Vincent Javaugue

Virginie Pascal

Sébastien Bender

Sarah Nasraddine

Mathilde Dargelos

Mehdi Alizadeh

Alexis Saintamand

Matthieu Filloux

Paco Derouault

Sabrina Bouyer

Estelle Desport

Arnaud Jaccard

Frank Bridoux

Michel Cogné
email: michel.cogne@unilim.fr

Christophe Sirac
email: christophe.sirac@unilim.fr

Keywords: Next generation sequencing, immunoglobulin, AL-amyloidosis, monoclonal gammopathy of renal significance

The diagnostic approach of monoclonal gammopathy of renal significance is based on the detection of a monoclonal immunoglobulin in the blood and urine, and the identification of the underlying clone through bone marrow and/or peripheral blood cytologic and flow cytometry analysis. However, the monoclonal component and its corresponding clone may be undetectable using these routine techniques. Since clone identification is the cornerstone for guiding therapy and assessing disease response, more sensitive methods are required. We recently developed a high-throughput sequencing assay from bone marrow mRNA encoding immunoglobulins (RACE-RepSeq). This technique provides both full-length V(D)J region (variable, diversity and joining genes that generate unique receptors as antigen receptors) of the monoclonal immunoglobulin and the dominant immunoglobulin repertoire. This allows analysis of mutational patterns, immunoglobulin variable gene frequencies and diversity due to somatic hypermutation. Here, we evaluated the diagnostic performance of RACE-RepSeq in 16 patients with monoclonal-associated kidney lesions, and low serum monoclonal immunoglobulin and free light chain levels at diagnosis. Bone marrow immunohistochemical analysis was negative in all 11 patients so tested and 7 of 12 patients had no detectable clone matching the kidney deposits using flow cytometry analysis. By contrast, RACE-RepSeq detected a dominant clonal light chain sequence of matched isotype with respect to kidney deposits in all patients. Thus, high throughput mRNA sequencing appears highly sensitive to detect subtle clonal disorders in monoclonal gammopathy of renal significance and suggest this novel approach could help improve the management of this kidney disease.

Translational Statement

High-throughput sequencing assay from bone marrow mRNA encoding immunoglobulins (RACE-RepSeq) is a very sensitive approach to detect small B/plasma cell clones in patients with monoclonal gammopathy of renal significance. This technique provides the full-length immunoglobulin variable domain sequences, allowing analysis of mutational patterns that may help to better understand mechanisms of immunoglobulin toxicity. Future studies are needed to evaluate the sensitivity of RACE-RepSeq analysis from bone marrow and blood samples for monitoring minimal residual disease, a new goal in the management of plasma cell disorders.

Introduction

In monoclonal gammopathy of renal significance (MRGS), renal lesions are independent of the tumor mass, but result from structural peculiarities in the variable (V) domain of the nephrotoxic immunoglobulin (Ig). [START_REF] Leung | The evaluation of monoclonal gammopathy of renal significance: a consensus report of the International Kidney and Monoclonal Gammopathy Research Group[END_REF][START_REF] Sirac | Animal models of monoclonal immunoglobulinrelated renal diseases[END_REF] The serum level of the pathogenic Ig is usually low, complicating the diagnosis and assessment of response to chemotherapy. The initial hematological workup in MGRS patients is based on serum and urine protein electrophoresis/immunofixation, measurement of serum free light chains (FLC), and characterization of the underlying clone in the bone marrow (BM) and/or peripheral blood.

