Cristina Garcia-Iriepa

Pauline Gosset

Romain Berraud-Pache

Madjid Zemmouche

Gregory Taupier

Dodzi Kokou

Pascal Dorkenoo

Jérémie Didier

Nicolas Léonard

Isabelle Ferré

Navizet

Cristina García-Iriepa

Grégory Taupier

Kokou Dodzi Dorkenoo

Pascal Didier

Jérémie Léonard

Nicolas Ferré

Isabelle Navizet

Simulation and Analysis of the Spectroscopic Properties of Oxyluciferin and Its Analogues in Water

Keywords: Oxyluciferin, TD-DFT, Molecular Dynamics, QM/MM, Absorption Spectra, Emission Spectra, Bioluminescence

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

exchange reactions. In this work, we present the simulation of the absorption and emission spectra of the possible natural oxyluciferin forms in water and some of their analogues considering both, the solvent/oxyluciferin interactions and the dynamical effects by using MD simulations and QM/MM methods. On the one hand, the absorption band shapes have been rationalized by analyzing the electronic nature of the transitions involved. On the other hand, the simulated and experimental emission spectra have been compared. In this case, an ultrafast excited state proton transfer (ESPT) occurs in oxyluciferin and its analogues, which impairs the detection of the emission from the protonated state by steady-state fluorescence spectroscopy. Transient absorption spectroscopy was used to evidence this ultrafast ESPT, and rationalize the comparison between simulated and experimental steady-state emission spectra. Finally, this work shows the suitability of the studied oxyluciferin analogues to mimic the corresponding natural forms in water solution, as an elegant way to block the desired inter-exchange reactions allowing the study of each oxyluciferin form separately.

INTRODUCTION

Bioluminescence is the process responsible for flashes of bright light coming out from a wide variety of living organisms. It results from the conversion of chemical energy into visible light, catalyzed by a protein. This natural phenomenon is present in several organisms such as bacteria, beetles, squid, worms, fungi, algae and jellyfish. [START_REF] Shimomura | Bioluminescence : Chemical Principles and Methods[END_REF][START_REF] Herring | Bioluminescence in Action[END_REF] In the last decades, the study of the bioluminescence mechanism as well as the spectroscopic properties of the emitter species have attracted considerable interest among scientists. In this regard, the bioluminescent process of the jellyfish (aequorin/green fluorescent protein system) [START_REF] Zimmer | Green Fluorescent Protein (GFP): Applications, Structure, and Related Photophysical Behavior[END_REF][START_REF] Tsien | The Green Fluorescent Protein[END_REF][START_REF] Remington | Green Fluorescent Protein: A Perspective[END_REF][START_REF] Baubet | Chimeric Green Fluorescent Protein-Aequorin as Bioluminescent Ca2+ Reporters at the Single-Cell Level[END_REF] and the one of fireflies [START_REF] Fraga | Firefly Luminescence: A Historical Perspective and Recent Developments[END_REF][START_REF] Kaskova | Lights: Luciferins, Luciferases, Their Mechanisms of Action and Applications in Chemical Analysis[END_REF] are the most studied systems up to now.

In particular, its large number of applications has motivated the extensive study of firefly bioluminescence. For instance, firefly bioluminescence has been used for rapid analysis of bacterial contamination, [START_REF] Griffiths | The Role of ATP Bioluminescence in the Food Industry: New Light on Old Problems[END_REF] in vivo bioluminescence imaging, [START_REF] Shinde | Luciferin Derivatives for Enhanced in Vitro and in Vivo Bioluminescence Assays[END_REF][START_REF] Greer | Imaging of Light Emission from the Expression of Luciferases in Living Cells and Organisms: A Review[END_REF] multiple gene expression and regulation, [START_REF] Kricka | Application of Bioluminescence and Chemiluminescence in Biomedical Sciences[END_REF][START_REF] Ohkuma | Detection of Luciferase Having Two Kinds of Luminescent Colour Based on Optical Filter Procedure: Application to an Enzyme Immunoassay[END_REF] monitoring of cell dynamics [START_REF] Shao | Photoactivable Bioluminescent Probes for Imaging Luciferase Activity[END_REF][START_REF] Doyle | In Vivo Bioluminescence Imaging for Integrated Studies of Infection[END_REF] and drug screening [START_REF] Fan | Bioluminescent Assays for High-Throughput Screening[END_REF] among others. Nowadays, the general bioluminescent mechanism in fireflies is well accepted and consists in an enzymatic multistep oxidation. [START_REF] White | The Chemi-and Bioluminescence of Firefly Luciferin: An Efficient Chemical Production of Electronically Excited States[END_REF][START_REF] Deluca | Firefly Luciferase[END_REF][START_REF] Navizet | The Chemistry of Bioluminescence: An Analysis of Chemical Functionalities[END_REF] First, an enzyme called luciferase binds a substrate known as luciferin and catalyzes the formation of luciferyl adenosine monophosphate. Afterwards, the emitter species, so-called oxyluciferin (OxyLH2), is generated in the first singlet-excited state by decomposition of a cyclic peroxide and release of CO2. Finally, oxyluciferin decays to the ground state by emitting light in the visible spectral window.

It is remarkable that for fireflies, the color emission can vary from yellow-green to red depending on diverse factors such as pH, [START_REF] Ando | Firefly Bioluminescence Quantum Yield and Colour Change by pH-Sensitive Green Emission[END_REF] the presence of bivalent metal ions, [START_REF] Wang | Quantum Yields and Quantitative Spectra of Firefly Bioluminescence with Various Bivalent Metal Ions[END_REF] temperature, [START_REF] Mochizuki | Robust Red-Emission Spectra and Yields in Firefly Bioluminescence against Temperature Changes[END_REF] luciferase mutations [START_REF] Wang | Impact of Site-Directed Mutant Luciferase on Quantitative Green and Orange/red Emission Intensities in Firefly Bioluminescence[END_REF] and luciferin structure modifications. [START_REF] Mofford | Aminoluciferins Extend Firefly Luciferase Bioluminescence into the Near-Infrared and Can Be Preferred Substrates over D -Luciferin[END_REF][START_REF] Reddy | Robust Light Emission from Cyclic Alkylaminoluciferin Substrates for Firefly Luciferase[END_REF] However, despite the large number of studies focused on understanding the origin of the emission color modulation, this mechanism remains unsolved. [START_REF] Hosseinkhani | Molecular Enigma of Multicolor Bioluminescence of Firefly Luciferase[END_REF][START_REF] Hirano | Spectroscopic Studies of the Light-Color Modulation Mechanism of Firefly (Beetle) Bioluminescence[END_REF][START_REF] Navizet | Color-Tuning Mechanism of Firefly Investigated by Multi-Configurational Perturbation Method[END_REF] Indeed, the fact that oxyluciferin can exist in six different chemical forms further complicates the rationalization of the emission color modulation. These six different chemical forms of oxyluciferin rise from a triple equilibrium of phenol and/or enol deprotonation together with a keto/enol tautomerization (Chart 1), being these equilibria sensitive to the solvent, pH or interactions with bases. [START_REF] Naumov | Spectral-Structural Effects of the Keto-Enol-Enolate and Phenol-Phenolate Equilibria of Oxyluciferin[END_REF][START_REF] Nath | Why Is Firefly Oxyluciferin a Notoriously Labile Substance?[END_REF][START_REF] Naumov | Structure and Spectroscopy of Oxyluciferin, the Light Emitter of the Firefly Bioluminescence[END_REF] Chart 1. Chemical equilibria leading to the six possible oxyluciferin forms.

