Ahmad Elbahnsi

Romain Retureau

Marc Baaden

Brigitte Hartmann
email: bhartman@ens-paris-saclay.fr

Christophe Oguey
email: oguey@u-cergy.fr

Holding the Nucleosome Together: A Quantitative Description of the DNA-Histone Interface in Solution

Keywords: nucleosome, sequence 601, DNA-histone interface, molecular dynamics, CHARMM36, Voronoi tessellation method, VLDM

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INTRODUCTION

The nucleosome is the fundamental building block of packaged DNA in eukaryotic cells. It consists of an octameric histone core, including two copies of histones H3, H4, H2A and H2B, around which 145-147 base-pairs (bp) of DNA are wrapped making 1.65 turns of a left-handed superhelix [START_REF] Luger | Nucleosomes: Structure and Function[END_REF][START_REF] Luger | Crystal structure of the nucleosome core particle at 2.8 A resolution[END_REF][START_REF] Luger | DNA binding within the nucleosome core[END_REF]. The term "gyre" is commonly used to denote a super-helix turn around the histone core, to distinguish it from turns of the DNA double-helix around its own axis. As molecular length, a gyre represents approximately 7 double helix turns.

Each histone comprises a folded domain (structured core) composed of 3 α-helices connected by short loops and an unstructured, flexible N-terminal domain called tail [START_REF] Gao | Histone H3 and H4 Nterminal tails in nucleosome arrays at cellular concentrations probed by magic angle spinning NMR spectroscopy[END_REF][START_REF] Zhou | Histone H4 K16Q mutation, an acetylation mimic, causes structural disorder of its N-terminal basic patch in the nucleosome[END_REF]. Moreover, H2A has a C-terminal tail. The two copies of each histone type are arranged symmetrically with respect to the dyad axis, which passes through the center of nucleosomal DNA.

In vivo, plethora of interplaying factors [START_REF] Hughes | Mechanisms underlying nucleosome positioning in vivo[END_REF] such as chaperones (reviewed in [START_REF] Gurard-Levin | Histone chaperones: assisting histone traffic and nucleosome dynamics[END_REF][START_REF] Park | Histone chaperones in nucleosome eviction and histone exchange[END_REF]), remodeling complexes (reviewed in [START_REF] Clapier | The biology of chromatin remodeling complexes[END_REF][START_REF] Längst | Chromatin Remodelers: From Function to Dysfunction[END_REF] ), histone variants (reviewed in [START_REF] Kamakaka | Histone variants: deviants?[END_REF][START_REF] Talbert | Histone variants on the move: substrates for chromatin dynamics[END_REF]), epigenetic modifications (reviewed in [START_REF] Tessarz | Histone core modifications regulating nucleosome structure and dynamics[END_REF][START_REF] Zentner | Regulation of nucleosome dynamics by histone modifications[END_REF]) and intrinsic, sequence dependent DNA properties [START_REF] Hughes | Mechanisms underlying nucleosome positioning in vivo[END_REF][START_REF] Eslami-Mossallam | Nucleosome dynamics: Sequence matters[END_REF][START_REF] Fernandez | Nucleosome positioning determinants[END_REF][START_REF] Hatakeyama | High-resolution biophysical analysis of the dynamics of nucleosome formation[END_REF][START_REF] Heddi | Intrinsic flexibility of B-DNA: the experimental TRX scale[END_REF][START_REF] Kaplan | The DNA-encoded nucleosome organization of a eukaryotic genome[END_REF][START_REF] Korber | Active nucleosome positioning beyond intrinsic biophysics is revealed by in vitro reconstitution[END_REF][START_REF] Liu | Determinants of nucleosome positioning and their influence on plant gene expression[END_REF][START_REF] Richmond | The structure of DNA in the nucleosome core[END_REF][START_REF] Struhl | Determinants of nucleosome positioning[END_REF][START_REF] Travers | The DNA sequence-dependence of nucleosome positioning in vivo and in vitro[END_REF][START_REF] Widom | Role of DNA sequence in nucleosome stability and dynamics[END_REF][START_REF] Xu | NMR studies of DNA support the role of pre-existing minor groove variations in nucleosome indirect readout[END_REF][START_REF] Yuan | Genome-scale identification of nucleosome positions in S. cerevisiae[END_REF] orchestrate the nucleosome positioning. Indeed, eukaryotic DNA transactions are intimately associated to the spatial and temporal distribution of nucleosomes within a specific genome, which regulates the DNA accessibility. Thus, assembly and disassembly of nucleosomes continuously occur in the nucleus.

In an attempt to better understand the physical and structural events underlying the disassembly mechanism, earlier studies examined the nucleosome behavior in response to an increased ionic strength, which concluded that the H2A-H2B dimers were released before the H3-H4 tetramer [START_REF] Burton | The interaction of core histones with DNA: equilibrium binding studies[END_REF][START_REF] Oohara | Spectroscopic studies on histone-DNA interactions. II. Three transitions in nucleosomes resolved by salt-titration[END_REF]. This global dissociation scheme, which may reproduce what occurs in the nucleus [START_REF] Arimura | Structural analysis of the hexasome, lacking one histone H2A/H2B dimer from the conventional nucleosome[END_REF][START_REF] Böhm | Nucleosome accessibility governed by the dimer/tetramer interface[END_REF][START_REF] Chen | Revealing transient structures of nucleosomes as DNA unwinds[END_REF], was confirmed by in vitro experiments using short nucleosome arrays reconstituted with various positioning DNA sequences, the Widom's 601 sequence [START_REF] Böhm | Nucleosome accessibility governed by the dimer/tetramer interface[END_REF][START_REF] Chen | Revealing transient structures of nucleosomes as DNA unwinds[END_REF][START_REF] Chen | Asymmetric unwrapping of nucleosomal DNA propagates asymmetric opening and dissociation of the histone core[END_REF][START_REF] Gansen | Nucleosome disassembly intermediates characterized by single-molecule FRET[END_REF][START_REF] Hoch | Protein-protein Förster resonance energy transfer analysis of nucleosome core particles containing H2A and H2A[END_REF][START_REF] Miyagi | Dynamics of nucleosomes assessed with time-lapse high-speed atomic force microscopy[END_REF] and sequences from sea urchin 5S ribosomal gene (sequence 5S) [START_REF] Chen | Revealing transient structures of nucleosomes as DNA unwinds[END_REF][START_REF] Hagerman | Chromatin stability at low concentration depends on histone octamer saturation levels[END_REF][START_REF] Lyubchenko | Nanoscale Nucleosome Dynamics Assessed with Time-lapse AFM[END_REF][START_REF] Park | A new fluorescence resonance energy transfer approach demonstrates that the histone variant H2AZ stabilizes the histone octamer within the nucleosome[END_REF][START_REF] Brower-Toland | Mechanical disruption of individual nucleosomes reveals a reversible multistage release of DNA[END_REF] or GUB [START_REF] Tomschik | Nucleosome dynamics as studied by single-pair fluorescence resonance energy transfer: a reevaluation[END_REF]. Time-lapse atomic force microscopy imaging [START_REF] Miyagi | Dynamics of nucleosomes assessed with time-lapse high-speed atomic force microscopy[END_REF], time-resolved small angle X-ray scattering [START_REF] Chen | Revealing transient structures of nucleosomes as DNA unwinds[END_REF][START_REF] Chen | Asymmetric unwrapping of nucleosomal DNA propagates asymmetric opening and dissociation of the histone core[END_REF], single-pair fluorescence resonance energy transfer [START_REF] Böhm | Nucleosome accessibility governed by the dimer/tetramer interface[END_REF][START_REF] Gansen | Nucleosome disassembly intermediates characterized by single-molecule FRET[END_REF][START_REF] Tomschik | Nucleosome dynamics as studied by single-pair fluorescence resonance energy transfer: a reevaluation[END_REF] or optical tweezers [START_REF] Brower-Toland | Mechanical disruption of individual nucleosomes reveals a reversible multistage release of DNA[END_REF][START_REF] Hall | High-resolution dynamic mapping of histone-DNA interactions in a nucleosome[END_REF] results are compatible with a disassembly first stage facilitated by transient unwrapping of one or both peripheral DNA regions; also, a DNA region at the vicinity of the dyad is the major barrier in the course of the nucleosome dissociation, but its length remains debated. Two studies of unzipping 601 and arbitrary DNA sequences in single nucleosomes [START_REF] Gansen | Nucleosome disassembly intermediates characterized by single-molecule FRET[END_REF][START_REF] Hall | High-resolution dynamic mapping of histone-DNA interactions in a nucleosome[END_REF] described a multi-step process and postulated that, in addition to the DNA center, two regions resisting the dissociation are symmetrically located around 40-50 bp from the dyad.

Elucidating the subtleties of nucleosome dissociation may be a pointless exercise without a precise knowledge of the DNA-histone interactions. The global organization of this interface emerged from the first high resolution X-ray structure of a nucleosome containing a DNA from the human α-satellite [START_REF] Luger | Nucleosomes: Structure and Function[END_REF][START_REF] Luger | Crystal structure of the nucleosome core particle at 2.8 A resolution[END_REF][START_REF] Luger | DNA binding within the nucleosome core[END_REF]. This analysis revealed that the DNA-histone interface was subdivided into 14 main sites (seven sites per half particle) spaced by ~10bp from center to center along the DNA (Figures 1-A and1-B).

