Breaking Charge Transfer of Visible Light Absorbing Systems: Zinc Dipyrrins[END_REF] Worthy of note, commercially available dyes were found to have much higher ΦF values than the compounds investigated in this study while being relatively insensitive to the polarity of the solvent.

Stability of 1-4

As an important parameter for an imaging agent, its stability in different solvents as well as upon light exposure was evaluated, as previous investigations showed that this could be critical for metal complexes. [START_REF] Basu | Investigation of photo-activation on Ruthenium(II)-arene complexes for the discovery of potential selective cytotoxic agents[END_REF][START_REF] Renfrew | Towards Light Activated Ruthenium-Arene (RAPTA-type) Prodrug Candidates[END_REF] The complexes were incubated in toluene, water and cell culture media (DMEM) and the corresponding absorbance spectra monitored over 48 h. As no significant difference (Figure S17-S28) was detected, the stability of the complexes was confirmed. In addition, the stability in toluene upon constant LED irradiation was assessed. Only minimal changes in the absorption spectra (Figure S29-S32) were recorded, demonstrating the photostability of the compounds. In comparison, commercially available dyes are also associated with a high chemical stability and stability upon light exposure. [START_REF] Loudet | BODIPY Dyes and Their Derivatives: Syntheses and Spectroscopic Properties[END_REF][START_REF] Benniston | Lighting the way ahead with boron dipyrromethene (Bodipy) dyes[END_REF][START_REF] Boens | Fluorescent indicators based on BODIPY[END_REF] Determination of the logP values of 1-4

The lipophilicity/hydrophilicity of the compounds was assessed by determination of the logP values between an aqueous phosphate buffer saline (PBS) phase and an octanol phase using the "shake-flask" method. [START_REF] Karges | Synthesis, Characterization, and Biological Evaluation of Red-Absorbing Fe (II) Polypyridine Complexes[END_REF][START_REF] Karges | Systematic investigation of the antiproliferative activity of a series of ruthenium terpyridine complexes[END_REF] All compounds were majorly present in the organic phase with logP values between 0.4 -1.1 (Table S4), indicative of their lipophilicity. This result was expected as the complexes are uncharged and bear lipophilic aromatic systems.

Cellular Uptake and Uptake Mechanism of 1-4

For further assessment, the cellular uptake of the compounds was investigated in human cervical carcinoma (HeLa) cells by measuring the amount of Zn inside the cells after subtracting the natural amount using inductively coupled plasma mass spectrometry (ICP-MS). Importantly, the investigations showed that all compounds were able to be internalized inside the cells. In agreement with the logP values, compound 4 was found to be able to reach a slightly higher concentration inside the cell compared to 1-3 after 4 h incubation (Figure S33). Furthermore, the internalisation mechanism of the compounds by blocking different pathways with metabolic (2-deoxy-D-glucose and oligomycin), cationic transporter (tetraethylammonium chloride) and endocytotic (ammonium chloride or chloroquine) inhibitors was investigated (Figure S35-S38). As these inhibitors had only a negligible effect on the uptake, the involvement of these pathways was excluded. The incubation at lower temperature (4 °C, Figure S34-S37) demonstrated a slightly lower uptake, which is caused by the temperature dependency of diffusion processes at a lower temperature. Overall, these results suggest an uptake by an energy-independent passive diffusion pathway for all compounds.

Cellular Localisation of 1-4

The cellular localisation was determined by confocal laser scanning microscopy in HeLa cells (Figure 5a) and comparison of the distribution pattern inside the cells with the commercial dyes MitoTracker Deep Red, LysoTracker Deep Red and Hoechst 33342.

As no significant congruency was detected, it indicates that the major localisation of the complexes 1-4 was not inside these organelles. For verification, the localisation of the complexes was also investigated by extraction of the major cellular organelles (i.e., nucleus, mitochondria, lysosome, cytoplasm) and quantification of the amount of Zn inside using ICP-MS. In agreement with the microscopy studies, all complexes 1-4 were found to be majorly present in the cytoplasm (Figure 5b). Worthy of note, structurally related bis(dipyrrinato)zinc(II) complexes as well as related Zn(II) porphyrin complexes previously investigated were also found to be mainly localising unselectively in the cell or with membrane proteins. [START_REF] Dąbrowski | Towards tuning PDT relevant photosensitizer properties: comparative study for the free and Zn2+ coordinated meso-tetrakis[2,6-difluoro-5-(N-methylsulfamylo)phenyl]porphyrin[END_REF][START_REF] Pavani | Effect of zinc insertion and hydrophobicity on the membrane interactions and PDT activity of porphyrin photosensitizers[END_REF]

(Photo-)cytotoxicity of 1-4

The cytotoxicity of the complexes in the dark as well as upon irradiation at 510 nm (20 min, 5.0 J/cm 2 ) and 540 nm (40 min, 9.5 J/cm 2 ) towards non-cancerous retinal pigment epithelium (RPE-1) and human cervical carcinoma (HeLa) cells was investigated (Table S5) as it was previously shown that this could be problematic for imaging agents. [START_REF] Alford | Toxicity of Organic Fluorophores Used in Molecular Imaging: Literature Review[END_REF] All complexes 1-4 were found to be non-toxic in the dark as well as upon irradiation (IC50 > 100 μM), which is an important requirement for an imaging probe as these agents should not trigger any kind of stress or toxicity for the cell.

Nanoparticle formation (3-NP) and characterisation

Despite the success of the investigated compounds to enter HeLa cells with a green fluorescence in the cytoplasm, the abilities of bis(dipyrrinato)zinc(II) complexes to act as imaging probes is limited due to 1) the quenching of the excited state in polar solvents, The generated nanoparticles 3-NP were found to be spherical shaped with an average size of 93 nm, as determined by dynamic light scattering (DLS, Figure 6a) and 104 nm, as determined by transmission electron microscopy (TEM, Figure 6b). Very importantly, the particles benefit from a high water solubility. The photophysical evaluation of 3-NP showed no significant difference in the absorption or emission spectra between 3 and 3-NP, indicating that the photophysical properties are not being influenced by the encapsulation. The nanoparticle were found to have a ΦF value of 0.26, which is in the same range than the complex alone in an apolar solvent. Overall, this encapsulation allows overcoming the quenching effects associated with bis(dipyrrinato)zinc(II)

complexes. The cytotoxicity of 3-NP in the dark as well as upon irradiation at 500 nm (16.7 min, 10.0 J/cm 2 ) in HeLa cells was investigated. As expected, 3-NP was found to be non-toxic (IC50 > 100 μM), as requested for an imaging agent. The release of the complex from the nanoparticle formulation was studied by dialysis against a PBS solution at physiological pH (7.4) and lowered pH (5.0). As expected, the formulation remains intact at physiological pH whereas the complex is released at low pH (Figure S38).