In order to simplify this scenario, some studies have proposed novel derivatives as an elegant way to block one or more of these chemical equilibria. For instance, the 5,5-dimethyloxyluciferin analogue has been used to block the keto/enol tautomerization [START_REF] Hirano | Spectroscopic Studies of the Light-Color Modulation Mechanism of Firefly (Beetle) Bioluminescence[END_REF][START_REF] Naumov | Structure and Spectroscopy of Oxyluciferin, the Light Emitter of the Firefly Bioluminescence[END_REF][START_REF] Branchini | Yellow-Green and Red Firefly Bioluminescence from 5,5-Dimethyloxyluciferin[END_REF][START_REF] Rebarz | Deciphering the Protonation and Tautomeric Equilibria of Firefly Oxyluciferin by Molecular Engineering and Multivariate Curve Resolution[END_REF] but, it is unstable upon photoexcitation. [START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF] For this reason, the 5,5-spirocyclopropyloxyluciferin (5,5-CprOxyLH in Chart 2) analogue has emerged as a best candidate to mimic the keto form of oxyluciferin. [START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF][START_REF] Ghose | Oxyluciferin Derivatives: A Toolbox of Environment-Sensitive Fluorescence Probes for Molecular and Cellular Applications[END_REF] Moreover, other oxyluciferin derivatives have been proposed in order to block the deprotonation/protonation equilibrium of the phenol and/or enol groups such as the 4-methoxyoxyluciferin (4-MeOxyLH in Chart 2) or 6'-methoxyoxyluciferin (6'-MeOxyLH in Chart 2). [START_REF] Hirano | Spectroscopic Studies of the Light-Color Modulation Mechanism of Firefly (Beetle) Bioluminescence[END_REF][START_REF] Rebarz | Deciphering the Protonation and Tautomeric Equilibria of Firefly Oxyluciferin by Molecular Engineering and Multivariate Curve Resolution[END_REF][START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF][START_REF] Ghose | Oxyluciferin Derivatives: A Toolbox of Environment-Sensitive Fluorescence Probes for Molecular and Cellular Applications[END_REF] This strategy enables the experimental study of the individual chemical forms of oxyluciferin that can otherwise not be isolated.

A different approach that can be followed to investigate the photochemical properties of the less stable oxyluciferin forms is to perform computational studies. In the past, computational studies have provided essential information about the bioluminescence mechanism of fireflies, [START_REF] Orlova | Theoretical Study of the Amazing Firefly Bioluminescence: The Formation and Structures of the Light Emitters[END_REF][START_REF] Navizet | Are the Bio-and Chemiluminescence States of the Firefly Oxyluciferin the Same as the Fluorescence State?[END_REF][START_REF] Hiyama | Theoretical Insights into the Effect of pH Values on Oxidation Processes in the Emission of Firefly Luciferin in Aqueous Solution[END_REF] emission color modulation, [START_REF] Navizet | Color-Tuning Mechanism of Firefly Investigated by Multi-Configurational Perturbation Method[END_REF][START_REF] Hiyama | Analysis of Photoexcitation Energy Dependence in the Photoluminescence of Firefly Luciferin[END_REF] relative stability of the different oxyluciferin forms [START_REF] Noguchi | Reverse Stability of Oxyluciferin Isomers in Aqueous Solutions[END_REF] as well as their pKa values. [START_REF] Hiyama | Theoretical Study of Firefly Luciferin pKa Values-Relative Absorption Intensity in Aqueous Solutions[END_REF][START_REF] Hiyama | Analysis of Oxyluciferin Photoluminescence Pathways in Aqueous Solutions[END_REF] This way, computational studies have become essential to learn more about this system. Here, we propose a simple approach to simulate the absorption and emission spectra of all possible oxyluciferin forms and some analogues in explicit water solution. For this aim, molecular dynamics have been performed to consider the explicit water molecules and QM/MM methods have been used to compute the excitation and emission energies. More precisely, this work presents three novelties compared to the computational studies reported up to now. First, the absorption and emission spectra have been simulated considering both the dynamical effect and the explicit interactions between the solvent and oxyluciferin by means of molecular dynamics. So far, the excitation and emission energies of some natural oxyluciferin forms have been computed in the gas phase, [START_REF] Orlova | Theoretical Study of the Amazing Firefly Bioluminescence: The Formation and Structures of the Light Emitters[END_REF][START_REF] Yang | Predictions of the Geometries and Fluorescence Emission Energies of Oxyluciferins[END_REF] considering the solvent implicitly with the polarizable continuum model (PCM), [START_REF] Hiyama | Vibronic Structures in Absorption and Fluorescence Spectra of Firefly Oxyluciferin in Aqueous Solutions[END_REF][START_REF] Coccia | Theoretical S1←S0 Absorption Energies of the Anionic Forms of Oxyluciferin by Variational Monte Carlo and Many-Body Green's Function Theory[END_REF] by microsolvation including 11 water molecules [START_REF] Falklöf | Distinguishing between Keto-Enol and Acid-Base Forms of Firefly Oxyluciferin through Calculation of Excited-State Equilibrium Constants[END_REF] or using a canonical ensemble of hydration structures obtained from first-principles molecular dynamics. [START_REF] Hiyama | The Effect of Dynamical Fluctuations of Hydration Structures on the Absorption Spectra of Oxyluciferin Anions in an Aqueous Solution[END_REF] Second, this work presents for the first time a detailed analysis of the electronic nature of the vertical transitions not only to the first singlet excited state but also to upper-lying excited states. This data is important to rationalize the shape of the absorption spectra, in some cases structured, and to reveal hidden transitions from an experimental point of view (e.g. embedded within a more intense and broad band). Third, a quite extensive study of the spectral properties of oxyluciferin and its analogues is reported as it covers a total of 12 derivatives: the six possible chemical forms of oxyluciferin and three analogues, considering the protonated and the deprotonated forms (Chart 2). Moreover, a comparative study of oxyluciferin and its analogues is presented to validate if they can be used to mimic natural oxyluciferin in water solution. It is remarkable that in all cases, a good agreement between the experimental and simulated spectra has been found using the strategy proposed in this study.

METHODS

Computational Methods.

Classical dynamic simulations have been performed with Amber14 48 in order to obtain a statistical number of snapshots to perform Quantum Mechanics/Molecular Mechanics (QM/MM) calculations to simulate the absorption (emission) spectra considering explicit water molecules.

For this aim, the structure of the compound optimized in the ground (excited) state has been selected as the starting point for MD simulations. The bond distances, angles, dihedrals and charges parameters for the oxyluciferin derivatives (geometry and charges of the minima in the ground and first excited states) have been derived from the QM calculations in gas phase and used as the starting point. The compound has been solvated with TIP3P water molecules [START_REF] Jorgensen | Comparison of Simple Potential Functions for Simulating Liquid Water[END_REF] within an octahedral box, ensuring a solvent shell of at least 10 Å around the molecule, corresponding to around 900 water molecules. Then, the system has been minimized, followed by a heating step from 100 K to 300 K in 20 ps. Afterwards, a production run of 10 ns has been performed with a 2 fs time step, under NPT conditions (300 K and 1 atm) and using periodic boundary conditions.