The interface is punctuated by arginines interacting at each DNA contact point, which contact the minor groove. These arginines are considered to be determinant for stabilizing the nucleosome, especially when their side chains penetrate the minor groove, making electrostatic and hydrophobic interactions with sugar atoms [START_REF] Wang | Sequence-dependent Kink-and-Slide deformations of nucleosomal DNA facilitated by histone arginines bound in the minor groove[END_REF]. Owing to well ordered DNA, histones and solvent components, another human α-satellite containing structure (1KX5) solved to 1.9 Å resolution, led to suggest that the limited collection of direct hydrogen bonds between DNA phosphate groups and arginines or lysines was supplemented by water molecules making interactions simultaneously with histone and DNA [START_REF] Davey | Solvent mediated interactions in the structure of the nucleosome core particle at 1.9 a resolution[END_REF]. In the same structure, specific ion-binding sites in histones and DNA were also identified, but they mainly contribute to nucleosomenucleosome interactions and not to the internal interface. A network of interactions, known as "sugar clamp", emerged from a structure of a 601 containing nucleosome [START_REF] Wu | Structural insight into the sequence dependence of nucleosome positioning[END_REF]. This network surrounds two TTAAA elements, symmetrically located at ±1.5 turn of the DNA center and combines electrostatic and hydrophobic contacts between DNA phosphate groups and sugars on one hand and H3-arginine, H3-leucine and H4proline on the other hand. Described as unique in the nucleosome [START_REF] Wu | Structural insight into the sequence dependence of nucleosome positioning[END_REF], the sugar clamp was suspected to promote the positioning power of these TTAAA elements [START_REF] Chua | The mechanics behind DNA sequencedependent properties of the nucleosome[END_REF]. Apart from the minor groove inserted arginines and the sugar clamp, it is commonly accepted that electrostatic interactions, comprising those mediated by water molecules, are the major elements responsible for maintaining the DNA wrapped around the histones [START_REF] Wang | Sequence-dependent Kink-and-Slide deformations of nucleosomal DNA facilitated by histone arginines bound in the minor groove[END_REF][START_REF] García-Pérez | Nonsequence-specific arginine interactions in the nucleosome core particle[END_REF][START_REF] West | Electrostatic interactions between arginines and the minor groove in the nucleosome[END_REF][START_REF] Yusufaly | Arginine-phosphate salt bridges between histones and DNA: intermolecular actuators that control nucleosome architecture[END_REF]. However, a previous exhaustive, careful analysis of protein-DNA X-ray structures highlighted the central importance of non-electrostatic contacts for complex formation and stability [START_REF] Luscombe | Amino acid-base interactions: a three-dimensional analysis of protein-DNA interactions at an atomic level[END_REF]. In the analyses of the DNA-histone interface, such interactions, only sparsely described, clearly deserve to be better investigated and quantified. Also, despite their essential contribution, X-ray structure analyses cannot answer all questions. The well known biases due to crystal growth conditions or lattice contacts, which are the major driving force for the formation of nucleosome crystals [START_REF] Travers | The DNA sequence-dependence of nucleosome positioning in vivo and in vitro[END_REF][START_REF] Wu | Structural insight into the sequence dependence of nucleosome positioning[END_REF][START_REF] Harp | Asymmetries in the nucleosome core particle at 2.5 A resolution[END_REF], are in fact not the only issue. First, 1KX5 [START_REF] Davey | Solvent mediated interactions in the structure of the nucleosome core particle at 1.9 a resolution[END_REF], is the only crystal to determine the structure of the H3 and H2B N-tails that pass between the two DNA double helices juxtaposed one above the other in the nucleosome as well as of the H4 and H2A tails that enter or exit on the upper or lower sides of the nucleosome (Figure 1-C). But even in this case, the tails were globally too poorly ordered to provide a precise map of their contacts with the DNA [START_REF] Davey | Solvent mediated interactions in the structure of the nucleosome core particle at 1.9 a resolution[END_REF]. Yet, modeling studies mentioned a substantial contribution of certain tail parts to the interface [START_REF] Erler | The Role of Histone Tails in the Nucleosome: A Computational Study[END_REF][START_REF] Shaytan | Coupling between Histone Conformations and DNA Geometry in Nucleosomes on a Microsecond Timescale: Atomistic Insights into Nucleosome Functions[END_REF] although another study [START_REF] Ettig | Dissecting DNA-histone interactions in the nucleosome by molecular dynamics simulations of DNA unwrapping[END_REF] suggested that the interactions with histone tails cannot explain the barriers described from experimental disassembly results. Second, the DNA super-helical wrapping brings phosphate groups of juxtaposed DNA gyres very close to each other. Because of electrostatic repulsion, such a proximity would not be possible without efficient shielding. H3 and H2B tails undoubtedly reduce this repulsion in surrounding DNA regions. Still, what happens in regions where the juxtaposed DNA gyres are not separated by histone tails has not been elucidated. Thus, our understanding of how the DNA is maintained by the histones remains incomplete. We therefore decided to study the nucleosome from all-atom molecular dynamics simulations with explicit water molecules and ions, focusing more on the DNA-histone interface in solution than on the nucleosome dynamics. To get a reference of a strong interface, we studied a nucleosome formed with the artificial sequence 601, known to generate a very stable complex [START_REF] Hatakeyama | High-resolution biophysical analysis of the dynamics of nucleosome formation[END_REF][START_REF] Thåström | Nucleosomal locations of dominant DNA sequence motifs for histone-DNA interactions and nucleosome positioning[END_REF]. In addition, our nucleosome models contains a large part of the histone tails. For the interface analysis, we used a tessellation method, called VLDM (Voronoi Laguerre Delauney for Macromolecules) initially developed for proteins [START_REF] Esque | VLDP web server: a powerful geometric tool for analysing protein structures in their environment[END_REF][START_REF] Esque | Comparative Analysis of Threshold and Tessellation Methods for Determining Protein Contacts[END_REF][START_REF] Esque | A novel evaluation of residue and protein volumes by means of Laguerre tessellation[END_REF] and adapted here to nucleic acids. Based on a representation of molecules by a collection of polyhedra filling space without overlaps or gaps, VLDM has the advantage of providing interface information by taking all the constituents into account -comprising the solvent-without resorting to any empirical or adjusted parameter.

Here, VLDM gave an exhaustive inventory of the interacting elements and quantitative account of the interactions in terms of contact area and time occurrence.

The DNA-histone contacts were characterized in this way at each DNA binding region. The respective contributions of the histone structured cores and tails to the interface were ascertained. A special attention was paid on the balance between electrostatic and hydrophobic contacts. Water molecules were analyzed in the interface context to identify their role in possibly supplementing the direct contacts between amino acids and nucleotides. The repulsions between superimposed, close DNA gyres were examined in relation to the presence of cations. In sum, the exhaustive description of the elements involved in the DNA-histone interface in solution provides new elements to better understand how the DNA is stabilized around the nucleosome.

MATERIALS AND METHODS

Nucleosome starting models

We built five models, all based on the folded domains of Xenopus laevis histones and the 601 sequence of 146 base-pairs (bp) from the nucleosome X-ray structure 3MVD [START_REF] Makde | Structure of RCC1 chromatin factor bound to the nucleosome core particle[END_REF]. The first model, SYS0, corresponds to the 3MVD structure without any change. The four other models, SYS1, SYS1-bis, SYS2 and SYS2-bis, differ by the length and conformations of the histone N-and C-terminal domains (called tails), hereafter described.

In addition to a structured, folded domain, each histone comprises a disordered N-terminal domain [START_REF] Gao | Histone H3 and H4 Nterminal tails in nucleosome arrays at cellular concentrations probed by magic angle spinning NMR spectroscopy[END_REF][START_REF] Zhou | Histone H4 K16Q mutation, an acetylation mimic, causes structural disorder of its N-terminal basic patch in the nucleosome[END_REF], the histone tail; H2A also contains a disordered C-terminal domain. Only short fragments of 5 to 8 amino acids corresponding to minimal N-tail roots were resolved in 3MVD. Full-length tails were observed in only one nucleosome X-ray structure, 1KX5 [START_REF] Davey | Solvent mediated interactions in the structure of the nucleosome core particle at 1.9 a resolution[END_REF], which is formed of a DNA sequence issued from the human α-satellite. In our models, the tails from 1KX5 were added to 3MVD after the 3MVD and 1KX5 histone folded parts, identical in both structures, were superimposed. Nevertheless, the 1KX5 N-tails were partially truncated to prevent their collapse and wrapping around the DNA as observed in modeling studies of 1KX5 involving full-length tails [START_REF] Shaytan | Coupling between Histone Conformations and DNA Geometry in Nucleosomes on a Microsecond Timescale: Atomistic Insights into Nucleosome Functions[END_REF][START_REF] Biswas | Role of histone tails in structural stability of the nucleosome[END_REF][START_REF] Roccatano | Structural flexibility of the nucleosome core particle at atomic resolution studied by molecular dynamics simulation[END_REF]. Indeed, such a folding disagrees with experimental results [START_REF] Zhou | Histone H4 K16Q mutation, an acetylation mimic, causes structural disorder of its N-terminal basic patch in the nucleosome[END_REF][START_REF] Dumuis-Kervabon | A chromatin core particle obtained by selective cleavage of histones by clostripain[END_REF] and with the idea that the tails may be easily accessible to proteins impacting nucleosome positioning (reviews: [START_REF] Hughes | Mechanisms underlying nucleosome positioning in vivo[END_REF][START_REF] Speranzini | Touch, act and go: landing and operating on nucleosomes[END_REF]) or extended when they interact with neighboring particles (review: [START_REF] Pepenella | Intra-and inter-nucleosome interactions of the core histone tail domains in higher-order chromatin structure[END_REF]). Thus, the N-tail amino acids more than 3.5 Å distant from DNA in 1KX5 were removed so that the kept regions coincide with the inaccessible parts as delimited by trypsin and clostripain digestions [START_REF] Dumuis-Kervabon | A chromatin core particle obtained by selective cleavage of histones by clostripain[END_REF][START_REF] Morales | Role of histone N-terminal tails and their acetylation in nucleosome dynamics[END_REF].

In addition, the tail conformations of the two copies of each histone in 1KX5 are different. For instance the RMSD (Root Mean Square Deviation) between the N-tail backbone heavy atoms of the two H3 copies is 12.2 Å. We took advantage of this heterogeneity to generate different models all including a symmetrically replicated tail conformation: SYS1 and SYS1-bis were built with truncated tails from 1KX5 chains A, B, C and D while SYS1 and SYS1-bis contained truncated tails from 1KX5 chains E, F, G and H.

Finally, the H2A C-tail from 1KX5 was integrated in two models, SYS1-bis and SYS2-bis. The composition of tails in our models is given in Table 1; more details in particular about the tail sequences are presented in Table S1.

