domain: info math

text:
Résumé

Les Rencontres R e édition ont eu lieu à Avignon du au juin et ont réuni près de participants pour présentations, tutoriels et invité.e.s.

Elles sont portées par la Société Française de Statistique (SFdS) et ont été co-organisée par l'Institut National de Recherche pour l'Agriculture, l'Alimentation et l'Environnement (INRAE) et Avignon Université (AU) avec le soutien de nombreux sponsors.

Cet événement en présentiel était très attendu par la communauté francophone R après une pause dû à la pandémie de COVID-, il a permis à la communauté de se retrouver, d'échanger et de partager sur le langage R et son écosystèmes. Il a réuni un public aussi bien débutants que confirmés et expérimentés issus de différents secteurs d'activités privé et public.

Un bilan très positifs

Cette édition a été plus que réussi et a permis de dynamiser la communauté R francophone autour de cette dernière. La qualités des présentations, la bienveillance de la communauté et les nombreux retours des participants sur l'organisation nous permettent d'attester du succès de cette édition.

Les partenaires financiers ont permis d'obtenir un budget équilibré ce qui a grandement joué dans l'organisation et la réussite des Rencontres R. Cela a permis de garantir un prix d'inscription comparable aux éditions précédentes.

La présence des sponsors lors de l'événement a amené de nombreux échanges et découvertes et les livres offerts par les éditeurs (Editions ENI, Editions Technip and Orphrys, Taylor and Francis et Springer) ont susité de la curiosité parmi les participants. Un remerciement particulier aux différents participants qui ont fait de cet événement un moment unique et chaleureux.

Supports numérique

Le comité d'organisation par l'intermédiaire de son président reste disponible pour de plus amples informations à jean-francois.rey@inrae.fr.

Language R

R is an open-source programming language and an IDE devoted to statistical analysis. It is available for different platforms (OS, Windows, Linux) and is completely free. In recent years, its growing success in public services (university, research centers, govermental agencies, ministers) as well as in the industry (pharmaceutic industry, informatics, energy, online commerce, marketing, food production etc.) made R an undispensable tool in data analysis. R environment is relying on a wide international and French-speaking community, which has its own journal, national and international conferences and a stack R. It is a language in active development (more available packages), which attracts people with a very different background (informaticians, statisticians, data scientists, biologists, researchers, economists, doctors. . . ) working in all economical sectors. It is a powerfull and complete software.

The «Rencontres R» is the biggest French event devoted to R, organized by the French Statistical Society. They give a perfect occassion to exchange and share ideas on different problems addressed by the community. Every year this meeting gathers between and participants, attracting both experienced users and beginners.

Organizing committee

List of Abstracts -Keynote talks

Tips pour combattre le syndrome de l'imposteur

Aurélie Vache

OVHcloud

Who has not once said the sentence: I feel like an impostor? I don't feel entitled to do this or do that? Some people are convinced that they do not deserve their success, despite the efforts they make to succeed. They often convince themselves that their success is not linked to their work, their personal accomplishment, but simply to luck or the work of others. In fact, they live permanently with a feeling of deception and constantly fear that someone will unmask them from one day to another.

In this talk we will return to what the impostor syndrome is, how it is reflected on a daily basis and we will see that it is not inevitable, on the contrary, that there are tips and tricks for the fight, overcome and improve.

Data science without the data

Rhian Davies

Jumping Rivers

As data scientists, we sometimes find ourselves faced with the daunting task of writing code without actually seeing the data we are working with. Whether it's due to data privacy concerns, limited access, or simply data that has not yet been collected, we often have to rely on incomplete or synthetic data to develop and test our code.

In a recent project, we worked on patient-level data. As such, the controls around the data and analysis (were rightfully) tightly controlled. We'll share how we used dummy data and mock-ups to inform code development, maintaining flexibility and adaptability in the face of changing data requirements. We'll also discuss the importance of and collaboration between developers and subject experts to ensure that code is developed with a deep understanding of the data domain By understanding these challenges and developing effective strategies for overcoming them, we can ensure that our code is robust, reliable, and effective, even in the absence of direct data access.

R dans l'univers de la Dataviz

Yan Holtz

Datadog

With a vast array of tools, use cases, and plot types available, it can be difficult to navigate the world of data visualization (dataviz). In this presentation, we'll explore how dataviz is used by various professions: research, data journalism, business intelligence, and others. We'll look at the different tools available, with a focus on R.

Although each tool has its strengths and weaknesses, R has become a popular choice among data scientists and analysts. We will explore R's position in the dataviz ecosystem, highlighting its advantages and limitations. We'll also look at d .js, a popular JavaScript library for creating graphics, and compare it to R in terms of its capabilities and limitations.