Cellular Uptake and Uptake Mechanism of 3-NP

The cellular internalisation and uptake mechanism of 3-NP was investigated by blocking different pathways through preincubation with various inhibitors (Figure S39). In addition to the pathways investigated for 1-4, the sodium-dependent multivitamin transporter (SMVT) as a potential internalisation mechanism was studied as biotin as well as many biotin conjugates are majorly taken up through this pathway. [START_REF] Vadlapudi | Biotin uptake by T47D breast cancer cells: functional and molecular evidence of sodium-dependent multivitamin transporter (SMVT)[END_REF] As the incubation with pantothenic acid and lipoic acid drastically decreased the amount of the compound inside the cell, the uptake through SMVT was assumed to be the primary pathway. In addition, the incubation with metabolic inhibitors as well as upon lower temperature (4°C) decreased the amount of the compound inside the cell, suggesting an energy-dependent mechanism. Furthermore, the incubation with endocytotic inhibitors also decreased the internalisation. Overall, these results suggest that 3-NP is majorly taken up through the SMVT transporter while minor amounts are taken up by endocytosis.

Cellular Localisation of 3-NP

The imaging properties of the particles 3-NP were further studied by determination of their localisation within HeLa cells using confocal laser scanning microscopy (Figure 7a) as well as ICP-MS after the corresponding organelle extraction (Figure 7b).

Interestingly, whereas complex 3 accumulated in the cytoplasm, its nanoformulation 3-NP was majorly found in the lysosomes and minorly in the surrounding cytoplasm. A recent study of the encapsulation of organic aromatic systems or structurally related bis(dipyrrinato)zinc(II) complexes with a DSPE-PEG2000-OCH3 matrix for photodynamic therapy, aggregating induced emission and 2-Photon imaging, were also found to be mainly localising in this organelle. [START_REF] Zang | Red-emissive azabenzanthrone derivatives for photodynamic therapy irradiated with ultralow light power density and two-photon imaging[END_REF][START_REF] Karges | Polymeric Encapsulation of Novel Homoleptic Bis(dipyrrinato) Zinc(II) Complexes with Long Lifetimes for Applications as Photodynamic Therapy Photosensitisers[END_REF]

3D Multicellular Tumor Spheroid Penetration of 3-NP

For further evaluation of the imaging properties of 3-NP, its fluorescence in 3D multicellular tumor spheroids (MCTS) was investigated. Worthy of note, the application of many investigated molecules for diagnosis or therapy is partially limited due to compromised delivery. [START_REF] Pluen | Role of tumor-host interactions in interstitial diffusion of macromolecules: Cranial vs. subcutaneous tumors[END_REF][START_REF] Netti | Role of Extracellular Matrix Assembly in Interstitial Transport in Solid Tumors[END_REF][START_REF] Friedrich | Spheroid-based drug screen: considerations and practical approach[END_REF][START_REF] Goodman | 3-D Tissue Culture Systems for the Evaluation and Optimization of Nanoparticle-Based Drug Carriers[END_REF] Therefore, in this study, the penetration of 3-NP towards large MCTS with a diameter of 800 μM was investigated by confocal laser scanning microscopy. As the luminescence signal was measurable at every depth (Figure S40), a full penetration of the particles was confirmed.

Selectivity assay of 3-NP between cancerous and non-cancerous cells

As 3-NP is mainly taken up through the SMVT transporter, these nanoparticles could show a significant cancer cell selectivity. The targeting effect of 3-NP was investigated by mixing transfected cancerous and non-transfected non-cancerous human lung cells.

More precisely, cancerous adenocarcinomic human alveolar basal epithelial cells (A549), which have a biotin receptor, were transfected with the fluorescence protein mCherry and mixed with non-cancerous human lung fibroblasts (HLF) cells. Not successfully transfected cells were removed using an antibiotic (G-418). Using confocal laser scanning microscopy (Figure 8), we could show that 3-NP was only internalised in the A549 cells, which were transfected with mCherry, demonstrating its ability as a cancer cell selective imaging agent.

CONCLUSIONS

In summary, we have prepared a series of bis(dipyrrinato)zinc(II) complexes as imaging probes and evaluated in-depth their photophysical and biological properties. The compounds were found to have a strong fluorescence in apolar environments whereas their fluorescence was quenched in polar solvents. The complexes were found to majorly accumulate in the cytoplasm of HeLa cells by passive diffusion. To overcome the limitation of these compounds (e.g., quenching of the excited state in water, water solubility problems and poor selectivity for cancer cells), the lead compound 3 was encapsulated in a polymer matrix with a biotin end group. The generated nanoparticles were found to have a bright luminescence, to selectively accumulate in the lysosomes of cancer cells and to fully penetrate 3D MCTS. As a benefit from the biotin targeting moiety, the nanoparticles selectively accumulated in cancerous cells over non-cancerous cells. Overall, we strongly believe that this class of compounds and their corresponding nanoparticles have a great potential as bright, selective imaging probes for cancer cells.

ASSOCIATED CONTENT

Supporting Information:

The Supporting Information is available free of charge on the ACS Publications website at DOI:

Figure 1 .