In our models, the pKa of histidines were calculated using an empirical equation expressing pKa as a function of electrostatic potential, hydrogen bonds and accessible surface area [START_REF] Krieger | Fast empirical pKa prediction by Ewald summation[END_REF]. The pKa of six histidines (H3-H39, H4-H75, H2A-H32, H2A-H82, H2B-H46 and H2B-H79) got values between 7.7 and 9.6. These histidines were therefore protonated. The role of this protonation was demonstrated through an additional simulation carried out with unprotonated histidines, in which irreversible breaking of DNA pairing was observed in several places. Such loss of base pairing, not retrieved either in nucleosome X-ray structures or during the other simulations, points out the importance of histidine protonation for maintaining the double helix integrity in nucleosome.

Molecular Dynamics Simulation Set-up

Molecular dynamics simulations were carried out with the CHARMM36 force field [START_REF] Hart | Optimization of the CHARMM additive force field for DNA: Improved treatment of the BI/BII conformational equilibrium[END_REF] with the CMAP correction (69) using the CHARMM (70) and NAMD 2.11 programs [START_REF] Phillips | Scalable molecular dynamics with NAMD[END_REF]. This force field was chosen for its ability to match with experimental NMR data collected on free B-DNAs [START_REF] Imeddourene | Simulations Meet Experiment to Reveal New Insights into DNA Intrinsic Mechanics[END_REF].

Each structure was immersed in a cubic box (side length of ~130 Å) filled with TIP3P water molecules (Jorgensen et al., 1983) preserving at least 10 Å of separation between the solute and the edges of the box.

Electric neutrality was achieved by the addition of Na + (73) and Cl -(74) reaching a final concentration of 150 mM as commonly used in in vitro nucleosome experimental studies. The location of ions was optimized using a Monte Carlo approach. The whole systems contain from ~220,000 to 270,000 atoms.

The water molecule and ion positions were first minimized by steepest descent and adapted basis Newton-Raphson (75) methods, keeping the solute fixed, in the CHARMM program. Then, in NAMD, an equilibration simulation progressively raised the temperature from 0 to 300K, by steps of 10K every 0.1ps.

During this heating process, harmonic restraints were applied to the atomic position RMSD of the protein backbones and DNA, using force constants of 1000 and 5000 kcal/mol/Å 2 , respectively. These restraints were then relaxed stepwise during a total of 1 ns equilibration using the NPT ensemble.

The production phase was also carried out in the NPT ensemble, at a temperature of 300K and a pressure of 1bar. Five simulations were performed starting from the models in Table 1 using the Langevin scheme [START_REF] Brooks | Deformable Stochastic Boundaries in Molecular-Dynamics[END_REF][START_REF] Feller | Constant-Pressure Molecular-Dynamics Simulation -the Langevin Piston Method[END_REF]. Periodic boundary conditions were used to avoid artifacts at the box boundaries. Infinite range Coulomb interactions were treated with the Particle Mesh Ewald approach [START_REF] Darden | Particle Mesh Ewald -an N.log(n) Method for Ewald Sums in Large Systems[END_REF]. For the van der Waals interactions, a switching function was applied at 10 Å and the cutoff was set to 12 Å. SHAKE was applied to constrain the bond lengths involving hydrogen atoms. The integration time steps were 2 fs and coordinates were saved every 1000 steps (2 ps). The duration of the simulations are 200ns for SYS0, SYS1 and SYS2 and 300ns for SYS1-bis and SYS2-bis.

Interface analysis

The interface between DNA and histones was analyzed by VLDM (Voronoi Laguerre Delaunay for Macromolecules), a software originally developed for proteins [START_REF] Esque | VLDP web server: a powerful geometric tool for analysing protein structures in their environment[END_REF][START_REF] Esque | Comparative Analysis of Threshold and Tessellation Methods for Determining Protein Contacts[END_REF][START_REF] Esque | A novel evaluation of residue and protein volumes by means of Laguerre tessellation[END_REF] and extended here to nucleic acids.

VLDM relies on a tessellation method, that is, a partition of space into a collection of polyhedra filling space without overlaps or gaps. The program builds the Delaunay tessellation and its Laguerre dual from a set of atomic data, each atom being characterized by its position in space and a weight depending on its van der Waals radius. For consistency with the simulations, the van der Waals radius values are those of CHARMM36. In the present analysis, only the heavy atoms of the solute and solvent were considered.

In this approach, a contact occurs whenever two atoms share a common face in the tessellation. The interface between two molecules or molecular groups is a polygonal surface, quantified by its area. The

Additional trajectory analyses

Root Mean Square Deviations (RMSDs) were computed with cpptraj (79) between the snapshots and the initial configuration derived from the X-ray structures before the heating and equilibration stages RMSDs were calculated on protein backbone heavy atoms or on all DNA heavy atoms.

Protein secondary structures were analyzed with DSSP [START_REF] Kabsch | Dictionary of protein secondary structure: pattern recognition of hydrogenbonded and geometrical features[END_REF]. The DNA base-pairing was examined with HBPLUS (81) using distance and angle cutoffs of 3.9 Å and 130° respectively. The electrostatic potential was calculated using APBS [START_REF] Baker | Electrostatics of nanosystems: Application to microtubules and the ribosome[END_REF] in PyMOL [START_REF] Schrödinger | The PyMOL Molecular Graphics System[END_REF], with the ion charges of CHARMM36 and ionic strength of 150mM. The cation occupancy was calculated with VolMap tools in VMD, using a grid of 1Å.

VMD [START_REF] Humphrey | VMD: visual molecular dynamics[END_REF] and PyMOL [START_REF] Schrödinger | The PyMOL Molecular Graphics System[END_REF] were used for visualization.

RESULTS

Overview of the simulations

Five nucleosome systems were built, based on the folded domains of Xenopus laevis histones and the 601 sequence of 146 base-pairs (bp) from the nucleosome X-ray structure 3MVD [START_REF] Makde | Structure of RCC1 chromatin factor bound to the nucleosome core particle[END_REF]. These initial systems differ by the length and conformations of the histone N-and C-terminal domains (tails) (Tables 1 andS1).

The first model, SYS0, corresponds to the 3MVD structure without any change. Histone N-and C-tail regions from the nucleosome X-ray structure 1KX5 [START_REF] Davey | Solvent mediated interactions in the structure of the nucleosome core particle at 1.9 a resolution[END_REF] where added to SYS0 in the four other models, SYS1, SYS1-bis, SYS2 and SYS2-bis according to the procedure described in the Materials and Methods section. The five systems were simulated with the CHARMM36 force field (68) for a total duration of 1.2µs.

Classical analyses were first performed mainly to ensure that the simulated nucleosomes were not subject to artefactual distortions but also to gain a first estimation of the consistency of the various simulations.

In all simulations, the stability of the radius of gyration (Rg) of the whole nucleosomes and DNA double helix (Figure S1) indicated that no histone-histone or DNA-histone disassembly occurred in the trajectories. The system with minimal tails, SYS0, expectedly showed slightly lower Rg values (Rg=36.2±0.3 Å, Figure S1) than the other systems containing longer tails (Rg=37.2±0.2 Å on average, Figure S1). The large Rg values, around 45 Å (see examples in Figure S1) obtained for the DNA in all simulations reflect its location at the periphery of the histone structured cores.

The overall stability of the histones was then assessed through the root mean square deviations (RMSD) calculated between the initial configuration and the simulated snapshots (see Materials and Methods). The octameric histone structured cores, first considered, remain globally close to their X-ray counterparts (RMSDs ~ 2 Å, Figure S2) in all the systems, with well-preserved α-helices and loops (example of H3 in Figure S3).

Unlike the structured cores, the tails included in SYS1, SYS1-bis, SYS2 and SYS2-bis deviate from those in the starting structure, which were extracted from 1KX5 tails (Figure S4). This is expected since the tail conformations in 1KX5 are largely shaped by the crystal contacts between neighboring nucleosomes [START_REF] Davey | Solvent mediated interactions in the structure of the nucleosome core particle at 1.9 a resolution[END_REF]. The highest RMSD values (> 10Å) are reached by H3 and H2B N-tails which are longer than any other tail (Tables 1 andS1). Although each initial model was built in such a way that the conformations of both copies of each histone type were strictly identical (see Materials and Methods), this pairwise symmetry is lost during the last part of the equilibration stage. The asymmetry is reflected by the RMSDs which are rarely identical for both copies of a given model (Figure S4). Also, the RMSDs of tails of the same species differ across different systems (Figure S4).

A finer understanding of the tail behavior emerges from additional analyses, discarding the first 50ns to limit the effect due to the earliest tail rearrangements. Focusing first on H3, H2A and H2B N-tails, we found that the atomic fluctuations, large and maximal at the extremities (atomic fluctuations up to 10 Å), decrease until reaching a plateau in the regions located before the structured cores and close to the DNA (atomic fluctuations below 5 Å) (Figure S5), alluding flexible extremities and stiff tail roots. However, low atomic fluctuations do not say anything about possible conformation variations across simulations. Systematic cross-RMSDs were thus calculated to better define which tail parts explore similar conformational landscapes. A satisfactory compromise was found for the amino acids 37-44 of H3, 12 or 11-16 of H2A or 26-34 of H2B since the mean cross-RMSDs between their structures are 2.6 ±1.1, 3.1 ±1.4, 2.9 ±1.1 Å, respectively. By comparison, the same analyses applied to the remaining N-tail parts led to mean cross-RMSDs of 14.6 ±6.0 Å for H3, 7.9 ±3.9 Å for H2A and 12.3 ±6.6 Å for H2B. By visual inspection, we found that these high values correspond to a large variety of conformations. Besides, in the latter regions, we observed the formation of a short α-helix involving amino acids 23-28 of H2B in SYS1 and SYS1-bis, as previously detected in an isolated H2A-H2B heterodimer by NMR [START_REF] Moriwaki | Solution structure of the isolated histone H2A-H2B heterodimer[END_REF].

Globally, the behavior of simulated H3, H2A and H2B tails is in full agreement with a NMR study of nucleosome in solution, which demonstrated that the tail structural disorder disappears beyond amino acid 37 of H3, 11 of H2A and 25 of H2B (5) (Table 2). Therefore, although the full range of tail polymorphism can only be partially sampled in the accessible simulation times, the molecular dynamics simulations adequately reproduce the contrast between flexible extremities and stiff roots in H3, H2A and H2B N-tails. The term "tail root" refers to the amino acids defined as such in Table 2.