Whether you are new to data visualization or an experienced practitioner, this presentation will provide you with valuable insights into the world of dataviz and equip you with the knowledge to choose the right tool for your needs.

Pastels, paillettes et packages pour accompagner la recherche avec R

Lise Vaudor

CNRS

For the last years, I work as a research support engineer in a geography laboratory.

To support my fellow researchers, teachers, students in their research and help them in their data analysis with R, I use various tools. The first and the one that benefits the greatest number consists in creating and offering educational materials (the pastels): blog, online tutorials, illustrations, and a regular internal training offer for the members of my laboratory. In certain specific cases, when the level of technicality required by an analysis is too high or when the need to explore the data in a user-friendly way is strong, I prefer to offer more "turnkey" tools such as shiny apps (glitter). Finally, in other cases, my contact is a more confirmed R user with a specific need and I analyze this need with him to develop a certain number of dedicated functions (packages).

I will explain what my job consists of on a daily basis, by presenting my personal background and enameling it with examples of collaborations.

Figure

  Figure . : Rencontres R

Remerciements

Le comité d'organisation tient à remercier les organisateurs de l'édition de pour leurs conseils et aides apportés ainsi que les différents comités, les bénévoles et les orateurs impliqués dans la réussite de l'événement. Nous tenons à remercier nos partenaires financier sans qui l'événement n'aurait pas pu se tenir : Le département MathNum INRAE, l'Institut National de la Statistique et des Etudes Economiques (INSEE), la société We would like to thank our financial partners without whom the event could not have been held: the MathNum INRAE departement, The National Institute of Statistics and Economic Studies (INSEE), the dreamR compagny, CBA informatique, the ThinkR compagny, the Research Federation of Mathematics Units of Marseille (FRUMAM), the Avignon Mathematics Laboratory (LMA), the METORI compagny, LDLC informatique, the Datastorm compagny, the ArData compagny, the SPE INRAE departement, the Makina Corpus compagny, the Avignon IUT of statistics ans Business Intelligence (STID), the Avignon Computer Science Laboratory (LIA), and of course the R Consortium.

A special thank you to the various participants who made this event a unique and warm moment.

The organizing committee through its president remains available for further information at jean-francois.rey@inrae.fr.

Report -EN Summary

The "Rencontres R" ("R Meetings") th edition took place in Avignon from June to , and brought together nearly participants for presentations, tutorials et guests.

They are affiliated with the French Statistical Society («Société Française de Statistique (SFdS)») and organised jointly by National Research Institut in Agriculture, Food and Enviroment (INRAE) and Avignon University (AU) with the support of many of sponsors. This face-to-face event was eagerly awaited by the French comunity of R users after a break due to the COVID-pandemic. It allowed the French-speaking R community to meet, exchange and share on the R language and its ecosystems. it brought together an audience of beginners and experienced people from different sectors of private and public activity.

A very positive outcome

This edition is more than successful and has made it possible to energize the French-speaking R community.

The quality of the presentations, the goodwill of the community and the numerous feedback from participants on the organization allow us to attest to the success of this edition.

The financial partners made it possible to obtain a balanced budget, which played a major role in the organization and success of the Rencontres R, guaranteeing a registration fee comparable to previous editions.

The presence of sponsors during the event sparked numerous discussions and discoveries and the books offered by the publishers (Editions ENI, Editions Technip and Orphrys, Taylor ans Francis and Springer) aroused curiosity.

Digital media

All information on this edition is available on the website https://rr .sciencesconf.org.

The presentation materials are freely accessible ans available on the GitHub repository and GitLab repository as well as viewable on the YouTube chanel @RencontresR.

Thanks

The organizing commitee would like to thank the organizers ot the edition for their advice ans assistance as well as the various committees, volunteers and speakers involved in the success of the event.

Useful Information

The opening of the th Rencontres R meetings took place on Wednesday June at : p.m. and closed on Friday June at : p.m. We offered tutorials before the conference on Wednesday, June morning from a.m. to p.m.

The conference took place in the south building of the Hannah Arendt campus (city center) of Avignon University (the glass building) on the st floor. Three rooms were at our disposal:

As well as the Main Hall between the two amphitheatres where the posters were displayed and some partners had a stand.

Lunches were taken in charge by the conference. They took place in University Restaurant CROUS a few steps from the University (see map below).

Gala diner took place on june at p.m. in the restaurant Le Bercail. The restaurant is situated on l'île de Barthelasse (Barthelasse Island) at Chem. des Canotiers, Avignon. To get there, a ferry boat was privatized for the occasion.