 1 Figure 1.Overview of reported bis(dipyrrinato)zinc(II) complexes.[START_REF] Sakamoto | New aspects in bis and tris (dipyrrinato) metal complexes: bright luminescence, self-assembled nanoarchitectures, and materials applications[END_REF][START_REF] Sazanovich | Structural Control of the Excited-State Dynamics of Bis(dipyrrinato)zinc Complexes: Self-Assembling Chromophores for Light-Harvesting Architectures[END_REF][START_REF] Trinh | Symmetry-Breaking Charge Transfer of Visible Light Absorbing Systems: Zinc Dipyrrins[END_REF][START_REF] Kusaka | An extremely bright heteroleptic bis (dipyrrinato) zinc (II) complex[END_REF][START_REF] Sakamoto | Fluorescent azadipyrrinato zinc (ii) complex: hybridisation with a dipyrrinato ligand[END_REF][START_REF] Lee | Enforced Effects of Side Group Substitution Position on Luminescence Properties; Synthesis of Bis(dipyrrinato)zinc Complex Derivatives[END_REF][START_REF] Sakamoto | Dissymmetric Bis(dipyrrinato)zinc(II) Complexes: Rich Variety and Bright Red to Near-Infrared Luminescence with a Large Pseudo-Stokes Shift[END_REF][22]

Figure 3 .

 3 Figure 3. Chemical structures of the bis(dipyrrinato)zinc(II) complexes investigated in this work.

  All of the chemicals were obtained from commercial sources and were used without further purification. Dulbecco's Modified Eagles Medium (DMEM), Dulbecco's Modified Eagles Medium supplemented with nutrient mixture F-12 (DMEM/F-12), Fetal Bovine Serum (FBS), Gibco Penicillin-Streptomycin-Glutamine (Penstrep), Dulbecco's Phosphate-Buffered Saline (PBS) were purchased from Fisher Scientific and Resazurin from ACROS Organics.

  small molecule structure solution program and refined with the SHELXL2018/3 program package 31 by full-matrix least-squares minimization on F 2 . Molecular graphics were created using Mercury 4.0. 32 The crystal data collections and structure refinement parameters are summarized in Tabs S1 and S2. CCDC 1937531 (for 3), CCDC 1937532 (for dipyrrin_1), CCDC 1937533 (for 2) and CCDC 1937534 (for 4) contain the supplementary crystallographic data for these compounds, and can be obtained free of charge from the Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif.

  quantum yield, F = fraction of light absorbed, I = integrated emission intensities, n = refractive index, A = absorbance of the sample at irradiation wavelength Stability in toluene/H 2 O/DMEM

1 . 10 4

 4 cells were seeded on 35 mm confocal dishes and allowed to adhere overnight. The cells were incubated with the complex (50 μM, 2% DMSO, v%) for 4 h at 37°C in the dark. After this time, the cells were washed with PBS. The cells were further incubated with Hoechst 33342 (5 μg/mL), LysoTracker Deep Red (500 nM) and MitoTracker Deep Red (500 nM) for 30 min at 37 °C in the dark. The cells were washed three times with PBS. Confocal images were taken with a LSM 880 (Carl Zeiss) laser scanning confocal microscope equipped with a GaAsP detector. The organelle trackers Hoechst 33342 (λex = 405 nm, λem = 410 -470 nm), MitoTracker Deep Red (λex = 633 nm, λem = 650 -720 nm) and LysoTracker Deep Red (λex = 633 nm, λem = 650 -720 nm) were excited and detected as recommended by the supplier. The investigated Zn complexes were detected under usage of their luminescence properties (λex = 514 nm, λem = 530 -610 nm).

  coordination with two substituted dipyrrinato molecules bound to the central Zn atom. The Zn -N bond distances are very similar for all complexes, they fall in the range 1.9583(10) -1.9642(10) for 2, 1.9621(11) -1.9690(11) Å for 3 and 1.956(4) -1.967(3) Å for 4. The bite angles N -Zn -N are all found in the narrow range 94.71(14) -95.35(15)° while the relative orientation of the bidentate ligands is nearly perpendicular in the three structures. Based on

Figure 4 .

 4 Figure 4. Spectroscopic properties of 1-4. a) Normalised absorption (solid line) and emission spectra (dashed line) in toluene. b) Fluorescence quantum yields of 1-4 plotted depending on the solvent (5% DMSO, v%) from non-polar to polar.

Figure 5 .

 5 Figure 5. a) Confocal luminescence image of HeLa cells incubated with compounds 1-4 (50 μM, 2% DMSO, v%) for 4 h at 37°C in the dark. The investigated Zn complexes were detected using their luminescence properties (Zn, λex = 514 nm, λem = 530 -610 nm). b) Sub-cellular distribution (Cell = Whole cell, Cyto. = Cytoplasm, Mito. = Mitochondria, Lys. = Lysosome, Nuc. = Nucleus) of 1-4 (25 μM, 2% DMSO, v%) in HeLa cells after 4 h incubation in the dark determined after organelle extraction and determination of the amount of Zn inside each organelle by ICP-MS.

2 )

 2 water solubility problems as well as 3) the lack of selectivity of the compounds for cancerous cells. To overcome these limitations, the lead compound 3 with the brightest fluorescence was encapsulated in the polymer matrix 1,2-distearoyl-sn-glycero-3phosphoethanolamine-N-[biotin(polyethylene glycol)-2000] ammonium salt (DSPE-PEG2000-biotin) to generate the corresponding nanoparticle formulation 3-NP with a biotin end group as a targeting moiety. Worthy of note, PEGylated phospholipids have been approved by US Food and Drug administration (FDA) for medical applications.

Figure 6 .

 6 Figure 6. Characterization of 3-NP. a) Size distribution determined by DLS. b) Transmission electron microscopy (TEM) image.

Figure 7 .

 7 Figure 7. a) Confocal luminescence image of HeLa cells incubated for 4 h with 3-NP (Zn, 10 μM, λex = 514 nm, λem = 530 -610 nm) and LysoTracker Deep Red (LTR, 500 nM, λex = 633 nm, λem = 650 -720 nm) at 37°C in the dark. The Pearson correlation coefficient of Zn and LTR was found to be 0.62 and the Mander's overlap coefficient 0.83. b) Sub-cellular distribution (Cell = Whole cell, Lys. = Lysosome, Cyto. = Cytoplasm, Mito. = Mitochondria, Nuc. = Nucleus) of 3-NP in HeLa cells after 4 h incubation in the dark determined after organelle extraction and determination of the amount of Zn inside each organelle by ICP-MS.

Figure 8 .