Unlike the N-tails of H3, H2A and H2B, the eight amino acids composing the simulated H4 N-tails are all stiff (atomic fluctuations below 4 Å; Figure S5). The RMSD analyses indicate that these tails adopt two main conformations, one in SYS1 and SYS1-bis (mean cross-RMSDs of 3.0 ±1.1 Å) and another, more variable, in SYS2 and SYS2-bis (mean cross-RMSDs of 6.5 ±3.3 Å). The DNA-histone interface analysis will provide more details on how the H4 tails are stabilized in two distinct conformations owing to differently located interactions with DNA. Concerning the H2A C-tail, most of its amino acids are especially flexible (Figure S5) and the whole tail is highly dynamic (mean cross-RMSDs of 13.2 ±7.2 Å); both features were observed by NMR [START_REF] Zhou | Histone H4 K16Q mutation, an acetylation mimic, causes structural disorder of its N-terminal basic patch in the nucleosome[END_REF]. Indeed, we will see that this tail does not engage sizeable contacts with DNA.

The last nucleosome component to be scrutinized is DNA. The RMSDs involving all the DNA heavy atoms stabilize around 2.6 ±0.5 Å on average, considering together the five simulations (Figure S2). This coherency across the systems indicates that the DNA overall structure is affected by neither the large tail truncations in SYS0 nor the presence of H2A C-terminal tails in SYS1-bis and SYS2-bis. Apart from 2 or 3 terminal base pairs, mainly unpaired, the base pair hydrogen bonds are well-conserved in all simulations (90% of simulation time, examples in Figure S6). By comparison, base pairing is maintained 99% of simulation time in the internal part of free DNA dodecamers related to the 601 sequence [START_REF] Imeddourene | Simulations Meet Experiment to Reveal New Insights into DNA Intrinsic Mechanics[END_REF]. This slight difference between free and bound DNAs may reflect the stress induced by the DNA wrapping around the histone core.

In sum, these preliminary analyses showed that all the simulated nucleosomes preserve the DNAhistone assembly without generating anomalous distortions. In addition, the behavior of histone structured cores and DNA is well preserved across the different simulations. Concerning the tails, an important point is that the proximity to the DNA clearly limits the structural variability of H3, H2A and H2B N-tail roots, which is consistent, at least for H3 and H2B, with their position sandwiched in a narrow passage between the two superimposed DNA gyres (Figure 1). In the next sections, devoted to interface analyses, the first 50ns were systematically discarded to guarantee a correct equilibration of the solvent (86) but also to discard early tail rearrangements from consideration.

DNA -histone interface

In this section, we present the direct contacts between DNA and either the histone structured cores or tails.

Contact maps from VLDM provide an identification of the nucleotides (Figure 2-A) and amino acids (example in Figure 2-B) participating in the interface as well as the evolution of the contacts over the trajectory time.

The interface was thus monitored in terms of both contact area (CA) and time occurrence. The nucleotide positions are specified in terms of Super Helix Location (SHL), that is, the number of helical turns separating a given base pair from the central base pair, SHL0. SHL is negative or positive for the 5' (Figure 1-B) or 3' DNA half sequence, respectively.

a) DNA -histone structured core contacts

The contact area (CA) associated to DNA -histone structured core is 2332±111 Å 2 on average. Calculated by histone type, CA av values range from 391±37 for H3 to 168±28 Å 2 for H2B, in the following decreasing order: CA av (H3) > CA av (H2A) > CA av (H4) > CA av (H2B) (Figure 3).

The correlation coefficients of CA av calculated at the amino acid level for equivalent copies across different systems (for instance: chain A in SYSn and SYSm) are equal to or higher than 0.9, confirming the consistency of the simulations, visible in Figure 3. Moreover, the interfaces associated to pairs of copies in the same system (for instance: chain A versus chain E in SYSn) also compare well (correlation coefficients > 0.9; see also Figure 3).

Concerning DNA, the contacts are clustered into separate blocks pertaining to nucleotides of both DNA strands (Figure 2-A). Successive blocks are separated by 8 to 12 base pairs (bp). The shift between the blocks in strands I and II indicates that the contacts occur across the minor grooves. The simulated interface reproduces the pattern known from nucleosome X-ray structures [START_REF] Luger | Crystal structure of the nucleosome core particle at 2.8 A resolution[END_REF][START_REF] Davey | Solvent mediated interactions in the structure of the nucleosome core particle at 1.9 a resolution[END_REF]. The comparison of the five datasets of CA av extracted from simulations and calculated at the nucleotide level led to very high values (≥ 0.95) of correlation coefficients, attesting that, as for the histone structured cores, both location and area of contacts are robust across the simulations. In particular, as evidenced by the similar CA av values, shortening the tails (SYS0 vs the four other systems) does not affect the DNA-histone structured core interface. As a practical consequence of such a coherence, the five molecular dynamics simulations were analyzed collectively, considering in addition both DNA strands together instead of each strand separately.

The CA av of the 14 DNA contact regions are symmetric with respect to the center, as shown in Figure 4-A.

These contacts were analyzed to examine the contribution of the different CA components. Roughly half of CA av values can be classified as electrostatic and hydrophobic contacts (Figure 4-B), the remaining contacts correspond to simple DNA/histone proximity. As expected, the phosphate groups, defined here by O5', O3', P, O1P and O2P atoms, are involved in 95% of the electrostatic contacts. The hydrophobic contacts implicate sugar carbon atoms and various amino acids but the largest CA av occur with the aliphatic chains of arginines that contact the minor grooves (both copies of H4-R45, H3-R49, H3-R63, H3-R83, H2A-R77 and H2A-R42; examples of H3-R63 and H2A-R42 in Figure 4-C; see also Figure S7). The complete information about the amino acids that participate to electrostatic or hydrophobic contacts at each SHL is furnished in Figure S7, which further illustrates the pairwise similarity of the interface spots with respect to the DNA center. Importantly, a large amount of contacts are present for a large part of the simulation time regardless the CA av values (examples in Figure 4-C; Figure S7). More precisely, 75% of contacts are observed in more than 90% of the snapshots. Such high occurrence rates in time reflect the remarkable stability of the interface between the DNA and histone structured cores (see also Figure 2).

A striking point of this analysis concerns the balance observed between electrostatic and hydrophobic CA av . Most SHLs are associated to extensive hydrophobic CA av , which are even predominant at SHLs ± 5.5, 2.5, 1.5 and 0.5 (Figure 4-B; detailed examples in Figure 4-C). So the mix of hydrophobic and electrostatic interactions previously described around SHLs ± 1.5 [START_REF] Wu | Structural insight into the sequence dependence of nucleosome positioning[END_REF] is a feature reproduced along the whole DNA, SHLs ± 4.5 being the only locations where electrostatic CA av prevail (Figure 4-B). At a global level, the area of hydrophobic (563±39 Å 2 ) and electrostatic (578±36 Å 2 ) contacts are equivalent. However, the most extensive hydrophobic components are associated to the (H3-H4) 2 structured core (Figures 4-A andS8). This may enlighten the early step of in vitro nucleosome reconstitution that occurs at high ionic strength. At 1.5 M, (H3-H4) 2 robustly binds the DNA exactly as in the complete nucleosome [START_REF] Hatakeyama | High-resolution biophysical analysis of the dynamics of nucleosome formation[END_REF] although the electrostatic contacts and consequently the DNA-histone interactions are expected to be weakened. Our results suggest that the substantial hydrophobic component of the (H3/H4) 2 interface complements the weakened electrostatic action so that this histone tetramer is at once appropriately anchored to the DNA.

b) DNA -histone tail interface

Here, we analyze SYS1, SYS2, SYS1-bis and SYS2-bis that contain elongated tails with respect to SYS0 (Table 1). The total CA av associated to the tails reaches 1887±425 Å 2 , a substantial quantity compared to the CA involving the structured cores (2332±111 Å 2 ). However, a large part of the DNA-tail interface is due to the N-tail roots of H3, H2A and H2B, as explained now.

The extremities of H3, H2A and H2B N-tails (defined in Table 2), the H4 N-tail and the H2A C-tail are involved in the interface with the DNA (Figures S9 andS10) but their conformational flexibility described in the Overview section causes significant CA av variations across the simulations and copies (Figure S9). In addition, their interaction with DNA generally corresponds to rather weak, unstable contacts (Figures S9 andS10). The H2A C-tail is an extreme case since its contribution to the interface is especially weak in terms of both area and time occurrence (Figure S10). In the H4 N-tail, a single arginine, located at the beginning of the simulated sequence (R17), interacts with DNA (Figure S10), binding the minor groove edge at either SHLs ± 2.5 in SYS1 and SYS1-bis or SHLs ± 1.5 in SYS2 and SYS2-bis. Finally, the first two or three N-tail amino acids of H3, H2A and H2B also contact DNA at best 60% of the time (Figure S10).

Overall, these results are in full accordance with previous simulations specifically dedicated to the tail behavior [START_REF] Erler | The Role of Histone Tails in the Nucleosome: A Computational Study[END_REF][START_REF] Shaytan | Coupling between Histone Conformations and DNA Geometry in Nucleosomes on a Microsecond Timescale: Atomistic Insights into Nucleosome Functions[END_REF]. They are also strongly supported by a NMR study demonstrating the enhanced flexibility of these tail regions [START_REF] Zhou | Histone H4 K16Q mutation, an acetylation mimic, causes structural disorder of its N-terminal basic patch in the nucleosome[END_REF], which is incompatible with any durable, stabilizing interaction with DNA. We thus decided not to further describe the interactions involving these tail extremities, also because complete sampling would require longer simulation times.

From now on, our analysis focuses on the roots of H3, H2B and H2A N-tails (defined in Table 2). The roots represent 65% of the total CA av between DNA and tails and, as seen before, they are characterized by low atomic fluctuations and restricted conformational variabilities. Except for H2B chain H N-tail root in SYS1, whose arginine R30 escaped from the DNA minor groove causing perturbations in this interface region, pairs of CA av across either simulations or copies are comparable (Figure 5, correlation coefficients calculated at the amino acid level from 0.76 to 0.85). SYS1, SYS2, SYS1-bis and SYS2-bis simulations were consequently analyzed together.