Build a complete machine learning pipeline with tidymodels

Antoine Bichat [1] , Julie Aubert [2] [ ]: Head of R&D project -Servier, [ ]: Research Engineer -INRAE Tidymodels brings together a set of packages facilitating the use of statistical learning methods (such as random forests, Bayesian linear models or not...) in a unified and "tidy" framework. This tutorial will show you how to use these packages to preprocess data, build, train and evaluate a model, optimize hyperparameters, and everything you need to know to complete a supervised statistical learning project from start to finish.

Statspatial : Spatial Analysis and Cartography with R

Kim Antunez [1] , Etienne Côme [2] [ ]: Statistician at INSEE, [ ]: CR at Université Gustave Eiffel

This tutorial offers an introduction to the main packages and tools available in the R ecosystem for manipulating spatial data and making thematic maps, with a focus on vector data.

A large place will be devoted to the handling of the sf package and the realization of simple geoprocessing. We will also cover making static thematic maps with ggplot and mapsf as well as using mapview to easily make interactive data mining maps. Finally, a short introduction to more specific tools and questions will complete this introduction: raster data, spatial smoothing, geocoding, Open Street Maps data, geographic networks, etc.

The majority of the workshop will be devoted to the application of the various concepts and tools on the annual database of road traffic injury accidents.

Spatial Ecosystem in R

Timothée Giraud CNRS R language has long been used to process spatial data. Several fairly recent packages (sf, terra, stars, etc.) have renewed the base allowing the implementation of spatial data processing.

Most of the current developments are forming a robust ecosystem which offers users most of the functionalities formerly reserved for Geographic Information Systems. During this presentation we will show a panorama of the spatial ecosystem of R.

We will illustrate this panorama through a series of examples mobilizing data of different natures (vector data and raster or raster data) from the free geographical database OpenStreetMap.

We will address in particular the acquisition, manipulation and mapping of geographic data, traditional geoprocessing operations as well as more advanced spatial analysis processing.

Poster List

RFLOMICS: Interactive web application for multi-omics data analysis

Audrey Hulot [1] , Delphine CHARIF [2]


Tayeb Chieb

Abdelkader Ladjarafi
email: ladjarafi@gmail.com

Billel Teyar

Jean-François Halet
email: jean-francois.halet@univ-rennes1.fr

Theoretical study of the effect of the solvent on the electronic properties of pyrimidopyrimidine derivatives

Keywords: heterocycles, pyrimidopyrimidine, quantum calculations, solvent effect, tautomeric forms

Quantum calculations were carried out on some pyrimidopyrimidine tautomeric heterocyclic derivatives both in gas phase and in solution in order to study the influence of the migration of hydrogen on their electronic and structural properties as well to highlight the effect of the solvent on the process of tautomerism. Results indicate that the electronic properties of the studied heterocycles are influenced by the nature of the solvent used. All tautomeric forms are more stable in H2O and DMSO than in CHCl3 and in the gas phase. The predominance of the di-ketone form in gas phase and solution is observed.

Introduction

Because of their importance and their widespread presence in living organisms, substances based on pyrimidopyrimidines or tetra-azanaphthalenes -organic heterobicyclic compounds with a skeleton consisting of two pyrimidine rings that are ortho-fused to each other at any position -have aroused the interest of researchers in organic and pharmacology chemistry for several decades [1][2][3][4][5][6][7][8][START_REF] Seio | Synthesis and fluorescence properties of nucleosides with pyrimidopyrimidine-type base moieties, fluorescent analogs of biomolecular building blocks, design and applications[END_REF][START_REF] Monier | [END_REF][11][12]. These molecules are well known for their important biological, biochemical and therapeutic properties as well as their rich and varied chemistry [START_REF] Monier | [END_REF][11][12][13][14][15][16][17][18][19][20][21][22][START_REF] Kitamura | Pyrimidopyrimidine derivatives, method for the production thereof, the pharmaceutical preparations containing them and their use as an anti-allergic agent[END_REF].