 8 Figure 8. Confocal luminescence image of a mixture of mCherry (λex = 633 nm, λem = 650 -

  Accession Codes: CCDC 1937531 (for 3), CCDC 1937532 (for dipyrrin_1), CCDC 1937533 (for 2) and CCDC 1937534 (for 4) contain the supplementary crystallographic data for these compounds, and can

be obtained free of charge from the Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif.

ACKNOWLEDGEMENT

This work was financially supported by an ERC Consolidator Grant PhotoMedMet to G.G. (GA 681679), has received support under the program "Investissements d'Avenir" launched by the French Government and implemented by the ANR with the reference ANR-10-IDEX-0001-02 PSL (G.G.), the National Science Foundation of China (Nos. 21525105 and 21778079 for H.C.) and the 973 Program (No. 2015CB856301 for H.C.).

Synthesis and Characterisation of 1-4

The dipyrrin ligands were prepared by reacting 2, 4-dimethylmethylpyrrole with the corresponding aldehydes in the presence of trifluoracetic acid. The dipyrrin scaffold was then oxidised with tetrachloro-p-benzoquinone. Finally, the complexes 1-4 were obtained by deprotonation of the dipyrrin ligands with Et3N and complexation with Zn(OAc)2 . 2H2O. All complexes were analysed using 1 H-and [START_REF] Sakamoto | New aspects in bis and tris (dipyrrinato) metal complexes: bright luminescence, self-assembled nanoarchitectures, and materials applications[END_REF] C-NMR and HRMS (Scheme S1, Figure S1-S12). The purity of the compounds was verified by elemental analysis.

X-ray crystallography

The Zn(II) complexes 2, 3 and 4 (Table S1-S2, Figure S13-S16) were characterized with singlecrystal X-ray diffraction studies. The crystal structures of complexes 3 and 4 were previously reported by Trinh et al. [START_REF] Trinh | Symmetry-Breaking Charge Transfer of Visible Light Absorbing Systems: Zinc Dipyrrins[END_REF] gilles.gasser@chimieparistech.psl.eu; Tel. +33 144275602.

ORCID:

Johannes Karges: 0000-0001-5258-0260

Olivier Blacque: 0000-0001-9857-4042

Hui Chao: 0000-0003-4153-5303

Gilles Gasser: 0000-0002-4244-5097

Synopsis:

Poor aqueous solubility and quenching effects in water are major drawbacks for some of the imaging agents currently investigated. Herein, we report the synthesis, characterization and biological evaluation of bis(dipyrrinato) Zinc(II) complexes.

Thanks to encapsulation in a polymer containing a targeting moiety, the formed nanoparticles were found to overcome these limitations and to selectively accumulate in the lysosomes of cancerous cells.


Anass El Aouni

Hussein Yahia

Khalid Daoudi

Khalid Minaoui

A Fourier approach to Lagrangian vortex detection

Keywords: Lagrangian coherent vortex, eddies, fluid dynamics, Fourier transform, frequency domain analysis

We study the transport properties of coherent vortices over a finite time duration. Here we reveal that such vortices can be identified based on frequency-domain representation of Lagrangian trajectories. We use Fourier analysis to convert particles' trajectories from their time domain to a presentation in the frequency domain. We then identify and extract coherent vortices as material surfaces along which particles' trajectories share similar frequencies. Our method identifies all coherent vortices in an automatic manner, showing high vortices' monitoring capacity. We illustrate our new method by identifying and extracting Lagrangian coherent vortices in different two-and three-dimensional flows.

I. INTRODUCTION

Lagrangian coherent structures (LCSs) are differentiated surfaces of trajectories in a dynamical system that exert a significant effect on adjacent trajectories over a time interval of interest [START_REF] Haller | Lagrangian coherent structures and mixing in two-dimensional turbulence[END_REF] . These LCSs are mainly structured into three physical shapes; hyperbolic LCSs which act as repelling or attracting neighboring material elements with locally the highest rate over a given finitetime interval, parabolic LCSs which serve as generalized jet cores and elliptic LCSs serve as generalized coherent vortex boundaries in finite-time unsteady flows. In the geophysical fluid dynamics community these structures are referred to as mesoscale and submesoscale filaments, jets and eddies [START_REF] Peacock | Introduction to focus issue: Lagrangian coherent structures[END_REF] .

In this paper, we are interested in physical structures of elliptical shapes. Such physical structures are omnipresent in the ocean and usually exhibit different properties to their surroundings. They are known to stir and mix surroundings water masses as well as by their ability to trap and carry fluid properties in coherent manner. In the present work, we are interested on those that live long-enough to be observed despite their environment's chaotic nature. These mesoscale eddies have an important role to play in climate change arising from their a) Electronic mail: anass.el-aouni@inria.fr; anas.elaouni@gmail.com influence on the circulation by transporting temperature and salinity. They are also known to contribute in maintaining the extra-tropical climate through the meridional transport of heat from the tropics to the poles [START_REF] Farneti | The role of mesoscale eddies in the rectification of the southern ocean response to climate change[END_REF] .

As these mesoscale eddies have a remarkable impact on the global circulation, their systematic and precise detection has been of great concern over the last two decades [START_REF] Karrasch | Automated detection of coherent lagrangian vortices in two-dimensional unsteady flows[END_REF][START_REF] Froyland | Transport in time-dependent dynamical systems: Finite-time coherent sets[END_REF][START_REF] Haller | Coherent lagrangian vortices: The black holes of turbulence[END_REF][START_REF] Haller | Defining coherent vortices objectively from the vorticity[END_REF] . Several vortex definitions have been introduced in the literature together with their automatic detection [START_REF] Zhang | Boosting techniques for physics-based vortex detection[END_REF][START_REF] Sahner | Vortex and strain skeletons in eulerian and lagrangian frames[END_REF][START_REF] Herrmann | A eulerian level set/vortex sheet method for twophase interface dynamics[END_REF] , most of these formulations are of Eulerian nature. These latter approaches make use of instantaneous velocity field to detect vortices boundaries. However, these vortices boundaries fail to coherently carry and transport their encircled water masses, instead, they stretch, deform and develop filaments [START_REF] Haller | An objective definition of a vortex[END_REF] .