Eight SHL spots are contacted by H3, H2B and H2A N-tail roots (Figure 6-A). Each H2B and H3 tail root contains 8 amino acids and binds to two juxtaposed double helices (Figures 6-A and-B) while the H2A tail root is shorter (4 amino acids) and interacts with only one double helix on the upper or lower side of the nucleosome (Figures 6-A andB). Accordingly, the H2A N-tail root contributes much less to the interface than the H3 and H2B ones (Figure 5). Nevertheless it maintains contacts with DNA during the whole simulation time (Figure S10), unlike the flexible H4 N-tail and the H2A C-tail (Figure S10). Each of the eight DNA binding spots combines electrostatic and hydrophobic contacts (Figures 6-C, 6-D and S11), most of them being very stable during the trajectories (Figures 6-D andS11). Overall, in the DNA-tail root interface considered here, the hydrophobic CA av is almost twice as large as the electrostatic CA av : 390 ± 45 Å 2 and 213 ± 39 Å 2 , respectively. This particular interface shows a more marked hydrophobic character than the DNA-structured core interface in which electrostatic and hydrophobic CA av are equivalent.

To conclude on this section, our analyses reveal the importance of hydrophobic contact areas that substantially complement the electrostatic interactions in the DNA-histone interface. Thus, the combination of very stable, direct electrostatic and hydrophobic contacts is the norm for each DNA point anchoring the histone structured core or H3, H2A and H2B N-tail roots. A noticeable contribution to the cohesion of the DNA super-helical wrapping is provided by the double contact made by each H3 and H2B N-tail root connecting two juxtaposed DNA gyres.

Interfacial water molecules

The water molecules were then scrutinized in the context of the DNA-histone interface. This investigation was motivated to a large extent by a previous analysis of hydrogen bonds in the nucleosome crystallographic structure 1KX5, that contained a 147bp DNA derived from the α human satellite [START_REF] Davey | Solvent mediated interactions in the structure of the nucleosome core particle at 1.9 a resolution[END_REF]. In this structure, the only one that allowed the unambiguous identification of a very large number of water molecules, 121 water molecules were found to bind simultaneously to DNA and histone structured cores. These water molecules either sustain direct DNA-histone hydrogen bonds or connect groups that are too distant or improperly oriented to authorize direct interactions. The authors of the crystallographic study concluded that direct and water mediated hydrogen bonds equally contribute to the nucleosome stability. The molecular dynamics runs gave us a good opportunity to probe, at least in silico, the existence of long-lived water mediated hydrogen bonds in solution.

Equivalent analyses were carried out on the simulated structures, using the same histone parts and criteria (D Donor-Acceptor ≤ 3.5 Å; Donor-Hydrogen-Acceptor angle > 90°) as Davey and coll. [START_REF] Davey | Solvent mediated interactions in the structure of the nucleosome core particle at 1.9 a resolution[END_REF], keeping in mind that our models differ from 1KX5 by the DNA length and sequence. In our simulations, we found a total of 51 amino acids involved in direct hydrogen bonds with DNA compared to 37 in 1KX5 (Table S2).

However, this difference is largely canceled by the fact that the water mediated connections between twelve amino acids and DNA in 1KX5 turn into direct hydrogen bonds in the simulated structures (Table S2). This observation illustrates that, compared to the solid state, DNA and histones in solution have the potential to undergo subtle, local rearrangements promoting direct electrostatic interactions. Overall, most amino acids involved in hydrogen bonds are retrieved in both experiment and simulation (Table S2).

Actually, according to VLDM, very few water molecules are trapped in the simulated interface. At SHLs ± 4.5, a water molecule links a phosphate group to S33 and I36 of H2B (Figure 7-A) in 74% of snapshots.

Another water molecule interacts with the O2 atom of thymine or cytosine in the minor groove floor at SHL -0.5 and R45 in H4 (Figure 7-B) in 40% of snapshots. Both DNA-water-I36 and DNA-water-R45 interactions reinforce direct contacts (Figure S7). Thus, an overwhelming majority of water molecules circulate all around the DNA and histones, without being confined at precise interface locations as observed in 1KX5. This mobility does not mean that water molecules do not intervene in the interface. By filling gaps between DNA and histone surfaces, reducing the repulsions between electronegative atoms or making transient hydrogen bonds, water molecules play an essential role in the nucleosome as well as in any DNA-protein complex [START_REF] Reddy | Do water molecules mediate protein-DNA recognition?[END_REF]. Nevertheless, the simulations provide a strong indication that stable, long-lived water molecules bridging DNA and histones are the exception rather than the rule in liquid solution.

Cations at DNA-histone and DNA-DNA interfaces

In a way similar to water molecules, ions could behave as interface mediators. Visual inspection revealed that some Na + spent a large part of the simulation time at the DNA-histone interface, contrary to Cl -that are, as expected, always far from the DNA. The existence of long-lived cation binding sites is also attested by Na + with low atomic fluctuations. By systematically calculating the distances between Na + and DNA (D Na+- DNA ) or histones (D Na+-histone ) for each ion in each snapshot, as well as distance distributions, we found that seven Na + reside close to both DNA and histones (D Na+-DNA ≤ 4 and D Na+-histone ≤ 5.5 Å, see examples of distance distributions in Figure S12) between 10 and 100% of the simulation time, but their location and time occurrence often vary across the simulations (Table S3). This disparity is not a reminiscence of the simulation starting points, since none of the observed interfacing ions is already in place at the beginning of the production phase; it may rather reflect an insufficient sampling or local structural differences modulating the electrostatic attractiveness for Na + . X-ray structures of nucleosome cannot help here, because they commonly contain divalent cations instead of Na + that, in any case, cannot be easily identified even in high resolution structures. Further specific investigations would be consequently required to determine which parameters influence the presence of ions at the interface. At the current stage, one can yet state that Na + are able to stay as much as 250 ns at a precise interface location (Table S3). Such situations are observed in particular at SHLs ± 4.5, ± 2.5 and -0.5, where one Na + can be enclosed in the DNA minor groove, close to an inserted arginine/ threonine couple (examples in Figures 7-C and-D).

Na + cations are also implicated in the stabilization of the DNA superhelical path that, strictly speaking, does not relate to the interface. Indeed, the wrapping of the DNA around the histones brings phosphate groups belonging to juxtaposed DNA gyres close to each other. The minimal distance that separates pairs of phosphate atoms (D P-P ) is especially short when two minor grooves face each other, dropping to 7 or 8 Å for certain pairs of superposed SHLs. Thus, the severe electrostatic repulsion induced by the proximity of negatively charged phosphate groups requires efficient shielding. Shielding is ensured by direct phosphate groups -amino acid interactions in the regions where H3 or H2B tail roots pass between the DNA gyres. To better understand what happens when close phosphate groups are not separated by histone tails, we carried out calculations of electrostatic potential and Na + density, expressed in terms of occupancy (see Materials and Methods).

Globally, there is a good correspondence between electronegative patches (Figures 8-A andB) and Na + occupancies (Figures 8-C andD). Cation (here Na + ) accumulation in the electronegative parts of minor grooves exposed to the solvent (Figures 8-C andD) were previously described for free DNAs (88-90) and are thus not surprising. More relevant for our purpose, Na + are also observed in interstices between DNA gyres devoid of histone tails, in particular in the gap extending between the two H2B tail contact points (Figure 8-C). This region is centered on an extremely narrow interstice between the juxtaposed minor grooves at SHLs -4 and +4 (D P-P = 7.3 ±1.0 Å), around the dyad axis, which generates a marked electronegative potential (Figure 8-A). Indeed, a substantial Na + occurrence is observed all along the gap, spreading from the first to the second H2B tail copy (Figure 8-C). Na + are not as prevalent in the gyre interstices delimited by the crossing points of H3 and H2B tails (Figure 8-D). In this region, the juxtaposed minor grooves are more distant (D P-P of 11±1.6 and 14.6 ±1.1 Å at SHLs -2/+6 and +2/-6, respectively) than for the facing SHLs -4 and +4 and, accordingly, produce a less strong electronegative potential (Figure 8-B). However, even in this case, Na + overflow from the minor groove and cover the closest phosphate groups at the top of the facing minor grooves (Figure 8-D).

The first part of the results presented above shows that Na + can penetrate the interface, at minor grooves where arginines and threonines are inserted. The second part reveals that a cloud of Na + intercalates in the narrow gaps between DNA gyres, weakening the repulsive electrostatic forces between close phosphate groups and complementing the shielding action of H3 and H2B tails. So, cations play an important role in stabilizing and preserving the DNA super-helix path.

DNA -histone contacts and DNA sequence

The last point relates to a possible sequence effect on the interface, which could play a role in the differential ability of the 5' and 3' halves of the 601 sequence to be maintained around the histone cores [START_REF] Hatakeyama | High-resolution biophysical analysis of the dynamics of nucleosome formation[END_REF][START_REF] Hall | High-resolution dynamic mapping of histone-DNA interactions in a nucleosome[END_REF][START_REF] Chua | The mechanics behind DNA sequencedependent properties of the nucleosome[END_REF]. The 601 sequence is not palindromic; most local contact patterns at SHL positions symmetric with respect to the DNA center (SHL-n in the DNA 5' half and SHL+n in the 3' half, n=0.5, 1.5, 2.5…) are composed of different sequences (given in Table S4), except for SHLs ± 1.5 where the TTAAA motif is found on both sides in a way satisfying symmetry. Now each SHL-n/SHL+n pair faces identical regions of histone copies and contacts the same amino acids. This particular feature offers the opportunity to finely probe to what extent the DNA sequence modulates the interface by comparing the SHL-n/histone and SHL+n/histone contacts.

The similarity within pairs of CAs measured from the point of view of either the histone copies (Figures 3 and5) or the SHL-n/SHL+n pairs (Figures 4,6, S7 and S11) is especially striking in the case of the histone structured cores (Figures 3, 4 andS7). The very symmetric contact profiles clearly argue against any sequence effect on the interface. The symmetry is confirmed by the excellent correlation existing between contacts measured in the 3' and 5' DNA halves, in terms of both electrostatic and hydrophobic CA with the protein core (Figure 9-A). Although slightly weaker, the correlation for the contacts with the tail roots is also convincing, taking into account the standard deviations (Figure 9-B). Thus, our simulations involving sequence 601 nucleosome do not support the idea that the DNA sequence significantly affects the contacts with the histones.