The pressure and temperature were assured by the Berendsen algorithm. [START_REF] Berendsen | Molecular Dynamics with Coupling to an External Bath[END_REF] Then, 100 snapshots have been extracted and the excitation (emission) energies computed at the QM/MM level of theory, being enough to achieve the spectral shape convergence (Figure S2 and S7). At least 7 electronic excited states have been considered for the excitation energy calculation to simulate the spectra up to 300 nm. In all cases, the absorption and emission spectra have been simulated as a convolution of the excitation and emission energies of 100 snapshots (full-width at half-maximum of 0.2 eV). The parameters have been updated as many times as needed by optimizing the system (derivative and explicit water molecules) at the QM/MM level to reach the spectral shape convergence of the simulated spectra. In particular, for each set of parameters, a corresponding 10 ns MD was performed, 100 snapshots were extracted and the QM/MM absorption (emission) spectra simulated. New parameters were adjusted from the QM/MM calculation and the process was performed with the new set of parameters until reaching the spectral shape convergence between the previous and the next simulated spectra (Figure S1). In order to further check the suitability of the oxyluciferin parameters we have compared the absorption and emission spectra simulated with and without optimizing oxyluciferin at the QM/MM level of theory. In all cases, similar spectra have been obtained (Figure S3 andS6).

For the QM/MM calculations, the water molecules were treated at the MM level whereas the oxyluciferin derivative was included in the QM region and its excitation (emission) energy computed at the B3LYP 51-53 /6-311g(2d,p) level of theory. This functional and basis set have been selected as they have been previously reported to be suitable for this chromophore. [START_REF] Hiyama | The Effect of Dynamical Fluctuations of Hydration Structures on the Absorption Spectra of Oxyluciferin Anions in an Aqueous Solution[END_REF][START_REF] Berraud-Pache | QM/MM Calculations on a Newly Synthesised Oxyluciferin Substrate: New Insights into the Conformational Effect[END_REF][START_REF] Min | A Time-Dependent Density Functional Theory Investigation on the Origin of Red Chemiluminescence[END_REF][START_REF] Min | Modification of Firefly Cyclic Amino Oxyluciferin Analogues Emitting Multicolor Light for OLED and near-Infrared Biological Window Light for Bioluminescence Imaging: A Theoretical Study[END_REF] Calculations in gas phase have been performed using Gaussin09 57 and the functional and basis set used for the QM/MM calculations. The QM/MM calculations have been carried out using a QM/MM coupling scheme [START_REF] Ferré | Approximate Electrostatic Interaction Operator for QM/MM Calculations[END_REF] between Gaussian09 57 and Tinker. [START_REF] Ponder | Software Tools for Molecular Design[END_REF] In particular, the interaction between the QM charge density (electrons and nuclei) and the external electrostatic potential of the MM part was computed by the electrostatic potential fitted (ESPF) method. [START_REF] Ferré | Approximate Electrostatic Interaction Operator for QM/MM Calculations[END_REF] The microiterations technique [START_REF] Melaccio | Unique QM/MM Potential Energy Surface Exploration Using Microiterations[END_REF] was used to converge the MM subsystem geometry for every QM minimization step.

Experimental Methods.

The model analogues 4-MeOxyLH and 6′-MeOxyLH were synthesized as previously reported. [START_REF] Rebarz | Deciphering the Protonation and Tautomeric Equilibria of Firefly Oxyluciferin by Molecular Engineering and Multivariate Curve Resolution[END_REF] The synthesis and characterization of the new spirocyclic analogues 5,5-CprOxyLH will be reported elsewhere. Stock solutions of all compounds were prepared in spectroscopic grade DMSO (Sigma-Aldrich) and stored in several aliquots at -20 °C to minimize the freeze-thaw cycle. They were further diluted 1000-fold to a final concentration of about a few μM in aqueous buffer with different pH. The buffers were prepared by diluting a 1 M stock solution in deionized, Milliporepurified water (18.2 MΩ). Buffered stock solutions were prepared as follows: 75 mM NaCl/20 mM KH2PO4/0.2 mM MgCl2 for pH ≤ 7.0 and 75 mM NaCl/25 mM Tris(hydroxymethyl)aminomethane (TRIS)/0.2 mM MgCl2 for pH > 7.0. The buffer was separated into several fractions, and their pH was adjusted by 250 mM HCl or by 250 mM NaOH separately at 20 °C.

Separate buffers at different pH were used in all spectral measurements. Absorption spectra were recorded with a Cary-4000 spectrometer (Agilent Technologies). Steadystate fluorescence spectra were recorded with Fluoromax 4 spectro-fluorometer (Horiba Jobin Yvon) with 2 nm excitation/emission slit and corrected for the instrumental response characteristics.

The femtosecond source is a regenerative amplifier that delivers 120 fs pulses at 800 nm with a repetition rate of 5kHz (''Spitfire'', Spectra-Physics, Mountain View, CA, USA). We generated 400 nm, 150 fs pump pulse by second harmonic generation in a BBO crystal. We obtained supercontinuum probe pulse (430 nm to 700 nm) by self phase modulation in a 2 mm thick sapphire crystal. Pump and probe beams were focused by the same 38.1 mm focal length off-axis parabolic mirror on a 1 mm thick quartz cuvette containing the sample. To excite a fresh volume of the sample for each pump pulse, the cuvette was moved continuously. Time and spectrally resolved differential transmission T/T (τ, λ) of the probe as a function of the pump-probe delay τ and the wavelength λ was measured in a standard pump-probe configuration. The spectra were obtained by dispersing the transmitted reference and probe beams in a spectrometer followed by a CCD detector (Roper, Trenton, NJ, USA). The analogues concentration was adjusted to obtain an optical density of 0.1 per mm at 400 nm (50 µM) to optimize the signal to noise ratio of the differential transmission of the probe.

RESULTS AND DISCUSSION

Here is presented a detailed analysis of the photochemical and photophysical properties of all the possible chemical forms of oxyluciferin and their analogues in explicit water (Chart 2). For this study we have selected the following oxyluciferin analogues: i) 5,5-CprOxyLH which presents a cyclopropyl substituent in the thiazole ring to block the keto-enol tautomerization; ii) 4-MeOxyLH which has a methoxy group in the enol moiety of the thiazole ring to block both the keto-enol tautomerization and enol deprotonation and iii) 6'-MeOxyLH that presents a methoxy group in the phenol moiety to block the phenol deprotonation. In particular, this section is divided into three parts: i) Analysis of the possible hydrogen bond interactions between each oxyluciferin derivative and the explicit water molecules by MD simulations. ii) Absorption spectrum simulation, comparison with the experimental data and analysis of the involved electronic transitions and iii) Simulation of the emission spectrum and comparison with the experimental steady-state and time resolved data to get insight into the chemical nature of the emitter specie.

Water/oxyluciferin hydrogen bond interactions.