Several works reported on the preparation of pyrimidopyrimidines and their derivatives have demonstrated their antiallergic [11,18,[START_REF] Smith | Pyrimidopyrimidine derivatives[END_REF], anti-cancer [12,21,[25][26][27][28], antihypertensive [15,29], anti-inflammatory utility and other antibiotic, antifungal and antibacterial properties [START_REF] Seio | Synthesis and fluorescence properties of nucleosides with pyrimidopyrimidine-type base moieties, fluorescent analogs of biomolecular building blocks, design and applications[END_REF][START_REF] Monier | [END_REF][11][12][30][31][32][33][34]. Some pyrimidopyrimidines have fungicidal activities whereas others have been reported to be agricultural compounds [START_REF] Hackler | fungicidaluse of pyridopyrimidine, pteridine, pyrimidopyrimidine, pyrimidopyridazine, and pyrimido-1,2,4-triazine derivatives[END_REF][36][START_REF] Freyne | Pyrido and pyrimidopyrimidine derivatives as anti-proliferative agents[END_REF][38][39][40][41][42][43]. Pyrimidopyrimidines also have veterinary uses as antibiotics, antiparasites, antipyretics, analgesics, analgesics and anti-inflammatories [44][45][46]. They also are found in meteorites, but scientists still do not know their origin [START_REF] Callahan | Proc. Natl. Acad. Sci. USA[END_REF].

In this work, we are interested in the theoretical study of the influence of proton transfer on the electronic and structural properties of one hypothetical pyrimidopyrimidine derivative, namely the pyrimido [5,4-d]pyrimidine-2,6(1H,3H,5H,7H)-dione C6N4H6O2 (PPD, Figure 1).

The choice of this derivative was guided by the large number of experimental studies on the pharmacological and therapeutic properties of the related heterocyclic dihydropyrimidinone (DHPM) and its derivatives [START_REF] De Fatima | [END_REF][49][50][51][52][53][54][55][56][57][58], arousing our interest in testing the possible fusion of two heterocycles and subsequently study its electronic properties. In this PPD species, the transfer of one proton can theoretically take place from the NH group to one of the neighboring oxygen atoms (having two lone electron pairs), thus leading to a mixture of several keto and enol forms called tautomers. Our interest then focused on the theoretical study in solution of different tautomeric pairs to prove the effect of the solvent on the tautomeric process encountered in this type of keto-enol tautomerization equilibrium. Three solvents of different polarity were chosen, those mostly used in synthesis which strongly differ by their dielectric constants, as chloroform (CHCl3), a non-polar solvent of low dielectric constant (e = 4.8), dimethylsulfoxide (C2H6OS, DMSO), a polar solvent of medium dielectric constant (e = 46.7) and water (H2O), a polar solvent of high dielectric constant (e = 80) -the gas phase is taken as a reference with a dielectric constant (e = 1). The different equilibria of the tautomers studied here are shown in Figure 1.

Computational Details

Calculations were carried out using three methods, namely the post-Hartree-Fock Moller-Plesset (MP2) method [59], and the density functional theory (DFT) method [60,61] using two functionals, the B3LYP (Becke's three-parameter exchange functional in conjunction with the Lee-Yang-Parr correlation functional [START_REF] Becke | [END_REF] with Lee-Yang-Parr [63]) and the M06-2X [64] correlational functional. In all cases, a 6-311++G** basis set was used [65].

Calculations were performed first in the gas phase and then in the presence of three solvents (CHCl3, DMSO, and H2O) using the polarizable continuum model (PCM) [66,[START_REF] Mennucci | Continuum Solvation Models in Chemical Physics[END_REF]. Atomic charges were analyzed using two different approaches, namely the Natural Bond Orbital (NBO) analysis [START_REF] Reed | [END_REF]69] and the atomic polar tensor (APT) analysis [70,71]. Full geometric optimizations were performed, followed by frequency calculations to ensure the nature of energy minima and transition states. All calculations were carried out with the Gaussian16 package [START_REF] Frisch | Revision B.01[END_REF]. Representation of molecular structures and molecular orbitals was performed using the GaussView [START_REF] Dennington | [END_REF] and ChemDraw programs [74]. The pyrimidopyrimidine tautomeric molecules studied here exhibit the phenomenon of tautomerism, which appears when one of the hydrogen atoms can occupy different positions in a given skeleton (H13, H14, H15, or H16 in Fig. 1). Migration of one of these hydrogen atoms create different structures in equilibrium, the composition of the tautomeric mixture of which is often difficult to determine experimentally [START_REF] Elguero | The Tautomerism of heterocycles[END_REF]. Knowledge of the relative stabilities of tautomeric forms of these heterocyclic molecules as well as the conversion from one form to another are important from the point of view of structural chemistry. Moreover, the geometric and electronic structures as well as the thermodynamical stability of the tautomeric forms provide a basis for understanding their biological activities and lead to the determination of their pharmacological properties.

