Jean-Charles Walter
email: jean-charles.walter@umontpellier.fr

Jérôme Rech

Nils-Ole Walliser

Jérôme Dorignac

Frédéric Geniet

John Palmeri

Andrea Parmeggiani

Jean-Yves Bouet
email: jean-yves.bouet@univ-tlse3.fr

Physical modeling of a sliding clamp mechanism for the spreading of ParB at short genomic distance from bacterial centromere sites

Bacterial ParB partitioning proteins involved in chromosomes and low-copy-number plasmid segregation have recently been shown to belong to a new class of CTP-dependent molecular switches. Strikingly, CTP binding and hydrolysis was shown to induce a conformational change enabling ParB dimers to switch between an open and a closed conformation. This latter conformation clamps ParB dimers on DNA molecules, allowing their diffusion in one dimension along the DNA. It has been proposed that this novel sliding property may explain the spreading capability of ParB over more than 10-Kb from parS centromere sites where ParB is specifically loaded. Here, we modeled such a mechanism as a typical reaction-diffusion system and compared this 'Clamping & sliding' model to the ParB DNA binding pattern from highresolution ChIP-sequencing data. We found that this mechanism can not account for all the in vivo characteristics, especially the long range of ParB binding to DNA. In particular, it predicts a strong effect of the presence of a roadblock on the ParB binding pattern that is not observed in ChIP-seq. Moreover, the rapid assembly kinetics observed in vivo after the duplication of parS sites is not easily explained by this mechanism. We propose that 'Clamping & sliding' might explain the ParB spreading pattern at short distances from parS but that another mechanism must apply for ParB recruitment at larger genomic distances.

Introduction

Faithful segregation of the full set of genetic information is essential for all living cells. In bacteria, segregation follows DNA replication and involves the separation and transportation of the new copies in opposite directions along the longitudinal cell axis [START_REF] Bouet | Mechanisms for chromosome segregation[END_REF]. Partition of chromosomes and low copy number plasmids mainly relies on ParABS systems (for reviews; [START_REF] Bouet | Plasmid Localization and Partition in Enterobacteriaceae[END_REF]Jalal and Le, 2020). They encode a Walker-type ATPase (ParA), a site-specific DNA binding protein (ParB), which binds to parS centromere sites. ParA action separates the ParB assemblies nucleated on parS sites, which are always located near the origin of replication, and actively relocates them toward opposite cell poles. This process ensures that every daughter cell receives at least one copy of the replicated DNA molecules.

Recent studies on the bacterial partition protein ParBs have revealed a new activity, namely CTP binding (Jalal et al., 2020;[START_REF] Osorio-Valeriano | ParB-type DNA Segregation Proteins Are CTP-Dependent Molecular Switches[END_REF][START_REF] Soh | Self-organization of parS centromeres by the ParB CTP hydrolase[END_REF]. ParB proteins thus now emerge as a new class of CTP-dependent molecular switches. ParB dimers bind specifically to a few short parS DNA sites, usually 16-bp, which subsequently nucleates the assembly of hundreds of ParB in its spatial vicinity. This leads to the formation of highly concentrated but dynamic nucleoprotein complexes of small size (~40 nm; [START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF], with ParB binding to non-specific DNA over large genomic distances (10-Kb; e.g. [START_REF] Breier | Whole-genome analysis of the chromosome partitioning and sporulation protein Spo0J (ParB) reveals spreading and origin-distal sites on the Bacillus subtilis chromosome[END_REF][START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF][START_REF] Lagage | Regional Control of Chromosome Segregation in Pseudomonas aeruginosa[END_REF][START_REF] Rodionov | Silencing of genes flanking the P1 plasmid centromere[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF], and visible as bright foci in fluorescent microscopy [START_REF] Breier | Whole-genome analysis of the chromosome partitioning and sporulation protein Spo0J (ParB) reveals spreading and origin-distal sites on the Bacillus subtilis chromosome[END_REF]Diaz et al., 2015, 4694;[START_REF] Erdmann | Intracellular localization of P1 ParB protein depends on ParA and parS[END_REF][START_REF] Lagage | Regional Control of Chromosome Segregation in Pseudomonas aeruginosa[END_REF][START_REF] Lim | Evidence for a DNA-relay mechanism in ParABS-mediated chromosome segregation[END_REF]. Biochemical studies in the presence of CTP and the resolution of a ParB-CDP co-crystal structure suggested that a sliding clamp mechanism controlled by CTP binding could be the basis for this clustering of numerous ParBs around the parS sequence (Jalal et al., 2020;[START_REF] Soh | Self-organization of parS centromeres by the ParB CTP hydrolase[END_REF].

The ParB-DNA nucleoprotein complex is involved in the intracellular positioning of the parS centromere sequences and their segregation toward opposite poles of the cell through its interaction with the cognate ParA ATPase (for reviews; [START_REF] Bouet | Plasmid Localization and Partition in Enterobacteriaceae[END_REF]Jalal and Le, 2020). CTP binding and hydrolysis control ParBs activities (Jalal et al., 2020;[START_REF] Osorio-Valeriano | ParB-type DNA Segregation Proteins Are CTP-Dependent Molecular Switches[END_REF][START_REF] Soh | Self-organization of parS centromeres by the ParB CTP hydrolase[END_REF]. Apo-ParB dimers adopt an open conformation and bind to DNA (Fig. 1A). Specific binding to parS activates CTP binding which induces a conformational change that converts ParB dimers in a close conformation (gate closure) entrapping double strand DNA. The clamped-ParB favor parS unbinding as suggested by (i) the steric clash for parS binding modeled in the CDP-bound ParB crystal structure [START_REF] Soh | Self-organization of parS centromeres by the ParB CTP hydrolase[END_REF] and (ii) the ParB release from parS in the presence of CTP (Jalal et al., 2020). Upon release from parS sites, clamped-ParB dimers would thus remain trapped along the DNA on either side of parS until the gate opens to release ParB from the DNA (Fig. 1B). [START_REF] Soh | Self-organization of parS centromeres by the ParB CTP hydrolase[END_REF] thus proposed that a ParB sliding clamp mechanism, restricting ParB around parS sites, might give rise to the characteristic ParB DNA binding distribution observed by ChIP assays (e.g. [START_REF] Breier | Whole-genome analysis of the chromosome partitioning and sporulation protein Spo0J (ParB) reveals spreading and origin-distal sites on the Bacillus subtilis chromosome[END_REF][START_REF] Rodionov | Silencing of genes flanking the P1 plasmid centromere[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF].

ParB was found to cluster around parS based (i) on gene silencing in the vicinity of parS [START_REF] Lynch | SopB protein-meditated silencing of genes linked to the sopC locus of Escherichia coli F plasmid[END_REF][START_REF] Rodionov | Silencing of genes flanking the P1 plasmid centromere[END_REF], (ii) on ParB binding over more than 10-Kb on both sides of parS [START_REF] Murray | The bacterial chromosome segregation protein Spo0J spreads along DNA from parS nucleation sites[END_REF][START_REF] Rodionov | Silencing of genes flanking the P1 plasmid centromere[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF], and (iii) on the lower accessibility of proteins around parS DNA [START_REF] Bouet | Molecular basis of the supercoil deficit induced by the mini-F plasmid partition complex[END_REF][START_REF] Lynch | SopB protein-meditated silencing of genes linked to the sopC locus of Escherichia coli F plasmid[END_REF]. However, the quantity of ParB dimers present in the cell is insufficient to continuously cover the observed spreading zone. Several groups have carried out physical modeling studies [START_REF] Broedersz | Condensation and localization of the partitioning protein ParB on the bacterial chromosome[END_REF][START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF][START_REF] Walter | Models of supercoiled DNA interacting with an anchored cluster of proteins: towards a quantitative estimation of chromosomal DNA supercoiling[END_REF][START_REF] Walter | Looping and clustering model for the organization of protein-DNA complexes on the bacterial genome[END_REF] to investigate the mechanisms that could account for the assembly of ParB (reviewed in [START_REF] Funnell | ParB Partition Proteins: Complex Formation and Spreading at Bacterial and Plasmid Centromeres[END_REF]Jalal and Le, 2020). By testing the '1D-spreading', 'Spreading & bridging' in the strong coupling limit, 'Looping & clustering' and 'Nucleation & caging' models on high-resolution ChIP-sequencing data, only the latter model based on low affinity but synergistic interactions actually fits best observations [START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF]. However, all these models were proposed before the finding that ParB can clamp and diffuse unidimensionally along the DNA.

Here, we developed a physical modeling approach based on the newly proposed sliding mechanism, namely 'Clamping & sliding', involving the loading of ParB at parS sites with subsequent free diffusion as a protein clamp along the DNA track followed by its unbinding upon clamp opening (Fig. 1B). We then compared this novel physical model with the ParB DNA binding pattern from high-resolution ChIP-sequencing data reported for the ParABS system of the F plasmid. Using recent estimations of the release and unbinding parameters (Jalal et al., 2020) and of the diffusion coefficient [START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF], we showed on the basis of the physical analysis with biological parameters of incoming and outgoing flux of ParB on DNA that 'Clamping & sliding' is not able to account for the number of particles found in vivo on the F plasmid, especially at large distances from parS. Importantly, one-dimensional translocation is expected to be interrupted by obstacles on DNA. The modeling of the effect of a roadblock failed to describe the ChIP-seq pattern obtained in the presence of a natural roadblock. Moreover, imaging ParB clusters using epifluorescence microscopy at high temporal resolution indicated a rapid recovery of ParB fluorescence intensity at the onset of DNA replication in contrast to the model prediction. These results suggest that the 'Clamping & sliding' model could only partly account for the assembly mechanism. We thus propose that a 'Clamping & sliding' mechanism might explain the ParB spreading pattern only at short distance but that another mechanism must apply for ParB binding at long distance.