Before starting with the study of the photochemical properties of oxyluciferin and its analogues it is important to analyze the hydrogen bond patterning between the water molecules and each solute due to its known influence on both the absorption and emission spectra. [START_REF] Noguchi | Reverse Stability of Oxyluciferin Isomers in Aqueous Solutions[END_REF][START_REF] Hiyama | The Effect of Dynamical Fluctuations of Hydration Structures on the Absorption Spectra of Oxyluciferin Anions in an Aqueous Solution[END_REF][START_REF] Ren | Predictions of the Electronic Absorption and Emission Spectra of Luciferin and Oxyluciferins Including Solvation Effects[END_REF] Starting with the natural forms of oxyluciferin, we have calculated the average number of hydrogen bond interactions between the water molecules and the heteroatoms, (O10, O11, N3, N3' in Figure 1) along the MD trajectory (Table S1). By analyzing the data, general patterns can be set: i) for both the phenolate (Figure 1A andB) and enolate (Figure 1C) derivatives, the deprotonated oxygen leads three to four hydrogen bonds with three/four different water molecules, ii) regarding to the keto group (Figure 1A), between one and two water molecules interact with the oxygen atom, both in the phenol and in the phenolate derivatives, whereas iii) the interaction of only one water molecule with the oxygen of the enol group (Figure 1B) or phenol group (Figure 1C) has been observed during half of the simulation time and iv) the two nitrogen atoms of the structure present in general one hydrogen bond interaction in less than half of the simulation time. Moreover, we have observed that during the simulation time there is a continuous interchange of the water molecules involved in the hydrogen bond interactions with the ones of the solvation sphere, as could be expected at room temperature. Similar hydrogen bond patterns have been found for the oxyluciferin analogues (Table S1). That is, neither the methyl nor the cyclopropyl substituents have a great influence on the hydrogen bond interactions. Just a slight difference was observed for phenolate-4-MeOxyL -and enolate-6'-MeOxyL -compared to their respective natural forms phenolate-enol-OxyLH -and phenol-enolate-OxyLH -. In these cases, the average number of hydrogen bonds found for the oxygen of the methoxy group is a bit larger than the one found for the corresponding natural forms, probably due to the inductive effect of the methyl group that increases the electronic charge of the oxygen atom.

Absorption Spectra Simulation.

Before starting the analysis of the absorption spectra and the electronic nature of the optically bright electronic transitions, it is essential to validate the procedure used to simulate the absorption spectra (see Methods section for details). For this aim, a comparison between the simulated and the experimental absorption spectra of the oxyluciferin analogues has been performed. We selected oxyluciferin analogues instead of the natural forms for this purpose as it is impossible to record experimentally the absorption spectra of each natural form in water due to the low thermal stability of some of them and, certainly, a mixture of different forms in equilibrium (Chart 1). [START_REF] Rebarz | Deciphering the Protonation and Tautomeric Equilibria of Firefly Oxyluciferin by Molecular Engineering and Multivariate Curve Resolution[END_REF][START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF] Hence, we simulated the absorption spectra for the protonated and deprotonated forms of 5,5-CprOxyLH, 4-MeOxyLH and 6'-MeOxyLH. By analyzing Figure 2, we can conclude that a good overall agreement between the simulated and experimental spectra has been achieved in terms of both the shape and the maximum absorption wavelengths. After the validation procedure, we have performed a systematic comparison between the simulated absorption spectra of the oxyluciferin analogues and their corresponding natural forms to give insight into their suitability to mimic oxyluciferin in water. In general, almost overlapped absorption spectra have been obtained (Figure S4). Only for phenol-5,5-CprOxyLH, the structured shape of the dominating absorption band is slightly different to the one simulated for phenol-keto-OxyLH2 but both share the same specificity compared to other chemical forms: absorption bands are significantly broader and more structured. Hence, we can conclude that, the electronic structures and absorption spectra of the proposed oxyluciferin analogues resemble to a great extent the ones of the natural forms in water. Moreover, we show that the introduction of the methyl and cyclopropyl groups do not modify the shape nor the maximum absorption wavelength of the absorption spectra.

In addition to the overall spectral shapes, we may analyze the electronic nature of the corresponding optically bright transitions. Starting with the keto derivatives, both phenol-keto-OxyLH2 and phenol-5,5-CprOxyLH present one of the broadest absorption bands (red line in Figure 2A). The computational data predict that the first three singlet excited states (S1, S2 and S3) are optically bright and close in energy such that the dominating low-energy absorption band actually corresponds to three different electronic transitions, explaining the width and structured shape of both the experimental and simulated spectra (Figure 3A and Figure S4C). To analyze the electronic nature of these three transitions, we have investigated the molecular orbitals involved in each one. The electronic nature of the S0-S1 transition corresponds to the previously reported transition responsible of oxyluciferin emission [START_REF] Navizet | Color-Tuning Mechanism of Firefly Investigated by Multi-Configurational Perturbation Method[END_REF][START_REF] Orlova | Theoretical Study of the Amazing Firefly Bioluminescence: The Formation and Structures of the Light Emitters[END_REF][START_REF] Nakatani | Red Light in Chemiluminescence and Yellow-Green Light in Bioluminescence: Color-Tuning Mechanism of Firefly, Photinus Pyralis, Studied by the Symmetry-Adapted Cluster-Configuration Interaction Method[END_REF][START_REF] Milne | Red-Shifting the Optical Response of Firefly Oxyluciferin with Group 15/16 Substitutions[END_REF] for which the electron density of the HOMO is mainly located on the benzothiazole moiety and the LUMO on the thiazolone part. The S0-S2 transition is similar to the S0-S1 transition but part of the electron density of the bonding orbital (HOMO-1) is located on the sulfur atom of the benzothiazole moiety. As a consequence of the close energy between S1 and S2, their electronic nature is largely mixed which can explain their similar absorption intensities as has been shown by computing their oscillator and transition dipole moment strengths (see Supporting Information and Figure S9). On the other hand, the S0-S3 transition is the most different one as it involves a local rearrangement of the electron density on the thiazolone part (Figure 3A). A similar picture has been observed for phenol-keto-OxyLH2 (Figure S5D). On the contrary, a simplest picture is observed for the corresponding deprotonated compounds phenolate-keto-OxyLH -and phenolate-5,5-CprOxyL -: the S0-S1 and S0-S2 transitions lead to two well-separated absorption bands (Figure 2A, S4D and S5A). Their electronic nature is similar to the ones found for the corresponding protonated derivatives.

A similar study has been done for phenol-enol-OxyLH2 and its two corresponding analogues phenol-4-MeOxyLH and enol-6'-MeOxyLH. Experimentally, the low-energy absorption band is similarly broad (red lines in Figure 2B and2C) but in this case no strong shoulders are observed like for the phenol-keto derivatives, although the band is slightly asymmetric. We predict that this band mostly corresponds to the S0-S1 transition, while the less intense S0-S2 transition is hidden in the tail (Figure 3B, S4A, S4E and S5B). In this case, the oscillator strength found for S2 is one order of magnitude smaller than the one of S1 (0.5 vs. 6 a.u), in contrast to the similar intensity found for these two transitions for the phenol-keto derivatives, as both states are quite far in energy and their electronic nature is not mixed (Figure S9). Hence, we can state that the general picture found for the phenol-enol derivatives is simpler than the one of the phenol-keto ones, since the S1 and S2 states are energetically more distant and S2 is not so intense. The electronic nature of S0-S1 and S0-S2 transitions are similar to the ones found for the phenol-keto derivatives.

Regarding to the deprotonated forms, the picture of phenolate-enol-OxyLH -and phenolate-4-MeOxyL -is very similar to the one found for their protonated forms: a broad band with the S0-S2 transition hidden in the tail (Figure 2B andS4B). For phenol-enolate-OxyLH -and enolate-6'-MeOxyL -the S1 and S2 states are even more distant in energy yielding to two well-separated bands.

(Figure 2C, S4F and S5C).

Hence, thanks to the computation of the vertical transitions we have been able to rationalize the different absorption band shapes found for oxyluciferin natural forms and its analogues and what is more, reveal hidden transitions which were experimentally embedded in high intense bands. Moreover, we have shown that the studied oxyluciferin analogues present almost identical absorption spectra and electronic nature of the vertical transitions than the corresponding natural forms in water (Figure 4A andB). It should be recalled that for the analogues under study, a quite good agreement between experimental and simulated absorption spectra has been achieved (Figure 2). In addition, the simulated absorption spectra of oxyluciferin natural forms in water are in line with the reported ones obtained by applying the multivariate curve resolution-alternating leastsquares (MCR-ALS) procedure (Figure 4B andC).