DISCUSSION and CONCLUSION

A nucleosome containing the high affinity 601 sequence was simulated by molecular dynamics in explicit solvent to study the DNA-histone interface in solution. The interface was analyzed by VLDM, a Voronoi based method, in terms of contact area and time occurrence.

The quantitative measurements of the interface at each contacted SHL clarify global features of DNA-histone interactions. That similar large and very stable contacts are observed for eight binding sites, from SHL -3.5 to SHL 3.5, indicates that all of these central SHLs are equally crucial for anchoring the H3 and H4 structured cores. At each of these SHL interfaces, one arginine, intimately interacting with the minor groove, represents nearly 20% of the contacts, and 7 to 9 other amino acids make the complement, the ensemble engaging long-lived contacts with the DNA. The situation is completely different near the DNA entry/exit, especially at SHLs ±6.5 where the contact areas with the histone structured domains are four times smaller than those at the central SHLs, and the contact occurrences are relatively low.

Concerning the tails, without surprise, the extremities of H3, H2A and H2B N-tails, as well as the whole H2A C-tail, are flexible and mobile as previously observed [START_REF] Zhou | Histone H4 K16Q mutation, an acetylation mimic, causes structural disorder of its N-terminal basic patch in the nucleosome[END_REF][START_REF] Erler | The Role of Histone Tails in the Nucleosome: A Computational Study[END_REF]; they have a minimal contribution to the DNA-histone interface. In contrast, the conformational variability of the roots of H3, H2B and H2A Ntails is considerably lower owing to intimate, long-lived interactions with the DNA. Indeed, these proximal histone regions contribute to 35% of the total DNA-histone contact area.

Considering together the structured domain and root tail contributions gives a global view of the contact distribution along both central and peripheral DNA regions. As expected, the central DNA region engages the more extensive and stable contacts, cumulating interactions with H3-H4 structured cores and H3 tail roots. However another noticeable region emerges around SHLs ±4.5, where large contacts with the H2A and H2B root tails substantially reinforce the interface involving the structured cores of the same histones. This observation remarkably parallels optical tweezer [START_REF] Brower-Toland | Mechanical disruption of individual nucleosomes reveals a reversible multistage release of DNA[END_REF][START_REF] Hall | High-resolution dynamic mapping of histone-DNA interactions in a nucleosome[END_REF] and FRET [START_REF] Gansen | Nucleosome disassembly intermediates characterized by single-molecule FRET[END_REF] experiments in which a strong resistance to the nucleosome disassembly was observed at 40-50 bp from the dyad, conjecturally related to a local strengthening of the DNA-histone interactions. Finally, the defective interactions are limited to the DNA extremities, in agreement with unwrapping or breathing studies (reviewed in [START_REF] Eslami-Mossallam | Nucleosome dynamics: Sequence matters[END_REF]. It is also interesting to note that post-translational modifications that affect amino acids contacting the DNA at SHLs ±6.5 (such as T45, R52, R53 or K56 of H3) enhance the DNA unwrapping [START_REF] Bowman | Post-Translational Modifications of Histones That Influence Nucleosome Dynamics[END_REF][START_REF] Lawrence | Lateral Thinking: How Histone Modifications Regulate Gene Expression[END_REF], very likely by further weakening this particular interface region.

Returning to the analysis of interactions, most interfaces at the SHLs contacting the histone structured cores and eventually tail roots combine both types of interactions, electrostatic and hydrophobic (Figures 4 and6). In several cases, in particular in the DNA-histone tail root interface, the hydrophobic interactions cover larger areas than the electrostatic contacts. Overall, the CA Electrostatic /CA Total and CA Hydrophobic /CA Total ratios are 0.23 and 0.27, respectively. These values stress the central importance of hydrophobic contacts in the nucleosome, which are too often relegated to a secondary role. Actually, the DNA-histone interface resembles those of other typical DNA-protein complexes, which the interfaces comprise two-thirds of nonelectrostatic contacts on average [START_REF] Luscombe | Amino acid-base interactions: a three-dimensional analysis of protein-DNA interactions at an atomic level[END_REF].

With very few exceptions, simulated water molecules fill space between DNA and histones in a fluid manner, without forming durable bridges between DNA and histones. The water mediated hydrogen bonds observed in 1KX5 [START_REF] Davey | Solvent mediated interactions in the structure of the nucleosome core particle at 1.9 a resolution[END_REF] either disappear or, more often, turn into direct hydrogen bonds owing to local adjustments of DNA and amino acids. In the context of a potential structuring role of water inferred from Xray structures of DNA-protein complexes [START_REF] Davey | Solvent mediated interactions in the structure of the nucleosome core particle at 1.9 a resolution[END_REF][START_REF] Luscombe | Amino acid-base interactions: a three-dimensional analysis of protein-DNA interactions at an atomic level[END_REF][START_REF] Schneider | Bioinformatic analysis of the protein/DNA interface[END_REF][START_REF] Sonavane | Cavities in protein-DNA and protein-RNA interfaces[END_REF], our simulations show that water molecules in interstices formed at the DNA-histone interface do not get durably trapped at any specific position. The various types of interaction, for example electrostatic shielding, polarization or hydrogen bonds, affect individual solvent molecules, including ions, mostly in a transitory way. The net effect is visible at the statistics level, through decreased fluctuation and diffusion near the DNA or protein surface.

Ions in solution play a decisive role in neutralizing electrostatic repulsions between solute charges. A typical case is the juxtaposed gyres of DNA wrapped around the nucleosome core: in particular in regions where two minor grooves face each other, some phosphate groups are very close to each other. The positively charged residues in H3 and H2B tail roots crossing the DNA belt between the gyres are undoubtedly essential for the super-helix formation and stability. Outside these contact points, the narrowest interstices between DNA gyres are the scene of Na + accumulation. At these locations, cations appear here as fundamental elements in the nucleosome cohesion, explaining why the DNA super-helical path exists notwithstanding phosphate groups in close proximity.

As a final point, the comparison of interfaces involving pairs of histone copies of the same kind bound to different DNA fragments did not supply evidence of any DNA sequence effect. At this stage, we cannot exclude that the DNA-histone interface could be affected by DNA sequences clearly unfavorable to nucleosome formation. However, our result echoes the findings of a recent experimental study that enabled to detect the DNA distortions during the sequential steps of the histone binding [START_REF] Hatakeyama | High-resolution biophysical analysis of the dynamics of nucleosome formation[END_REF]. During the reconstitution process, the DNA sequence clearly influences the early events of nucleosome formation. So, the (H3/H4) 2 tetramer binds more efficiently the 5' than the 3' side of the central part of the 601 sequence.

Introducing mutations in the 5' side of the 601 sequence also decreases the nucleosome reconstitution efficiency. However, no difference persists once the complexes stabilized.

As provided by our simulations in explicit solvent and VLDM, the detailed DNA-histone interface in solution can serve as a reference, very comparable to the other nucleoprotein complexes [START_REF] Luscombe | Amino acid-base interactions: a three-dimensional analysis of protein-DNA interactions at an atomic level[END_REF]. From a practical point of view, our results can be used as biophysical background to interpret experiments on nucleosome assembly or disassembly, as well as to anticipate the structural consequences of epigenetic modifications or histone mutations. The intricate and cohesive interface depicted here shed light on how the DNA super-helical wrapping and induced distortions are maintained around the histones in the nucleosome.

On the other hand, the nature of this interface raises the issue of completely dissociating DNA from histones, a process that implies breaking extensive and robust contacts and may thus be a hard, energy consuming task. This concern is supported, in particular, by the rarity of spontaneous partial unwrapping of DNA peripheral regions [START_REF] Tomschik | Nucleosome dynamics as studied by single-pair fluorescence resonance energy transfer: a reevaluation[END_REF], still supposed to be the initial step of nucleosome unfolding. It was proposed that, in cell, remodeling factors are required to favor the occurrence of such breathing and to achieve the nucleosome disassembly [START_REF] Miyagi | Dynamics of nucleosomes assessed with time-lapse high-speed atomic force microscopy[END_REF]. The strong DNA-histone interface described here could further explain why the action of remodeling factors is essential in the dynamical positioning of nucleosomes along eukaryotic genomes. The data were extracted and averaged from the four simulations, SYS1, SYS2, SYS1-bis and SYS2-bis. The composition of the N-terminal tails is given for the four histone types in the five models studied here: SYS0 is the 3MVD structure, and SYS1, SYS1-bis, SYS2 and SYS2-bis each contain added amino acids taken from the specified chains of 1KX5. The H2A C-tail from 1KX5 was integrated in SYS1-bis and SYS2bis. The tail sequences are detailed in Table S1.

Figures

Tables

Q K K D G K K* R R K T R K E S Y
In SYS1, SYS2, SYS1-bis and SYS2-bis simulations, the N-tails of H3, H2A and H2B showed flexible extremities sampling various conformations (amino acids in italic) and stiff roots covering a much more limited conformation landscape (amino acids in bold). The stars indicate the limit between those two regions according to a NMR study [START_REF] Zhou | Histone H4 K16Q mutation, an acetylation mimic, causes structural disorder of its N-terminal basic patch in the nucleosome[END_REF].

  interface can be analyzed according to the nature of contacts. Hydrophobic contacts correspond to carbon atoms exclusively (C-C). Electrostatic contacts involve N and O atoms (N-N, N-O or O-O), excluding repulsive interactions between two donors or two acceptors; hydrogen bonds and salt bridges belong to this category. A third type reports a proximity of N-C or O-C atoms. VLDM analyses of simulated nucleosomes were carried out on snapshots extracted every 250ps from the trajectories, discarding the first 50ns. To avoid open or distorted polyhedra in the Laguerre tessellation, an 8 Å thick water layer around the solute was taken together with the solute as input to VLDM. Examination of the distances characterizing the hydrophobic or electrostatic contacts showed maximal distribution peaks at 4Å (from 3.5 to 6Å) for C-C contacts and 2.5Å (from 2.5 to 5 Å) for N-O, N-N and O-O contacts.