The 'Clamping & sliding' model

The DNA is modeled as a circular filament of length L displaying N discrete sites of 16-bp (footprint of a ParB dimer; [START_REF] Bouet | Molecular basis of the supercoil deficit induced by the mini-F plasmid partition complex[END_REF]. ParB dimers (i) bind specifically to parS, (ii) switch conformation and bind CTP, (iii) convert to protein clamps, (iv) release parS, (v) freely diffuse on DNA, and (vi) unbind DNA by opening the clamp (Fig. 1B). Therefore, ParB can only enter the filament by a unique loading gate, parS, at the binding rate B (Fig. 1B). It is released from parS at the rate R, and it can diffuse unidimensionally on the DNA, with a diffusion coefficient D, but cannot cross parS, which is always occupied by ParB. Lastly, ParB clamps exit anywhere from the circular DNA filament by opening the clamp at the unbinding rate U. We thus modeled the 'Clamping & sliding' mechanism by a typical reaction-diffusion model. Remarkably, in contrast to all other models at thermodynamic equilibrium previously proposed (see above), this steady-state sliding mechanism is out-of-equilibrium. Indeed, it displays a stationary flux of ParBs emerging only from parS and diffusing unidimensionally away before eventually detaching stochastically. Note that in the framework of the 'Clamping & sliding' model, ParB non-specific DNA binding activity is not relevant due to the high concentration of competitor DNA represented by the nucleoid.

In the cells, ParB dimers can be found in two states: (i) assembled in clusters around parS, which represents over 90% of ParB present in the cell and (ii) freely diffusing over the nucleoid corresponding to the remaining ParB population, which acts as a reservoir for the clusters [START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF]. These two populations may represent the sliding clamps and the free ParBs, respectively. Therefore, in the stationary regime, the flux balance of ParB in the partition complex is J in = J out , with the current of incoming particles J in = R, and outgoing particles J out = N a U, where N a is the number of particles clamped to DNA. This leads to the conservation of the average number N a of clamped-ParB, imposed by the two rates, R and U:

N a = R/U.
(1)

The time-dependent density ߩሺ,ݔ ݐሻ of ParB can be described by the following equation:

߲ߩሺ௫,௧ሻ డ௧ ൌ ,ݔ߂ߩሺܦ ݐሻ ܴߜሺݔሻ െ ߩሺ,ݔ ݐሻܷ.
(2)

The stationary solution for a coarse grained number density, n(x)=ߩሺݔሻߜ,ݔ is readily obtained from the steady-state solution of Eq.(2) as:

݊ሺݔሻ ൌ ߞ݁ ି௫/ஜ ݔߜ (3)
where δx = 16-bp corresponds to one footprint of ParB, ߤ ൌ ඥܷ/ܦ is the characteristic length scale (in bp) of the decay corresponding to the typical distance performed by ParB by diffusion before detachment, and ݔߜߞ ൌ ܴ/ሺ2ඥܷܦሻ ݔߜ is the overall amplitude of the density decay, which linearly increases with the release rate R. In the next section, we applied this formalism using biologically determined parameter values.

Results

Simulation of the "Clamping & sliding' model using biological parameters for ParB.

Our physical modeling for the 'Clamping & sliding' mechanism contains four kinetic parameters (see Fig. 1B). We estimated these parameters as follows: (i) the ParB F binding to parS F , B, with an apparent association constant K on ~2ൈ10 5 mol -1 s -1 (K D ~ 2 nM) [START_REF] Ah-Seng | Dual role of DNA in regulating ATP hydrolysis by the SopA partition protein[END_REF], is extremely rapid since ParB intracellular concentration is well above the K D [START_REF] Bouet | Probing plasmid partition with centromere-based incompatibility[END_REF], and several orders of magnitude higher than the other model rates (see below). We thus considered that (i) parS is always occupied by a ParB dimer; (ii) the apparent diffusion coefficient of ParB F clustered around parS F , D ~ 0.05 µm 2 s -1 (or ~ 4.3ൈ10 5 bp 2 s -1 ) as measured by super-resolution microscopy [START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF]; (iii) the release of ParB from parS, R ~ 0.1 s -1 (Fig. S1A-B); and (iv) the unbinding of the ParB clamp from non-specific DNA U ~ 0.017 s -1 (Fig. S1A,C). These two latter parameters were obtained from the in vitro biochemical experiments, performed with ParB of Caulobacter crescentus (ParB Ccre ) in the presence of CTP (Fig. 2A in Jalal et al., 2020), by calculating the kinetic rates using nonlinear regression fits (see Fig. S1 and Methods). Eq.( 2) contains only the three kinetic parameters D, R and U. We accounted for the assumption that the parS site is always occupied by a ParB with a constant source term, i.e. a delta-function δ(x).

We introduced these parameter values in our physical model to estimate the number of attached particles, and we obtained, from Eq.(1), an average number of clamped ParB dimers per DNA molecule N a = R/U ~6. This value is much lower than the N a ~250 observed experimentally [START_REF] Adachi | Subcellular positioning of F plasmid mediated by dynamic localization of SopA and SopB[END_REF][START_REF] Bouet | Probing plasmid partition with centromere-based incompatibility[END_REF] indicating that the biological values for R and U are too small to account for the number of ParB per cluster. Despite this discrepancy, we further plotted, in Fig. 2A, the theoretical prediction for the profile given by Eq. (3) and, as a benchmark, the corresponding Monte Carlo simulations. These simulations consist of a discrete version of the 'Clamping & sliding' model applied on a DNA molecule of length L = 60-Kb, which corresponds to the actual size of the F plasmid used for the ChIP-sequencing data (see Methods), giving N = 3750 non-overlapping ParB sites. From Eq. (3), the exponential decay has a characteristic length µ ~5.1-Kb and an amplitude ߞ ~6ൈ10 -4 bp -1 , leading to the coarse-grained amplitude ݔߜߞ ~10 -2 . As shown in Fig. 2A, analytic and numerical predictions are both in excellent agreement with each other but result in a very low density profile around parS unable by far to describe the ChIP-seq profile (red symbols). N a , the total number of ParB on DNA, corresponds to the integral of n(x). Thus, a low value of N a ~6 implies directly a low value of the overall amplitude of n(x) with the biological parameters from Fig. S1 compared to the ChIP-seq data. Also, the slopes of the decay are different: the ChIP-seq profile displays a slower decay, suggesting a larger characteristic length ߤ (discussed in the next paragraph). Since the parameters arise from in vitro assays performed on a small and constrained DNA molecule (Jalal et al., 2020), the value of U may be overestimated compared to in vivo conditions (see discussion).

Another approach is to fit directly the U and R parameters, the DNA unbinding and parS releasing rates, respectively, from the in vivo ParB DNA binding profile. We estimated these two values (D remains the same) leading to the best match with the ChIP-seq profile. In Fig. 2B, the ChIP-seq data (red symbols) display an enrichment at parS F since these sites are saturated due to the large value of the binding rate B. The drop on both ends of parS F and the subsequent slower decay can be interpreted in the framework of the 'Clamping & sliding' model described by Eq.( 3). We fitted the right side of the ChIP-seq profile (in the range 0 < x < 11-Kb) because the decay is longer with no roadblock hindering ParB binding compared to the left side. We used the functional form n(x)= Ae -Bx and first fitted the parameter B = 1.04 (+/-0.1)ൈ10 -4 bp -1 , which is related to ߤ in Eq.( 3) as ܤ ൌ 1/ߤ ൌ ඥܷ/.ܦ Assuming D = 4.3ൈ10 5 bp 2 s -1 , we obtained U = 4.7 (+/-0.5) ൈ10 -3 s -1 . This fitted value is ~4-times lower than the in vitro estimate from Fig. S1C. Also, note that the optimized value of µ ~9.6-Kb is ~2-times higher than the value estimated in the previous paragraph with the in vitro parameters (Fig. 2A). Second, we fitted A = 0.34 (+/-0.1), which is related to ߞ in Eq.( 3) as ܣ ൌ ݔߜߞ ൌ ߜ/ݔሺ2√ܷܦሻ. Assuming the previous values of D and U, this leads to R = 1.91 (+/-0.05) s -1 , a value ~20-times higher than the in vitro one. This theoretical prediction provides a symmetrical ParB DNA binding pattern with respect to parS (black diamonds, Fig. 2B), which also fits with the decrease observed experimentally on the left side of parS F up to ~3-Kb, beyond which the RepE/IncC protein-DNA complexes interfere with ParB F binding (see below).

Using these optimized values, the number of particles N a clamped to DNA at the steadystate is N a = R/U ~400, which is roughly compatible with the estimation of ~250 ParB F in average per cluster at parS F [START_REF] Adachi | Subcellular positioning of F plasmid mediated by dynamic localization of SopA and SopB[END_REF][START_REF] Bouet | Probing plasmid partition with centromere-based incompatibility[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF]. Also, these values are, to a first approximation, within the biological range of ParB release time from parS ߬ R ~1/R ~0.5 sec and ParB unbinding time from the DNA ߬ U ~1/U ~213 sec.

Modeling the ParB binding pattern in the presence of roadblock

The ParB binding pattern is strongly impaired by protein-DNA complexes acting as roadblocks which strongly reduce the ParB density on DNA beyond these sites as observed both in vivo and in vitro [START_REF] Breier | Whole-genome analysis of the chromosome partitioning and sporulation protein Spo0J (ParB) reveals spreading and origin-distal sites on the Bacillus subtilis chromosome[END_REF]Jalal et al., 2020;[START_REF] Rodionov | Silencing of genes flanking the P1 plasmid centromere[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF]. In order to test the prediction of roadblocks in the 'Clamping and sliding' model, we simulated the effect of a protein-DNA complex present at ~3-Kb on the left side of parS which mimics the position of the RepE-IncC roadblock present on the plasmid F [START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF]. This roadblock is modeled by a no-flux boundary condition, i.e. particles cannot cross the barrier. By applying the fitted parameters as for the case without roadblock (see previous section), we obtained the curve presented in Fig. 2B (blue circles). The symmetry between the left and right sides is clearly broken with ParB particles accumulating between parS and the roadblock as shown by the formation of a plateau starting immediately after the parS site. The ParB that are released on the left side of parS are diffusing on an isolated domain of DNA of ~3-Kb comprised between the two diffusion barriers constituted by parS and the roadblock. ParB dimers only detach from the DNA segment at the unbinding rate U. Given that ParB diffuses with a diffusion coefficient D = 4.3ൈ10 5 bp 2 s -1 , the size of the isolated domain is covered in a few seconds. In a first approximation, if we consider that ParB is uniformly released from DNA between parS and the roadblock, we conclude that R/(2U) ~200 ParB are homogeneously distributed over 3-Kb, giving a nearly saturated average density ~1. The modeling is thus in clear contrast with the observed data from ChIP-sequencing showing that (i) the ParB binding pattern is nearly symmetrical on both sides of parS F and (ii) no plateau is observed between parS and the roadblock. This finding may also argue for an unbinding rate higher than the model prediction.