Emission Spectra Simulation.

In this section we analyzed and rationalized the shape of the emission spectra of the three selected oxyluciferin analogues and their corresponding natural forms (Chart 2) in both their protonated and deprotonated forms. With this aim, a combined experimental and computational study has been performed.

Starting with the experimental results, the steady-state emission spectra recorded at pH 5 and 11 are quite similar for all oxyluciferin analogues (dotted red and blue lines in Figure 5). In particular for 5,5-CprOxyLH and 6'-MeOxyLH the spectra at the two pH overlap while for 4-MeOxyLH only an additional weak, higher-energy band arises at pH 5 (Figure 5B). This is due to the fact that the protonated species are very short-lived in the excited state due to Excited State Proton Transfer (ESPT) and hence, essentially only the emission of the deprotonated forms is observed by steadystate fluorescence, even if the protonated form is excited. To evidence the ultrafast ESPT, femtosecond pump-probe measurements were performed on 5,5-CprOxyLH in two different pH conditions: the compound was dissolved in aqueous buffer at pH=5, where it is protonated, and at pH=11 where it is deprotonated (pKa~7.8). To extract the characteristic time associated to the proton transfer we analyzed the positive contribution associated to the stimulated emission (SE) of the protonated and the deprotonated forms for both pH. At pH=11, the compound is already deprotonated in the ground state. The positive band around 610 nm rapidly red shifts to 640 nm that is the fluorescence emission wavelength of the deprotonated compound (see Figure 6). Hence the 640 nm band is attributed to SE from the S1 state, and its early red shift is attributed to the rapid Stokes shift due to vibrational and/or solvent relaxation (<1 ps), according to the fit of the 640 nm kinetic trace (Figure 6). After this initial rapid Stokes shift, the signal decays with a monoexponential behavior on the 700 ps time scale in excellent agreement with the measured fluorescence lifetime. [START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF] At pH=5, the early band centered at 525 nm is interpreted as SE, in agreement with the known 525 nm emission of the protonated species. [START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF] This band rapidly decays while the 640 nm band characterizing the emission of the deprotonated form rises. This spectral evolution is a clear signature of the ESPT.

The rise and decay of the 640 nm and 525 nm bands, respectively, are displayed in Figure 6. Their global fit confirms that both kinetics evolve simultaneously and bi-exponentially with the same 1.5 ps and 16.5 ps time constants. Finally, after formation of the deprotonated form in the excited state, the latter decays on the same time scale as at pH=11. Next, we compared the emission wavelengths maxima of the three analogues previously measured by steady-state spectroscopy with the simulated spectra. [START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF] For all the analogues, the simulated spectra of the deprotonated forms match the steady-state fluorescence spectra recorded at pH 11 (thick and dotted blue lines in Figure 5) with maxima (620, 554 and 551 nm for 5,5-CprOxyLH, 4-MeOxyLH and 6'-MeOxyLH, respectively) in good agreement with the values recorded by steady-state fluorescence (640, 560 and 550 nm for 5,5-CprOxyLH, 4-MeOxyLH and 6'-MeOxyLH, respectively). [START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF] Regarding the simulated emission spectra of the protonated forms (thick red lines in Figure 5), they cannot be directly compared, for all compounds, with the steadystate fluorescence spectra recorded at pH 5 due to efficient ESPT, as demonstrated for 5,5-CprOxyLH by pump probe experiments. In the case of 4-MeOxyLH, time-correlated single photon counting measurements evidenced a slower ESPT dynamics (240 ps) [START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF] , which is slow enough to enable monitoring the steady-state emission from the protonated form (Figure 5B, red dotted line band centered at 450 nm). Moreover, it was possible to experimentally determine the emission maxima of two analogues of the (protonated) phenol-enol and phenol-keto forms (6'-Me-5,5-CprOxyL and 4,6'-DMeOxyL, where no ESPT reaction can occur due to replacement of the labile H atom by a methyl group) at 525 and 445 nm, respectively. [START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF] The simulated emission maxima of 5,5-CprOxyLH, 4-MeOxyLH and 6'-MeOxyLH, are obtained at 526, 439 and 430 nm, respectively thus fully in line with the experimental spectra.

Besides, we have also simulated the emission spectra of the six possible forms of oxyluciferin and compared with the ones simulated for the analogues (Figure S8). In all cases, almost overlapping spectra have been obtained, with less than 0.1 eV differences in the emission maxima. Hence, we can state that all the analogues studied in this work are representative of oxyluciferin in water in terms of the emission properties.

So, thanks to this analysis we have been able to rationalize the emission spectra experimentally recorded for the three analogues at pH 5 and 11, demonstrating an efficient ESPT for their protonated forms. Moreover, we have provided by simulation the emission spectra of the protonated forms of 5,5-CprOxyLH and 6'-MeOxyLH, impossible to record by steady-state experiments. Finally, we have shown that the analogues under study are representative of oxyluciferin natural forms in water (Figure 7A andB). It should be remarked that it has been achieved a good agreement between the spectra obtained by applying the MCR-ALS procedure [START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF] and the simulated emission spectra of the six possible forms of Oxyluciferin (Figure 7B andC).

CONCLUSIONS

In this study, we have presented a simple procedure to simulate the absorption and emission spectra of oxyluciferin and their analogues in water solution. It has to be noted that this procedure considers at the same time the interaction of the solute with explicit water molecules and dynamical effects, giving a more realistic picture of the system. In general, a quite good agreement between the simulated and experimental spectra has been obtained hence, independently confirming the MCR-ALS results for oxyluciferin in water.

By analyzing the electronic nature of the vertical transitions, we have found a main difference between phenol-enol and phenol-keto derivatives: the absorption band of the phenol-keto derivatives was found to be structured, since three optically bright excited states are close in energy; whereas for the phenol-enol derivatives an intense band corresponding to the S0-S1 transition has been found with a high-energy tail corresponding to S0-S2. Moreover, it has been shown the key importance of including the dynamical effect for phenol-keto derivatives to reproduce the structured absorption spectra experimentally found.

Regarding to the emission spectra, we have demonstrated an ultrafast ESPT for the neutral phenolketo form, not time-resolved in our previous work. [START_REF] Ghose | Emission Properties of Oxyluciferin and Its Derivatives in Water: Revealing the Nature of the Emissive Species in Firefly Bioluminescence[END_REF] In addition, we report the simulated emission spectra of the protonated forms, being impossible to record them by steady-state experiments.

Finally, thanks to the comparative study of the photochemical properties of the natural oxyluciferin forms and their corresponding analogues presented in this work, we have demonstrated that these analogues can be used to mimic natural oxyluciferin forms, at least in water solution. This will promote future studies focused on rationalizing the emission color modulation in fireflies, as the possible chemical equilibria can be smartly blocked by using these derivatives.

Chart 2 .

 2 Chemical structures of the six possible chemical forms of oxyluciferin and the corresponding analogues under study.

Figure 1 .

 1 Figure 1. Graphical representation of the hydrogen bond pattern for selected MD snapshots of A) phenolate-keto-OxyLH -, B) phenolate-enol-OxyLH -and C) phenol-enolate-OxyLH -. Atom labelling for

Figure 2 .