Figure 1 :

 1 Figure 1: Overviews of the nucleosome.

Figure 2 :

 2 Figure 2: Contact area maps of the DNA-histone interface.

Figure 3 :

 3 Figure 3: Interface between the histone structured cores and DNA across simulations.

Figure 4 :

 4 Figure 4: Interface between DNA and the histone structured cores.

Figure 5 :

 5 Figure 5: Interfaces involving the H3, H2A and H2B tail roots and DNA across simulations.

Figure 6 :

 6 Figure 6: Interface between DNA and the tail roots of H3, H2A and H2B.

Figure 7 :

 7 Figure 7: Water molecules and Na + cations trapped in the DNA-histone structured core interface.

Figure 8 :

 8 Figure 8: Electrostatic potentials and Na + occupancy.

Figure 9 :

 9 Figure 9: Comparison between the contacts involving the 5' and 3' halves of the 601 sequence and the histones.

Table 1 :

 1 Histone tails composition of the models.

		SYS0	SYS1 and SYS1-bis	SYS2 and SYS2-bis
	H3 N tail	37 → 44	24 → 44 from chain A	24 → 44 from chain E
	H4 N tail	20 → 24	17 → 24 from chain B	17 → 24 from chain F
	H2A N tail	12 → 16	9 → 16 from chain C	9 → 16 from chain G
	H2B N tail	28 → 34	19 → 34 from chain D	19 → 34 from chain H
		SYS0	SYS1 and SYS2	SYS1-bis and SYS2-bis
	H2A C tail	No C-tail	No C-tail	119 → 128 from chain C

Table 2 :

 2 Flexible extremities and stiff roots of histone tails.

	H3	24 25 26 27 28 29 30 31 32 33 34 35 36 37
	N-tails

* 38 39 40 41 42 43 44

A A R K S A P A T G G V K K* P H R Y R P G H2A

9

10 11*

12 13 14 15 16

		K T R* A K A K T
	H2B	19 20 21 22 23 24 25*

26 27 28 29 30 31 32 33 34

Acknowlegments

The authors thank Dr Liliane Mouawad (Institut Curie, Centre universitaire d'Orsay, Orsay) for helpful advices.

Fundings

This work was funded by the PEPS (Projets Exploratoires Pluridisciplinaires) program of CNRS. The simulations were carried out on the GENCI-CEA platform. This study was further supported by the "Initiative d'Excellence" program from the French State (Grant "DYNAMO", ANR-11-LABX-0011-01 and ANR-11-EQPX-0008)".

Supporting Information

Table S3: Time occurrence of Na + cations at the DNA histone interface. Table S4 : DNA sequences at contacted SHLs.

Corresponding authors:

Brigitte

Author Contributions

Ahmad Elbahnsi performed the simulations and contributed to the development of VLDM with Christophe Oguey. Ahmad Elbahnsi initiated the analyses that were completed and supplemented by Romain Retureau, also in charge of most figures. Marc Baaden participated in discussions and in rereading the manuscript.

Christophe Oguey and Brigitte Hartmann, the project leaders, wrote the manuscript.


Anis Amazigh Hamza

Iyad Dayoub

Ihsen Alouani

Abderrahmane Amrouche

Enabling User Relaying In MCM-NOMA Under Doubly Selective Channels Using Iterative Interference Cancellation Schemes For Wireless IoT Networks

Keywords: Interference, IoT, non-orthogonal multiple access, cooperative relaying, doubly selective channels, multi-carrier modulation

Cell-edge users of the future cellular internet of things (IoT) with massive IoT sensors can suffer from extremely severe channel conditions, especially under very high-speed scenarios. In this paper, we present a performance improvement method for cell-edge users of multi-carrier modulation (MCM)-based non-orthogonal multiple access (NOMA) downlink systems. To this end, we consider the implementation of cooperative user relaying NOMA (CUR-NOMA) and derive its lower bound E2E-BER under doubly selective channels. In addition, the imperfect successive interference cancellation (SIC) process is analyzed, wherein two interference cancellation schemes are combined to remove the NOMA induced inter-user interference (IUI) and the doubly selective channel induced inter-carrier interference (ICI).

I. INTRODUCTION

NOMA has been recognized as a promising multiple access scheme for future wireless networks such as the emerging cellular IoT with massive IoT sensors and devices [START_REF] Chen | Energy-Efficient Power Allocation and Splitting for mmWave Beamspace MIMO-NOMA With SWIPT[END_REF] as it effectively improves the network performance [START_REF] Maraqa | A survey of rate-optimal power domain noma with enabling technologies of future wireless networks[END_REF]. The majority of papers on NOMA and cooperative communication networks assume a perfect SIC (P-SIC). However, due to hardware limitations and channel estimation errors, SIC is imperfect in most practical cases. In this regard, recent papers are pushing the research ahead by going beyond that assumption. Recently, some works tried to investigate the performance degradation that occurs due to imperfect SIC (I-SIC) [START_REF] Kara | On the error performance of cooperative-NOMA with statistical CSIT[END_REF]- [START_REF] Hamza | On the error rate performance of full-duplex cooperative NOMA in wireless networks[END_REF]. However, the existing contributions exploring CUR-NOMA mainly focus on single-carrier scenarios. On the other hand, Multi-Carrier Modulation (MCM) based transmission has proven better performances for combating the hostility of wireless channels and providing high data rates. Few studies can be found on MCM-NOMA systems. In [START_REF] Maatouk | Frequencydomain NOMA with two sets of orthogonal signal waveforms[END_REF] a NOMA scheme based on combining the orthogonal frequency division multiple access (OFDMA) with the multi-carrier code division multiple acces (MC-CDMA) is proposed. The resource allocation for MCM-NOMA systems with I-SIC is studied and optimized in [START_REF] Chen | Optimal resource allocation for multicarrier NOMA in short packet communications[END_REF]. The authors of [START_REF] Li | Outage Probability Analysis for the Multi-Carrier NOMA Downlink Relying on Statistical CSI[END_REF] derived the expressions of the outage probability of MCM-NOMA downlink system.

The common aspect of all the aforementioned studies is that they assume that each subcarrier undergoes quasi-static non-selective flat fading. However, this assumption does not hold in very highspeed scenarios, where the wireless channels are highly doubly selective [START_REF] Saideh | Efficient Equalization for FBMC-OQAM Under Doubly Selective Channel Estimation Errors[END_REF]. Moreover, the high mobility nature of the modern transportation systems leads to more interference and channel estimation errors which affect link reliability and system robustness [START_REF] Alalewi | On 5G-V2X use cases and enabling technologies: A comprehensive survey[END_REF]. The implementation of NOMA with different MCM waveforms under doubly selective scenarios is investigated in [START_REF] Saideh | Joint interference cancellation for multi-carrier modulation-based non-orthogonal multiple access[END_REF] wherein an efficient joint interference cancellation scheme is proposed. However, in [START_REF] Saideh | Joint interference cancellation for multi-carrier modulation-based non-orthogonal multiple access[END_REF] we considered only the signal processing applied at the level of the cell-center user. In practice, the challenges facing the cell-edge user are more interesting since it experiences poor channel conditions compared to the cell-center user.

To the best of our knowledge, there is a lack of works investigating the performance of cell-edge users of CUR-MCM-NOMA with I-SIC under doubly selective environments. Motivated to develop this treatise, we present in this paper a performance improvement method for cell-edge users of MCM-NOMA systems with I-SIC under doubly selective wireless channels. In this regard, we consider the implementation of a cooperative user relaying scheme, wherein the cell-center user exploits the joint interference cancellation scheme to serve as DF relay and assist the communication between the cell-edge user and the base station (BS). Furthermore, we combine it with an efficient interference cancellation scheme applied by the cell-edge user to reduce the inter-carrier interference (ICI) caused by the doubly selective nature of the relaying link.

II. SYSTEM MODEL

We consider a downlink single-cell MCM-based CUR-NOMA network, in which a BS transmits data to a set of users. The set of users is divided into two subsets based on the channel strength of the users, i.e., the set of users with strong channel gains are called cell-center users (CUs), and the set of users with low channel gains are referred to as cell-edge users (EUs). As in [START_REF] Zhang | Performance analysis and optimization in downlink NOMA systems with cooperative full-duplex relaying[END_REF], we assume that only two users are multiplexed on the same resource block (RB), wherein one is selected from the CUs and the other is selected from the EUs. Different from previous works, we assume all the wireless links to be doubly selective channels. Furthermore, we assume the worst-case scenario for the EU wherein the direct link between the BS and the EU is blocked by some obstacles, and hence it cannot be used to convey the information as it is shown in Fig. 1. The CU is used as a DF relay to assist the communication between the BS and the EU and the MCM-based CUR-NOMA consists of two phases, as described in the following. During the direct transmission phase, the BS transmits the superposed MCM-NOMA signal to the CU. According to [START_REF] Saideh | Joint interference cancellation for multi-carrier modulation-based non-orthogonal multiple access[END_REF] the transmitted signal ( ) consisting of subcarriers and timesymbols can be expressed as

( ) = =1 =1 , ( ) , (1)
, = √ , , + √ 1 - , , (2)
, , , , , are the transmitted symbols at the th subcarrier and the th time symbol of the EU and the CU, respectively. We assume that the powers of the transmitted symbols are normalized such that the average power of the NOMA signal = 1 . denotes the PA coefficient of the EU.

, ( ) denotes the basis pulses which are, essentially, time and frequency shifted versions of the prototype filter ( ):

, ( ) = ( -) 2 ( - ) (3)
After conducting the sampling process, (1) can be written in a matrix form as [START_REF] Nissel | OFDM and FBMC-OQAM in Doubly-Selective Channels: Calculating the Bit Error Probability[END_REF] = (4)

= [ 1,1 2,1 ... ,1 1,2 ... , ] (5)
= [ 1,1 2,1 ... ,1 1,2 ... , ] (6)
The vector , ∈ C ×1 represents the sampled basis pulses of (3) and builds the transmit matrix ∈ C × while ∈ C ×1 stacks all the transmitted data symbols in one vector. denotes the number of samples of the whole transmission block. Hence, the demodulated signal at the CU can be written in a matrix form as

= + (7) = ( 8
)
where is called the system transmission matrix (STM), ∈ C × denotes the demodulation matrix, the Gaussian distributed noise at the CU is denoted by ∈ C ×1 , and ∈ C × is the doubly-selective channel convolution matrix at the CU. As stated before, the CU performs I-SIC, where it first detects the EU data while considering as interference.