We reasoned that the unbinding rate could be higher when encountering a roadblock. Indeed, we found that the ParB unbinding kinetics is best fit with a two phase exponential decay giving two unbinding rates (Fig. S1C-D). We used the lowest rate (K off_slow = 0.017 s -1 ) in our initial modeling since it may best correspond to the physiological condition on large parS DNA molecules. However, ParBs may reach saturation in front of a roadblock, as it occurs in vitro on the small DNA probe with fixed ends (see legend Fig. S1C), leading to an initial fast unbinding rate (K off_fast = 0.086 s -1 ). We therefore tested the higher rate U = 0.086 s -1 on the left side of parS in front of the roadblock. With this value, 20-times higher than on the right side of parS F , the pattern is not saturated but still displayed a plateau (Fig. S2). This contrasts with the ChIP-seq data indicating that the 'Clamping & Sliding' mechanism is not able to account for a barrier to ParB clamps diffusion. Moreover, this data points out that sliding by free, unidimensional diffusion as a clamp would be highly dependent on any obstacle present along the DNA. These obstacles, mainly proteins stably bound to DNA, would strongly impair ParB sliding by reducing its diffusion away from parS. Notably, ChIP-seq data present ParB DNA binding patterns with dips and peaks, i.e. indicating numerous obstacles along the DNA (Fig. 2B, and e.g. [START_REF] Baek | Chromosome segregation proteins of Vibrio cholerae as transcription regulators[END_REF][START_REF] Bohm | Chromosome organization by a conserved condensin-ParB system in the actinobacterium Corynebacterium glutamicum[END_REF][START_REF] Breier | Whole-genome analysis of the chromosome partitioning and sporulation protein Spo0J (ParB) reveals spreading and origin-distal sites on the Bacillus subtilis chromosome[END_REF][START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF][START_REF] Lagage | Regional Control of Chromosome Segregation in Pseudomonas aeruginosa[END_REF]. Such a pattern indicating a recovery of the ParB binding signal after strong dips could not be easily explained by the sliding clamp mechanism (see discussion).

Based on these observations, we thus conclude that the 'Clamping & sliding' model does not apply on a large range of genomic coordinates, and thus could not account fully for the assembly of the partition complex. In the following sections, we will only consider the value of the parameters R and U obtained in vitro.

ParB clusters reassemble faster than predicted by 'Clamping & sliding'

A 'Clamping & sliding' based-mechanism is expected to have an important effect on the de novo assembly kinetics of ParB. In this model, the ParB dimers load sequentially from a unique source, parS (Fig. 1B), thus their accumulation in clusters should increase progressively at the rate R up to a stationary population average. ParB are considered in steady state within the clusters, as suggested by fluorescence microscopy and fluorescence recovery after photobleaching (FRAP) experiments [START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF][START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF], except when the centromere site is replicated. DNA replication is highly processive, with a rate of DNA unwinding and duplication of ~1-Kb per second [START_REF] Kelman | DNA polymerase III holoenzyme: structure and function of a chromosomal replicating machine[END_REF]. ParB clamped at and around parS should therefore be unloaded or dispersed very rapidly when DNA polymerase III holoenzyme crosses over the ParB binding zone. ParB binding to newly duplicated parS sites is fast but its release as a sliding clamp on either side is limited by the rate R ~ 0.1 s -1 (see note in Figure S1B). In the framework of the 'Clamping & sliding' model, it would take ~20-40 min to load an average stationary population of ~125-250 ParB after the passage of the replication fork. We argue that, at the onset of replication, such disassembly/reassembly of ParB should be visible in time-lapse epi-fluorescence microscopy.

To test this expectation, we recorded ParB clusters in an E. coli strain carrying a 11.1-Kb mini-F plasmid expressing the fully functional ParB F -mTq2 fusion protein [START_REF] Diaz | Imaging centromere-based incompatibilities: Insights into the mechanism of incompatibility mediated by low-copy number plasmids[END_REF]. Images were taken every 5 sec for a period of 10 min and displayed as kymographs (Fig. 3A). Since the generation time in our growth condition is ~100 min, only a few cells would undergo mini-F replication during the recorded period. Note that mini-F replication occurs within 10 sec, and about once per cell cycle and per plasmid (present at ~two copies per chromosome; [START_REF] Frame | The number of sex factors per chromosome in Escherichia coli[END_REF]). In addition, subsequent DNA segregation, visible as ParB F foci splitting, occurs within less than five min after replication [START_REF] Onogi | Behavior of sister copies of mini-F plasmid after synchronized plasmid replication in Escherichia coli cells[END_REF]. In spite of this difficulties, we were able to observe three trajectories (labelled #1, 2 & 5; Fig. 3A) displaying ParB foci more than 5 min before their splitting (indicated by arrows). We observed that (i) the ParB fluorescence intensity is maximal just before the splitting event and lasts at this level between one to three minutes, (ii) the ParB signals do not decrease importantly over the time course recorded. Only a few frames display transiently low signals (indicated by black circles) that last less than two frames (10 sec). However, we could not exclude that these transient faint signals were due to out of focus variation of the ParB foci since such decrease were also observed independently of a splitting event as observed for trajectories #3 and #4 (indicated by white circles).

To gain quantitative insights, we measured and plotted the relative fluorescence intensity signal within ParB foci over time (Fig. 3B and Fig. S3). We found that it displayed two clear transitions between three stable states: (i) the fluorescence intensity from the ParB focus is maximal before the splitting event (S) and is then nearly equally divided between the two newly formed foci; (ii) less than 5 min before the splitting event, the fluorescence intensity decreased abruptly (D) over 10-15 sec, followed by an increase (A) over a short period of time (20-50 sec). We propose that this decrease might correspond to the passage of the replication fork over parS removing ParB from the DNA, and that the immediate increase might correspond to the loading of ParB at parS for de novo assembly of the partition complexes. Note that, at this stage, a doubling of the ParB fluorescence intensity is not expected since most of ParBs were already present in foci before duplication [START_REF] Lim | Evidence for a DNA-relay mechanism in ParABS-mediated chromosome segregation[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF]. We found that the observed experimental loading time (<60 sec) is much shorter than the time predicted by the 'Clamping & sliding' model (~20-40 min; see above). Rather, it is of the same order of magnitude as the time needed for two foci to exchange their ParB (~90 sec; see [START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF][START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF], confirming the fast dynamics of ParB within clusters. This result suggests that the disassembly of the partition complexes is carried out too rapidly to be explained by the 'Clamping & sliding' model.

Discussion

The understanding of the molecular mechanism responsible for the assembly of the bacterial partition complex has resisted three decades of biochemical and molecular studies performed on several ParABS systems from chromosomes and low-copy-number plasmids. How a few ParB molecules bound to parS sites cluster hundreds of other ParB in the vicinity of parS in a selfassembled high molecular weight structure that serves to position and actively segregate DNA molecules remains puzzling. The recent finding that ParB proteins belong to a new class of CTPdependent molecular switches has opened new avenues of research to decipher this mechanism. In particular, ParB dimers switch from an opened conformation prone to DNA binding to a close conformation (clamp) upon parS and CTP binding that enable the release and sliding away from parS as a protein clamp (Jalal et al., 2020;[START_REF] Soh | Self-organization of parS centromeres by the ParB CTP hydrolase[END_REF]. In this work, we propose a physical model, namely 'Clamping & sliding', for this newly proposed CTP-based ParB clamping mechanism.

First, we considered that the parS binding site is always occupied by ParB as suggested from the high affinity constants in the nM to 100 nM range for all ParBs investigated (e.g. [START_REF] Ah-Seng | Dual role of DNA in regulating ATP hydrolysis by the SopA partition protein[END_REF][START_REF] Funnell | The P1 plasmid partition complex at parS: II. Analysis of ParB protein binding activity and specificity[END_REF][START_REF] Taylor | Specific and non-specific interactions of ParB with DNA: implications for chromosome segregation[END_REF]. Therefore, clamped-ParBs must slide away from parS by free diffusion since (i) they could not cross back over this strong binding site and (ii) they are pushed away by the successive loading at parS. Second, the 'Clamping & sliding' model contains three kinetic parameters (see Eq.( 2)), which were inferred from biochemical, microscopy and ChIP-seq analyses. The apparent diffusion coefficient, D, was estimated from single molecule live imaging assays [START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF]. ParB F displays two distinct diffusion modes depending on whether it belongs to a mobile or a clustered fraction. In the mobile fraction (not positioned at parS sites), D ~ 0.7 μm 2 s -1 which typically corresponds to the diffusion of non-specific DNA binding proteins (Garza de [START_REF] Garza De Leon | Tracking Low-Copy Transcription Factors in Living Bacteria: The Case of the lac Repressor[END_REF][START_REF] Stracy | Single-molecule imaging of UvrA and UvrB recruitment to DNA lesions in living Escherichia coli[END_REF][START_REF] Stracy | Single-molecule and super-resolution imaging of transcription in living bacteria[END_REF]. In the fractions clustered around the parS F site, D was estimated at ~ 0.05 µm 2 s -1 [START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF]. We used this latter estimate for D as it corresponds to the ParB F (~ 95%) present in the vicinity of ParS F while the former corresponds to the ParB F that are freely diffusing over the nucleoid. Here, we only considered that ParB clamps move unidimensionally and passively by free diffusion along DNA. Indeed, despite that the role of CTP hydrolysis is not known, it is unlikely that it could provide energy for translocation since (i) the hydrolysis rate is as low as ~40 CTP hydrolyzed per hour, and (ii) ParB still accumulates on DNA in the presence of a non-hydrolysable CTP analog (Jalal et al., 2020;[START_REF] Osorio-Valeriano | ParB-type DNA Segregation Proteins Are CTP-Dependent Molecular Switches[END_REF][START_REF] Soh | Self-organization of parS centromeres by the ParB CTP hydrolase[END_REF].