On the other hand, Lagrangian methods are powerful tools to identify coherent vortices because they take into account the time-evolution of particles trajectories [START_REF] Haller | Coherent lagrangian vortices: The black holes of turbulence[END_REF][START_REF] Haller | Defining coherent vortices objectively from the vorticity[END_REF][START_REF] Haller | An objective definition of a vortex[END_REF] . Various dynamical systems approaches have been presented to define a Lagrangian transport characterization of nonautonomous chaotic dynamical systems. These approaches generally fall into different classes. Probabilistic techniques, which study the evolution of probability densities and almost-invariant sets [START_REF] Froyland | Transport in time-dependent dynamical systems: Finite-time coherent sets[END_REF][START_REF] Froyland | An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems[END_REF][START_REF] Froyland | Three-dimensional characterization and tracking of an agulhas ring[END_REF][START_REF] Crawford | Mesoscale eddies dominate surface phytoplankton in northern gulf of alaska[END_REF][START_REF] Sobel | Methods of calculating transport across the polar vortex edge[END_REF][START_REF] Nash | An objective determination of the polar vortex using ertel's potential vorticity[END_REF] and geometric approaches which use invariant manifolds. In 6 the authors introduce a variational principle for coherent material vortices, where vortices' boundaries are sought as elliptic LCSs. This method has been reformulated such that it can be solved via the variational level set methodology [START_REF] Hadjighasem | Level set formulation of twodimensional lagrangian vortex detection methods[END_REF] . These stretching-based variational methods rely on a precise computation of Cauchy-green tensor and its invariants, which requires accurate differentiation of particles trajectories with respect to their initial positions [START_REF] Hadjighasem | A critical comparison of lagrangian methods for coherent structure detection[END_REF] . In 19 authors propose a clustering approach to identify coherent structures by classifying fluid particles into coherent and incoherent. This was done by defining coherent structures as a group of trajectories that conserve short distances between themselves relative to the others outside the structure. This method is simple to implement and requires no differentiation of particles trajectories with respect to their initial positions but it requires a well-defined eigengap [START_REF] Hadjighasem | A critical comparison of lagrangian methods for coherent structure detection[END_REF] . In 7 , authors use the Lagrangian-averaged vorticity deviation to identify Lagrangian coherent vortices. This is based on a unique decomposition of gradient deformation into product of pure strain and pure rotation gradients [START_REF] Haller | Dynamic rotation and stretch tensors from a dynamic polar decomposition[END_REF] . This approach is simple and low in computation cost but it requires large computational domain for spatial mean of vorticity [START_REF] Hadjighasem | A critical comparison of lagrangian methods for coherent structure detection[END_REF] .

Our approach is based on frequency-domain analysis of Lagrangian particle trajectories. We represent particles' trajectory in their frequency domain in the form of sinusoid. Each sinusoid is characterized by an amplitude which gives information about vortex's diameter, and frequency which measures the vortex-turn-over time. The latter part yields an objective measure of material rotation. We define Lagrangian coherent vortex as closed material surfaces in which fluid parcels complete the same rotation. In the frequency domain, this turns out to be filled with tubular level-sets of particle trajectories' frequencies. This paper is organized as follows: Section II describes the setup and outlines the main computational tool. Section III discusses and illustrates particles trajectories in time domain. Section IV describes the representation of particles' trajectories in frequency domain. Section V presents and details our new approach. Section VI illustrate our method via different fluid simulations. Conclusion is drawn in the last Section.

II. SET-UP

We consider a time-dependent smooth vector field:

v(x, t), x ∈ R 3 , t ∈ [α, β] (1)
and its associated ordinary differential equation:

ẋ = v(x, t), x ∈ R 3 , t ∈ [α, β] (2)
where v a smooth velocity field defined on a domain:

U(t) ⊂ R 3 , U = t∈[α,β] U(t) ⊂ R 3 × [α, β] (3)
The flow map is defined as the map that takes a particle from its initial location x 0 at time t 0 to its location x t at time t:

F t t0 (x 0 ) := x(t, t 0 , x 0 ), α ≤ t 0 ≤ t ≤ β, (4)
x(t, t 0 , x 0 ) denoting the trajectory of Eq.2 passing through a point x 0 at time t 0 .

Consider a material domain (defined by a set of fluid particles) M(t 0 ) advected by the flow. Its image at time t can be expressed in term of the flow map as M(t) = F t t0 (M(t 0 )).

III. METHOD

The general outline of the proposed method is as follows. We produce particle trajectories by solving the ordinary differential equation 2. Next we apply the Fourier transform over each particle trajectory to represent them in the frequency domain. Finally, we seek coherent vortices as material surfaces along which particles' trajectories share similar frequencies' components.

A. Frequency-domain analysis and Fourier transform

Frequency-domain analysis is a cornerstone of signal and system analysis. While time-domain analysis shows how a mathematical function or a signal changes over time, frequency-domain analysis shows how their energies are distributed over a domain of frequencies. Furthermore, observing a given system from frequency point of view usually gives an intuitive interpretation of the qualitative behavior of the system. A given signal contains frequency information that can be computed using the Fourier transform which is recalled below.

If x(t) is any signal, either integrable or of finite energy, the Fourier transform of x, F(x)(ξ), measures, at each frequency ξ, the amount of periodic information contained in x, by simply evaluating < x | e -iξ > through an integral computation:

F(x)(ξ) = R x(t)e -itξ dt (5)
with the frequency spectrum defined as the mapping

ξ → |F(ξ)| 2 . If x(t) is absolutely integrable on ] -∞, +∞[, then F(ξ) is continuous, F(ξ) → 0 as ξ → ±∞, and
|F(ξ)| ≤ +∞ -∞ |x(t)|dt (6)
In present scenario, continuous signals are sampled and quantized to get a discrete domain signal. This approach helps employing a variety of digital signal processing operations. In signal processing field, the Fourier transform is mostly computed using the Fast Fourier Transform algorithm (FFT).