However, depending on the type of MGRS-renal disorder, these techniques may fail to detect the nephrotoxic Ig and to identify the pathogenic clone. A previous study found that 1.9% of patients with AL-amyloidosis had no detectable abnormality using all available assays. [START_REF] Katzmann | Screening Panels for Detection of Monoclonal Gammopathies[END_REF] Moreover, 20% of patients with AL-amyloidosis display FLC levels below the threshold required for evaluation of hematological response, i.e. difference between involved and uninvolved FLC (dFLC) <50 mg/L, making therapeutic management a challenge. Recent studies suggest that such cases could be evaluated with the so called low-dFLC partial response criteria defined as achieving dFLC <10 mg/L in patients with a baseline dFLC between 20 and 50 mg/L. [START_REF] Dittrich | AL amyloidosis patients with low amyloidogenic free light chain levels at first diagnosis have an excellent prognosis[END_REF][START_REF] Milani | Patients with light-chain amyloidosis and low free light-chain burden have distinct clinical features and outcome[END_REF] However, in our experience, around 2% of patients with AL-amyloidosis are still not evaluable. As compared to AL amyloidosis, less patients with type I cryoglobulinemic glomerulonephritis (~90%) or immunotactoid glomerulonephritis (~70%) have a detectable monoclonal component. The detection rate is noticeably low in patients with proliferative glomerulonephritis with monoclonal Ig deposits (PGNMID), 80% of whom, have a negative hematologic evaluation. [START_REF] Sidana | Clinical presentation and outcomes of patients with type 1 monoclonal cryoglobulinemia[END_REF][START_REF] Javaugue | Results of a nation-wide cohort study suggest favorable long-term outcomes of clone-targeted chemotherapy in immunotactoid glomerulopathy[END_REF][START_REF] Bhutani | Hematologic characteristics of proliferative glomerulonephritides with nonorganized monoclonal immunoglobulin deposits[END_REF] Thus, more sensitive diagnostic techniques are required to guide therapeutic strategy in MGRS.

Next generation sequencing (NGS) has improved Ig sequencing and detection of minimal residual disease in B-cell malignancies. [START_REF] Perrot | Minimal residual disease negativity using deep sequencing is a major prognostic factor in multiple myeloma[END_REF] The sensitivity of Ig sequencing on DNA is limited by the number of cells and thus may be insufficient to detect subtle clones in MGRS patients. [START_REF] Van Dongen | Design and standardization of PCR primers and protocols for detection of clonal immunoglobulin and T-cell receptor gene recombinations in suspect lymphoproliferations: report of the BIOMED-2 Concerted Action BMH4-CT98-3936[END_REF] We recently developed a high-throughput sequencing assay from BM mRNA encoding Igs (RACE-RepSeq). This RNA sequencing method has the ability to catch any Ig transcript, even after heavy somatic hypermutation , and to give access to the full-length V(D)J sequence. [START_REF] Bender | Immunoglobulin variable domain highthroughput sequencing reveals specific novel mutational patterns in POEMS syndrome[END_REF] Additionally, it has a strong sensitivity for the detection of small plasma cell clones, since every single secreting cell produces hundreds of copies of the Ig transcripts. [START_REF] Bender | Immunoglobulin variable domain highthroughput sequencing reveals specific novel mutational patterns in POEMS syndrome[END_REF][START_REF] Shi | Transcriptional profiling of mouse B cell terminal differentiation defines a signature for antibody-secreting plasma cells[END_REF] The aim of this study was to assess the clone detection rate of RACE-RepSeq in patients with MGRS-associated renal lesions with low monoclonal Ig and FLC levels, in order to evaluate its value in the diagnostic workup of MGRS.

Methods

Written informed consent was obtained from all study participants, recruited between 2015 and 2020. Kidney biopsy and bone marrow analysis were performed in all cases (Supplementary Methods).

A detailed protocol of RACE-RepSeq analysis is available in supplementary data.