Results and discussion

Stability of pyrimidopyrimidines

First, a conformational study was performed at the B3LYP/6-311++G** level on compounds 2-6 containing OH group(s) to find their most stable structures, varying the H-O-C-N dihedral angle (t) from 0 to 180°. The most stable forms -we checked that they were genuine minima on the potential energy surface -and their relative free energies are shown in Fig. 2 (see also Fig. S1, Supporting Information). Two possible conformations were found for 2 and 3 (t = 0° (2a and 3a) and 180° (2b and 3b)). For compounds 4, 5, and 6, the variation of the dihedral angles H14-O12-C2-N3, H13-O12-C2-N1, H16-O11-C9-N10 and H15-O11-C9-N8 from 0° to 180° leads to four possible conformers (two being identical in the case of 4 and 5). It turns out that conformers a are substantially more stable than the other ones (see Fig. 2). Therefore, in the following compounds (2a, 3a, 4a, 5a and 6a) will subsequently be called 2, 3, 4, 5, and 6, respectively.

ΔG2a=0.0 kcal.mol -1 ΔG2b=7.1 kcal.mol -1 ΔG3a=0.0 kcal.mol -1 ΔG3b=8.5 kcal.mol -1

ΔG4a=0.0 kcal.mol -1 ΔG4b=7.2 kcal.mol -1 ΔG4c=15.1 kcal.mol Full optimization calculations of geometries of the gas phase most stable conformers, hereafter denominated 2, 3, 4, 5, and 6, were also carried out in the presence of three solvents (CHCl3, DMSO and H2O) at the MP2, B3LYP and M06-2X levels of theory for comparison (Fig. 3). The analysis of relative Gibbs free energies obtained in the gas phase and in solution by the three methods indicates the same order of stability (Table 1, see also Table S1, Supporting Information). The conformal compound di-ketone 1 is thermodynamically the most stable species in both gas phase and solution, followed by the tautomers enols 2 and 3 then followed by the dienols 4, 5, and 6, respectively. The stability of the compounds obtained is then in the following decreasing order: 1 > 2 > 3 > 4 > 5 > 6. Compound 6 is therefore the least stable. This result indicates that the migration of a hydrogen atom from the imine function to an oxygenated substituent decreases the stability of the compounds. This decrease becomes more significant in the case of dienols 4, 5, and 6 where there is a transfer of two protons.

Table 1

Relative Gibbs free energies DDG ‡ (in kcal.mol -1 ) in gas phase and in solvent calculated using M06-2X, B3LYP (underlined) and MP2 (values in brackets) methods.

a DG ! "#$-!& -D𝐺 ' "#$(!& ; b DG ! )*+,--D𝐺 ' )*+,-; c DG ! "-! -D𝐺 ' "-! .

Table 2

Solvation free energies (DGsol ‡ , in kcal.mol -1 ) calculated using M06-2X, B3LYP (underlined) and MP2 (in brackets) methods.

a DGsol ‡ = DG ./.0 ! "#$-!& -DG 123 "#$(!& ; b DGsol ‡ = DG ./.0 ! )*+,--DG 123 )*+,-; c DGsol ‡ = DG ./.0 ! "-! -DG 123 "-! .
Results shown in Table 2, indicate that the solvation free energies of the studied species increase when passing from the gaseous state to the solution state and this for the three methods. Thus, the tautomers are more stable in water than in DMSO and then in CHCl3, respectively. This increase is proportional, as expected, to the polarity of solvents used. By way of example using the M06-2X method, the presence of solvent provides compound 1 with a solvation free energy gain of 10.0, 14.0 and 14.1 kcal/mol in CHCl3, DMSO and H2O, respectively. The solvation free energy results obtained by the M06-2X and B3LYP methods are similar for each solvent. For example, for compound 5 the energy gain between the gas phase and the three solvents is respectively 6.5, 9. 2 ) for all phases (see Table S1, Supporting Information). The comparison carried out between the Gibbs free energies obtained with the M06-2X and those obtained with B3LYP in gas phase and in solution gave a very good correlation between both methods with determination coefficients R 2 ≃1 (Fig. 4).

Fig. 4.

Correlation diagram between B3LYP computed and M06-2X computed Gibbs free energies in gas phase and in solution.

Orbital analysis

The computed energies of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), as well as the HOMO-LUMO gap of compounds 1-6 obtained after complete optimization in the gas phase and in solution using the three methods M06-2X, B3LYP and MP2 are given in Table 3 and Fig. 5. Results indicate that the solvent hardly influences the HOMO-LUMO gaps. Comparable HOMO-LUMO gaps are computed in both phases and for the three methods for all compounds. The nodal properties of the HOMO and LUMO of 1-6 are shown in Fig. 5.