IV. VORTEX TRAJECTORY IN TIME DOMAIN

Various studies have addressed the problem of coherent vortex identification based on their observed trajectories. In 21 authors use trajectory length to reveal the time t 0 positions of geometrical structures of different dynamics. This method is supposed to display an imprint of Lagrangian coherent structures as shown in [START_REF] Mancho | Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems[END_REF] . However, no existing mathematical link between that method's features and material coherent structures [START_REF] Hadjighasem | A critical comparison of lagrangian methods for coherent structure detection[END_REF] . Moreover, the scalar field does not allow the identification and extraction of coherent vortices to allow their tracking and monitoring. Another method which make use of particles' trajectories to identify coherent structures is presented in [START_REF] Hadjighasem | Spectral-clustering approach to lagrangian vortex detection[END_REF] . This has been done by classifying fluid particles into coherent and incoherent classes. In their work, a coherent structure is described as a set of Lagrangian trajectories that keep minimum distances between themselves with respect to others outside the structure. This method is simple to implement but it has no link between material vortices and the expected spinning motion [START_REF] Hadjighasem | A critical comparison of lagrangian methods for coherent structure detection[END_REF] .

Particle trajectory produced by integrating the velocity vector field in a vortex results into a loopy curve as shown in Fig. 1, this figure shows trajectories' examples of particles initialized within a given vortex. Thus, a natural way to identify vortex is by locating particle trajectories which exhibit spinning motion. By neglecting the vertical components, particles' trajectories in Fig. 1 result in continuous curves which intersects themselves and create closed-curves segments each time a particle returns to one of its former positions after some time such as in Fig. 1. Based on this idea, authors in [START_REF] Dong | A scheme to identify loops from trajectories of oceanic surface drifters: an application in the kuroshio extension region[END_REF] proposed a scheme to identify loops from trajectories of oceanic surface drifters. Such method indeed does identify the existence of eddies, but it does not identify vortices' cores or boundaries. Moreover this method leaves several unaddressed vortices cases where particle trajectories don't intersect themselves; for example, given a vortex with zero-translating speed and radial flow where the particles' trajectories converge according to inflow. x 0 (t 0 )

x 0 x 1 (t 0 ) x 1 (t f ) (b)
FIG. 1: a) Two particle trajectories within the same vortex: particle within the vortex boundary is presented in black while red color refers to a particle near by the vortex center. b) Their projection on the (x, y) coordinates.

V. PARTICLE TRAJECTORY IN FREQUENCY DOMAIN

In spatial domain, particle's trajectory coordinates are represented by the Euclidean components x, y and z, where they refer to their spatial position. In this domain, a particle trajectory within a vortex results into a loopy curve as shown in Fig. 1. Its time projection over each of the spatial direction results into three components responsible for plane and vertical movement as shown in Fig. 2. The x and y components of this particle trajectory show the emergence of an attractor associated with vortical motion. They display periodic motions characterized by a frequency presenting the vortex-turn-over time and an amplitude referring to the vortex radius.

In frequency domain, particle trajectory coordinates refer to frequencies ξ x , ξ y and ξ w in each of the spatial directions. Fig. 3 shows the frequency-domain representation of particles' trajectories in the Fig. 1 after excluding the drift effect. Both red and black trajectories share the same frequency but have different amplitudes. Here we see that analyzing such trajectories in their frequency domain representation is more easier than analyzing their general shapes. x 0,x (t f )

x 1,x (t 0 ) FIG. 2: Two particle trajectories within the same vortex: particle within the vortex boundary is presented in black while red color refers to a particle near by the vortex center.

x 1,x (t f ) (a

VI. DEFINING VORTICES FROM THEIR FREQUENCY COMPONENTS

We seek to identify Lagrangian coherent vortices as closed material surfaces along which particles exhibit similar rotations around the same axis, and over a finite time interval. In the frequency domain representation, we seek Lagrangian coherent vortices as closed material surfaces along which particle trajectories share similar frequencies' components. In this domain, a particle trajectory can further be divided into two components; frequency which gives information about uniformly rotating flow, and the amplitude that describes the radius of a rotating particle. Based on the fact that fluid particles along vortex boundary exhibit the same polar rotation, we only consider the first part which gives information about the frequency of particle trajectory.

We define Lagrangian Trajectory Frequency LT F as:

LT F ξ (x(•, t 0 , x 0 )) := tn t0 x(t, x 0 )e -itξ dt, ξ = (ξ 1 , ξ 2 , . . . ξ n ) (7)
where ξ = (ξ 1 , ξ 2 , . . . ξ n ) presents a vector of n frequencies of a given particle trajectory. With that, we are computing n points of a one-dimensional Fourier transform. For a particle fluid starting from x 0 , the LT F ξ (x 0 ) vector field is a dynamically consistent measure of material rotation; the average frequency weighted by the frequency spectrum:

ξ = ξn ξ0 ξdξ|LT F ξ | 2 ξn ξ0 dξ|LT F ξ | 2 (8)
gives a consistent measure of particle rotation, whereas, the mean average of the magnitude |LT F ξ (x 0 )| is a consistent measure of particle rotation radius.

We now use the LT F to identify closed material surface along which fluid parcels experience the same rotation over a time interval [t 0 t f ]. Time t 0 positions of such material tubes are tubular level ( a convex contour in two dimensions, and a convex cylindrical or cone-shaped set in three dimensions) surfaces of the scalar ξ, with ξ → |LT F ξ (x 0 )| 2 . By definition, the |LT F ξ (x 0 )| map allows the identification of vortex' center as inner most member of the vortex with the minimal rotation radius.

We summarize the LT F-based vortex identification in the following definition, with its geometry illustrated in Fig. 4-a. Definition 1 For a given time interval [t 0 , t f ]:

• 1) Lagrangian coherent vortex is an evolving mate- rial domain U = t∈[t0,t f ] U(t) ⊂ R 3 × [t 0 , t f ] such that U(t 0 ) ⊂ R 3 is filled with a nested family of tubular level surfaces of ξ = ξn ξ 0 ξdξ|LT F ξ | 2 ξn ξ 0 dξ|LT F ξ | 2 . • 2) The boundary B = t∈[t0,t f ] B(t) ⊂ R 3 × [t 0 , t f ]
of U is the outermost closed material surface of ξ in U(t 0 ).