 2 Figure 2. Normalized simulated absorption spectra (thick lines) of both the protonated (red lines) and

Figure 3 .

 3 Figure 3. Simulated absorption spectra and molecular orbitals involved in each optically bright electronic transition of A) phenol-5,5-CprOxyLH, B) phenol-4-MeOxyLH and C) phenolate-4-MeOxyL -.

34

 34

Figure 4 .

 4 Figure 4. Summary graph of the simulated absorption spectra considering explicit water molecules of A)

Figure 5 .

 5 Figure 5. Normalized simulated emission spectra (thick lines) of both the protonated (red lines) and

Figure 6 .

 6 Figure 6. Left Panel: Results of a pump-probe experiment performed on 5,5-CprOxyLH in water at

Figure 7 .

 7 Figure 7. Summary graph of the simulated emission spectra considering explicit water molecules of A)

  ASSOCIATED CONTENT Supporting Information. Computational Details. Absorption Spectrum Shape Convergence. Analysis of Hydrogen Bond Interactions. Simulated Absorption Spectra Optimizing the Chromophore. Simulated Absorption Spectra of Oxyluciferin Forms and Analogues. Analysis of the Absorption Spectra of Oxyluciferin Analogues. Simulated Emission Spectra Optimizing the Chromophore. Emission Spectrum Shape Convergence. Simulated Emission Spectra of Oxyluciferin Forms and Analogues. Emission Oscillator Strength. Relative Absorption Intensity for Phenol-keto and Phenol-enol Derivatives.

AUTHOR INFORMATION

Corresponding Author *E-mail: cristina.garciairiepa@u-pem.fr *E-mail: isabelle.navizet@u-pem.fr ACKNOWLEDGMENT All authors are grateful to the French Agence Nationale de la Recherche (grant ANR-BIOLUM ANR-16-CE29-0013). Cristina García-Iriepa acknowledge Fundación Ramón Areces for a postdoctoral fellowship.


Eitan Altman
email: eitan.altman@inria.fr

Tania Jimenez
email: tania.jimenez@univ-avignon.fr

Tania Jim

Measuring Audience Retention in YouTube

come

INTRODUCTION

In beginning of 2017, the average number YouTube videos from mobiles exceeded 1,000,000,000 views and Google's annual revenue generated from YouTube reached $4,000,000,000. This has generated a huge research interest in understanding the video propagation in the Internet and how this propagation can be accelerated.

The propagation of YouTube videos can be observed and studied at various levels:

• We have proposed in [START_REF] Richier | Bio-Inspired Models for Characterizing YouTube Viewcount[END_REF] an epidemic approach for the analysis of the propagation of a single video based on the viewcount process of the video. We studied the viewcount of 80,000,000 videos and showed that over 90% of them can be classified within one of the six epidemic models with a mean square error below 5%. Note that the viewcount information over a video is quite often available (it becomes unavailable if the creator does not wish it to be, or if there are copyright issues).

• The level of Internet Memes. The word Meme was coined in page 192 of [START_REF] Dawkins | The Selfish Gene[END_REF] as a concept for describing evolutionary principles and for explaining the spread of ideas and cultural phenomena. Internet memes are a subset of this general meme concept specific to the culture and environment of the Internet. Wikipedia cites from [START_REF] Solon | Richard Dawkins on the internet's hijacking of the word 'meme[END_REF] Dawkins who characterized an Internet meme as being a "meme deliberately altered by human creativity, distinguished from biological genes and Dawkins' pre-Internet concept of a meme". In [START_REF] Altman | Social Networks: A Cradle of Globalized Culture in the Mediterranean Region[END_REF] we studied three videos at the level of both Internet meme as well as of the viewcount processes related to each video. To get information about the propagation of Internet memes we used Google trends.

• Marketing or advertisement policies play an important role in accelerating the propagation of a video. The study of this propagation aspect includes game theoretical approaches that answer the question of which content to advertize, how much and/or how long to invest in advertizing [START_REF] Altman | A stochastic game approach for competition over popularity in social networks[END_REF], when to post or to advertize a content (time of the day or of the week) [START_REF] Altman | The Ordered Timeline Game: Strategic Posting Times Over a Temporally Ordered Shared Medium[END_REF].

• The protocols used in the recommendation system of YouTube and/or of other social networks over which videos are posted have of course an impact on the propagation of YouTube videos. In [START_REF] Portilla | A Study of YouTube recommendation graph based on measurements and stochastic tools[END_REF] we carried out an empirical analysis of some stability properties of the stochastic processes that describes the sequence of videos that are watched by a YouTube user who moves from watching one video after another according to respective recommended videos.

Analysis of audience retention has many applications. It can be used to assist with the decision of where on a video, or on what video to place advertissements so as to maximize their visibility. Such a use in the context of audience of television programs has been patented in [START_REF] Vinson | System and method for measurinng television audience engagement[END_REF].

Audience retention information can also be used in the context of caching [START_REF] Maggi | Adapting Caching to Audience Retention Rate: Which Video Chunk to Store?[END_REF] as it can help identifying what parts of a video would benefit most from caching (as we receive information on the percentage of viewers that watch the video in each point in time).

The audience retention provides a measure for the potential of a video to become epidemic. It can thus be used to select videos for uploading. Companies who produce videos further use it as a feedback on videos they create and then when possible, the video can be modified by using the insight from data on public retention, see [START_REF] Fishman | Understanding Audience Retention[END_REF].

A video version of our work can be found at https://www.youtube.com/watch?v=H3EWUt1Ot5A

EGO NETWORK APPROACH

Models for analysing popularity of videos and the virality of content have often been studied and analysed based on statistical inference from large sets of videos. The larger the set is, the more precise are the conclusions that can be drawn on a randomly chosen "representative" of the group. Yet popularity and virality can also be studied within the framework of Ego networks [START_REF] Valerio Arnaboldi | Analysis of Ego Network Structure in Online Social Networks[END_REF][START_REF] Cook | [END_REF] in which the object of the study may be a given small set of videos or of video channels, which need not be representative of all videos, nor even of all videos within some communities. This small set may contain a single interesting element which we may wish to study. In the video "Why study Ego networks" [START_REF] Cook | [END_REF], the author presents various examples where one is interested in the study of a small subnetwork centered around an individual called EGO, rather than in the study of the entire network. User retention is a particular property of a video that is interesting to study in the Ego network framework. Such a study enables video creators to increase the popularity of a particular video or of a set of videos. Some measurements of retention are of course needed, but in the case of YouTube they are indeed made available to the video creator by YouTube.

AUDIENCE RETENTION AND ENGAGE-MENT GRAPHS

A key indicator for audience engagement is the average retention, this is the time spent by a viewer watching the video averaged over all viewers. The data on audience retention of a video is not publicly available and is provided by YouTube only to the video creator. We have therefore spent an effort in creating some 1000 videos since 2012 which have received more than a million of views. More precisely, we shall use in this work videos from the following YouTube channels:

1. altman2208, 129 videos, 76736 views, 102 subscribers, 2. eitan meir altman, 410 videos, 869000 views, 970 subscribers 3. eitan m. altman, 189 videos, 69534 views, 170 subscribers 4. musicandshows eitan, 274 videos, 26899 views, 30 subscribers Note: we did not attempt to study the generic retention in a "typical" video. Neither did we restrict the study to a specific community. We used the Ego network approach since we are interested in understanding how can a video creator use retention data made available to him by YouTube in order to increase the popularity of his video channels.