ˆ = M 1 √ (9)
where ˆ denotes the data of the EU detected during the first step of the SIC and M (.) is the quantization operator, that is, nearest neighbor detection while is the adopted equalizer that estimates the EU data at the CU. To detect the data of the CU, ˆ will be removed from the received signal as follows

ˆ = M 1 √ 1 - -ˆ √ ˆ ( 10
)
where denotes the adopted equalizer that estimates the data of the CU and the hat1 in ˆ denotes the estimated STM. During the cooperative phase, the CU regenerates the data of the EU detected in the first step of the I-SIC process and forwards it to the EU with an average power ˆ = . Similar to the direct phase the received signal at the EU after the sampling is given by

= ˆ + (11) = ( 12
)
The EU can then detect its signal directly as follows

ˆ = M ( 13
)
Note that we assume both wireless links and to be doubly selective. Therefore, the received power will be spread across the non-diagonal elements of the STM matrices, and hence the use of a simple one-tap equalizer2 = ( ˆ ) -1 , where ( ˆ ) gives the diagonal matrix of ˆ , might not be sufficient.

III. EQUALIZATION AND INTERFERENCE CANCELLATION

To reduce the interference during the direct transmission phase, the CU will exploit the joint iterative algorithm which consist of the following steps. Initially, the CU detects the data of the EU and then his own data by considering a one-tap equalizer at (9) and [START_REF] Saideh | Efficient Equalization for FBMC-OQAM Under Doubly Selective Channel Estimation Errors[END_REF], respectively. After that, it starts to remove the interference at each iteration as follows

¯ ( ) = - ( ˆ ) √ ˆ ( -1) -ˆ √ 1 -ˆ ( -1) (14)
ˆ ( ) = M 1 √ ( ˆ ) -1 ¯ ( ) (15)
¯ ( ) = - ( ˆ ) √ 1 -ˆ ( -1) -ˆ √ ˆ ( ) (16)
ˆ ( ) = M 1 √ 1 - ( ˆ ) -1 ¯ ( ) (17)
where ( ˆ ) zeros the diagonal elements of ˆ while keeping the off-diagonal elements. The term ¯ ( ) in ( 14) denotes the result of the th iteration which is used to detect the data of the EU at the CU, whereas ¯ ( ) in ( 16) is used to detect the data of the CU. Note that in ( 14) the ( -1)th detected data of the EU, is used to remove the off-diagonal induce ICI while the ( -1)th detected data of the CU is used to cancel the NOMA induced IUI. ( 16) follows the same principle as [START_REF] Nissel | Doubly-selective channel estimation in FBMC-OQAM and OFDM systems[END_REF] (see Algorithm 1 at top of the next page).

After 1 iterations the CU is ready to transmit the detected ˆ ( 1 ) to the edge user. However, due to the doubly selective nature of the cooperative link, new ICI will occur at the received signal of the EU. The received signal at the EU can be written in a matrix form exactly as in [START_REF] Alalewi | On 5G-V2X use cases and enabling technologies: A comprehensive survey[END_REF] by just replacing ˆ by ˆ ( 1 ) since in here, we assume that the CU has already applied the joint iterative algorithm before starting the cooperative phase. To cancel the new ICI we propose a simple, yet effective, interference cancellation scheme. Similar to Algorithm 1: Interference Cancellation at the CU Initialize: = 0; 1) Detect the EU data using (9)

2) Detect the CU data using (10) for ( < Number of iterations " 1 " ) do 1) ++ 2) Remove the ICI and the IUI induced interference using ( 14)

3) Improve estimation of ˆ ( +1) , enabled by reduced interference at the pilot positions. 4) Detect the EU data using (15) 5) Remove the ICI and the IUI induced interference using (16) 6) Detect the CU data using ( 17) end for the one suggested in [START_REF] Molisch | Iterative methods for cancellation of intercarrier interference in OFDM systems[END_REF] for OFDM waveforms, the ICI is canceled as follows

¯ ( ) = - ( ˆ ) ˆ ( -1) (18) ˆ ( ) = M ( ˆ ) -1 ¯ ( ) (19)
First, the edge user detects its signal using ( 13) adopting the one-tap equalizer = ( ˆ ) -1 . Then, it applies the iterative scheme using ( 18) and ( 19) (see Algorithm 2 below).

The most computationally expensive part of each algorithm is the vector-matrix multiplication used in the interference calculations (see ( 14), ( 16), and ( 18)). Therefore, the computational complexity of each algorithm is O ( ) 2 per iteration 3 which is very low compared to the full block MMSE that requires O ( 2) 3 operations as it has to invert a matrix of a size of 2 × 2 .

Algorithm 2: Interference Cancellation at the EU Initialize: = 0; 1) Detect the EU data using (13) for ( < Number of iterations " 2 " ) do 1) ++ 2) Remove the ICI induced interference using (18) 3) Improve estimation of ˆ ( +1) . 4) Detect the EU data using (19) end for

IV. THE END-TO-END BIT ERROR RATE

An erroneous detection can occur either in the direct transmission phase or the cooperative phase. Thus, the end-to-end bit error rate (E2E-BER) can be expressed as:

E2E-BER = 1 -(1 -SIC )(1 -coop ) (20)
Where SIC denotes the probability that 2 is detected erroneously during the SIC process of the direct phase, and coop denotes the probability that the EU detects erroneously the relaying link signal during the cooperative phase. The lower bound E2E-BER that describes the best performance that could be achieved under doubly selective channels would be the E2E-BER of the theoretical doubly 3 Only 4 iterations are needed for each algorithm (see Fig. 3). flat channels with zero NOMA induced IUI. Hence, using [13, Eq. ( 15)-( 18)], it can be shown that the lower bound E2E-BER that can be achieved by CUR-MCM-NOMA system is given by

E2E-BER min = 1 -1 - 2 1 1 - 2 2 (21)
Where ( ) is given in Table 1, 1 denotes the average channel gain of , 2 denotes the average channel gain of and is the average noise power.

V. NUMERICAL RESULTS

In this section, we simulate the performance of the proposed CUR-MCM-NOMA system while taking the performance of the P-SIC with perfect channel state information (CSI) as a benchmark for comparison. In all scenarios, we conduct Monte Carlo simulations with 1500 repetitions while considering a channel model with Vehicular-A power delay profile and Jakes Doppler model. Unless stated otherwise, we assume imperfect CSI and consider the auxiliary symbol channel estimation method [START_REF] Nissel | Doubly-selective channel estimation in FBMC-OQAM and OFDM systems[END_REF] with the simulation parameters shown in Table 2. Fig. 2 shows the E2E-BER over SNR for CUR-MCM-NOMA. It can be seen that our proposed scheme performs close to Bit Error Rate

10 -3 L 1 =0, L 2 =0 L 1 =4, L 2 =0 L 1 =0, L 2 =4 L 1 =4, L 2 =4
Perfect SIC CU's speed=200 Km/h EU's speed=100 Km/h Fig. 4. The E2E-BER vs the speed of each user for SNR = 30 dB.

Bit Error Rate

L 1 =0, L 2 =0 L 1 =4, L 2 =0 L 1 =0, L 2 =4 L 1 =4, L 2 =4
Perfect SIC Fig. 5. The E2E-BER vs the PA coefficient for SNR = 30 dB.

the P-SIC and can even outperform the P-SIC in the high-SNR regime (SNR ≈ 35dB) or when perfect CSI is provided. Because our scheme reduces the ICI and IUI, whereas the P-SIC cancels only the IUI and keeps the ICI. Another advantage of the proposed iterative algorithms is their fast convergence. This fact is presented in Fig. 3, where we can note how after 3 to 4 iterations both algorithms will converge. One can also note that when both algorithms are combined ( 1 = 4), the performance gain is significantly increased even when imperfect CSI is assumed. Fig. 4 shows the E2E-BER versus the speed of each user for CUR-MCM-NOMA with perfect CSI. Interestingly, we note that when the EU's speed is very high ( ≥ 300km/h), our proposed scheme ( 1 = 4, 2 = 4) outperforms the P-SIC case. Moreover, when the speed of the CU is very high, the performance of the proposed scheme is very close to that of the P-SIC. Finally, It is worth noting that as with any NOMA-based system, appropriate PA coefficient selection is very crucial, as shown in Fig. 5.

VI. CONCLUSION

This paper presented a performance improvement method for EUs under very high-speed mobility using cooperative MCM-NOMA technology combined with two efficient interference cancellation schemes. The obtained results have shown that the introduced scheme provides a significant performance improvement while maintaining a fast convergence rate, hence low complexity.

Fig. 1 .

 1 Fig. 1. The system model of the considered very high speed scenario.

Fig. 2 .

 2 Fig. 2. The E2E-BER vs SNR.

4 Fig. 3 .

 43 Fig. 3. The E2E-BER vs Number of Iterations 2 for SNR = 30 dB.

TABLE 1

 1

	. f(x) for different modulation orders
	( )								modulation order	waveform
	( ) = 1 2 -1 2 √ 1+ 2					4-QAM	FBMC-OQAM
	( ) = 1 2 -	8	√	3 1+ 10 -	8 √	6 9+ 10 +	8 √	5 25+ 10	16-QAM	FBMC-OQAM

TABLE 2 .

 2 Simulation parameters

	Parameter		value
	Number of symbols		30
	Number of subcarriers		24
	Number of auxilary symbols (per pilot)	4
	Pilot-to-data power offset (	)	4.685
	Pilot overhead		11.1%
	Subcarrier spacing		15 kHz
	Carrier frequency		2.5 GHz
	Waveform		FBMC-OQAM
	Modulation order		4
	Channel gain 1		0.8
	Channel gain 2		0.6
	CU's speed		200 km/h
	EU's speed		100 km/h
	PA coeffecient		0.9

As mentioned in[START_REF] Nissel | Doubly-selective channel estimation in FBMC-OQAM and OFDM systems[END_REF], it is computationally more efficient to directly estimate ˆ compared to the channel transfer function ˆ .

Note that a one-tap equalizer requires O ( ) computational complexity since only the diagonal of ∈ C × is inverted.