Briefly, total mRNA was extracted from BM samples (1 mL) using Tempus TM spin RNA isolation kit (Thermofisher), retrotranscribed to cDNA and then amplified using the 5'RACE PCR technique with unique molecular identifiers (UMI) to reduce amplification errors and quantitative bias. Two specific primers located in the 3' part of the constant domain of λand -LC genes and a generic adapter in 5' allowed the amplification of the entire corresponding repertoire of the patient (Figure 1 and Supplementary Table S1). An asymmetric paired-end sequencing strategy is employed to obtain full length V(D)J sequences on an Illumina MiSeq sequencer using a MiSeq Reagent Kit v2 (500 cycles). On average, the capacity of this kit allows exploring the simultaneous -and λ-LC repertoires of 30 patients with ~300,000 raw reads per sample. Ig repertoire analysis was done using IMGT/HighV-QUEST (http://imgt.org/) and Vidjil software. [START_REF] Alamyar | IMGT® Tools for the Nucleotide Analysis of Immunoglobulin (IG) and T Cell Receptor (TR) V-(D)-J Repertoires, Polymorphisms, and IG Mutations: IMGT/V-QUEST and IMGT/HighV-QUEST for NGS[END_REF][START_REF] Duez | Vidjil: A Web Platform for Analysis of High-Throughput Repertoire Sequencing[END_REF]

Results

We retrospectively analyzed 16 patients with biopsy-proven MGRS-associated renal disease (Table 1 and2), and 8 control patients with immune thrombocytopenia and no evidence of monoclonal gammopathy by serum immunofixation (Supplementary Table S2). Eleven patients had dFLC <20 mg/L and 5 had negative serum/urine immunofixation at diagnosis.

Search for a plasma/B-cell clone was negative in 11/11 patients using immunohistochemistry analysis of BM biopsies. Flow cytometry analysis was negative on peripheral lymphocytes in 8/9 tested patients and on BM in 5/12 patients. Finally, 2 patients had neither monoclonal gammopathy nor detectable clonal disorder. Amongst 10 patients with λ AL-amyloidosis, BM flow cytometry analysis (n=7), allowed detection of a plasma cell clone in 2 patients (cases ALλ2 and ALλ10) and of a small lymphoplasmacytic clone in 2 patients (cases ALλ5 and ALλ7). However, the clone found in patient ALλ2 was of κ isotype, thus not related to glomerular λ LC deposits. Flow cytometry analysis detected an abnormal plasma cell population in patient ALλ4, but failed to confirm monoclonality. Among 3 patients with κ AL-amyloidosis, BM flow cytometry disclosed a lymphoplasmacytic clone in 1 of 2 tested patients (case ALκ3). No clone matching the renal deposits was found in the 2 patients with type 1 cryoglobulinemic glomerulonephritis, and the patient with immunotactoid glomerulopathy showed a small detectable κ-CLL clone using blood and BM flow cytometry (Table 1 and2).

Using RACE-ReqSeq, an IGLV or IGKV abundant clonotype was readily identified in all patients, ranging from 14 to 86% of total clonotypes. By contrast, the most abundant clonotype in the control group was between 0.2 to 3.4% of total clonotypes (Figure 2). Besides the high detection rate of clonal LC sequences, RACE-RepSeq also provides the full-length VJ region of the LCs. Figure 3 depicts clonal sequences obtained in patients with MGRS in comparison with the corresponding germline sequences.

Discussion

The present study indicates that RACE-RepSeq is highly sensitive to detect subtle B/plasma-cell clones in patients with MGRS. This novel approach appears promising for optimizing management and refining assessment of the hematologic response in MGRS subtypes associated with low clonal identification rate.