In the computational world, we relax the convexity strictness for closed material surface of ξ. The first reason for this convexity relaxation consists of allowing small tangential filamentation even at time t 0 of vortices' boundaries (Fig. 4-b-1). The second reason consists of the nature of multi-scale data, such data shows the presence of small-scale vortices nearby the boundaries of big-scale vortices (Fig. 4-b-2). The third reason consists of the representation of vortices' boundaries by discrete polygons (Fig. 4-b-3). At the initial time t 0 , the definition 1 identify Lagrangian coherent vortices with a simple geometry, that is by defining a parameter of maximal convexity deficiency d max to allow the relaxation of convexity strictness. This enables capturing filamented parts that rotate together with the vortices without a global breakaway. The aforementioned definition allows the identification of vortex boundary B(t 0 ) which has convexity deficiency less than the maximal limit d max . We define the convexity deficiency of a closed curve in the plane as the ratio of the area difference between the curve and its convex hull to the area enclosed by the curve as: d = A(Conv(B(t0)))-A(B(t0))

A(B(t0))

. Fig. 4-b shows the geometrical view of the convexity deficiency parameter, the dark red color refers to the difference between the curve (in white) and its convex hull. Lagrangian vortices, as well as their boundaries are materials objects 7 . Thus, their position at a given time t is only determined by Lagrangian advection:

C 3 (t 0 ) C 3 (t) B(t0 ) B(t) U(t0 ) U(t) C 2 (t 0 ) C 2 (t) C 1 (t 0 ) C 1 (t) (a)
U(t) = F t t0 (U(t 0 )), B(t) = F t t0 (B(t 0 )), t ∈ [t 0 , t f ] ( 9
)
The LT F-vortex approach differs from the previous definitions, it does define vortex based on the frequency domain representation of their trajectories; more precisely, it defines vortices as closed material surfaces along which particles share similar frequencies. Our method does not require differentiation of particles trajectories with respect to their initial positions, thus, it does not require advection of high-density grids. LT F-vortex might show material filament, but by definition, they will all rotate together with the vortex without breaking away.

VII. EXPERIMENTS

Here we present numerical results that confirm our theoretical predictions regarding the identification of coherent vortices. Where u • ∇u is the inertial term which characterizes Navier-Stokes equation, and is responsible for the transfer of kinetic energy in the turbulent cascade. ∇p is the pressure gradients which guarantee the incompressibility of the flow, and [START_REF] Haller | Lagrangian coherent structures and mixing in two-dimensional turbulence[END_REF] Re ∆u is the the dissipative viscous term. We further assume periodic boundary conditions and use a standard pseudo-spectral method with 512 modes in each direction and 2/3 dealiasing to solve the above Navier-Stokes equation with Reynolds number Re = 10 4 on the time interval t ∈ [0, 1000]. The model is parametrized by the pressure function p : D × [a, b] → R, with no external forcing (f = 0). We initialize the system with the vorticity of two adjacent vortices perturbed by a random uniform distribution:

∂u t + u • ∇u = -∇p + 1 Re ∆u + f, (x, t) ∈ D × [a, b], ∇ • u = 0, (x, t) ∈ D × [a, b], u j dx = 0, (x, t) ∈ D × [a, b], j = 1, 2, u = u * , (x, t) ∈ D × {0} (10)
ω| t0 = exp( (x -π) 2 + (y -2π -π/4) 2 0.2 ) -exp( (x -π -π/4) 2 + (y -π -π/4) 2 0.8 ) (11)
We use the vorticity stream formulation [START_REF] Spotz | High-order compact scheme for the steady stream-function vorticity equations[END_REF] for implementation and get back velocity and pressure from the stream function. The flow integration is then carried out over the interval t ∈ [400, 1000], in which the turbulent flow is under fully developed turbulence, by a fourthorder Runge-Kutta method with variable step-size.

Fig. 5 shows the forward simulated velocity field, obtained at four distinct time points by the Navier-Stokes model 10, over two-dimensional plane defined by: the angle of the inner and outer rings of a two dimensional torus.

To construct the LT F ξ vector field, we consider an initial grid of 512×512 points and integrate the simulated velocity field over the time interval [t 0 = 400, t f = 1000] using the eq.2.

Fig. 6(a) shows coherent vortices extracted from the ξ map computed from the simulated realization of fluid velocity of the model 10, with d max = 10 -3 . For each vortex we extract the inner-and outer-most cores. Fig. 6(c) shows their final position under Lagrangian advection as well as trajectories of particles initialized on their boundaries. These Lagrangian vortices maintain their coherency, they don't stretch or fold. The complete advection sequence over the time interval [t 0 = 400, t f = 1000] is illustrated in the movie M1.

B. Two-dimensional eddies in satellite altimetry

Here, we use sea surface velocity data to illustrate the detection of Lagrangian vortices. This velocity data is derived from satellite altimetry under the geostrophic approximation where sea-surface height η(ϕ, θ, t) serves as a non-canonical Hamiltonian for surface velocities in the (ϕ, θ) longitude-latitude coordinate system. The evolution of fluid particles satisfies

φ(ϕ, θ, t) = - g R 2 f (θ) cos θ ∂ θ η(ϕ, θ, t) θ(ϕ, θ, t) = g R 2 f (θ) cos θ ∂ ϕ η(ϕ, θ, t) (12)
where g is the constant of gravity, R is the mean radius of the Earth and f (θ) = 2Ω sin θ is the Coriolis effect, with Ω denoting the Earth's mean angular velocity. This data is produced by Ssalto/duacs multi-mission sea level products provided by AVISO (CLS/Archiving, Validation, and Interpretation of Satellite Oceanographic data) 24 with a spatial resolution of 1/4 • and temporal resolution of 7 days.

We chose the region of the Agulhas leakage in the Southern Ocean, spans from [-28 • N, -4 • W ] and (See the supplemental movie M2 for the complete advection sequence of these vortices.)