Inspired by [START_REF] Fishman | Understanding Audience Retention[END_REF], we decompose the engagement graph into three disjoint parts: the beginning, the core and the end of the video. We use the corresponding terms "nose", "body" and "tail" from [START_REF] Fishman | Understanding Audience Retention[END_REF]. As an example consider Figure 1. According to [START_REF] Fishman | Understanding Audience Retention[END_REF], there is a "large drop in engagement at the very beginning of the video, a steady drop over the middle section, and then another inflection point at the end." In [START_REF] Fishman | Understanding Audience Retention[END_REF] the nose is defined as the first 2% of the video.

In a retention graph, such as in Figure 1, the horizontal axis represents the time t and the vertical axis x(t) represents the percentage of persons that watched the video at time t.

Let X be the maximum fraction that watched the video at any time. YouTube allows for X to be greater than 100%. This may occur if after viewing the video viewers drag the cursor to the beginning and watches the video (or parts of it) again. This is well illustrated in Figure 2.

The average fraction of persons watching the video is given by

X = 1 T T 0 x(t)dt
where T is the duration of the video. But X is also the ratio between the duration of the video watched, averaged over all those who watched the video, and the video length. It is called in YouTube the "average percentage viewed" (APV).

The average duration T that the video is watched is given by

T = T 0 x(t)dt = T X. (1)
It is called in YouTube the "Average View Duration" (AVD). In the first figure, the percentage of viewers reaches 170%. YouTube explain this phenomenon by the fact that many viewers watched the video several times consecutively by dragging the mouse to the beginning. The non-monotonicity then could be due to dragging the mouse not all the way to the beginning, which results in points further away from the beginning to be viewed more than the beginning of the video. Note also that in Figure 2, the average video watching time T exceeds the video duration.

Figure 3 is the curve corresponding to one of 14 videos that we filmed in a concert. The U form of the nose is common to all of them and is due to the fact that each of them starts with around one minute of introduction in which the conductor explains briefly the piece that is about to be played. A fraction of the viewers were not interested in the introduction but were interested in the piece. Thus the U shape of the nose. This video has an APV of 46%, its duration is 4'48" and 304 views. We filmed the concert at the end of January 2016. The retention curve in Figure 1 corresponds to one of 14 videos we took a year later with the same orchestra and at the same place. This time we did not include the introductions in the videos and the U type noses have disappeared as can be seen in the curve.

IMPROVING VIDEO RETENTION

We begin by discussing the shortening of videos. We shall provide below conditions under which shortening the video would result in an increase in its average fraction of viewers.

As an example, we reduced a video of 5'51" on the French song "sur le pont d'Avignon" (see Figure 7) to 2'33" (see Figure 8). As a result, we observed an increase in the averaged percentage viewed from 16% to 42%. The average viewed duration increased by 2.625 while the whole video duration decreased by 2.294. Thus not only did the average viewed time increase, it increased by a factor which is lower bounded by the amount by which we shortened the video.

We shall say that a video is causal if the corresponding x(t) is causal, i.e., it is only a function of events that occurred before t. By truncating a causal video defined on [0, T ] to [0, S] for some S < T , the retention curve of the shorter video is the same as the initial one restricted to [0, S).

Theorem 2. Assume that we shorten a causal video 1 to obtain a new video 2. (i) We have

X2 = 1 T2 T 2 = 1 T2 T 1∆ ≤ T1 T2 X1 (2)
where ∆ ≤ 1 is given by (

∆ = T 2 0 x(t)dt
) 3
Proof.

(i) Follows from direct computations. (ii) The first part follows from

dT dT = d T 0 x(t)dt dT = x(T ) > 0
The second statement follows from

dX dT = dT -1 T 0 x(t)dt dT = T -1 (x(T ) -X(T )) = 1 T 2 T 0 (x(T ) -x(t))dt < 0 (4)
The first part of the theorem identified the best possible retention curve x(t) under which it is worthwhile shortening a video from T1 to T2. This was done by upper bounding ∆. If x(t) = 0 for T2 < t < T1 then the AVD (1) is the same for the originall and the shorteneed video while the correspondingn duration was shortened by a factor of T2/T1.

Similarly, the performance of the worse possible retention curve for shortening a video can be obtained by lower bounding ∆.

Lemma 3. Assume that we shorten a causal video 1 to obtain a new video 2. Assume that x1 is strictly positive and is decreasing. Then ∆ ≥ T 2 T 1 .

Proof.

∆ ≥ T 2 0 x(T2)dt T 2 0 x(T2)dt + T 1 T 2 x(t)dt ≥ T 2 0 x(T2)dt T 2 0 x(T2))dt + T 1 T 2 x(T2)dt = T2 T1
Note that if x(t) is strictly decreasing then the last inequality is strict. This is thus an alternative way to obtain (3). We shall next identify a retention that approaches this bound.

Lemma 4. Consider a causal video of duration T1 which we wish to shorten to a duration T2. Assume that x(0) = 1.

Assume that the retention curve has two phases such that for some 0 ≤ < 1: (i) it has a nose of duration T1 < T2, and (ii) it has a flat body, i.e. x(t) = x(T1) is constant over all t ≥ T . Then

|X1 -X2| ≤ log(T1/T2) (5)
Proof.

It follows from (4) that

dX dT = T 0 (x(T ) -x(t))dt T 2 = T 0 (x(T ) -x(t))dt T 2
We conclude that

dX dT ≤ T
Multiplying by dT both sides and integrating between T2 and T1 we get [START_REF] Fishman | Understanding Audience Retention[END_REF].

The take away message from Theorem 2 is that shortening a video from duration T1 to duration T2 may improve its APV (Absolute Percentage Viewed) by a factor of up to T1/T2. The smaller x(t) is in the interval [T2, T1] the more we approach this improvement. Thus shortening videos is more appealing when the percentage of those that saw the video till the end is small. On the contrary, there is little gain in shortening a video if a large part of its body has a constant retention.

Figure 4: Relative audience retention

We used this intuition on the video of a Palestinian painter, Nabil Anani (https://youtu.be/ZLHFWHwYqzI), which we had prepared in the form of a slide show. Its APV is 29, its duration is 6'16" and it has 1516 views. At time 4'20", my piano music which accompanies the show sounds as if this were the end. We were therefore suggested recently to shorten the video (which is possible on YouTube). However, its retention curve 4 shows that 22% of its viewers saw the video till its end. In view of the above take away message, we decided not to do any change in the video.

Equation (3) states that under quite standard conditions, shortening a video will increase its APV index. Thus in terms of the APV, shorter videos have an advantage over longer ones. For reasons that will be stated later this is not a desired feature. In the next section we shall describe both YouTube's as well as our own proposed solution to this problem.

OTHER RETENTION MEASURES

To fairly compare the retention of videos of different lengths, YouTube introduced a measure called the "relative audience retention" (RAR). It consists in normalizing each point x(t) of the original video by the value Ave(t) obtained by averaging xi(t) over all similar YouTube videos i ∈ I which have the same duration T as the original video. At each point in time the relative retention can take a value between 1 and 5 where the integers 1-5 correspond respectively to the categories "Low, below average, average, above average, High". These categories remind of the MOS measure for the quality of experience as standardized by the ITU.

In Figure 5 we see the relative retention curve corresponding to the absolute audience retention of Figure 4. Recall that it was suggested to end that video at time 4'20" and that we decided not to follow the suggestion since the benefit would have been small. The RAR seems definitely to agree with our decision and as we see in the Figure, it is precisely at the time that was suggested for ending the video that the rating moves steadily from 3 (average) to 4 (above average).