Besides its sensitivity to detect clones, RACE-RepSeq offers other interesting perspectives. It requires less than 1 ml of total BM aspirate to depict the dominant B-cell repertoire with characterization of full-length V(D)J region of produced Ig. Since sequence peculiarities in the V domain of monoclonal Igs govern their toxicity, RACE-RepSeq may help identify specific molecular profiles in various MGRS-related disorders. However, we cannot formally conclude that the most abundant LC sequence is the pathogenic one, especially when two clones of the same isotype were found (patients AL1 and AL3). Mass spectrometry (MS) analysis of laser-capture dissected deposits remains the only tool to gain information on the real deposited LC, but usually results in low V domain coverage or fails to identify any V gene due to the high mutation rate. [START_REF] Dasari | Proteomic detection of immunoglobulin light chain variable region peptides from amyloidosis patient biopsies[END_REF] Coupling RACE-RepSeq for full-length sequence information and MS analysis would be a powerful tool to definitely identify the pathogenic Ig. RACE-RepSeq could also be useful to assess the monoclonal origin of certain renal diseases in which immunofluorescence studies lack enough sensitivity to differentiate between monoclonal and oligoclonal Ig deposits, such as monotypic fibrillary glomerulonephritis, IgA nephropathy with λ-LC restriction or PGNMID. [START_REF] Said | DNAJB9-positive monotypic fibrillary glomerulonephritis is not associated with monoclonal gammopathy in the vast majority of patients[END_REF][START_REF] Vignon | The clinicopathologic characteristics of kidney diseases related to monotypic IgA deposits[END_REF][START_REF] Aucouturier | A Fresh Perspective on Monoclonal Gammopathies of Renal Significance[END_REF][START_REF] Bridoux | Proliferative glomerulonephritis with monoclonal immunoglobulin deposits: a nephrologist perspective[END_REF] The prognosis of MGRS depends on the rapid achievement of deep hematologic response, whose accurate assessment often remains a challenge. [START_REF] Leung | The evaluation of monoclonal gammopathy of renal significance: a consensus report of the International Kidney and Monoclonal Gammopathy Research Group[END_REF] In multiple myeloma, negative

Acknowledgments:

  This work was supported by grants from AREN Poitou-Charentes, Comité d'Orientation de la Recherche en Cancérologie (CORC), Limousin committees of Ligue nationale contre le cancer, Agence régionale de la santé, Institut Universitaire de France, "Association Française contre l'Amylose" and "Fondation Française pour la Recherche contre le Myélome et les Gammapathies" The authors acknowledge A. Rinsant and A. Lehericy for their technical assistance and E. Guerin of the sequencing technical plateform of CHU Limoges.

Figure 1 .

 1 Figure 1. Schematic representation of the high-throughput sequencing assay from bone marrow mRNA encoding immunoglobulins performed in patient ALλ8.

Figure 2 .

 2 Figure 2. (a) Most abundant clonotype found in each patient. (b) Comparison between MGRS and immune thrombocytopenia patients (***p<0.001, Mann-Whitney test).

Figure 3 .

 3 Figure 3. Deduced amino acid sequences of the monoclonal λ (a) and κ (b) light chain VJ domains in MGRS patients compared to germline sequences according to IMGT numbering. Mutated amino acids are highlighted in green.

minimal residual disease measured by next generation flow cytometry (NGF) or NGS on DNA has been shown to correlate with improved progression-free and overall survival. [START_REF] Perrot | Minimal residual disease negativity using deep sequencing is a major prognostic factor in multiple myeloma[END_REF][START_REF] Flores-Montero | Next Generation Flow for highly sensitive and standardized detection of minimal residual disease in multiple myeloma[END_REF] Our results suggest that RACE-RepSeq may represent a valuable additional tool to precisely assess hematologic response and thus guide duration and adaptation of chemotherapeutic regimens. A precise comparison of the sensitivity of RACE-RepSeq with those of NGF and NGS on DNA deserves to be carried out. Additionally, in unpublished preliminary data including 14 patients with MGRS, RACE-RepSeq shows a significant correlation between BM and blood for the detection of clonal plasma cells. Thus, this technique could represent an additional noninvasive tool for frequent monitoring of the hematologic response. In clinical practice, the value of RACE-RepSeq, appears promising although it remains to be evaluated in larger cohorts of MGRS patients. (24) Abbreviations : AL, light-chain amyloidosis, CryoIgGλ, Type 1 cryoglobulinemic glomerulonephritis with IgGλ deposits; dFLC, difference between lambda and kappa light chain serum concentrations; IHC, immunohistochemistry analysis of bone marrow biopsy; LP clone, lymphoplasmacytic clone; MGRS, monoclonal gammopathy of renal significance; M-spike, monoclonal spike; ND, not done; PC clone, plasma cell clone; RACE-RepSeq analysis, high-throughput sequencing assay from bone marrow mRNA encoding immunoglobulins; SPEP, serum protein electrophoresis; UPEP, urine protein electrophoresis *Normal range (Freelite™) : kappa = 3.3-19.4 mg/L, lambda = 5.71-26.3 mg/L, kappa/lambda ratio = 0.26-1.65 **Detection of less than 0.2% of abnormal plasma cells (CD19-, CD56-, CD27+, CD28+, CD81+, CD117-) without monotypic appearance