[-35 • N, 9 • W ]. This region is well known for its longlived propagating eddies that carry water properties from the Indian ocean far into the South Atlantic [START_REF] Ruijter | Indianatlantic interocean exchange: Dynamics, estimation and impact[END_REF] . In this study, we chose the time period between 11/11/2006 and 11/1/2007. We integrate the AVISO data set (eq.12) over the period of time between t 0 = 11 November 2006 and t f = t 0 +90 days over an initial grid of particles with step size ∆x 0 = 1/50 • . We show in Fig. 7(a) ξ map computed from the satellite velocity field (eq.12). In the same image, we show coherent eddies' boundaries as they are extracted from the ξ map, with d max = 10 -3 . In Fig. 7(c) , we show their initial and final position under Lagrangian advection as well as trajectories of particles initialized on their boundaries. These eddies remain coherent. The complete advection sequence over the time interval [0, 90] is illustrated in the movie M2. As anticipated there is moderate tangential filamentation in the material eddy boundary, but precisely no breakaway from the rotating eddy core. This high degree of material coherence in the presence of the complex surrounding material mixing demonstrates the efficiency of LT F-based vortex detection.

VIII. CONCLUSION

We have presented an objective frequency-domain definition of Lagrangian coherent vortices as closed material surfaces along which fluid parcels exhibit the same intrinsic rotation. This coherent material is obtained from the frequency-domain representation of their trajectories. This intrinsic rotating material is expressible as material tube characterized by a given frequency. We illustrated our new method on different two-and three-dimensional flows. Our results show that vortices boundaries obtained are sharply defined. Our approach does not require advection of high-density material grids, a generally an expensive computational procedure in Lagrangian coherence calculations. Moreover, it is suitable to applications to float data. See the supplemental movie M1 for the complete advection sequence of the vortices in Fig. 6.

See the supplemental movie M2 for the complete advection sequence of the vortices in Fig. 7.

See the supplemental movie M3 and M4 for the complete advection sequence of the vortices in Fig. 8.

FIG. 3 :

 3 FIG.3: Frequency-domain representation of the particles' trajectories in Fig.1, red an black trajectories have the same frequency but different amplitudes: a) within x-direction, b) within y-direction.

FIG. 4 :

 4 FIG. 4: a) Initial and time t positions of a coherent Lagrangian vortex U(t) and a Lagrangian vortex boundary B(t). Also shown within U(t) are different vortex cores C(t) along which particles' trajectories share different frequencies over the time interval [t 0 , t]. b) An example of a closed material line which profits from the relaxation of convexity to small convexity deficiency. Dark red area indicates the area difference between the closed material line and its convex hull: (1) minor tangential filamentation, (2) deformation by smaller-scale vortices, (3) discrete approximation of a convexity.

A. Two-dimensional examples 1 .

 1 Direct numerical simulation of two-dimensional turbulence We solve numerically the Navier-Stokes PDE model for the time evolution of 2 components of the velocity, u : D → R 2 of an incompressible fluid on a torus, D = [0, 2π] × [0, 2π]. This can be expressed as:

FIG. 5 :

 5 FIG. 5: Time evolution of the magnitude of fluid velocity, governed by the Navier-Stokes model 10, over two spatial dimensions: the angle of the inner ring (horizontal axis) and outer ring (vertical axis) of a two dimensional torus. Angles are expressed in radians. Velocities are evaluated at times t = (0, 400, 700, 1000).

FIG. 6 :

 6 FIG. 6: a) Lagrangian coherent vortices at time t 0 extracted from the velocity field generated by Navier-Stokes model 10 using definition.1 with the ξ map shown in background, with ξ → |LT F ξ (x 0 )| 2 . b) The mean-average magnitude |LT F ξ (x 0 )| highlighting areas of spinning movement. c) Their initial and final positions under Lagrangian advection as well as different particles initialized within their boundaries and their trajectories in Dashed line. (See the supplemental movie M1 for the complete advection sequence of these vortices.)

FIG. 7 :

 7 FIG. 7: a) Coherent eddies at time t 0 extracted using definition.1 with the ξ map shown in background, with ξ → |LT F ξ (x 0 )| 2 . b) The mean-average magnitude |LT F ξ (x 0 )| highlighting areas of spinning movement. c) Their initial and final positions under Lagrangian advection as well as different particles initialized within their boundaries and their trajectories in Dashed line.(See the supplemental movie M2 for the complete advection sequence of these vortices.)

1 .

 1 Three-dimensional Agulhas eddies in a data-assimilating circulation model Here, we apply LT F-based vortex extraction to a three-dimensional unsteady velocity field set obtained from the Southern Ocean state estimation (SOSE) model 26 . The domain of the data set lies in the area of the Agulhas leakage in the Southern Ocean. Our Lagrangian study covers a period of T = 30 days, ranging from t 0 = 15 May 2006 to t = 15 june 2006. As in 7 , we select the computational domain bounded by longitudes [11 • E, 16 • E], latitudes [37 • S, 33 • S] and depths [7, 2000] m. We compute the LT F vector field over a uniform grid of 150 × 160 × 60 points, and identify a Lagrangian coherent eddy. Fig.8(a) shows the three-dimensional field of the ξ, ξ → |LT F ξ (x 0 )| 2 with the eddy's inner-and outer-most cores enclosed with red color. Particles belonging to these two cores have different frequencies. Fig.8(b) shows the initial position of a coherent Lagrangian eddy boundary (dark red), extracted as level sets of ξ. Also shown is a nearby |LT F ξ | level surface outside the eddy boundary, illustrating the complexity of the near-surface mixing region enclosing the eddy. Fig.8(c) gives a full view of the Lagrangian eddy. Fig.8(d) shows the initial and the materially advected position of the eddy at the final time t f = t 0 + 30 days, whereas, Fig.8(e) shows the initial and final position of different eddy's layers presented in different colors.

FIG. 8 :

 8 FIG. 8: a) A three-dimensional field of the ξ, with ξ → |LT F ξ (x 0 )| 2 . b) Representative level surfaces of ξ. The dark red surface is extracted using the definition.1, marking the vortex boundary for a mesoscale coherent Lagrangian eddy, extending from 7 m down to 600 m in depth. The green surface is a nearby level surface of |LT F ξ | outside the Lagrangian eddy region. c) Full view of the Lagrangian eddy boundary. d) The initial and the advected eddy boundary 30 days later, whereas e) shows initial and the advected images of different layers of the eddy. (See the supplemental movie M3 and M4 for the complete advection sequence of these vortices.)

ACKNOWLEDGMENTS

This work was supported by the French-Moroccan PHC-Toubkal project n • TBK/16-24 and PPR2-6 project. A. El Aouni is partially supported by the French project BOOSTE TON DOC 2019.