Both HOMOs and LUMOs are heavily distributed over the heterocycles. The Fig. 5 shows that the HOMO and LUMO of the tautomers 1-6 are mainly localized on the two pyrimidine rings, involving an important contribution of the nitrogen atoms, indicating that the two pyrimidine rings of the tautomers 1-6 are good π electrons donors. As expected, the HOMO-LUMO gaps obtained by MP2 are largely overestimated and practically twice those obtained with B3LYP and larger by 2.5 eV on average than those obtained with the M06-2X method. As expected, this is due to the positive energies of the MP2-computed LUMOs. Positive values for the LUMOs mean that the attachment of an electron to these molecules is energetically unfavorable. We also observe, with all methods, that compound 1 with the lowest LUMO energy has a strong electronic affinity in the gas phase and in solution, revealing its important electro-acceptor character. 10 M06-2X-, B3LYP-and MP2-computed energies of the HOMO and LUMO (in eV), HOMO-LUMO gaps (in eV) in gas phase and in the solvents CHCl3 (underlined), DMSO (in italics) and H2O (in brackets).

Free energy profiles

For the research of the transition states connecting two tautomers, we used the B3LYP method (Table 4), which is considerably less costly in computation time compared to M06-2X especially to MP2 and which is known to qualitatively reproduce the free energy profiles of reactions in organic chemistry and find in particular the main intermediate and transition states. The results given in Table 4 show that the presence of the solvent has a little influence on the free energy barriers which increase slightly by approximately 2 kcal/mol on going from the gas phase to solution. The values of the free energy barriers in water (H2O) are close to those obtained in DMSO. Vibration frequency calculations of the transition states obtained

M06-2X

EHOMO / ELUMO -6.50 / -0.38 -6.37 /-0.14 -6.33 /-0.07 (-6.33)/(-0.07) -6.10 / -0.33 -6.05 /-0.13 -6.05 /-0.07 (-6.05)/(-0.07) -6.03 / -0.30 -5.99 /-0.11 -6.01 /-0.06 (-6.01)/(-0.06) -5.72 / -0.16 -5.76 /-0.08 -5.79 /-0.07 (-5.79)/(-0.07) -5.60 / -0.12 -5.67 /-0.06 -5. [START_REF] Dennington | [END_REF] (Fig. 6) show that each of these states has a significant imaginary frequency (see Table S2 and Fig. S2, Supporting Information), which means that they really represent energy maxima.

Table 4

Gibbs free energies (GTS ‡ in kcal.mol -1 ) and free energy barriers (DGa↔b, in kcal.mol -1 ) of the transition states in gas phase and in solution, obtained by the B3LYP method.

. gas.

. Thermodynamic quantities obtained of tautomeric couples were calculated at T = 298.15 K and under 1 atm pressure (Table 5) for compounds 1-6. In order to determine the influence of the solvent on the reactivity, the effect of the solvent was taken into account using the PCM model [START_REF] Mennucci | Continuum Solvation Models in Chemical Physics[END_REF][START_REF] Reed | [END_REF] which has been widely used for the study of a large variety of molecules in solution and in the prediction of solvent effects [START_REF] Mennucci | [END_REF][START_REF] Cammi | Molecular Response Functions for the Polarizable Continuum Model[END_REF][START_REF] Ladjarafi | [END_REF][79]. Let us remind that the PCM model does not consider the presence of explicit solvent molecules; therefore, specific solute-solvent interactions are not defined and the studied solvation effects arise only from mutual solutesolvent electrostatic polarization.

Thermodynamic quantities of tautomeric couples

Equilibrium constants were calculated as the difference between the Gibbs free energies (or free enthalpies) ΔG°ab of the two tautomers a and b (a, b = 1-6) by the relation:

DG°ab = DH°ab -TDS°ab (1)
where DH°ab et DS°ab represent the difference of the standard enthalpy and the standard entropy of the two forms a and b, respectively.

The equilibrium constant was calculated with the relation:

Kab = exp (-DG°ab/RT) (2)
where R is the ideal gas constant and T is the absolute temperature. Results in the gas phase and in all solvents correctly confirm the predominance of 1 against 2, 2 against 4 and 6, and 3 against 5 and 6. They also indicate that the solvents H2O and DMSO and to a lesser extent CHCl3 increase the variation of the enthalpy and the Gibbs free energy compared to the gas phase. The values of the variation of entropy in the gas phase are all negative, indicating that the disorder of these systems decreased going from one tautomer to another.

Dipole moments

Dipole moments obtained for compound 2, 3, 4 and 6 via the M06-2X, B3LYP and MP2 methods increase from the gas phase to solution (Table 6). This change is mainly due to the modification of the electronic distribution. They are sensitive to solvent polarity, being, as expected, higher in water (H2O) than in DMSO and in the less polar solvent CHCl3.