Figure 5: Relative audience retention

In Figure 6 we see the relative retention curve corresponding to the absolute audience retention of Figure 3. We see the same phenomenon again: there are above 30% of the audience who watch the video till the end, and in parallel we see that the RAR has a long increase in its score which gradually improves from 2.6 to 4 (it then decreases again during the tail part). In contrast, consider the retention curve of the French song "Sur le pont d'Avignon" in Figure 7. Its duration is 5'51", and it has an APV of 16%. We observe that a fraction of less than 5% of viewers saw it till the end. We thus decided to create a shorter version. The new short version is 2'33" long and its APV has increased to 40% or more (details are given below as well as the relative retention curve).

A NEW LOYALTY MEASURE OF RE-TENTION

In the previous Section we have shown that videos whose fraction of the audience that remain till the end of the video is large benefit little from shortening. Moreover, some evidence from our videos showed that they get better RAR ranking.

This motivates us to introduce a new audience retention measure which is the fraction of the audience that remain till the end of the body of the video. We call this the loyal percentage. Here are other advantages to judge video's quality using this index.

1. When a person creates a video he often intends it to be viewed from the beginning to the end. Often the punchline comes in the end.

2. If a large part of the audience leaves before the end then they will probably not be interested in the next video by the same author.

3. In the context of real time broadcasts, if a person leaves before the end then she is more likely to switch to another program broadcasted at the same time and the chances to come back to watch the original channel are smaller.

4. In the context of YouTube, the video creator can embed at the 20 last seconds of the video personal recommendations to other videos or playlists. This option is called "End Screen". If a person leaves before the end of the video then he does not have access to these recommendations. Hence again the importance of this performance measure. We finally present an example that shows that the APV and RAR may not be consistent. We say that two indexes are consistent if a better performance according to one of them results in a better performance with respect to the other. While all three versions have almost identical APV performance (with the one of Feb 21 having the highest APV of 43% and that of March 2 the lowest at 40%), the relative retention ranking has quite a surprising behavior.

The RAR ranking of the period ending at March 2 (which has the lowest APV among the three) is almost uniformly higher in its relative ranking by 1 point with respect to that of the period ending at 21 January (which has the highest APV). The RAR ranking of the period ending at January 20 is in between the two others.

We thus have in this example a complete inversion between the rankings of the RAR and that of the APV. More- over, very small variations between the Absolute retention curves result in high variations between the RAR curves. Now if we compare the APV with our loyalty measure, we see that the order of preference is the same under both rankings, and furthermore the percentage difference in the performance of the video in different periods is close under the two performance measures.

We conclude that in this example there seems to be consistency between the APV and the loyalty measure where as there is no consistency between these measures and the RAR index. Note that these conclusions were only obtained for a single example and may not hold in other ones. Further experimentation would be needed on a larger set of videos in order to understand the relation between the differnt performance measures.

PERSPECTIVES AND FUTURE WORK

In this paper, we have studied the relation between various measures of audience retention both existing ones as well as a new proposed measure, and have shown how such measures can be used in order to help deciding when a video can profit in being shortened. The measures include averaged ones in the form of a single number, such as the APV, the AVD, and the loyalty measure, as well as dynamic measures such as the retention curve or the relative audience retention.

Shortening videos

We have provided assumptions under which shortening a video seems beneficial or not. Of course there may encounter other user behavior that would provide different conclusions under different assumptions. As an example, The causality definition does not consider the effect that the knowledge of video duration has on the audience, i.e., if someone knows that there is still 1 min of video instead of 1 hour, this knowledge will most probably influence her/his interest to watch further or not. It would be interesting to study the influence of the information on the remaining video duration on the abandonment process of viewers.

Where as this work along with previous work cited here provide some understanding of when shortening of a video has a potential to improve the retention, a more systematic supervised learning may be necessary if one wishes to come up with automatic decision rules. A statistical analysis would then be required to evaluate the false positive and false negative probabilities induced by learning rules.

In and Beyond the Ego network framework

In this work, we adopted the Ego network approach of studying a restricted small number of videos rather than a random video. This allows one to predict virality of a video at a very early stage. A high degree of audience retention would indicate that the video has a potential of growing viral even if the number of viewers of the video or the rate of views of the video is small. The relation between number of views, the rate of new views and retention measures is the subject of future work.

Even within the Ego network framework, many empirical results over retention measures have been obtained by using observations from a couple of videos to make an argument. While this approach allows us to get some insight on retention (e.g. discover phenomena such as getting non monotone retention curve), a statistical study can further provide more quantitative understanding of retention.

Other directions

There are several ways to pursue the research on audience retention. We plan to study the relation between the audience retention and network operation characteristics such as losses, throughputs and delay. These can be inserted to videos and their impact on the audience retention can then be measured using the tools that were discussed in this paper. This can be used to optimize the network resource allocation, and in particular the video resolution, caching decisions as well as radio channel characteristics (modulation and error correction).

Since the relative audience retention is obtained by a normalising procedure over a large number of videos, this opens the door for a statistical analysis that may take into account the video category (eg sport, music, travel etc...). This, as well as the analysis of the structure of a typical retention curve are properties that go beyond the Ego network framework.

Yet using this framework is of interest not only because video creators have access to the retention curve, but also because it allowed us to obtain properties that hold for any causal video, such as the impact of making the video shorter.

1

 1

Figure 1 :

 1 Figure 1: A typical behavior of the engagement graph

Figure 2 :

 2 Figure 2: The case of X larger than 100%.

Figure 3 :Figure 2 and

 32 Figure 3: A non monotone nose

T 1 0

 1 x(t)dt The inequality in (2) becomes an equality and ∆ = 1 if x(t) = 0 for all T2 < t < T1. (ii) Assume that x1 is strictly positive and strictly decreasing. Then T 1 > T 2 and X1 < X2.

Figure 6 :

 6 Figure 6: Relative audience retention

Figure 7 :

 7 Figure 7: Reflative audience retention

Figure 8 :

 8 Figure 8: Absolute audience retention for the shortened video of "Sur le Pont d'Avignon" from Jan 5 till Feb 20, 2017.

Figure 9 :

 9 Figure 9: Absolute audience retention for the shortened video of "Sur le Pont d'Avignon" from Jan 5 till Feb 21 2017.

Figure 10 :

 10 Figure 10: Absolute audience retention for the shortened video of "Sur le Pont d'Avignon" from Jan 5 till March 2nd 2017.

Figure 8 -

 8 10 correspond to the shortened video of the song "Sur le Pont d'Avignon" which we had created on January 5th, 2015. The duration of the video is 2'33". We checked the AVP obtained by the same video in three different periods: Jan 5 till Feb 20, Jan 5 till Feb 21 and Jan 5 till March 2, 2017. The corresponding RAR are 42%, 43% and 40% (and the number of viewers increased from 61 to 70). Figures 11-13 are the corresponding RAR ranking.

Figure 11 :

 11 Figure 11: Relative audience retention for the shortened video of "Sur le Pont d'Avignon" from the period Jan 5 till Feb 20, 2017.

Figure 12 :

 12 Figure 12: Relative audience retention for the shortened video of "Sur le Pont d'Avignon" from Jan 5 till Feb 21 2017.

Figure 13 :

 13 Figure 13: Relative audience retention for the shortened video of "Sur le Pont d'Avignon" from Jan 5 till March 2nd 2017.

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