Disclosures: All the authors declared no competing interests

Supplementary Material

Accepted manuscript / Final version


Romain Bachelard

Cristel Chandre

Duccio Fanelli

Xavier Leoncini

Stefano Ruffo

Abundance of regular orbits and out-of-equilibrium phase transitions in the thermodynamic limit for long-range systems

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The vast majority of phenomena observed in nature results from complex interactions present in large assemblies of elementary constituents. A widespread observation is the emergence of regular trajectories despite the complexity of the underlying network of couplings. A successful approach to describe the collective behaviour of large assemblies of particles is traditionally provided by statistical mechanics. The theoretical foundation of equilibrium statistical mechanics relies upon the hypothesis of ergodicity, i.e. the agreement of time with ensemble averages. Arguing for an effective degree of global mixing of dynamical trajectories in phase space implies ergodicity and thus the validity of statistical mechanics [START_REF] Gallavotti | Statistical Mechanics: A short treatise[END_REF]. Thermodynamic behaviour is obtained in the limit where the number of degrees of freedom goes to infinity, which offers innumerable pathways to chaos. Indeed, in this limit regular regions (invariant tori) do not possess enough dimensions in phase space to prevent trajectories from spreading, while the largest fraction of the phase space is occupied by chaotic motion, hence sustaining mixing [START_REF] Froeschlé | [END_REF].

However, the fact that any weak nonlinearity would imply ergodicity has been vigorously debated since the pioneering work of Fermi, Pasta and Ulam (FPU) [START_REF] Fermi | Studies of non linear problems[END_REF] on the dynamics of oscillators interacting via short-range couplings. Contrary to expectations, the celebrated FPU chain exhibits a recurrent behavior on very long times, violating ergodicity. Nowadays, there is a growing evidence that, for generic initial conditions, the relaxation time to equilibrium remains finite in the thermodynamic limit [START_REF] Gallavotti | The Fermi-Pasta-Ulam problem: a status report[END_REF]. On the contrary, the question of relaxation to equilibrium is still open when long-range forces come into play [START_REF]Dynamics and thermodynamics of systems with long-range interactions[END_REF][START_REF]Dynamics and thermodynamics of systems with long-range interactions: Theory and Experiments[END_REF]. Indeed, systems with long-range interactions have been shown to display an extremely slow relaxation to equilibrium. More specifically, out-of-equilibrium metastable regimes have been identified, where the system gets trapped before eventually attaining its asymptotic state [7,8]. The equilibration time increases with system size and formally diverges in the thermodynamic limit, leading to a breaking of ergodicity. For gravitational systems the approach to equilibrium has never been proven and seems problematic. Galaxies could therefore represent the most spectacular example of such far-fromequilibrium processes [9], but analogous phenomena have also been reported in fundamental problems of plasma physics [START_REF] Elskens | Microscopic Dynamics of Plasmas and Chaos[END_REF]. These metastable states have been termed Quasi-Stationary States (QSS) in the literature and often represent the solely accessible experimental regimes (e.g., in the Free Electron Laser [START_REF] Barré | [END_REF]). A relevant feature of long-range systems is the self-consistent nature of the interaction of a particle with its "local field", which itself results from the combined action of all the other particles or of an "external field" like for wave-particle systems [START_REF] Elskens | Microscopic Dynamics of Plasmas and Chaos[END_REF]. It is exactly this self-consistency which finally entails the widespread regularity of the motion. Besides that, one observes the presence of classes of initial states that show different time evolutions.