Compound 3 is the most polar species, leading to a clear dissymmetry in the atomic charge distribution, resulting in a high polarity of the N-H and O-H bonds. While compound 4 is the least polar, note that, as expected, symmetric compounds 1 and 5 are non-polar, so they have a better distribution of partial atomic charges. for the studied compounds (Table 7) confirm these interatomic distance variations. Each pair of bond atoms C2-O12 (for 1), C9-O11 (for 1-3), C2-N1 (for 1 and 5), C2-N3 (for 2, 4 and 6), C9-N8 (for 5 and 6) and C9-N10 (for 4), sharing a half-bond π, plus a σ bond giving these bonds a bonding index of approximately 1.6.

Analysis of atomic charges

A study of the atomic charges in compounds 1-6 was carried out using two different approaches (NBO and APT (Atomic Polar Tensor)). In order to facilitate the comparison with the different tautomers studied, only the most representative values are gathered in Tables S3-S6 (Supporting Information). The atom numbering is as shown in Fig. 1. The results obtained with M06-2X, B3LYP and MP2 methods indicate that intra-and extra-cyclic heteroatoms (N1, N3, N8, N10, O11 and O12) of all the compounds bear significant negative charges, due to their high electronegativity and therefore their high potential reactivity. The atomic charge of these and MP2) going from the gas phase to the solvents CHCl3, DMSO and finally H2O. This increase in atomic charges is proportional to the solvent polarity [81,82]. On the other hand, the atomic charges computed with the NBO approach show little dependence on the effect of the solvents between 0.002 and 0.006 Å. Nevertheless, the NBO and APT approaches show that the atoms C2 and C9 are the most positively charged, indicating therefore they are the sites for nucleophilic attack. On the contrary, the atoms O11 and O12 are the most negatively charged thus representing electrophilic attack sites.

Conclusion

This theoretical study on the electronic and structural properties of some tautomeric pyrimidopyrimidine derivatives in gas phase and in solution in order to study the influence of the migration of hydrogen atoms allowed us to determine the most stable structures and to study possible equilibrium displacements. The results obtained with the M06-2X, B3LYP, and MP2 methods showed that the keto forms are more stable than the enol ones. Additionally, the large HOMO-LUMO gap computed for compound 1 shows its high kinetic stability and, consequently must the least reactive compound of the series. The influence of the solvents on the geometry, the stability and the reactivity, was discussed. Wiberg bond indices computed for the studied compounds confirm the interatomic distances variations of each pair of bond atoms. The atomic charges computed with the APT approach reflect their dependence on the effect of the solvents unlike the NBO analysis.

Fig. 1 .

 1 Fig. 1. Different tautomer equilibria studied in this article.

1 Fig. 2 .

 12 Fig. 2. Possible conformers of tautomeric forms 2-6 and their relative free energy computed in gas phase computed at the B3LYP/6-311++G** DFT level.

Fig. 3 .

 3 Fig. 3. Optimized structures of compounds 1-6 computed at the B3LYP/6-311++G** level in gas phase.

  1 and 9.2 kcal/mol with M06-2X and B3LYP, respectively. The solvation free energy values obtained in the solvent H2O are very close to those obtained in DMSO. The solvation free energies results obtained from the three methods are proportional to each other. For example, the proportionality coefficient calculated by dividing the values obtained in MP2 by those obtained in B3LYP is the same (99.728 × 10 -

Fig. 5 . 1 < 2 < 3 < 4 < 5 < 6 , 1 > 2 > 3 > 4 > 5 > 6 .

 5123456123456 Fig. 5. Energies and nodal properties of the HOMO and LUMO of the tautomers 1-6 computed at the B3LYP/6-311++G** DFT level in the gas phase. Contour isodensity values: ±0.05 (e/bohr 3 ) 1/2 .

Fig. 6 .

 6 Fig. 6. Energy profile of the tautomers 1-6 studied in the gas phase.

Free

  TS1↔2 -377 597.40 46.0 -377 376.79 47.8 -377 609.02 48.7 -377 609.18 48.3 TS1↔3 -377 595.25 48.2 -377 376.62 49.9 -377 606.85 50.8 -377 607.04 50.7 TS2↔4 -377 584.93 45.8 -377 375.36 48.6 -377 593.85 49.4 -377 593.99 48.4 TS2↔6 -377 581.53 49.2 -377 375.34 51.7 -377 591.10 52.2 -377 591.24 51.3

			CHCl3	DMSO	. H2O .
	G 45 "↔$ 123	∆𝐺 2↔7 123	𝐺 45 "↔$ ./.0 !	∆𝐺 2↔7 ./.0 !	𝐺 2↔7 8"59	∆𝐺 2↔7 8"59	𝐺 2↔7 / % 9	∆𝐺 2↔7 / % 9

TS3↔5 -377 582.80 38.0 -377 375.79 41.9 -377 592.58 44.2 -377 592.71 44.3 TS3↔6 -377 581.21 39.6 -377 375.69 44.0 -377 590.20 46.6 -377 590.34 46.7 ∆𝐺 2↔7 3:2:; = G 45 "↔$ 3:2:; -𝐺 2 3:2:; (state = gas, CHCl3, DMSO, H2O).