The release rate of ParB from parS, R, is the limiting step for creating the flux of ParB clamps on parS-proximal DNA. It occurs at the same rate on both sides of parS as indicated from all the ChIP assays performed with numerous ParB belonging to plasmids and chromosomes [START_REF] Baek | Chromosome segregation proteins of Vibrio cholerae as transcription regulators[END_REF][START_REF] Breier | Whole-genome analysis of the chromosome partitioning and sporulation protein Spo0J (ParB) reveals spreading and origin-distal sites on the Bacillus subtilis chromosome[END_REF][START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF][START_REF] Donczew | ParA and ParB coordinate chromosome segregation with cell elongation and division during Streptomyces sporulation[END_REF][START_REF] Lagage | Regional Control of Chromosome Segregation in Pseudomonas aeruginosa[END_REF][START_REF] Rodionov | Silencing of genes flanking the P1 plasmid centromere[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF]. The value estimated from the in vitro data (Fig. S1B and Jalal et al., 2020) is 20-times lower than the best fit from the in vivo ChIP-seq data using the 'Clamping & sliding' model (Fig. 2B). This difference may arise from the small linear and closed DNA fragment used in the in vitro assay (Jalal et al., 2020) compared to the ~60-kb circular DNA for the ChIP-seq. However, the release rate from the former is nicely fitted by a one phase association curve (Fig. S1B) indicating that it is not perturbed by steric hindrance when the DNA probe becomes saturated. This suggests that R remains constant from the initial loading stage after replication to the equilibrium state with one ParB clamp released every 20 sec on each side of parS. We cannot exclude that the release rate R is intrinsic to each ParB from different systems, and that it could thus be slightly different between ParB F and ParB Ccre .

The unloading rate of the clamped-ParB from DNA, U, was estimated from the in vitro unbinding curve (Jalal et al., 2020) which displays a two phase exponential decay (Fig. S1C). These two unbinding rates may be explained by the small size of the closed DNA probe since, at the beginning of the decay measurement, the DNA probe is saturated with one ParB every 16-bp (see note in Fig. S3) which might lead to an initial fast unbinding and to a slow one when the probe becomes unsaturated. We used the lower rate (~0.017 s -1 ) in the modeling (Fig. 2A) since in vivo and in the absence of roadblock, ParB diffuses rapidly on large DNA molecules, plasmids or chromosomes. However, to better fit the ChIP-seq data, the unbinding rate has to be set fourtimes lower (Fig. 2B). This optimized value corresponds to ParB clamps opening after ~3.6 min on average. It was previously shown that ParBs are highly dynamic with a residence time inside the clusters of ~100 sec, i.e. an exchange rate of 0.01 s -1 [START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF][START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF], a value in favor of the in vitro estimate (Fig. S1C and Jalal et al., 2020).

Although the values for these three parameters sound biologically relevant, it is important to note that our modeling is performed on ideal DNA, i.e. on a naked DNA without any protein bound to it. The ParB clamp harbors a central hole enabling the DNA molecule to pass through but not protein-DNA complexes [START_REF] Soh | Self-organization of parS centromeres by the ParB CTP hydrolase[END_REF]. The unidimensional diffusion on a filament is a physical process that is completely interrupted by roadblocks such as any protein bound stably to the DNA. The 'Clamping & sliding' model predicts that ParB clamps would accumulate rapidly between the roadblock and parS leading to a plateau, saturated or not, depending on the high or low value of the unbinding rate used (Fig. S2). Strikingly, this result is in stark contrast with the nearly symmetrical decreasing pattern observed on either side of parS up to the roadblock (Fig. 2B). On bacterial genomes, with an average density of one gene every Kb and with numerous transcriptional regulators bound to promoter regions [START_REF] Browning | The regulation of bacterial transcription initiation[END_REF], the probability of having obstacles over the > 15-Kb of ParB binding pattern around parS is very high. Therefore, it is very unlikely that numerous ParB could cover a large genomic distance by free diffusion without being halted many times before unloading from DNA. With a 'Clamping & sliding' scenario, a higher density of ParB close to parS with an important decrease at each locus bound by a protein is rather expected, but never observed in ChIP-seq data (e.g. [START_REF] Baek | Chromosome segregation proteins of Vibrio cholerae as transcription regulators[END_REF][START_REF] Breier | Whole-genome analysis of the chromosome partitioning and sporulation protein Spo0J (ParB) reveals spreading and origin-distal sites on the Bacillus subtilis chromosome[END_REF][START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF][START_REF] Donczew | ParA and ParB coordinate chromosome segregation with cell elongation and division during Streptomyces sporulation[END_REF][START_REF] Lagage | Regional Control of Chromosome Segregation in Pseudomonas aeruginosa[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF]. Also, dips in the ParB DNA binding pattern corresponding to promoter regions with transcriptional regulator binding sites have been previously described both on plasmids and chromosome DNA [START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF]. Notably, a strong dip in the ParB binding pattern was observed ~1-Kb on the right side of parS F inserted on the E. coli chromosome, corresponding to the presence of a promoter in reverse orientation relative to ParB diffusion. This suggests that transcription is preventing the diffusion of ParB clamps. However, after this important decrease in intensity, the ParB DNA binding signal fully recovered. Such a behavior with dips and peaks is incompatible with a sliding mechanism over large genomic distances.

Partition complexes are in a stationary state most of the time involving > 90% of ParB, but not at the onset of parS replication. By fast time-lapse microscopy of fluorescently-tagged ParB, we were able to observe some splitting events corresponding to the plasmid segregation step, and to detect variations of fluorescent intensity that might correspond to the replication of parS F : a rapid drop in the foci intensity followed by its progressive increase to a higher level than the initial one. This temporal pattern might account for the fast disassembly of partition complexes followed by their progressive reassembly, respectively (Fig. 3B). From our measurements, it takes between 20 and 50 sec to reassemble partition complexes (Fig. 3B andS3). The rate obtained from the in vitro data with only one ParB release every 10 sec (R ~ 0.1 s -1 ; Fig. S1B and Jalal et al., 2020) would be too slow to account for the reassembly with only 2 to 5 ParB loaded in this period of time (requiring 40 min to load 250 ParB).

A fundamental difference in the 'Clamping & sliding' model is the directed flux of ParB, making the system out-of-equilibrium, by contrast to all other ParB assembly models to date. Indeed, these latter explain the formation of ParBS assembly in the framework of thermodynamic equilibrium: '1D-spreading', 'Spreading & bridging' in the strong coupling limit, 'Looping & clustering' and 'Nucleation & caging' [START_REF] Broedersz | Condensation and localization of the partitioning protein ParB on the bacterial chromosome[END_REF][START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF][START_REF] Walter | Looping and clustering model for the organization of protein-DNA complexes on the bacterial genome[END_REF]. This notable difference comes from the fact that ParB can only be loaded at parS, giving rise to a flux of ParB from parS to genomic regions away from parS. This leads to a severe limitation for this model: ParB would have to cover long genomic distances by unidimensional diffusion to account for the coverage observed in ChIP-seq experiments. On the contrary, models at thermodynamic equilibrium are based on ParB interactions with other ParB and with DNA in the spatial vicinity of the parS sites. These ingredients could give rise to droplet formation (via phase transition), which is a mechanism known to quickly create a high concentration region at a targeted cellular location without requiring a membrane [START_REF] Hyman | Liquid-liquid phase separation in biology[END_REF], and which has been recently proposed to occur for partition complex assembly [START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF].

In summary, the 'Clamping & sliding' model is unable to describe the overall ParB DNA binding pattern with previously determined parameters and only approximately with best fitted parameters. Moreover, it does not to account for several main aspects of this assembly: (i) the rapid turnover of ParB between clusters, (ii) the absence of accumulation of ParB in front of a roadblock, and (iii) the recovery of the ParB binding after strong dips in the profile. Also, the presence of numerous proteins bound along the DNA would prevent ParB clamps from diffusing rapidly to large genomic distances from parS centromere sites. For these reasons, the 'Clamping & sliding' model alone is not a plausible physical mechanism for explaining the partition complex assembly mechanism. We rather propose the possibility that a combination of two mechanisms is at play for the assembly of the higher-order nucleoprotein ParBS complex: one occurring at short distance), namely 'Clamping & sliding', and one at long distance. We speculate that the distance covered by diffusing clamped-ParB is of the order of the distance between two genes, i.e. ~1-Kb. Indeed, this distance corresponds to average transcription units that would induce barriers for diffusing clamped-ParB with both transcription factors and RNA polymerases. We therefore envision that ParB clamps only accumulate in the close vicinity of parS sites. These ParBs have undergone a conformational change, occurring possibly (i) with the transition between the parS-bound and the sliding clamp states [START_REF] Soh | Self-organization of parS centromeres by the ParB CTP hydrolase[END_REF] and/or (ii) with the stimulation of CTP hydrolysis switching ParB from the closed (CTP-bound) to the open conformation (apo/CDP-bound) [START_REF] Osorio-Valeriano | ParB-type DNA Segregation Proteins Are CTP-Dependent Molecular Switches[END_REF], that modify the N-terminal domain involved in ParB-ParB dimer interactions [START_REF] Osorio-Valeriano | ParB-type DNA Segregation Proteins Are CTP-Dependent Molecular Switches[END_REF][START_REF] Surtees | P1 ParB domain structure includes two independent multimerization domains[END_REF] rendering these ParB prone to interact with other ParBs. The accumulation of these numerous proned-ParB would increase the number of nucleation points that can further recruit most of the intracellular ParBs into a highly concentrated cluster with a phase transitionlike mechanism [START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF]. The 'Nucleation & caging' model remains attractive to explain the ParB-ParB interactions occurring at long-distance (> ~1-Kb) as it currently best describes the ParB DNA binding pattern [START_REF] Debaugny | A conserved mechanism drives partition complex assembly on bacterial chromosomes and plasmids[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF]. Such a combination of two modes of actions, 'Clamping & sliding' and 'Nucleation & caging' is also compatible with the recent study that reveals the droplet-like behavior of the ParB assemblies [START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF]. Further work is needed to provide new insights into this higher-order nucleoprotein complex that drives the segregation of the bacterial DNA.