The development of a systematic theoretical treatment of the QSS, which would enable us to unravel the puzzle of their ubiquity, is still an open problem. Both the selfconsistent nature of the interaction and the strong dependence on the initial condition suggest that QSS could be related to the presence of some type of regular motion. The traditional approach to clarifying the emergence of regular trajectories is based on the following result : If the Hamiltonian system under scrutiny is close to integrable, Kolmogorov-Arnold-Moser [12] theory proves that the phase space is filled with invariant tori on which the motion is quasi-periodic. In this framework, however, increasing the number of particles enhances the contribution of chaotic trajectories [START_REF] Froeschlé | [END_REF], in stringent contradiction with the observation that QSS prevail in the large N limit. Therefore the aforementioned scenario cannot be invoked to explain the presence of regular motion in systems with long range interactions.

A first purpose of this Letter is to put forward a different interpretative framework. We argue that tori can form in phase space also as a result of a self-consistent interaction in the thermodynamic limit. As we shall demonstrate, while for a small number N of degrees of freedom the single particle motion of a paradigmatic system with long-range interaction is erratic, the trajecto-ries become more and more regular as N → ∞. These trajectories arise from a low-dimensional time dependent effective Hamiltonian.

For systems with long-range interactions, the dependence on the initial condition can materialize in the form of a true out-of-equilibrium phase transition: By varying some crucial parameters of the initial state, one observes a convergence towards asymptotic states (in the limit N → ∞) with different macroscopic properties (e.g., homogeneous/inhomogeneous) [13]. This phase transition has been interpreted by resorting to Lynden-Bell's theory of "violent relaxation" [14], which is briefly discussed in the following with reference to a specific model. As a second purpose of this Letter, we provide a microscopic, dynamical, interpretation of this transition in terms of a sharp modification of the properties of the single particle orbits, corresponding to a change of the effective Hamiltonian. Our ultimate aim is to suggest a unifying picture that potentially applies to systems where collective, organized, phenomena emerge from the globally coupled sea of individual components.

The Hamiltonian Mean Field (HMF) model [15] is widely referred to as the benchmark for long-range systems and analysed for pedagogical reasons. The model, which describes the evolution of N particles coupled through an equally strong, attractive, cosine interaction is specified by the following Hamiltonian:

H = N i=1   p 2 i 2 + 1 2N N j=1 (1 -cos (θ i -θ j ))   , (1)
where θ i and p i label respectively the position of particle i on the unit circle and its corresponding momentum. Note that Hamiltonian (1) can also be seen as a simplified version of the gravitational [16] or plasma [17] sheets model when considering only the first harmonic in the Fourier expansion of the potential. To characterize the behaviour of the system, it is convenient to introduce the "magnetization" M = 1 N ( cos θ i , sin θ i ) = M (cos φ, sin φ), which quantifies the degree of spatial bunching of the particles (homogeneity vs. inhomogeneity). We here consider water-bag initial conditions, consisting of particles uniformly distributed in a rectangle [-θ 0 , θ 0 ] × [-p 0 , p 0 ] in the (θ, p)-plane. These states bear a magnetization M = M 0 = sin(θ 0 )/θ 0 , the associated energy per particle being U = p 2 0 /6 + (1 -M 2 0 )/2. When performing numerical simulations, starting from the water-bag initial condition, the system gets usually frozen in a QSS of the type discussed above [7,8].

The individual particle i obeys the following equations of motion ṗi = -M sin (θ i -φ) and θi = p i , where M and φ are functions of all the positions of the particles. Numerical observations suggest that, for large enough values of N , both the magnetization M and the phase φ develop a specific oscillatory time dependence. Hence, the single particle motion turns out to be governed by a time dependent one degree of freedom (often referred to as a one and a half degree of freedom) effective Hamiltonian. This justifies the investigation of the phase space properties of the QSS using a technique inspired by that of Poincaré sections. In particular, we consider the time average M (after a transient) and record the positions and momenta of a few selected particles (θ i , p i ) when M (t) = M and dM/dt > 0 (since M typically shows an oscillatory behavior). The resulting stroboscopic sections are displayed in Fig. 1. Two different phase space structures are found depending on the choice of the initial pair (M 0 , U ), one with monocluster and the other with a bicluster. The monocluster QSS displays a nonzero magnetization (inhomogeneous phase), while the bicluster QSS has a small residual magnetization (homogeneous phase).