Table 5

 5 Variation of enthalpy (ΔH°, in kcal/mol), entropy (ΔS°, in cal/mol.K), Gibbs free energy

	(ΔG°ab, in kcal/mol) and equilibrium constant (Kab) in gas phase and in solution, calculated
	via B3LYP and MP2 (in brackets) methods.
		ΔH°	ΔS°	ΔG°ab	Kab	ΔH°	ΔS°	ΔG°ab	Kab
			Gas			CHCl3
	1↔2 11.99 (10.17)	-1.45 (-4.00)	12.68 (11.48)	5.08 10 -10 (3.86 10 -9 )	13.72 (10.85)	1.73 (1.20)	13.60 (11.03)	1.07 10 -10 (8.16 10 -9 )
	1↔3	13.59 (11.43)	-0.98 (-2.95)	14.12 (12.53)	4.47 10 -11 (6.56 10 -10 )	15.36 (13.14)	-0.88 (2.98)	15.88 (11.89)	2.27 10 -12 (1.93 10 -9 )
	2↔4	13.12 (11.46)	-0.51 (-2.74)	13.55 (12.39)	1.16 10 -10 (8.23 10 -10 )	14.79 (12.41)	-1.11 (-2.04)	15.53 (13.25)	4.14 10 -12 (1.95 10 -10 )
	2↔6	15.70 (13.49)	-1.53 (-1.73)	16.39 (14.21)	9.64 10 -13 (3.81 10 -11 )	16.88 (14.02)	-2.91 (-1.16)	18.03 (14.62)	6.10 10 -14 (1.92 10 -11 )
	3↔5	13.87 (11.64)	-1.68 (-0.33)	14.60 (12.04)	1.97 10 -11 (1.48 10 -9 )	15.62 (12.73)	-1.09 (-0.98)	16.25 (13.32)	1.22 10 -12 (1.71 10 -10 )
	3↔6	14.11 (12.23)	-2.00 (-2.77)	14.95 (13.16)	1.10 10 -11 (2.24 10 -10 )	15.24 (12.74)	-0.31 (-2.93)	15.74 (13.76)	2.89 10 -12 (8.12 10 -11 )

Table 6

 6

	Dipolar moments (in Debye) in the gas phase and solution, calculated via M06-2X, B3LYP
	(underlined) and MP2 (in brackets) methods.
	Cmpd	1	2	3	4	5	6
		0.00	3.63	3.96	0.41	0.00	3.20
	gas	0.00	3.66	3.96	0.38	0.00	3.25
		(0.00)	(3.71)	(4.30)	(0.46)	(0.00)	(3.22)

from a single bond to a double bond upon migration of the protons (H13, H14 or H15, H16) to one of the oxygen atoms O12 or O11, respectively. Wiberg bond indices (WBI)

[80]

computed

Table 7

 7

	Interatomic distances (Å) and Wiberg bond indices (in parenthesis) obtained via the B3LYP
	method.
	Cmpd	1	2	3	4	5	6
	N1-H13	1.007 (0.806)	1.007 (0.804)	2.400 (0.008)	1.007 (0.805)	2.304 (0.008)	1.008 (0.808)
	N3-H14	1.009 (0.805)	2.288 (0.008)	0.992 (0.802)	2.279 (0.009)	1.009 (0.806)	2.280 (0.006)
	N8-H15	1.007 (0.806)	1.007 (0.807)	0.994 (0.808)	1.007 (0.806)	2.304 (0.008)	2.304 (0.009)
	N10-H16	1.009 (0.805)	1.009 (0.806)	0.994 (0.804)	2.279 (0.008)	1.009 (0.802)	1.009 (0.806)
	C2-O12	1.218 (1.633)	1.350 (1.033)	1.353 (1.033)	1.350 (1.034)	1.351 (1.030)	1.350 (1.029)
	C9-O11	1.218 (1.633)	1.219 (1.630)	1.218 (1.634)	1.350 (1.034)	1.351 (1.030)	1.351 (1.034)

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:xxx.