Material and methods

Monte Carlo simulations

The simulations data presented in Fig. 2A-B are performed with a sequential Monte Carlo (MC) algorithm. The 60-Kb plasmid is modeled by a filament of N = 3750 sites, each of length δx = 16-bp corresponding to the footprint of a ParB protein. The MC time step is defined as a sweep of the N sites of the filament representing the plasmid. When a site is chosen during the MC sweep, different events can occur: if the site is occupied by ParB, this protein (i) can disappear from the lattice at a rate U or (ii) diffuses with a diffusion constant D; if the chosen site is neighbor to either side of parS, a ParB protein is injected at a rate R. We need to perform simulation on a sufficiently small time interval to prevent the appearance of numerical approximations (the analytic solution Eq.( 3) are used as a benchmark for the simulations). We used an integration time δt corresponding to the time needed for ParB to diffuse over one site (i.e. a random step to the right or to the left), thus ݐߜ ൌ ݔߜ 2 .ܦ/ The corresponding release rate R and unbinding rate U on the time interval ݐߜ become ݐߜܴ ൌ ݔߜܴ ଶ ܦ/ and ݐߜܷ ൌ ݔߜܷ ଶ ,ܦ/ respectively. Thus, we need D Monte Carlo iterations to perform an evolution of the system during one second. It is important to remark that, in the absence of interactions between ParB proteins, several particles may be present on the same site. Thus, when a site is chosen, we update all particles on the site.

We define a no-flux boundary condition at parS, so that parS acts as a barrier for diffusion. Thus a particle is released with a probability one-half on either side of parS, i.e. the total release rate R becomes R/2 on each side (this is reflected in Eq.( 3)).

In Fig. 2B, the roadblock is defined as an additional barrier located at 3-Kb from parS F (left side); thus particles that are released on the left side of parS F have to evolve on an isolated filament of ~3-Kb between the roadblock and parS. As the diffusion is fast (D = 4.3ൈ10 5 bp 2 s -1 ) with respect to 3-Kb, a good approximation of the distribution is thus a plateau, whose height depends on ߜ/ܴݔሺ2ܷ ൈ 3ܾܭሻ. In order to prevent occupancy to become larger than one in the case of large density, we do not allow new particles to be released when the filament is completely covered (i.e. saturation).

The initial configuration of the system is empty. Before starting the sampling of the ParB density, we ensure that stationarity is reached. It is helpful to realize that the typical time needed to fill the system at the average stationary value of particles R/U corresponds to the release time 1/R times R/U, thus 1/U corresponds to the average time to fill the system (starting from an empty configuration). It corresponds also to the typical time to replace all the particles of the system in the stationary state, thus it corresponds to the correlation time. In Fig. 2A and 2B, we perform 10ൈ1/U seconds of evolution before starting the sampling in order to ensure stationarity. Subsequently, the sampling was started and, to ensure decorrelation of the system between two samplings, samplings were spaced by a time 1/U. Averages were performed over 50.000 independent samplings in Fig. 2A and2B.

We finally note that we can also solve the version of Eq.( 2) with a discrete space formulation, i.e. the same framework as Monte Carlo simulations. The discrete solution is the same as Eq.( 3) in the limit of ݔߜ/ߤ ب 1, i.e. when the characteristic length ߤ of the profile is much larger than the microscopic length ݔߜ of the system. This limit is satisfied for all the simulations of the paper (ߤ~few Kb and ߜ-61=ݔbp), namely in Fig. 2A where we observed the excellent agreement between both approaches.

Release (R) and unbinding (U) parameters

The kinetic studies for ParB binding and unbinding in the presence of CTP, assayed by bio-layer interferometry (BLI), were performed elsewhere (Fig. 2A in Jalal et al., 2020). To summarize briefly, the measurement of ParB-DNA interaction was monitored by the wavelength shifts during association or dissociation of ParB to a 169-bp biotinylated double-stranded parS DNA bound at both ends to the sensor surface (no free end). The reactions were measured with and without 1 µM of ParB (dimer) for 120 sec for association and dissociation phases, respectively.

The uploaded data were analyzed using GraphPad Prism 8© to determine the ParB kinetics parameters. The rates of ParB release from parS (R) and ParB unbinding from DNA (U) were calculated using the nonlinear regression fitting tools 'One phase association' and 'One phase decay' or 'Two phase decay', respectively (Fig. S1).

ChIP-sequencing data

High-resolution ChIP-sequencing were from a previous study with 50ൈ10 6 reads per library [START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF]. Data from the ~60-Kb plasmid F derivative (pOX38B) grown in E. coli cells displayed the average number of reads (first nucleotide of each DNA fragment sequenced) per 100-bp windows. The signal is normalized to 1 by averaging the number of reads over the centromere sequence parS F (550-bp carrying the 12×43-bp repeat sequences; [START_REF] Pillet | Centromere binding specificity in assembly of the F plasmid partition complex[END_REF]. The drop on the left side corresponds to the RepE/incC roadblock [START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF].

Bacterial strain and growth condition E. coli K-12 strain DLT1215 [START_REF] Bouet | Concerted action of plasmid maintenance functions: partition complexes create a requirement for dimer resolution[END_REF], carrying the reporter mini-F plasmids pJYB249 [START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF], were grown at 30°C in M9-Gly (M9 minimal medium supplemented with 0.4% glycerol, 1 mM MgSO 4 , 0.1 mM CaCl 2 , 40 µg.ml -1 thymine, 20 µg.ml -1 leucine and 1 µg.ml -1 thiamine) with a generation time of ~100 min allowing the visualization of 1 to 3 plasmids per cell.

Microscopy and image analyses

Mid-exponential phase bacterial cultures were sampled, concentrated 5-times by centrifugation and resuspension in M9-Gly, and 0.7 μl was deposited onto slides coated with 1% agarose buffered in M9 solution. Samples were visualized at 30°C as previously described [START_REF] Diaz | Imaging centromere-based incompatibilities: Insights into the mechanism of incompatibility mediated by low-copy number plasmids[END_REF], with images taken every 5 seconds over 10 minutes periods. Nis Elements AR software (Nikon) was used for image capture and editing. Kymographs were generated using the "MultipleKymograph" plugin (ImageJ software). Foci detection and integrated fluorescence were measured using "Trackmate" plugin [START_REF] Tinevez | TrackMate: An open and extensible platform for single-particle tracking[END_REF] in Fiji software. A-Open and close conformations mediated by CTP and parS DNA. ParB is a homodimer composed of a C-terminal dimerization domain (orange) link to the central (light blue) and Nterminal (dark blue) domains by a flexible linker (red). The central domain contains the two DNA binding motifs for parS binding [START_REF] Sanchez | Insight into centromere-binding properties of ParB proteins: a secondary binding motif is essential for bacterial genome maintenance[END_REF]. The N-terminal part contains the ParA interaction domain, the arginine-like motif, the CTP binding motif and the multimerization domain [START_REF] Ah-Seng | Dual role of DNA in regulating ATP hydrolysis by the SopA partition protein[END_REF][START_REF] Soh | Self-organization of parS centromeres by the ParB CTP hydrolase[END_REF][START_REF] Surtees | P1 ParB domain structure includes two independent multimerization domains[END_REF]. In the presence of parS and CTP, ParB dimer forms a clamp around the DNA. C). Clamping promotes its release from parS allowing the ParB clamp to slide away from parS by diffusion (step S), and to free the parS loading site for other rounds of loading. The parameters used in the physical modeling B, R, D and U correspond to the parS binding (B), ParB clamping (C) and release (R), free diffusion during sliding (S), and DNA unbinding (U) steps, respectively. Note that (i) R is the total release rate from parS which is, at first approximation, equal on either side of parS (not represented on the schema for simplicity), therefore ParB clamps are loaded on each side at a rate R/2, and (ii) the stage at which CTP hydrolysis is occurring was not determined in the original clamping models (Jalal et al., 2020;[START_REF] Soh | Self-organization of parS centromeres by the ParB CTP hydrolase[END_REF]. [START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF] display the average number of reads per 100-bp windows as a function of the genomic coordinates (red circles). Note that the signal at parS F (coordinate 0) is normalized to 1 by averaging the number of reads over the 550-bp centromere locus.

B-

A-The model from Eq.(2) contains the three kinetic parameters D = 4.3ൈ10 5 bp 2 s -1 [START_REF] Guilhas | ATP-Driven Separation of Liquid Phase Condensates in Bacteria[END_REF], R = 0.1 s -1 and U = 0.017 s -1 (Fig. S1B-D). The exact solution (black line) and the Monte Carlo (MC; blue dots) simulation are plotted on the right side of parS F . The theoretical prediction is compared to the ChIP-seq data [START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF]. The large discrepancy between the model and the ChIP-seq data is essentially due to the amplitude (value of R).