The monocluster QSS can be ideally mapped onto a collection of weakly interacting pendula. As revealed by our stroboscopic analysis, particles evolve on regular tracks, which are approximately one-dimensional, though they do manifest a degree of local diffusion (thick- Ignoring the time dependence of δM and using a reduced model of test particles in the external field M (t), one obtains stroboscopic sections which are qualitatively and quantitatively similar to the ones reported in this figure, with the unique difference that the thickness is zero [18]. Considering a torus with action J ≈ 1.9, we plot in panel d) its variance ∆J computed over a time interval ∆t = 300 as a function of N . The scaling 1/N (dotted line) looks accurate over a wide range of N values. ness). For the bicluster QSS, the Poincaré section shows a phase portrait which closely resembles the one obtained for a particle evolving in the potential of two contrapropagating waves. These latter interact very weakly, as the associated propagation velocities appear rather different.

In order to get a quantitative estimate of the thickness of the tori as a function of the total number of particles, we focus on the monocluster QSS. Panels a), b) and c) in Fig. 2 display the single particle phase space for increasing values of N . A clear trend towards integrability is observed as quantified in panel d), where the thickness is plotted versus N .

Summing up, we have assessed that the single particle motion of a typical long-range interacting system becomes progressively more regular as the number of particles is increased. This is at variance with what happens for systems with short-range interactions and provides a different interpretation of the abundance of regular motion in long-range dynamics. Besides that, we have seen that the features of the single particle motion depend on the choice of the initial condition. A natural question then arises: what is the link between the macroscopic properties of the different QSS with the change observed in the single particle dynamics? Anticipating the answer, we will see that this is related to a bifurcation occurring in the effective Hamiltonian.

In the thermodynamic limit, the evolution of the single particle distribution function f (θ, p, t) is governed by the Vlasov equation [19]. This equation also describes the mean-field limit of wave-particle interacting systems [START_REF] Elskens | Microscopic Dynamics of Plasmas and Chaos[END_REF]. It can be reasonably hypothesized that QSS are stationary stable solutions of the Vlasov equation [8]. Following these lines, a maximum entropy principle, previously developed in the astrophysical context by Lynden-Bell [14], allowed one to predict [13,[START_REF] Chavanis | [END_REF], for the HMF model, the occurrence of out-of-equilibrium phase transitions, separating distinct macroscopic regimes (magnetized/demagnetized) by varying selected control parameters which represent the initial condition.

The central idea of Lynden-Bell's approach consists in coarse-graining the microscopic one-particle distribution function f (θ, p, t) by introducing a local average in phase space. Starting from a water-bag initial profile, with a uniform distribution f 0 , a fermionic-like entropy can be associated with the coarse grained profile f , namely

s[ f ] = -dpdθ f f 0 ln f f 0 + 1 - f f 0 ln 1 - f f 0 .
(2) The corresponding statistical equilibrium, which applies to the relevant QSS regimes, is hence determined by maximizing such an entropy, while imposing the conservation of the Vlasov dynamical invariants, namely energy, momentum and norm of the distribution. The analysis reveals the existence of an out-of-equilibrium phase transition from a magnetized to a demagnetized phase [13,[START_REF] Chavanis | [END_REF].