B-Simulation in the absence (black diamonds) and in the presence (blue circles) of a roadblock compared to the Chip-seq data with the optimized parameters (R = 1.91 s -1 and U = 0.0047 s -1 ). The two strong roadblocks (green bars), present at ~3-kb and 4.5-kb on the left of parS F , are due to the binding of RepE to incC and ori2 iterons, respectively [START_REF] Rodionov | Silencing of genes flanking the P1 plasmid centromere[END_REF][START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF]. Note that the two MC simulations overlap on the right side of parS F . B-Intensity of ParB F fluorescent foci before and after the splitting event. The integrated ParB F -mTq2 fluorescence (thin line) from trajectory number 2 in panel A is measured every frame and plotted over time with color data points. The signal was subtracted from the average background level, and normalized to 1 from the average intensity before the drop in intensity (blue dots). Dots and triangles represent the intensity of one and two ParB foci before and after the splitting event (S), respectively. R and L represent the putative centromere replication and the ParB loading steps, respectively. The blue, brown, red and blue horizontal bars represent the mean values of the normalized ParB foci intensity of the one focus before R, before S and the 2 foci after S, respectively. The orange and green bars represent the linear regression of the variation in ParB fluorescent intensities. Note that replication (R) is inferred from the increase in foci intensity that (i) occurs between the 2 plateau of ParB fluorescence mean intensity and (ii) takes place less than 5 min before splitting (S).

Figures legends Figure 1 :

 legends1 Figures legends

  Schematic for 'Clamping & sliding' model displaying the five key steps. The open conformation of ParB dimer enables DNA binding. Upon specific binding to parS centromere (step B), ParB undergoes a conformational change promoting CTP binding which subsequently induces ParB to form a clamp around parS (step

Figure 2 :

 2 Figure 2: Comparisons between the 'Clamping & sliding' model and the ParB F DNA binding pattern. High-resolution ChIP-sequencing data from a previous study[START_REF] Sanchez | Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus[END_REF] display the average number of reads per 100-bp windows as a function of the genomic coordinates (red circles). Note that the signal at parS F (coordinate 0) is normalized to 1 by averaging the number of reads over the 550-bp centromere locus.

Figure 3 :

 3 Figure 3: ParB foci do not disassemble for a long time before splitting events.

  Figure1

Acknowledgements

We thank all members of the Gedy team for fruitful discussions, and P. Rousseau, B. Ton-Hoang and M. Campos for critical reading of the manuscript. This work was supported by Agence Nationale pour la Recherche (ANR-14-CE09-0025-01), CNRS 80Prime (Numacoiled) grant, CNRS "Modélisation pour le Vivant" Grant (CoilChrom), and by the LabEx NUMEV (ANR 2011-LABX-076) within the I-SITE MUSE.


Dr Monique Calvo-Dahlborg
email: monique.calvo-dahlborg@univ-rouen.fr

Ulf Dahlborg

Stephen G R Brown

Jean Juraszek

M Calvo

Influence of the electronic polymorphism of Ni on the classification and design of high entropy alloys

Keywords: HEA, e/a, phases, design, magnetism, Brillouin HEA, e/a, phases, Ni, design, magnetism, Brillouin

According to a recent Hume-Rothery approach, the electron concentration, e/a, and the average radius can be used to identify the domain of stability of HEAs and to estimate the phases that may occur in the alloy. The present study investigates the influence of the electronic polymorphism of nickel on the efficiency of the classification and on the design of HEAs for magnetic applications. Many different compositions were used, based on 4 to 7 elements out of a total 13 different elements (Co, Cr, Fe, Ni, Al, Cu, Pd, Ti, Mn, V, Nb, Sn, Ru). Phases have been determined by X-ray and neutron diffraction as well as in some cases high energy X-ray diffraction. The e/a for the constituent elements is calculated according to Massalski. The two polymorphic electronic structure of nickel, namely (e/a)Ni=1 or (e/a)Ni=2 are considered. The average e/a for the alloy is calculated assuming a solid solution case. The electronic structure [Ar] 3d9 4s1 seems to be more appropriate for the classification of HEAs. Based on a Self-Organizing Map predictions are made for the average magnetic moment at saturation for this electronic structure of Ni. Non-saturated values and data from the literature are compared with the predictions. The consequences of such results when modelling the structure and properties of Ni containing HEAs are presented, in particular the consideration of the shape and transformation of the Brillouin zone.

Groupe de Physique des Matériaux

Introduction

The term "high entropy alloys" (HEAs) was introduced in 2004 to define alloys with multiple principal elements in equimolar or near-equimolar ratios [START_REF] Yeh | Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes[END_REF][START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF]. A new metallurgy was born by considering not one or two main elements with the others as minor additions but with all elements solidifying on an equal footing from the liquid state. In the first paper [START_REF] Yeh | Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes[END_REF] stress was put on the apparent simple structure of HEAs, within a solid solution approach, having face centred cubic (fcc), body centred cubic (bcc) and hexagonal closed packed (hcp) structures and that these new systems exhibit extraordinary properties. In the second paper [START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF] the feasibility of multi-major component alloys with a large number of elements was demonstrated as well as the multiphase single structure of the CoCrFeNiMn alloy (also called the Cantor alloy) which was shown to have two fcc phases, one Fe-rich and one Fe-poor.

Since 2004, many reports have determined the structures and microstructures of HEAs. It has been found that in many cases a HEA has a structure and a microstructure that are very different from ordinary alloys. Furthermore, the multi-phase single simple structure was confirmed for many of the HEAs investigated [START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF][START_REF] Senkov | Accelerated exploration of multiprincipal element alloys with solid solution phases[END_REF][START_REF] Guo | More than entropy in high-entropy alloys : Forming solid solutions or amorphous phase[END_REF][START_REF] Singh | Decomposition in multicomponent AlCoCrCuFeNi high-entropy alloy[END_REF][START_REF] Manzoni | Phase separation in equiatomic AlCoCrFeNi highentropy alloy[END_REF][START_REF] Wang | Solid solution or intermetallics in a high-entropy alloy[END_REF][START_REF] Dahlborg | Structure of some CoCrFeyNi and CoCrFeNiPdx multicomponent HEA alloys by diffraction techniques[END_REF][START_REF] Guo | More than entropy in high-entropy alloys: Forming solid solutions or amorphous phase[END_REF]. Following the definition of Yeh et al [START_REF] Yeh | Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes[END_REF] the domain of existence of HEAs as compared to Bulk Metallic Glasses (BMGs) and intermetallics was derived using the entropy of mixing versus the atomic mismatch, delta, as quantitative parameters [START_REF] Guo | More than entropy in high-entropy alloys : Forming solid solutions or amorphous phase[END_REF].

During the last 15 years, following the expectation of "extraordinary properties" mentioned by Yeh et al [START_REF] Yeh | Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes[END_REF] many reports have also reviewed the potential use of these alloys for special kinds of technological applications [START_REF] Gao | High-Entropy alloys: Fundamentals and applications[END_REF][START_REF] Zhang | Microstructures and properties of high-entropy alloys[END_REF][START_REF] Miracle | A critical review of high entropy alloys and related concepts[END_REF][START_REF] Miracle | Exploration and Development of High Entropy Alloys for Structural Applications[END_REF][START_REF] Senkov | Accelerated exploration of multiprincipal element alloys with solid solution phases[END_REF]. The determination of the phase composition after solidification as well as their stability under external influence has been an important issue as the phase composition might strongly affect the mechanical efficiency of the alloy and thus the potential applications. Generally, fcc phases favour ductility and bcc phases favour strength and hardness. Numerous attempts using both semi-empirical and thermodynamic parameter(s) have been developed in order to understand the stability of solid solutions in some HEAs and the formation of unwanted intermetallic phases during use [START_REF] Otto | Relative effects of enthalpy and entropy on the phase stability of equiatomic high-entropy alloys[END_REF][START_REF] Poletti | Electronic and thermodynamic criteria for the occurrence of high entropy alloys in metallic systems[END_REF][START_REF] Toda-Caraballo | A criterion for formation of high entropy alloys based on lattice distortion[END_REF][START_REF] Kozak | Single-phase high-entropy alloysan overview[END_REF][START_REF] King | Predicting the formation and stability of single phase high-entropy alloys[END_REF]. Scientific issues and challenges have been discussed including aspects of definition/terminology, phase formation, microstructure and phase stability, strengthening mechanisms, and high temperature properties, as well as the density and cost [20 -22].

Recently a classification of HEAs was proposed using the average value of the electron concentration or number of itinerant electrons per atom (e/a) in the alloy and the average atomic radius calculated for a 12-atoms atomic neighbourhood [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF]. The importance of the number of itinerant electrons per atom to understand and predict the phases in solid solutions has been stressed as early as 1966 [START_REF]Structure of metals[END_REF]. In [START_REF] Massalski | Comments Concerning Some Features of Phase Diagrams and Phase Transformations[END_REF] the use of e/a is discussed by Massalski for the determination of crystalline Hume-Rothery phases as opposed to an approach using the total number of valence electrons (VEC) (which includes the d electrons). In [START_REF] Dong | The concept of the approximants of quasicrystals[END_REF] Dong proposed a successful criterion based on a value of e/a and the radius for predicting approximants in quasicrystals. By applying these approaches for HEAs as solid solutions a classification and design methodology was proposed in [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF]. From the variation of the average magnetic moment per atom and the hardness as a function of e/a and the 12 atoms radius, three domains could be identified in [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF]: one domain containing alloys with cubic close-packed (fcc) structure, a second one containing complex and/or mixt structures and a third one containing structures belonging to the body-centered cubic (bcc) structural family. Since the work of Cantor [START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF] several publications have reported studies of alloys based on addition of a fifth, sixth or seventh element to a CoCrFeNi base alloy, thus replacing Mn by one, two or three other elements. However, it is widely reported that the electronic structure of Ni can be either [Ar] 3d 8 4s 2 or

[Ar] 3d 9 4s 1 , thus, with the configuration per shell 2, 8, 16, 2 or 2, 8, 17, 1 [START_REF] Scerri | The periodic table: its story and its significance[END_REF]. Accordingly, Ni has one or two itinerant electrons per atom i.e. e/a for Ni can be either 1 or 2, whereas VEC is equal to 10 for both configurations. Because of the resulting influence on bonding, this duality has as consequence on the polymorphism of Ni in alloys, e.g. in amorphous alloys [START_REF] Lashmore | Polymorphism of Nickel-Phosphorus Metallic Glasses[END_REF]. In the present work the influence of the polymorphism of Ni on the classification previously presented in [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF] is investigated for the same set of alloys but excluding the ones not containing Ni.