We here reinterpret the transition in a purely dynamical framework, as a bifurcation from a monocluster QSS to a bicluster QSS. Aiming at shedding light on this issue, we proceed as follows: For fixed M 0 and N , we gradually increase the energy U and compute the Poincaré sections, as discussed above. We then analyze the recorded sections by identifying the number of resonances and measuring the associated width and position (both calculated in the p direction). Results for M 0 = 0.6 are displayed in the lower panel of Fig. 3: the shaded region, bounded by the dashed lines, quantifies the width of the resonances. As anticipated, one can recognize the typical signature of a bifurcation pattern. Repeating the above analysis for different values of the initial magnetization M 0 , allows us to draw a bifurcation line in the parameter space (M 0 , U ). In the upper panel of Fig. 3 we report both this bifurcation (full) and the Lynden-Bell phase transition (dashed) lines [13]. The two profiles resemble each other qualitatively, and even quantitatively for small M 0 . The change of the bifurcation type from sub-critical to supercritical signaled by the opening of a gap in the resonance plot could be associated with the change of the order of the phase transition from first to second. These results strongly corroborate our claim that the out-of-equilibrium phase transition from magnetized to non magnetized QSS corresponds to a bifurcation in the single particle dynamics.

The phenomena here discussed are also found (data not shown) in the context of Hamiltonian models deputed to describe the interaction between a beam of charged particles and a set of self-consistently evolving waves [21]. A large number of relevant applications, facing the longrange nature of the couplings among constituents, fall within this rather wide category, ranging from plasma models, traveling wave tubes, and free electron lasers [22]. More concretely, the same tendency towards integrability is found in these latter settings, a fact which holds promise to eventually result in novel insight, potentially relevant for the functioning of the devices.

In summary, investigating the dynamics of systems with long range interactions allowed us to come to an interesting and general conclusion: a universal trend to organization/integrability of the single particle dynamics is found as the system size (number of particles) is increased. Phase space structures are identified and interpreted as invariant tori of a time dependent one degree of freedom Hamiltonian. Based on this analysis, the outof-equilibrium statistical phase transitions is here understood from the viewpoint of microscopic dynamics, as a bifurcation occurring in the effective Hamiltonian which describes the single particle dynamics.

Figure 1 :

 1 Figure1: Poincaré sections of a few selected particles of one trajectory of Hamiltonian (1) for N = 2 × 10 5 in the QSS regime for two different water-bag initial conditions : (M0, U ) = (0.6, 0.54) (upper panel) and (M0, U ) = (0.6, 0.88) (lower panel). The former returns a single cluster, which gives a non-zero magnetization QSS (MQSS ≈ 0.5), while the latter shows two symmetric clusters, which produce a QSS with a small magnetization. In the bicluster regime (lower panel) the presence of a large set of rotational tori implies a substantially lower magnetization level, whereas the librational tori around the two clusters are responsible for the residual magnetization. The color code corresponds to the values of the action variable associated with individual particles.

Figure 2 :

 2 Figure 2: Poincaré sections of a few selected particles of one trajectory of Hamiltonian (1), when the system size is varied (for M0 = 0.6 and U = 0.54). The thickness of the tori decreases as N is increased (see text). For large enough values of N , the magnetization M is found numerically to approximately scale as M (t) ≈ M + δM (t) cos ωt, with |δM | ≪ M and |∂tδM | ≪ ω|M |. Ignoring the time dependence of δM and using a reduced model of test particles in the external field M (t), one obtains stroboscopic sections which are qualitatively and quantitatively similar to the ones reported in this figure, with the unique difference that the thickness is zero[18]. Considering a torus with action J ≈ 1.9, we plot in panel d) its variance ∆J computed over a time interval ∆t = 300 as a function of N . The scaling 1/N (dotted line) looks accurate over a wide range of N values.

Figure 3 :

 3 Figure 3: Upper panel: Phase diagram in the control parameter plane (M0, U ) of the out-of-equilibrium phase transition of the HMF model from a magnetized to a demagnetized phase. The solid curve pinpoints the position of the bifurcation from the monocluster to the bicluster QSS. The dashed line stands for the theoretical prediction based on Lynden-Bell's violent relaxation theory (see text). The star refers to the tricritical point separating first from second order phase transitions. Lower panel: The bifurcation is monitored as function of U , for M0 = 0.6. The grey zones highlight the width of the resonances.