Materials and Methods

In order to correlate the composition and the crystalline structure of phases, several different compositions were investigated. All alloys contained Co, Fe and Ni as base elements. To this base alloy one or more elements including Al, Cr, Cu, Mn, Nb, Pd, Ru, Ti, and V were added in different proportions. The crystalline structure of the phases that were found in these alloys as well as the corresponding lattice constants have earlier been determined by high resolution diffraction techniques [START_REF] Dahlborg | Structure of some CoCrFeyNi and CoCrFeNiPdx multicomponent HEA alloys by diffraction techniques[END_REF][START_REF] Cornide | The Minerals[END_REF][START_REF] Dahlborg | [END_REF][START_REF] Cornide | [END_REF][START_REF] Calvo-Dahlborg | Structural and microstructural characterization of CoCrFeNiPd High Entropy Alloys[END_REF]. All investigated alloys have been produced by arc-melting and re-melted several times for homogenization and for minimization of oxides. The melts were cooled by a rapid quench in a copper mould. All diffraction studies were performed on as-cast materials. For comparison, a few results on alloys containing refractory elements taken from the literature have also been included for illustrative purposes [START_REF] Gao | High-Entropy alloys: Fundamentals and applications[END_REF].

When aiming to classify alloys according to their phase content one immediately faces a problem. How are the phases determined? Diffraction techniques are most commonly used but the different diffraction techniques have different resolutions. The investigations of a CoCrFeNiPd alloy clearly illustrates this [START_REF] Dahlborg | Structure of some CoCrFeyNi and CoCrFeNiPdx multicomponent HEA alloys by diffraction techniques[END_REF]. This alloy, in as-cast condition originally, was determined from laboratory X-ray measurements (XRD) to be monophase with a simple fcc structure. Neutron diffraction measurements (ND) later suggested the alloy to be at least twophase while high-energy X-ray diffraction (HEXRD) found it to be at least four-phase. All phases were of simple fcc structure. It can thus be concluded that this alloy is multiphase but single-structured [START_REF] Calvo-Dahlborg | Structural and microstructural characterization of CoCrFeNiPd High Entropy Alloys[END_REF]. This has been found to be the case for most HEAs. Moreover, the CoCrFeNiMn was indeed reported to be two-phase in the original work of Cantor [START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF].

In the present work results obtained using the three above-mentioned experimental diffraction techniques are used to classify alloys according to their structural content. The ND measurements were performed on the D20 diffractometer at Institute Laue-Langevin (ILL), Grenoble (France), the HEXRD measurements on the ID22 diffractometer at the European Synchrotron Research Facility (ESRF), Grenoble (France) and the standard XRD measurements were performed with a Cobalt source at GPM, University of Rouen, France. As all samples were found to have a pronounced texture, they were rotated during the measurements, whenever possible. The structure and the lattice constants of the phases present in the alloys were determined by a refinement procedure. Magnetic characteristics of the samples were determined by measuring the average magnetic moment per atom using a MPMS-5XL SQUID magnetometer as a function of applied magnetic field and temperature.

As in [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF], Self-Organising Map (SOMs) [START_REF] Kohonen | Self-Organized Formation of Topologically Correct Feature Maps[END_REF][START_REF] Alhoniemi | Biennial report[END_REF][START_REF] Ong | Data Mining Using Self-Organizing Kohonen maps: A Technique for Effective Data Clustering & Visualisation[END_REF][START_REF] Yin | [END_REF] were determined to try to predict the existing phases as well as the average magnetic moment at 5K at saturation, M 5K-Sat , in a HEA according to its e/a and radius values. SOMs are commonly used to reveal trends in multiple parameter characterization in various domains. In [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF] the applicability of a standard SOM to HEAs with 15 parameters was demonstrated. In the present work only three parameters were considered, namely the e/a value of the alloy, its average radius and M 5K-Sat . However, in the current work a 'visualisation-induced self-organising map' (ViSOM) has been used [START_REF] Yin | [END_REF]. The ViSOM procedure followed here is a modification of the standard SOM algorithm used in [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF] and is described below.

The 3 parameters are defined as the vector ALLOY and are stored in a Nx3 array, N being the number of compositions. SOMs provide a methodology for representing such data in a lower dimensional format, usually in the form of a 2D map [START_REF] Kohonen | Self-Organized Formation of Topologically Correct Feature Maps[END_REF]. SOMs are widely described in the general literature so only a brief overview is provided here [START_REF] Kohonen | Self-Organized Formation of Topologically Correct Feature Maps[END_REF][START_REF] Alhoniemi | Biennial report[END_REF][START_REF] Ong | Data Mining Using Self-Organizing Kohonen maps: A Technique for Effective Data Clustering & Visualisation[END_REF][START_REF] Yin | [END_REF]. Initially each parameter in the vector ALLOY is individually scaled from 0 to 1 (e.g. all average radii are scaled between 0 and 1 etc.). A training vector is then selected. The training vector consists of a subset SUB containing the 3 parameters from ALLOY. SUB is represented by a 20x20 grid of nodes, which forms the map, where each node initially possesses its own uniformly distributed random values, from U(0,1), for each of the 3 parameters. Construction of the SOM proceeds by repetition of the following 'time' steps, t.

1: Randomly select an alloy (1 to N), ALLOY x .

2: Determine the node that most closely matches the chosen alloy, SUB w , i.e. the node that minimizes the expression, . This 'winning node' is now defined as the best matching unit or BMU.

3: Next, the neighbourhood of the BMU is determined. The neighbourhood is made up of those nodes, SUB n , that are within a fixed distance of the BMU. This fixed distance decreases as more steps are performed so that BMU neighbourhoods become smaller as t increases.

4:

The winning node is updated in the standard way using

(1)

5: In the neighbourhood nodes, n, the value of SUB n is adjusted to move towards ALLOY x using a modified form of equation 1,

where d wn is the distance between the nodes in the data space,  wn is the distance between the nodes on the unit grid,  is a prespecified resolution parameter, η(w,n,t-1) is a neighbourhood function that decreases monotonically with distance and θ(t) is a weighting function, the socalled learning rate, which decreases linearly with increasing time steps, t.

6:

The process is then repeated N times until the fixed distance defining the BMU neighbourhood becomes just one node.

Once this process is finished the BMU for each alloy is then determined and the alloys are placed on the map at their BMU locations. The ViSOM is very similar to the standard SOM approach but it constrains the contraction within the updating neighbourhood and unlike standard SOM it provides quantitative inter-point distances on the final map. Rothery phases [START_REF]Structure of metals[END_REF][START_REF] Massalski | Comments Concerning Some Features of Phase Diagrams and Phase Transformations[END_REF] and the approximants of quasicrystals [START_REF] Dong | The concept of the approximants of quasicrystals[END_REF], and to some extent also their physical properties, two other requirements were needed: 1) to combine e/a calculation with the calculated average radius of the elements in the alloy and 2) the validity of the solid solution approximation. The last condition is solved by both the definition of HEAs by Yeh [START_REF] Yeh | Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes[END_REF] and the classification of HEAs as a function of the entropy of mixing and the mismatch [START_REF] Guo | More than entropy in high-entropy alloys : Forming solid solutions or amorphous phase[END_REF]. As pointed out in [START_REF] Hume-Rothery | The freezing points, melting points and solid solubility limits of the alloys of silver and copper with the elements of the Bsub-groups[END_REF] the exact "atomic diameter" of an element is always difficult to define. The atomic distances used in Hume-Rothery approach are defined [START_REF] Hume-Rothery | Atomic diameters, atomic volumes, and solid Solubility relations in alloys[END_REF] as the closest distances of approach of atoms in the crystals of the elements. The radii given by Teatum in [START_REF] Teatum | Compilation of calculated data useful in predicting metallurgical behavior of the elements in binary alloy systems[END_REF] were chosen for the classification in [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF] as they are reported for a coordination number of 12. Furthermore, the Teatum radii were calculated from a combination of the observed interatomic distances in fcc, hcp and bcc structures. Assuming in a first approximation that all constituent phases can be considered as solid solutions, all characteristics have been calculated from linear combinations of the characteristics of the individual constituting elements weighted by their atomic percentages. Providing error bars for all of the calculated data is acknowledged as challenging as all quantities are obtained by using commonly available databases of physics, such as the radius for bcc, fcc and hcp structure [START_REF] Teatum | Compilation of calculated data useful in predicting metallurgical behavior of the elements in binary alloy systems[END_REF].

The alloy average values of e/a have been calculated as in [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF] according to Massalski [START_REF] Massalski | Comments Concerning Some Features of Phase Diagrams and Phase Transformations[END_REF], that is e/a is equal to the electrons of the outside layer. For example, in the case of Zn, with [Ar] 3d 10 4s 2 as electronic structure, there are 2 electrons in the outside layer and 10 d electrons. Thus, e/a=2. In the following e/a2 or eoa2 will stand for the e/a calculated with the value 2 for Ni and e/a1 or eoa1 the one for a value of 1. In [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF] the e/a values are e/a2. In order to investigate the influence of the polymorphism of nickel for the case of HEAs Fig. 1 presents the variation of e/a1 and e/a2 as a function of the average radius, with an indication of the phases identified by diffraction.

It can be seen that for both choices of e/a the alloys containing solely fcc phases are found at low e/a values and that those with bcc phases are found especially at large e/a and r values.

It is clearly seen that there is a range of e/a values in which one finds alloys consisting of one or several fcc phases (Domain I), another range within which one finds alloys consisting of mainly bcc phases (Domain III) and a third range in which alloys consisting of phases with more complex structures are found (Domain II). For both configurations of Ni (Fig. 1a andb) the three Domains identified to separates the three types of alloys, namely fcc, mixed and bcc, are indicated by the horizontal bars.

In Fig. 1a From observation of Fig. 1, a and b, the following conclusions can be drawn:

1) The domain borders identified for e/a2 do not properly separate alloys according to the phases present. There is no clear division between fcc, mixed and bcc types.

2) The value 1 for Ni e/a ([Ar] 3d9 4s1) seems to be more suitable to obtain a correct separation of the three Domains: Accordingly, the e/a1 values were used in the following.

Mapping of the experimental results

The target physical property used for a classification based on e/a and r in [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF] was the value of the measured average magnetic moment per atom at saturation at 5K, M 5K-Sat . This value is obtained from the variation of M as a function of the applied magnetic field as presented in Fig. 2 for a Pd HEA alloy. The variation of M 5K-Sat is shown as a function of e/a1 in Fig. 3. The vertical bars indicate the separation between the identified domains in Fig. 1a.

As expected from [START_REF]Structure of metals[END_REF] and [START_REF] Massalski | Comments Concerning Some Features of Phase Diagrams and Phase Transformations[END_REF] for Hume-Rothery crystalline phases, and already shown with less resolution as a function of e/a2 in [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF], there is a significant variation of M 5K-Sat as a function of the electronic structure of the corresponding HEAs. The set of values presented in Fig. 3 has been used in the following to build a Self-Organising Map and to derive a prediction program for M 5K-Sat for any HEA, given its e/a1 and its radius.

As already used in [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF] with e/a2, the Self-Organising Map (SOM) approach can be used to get a picture of several combined parameters. It is important to stress here that contrary to the {e/a;r} approach based on considerations on the electronic structure of alloys as detailed in [START_REF]Structure of metals[END_REF][START_REF] Massalski | Comments Concerning Some Features of Phase Diagrams and Phase Transformations[END_REF], the SOM approach belongs to data mining.

Comparison of the prediction of the average magnetic moment per atom with e/a1, design maps

In [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF] Self-Organizing Maps were built and used to predict the existing phases of HEAs according to their e/a and radius values. Whereas the values 2 was used for Ni in [START_REF] Calvo-Dahlborg | Hume-Rothery for HEA classification and selforganizing map for phases and properties prediction[END_REF], the e/a1 values of Fig. 3 are used here to make another prediction program, PredictorE1. Two sets of alloys have been used for comparison with experiments: i) alloys for which no saturation of magnetization could be recorded on M(H) loops until 2T (Fig. 6).

For clarity, Fig. 5 gives a representative example of the variation M(H) of the average magnetic moment per atom, M, as a function of the applied field H for such alloys. In Fig. 2, the alloy has reached a saturated magnetic state and a value at saturation, M 5Ksat , can be estimated. In Fig. 5, no saturation could be observed in the interval of the experiment. Thus, the reported experimental value of M in is just the maximum value at 5K for a maximum field value of μ 0 =2 T.

Fig. 6 present the values obtained through the PredictorE1 in the same frame as Fig. 3, the map data being represented by small stars, the non-saturated values by large down triangles and the results of PredictorE1 with indication of the phases present.

It can be observed the following:

1) The non-saturated values are outside the range of the curve from Fig. 2 and much lower.

2) The values predicted by the PredictorE1 program are all located within the set of values presented in Fig. 2.

3) As would be expected, these values are all above the values determined for the nonsaturated samples.

4) The identified phases correspond to the three Domains identified before.

Again, it should be stressed that the PredictorE1 program is based on data mining approach whereas the value at saturation is a magnetic characteristic.

ii) alloys from literature produced in the same way [START_REF] Gao | High-Entropy alloys: Fundamentals and applications[END_REF].

Fig. 7 presents the values of M 5K-Sat predicted for alloys for which the identified phases were reported in the literature [START_REF] Gao | High-Entropy alloys: Fundamentals and applications[END_REF]. The predicted values for these alloys are within the range of values of the map presented in Fig. 2. Furthermore, the values are for most within the Domain corresponding to the identified phases. It should be noted that details on the identification of the phases was not always given in the literature. It was thus impossible to check the accuracy of the determination and/or whether other minor phases were present.

The summary of observations from both Fig. 6 and 7 are the following:

1) As expected, the predicted values for saturation of the non-saturated values are within the range of the map and within the Domains corresponding to the identified phases.

2) The predicted values for samples from the literature for which the identified phases were reported are within the range of the map and within the Domains corresponding to the identified phases.

Brillouin zone

The influence of the electron concentration e/a is known since [START_REF]Structure of metals[END_REF] to determine the phases present in an alloy following the 3 Hume-Rothery rules. In the iconic works [START_REF] Jones | Applications of the Bloch theory to the study of alloys and of the properties of bismuth[END_REF][START_REF] Jones | The effect of electron concentration on the lattice spacings in magnesium solid solutions[END_REF] Jones is

showing for a large series of binary systems that the value e/a=1.85 is identified as the change in the c/a ratio of the hcp structure and linked to the overlapping of some faces of the Brillouin zone. The same observation has been used in [START_REF] Dong | The concept of the approximants of quasicrystals[END_REF] for the approximants of quasicrystals.

In the case of HEAs, the Teatum radius associated with the electron concentration e/a1 in the solid solution approximation is showing that the phases present as well as certain properties can be similarly predicted. The value 1.85 corresponds to a e/a1 value equal to 1.65. In order to investigate if such value has a meaning in the case of HEAs a characteristic related to an abrupt change of property was chosen: the magnetic transition as recorded in an M(T) curve by SQUID. Fig. 8 present the variation of T m as a function of e/a1. T m is determined by the zero point of the second derivative of the M(T) curve. Thus, it is not the Curie temperature defining the start of the zero magnetization. Two lines were drawn between the fcc points and between the bcc points. The value 1.65 corresponds to the intersection between the two lines.

From the observation of Fig. 8 it is clear that the value e/a1=1.65 is also corresponding to an abrupt change in the phases present in HEAs. Moreover, similarly to what has been the case for the design and interpretation of quasicrystals the Brillouin zone has to be introduced in calculations and modelling of HEAs in addition to the electron concentration e/a and the radius.

Conclusions

1) When an HEA composition contains nickel its electronic structure or [Ar] 3d 9 4s 1 should be considered instead of [Ar] 3d 8 4s 2 .

Thus, the domains defining the presence of phases are the following: 2) A Self-Organizing Map approach can be built to predict phases present in a HEA according to its e/a and radius values.

3) For the [Ar] 3d 9 4s 1 electronic structure of Ni, the experimental and the predicted values are consistent within the range of the map.

4) Based on this map the average magnetic moment at saturation can be predicted for HEAs as well as the Domain defining the presence of particular phases. This last result is of great importance and potential use for future applications involving the magnetic properties of HEAs 5) As has been the case for the design and understanding of Hume Rothery phases and quasicrystals the Brillouin zone of HEA systems has to be investigated in their calculations and modelling.

Figure captions

*Credit Author Statement

3. Results and discussion 3 . 1 .

 31 Comparison of the distribution of the existing phases in the {e/a1;r] and {e/a2;r} frames From the two previous uses of e/a to predict crystalline phases in alloys, e.g. the Hume

  Fig.1a and 1b, contains alloys for which the phases identified are of mixed type. This classification is very useful to clearly distinguish the domain of occurrence of intermetallic phases which will evolve and/or stabilize upon heating leading to deterioration of specific HEAs properties.

3 )

 3 Domain II contains phases of mixed types and/or intermetallic phases.

Fig. 4 , 3 :

 43 a to c presents the results arising from the selective choice of e/a, r and M. The maps show the locations of the different alloys (labelled as L1, L2 etc.). It can be observed is that the alloys are always at the same positions on the maps shown in figures 4, a to c. In each figure the thick doted lines represent the values of the frontiers between domains as identified in Fig. 1a, namely e/a1= 1.53 and 1.88 for the alloys. The representation produced by the SOM and the identified domains in figure 4a, b and c, corresponding respectively to e/a, r and M, are correlated to what is observed in Fig. Alloys with e/a1<1.53 of Domain I are in the blue regions of the three figures, alloys with e/a1>1.88 of Domain III in the red regions.

  e/a (Ni=1) < 1.88: mixed phases, Domain III e/a (Ni=1) > 1.88: bcc.

Fig. 1 :

 1 Fig. 1: a) e/a1 b) e/a2 as a function of the average Teatum radius. The symbols indicate the types of phases identified by diffraction: ■ fcc, ○ bcc, hcp, ▲L21,  B2, + Sigma or Tetragonal,  mixed,  not determined. The horizontal bars determine the domains I, II and III containing fcc, and bcc phases.

Fig. 2 :Fig. 3 :

 23 Fig.2: Representative example of an HEA exhibiting ferromagnetic properties with saturation of magnetization on the M(H) loops at 5 K.

Fig. 4 :Fig. 5 :Fig. 6 :

 456 Fig.4: SOM results for (a) e/a, (b) r, (c) M 5K-Sat when SUB = e/a and r. On the colour scale red corresponds to the maximum value and blue to the minimum.

Fig. 7 :

 7 Fig. 7: Average magnetic moment per atom at 5K, M 5K , as a function of e/a1 for alloys in the literature. () Values of the map from Fig. 2. Values after PredictorE1: The symbols indicate the types of phases identified by diffraction: ■ fcc, ○ bcc, hcp, ▲L21,  B2,  not determined. The horizontal bars determine the domains I, II and III containing fcc, and bcc phases.

Fig. 8 :Highlights

 8 Fig.8: The temperature of magnetic transition T m as a function of e/a1 for the alloys of Fig.3for which it was possible to record until 400K. The symbols indicate the types of phases identified by diffraction: ■ fcc, ○ bcc,T,  B2, not determined.

Acknowledgments:

SGRB acknowledges the support of the Welsh Government and Welsh European Funding Office in the Combinatorial Metallurgy (COMET) Smart Expertise Project. The samples have been prepared within the FP7 European project AccMet NMP4-LA-2011-263206.

Declaration of interests

☒ 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.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

*Declaration of Interest Statement